“Frontmatter” Structural Engineering Handbook Ed. Chen WaiFah Boca Raton: CRC Press LLC, 1999
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“Frontmatter” Structural Engineering Handbook Ed. Chen WaiFah Boca Raton: CRC Press LLC, 1999
Structural Engineering Contents
1
Basic Theory of Plates and Elastic Stability
2
Structural Analysis
3
Structural Steel Design1
4
Structural Concrete Design2
5
Earthquake Engineering
Charles Scawthorn
6
Composite Construction
Edoardo Cosenza and Riccardo Zandonini
7
ColdFormed Steel Structures
8
Aluminum Structures
9
Timber Structures
10
Bridge Structures
11
Shell Structures
12
Multistory Frame Structures
13
Space Frame Structures
14
Cooling Tower Structures
Phillip L. Gould and Wilfried B. Krätzig
15
Transmission Structures
Shujin Fang, Subir Roy, and Jacob Kramer
Eiki Yamaguchi
J.Y. Richard Liew,
N.E. Shanmugam, and
C.H. Yu
E. M. Lui Amy Grider and Julio A. Ramirez and Young Mook Yun
WeiWen Yu
Maurice L. Sharp
Kenneth J. Fridley Shouji Toma, Lian Duan, and WaiFah Chen Clarence D. Miller
15B Tunnel Structures
J. Y. Richard Liew and T. Balendra and W. F. Chen
Tien T. Lan
Birger Schmidt, Christian Ingerslev, Brian Brenner, and J.N. Wang
16 PerformanceBased Seismic Design Criteria For Bridges LianDuanandMarkReno 1999 by CRC Press LLC
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17
Effective Length Factors of Compression Members
18
Stub Girder Floor Systems
19
Plate and Box Girders
20
Steel Bridge Construction
21
Basic Principles of Shock Loading
22
Welded Connections
23
Composite Connections
24
Fatigue and Fracture
25
Underground Pipe
26
Structural Reliability3
27
Passive Energy Dissipation and Active Control
28
An Innnovative Design For Steel Frame Using Advanced Analysis4 W. F. Chen
29
Welded Tubular Connections—CHS Trusses
Reidar Bjorhovde
Mohamed Elgaaly Jackson Durkee
1999 by CRC Press LLC
O.W. Blodgett and D.K. Miller
O.W. Blodgett and D. K. Miller Roberto Leon
Robert J. Dexter and John W. Fisher J. M. Doyle and S.J. Fang D. V. Rosowsky
30 Earthquake Damage to Structures
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Lian Duan and W.F. Chen
T.T. Soong and G.F. Dargush
Peter W. Marshall
Mark Yashinsky
SeungEock Kim and
Yamaguchi, E. “Basic Theory of Plates and Elastic Stability” Structural Engineering Handbook Ed. Chen WaiFah Boca Raton: CRC Press LLC, 1999
Basic Theory of Plates and Elastic Stability 1.1 1.2 1.3
Eiki Yamaguchi Department of Civil Engineering, Kyushu Institute of Technology, Kitakyusha, Japan
1.1
Introduction Plates
Basic Assumptions • Governing Equations • Boundary Conditions • Circular Plate • Examples of Bending Problems
Stability
Basic Concepts • Structural Instability Walled Members • Plates
•
Columns
•
Thin
1.4 Defining Terms References Further Reading
Introduction
This chapter is concerned with basic assumptions and equations of plates and basic concepts of elastic stability. Herein, we shall illustrate the concepts and the applications of these equations by means of relatively simple examples; more complex applications will be taken up in the following chapters.
1.2 1.2.1
Plates Basic Assumptions
We consider a continuum shown in Figure 1.1. A feature of the body is that one dimension is much smaller than the other two dimensions: t 0 : unstable Equation 1.41 implies that as P increases, the state of the system changes from stable equilibrium to unstable equilibrium. The critical load is kL, at which multiple equilibrium positions, i.e., θ = 0 and θ 6 = 0, are possible. Thus, the critical load serves also as a bifurcation point of the equilibrium path. The load at such a bifurcation is called the buckling load. 1999 by CRC Press LLC
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FIGURE 1.10: Rigid bar AB with a spring. For the present system, the buckling load of kL is stability limit as well as neutral equilibrium. In general, the buckling load corresponds to a state of neutral equilibrium, but not necessarily to stability limit. Nevertheless, the buckling load is often associated with the characteristic change of structural behavior, and therefore can be regarded as the limit state of serviceability. Linear Buckling Analysis
We can compute a buckling load by considering an equilibrium condition for a slightly deformed state. For the system of Figure 1.10, the moment equilibrium yields P L sin θ − (kL sin θ )(L cos θ ) = 0 Since θ is infinitesimal, we obtain
Lθ (P − kL) = 0
(1.42) (1.43)
It is obvious that this equation is satisfied for any value of P if θ is zero: θ = 0 is called the trivial solution. We are seeking the buckling load, at which the equilibrium condition is satisfied for θ 6 = 0. The trivial solution is apparently of no importance and from Equation 1.43 we can obtain the following buckling load PC : (1.44) PC = kL A rigorous buckling analysis is quite involved, where we need to solve nonlinear equations even when elastic problems are dealt with. Consequently, the linear buckling analysis is frequently employed. The analysis can be justified, if deformation is negligible and structural behavior is linear before the buckling load is reached. The way we have obtained Equation 1.44 in the above is a typical application of the linear buckling analysis. In mathematical terms, Equation 1.43 is called a characteristic equation and Equation 1.44 an eigenvalue. The linear buckling analysis is in fact regarded as an eigenvalue problem.
1.3.2 Structural Instability Three classes of instability phenomenon are observed in structures: bifurcation, snapthrough, and softening. We have discussed a simple example of bifurcation in the previous section. Figure 1.11a depicts a schematic loaddisplacement relationship associated with the bifurcation: Point A is where the 1999 by CRC Press LLC
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bifurcation takes place. In reality, due to imperfection such as the initial crookedness of a member and the eccentricity of loading, we can rarely observe the bifurcation. Instead, an actual structural behavior would be more like the one indicated in Figure 1.11a. However, the bifurcation load is still an important measure regarding structural stability and most instabilities of a column and a plate are indeed of this class. In many cases we can evaluate the bifurcation point by the linear buckling analysis. In some structures, we observe that displacement increases abruptly at a certain load level. This can take place at Point A in Figure 1.11b; displacement increases from UA to UB at PA , as illustrated by a broken arrow. The phenomenon is called snapthrough. The equilibrium path of Figure 1.11b is typical of shelllike structures, including a shallow arch, and is traceable only by the finite displacement analysis. The other instability phenomenon is the softening: as Figure 1.11c illustrates, there exists a peak loadcarrying capacity, beyond which the structural strength deteriorates. We often observe this phenomenon when yielding takes place. To compute the associated equilibrium path, we need to resort to nonlinear structural analysis. Since nonlinear analysis is complicated and costly, the information on stability limit and ultimate strength is deduced in practice from the bifurcation load, utilizing the linear buckling analysis. We shall therefore discuss the buckling loads (bifurcation points) of some structures in what follows.
1.3.3
Columns
Simply Supported Column
As a first example, we evaluate the buckling load of a simply supported column shown in Figure 1.12a. To this end, the moment equilibrium in a slightly deformed configuration is considered. Following the notation in Figure 1.12b, we can readily obtain w00 + k 2 w = 0 where k2 =
P EI
(1.45)
(1.46)
EI is the bending rigidity of the column. The general solution of Equation 1.45 is w = A1 sin kx + A2 cos kx
(1.47)
The arbitrary constants A1 and A2 are to be determined by the following boundary conditions: w w
= 0 at x = 0 = 0 at x = L
(1.48a) (1.48b)
Equation 1.48a gives A2 = 0 and from Equation 1.48b we reach A1 sin kL = 0
(1.49)
A1 = 0 is a solution of the characteristic equation above, but this is the trivial solution corresponding to a perfectly straight column and is of no interest. Then we obtain the following buckling loads: PC = 1999 by CRC Press LLC
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n2 π 2 EI L2
(1.50)
1999 by CRC Press LLC
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FIGURE 1.11: Unstable structural behaviors.
FIGURE 1.12: Simplysupported column.
Although n is any integer, our interest is in the lowest buckling load with n = 1 since it is the critical load from the practical point of view. The buckling load, thus, obtained is PC =
π 2 EI L2
(1.51)
which is often referred to as the Euler load. From A2 = 0 and Equation 1.51, Equation 1.47 indicates the following shape of the deformation: w = A1 sin
πx L
(1.52)
This equation shows the buckled shape only, since A1 represents the undetermined amplitude of the deflection and can have any value. The deflection curve is illustrated in Figure 1.12c. The behavior of the simply supported column is summarized as follows: up to the Euler load the column remains straight; at the Euler load the state of the column becomes the neutral equilibrium and it can remain straight or it starts to bend in the mode expressed by Equation 1.52. 1999 by CRC Press LLC
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Cantilever Column
For the cantilever column of Figure 1.13a, by considering the equilibrium condition of the free body shown in Figure 1.13b, we can derive the following governing equation: w00 + k 2 w = k 2 δ
(1.53)
where δ is the deflection at the free tip. The boundary conditions are w w0 w
= 0 at x = 0 = 0 at x = 0 = δ at x = L
(1.54)
FIGURE 1.13: Cantilever column.
From these equations we can obtain the characteristic equation as δ cos kL = 0 1999 by CRC Press LLC
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(1.55)
which yields the following buckling load and deflection shape: PC
=
w
=
π 2 EI 2 4L
δ 1 − cos
(1.56)
πx
(1.57)
2L
The buckling mode is illustrated in Figure 1.13c. It is noted that the boundary conditions make much difference in the buckling load: the present buckling load is just a quarter of that for the simply supported column. HigherOrder Differential Equation
We have thus far analyzed the two columns. In each problem, a secondorder differential equation was derived and solved. This governing equation is problemdependent and valid only for a particular problem. A more consistent approach is possible by making use of the governing equation for a beamcolumn with no laterally distributed load: EI wI V + P w 00 = q
(1.58)
Note that in this equation P is positive when compressive. This equation is applicable to any set of boundary conditions. The general solution of Equation 1.58 is given by w = A1 sin kx + A2 cos kx + A3 x + A4
(1.59)
where A1 ∼ A4 are arbitrary constants and determined from boundary conditions. We shall again solve the two column problems, using Equation 1.58. 1. Simply supported column (Figure 1.12a) Because of no deflection and no external moment at each end of the column, the boundary conditions are described as w w
= 0, = 0,
w00 = 0 at x = 0 w00 = 0 at x = L
(1.60)
From the conditions at x = 0, we can determine A2 = A4 = 0
(1.61)
Using this result and the conditions at x = L, we obtain
sin kL −k 2 sin kL
L 0
A1 A3
=
0 0
(1.62)
For the nontrivial solution to exist, the determinant of the coefficient matrix in Equation 1.62 must vanish, leading to the following characteristic equation: k 2 L sin kL = 0
(1.63)
from which we arrive at the same critical load as in Equation 1.51. By obtaining the corresponding eigenvector of Equation 1.62, we can get the buckled shape of Equation 1.52. 1999 by CRC Press LLC
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2. Cantilever column (Figure 1.13a) In this column, we observe no deflection and no slope at the fixed end; no external moment and no external shear force at the free end. Therefore, the boundary conditions are w = 0, w 00 = 0,
w000
w0 = 0 + k 2 w0 = 0
at x = 0 at x = L
(1.64)
Note that since we are dealing with a slightly deformed column in the linear buckling analysis, the axial force has a transverse component, which is why P comes in the boundary condition at x = L. The latter condition at x = L eliminates A3 . With this and the second condition at x = 0, we can claim A1 = 0. The remaining two conditions then lead to 1 1 A2 0 (1.65) = A4 0 k 2 cos kL 0 The smallest eigenvalue and the corresponding eigenvector of Equation 1.65 coincide with the buckling load and the buckling mode that we have obtained previously in Section 1.3.3. Effective Length
We have obtained the buckling loads for the simply supported and the cantilever columns. By either the second or the fourthorder differential equation approach, we can compute buckling loads for a fixedhinged column (Figure 1.14a) and a fixedfixed column (Figure 1.14b) without much difficulty: PC
=
PC
=
π 2 EI (0.7L)2 π 2 EI (0.5L)2
for a fixed  hinged column for a fixed  hinged column
(1.66)
For all the four columns considered thus far, and in fact for the columns with any other sets of boundary conditions, we can express the buckling load in the form of PC =
π 2 EI (KL)2
(1.67)
where KL is called the effective length and represents presumably the length of the equivalent Euler column (the equivalent simply supported column). For design purposes, Equation 1.67 is often transformed into σC =
π 2E (KL/r)2
(1.68)
where r is the radius of gyration defined in terms of crosssectional area A and the moment of inertia I by r I (1.69) r= A For an ideal elastic column, we can draw the curve of the critical stress σC vs. the slenderness ratio KL/r, as shown in Figure 1.15a. 1999 by CRC Press LLC
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FIGURE 1.14: (a) Fixedhinged column; (b) fixedfixed column. For a column of perfectly plastic material, stress never exceeds the yield stress σY . For this class of column, we often employ a normalized form of Equation 1.68 as 1 σC = 2 σY λ where λ=
1 KL π r
r
(1.70)
σY E
(1.71)
This equation is plotted in Figure 1.15b. For this column, with λ < 1.0, it collapses plastically; elastic buckling takes place for λ > 1.0. Imperfect Columns
In the derivation of the buckling loads, we have dealt with the idealized columns; the member is perfectly straight and the loading is concentric at every crosssection. These idealizations help simplify the problem, but perfect members do not exist in the real world: minor crookedness of shape and small eccentricities of loading are always present. To this end, we shall investigate the behavior of an initially bent column in this section. We consider a simply supported column shown in Figure 1.16. The column is initially bent and the initial crookedness wi is assumed to be in the form of wi = a sin
πx L
(1.72)
where a is a small value, representing the magnitude of the initial deflection at the midpoint. If we describe the additional deformation due to bending as w and consider the moment equilibrium in 1999 by CRC Press LLC
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FIGURE 1.15: (a) Relationship between critical stress and slenderness ratio; (b) normalized relationship.
FIGURE 1.16: Initially bent column. this configuration, we obtain
πx (1.73) L where k 2 is defined in Equation 1.46. The general solution of this differential equation is given by w00 + k 2 w = −k 2 a sin
w = A sin 1999 by CRC Press LLC
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πx πx P /PE πx a sin + B cos + L L 1 − P /PE L
(1.74)
where PE is the Euler load, i.e., π 2 EI /L2 . From the boundary conditions of Equation 1.48, we can determine the arbitrary constants A and B, yielding the following loaddisplacement relationship: w=
πx P /PE a sin 1 − P /PE L
(1.75)
By adding this expression to the initial deflection, we can obtain the total displacement wt as wt = wi + w =
a πx sin 1 − P /PE L
(1.76)
Figure 1.17 illustrates the variation of the deflection at the midpoint of this column wm .
FIGURE 1.17: Loaddisplacement curve of the bent column.
Unlike the ideally perfect column, which remains straight up to the Euler load, we observe in this figure that the crooked column begins to bend at the onset of the loading. The deflection increases slowly at first, and as the applied load approaches the Euler load, the increase of the deflection is getting more and more rapid. Thus, although the behavior of the initially bent column is different from that of bifurcation, the buckling load still serves as an important measure of stability. We have discussed the behavior of a column with geometrical imperfection in this section. However, the trend observed herein would be the same for general imperfect columns such as an eccentrically loaded column.
1.3.4
ThinWalled Members
In the previous section, we assumed that a compressed column would buckle by bending. This type of buckling may be referred to as flexural buckling. However, a column may buckle by twisting or by a combination of twisting and bending. Such a mode of failure occurs when the torsional rigidity of the crosssection is low. Thinwalled open crosssections have a low torsional rigidity in general and hence are susceptible of this type of buckling. In fact, a column of thinwalled open crosssection usually buckles by a combination of twisting and bending: this mode of buckling is often called the torsionalflexural buckling. 1999 by CRC Press LLC
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A bar subjected to bending in the plane of a major axis may buckle in yet another mode: at the critical load a compression side of the crosssection tends to bend sideways while the remainder is stable, resulting in the rotation and lateral movement of the entire crosssection. This type of buckling is referred to as lateral buckling. We need to use caution in particular, if a beam has no lateral supports and the flexural rigidity in the plane of bending is larger than the lateral flexural rigidity. In the present section, we shall briefly discuss the two buckling modes mentioned above, both of which are of practical importance in the design of thinwalled members, particularly of open crosssection. TorsionalFlexural Buckling
We consider a simply supported column subjected to compressive load P applied at the centroid of each end, as shown in Figure 1.18. Note that the x axis passes through the centroid of every crosssection. Taking into account that the crosssection undergoes translation and rotation as illustrated in Figure 1.19, we can derive the equilibrium conditions for the column deformed slightly by the torsionalflexural buckling EIy ν I V + P ν 00 + P zs φ 00 = 0 EIz w I V + P w00 − P ys φ 00 = 0 EIw φ I V + P rs2 φ 00 − GJ φ 00 + P zs ν 00 − P ys w 00 = 0 where ν, w φ EIw GJ ys , zs and
= = = = =
(1.77)
displacements in the y, zdirections, respectively rotation warping rigidity torsional rigidity coordinates of the shear center Z EIy
=
EIz
=
rs2
=
ZA A
y 2 dA z2 dA
(1.78)
Is A
where = polar moment of inertia about the shear center Is A = crosssectional area We can obtain the buckling load by solving the eigenvalue problem governed by Equation 1.77 and the boundary conditions of ν = ν 00 = w = w00 = φ = φ 00 = 0 at x = 0, L
(1.79)
For doubly symmetric crosssection, the shear center coincides with the centroid. Therefore, ys , zs , and rs vanish and the three equations in Equation 1.77 become independent of each other, if the crosssection of the column is doubly symmetric. In this case, we can compute three critical loads as follows: 1999 by CRC Press LLC
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FIGURE 1.18: Simplysupported thinwalled column.
FIGURE 1.19: Translation and rotation of the crosssection.
PyC
=
PzC
=
PφC
=
π 2 EIy L2 2 π EIz L2 1 π 2 EIw GJ + rs2 L2
(1.80a) (1.80b) (1.80c)
The first two are associated with flexural buckling and the last one with torsional buckling. For all cases, the buckling mode is in the shape of sin πLx . The smallest of the three would be the critical load of practical importance: for a relatively short column with low GJ and EIw , the torsional buckling may take place. When the crosssection of a column is symmetric with respect only to the y axis, we rewrite Equation 1.77 as EIy ν I V + P ν 00 = 0
(1.81a)
EIz w I V + P w00 − P ys φ 00 = 0 EIw φ I V + P rs2 − GJ φ 00 − P ys w 00 = 0
(1.81b) (1.81c)
The first equation indicates that the flexural buckling in the x − y plane occurs independently and the corresponding critical load is given by PyC of Equation 1.80a. The flexural buckling in the x − z plane and the torsional buckling are coupled. By assuming that the buckling modes are described by πx w = A1 sin πx L and φ = A2 sin L , Equations 1.81b,c yields
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P − PzC −P ys
rs2
−P ys P − PφC
A1 A2
=
0 0
(1.82)
This eigenvalue problem leads to f (P ) = rs2 P − PφC (P − PzC ) − (P ys )2 = 0
(1.83)
The solution of this quadratic equation is the critical load associated with torsionalflexural buckling. Since f (0) = rs2 PφC PzC > 0, f (PφC = −(P ys )2 < 0, and f (PzC ) = −(P ys )2 < 0, it is obvious that the critical load is lower than PzC and PφC . If this load is smaller than PyC , then the torsionalflexural buckling will take place. If there is no axis of symmetry in the crosssection, all the three equations in Equation 1.77 are coupled. The torsionalflexural buckling occurs in this case, since the critical load for this buckling mode is lower than any of the three loads in Equation 1.80. Lateral Buckling
The behavior of a simply supported beam in pure bending (Figure 1.20) is investigated. The equilibrium condition for a slightly translated and rotated configuration gives governing equations for the bifurcation. For a crosssection symmetric with respect to the y axis, we arrive at the following equations: EIy ν I V + Mφ 00 = 0
(1.84a)
IV
EIz w = 0 EIw φ I V − (GJ + Mβ) φ 00 + Mν 00 = 0 where β=
1 Iz
(1.84b) (1.84c)
Z n A
o y 2 + (z − zs )2 zdA
(1.85)
FIGURE 1.20: Simply supported beam in pure bending.
Equation 1.84b is a beam equation and has nothing to do with buckling. From the remaining two equations and the associated boundary conditions of Equation 1.79, we can evaluate the critical load for the lateral buckling. By assuming the bucking mode is in the shape of sin πLx for both ν and φ, we obtain the characteristic equation M 2 − βPyC M − rs2 PyC PφC = 0
(1.86)
The smallest root of this quadratic equation is the critical load (moment) for the lateral buckling. For doubly symmetric sections where β is zero, the critical moment MC is given by q MC = rs2 PyC PφC =
1999 by CRC Press LLC
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s
π 2 EIy L2
GJ +
π 2 EIw L2
(1.87)
1.3.5
Plates
Governing Equation
The buckling load of a plate is also obtained by the linear buckling analysis, i.e., by considering the equilibrium of a slightly deformed configuration. The plate counterpart of Equation 1.58, thus, derived is ∂ 2w ∂ 2w ∂ 2w + Ny 2 = 0 (1.88) D∇ 4 w + N x 2 + 2N xy ∂x∂y ∂x ∂y The definitions of N x , N y , and N xy are the same as those of Nx , Ny , and Nxy given in Equations 1.8a through 1.8c, respectively, except the sign; N x , N y , and N xy are positive when compressive. The boundary conditions are basically the same as discussed in Section 1.2.3 except the mechanical condition in the vertical direction: to include the effect of inplane forces, we need to modify Equation 1.18 as ∂w ∂w + Nns = Sn (1.89) Sn + Nn ∂n ∂s where Z Nn
=
Nns
=
Zz z
σn dz τns dz
(1.90)
Simply Supported Plate
As an example, we shall discuss the buckling load of a simply supported plate under uniform compression shown in Figure 1.21. The governing equation for this plate is D∇ 4 w + N x
∂ 2w =0 ∂x 2
(1.91)
and the boundary conditions are w
=
0,
w
=
0,
∂ 2w = 0 along x = 0, a ∂x 2 ∂ 2w = 0 along y = 0, b ∂y 2
FIGURE 1.21: Simply supported plate subjected to uniform compression. 1999 by CRC Press LLC
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(1.92)
We assume that the solution is of the form w=
∞ ∞ X X
Amn sin
m=1 n=1
mπ x nπ x sin a b
(1.93)
where m and n are integers. Since this solution satisfies all the boundary conditions, we have only to ensure that it satisfies the governing equation. Substituting Equation 1.93 into 1.91, we obtain " # 2 2 n2 N x m2 π 2 4 m + 2 − (1.94) =0 Amn π D a2 a2 b Since the trivial solution is of no interest, at least one of the coefficients amn must not be zero, the consideration of which leads to 2 π 2D b n2 a Nx = 2 (1.95) m + a mb b As the lowest N x is crucial and N x increases with n, we conclude n = 1: the buckling of this plate occurs in a single halfwave in the y direction and kπ 2 D b2
(1.96)
1 N xC π 2E =k 2 t 12(1 − ν ) (b/t)2
(1.97)
1a 2 b k= m + a mb
(1.98)
N xC = or σxC = where
Note that Equation 1.97 is comparable to Equation 1.68, and k is called the buckling stress coefficient. The optimum value of m that gives the lowest N xC depends on the aspect ratio a/b, as can be realized in Figure 1.22. For example, the optimum m is 1 for a square plate while it is 2 for a plate of a/b = 2. For a plate with a large aspect ratio, k = 4.0 serves as a good approximation. Since the aspect ratio of a component of a steel structural member such as a web plate is large in general, we can often assume k is simply equal to 4.0.
1.4
Defining Terms
The following is a list of terms as defined in the Guide to Stability Design Criteria for Metal Structures, 4th ed., Galambos, T.V., Structural Stability Research Council, John Wiley & Sons, New York, 1988. Bifurcation: A term relating to the loaddeflection behavior of a perfectly straight and perfectly centered compression element at critical load. Bifurcation can occur in the inelastic range only if the pattern of postyield properties and/or residual stresses is symmetrically disposed so that no bending moment is developed at subcritical loads. At the critical load a member can be in equilibrium in either a straight or slightly deflected configuration, and a bifurcation results at a branch point in the plot of axial load vs. lateral deflection from which two alternative loaddeflection plots are mathematically valid. Braced frame: A frame in which the resistance to both lateral load and frame instability is provided by the combined action of floor diaphragms and structural core, shear walls, and/or a diagonal K brace, or other auxiliary system of bracing. 1999 by CRC Press LLC
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FIGURE 1.22: Variation of the buckling stress coefficient k with the aspect ratio a/b.
Effective length: The equivalent or effective length (KL) which, in the Euler formula for a hingedend column, results in the same elastic critical load as for the framed member or other compression element under consideration at its theoretical critical load. The use of the effective length concept in the inelastic range implies that the ratio between elastic and inelastic critical loads for an equivalent hingedend column is the same as the ratio between elastic and inelastic critical loads in the beam, frame, plate, or other structural element for which buckling equivalence has been assumed. Instability: A condition reached during buckling under increasing load in a compression member, element, or frame at which the capacity for resistance to additional load is exhausted and continued deformation results in a decrease in loadresisting capacity. Stability: The capacity of a compression member or element to remain in position and support load, even if forced slightly out of line or position by an added lateral force. In the elastic range, removal of the added lateral force would result in a return to the prior loaded position, unless the disturbance causes yielding to commence. Unbraced frame: A frame in which the resistance to lateral loads is provided primarily by the bending of the frame members and their connections.
References [1] Chajes, A. 1974. Principles of Structural Stability Theory, PrenticeHall, Englewood Cliffs, NJ. [2] Chen, W.F. and Atsuta, T. 1976. Theory of BeamColumns, vol. 1: InPlane Behavior and Design, and vol. 2: Space Behavior and Design, McGrawHill, NY. [3] Thompson, J.M.T. and Hunt, G.W. 1973. A General Theory of Elastic Stability, John Wiley & Sons, London, U.K. [4] Timoshenko, S.P. and WoinowskyKrieger, S. 1959. Theory of Plates and Shells, 2nd ed., McGrawHill, NY. [5] Timoshenko, S.P. and Gere, J.M. 1961. Theory of Elastic Stability, 2nd ed., McGrawHill, NY. 1999 by CRC Press LLC
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Further Reading [1] Chen, W.F. and Lui, E.M. 1987. Structural Stability Theory and Implementation, Elsevier, New York. [2] Chen, W.F. and Lui, E.M. 1991. Stability Design of Steel Frames, CRC Press, Boca Raton, FL. [3] Galambos, T.V. 1988. Guide to Stability Design Criteria for Metal Structures, 4th ed., Structural Stability Research Council, John Wiley & Sons, New York.
1999 by CRC Press LLC
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Richard Liew, J.Y.; Shanmugam, N.W. and Yu, C.H. “Structural Analysis” Structural Engineering Handbook Ed. Chen WaiFah Boca Raton: CRC Press LLC, 1999
Structural Analysis
J.Y. Richard Liew, N.E. Shanmugam, and C.H. Yu Department of Civil Engineering The National University of Singapore, Singapore
2.1
2.1 Fundamental Principles 2.2 Flexural Members 2.3 Trusses 2.4 Frames 2.5 Plates 2.6 Shell 2.7 Influence Lines 2.8 Energy Methods in Structural Analysis 2.9 Matrix Methods 2.10 The Finite Element Method 2.11 Inelastic Analysis 2.12 Frame Stability 2.13 Structural Dynamic 2.14 Defining Terms References Further Reading
Fundamental Principles
Structural analysis is the determination of forces and deformations of the structure due to applied loads. Structural design involves the arrangement and proportioning of structures and their components in such a way that the assembled structure is capable of supporting the designed loads within the allowable limit states. An analytical model is an idealization of the actual structure. The structural model should relate the actual behavior to material properties, structural details, and loading and boundary conditions as accurately as is practicable. All structures that occur in practice are threedimensional. For building structures that have regular layout and are rectangular in shape, it is possible to idealize them into twodimensional frames arranged in orthogonal directions. Joints in a structure are those points where two or more members are connected. A truss is a structural system consisting of members that are designed to resist only axial forces. Axially loaded members are assumed to be pinconnected at their ends. A structural system in which joints are capable of transferring end moments is called a frame. Members in this system are assumed to be capable of resisting bending moment axial force and shear force. A structure is said to be two dimensional or planar if all the members lie in the same plane. Beams are those members that are subjected to bending or flexure. They are usually thought of as being in horizontal positions and loaded with vertical loads. Ties are members that are subjected to axial tension only, while struts (columns or posts) are members subjected to axial compression only. 1999 by CRC Press LLC
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2.1.1
Boundary Conditions
A hinge represents a pin connection to a structural assembly and it does not allow translational movements (Figure 2.1a). It is assumed to be frictionless and to allow rotation of a member with
FIGURE 2.1: Various boundary conditions.
respect to the others. A roller represents a kind of support that permits the attached structural part to rotate freely with respect to the foundation and to translate freely in the direction parallel to the foundation surface (Figure 2.1b) No translational movement in any other direction is allowed. A fixed support (Figure 2.1c) does not allow rotation or translation in any direction. A rotational spring represents a support that provides some rotational restraint but does not provide any translational restraint (Figure 2.1d). A translational spring can provide partial restraints along the direction of deformation (Figure 2.1e).
2.1.2
Loads and Reactions
Loads may be broadly classified as permanent loads that are of constant magnitude and remain in one position and variable loads that may change in position and magnitude. Permanent loads are also referred to as dead loads which may include the self weight of the structure and other loads such as walls, floors, roof, plumbing, and fixtures that are permanently attached to the structure. Variable loads are commonly referred to as live or imposed loads which may include those caused by construction operations, wind, rain, earthquakes, snow, blasts, and temperature changes in addition to those that are movable, such as furniture and warehouse materials. Ponding load is due to water or snow on a flat roof which accumulates faster than it runs off. Wind loads act as pressures on windward surfaces and pressures or suctions on leeward surfaces. Impact loads are caused by suddenly applied loads or by the vibration of moving or movable loads. They are usually taken as a fraction of the live loads. Earthquake loads are those forces caused by the acceleration of the ground surface during an earthquake. A structure that is initially at rest and remains at rest when acted upon by applied loads is said to be in a state of equilibrium. The resultant of the external loads on the body and the supporting forces or reactions is zero. If a structure or part thereof is to be in equilibrium under the action of a system 1999 by CRC Press LLC
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of loads, it must satisfy the six static equilibrium equations, such as P P P Fx = 0, Fy = 0, Fz = 0 P P P My = 0, Mz = 0 Mx = 0,
(2.1)
The summation in these equations is for all the components of the forces (F ) and of the moments (M) about each of the three axes x, y, and z. If a structure is subjected to forces that lie in one plane, say xy, the above equations are reduced to: X X X Fy = 0, Mz = 0 (2.2) Fx = 0, Consider, for example, a beam shown in Figure 2.2a under the action of the loads shown. The
FIGURE 2.2: Beam in equilibrium. reaction at support B must act perpendicular to the surface on which the rollers are constrained to roll upon. The support reactions and the applied loads, which are resolved in vertical and horizontal directions, are shown in Figure 2.2b. √ From geometry, it can be calculated that By = 3Bx . Equation 2.2 can be used to determine the magnitude of the support reactions. Taking moment about B gives 10Ay − 346.4x5 = 0 from which Equating the sum of vertical forces,
P
Ay = 173.2 kN. Fy to zero gives
173.2 + By − 346.4 = 0 and, hence, we get By = 173.2 kN. Therefore,
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√ Bx = By / 3 = 100 kN.
Equilibrium in the horizontal direction,
P
Fx = 0 gives,
Ax − 200 − 100 = 0 and, hence, Ax = 300 kN. There are three unknown reaction components at a fixed end, two at a hinge, and one at a roller. If, for a particular structure, the total number of unknown reaction components equals the number of equations available, the unknowns may be calculated from the equilibrium equations, and the structure is then said to be statically determinate externally. Should the number of unknowns be greater than the number of equations available, the structure is statically indeterminate externally; if less, it is unstable externally. The ability of a structure to support adequately the loads applied to it is dependent not only on the number of reaction components but also on the arrangement of those components. It is possible for a structure to have as many or more reaction components than there are equations available and yet be unstable. This condition is referred to as geometric instability.
2.1.3
Principle of Superposition
The principle states that if the structural behavior is linearly elastic, the forces acting on a structure may be separated or divided into any convenient fashion and the structure analyzed for the separate cases. Then the final results can be obtained by adding up the individual results. This is applicable to the computation of structural responses such as moment, shear, deflection, etc. However, there are two situations where the principle of superposition cannot be applied. The first case is associated with instances where the geometry of the structure is appreciably altered under load. The second case is in situations where the structure is composed of a material in which the stress is not linearly related to the strain.
2.1.4
Idealized Models
Any complex structure can be considered to be built up of simpler components called members or elements. Engineering judgement must be used to define an idealized structure such that it represents the actual structural behavior as accurately as is practically possible. Structures can be broadly classified into three categories: 1. Skeletal structures consist of line elements such as a bar, beam, or column for which the length is much larger than the breadth and depth. A variety of skeletal structures can be obtained by connecting line elements together using hinged, rigid, or semirigid joints. Depending on whether the axes of these members lie in one plane or in different planes, these structures are termed as plane structures or spatial structures. The line elements in these structures under load may be subjected to one type of force such as axial force or a combination of forces such as shear, moment, torsion, and axial force. In the first case the structures are referred to as the trusstype and in the latter as frametype. 2. Plated structures consist of elements that have length and breadth of the same order but are much larger than the thickness. These elements may be plane or curved in plane, in which case they are called plates or shells, respectively. These elements are generally used in combination with beams and bars. Reinforced concrete slabs supported on beams, boxgirders, plategirders, cylindrical shells, or water tanks are typical examples of plate and shell structures. 3. Threedimensional solid structures have all three dimensions, namely, length, breadth, and depth, of the same order. Thickwalled hollow spheres, massive raft foundation, and dams are typical examples of solid structures. 1999 by CRC Press LLC
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Recent advancement in finite element methods of structural analysis and the advent of more powerful computers have enabled the economic analysis of skeletal, plated, and solid structures.
2.2
Flexural Members
One of the most common structural elements is a beam; it bends when subjected to loads acting transversely to its centroidal axis or sometimes by loads acting both transversely and parallel to this axis. The discussions given in the following subsections are limited to straight beams in which the centroidal axis is a straight line with shear center coinciding with the centroid of the crosssection. It is also assumed that all the loads and reactions lie in a simple plane that also contains the centroidal axis of the flexural member and the principal axis of every crosssection. If these conditions are satisfied, the beam will simply bend in the plane of loading without twisting.
2.2.1
Axial Force, Shear Force, and Bending Moment
Axial force at any transverse crosssection of a straight beam is the algebraic sum of the components acting parallel to the axis of the beam of all loads and reactions applied to the portion of the beam on either side of that crosssection. Shear force at any transverse crosssection of a straight beam is the algebraic sum of the components acting transverse to the axis of the beam of all the loads and reactions applied to the portion of the beam on either side of the crosssection. Bending moment at any transverse crosssection of a straight beam is the algebraic sum of the moments, taken about an axis passing through the centroid of the crosssection. The axis about which the moments are taken is, of course, normal to the plane of loading.
2.2.2
Relation Between Load, Shear, and Bending Moment
When a beam is subjected to transverse loads, there exist certain relationships between load, shear, and bending moment. Let us consider, for example, the beam shown in Figure 2.3 subjected to some arbitrary loading, p.
FIGURE 2.3: A beam under arbitrary loading.
Let S and M be the shear and bending moment, respectively, for any point ‘m’ at a distance x, which is measured from A, being positive when measured to the right. Corresponding values of shear and bending moment at point ‘n’ at a differential distance dx to the right of m are S + dS and M + dM, respectively. It can be shown, neglecting the second order quantities, that p= 1999 by CRC Press LLC
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dS dx
(2.3)
and
dM (2.4) dx Equation 2.3 shows that the rate of change of shear at any point is equal to the intensity of load applied to the beam at that point. Therefore, the difference in shear at two crosssections C and D is Z xD pdx (2.5) SD − SC = S=
xC
We can write in the same way for moment as MD − MC =
2.2.3
Z
xD
xC
Sdx
(2.6)
Shear and Bending Moment Diagrams
In order to plot the shear force and bending moment diagrams it is necessary to adopt a sign convention for these responses. A shear force is considered to be positive if it produces a clockwise moment about a point in the free body on which it acts. A negative shear force produces a counterclockwise moment about the point. The bending moment is taken as positive if it causes compression in the upper fibers of the beam and tension in the lower fiber. In other words, sagging moment is positive and hogging moment is negative. The construction of these diagrams is explained with an example given in Figure 2.4.
FIGURE 2.4: Bending moment and shear force diagrams.
The section at E of the beam is in equilibrium under the action of applied loads and internal forces acting at E as shown in Figure 2.5. There must be an internal vertical force and internal bending moment to maintain equilibrium at Section E. The vertical force or the moment can be obtained as the algebraic sum of all forces or the algebraic sum of the moment of all forces that lie on either side of Section E. 1999 by CRC Press LLC
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FIGURE 2.5: Internal forces. The shear on a crosssection an infinitesimal distance to the right of point A is +55 k and, therefore, the shear diagram rises abruptly from 0 to +55 at this point. In the portion AC, since there is no additional load, the shear remains +55 on any crosssection throughout this interval, and the diagram is a horizontal as shown in Figure 2.4. An infinitesimal distance to the left of C the shear is +55, but an infinitesimal distance to the right of this point the 30 k load has caused the shear to be reduced to +25. Therefore, at point C there is an abrupt change in the shear force from +55 to +25. In the same manner, the shear force diagram for the portion CD of the beam remains a rectangle. In the portion DE, the shear on any crosssection a distance x from point D is S = 55 − 30 − 4x = 25 − 4x which indicates that the shear diagram in this portion is a straight line decreasing from an ordinate of +25 at D to +1 at E. The remainder of the shear force diagram can easily be verified in the same way. It should be noted that, in effect, a concentrated load is assumed to be applied at a point and, hence, at such a point the ordinate to the shear diagram changes abruptly by an amount equal to the load. In the portion AC, the bending moment at a crosssection a distance x from point A is M = 55x. Therefore, the bending moment diagram starts at 0 at A and increases along a straight line to an ordinate of +165 kft at point C. In the portion CD, the bending moment at any point a distance x from C is M = 55(x + 3) − 30x. Hence, the bending moment diagram in this portion is a straight line increasing from 165 at C to 265 at D. In the portion DE, the bending moment at any point a distance x from D is M = 55(x + 7) − 30(X + 4) − 4x 2 /2. Hence, the bending moment diagram in this portion is a curve with an ordinate of 265 at D and 343 at E. In an analogous manner, the remainder of the bending moment diagram can be easily constructed. Bending moment and shear force diagrams for beams with simple boundary conditions and subject to some simple loading are given in Figure 2.6.
2.2.4
FixEnded Beams
When the ends of a beam are held so firmly that they are not free to rotate under the action of applied loads, the beam is known as a builtin or fixended beam and it is statically indeterminate. The bending moment diagram for such a beam can be considered to consist of two parts, namely the free bending moment diagram obtained by treating the beam as if the ends are simply supported and the fixing moment diagram resulting from the restraints imposed at the ends of the beam. The solution of a fixed beam is greatly simplified by considering Mohr’s principles which state that: 1. the area of the fixing bending moment diagram is equal to that of the free bending moment diagram 2. the centers of gravity of the two diagrams lie in the same vertical line, i.e., are equidistant from a given end of the beam The construction of bending moment diagram for a fixed beam is explained with an example shown in Figure 2.7. P Q U T is the free bending moment diagram, Ms , and P Q R S is the fixing 1999 by CRC Press LLC
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FIGURE 2.6: Shear force and bending moment diagrams for beams with simple boundary conditions subjected to selected loading cases.
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FIGURE 2.6: (Continued) Shear force and bending moment diagrams for beams with simple boundary conditions subjected to selected loading cases.
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FIGURE 2.6: (Continued) Shear force and bending moment diagrams for beams with simple boundary conditions subjected to selected loading cases.
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FIGURE 2.7: Fixedended beam. moment diagram, Mi . The net bending moment diagram, M, is shown shaded. If As is the area of the free bending moment diagram and Ai the area of the fixing moment diagram, then from the first Mohr’s principle we have As = Ai and 1 W ab × ×L 2 L
=
MA + MB
=
1 (MA + MB ) × L 2 W ab L
(2.7)
From the second principle, equating the moment about A of As and Ai , we have, MA + 2MB =
W ab 2 2 + 3ab + b 2a L3
(2.8)
Solving Equations 2.7 and 2.8 for MA and MB , we get MA
=
MB
=
W ab2 L2 W a2b L2
Shear force can be determined once the bending moment is known. The shear force at the ends of the beam, i.e., at A and B are SA
=
SB
=
Wb MA − MB + L L MB − MA Wa + L L
Bending moment and shear force diagrams for some typical loading cases are shown in Figure 2.8.
2.2.5 Continuous Beams Continuous beams, like fixended beams, are statically indeterminate. Bending moments in these beams are functions of the geometry, moments of inertia and modulus of elasticity of individual members besides the load and span. They may be determined by Clapeyron’s Theorem of three moments, moment distribution method, or slope deflection method. 1999 by CRC Press LLC
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FIGURE 2.8: Shear force and bending moment diagrams for builtup beams subjected to typical loading cases.
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FIGURE 2.8: (Continued) Shear force and bending moment diagrams for builtup beams subjected to typical loading cases.
An example of a twospan continuous beam is solved by Clapeyron’s Theorem of three moments. The theorem is applied to two adjacent spans at a time and the resulting equations in terms of unknown support moments are solved. The theorem states that A1 x1 A2 x2 + (2.9) MA L1 + 2MB (L1 + L2 ) + MC L2 = 6 L1 L2 in which MA , MB , and MC are the hogging moment at the supports A, B, and C, respectively, of two adjacent spans of length L1 and L2 (Figure 2.9); A1 and A2 are the area of bending moment diagrams produced by the vertical loads on the simple spans AB and BC, respectively; x1 is the centroid of A1 from A, and x2 is the distance of the centroid of A2 from C. If the beam section is constant within a
FIGURE 2.9: Continuous beams. 1999 by CRC Press LLC
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span but remains different for each of the spans, Equation 2.9 can be written as L2 A1 x1 A2 x2 L1 L1 L2 + 2MB + =6 + + MC MA I1 I1 I2 I2 L1 I1 L2 I2
(2.10)
in which I1 and I2 are the moments of inertia of beam section in span L1 and L2 , respectively.
EXAMPLE 2.1:
The example in Figure 2.10 shows the application of this theorem. For spans AC and BC
FIGURE 2.10: Example—continuous beam.
MA × 10 + 2MC (10 + 10) + MB × 10 # " 2 1 3 × 250 × 10 × 5 2 × 500 × 10 × 5 + =6 10 10 Since the support at A is simply supported, MA = 0. Therefore, 4MC + MB = 1250
(2.11)
Considering an imaginary span BD on the right side of B, and applying the theorem for spans CB and BD ×2 MC × 10 + 2MB (10) + MD × 10 = 6 × (2/3)×10×5 10 MC + 2MB = 500 (because MC = MD ) Solving Equations 2.11 and 2.12 we get MB MC 1999 by CRC Press LLC
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= =
107.2 kNm 285.7 kNm
(2.12)
Shear force at A is SA = Shear force at C is
MA − MC + 100 = −28.6 + 100 = 71.4 kN L
MC − MB MC − MA + 100 + + 100 = L L = (28.6 + 100) + (17.9 + 100) = 246.5 kN
SC
Shear force at B is
SB =
MB − MC + 100 = −17.9 + 100 = 82.1 kN L
The bending moment and shear force diagrams are shown in Figure 2.10.
2.2.6
Beam Deflection
There are several methods for determining beam deflections: (1) momentarea method, (2) conjugatebeam method, (3) virtual work, and (4) Castigliano’s second theorem, among others. The elastic curve of a member is the shape the neutral axis takes when the member deflects under load. The inverse of the radius of curvature at any point of this curve is obtained as M 1 = R EI
(2.13)
in which M is the bending moment at the point and EI is the flexural rigidity of the beam section. 2 Since the deflection is small, R1 is approximately taken as ddxy2 , and Equation 2.13 may be rewritten as: d 2y (2.14) M = EI 2 dx In Equation 2.14, y is the deflection of the beam at distance x measured from the origin of coordinate. The change in slope in a distance dx can be expressed as Mdx/EI and hence the slope in a beam is obtained as Z B M (2.15) dx θB − θA = A EI Equation 2.15 may be stated as the change in slope between the tangents to the elastic curve at two points is equal to the area of the M/EI diagram between the two points. Once the change in slope between tangents to the elastic curve is determined, the deflection can be obtained by integrating further the slope equation. In a distance dx the neutral axis changes in direction by an amount dθ. The deflection of one point on the beam with respect to the tangent at another point due to this angle change is equal to dδ = xdθ , where x is the distance from the point at which deflection is desired to the particular differential distance. To determine the total deflection from the tangent at one point A to the tangent at another point B on the beam, it is necessary to obtain a summation of the products of each dθ angle (from A to B) times the distance to the point where deflection is desired, or Z B Mx dx (2.16) δB − δA = EI A The deflection of a tangent to the elastic curve of a beam with respect to a tangent at another point is equal to the moment of M/EI diagram between the two points, taken about the point at which deflection is desired. 1999 by CRC Press LLC
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Moment Area Method
Moment area method is most conveniently used for determining slopes and deflections for beams in which the direction of the tangent to the elastic curve at one or more points is known, such as cantilever beams, where the tangent at the fixed end does not change in slope. The method is applied easily to beams loaded with concentrated loads because the moment diagrams consist of straight lines. These diagrams can be broken down into single triangles and rectangles. Beams supporting uniform loads or uniformly varying loads may be handled by integration. Properties of M diagrams designers usually come across are given in Figure 2.11. some of the shapes of EI
FIGURE 2.11: Typical M/EI diagram.
It should be understood that the slopes and deflections that are obtained using the moment area theorems are with respect to tangents to the elastic curve at the points being considered. The theorems do not directly give the slope or deflection at a point in the beam as compared to the horizontal axis (except in one or two special cases); they give the change in slope of the elastic curve from one point to another or the deflection of the tangent at one point with respect to the tangent at another point. There are some special cases in which beams are subjected to several concentrated loads or the combined action of concentrated and uniformly distributed loads. In such cases it is advisable to separate the concentrated loads and uniformly distributed loads and the moment area method can be applied separately to each of these loads. The final responses are obtained by the principle of superposition. For example, consider a simply supported beam subjected to uniformly distributed load q as shown in Figure 2.12. The tangent to the elastic curve at each end of the beam is inclined. The deflection δ1 of the tangent at the left end from the tangent at the right end is found as ql 4 /24EI . The distance from the original chord between the supports and the tangent at right end, δ2 , can be computed as ql 4 /48EI . The deflection of a tangent at the center from a tangent at right end, δ3 , is determined in ql 4 5 ql 4 . The difference between δ2 and δ3 gives the centerline deflection as 384 this step as 128EI EI .
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FIGURE 2.12: Deflectionsimply supported beam under UDL.
2.2.7
Curved Flexural Members
The flexural formula is based on the assumption that the beam to which bending moment is applied is initially straight. Many members, however, are curved before a bending moment is applied to them. Such members are called curved beams. It is important to determine the effect of initial curvature of a beam on the stresses and deflections caused by loads applied to the beam in the plane of initial curvature. In the following discussion, all the conditions applicable to straightbeam formula are assumed valid except that the beam is initially curved. Let the curved beam DOE shown in Figure 2.13 be subjected to the loads Q. The surface in which the fibers do not change in length is called the neutral surface. The total deformations of the fibers between two normal sections such as AB and A1 B1 are assumed to vary proportionally with the distances of the fibers from the neutral surface. The top fibers are compressed while those at the bottom are stretched, i.e., the plane section before bending remains plane after bending. In Figure 2.13 the two lines AB and A1 B1 are two normal sections of the beam before the loads are applied. The change in the length of any fiber between these two normal sections after bending is represented by the distance along the fiber between the lines A1 B1 and A0 B 0 ; the neutral surface is represented by NN1 , and the stretch of fiber P P1 is P 1P10 , etc. For convenience it will be assumed that the line AB is a line of symmetry and does not change direction. The total deformations of the fibers in the curved beam are proportional to the distances of the fibers from the neutral surface. However, the strains of the fibers are not proportional to these distances because the fibers are not of equal length. Within the elastic limit the stress on any fiber in the beam is proportional to the strain of the fiber, and hence the elastic stresses in the fibers of a curved beam are not proportional to the distances of the fibers from the neutral surface. The resisting moment in a curved beam, therefore, is not given by the expression σ I /c. Hence, the neutral axis in a curved beam does not pass through the centroid of the section. The distribution of stress over the section and the relative position of the neutral axis are shown in Figure 2.13b; if the beam were straight, the stress would be zero at the centroidal axis and would vary proportionally with the distance from the 1999 by CRC Press LLC
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FIGURE 2.13: Bending of curved beams. centroidal axis as indicated by the dotdash line in the figure. The stress on a normal section such as AB is called the circumferential stress. Sign Conventions
The bending moment M is positive when it decreases the radius of curvature, and negative when it increases the radius of curvature; y is positive when measured toward the convex side of the beam, and negative when measured toward the concave side, that is, toward the center of curvature. With these sign conventions, σ is positive when it is a tensile stress. Circumferential Stresses
Figure 2.14 shows a free body diagram of the portion of the body on one side of the section; the equations of equilibrium are applied to the forces acting on this portion. The equations obtained are Z X σ da = 0 (2.17) Fz = 0 or Z X Mz = 0 or M = yσ da (2.18) Figure 2.15 represents the part ABB1 A1 of Figure 2.13a enlarged; the angle between the two sections AB and A1 B1 is dθ . The bending moment causes the plane A1 B1 to rotate through an angle 1dθ, thereby changing the angle this plane makes with the plane BAC from dθ to (dθ + 1dθ ); the center of curvature is changed from C to C 0 , and the distance of the centroidal axis from the center of curvature is changed from R to ρ. It should be noted that y, R, and ρ at any section are measured from the centroidal axis and not from the neutral axis. It can be shown that the bending stress σ is given by the relation 1 y M (2.19) 1+ σ = aR ZR+y in which Z=−
1 a
Z
y da R+y
σ is the tensile or compressive (circumferential) stress at a point at the distance y from the centroidal axis of a transverse section at which the bending moment is M; R is the distance from the centroidal 1999 by CRC Press LLC
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FIGURE 2.14: Freebody diagram of curved beam segment.
FIGURE 2.15: Curvature in a curved beam. axis of the section to the center of curvature of the central axis of the unstressed beam; a is the area of the crosssection; Z is a property of the crosssection, the values of which can be obtained from the expressions for various areas given in Table 2.1. Detailed information can be obtained from [51].
EXAMPLE 2.2:
The bent bar shown in Figure 2.16 is subjected to a load P = 1780 N. Calculate the circumferential stress at A and B assuming that the elastic strength of the material is not exceeded. We know from Equation 2.19 M 1 y P + 1+ σ = a aR ZR+y
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TABLE 2.1
Analytical Expressions for Z
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TABLE 2.1
Analytical Expressions for Z (continued)
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TABLE 2.1
Analytical Expressions for Z (continued)
From Seely, F.B. and Smith, J.O., Advanced Mechanics of Materials, John Wiley & Sons, New York, 1952. With permission.
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FIGURE 2.16: Bent bar. in which a = area of rectangular section = 40 × 12 = 480 mm2 R = 40 mm yA = −20 yB = +20 P = 1780 N M = −1780 × 120 = −213600 N mm From Table 2.1, for rectangular section R+c R loge h R−c = 40 mm = 20 mm
Z
=
h c Hence,
−1 +
40 + 20 40 loge = 0.0986 Z = −1 + 40 40 − 20
Therefore,
2.3
σA
=
1780 480
+
−213600 480×40
σB
=
1780 480
+
−213600 480×40
−20 1 0.0986 40−20
1+
20 1 0.0986 40+20
1+
= 105.4 N mm2 (tensile) = −45 N mm2 (compressive)
Trusses
A structure that is composed of a number of bars pin connected at their ends to form a stable framework is called a truss. If all the bars lie in a plane, the structure is a planar truss. It is generally assumed that loads and reactions are applied to the truss only at the joints. The centroidal axis of each member is straight, coincides with the line connecting the joint centers at each end of the member, and lies in a plane that also contains the lines of action of all the loads and reactions. Many truss structures are three dimensional in nature and a complete analysis would require consideration of the full spatial interconnection of the members. However, in many cases, such as bridge structures and simple roof systems, the threedimensional framework can be subdivided into planar components for analysis as planar trusses without seriously compromising the accuracy of the results. Figure 2.17 shows some typical idealized planar truss structures. 1999 by CRC Press LLC
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FIGURE 2.17: Typical planar trusses.
There exists a relation between the number of members, m, number of joints, j , and reaction components, r. The expression is m = 2j − r
(2.20)
which must be satisfied if it is to be statically determinate internally. The least number of reaction components required for external stability is r. If m exceeds (2j − r), then the excess members are called redundant members and the truss is said to be statically indeterminate. Truss analysis gives the bar forces in a truss; for a statically determinate truss, these bar forces can be found by employing the laws of statics to assure internal equilibrium of the structure. The process requires repeated use of freebody diagrams from which individual bar forces are determined. The method of joints is a technique of truss analysis in which the bar forces are determined by the sequential isolation of joints—the unknown bar forces at one joint are solved and become known bar forces at subsequent joints. The other method is known as method of sections in which equilibrium of a part of the truss is considered.
2.3.1
Method of Joints
An imaginary section may be completely passed around a joint in a truss. has become a P The jointP free body in equilibrium under the forces applied to it. The equations H = 0 and V = 0 may be applied to the joint to determine the unknown forces in members meeting there. It is evident that no more than two unknowns can be determined at a joint with these two equations. 1999 by CRC Press LLC
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EXAMPLE 2.3:
A truss shown in Figure 2.18 is symmetrically loaded, and it is sufficient to solve half the truss by considering the joints 1 through 5. At Joint 1, there are two unknown forces. Summation of the
FIGURE 2.18: Example—methods of joints, planar truss.
vertical components of all forces at Joint 1 gives 135 − F12 sin 45 = 0 which in turn gives the force in the member 12, F12 = 190.0 kN (compressive). Similarly, summation of the horizontal components gives F13 − F12 cos 45◦ = 0 Substituting for F12 gives the force in the member 13 as F13 = 135 kN (tensile). 1999 by CRC Press LLC
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Now, Joint 2 is cut completely and it is found that there are two unknown forces F25 and F23 . Summation of the vertical components gives F12 cos 45◦ − F23 = 0. Therefore, F23 = 135 kN (tensile). Summation of the horizontal components gives F12 sin 45◦ − F25 = 0 and hence F25 = 135 kN (compressive). After solving for Joints 1 and 2, one proceeds to take a section around Joint 3 at which there are now two unknown forces, namely, F34 and F35 . Summation of the vertical components at Joint 3 gives F23 − F35 sin 45◦ − 90 = 0 Substituting for F23 , one obtains F35 = 63.6 kN (compressive). Summing the horizontal components and substituting for F13 one gets −135 − 45 + F34 = 0 Therefore, F34 = 180 kN (tensile). The next joint involving two unknowns is Joint 4. When we consider a section around it, the summation of the vertical components at Joint 4 gives F45 = 90 kN (tensile). Now, the forces in all the members on the left half of the truss are known and by symmetry the forces in the remaining members can be determined. The forces in all the members of a truss can also be determined by making use of the method of section.
2.3.2
Method of Sections
If only a few member forces of a truss are needed, the quickest way to find these forces is by the Method of Sections. In this method, an imaginary cutting line called a section is drawn through a stable and determinate truss. Thus, a section subdivides the truss into two separate parts. Since the entire truss is in equilibrium, any part of it must also be in equilibrium. Either P of the two parts P of the P truss can be considered and the three equations of equilibrium Fx = 0, Fy = 0, and M = 0 can be applied to solve for member forces. The example considered in Section 2.3.1 (Figure 2.19) is once again considered. To calculate the force in the member 35, F35 , a section AA should be run to cut the member 35 as shown in the figure. It is only required to consider the equilibrium of one of the two parts of the truss. In this case, the portion of the truss on the left of the section is considered. The left portion of the truss as shown in Figure 2.19 is in equilibrium under the action of the forces, namely, the external and internal forces. Considering the equilibrium of forces in the vertical direction, one can obtain 135 − 90 + F35 sin 45◦ = 0 1999 by CRC Press LLC
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FIGURE 2.19: Example—method of sections, planar truss. Therefore, F35 is obtained as
√ F35 = −45 2 kN
The negative sign indicates that the member force is compressive. This result is the same as the one obtained by the Method of Joints. The other memberP forces cut by the section can be obtained by considering the other equilibrium equations, namely, M = 0. More sections can be taken in the same way so as to solve for other member forces in the truss. The most important advantage of this method is that one can obtain the required member force without solving for the other member forces.
2.3.3
Compound Trusses
A compound truss is formed by interconnecting two or more simple trusses. Examples of compound trusses are shown in Figure 2.20. A typical compound roof truss is shown in Figure 2.20a in which
FIGURE 2.20: Compound truss.
two simple trusses are interconnected by means of a single member and a common joint. The compound truss shown in Figure 2.20b is commonly used in bridge construction and in this case, 1999 by CRC Press LLC
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three members are used to interconnect two simple trusses at a common joint. There are three simple trusses interconnected at their common joints as shown in Figure 2.20c. The Method of Sections may be used to determine the member forces in the interconnecting members of compound trusses similar to those shown in Figure 2.20a and b. However, in the case of cantilevered truss, the middle simple truss is isolated as a free body diagram to find its reactions. These reactions are reversed and applied to the interconnecting joints of the other two simple trusses. After the interconnecting forces between the simple trusses are found, the simple trusses are analyzed by the Method of Joints or the Method of Sections.
2.3.4
Stability and Determinacy
A stable and statically determinate plane truss should have at least three members, three joints, and three reaction components. To form a stable and determinate plane truss of ‘n’ joints, the three members of the original triangle plus two additional members for each of the remaining (n−3) joints are required. Thus, the minimum total number of members, m, required to form an internally stable plane truss is m = 2n−3. If a stable, simple, plane truss of n joints and (2n−3) members is supported by three independent reaction components, the structure is stable and determinate when subjected to a general loading. If the stable, simple, plane truss has more than three reaction components, the structure is externally indeterminate. That means not all of the reaction components can be determined from the three available equations of statics. If the stable, simple, plane truss has more than (2n − 3) members, the structure is internally indeterminate and hence all of the member forces cannot be determined from the 2n available equations of statics in the Method of Joints. The analyst must examine the arrangement of the truss members and the reaction components to determine if the simple plane truss is stable. Simple plane trusses having (2n − 3) members are not necessarily stable.
2.4
Frames
Frames are statically indeterminate in general; special methods are required for their analysis. Slope deflection and moment distribution methods are two such methods commonly employed. Slope deflection is a method that takes into account the flexural displacements such as rotations and deflections and involves solutions of simultaneous equations. Moment distribution on the other hand involves successive cycles of computation, each cycle drawing closer to the “exact” answers. The method is more labor intensive but yields accuracy equivalent to that obtained from the “exact” methods. This method, however, remains the most important handcalculation method for the analysis of frames.
2.4.1
Slope Deflection Method
This method is a special case of the stiffness method of analysis, and it is convenient for hand analysis of small structures. Moments at the ends of frame members are expressed in terms of the rotations and deflections of the joints. Members are assumed to be of constant section between each pair of supports. It is further assumed that the joints in a structure may rotate or deflect, but the angles between the members meeting at a joint remain unchanged. The member forcedisplacement equations that are needed for the slope deflection method are written for a member AB in a frame. This member, which has its undeformed position along the x axis is deformed into the configuration shown in Figure 2.21. The positive axes, along with the positive memberend force components and displacement components, are shown in the figure. 1999 by CRC Press LLC
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FIGURE 2.21: Deformed configuration of a beam.
The equations for end moments are written as MAB
=
MBA
=
2EI (2θA + θB − 3ψAB ) + MF AB l 2EI (2θB + θA − 3ψAB ) + MF BA l
(2.21)
in which MFAB and MFBA are fixedend moments at supports A and B, respectively, due to the applied load. ψAB is the rotation as a result of the relative displacement between the member ends A and B given as 1AB yA + yB (2.22) = ψAB = l l where 1AB is the relative deflection of the beam ends. yA and yB are the vertical displacements at ends A and B. Fixedend moments for some loading cases may be obtained from Figure 2.8. The slope deflection equations in Equation 2.21 show that the moment at the end of a member is dependent on member properties EI , dimension l, and displacement quantities. The fixedend moments reflect the transverse loading on the member.
2.4.2
Application of Slope Deflection Method to Frames
The slope deflection equations may be applied to statically indeterminate frames with or without sidesway. A frame may be subjected to sidesway if the loads, member properties, and dimensions of the frame are not symmetrical about the centerline. Application of slope deflection method can be illustrated with the following example.
EXAMPLE 2.4:
Consider the frame shown in Figure 2.22. subjected to sidesway 1 to the right of the frame. Equation 2.21 can be applied to each of the members of the frame as follows: Member AB: MAB 1999 by CRC Press LLC
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=
2EI 20
31 2θA + θB − + MF AB 20
FIGURE 2.22: Example—slope deflection method.
θA = 0,
MBA
=
MFAB
=
2EI 31 2θB + θA − + MF BA 20 20 MF BA = 0
Hence, MAB
=
MBA
=
2EI (θB − 3ψ) 20 2EI (2θB − 3ψ) 20
in which ψ=
(2.23) (2.24)
1 20
Member BC: MBC
=
MCB
=
MFBC
=
MF CB
=
2EI (2θB + θC − 3 × 0) + MF BC 30 2EI (2θC + θB − 3 × 0) + MF CB 30 40 × 10 × 202 − = −178 ftkips 302 40 × 102 × 20 − = 89 ftkips 302
Hence, MBC
=
MCB
=
Member CD:
1999 by CRC Press LLC
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MCD
=
MDC
=
MF CD
=
2EI (2θB + θC ) − 178 30 2EI (2θC + θB ) + 89 30
2EI 31 2θC + θD − + MF CD 30 30 2EI 31 2θD + θC − + MF DC 30 30 MF DC = 0
(2.25) (2.26)
Hence, MDC
=
MDC
=
2EI θC − 3 × 30 2EI θC − 3 × 30
2 ψ = 3 2 ψ = 3
2EI (2θC − 2ψ) 30 2EI (θC − 2ψ) 30
(2.27) (2.28)
Considering moment equilibrium at Joint B X MB = MBA + MBC = 0 Substituting for MBA and MBC , one obtains EI (10θB + 2θC − 9ψ) = 178 30 or 10θB + 2θC − 9ψ =
267 K
(2.29)
where K = EI 20 . Considering moment equilibrium at Joint C X MC = MCB + MCD = 0 Substituting for MCB and MCD we get 2EI (4θC + θB − 2ψ) = −89 30 or θB + 4θC − 2ψ = −
66.75 K
(2.30)
Summation of base shear equals to zero, we have X H = HA + HD = 0 or
MCD + MDC MAB + MBA + =0 1AB 1CD
Substituting for MAB , MBA , MCD , and MDC and simplifying 2θB + 12θC − 70ψ = 0
(2.31)
Solution of Equations 2.29 to 2.31 results in θB
=
θC
=
and ψ= 1999 by CRC Press LLC
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42.45 K 20.9 K 12.8 K
(2.32)
Substituting for θB , θC , and ψ from Equations 2.32 into Equations 2.23 to 2.28 we get, MAB MBA MBC MCB MCD MDC
2.4.3
= = = = = =
10.10 ftkips 93 ftkips −93 ftkips 90 ftkips −90 ftkips −62 ftkips
Moment Distribution Method
The moment distribution method involves successive cycles of computation, each cycle drawing closer to the “exact” answers. The calculations may be stopped after two or three cycles, giving a very good approximate analysis, or they may be carried on to whatever degree of accuracy is desired. Moment distribution remains the most important handcalculation method for the analysis of continuous beams and frames and it may be solely used for the analysis of small structures. Unlike the slope deflection method, this method does require the solution to simultaneous equations. The terms constantly used in moment distribution are fixedend moments, unbalanced moment, distributed moments, and carryover moments. When all of the joints of a structure are clamped to prevent any joint rotation, the external loads produce certain moments at the ends of the members to which they are applied. These moments are referred to as fixedend moments. Initially the joints in a structure are considered to be clamped. When the joint is released, it rotates if the sum of the fixedend moments at the joint is not zero. The difference between zero and the actual sum of the end moments is the unbalanced moment. The unbalanced moment causes the joint to rotate. The rotation twists the ends of the members at the joint and changes their moments. In other words, rotation of the joint is resisted by the members and resisting moments are built up in the members as they are twisted. Rotation continues until equilibrium is reached—when the resisting moments equal the unbalanced moment—at which time the sum of the moments at the joint is equal to zero. The moments developed in the members resisting rotation are the distributed moments. The distributed moments in the ends of the member cause moments in the other ends, which are assumed fixed, and these are the carryover moments. Sign Convention
The moments at the end of a member are assumed to be positive when they tend to rotate the member clockwise about the joint. This implies that the resisting moment of the joint would be counterclockwise. Accordingly, under gravity loading condition the fixedend moment at the left end is assumed as counterclockwise (−ve) and at the right end as clockwise (+ve). FixedEnd Moments
Fixedend moments for several cases of loading may be found in Figure 2.8. Application of moment distribution may be explained with reference to a continuous beam example as shown in Figure 2.23. Fixedend moments are computed for each of the three spans. At Joint B the unbalanced moment is obtained and the clamp is removed. The joint rotates, thus distributing the unbalanced moment to the Bends of spans BA and BC in proportion to their distribution factors. The values of these distributed moments are carried over at onehalf rate to the other ends of the members. When equilibrium is reached, Joint B is clamped in its new rotated position and Joint C is released afterwards. Joint C rotates under its unbalanced moment until it reaches equilibrium, the rotation causing distributed moments in the Cends of members CB and CD and their resulting carryover 1999 by CRC Press LLC
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FIGURE 2.23: Example—continuous beam by moment distribution. moments. Joint C is now clamped and Joint B is released. This procedure is repeated again and again for Joints B and C, the amount of unbalanced moment quickly diminishing, until the release of a joint causes negligible rotation. This process is called moment distribution. The stiffness factors and distribution factors are computed as follows: DFBA
=
DFBC
=
DFCB
=
DFCD
=
KBA P K KBC P K KCB P K KCD P K
I /20 I /20 + I /30 I /30 = I /20 + I /30 I /30 = I /30 + I /25 I /25 = I /30 + I /25 =
= 0.6 = 0.4 = 0.45 = 0.55
Fixedend moments MFAB MFBA
= −50 ftkips; = 50 ftkips;
MF BC MF CB
= −150 ftkips; = 150 ftkips;
MF CD MF DC
= −104 ftkips = 104 ftkips
When a clockwise couple is applied at the near end of a beam, a clockwise couple of half the magnitude is set up at the far end of the beam. The ratio of the moments at the far and near ends is defined as carryover factor, and it is 21 in the case of a straight prismatic member. The carryover factor was developed for carrying over to fixed ends, but it is applicable to simply supported ends, which must have final moments of zero. It can be shown that the beam simply supported at the far end is only threefourths as stiff as the one that is fixed. If the stiffness factors for end spans that are simply supported are modified by threefourths, the simple end is initially balanced to zero and no carryovers are made to the end afterward. This simplifies the moment distribution process significantly. 1999 by CRC Press LLC
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FIGURE 2.24: Example—nonsway frame by moment distribution.
Moment Distribution for Frames
Moment distribution for frames without sidesway is similar to that for continuous beams. The example shown in Figure 2.24 illustrates the applications of moment distribution for a frame without sidesway. EI /20 = 0.25 DFBA = EI EI 2EI 20 + 20 + 20 Similarly DFBE MFBC MFBE
= 0.50; = −100 ftkips; = 50 ftkips;
DFBC MF CB MF EB
= 0.25 = 100 ftkips = −50 ftkips.
Structural frames are usually subjected to sway in one direction or the other due to asymmetry of the structure and eccentricity of loading. The sway deflections affect the moments resulting in unbalanced moment. These moments could be obtained for the deflections computed and added to the originally distributed fixedend moments. The sway moments are distributed to columns. Should a frame have columns all of the same length and the same stiffness, the sidesway moments will be the same for each column. However, should the columns have differing lengths and/or stiffness, this will not be the case. The sidesway moments should vary from column to column in proportion to their I / l 2 values. 1999 by CRC Press LLC
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The frame in Figure 2.25 shows a frame subjected to sway. The process of obtaining the final moments is illustrated for this frame. The frame sways to the right and the sidesway moment can be assumed in the ratio 400 300 : (or) 1 : 0.7 202 202 Final moments are obtained by adding distributed fixedend moments and tributed assumed sidesway moments.
2.4.4
13.06 2.99
times the dis
Method of Consistent Deformations
The method of consistent deformations makes use of the principle of deformation compatibility to analyze indeterminate structures. This method employs equations that relate the forces acting on the structure to the deformations of the structure. These relations are formed so that the deformations are expressed in terms of the forces and the forces become the unknowns in the analysis. Let us consider the beam shown in Figure 2.26a. The first step, in this method, is to determine the degree of indeterminacy or the number of redundants that the structure possesses. As shown in the figure, the beam has three unknown reactions, RA , RC , and MA . Since there are only two equations of equilibrium available for calculating the reactions, the beam is said to be indeterminate to the first degree. Restraints that can be removed without impairing the loadsupporting capacity of the structure are referred to as redundants. Once the number of redundants is known, the next step is to decide which reaction is to be removed in order to form a determinate structure. Any one of the reactions may be chosen to be the redundant provided that a stable structure remains after the removal of that reaction. For example, let us take the reaction RC as the redundant. The determinate structure obtained by removing this restraint is the cantilever beam shown in Figure 2.26b. We denote the deflection at end C of this beam, due to P , by 1CP . The first subscript indicates that the deflection is measured at C and the second subscript that the deflection is due to the applied load P . Using the moment area method, it can be shown that 1CP = 5P L3 /48EI . The redundant RC is then applied to the determinate cantilever beam, as shown in Figure 2.26c. This gives rise to a deflection 1CR at point C the magnitude of which can be shown to be RC L3 /3EI . In the actual indeterminate structure, which is subjected to the combined effects of the load P and the redundant RC , the deflection at C is zero. Hence the algebraic sum of the deflection 1CP in Figure 2.26b and the deflection 1CR in Figure 2.26c must vanish. Assuming downward deflections to be positive, we write (2.33) 1CP − 1CR = 0 or
RC L3 5P L3 − =0 48EI 3EI
from which
5 P 16 Equation 2.33, which is used to solve for the redundant, is referred to as an equation of consistent of deformation. reactions by applying Once the redundant RC has been evaluated, one can determine the remaining P the equations of equilibrium to the structure in Figure 2.26a. Thus, Fy = 0 leads to RC =
RA = P − 1999 by CRC Press LLC
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11 5 P = P 16 16
FIGURE 2.25: Example—sway frame by moment distribution.
1999 by CRC Press LLC
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FIGURE 2.25: (Continued) Example—sway frame by moment distribution.
1999 by CRC Press LLC
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FIGURE 2.26: Beam with one redundant reaction. and
P
MA = 0 gives
PL 5 3 − PL = PL 2 16 16 A free body of the beam, showing all the forces acting on it, is shown in Figure 2.26d. The steps involved in the method of consistent deformations are: MA =
1. The number of redundants in the structure is determined. 2. Enough redundants are removed to form a determinate structure. 3. The displacements that the applied loads cause in the determinate structure at the points where the redundants have been removed are then calculated. 4. The displacements at these points in the determinate structure due to the redundants are obtained. 5. At each point where a redundant has been removed, the sum of the displacements calculated in Steps 3 and 4 must be equal to the displacement that exists at that point in the actual indeterminate structure. The redundants are evaluated using these relationships. 6. Once the redundants are known, the remaining reactions are determined using the equations of equilibrium. Structures with Several Redundants
The method of consistent deformations can be applied to structures with two or more redundants. For example, the beam in Figure 2.27a is indeterminate to the second degree and has two redundant reactions. If we let the reactions at B and C be the redundants, then the determinate structure obtained by removing these supports is the cantilever beam shown in Figure 2.27b. To this determinate structure we apply separately the given load (Figure 2.27c) and the redundants RB and RC one at a time (Figures 2.27d and e). Since the deflections at B and C in the original beam are zero, the algebraic sum of the deflections in Figures 2.27c, d, and e at these same points must also vanish. Thus, 1BP − 1BB − 1BC 1CP − 1CB − 1CC 1999 by CRC Press LLC
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= =
0 0
(2.34)
FIGURE 2.27: Beam with two redundant reactions.
1999 by CRC Press LLC
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It is useful in the case of complex structures to write the equations of consistent deformations in the form 1BP − δBB RB − δBC RC 1CP − δCB RB − δCC RC
= =
0 0
(2.35)
in which δBC , for example, denotes the deflection at B due to a unit load at C in the direction of RC . Solution of Equation 2.35 gives the redundant reactions RB and RC .
EXAMPLE 2.5:
Determine the reactions for the beam shown in Figure 2.28 and draw its shear force and bending moment diagrams. It can be seen from the figure that there are three reactions, namely, MA , RA , and RC one more than that required for a stable structure. The reaction RC can be removed to make the structure determinate. We know that the deflection at support C of the beam is zero. One can determine the deflection δCP at C due to the applied load on the cantilever in Figure 2.28b. The deflection δCR at C due to the redundant reaction on the cantilever (Figure 2.28c) can be determined in the same way. The compatibility equation gives δCP − δCR = 0 By moment area method, δCP
=
δCR
=
1 20 2 20 ×2×1+ × ×2× ×2 EI 2 EI 3 1 60 2 40 ×2×3+ × ×2× ×2+2 + EI 2 EI 3 1520 = 3EI 1 4RC 2 64RC × ×4× ×4= 2 EI 3 3EI
Substituting for δCP and δCR in the compatibility equation one obtains 1520 64RC − =0 3EI 3EI from which RC = 23.75 kN ↑ By using statical equilibrium equations we get RA = 6.25 kN ↑ and MA = 5 kNm. The shear force and bending moment diagrams are shown in Figure 2.28d. 1999 by CRC Press LLC
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FIGURE 2.28: Example 2.5.
1999 by CRC Press LLC
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2.5 2.5.1
Plates Bending of Thin Plates
When the thickness of an object is small compared to the other dimensions, it is called a thin plate. The plane parallel to the faces of the plate and bisecting the thickness of the plate, in the undeformed state, is called the middle plane of the plate. When the deflection of the middle plane is small compared with the thickness, h, it can be assumed that 1. There is no deformation in the middle plane. 2. The normal of the middle plane before bending is deformed into the normals of the middle plane after bending. 3. The normal stresses in the direction transverse to the plate can be neglected. Based on these assumptions, all stress components can be expressed by deflection w0 of the plate. is a function of the two coordinates (x, y) in the plane of the plate. This function has to satisfy a linear partial differential equation, which, together with the boundary conditions, completely defines w0 . Figure 2.29a shows a plate element cut from a plate whose middle plane coincides with the xy plane. The middle plane of the plate subjected to a lateral load of intensity ‘q’ is shown in Figure 2.29b. It can be shown, by considering the equilibrium of the plate element, that the stress resultants are given as 2 ∂ w ∂ 2w +ν 2 Mx = −D ∂x 2 ∂y 2 ∂ w ∂ 2w + ν My = −D ∂y 2 ∂x 2
w0
where Mx and My Mxy and Myx Qx and Qy Vx and Vy R D E
= = = = = = =
1999 by CRC Press LLC
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Mxy
=
Vx
=
Vy
=
Qx
=
Qy
=
R
=
∂ 2w ∂x∂y 3 3 ∂ w ∂ w + (2 − ν) ∂x 3 ∂x∂y 2 ∂ 3w ∂ 3w + (2 − ν) ∂y 3 ∂y∂x 2 2 ∂ ∂ w ∂ 2w −D + ∂x ∂x 2 ∂y 2 2 ∂ ∂ w ∂ 2w −D + ∂y ∂x 2 ∂y 2 −Myx = D(1 − ν)
2D(1 − ν)
∂ 2w ∂x∂y
bending moments per unit length in the x and y directions, respectively twisting moments per unit length shearing forces per unit length in the x and y directions, respectively supplementary shear forces in the x and y directions, respectively corner force Eh3 , flexural rigidity of the plate per unit length 12(1−ν 2 ) modulus of elasticity
(2.36) (2.37) (2.38) (2.39) (2.40) (2.41)
FIGURE 2.29: (a) Plate element; (b) stress resultants.
ν
= Poisson’s Ratio The governing equation for the plate is obtained as ∂ 4w ∂ 4w q ∂ 4w + 2 + = 4 2 2 4 D ∂x ∂x ∂y ∂y
(2.42)
Any plate problem should satisfy the governing Equation 2.42 and boundary conditions of the plate.
2.5.2
Boundary Conditions
There are three basic boundary conditions for plate problems. These are the clamped edge, the simply supported edge, and the free edge. 1999 by CRC Press LLC
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Clamped Edge
For this boundary condition, the edge is restrained such that the deflection and slope are zero along the edge. If we consider the edge x = a to be clamped, we have (w)x=a = 0
∂w ∂x
x=a
=0
(2.43)
Simply Supported Edge
If the edge x = a of the plate is simply supported, the deflection w along this edge must be zero. At the same time this edge can rotate freely with respect to the edge line. This means that (w)x=a = 0;
∂ 2w ∂x 2
x=a
=0
(2.44)
Free Edge
If the edge x = a of the plate is entirely free, there are no bending and twisting moments or vertical shearing forces. This can be written in terms of w, the deflection as
2.5.3
∂ 2w ∂ 2w + ν ∂x 2 ∂y 2
x=a
∂ 3w ∂ 3w + (2 − ν) 3 ∂x ∂x∂y 2
=0
x=a
=0
(2.45)
Bending of Simply Supported Rectangular Plates
A number of the plate bending problems may be solved directly by solving the differential Equation 2.42. The solution, however, depends on the loading and boundary condition. Consider a simply supported plate subjected to a sinusoidal loading as shown in Figure 2.30. The differential
FIGURE 2.30: Rectangular plate under sinusoidal loading.
Equation 2.42 in this case becomes πx πy ∂ 4w ∂ 4w qo ∂ 4w sin sin + 2 + = 4 2 2 4 D a b ∂x ∂x ∂y ∂y 1999 by CRC Press LLC
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(2.46)
The boundary conditions for the simply supported edges are w
= 0,
w
= 0,
∂ 2w = 0 for x ∂x 2 ∂ 2w = 0 for y ∂y 2
= 0 and x = a = 0 and y = b
(2.47)
The deflection function becomes
πy πx sin (2.48) a b which satisfies all the boundary conditions in Equation 2.47. w0 must be chosen to satisfy Equation 2.46. Substitution of Equation 2.48 into Equation 2.46 gives w = w0 sin
π4
1 1 + 2 a2 b
2 w0 =
qo D
The deflection surface for the plate can, therefore, be found as w=
π 4D
qo 1 a2
+
1 b2
πx πy sin a b
2 sin
(2.49)
Using Equations 2.49 and 2.36, we find expression for moments as πy qo ν 1 πx sin + sin Mx = 2 2 2 a b a b π 2 a12 + b12 qo 1 πy ν πx + 2 sin sin My = 2 2 a b a b π 2 a12 + b12 Mxy
=
πy πx qo (1 − ν) cos cos 2 a b π 2 a12 + b12 ab
(2.50)
Maximum deflection and maximum bending moments that occur at the center of the plate can be written by substituting x = a/2 and y = b/2 in Equation 2.50 as wmax (Mx )max (My )max
=
π 4D
=
π2
=
π2
qo 1 a2
+
qo 1 a2
+
1 b2
qo 1 a2
+
1 b2
1 b2
2
2 2
(2.51)
ν 1 + 2 a2 b 1 ν + 2 2 a b
If the plate is square, then a = b and Equation 2.51 becomes
1999 by CRC Press LLC
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wmax
=
qo a 4 4π 4 D 0
(Mx )max
=
(My )max =
(1 + ν) qo a 2 4π 2
(2.52)
If the simply supported rectangular plate is subjected to any kind of loading given by q = q(x, y)
(2.53)
the function q(x, y) should be represented in the form of a double trigonometric series as q(x, y) =
∞ ∞ X X
qmn sin
m=1 n=1
mπ x nπy sin a b
(2.54)
in which qmn is given by qmn =
Z
4 ab
a
Z
0
b
q(x, y) sin
0
nπy mπ x sin dxdy a b
(2.55)
From Equations 2.46, 2.53, 2.54, and 2.55 we can obtain the expression for deflection as w=
∞ ∞ mπ x qmn nπy 1 XX sin 2 2 sin 2 a b π 4D m=1 n=1 m2 + n2 a b
(2.56)
If the applied load is uniformly distributed of intensity qo , we have q(x, y) = qo and from Equation 2.55 we obtain qmn =
4qo ab
Z
a
Z
b
sin 0
0
nπy 16qo mπ x sin dxdy = 2 a b π mn
(2.57)
in which ‘m’ and ‘n’ are odd integers. qmn = 0 if ‘m’ or ‘n’ or both of them are even numbers. We can, therefore, write the expression for deflection of a simply supported plate subjected to uniformly distributed load as ∞ ∞ nπy 16qo X X sin mπa x sin b (2.58) w= 6 2 2 2 π D m n m=1 n=1 mn + b2 a2 where m = 1, 3, 5, . . . and n = 1, 3, 5, . . . The maximum deflection occurs at the center and it can be written by substituting x = y = b2 in Equation 2.58 as wmax
∞ ∞ 16qo X X = 6 π D m=1 n=1
a 2
and
m+n
(−1) 2 −1 2 2 n2 mn m + a2 b2
(2.59)
Equation 2.59 is a rapid converging series and a satisfactory approximation can be obtained by taking only the first term of the series; for example, in the case of a square plate, wmax =
qo a 4 4qo a 4 = 0.00416 D π 6D
Assuming ν = 0.3, we get for the maximum deflection wmax = 0.0454 1999 by CRC Press LLC
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qo a 4 Eh3
FIGURE 2.31: Typical loading on plates and loading functions.
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FIGURE 2.31: (Continued) Typical loading on plates and loading functions.
FIGURE 2.32: Rectangular plate.
The expressions for bending and twisting moments can be obtained by substituting Equation 2.58 into Equation 2.36. Figure 2.31 shows some loading cases and the corresponding loading functions. The above solution for uniformly loaded cases is known as Navier solution. If two opposite sides (say x = 0 and x = a) of a rectangular plate are simply supported, the solution taking the deflection function as ∞ X mπ x Ym sin (2.60) w= a m=1
1999 by CRC Press LLC
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can be adopted. This solution was proposed by Levy [53]. Equation 2.60 satisfies the boundary 2 conditions w = 0 and ∂∂xw2 = 0 on the two simply supported edges. Ym should be determined such that it satisfies the boundary conditions along the edges y = ± b2 of the plate shown in Figure 2.32 and also the equation of the deflection surface ∂ 4w ∂ 4w qo ∂ 4w +2 2 2 + = 4 D ∂x ∂x ∂y ∂y 4
(2.61)
qo being the intensity of uniformly distributed load. The solution for Equation 2.61 can be taken in the form w = w1 + w2
(2.62)
for a uniformly loaded simply supported plate. w1 can be taken in the form qo 4 x − 2ax 3 + a 3 x w1 = 24D
(2.63)
representing the deflection of a uniformly loaded strip parallel to the x axis. It satisfies Equation 2.61 and also the boundary conditions along x = 0 and x = a. The expression w2 has to satisfy the equation ∂ 4 w2 ∂ 4 w2 ∂ 4 w2 + 2 + =0 ∂x 4 ∂x 2 ∂y 2 ∂y 4
(2.64)
and must be chosen such that Equation 2.62 satisfies all boundary conditions of the plate. Taking w2 in the form of series given in Equation 2.60 it can be shown that the deflection surface takes the form wψ =
∞ q a4 X mπy qo 4 o x − 2ax 3 + a 3 x + Am cosh 24D 24D a m=1
mπy mπy mπy sinh + Cm sinh + Bm a a a mπy mπy mπ x +Dm cosh sin a a a
(2.65)
Observing that the deflection surface of the plate is symmetrical with respect to the x axis, we keep in Equation 2.65 only an even function of y; therefore, Cm = Dm = 0. The deflection surface takes the form ∞ q a4 X mπy qo 4 o 3 3 x − 2ax + a x + Am cosh w= 24D 24D a m=1 mπy mπ x mπy sinh sin +Bm a a a
(2.66)
Developing the expression in Equation 2.63 into a trigonometric series, the deflection surface in Equation 2.66 is written as ∞ mπy mπx mπy 4 mπy qo a 4 X + Bm sin sin + Am cosh (2.67) w= 5 5 D a a a a π m m=1
Substituting Equation 2.67 in the boundary conditions w = 0, 1999 by CRC Press LLC
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∂ 2w =0 ∂y 2
(2.68)
one obtains the constants of integration Am and Bm and the expression for deflection may be written as ∞ 2αm y 1 αm tanh αm + 2 4qo a 4 X cosh 1− w= 5 5 2 cosh αm b π D m m=1,3,5... 2αm y mπ x 2y αm sinh sin (2.69) + 2 cosh αm b b a b in which αm = mπ 2a . Maximum deflection occurs at the middle of the plate, x = a2 , y = 0 and is given by
4qo a 4 w= 5 π D
∞ X m=1,3,5...
m−1
(−1) 2 m5
αm tanh αm + 2 1− 2 cosh αm
(2.70)
Solution of plates with arbitrary boundary conditions are complicated. It is possible to make some simplifying assumptions for plates with the same boundary conditions along two parallel edges in order to obtain the desired solution. Alternately, the energy method can be applied more efficiently to solve plates with complex boundary conditions. However, it should be noted that the accuracy of results depends upon the deflection function chosen. These functions must be so chosen that they satisfy at least the kinematics boundary conditions. Figure 2.33 gives formulas for deflection and bending moments of rectangular plates with typical boundary and loading conditions.
2.5.4
Bending of Circular Plates
In the case of symmetrically loaded circular plate, the loading is distributed symmetrically about the axis perpendicular to the plate through its center. In such cases, the deflection surface to which the middle plane of the plate is bent will also be symmetrical. The solution of circular plates can be conveniently carried out by using polar coordinates. Stress resultants in a circular plate element are shown in Figure 2.34. The governing differential equation is expressed in polar coordinates as d 1 d dw q 1 d r r = (2.71) r dr dr r dr dr D in which q is the intensity of loading. In the case of uniformly loaded circular plates, Equation 2.71 can be integrated successively and the deflection at any point at a distance r from the center can be expressed as w=
r C1 r 2 qo r 4 + + C2 log + C3 64D 4 a
(2.72)
in which qo is the intensity of loading and a is the radius of the plate. C1 , C2 , and C3 are constants of integration to be determined using the boundary conditions. For a plate with clamped edges under uniformly distributed load qo , the deflection surface reduces to 2 qo 2 a − r2 (2.73) w= 64D The maximum deflection occurs at the center where r = 0, and is given by w= 1999 by CRC Press LLC
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qo a 4 64D
(2.74)
FIGURE 2.33: Typical loading and boundary conditions for rectangular plates. Bending moments in the radial and tangential directions are respectively given by i qo h 2 a (1 + ν) − r 2 (3 + ν) Mr = 16 h i qo 2 a (1 + ν) − r 2 (1 + 3ν) Mt = 16
(2.75)
The method of superposition can be applied in calculating the deflections for circular plates with 1999 by CRC Press LLC
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FIGURE 2.34: (a) Circular plate; (b) stress resultants. simply supported edges. The expressions for deflection and bending moment are given as follows: qo (a 2 − r 2 ) 5 + ν 2 a − r2 w = 64D 1+ν wmax
=
Mr
=
Mt
=
5 + ν qo a 4 64(1 + ν) D qo (3 + ν)(a 2 − r 2 ) 16 h i qo 2 a (3 + ν) − r 2 (1 + 3ν) 16
(2.76)
(2.77)
This solution can be used to deal with plates with circular holes at the center and subjected to concentric moment and shearing forces. Plates subjected to concentric loading and concentrated loading also can be solved by this method. More rigorous solutions are available to deal with irregular loading on circular plates. Once again energy method can be employed advantageously to solve circular plate problems. Figure 2.35 gives deflection and bending moment expressions for typical cases of loading and boundary conditions on circular plates. 1999 by CRC Press LLC
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FIGURE 2.35: Typical loading and boundary conditions for circular plates.
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FIGURE 2.35: (Continued) Typical loading and boundary conditions for circular plates.
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2.5.5 Strain Energy of Simple Plates The strain energy expression for a simple rectangular plate is given by ( 2 Z Z ∂ 2w ∂ 2w D + Uψ = 2 ∂x 2 ∂y 2 area " 2 2 #) ∂ 2w ∂ 2w ∂ w dxdyψ −2(1 − ν) − ∂x∂y ∂x 2 ∂y 2
(2.78)
Suitable deflection function w(x, y) satisfying the boundary conditions of the given plate may be chosen. The strain energy, U , and the work done by the given load, q(x, y), Z Z (2.79) W =− q(x, y)w(x, y)dxdyψ area can be calculated. The total potential energy is, therefore, given as V = U + W . Minimizing the total potential energy the plate problem can be solved. " 2 2 # ∂ w ∂ 2w ∂ 2w − ∂x∂y ∂x 2 ∂y 2 The term is known as the Gaussian curvature. If the function w(x, y) = f (x) · φ(y) (product of a function of x only and a function of y only) and w = 0 at the boundary are assumed, then the integral of the Gaussian curvature over the entire plate equals zero. Under these conditions U=
D 2
Z Z
∂ 2w ∂ 2w + ∂x 2 ∂y 2
area
2 dxdy
If polar coordinates instead of rectangular coordinates are used and axial symmetry of loading and deformation is assumed, the equation for strain energy, U , takes the form ( ) 2 Z Z 2(1 − ν) ∂w ∂ 2 w ∂ 2 w 1 ∂w D + − (2.80) rdrdθψ U= 2 r ∂r r ∂r ∂r 2 ∂r 2 area and the work done, W , is written as
Z Z W =−
area
qwrdrdθψ
(2.81)
Detailed treatment of the Plate Theory can be found in [56].
2.5.6
Plates of Various Shapes and Boundary Conditions
Simply Supported Isosceles Triangular Plate Subjected to a Concentrated Load
Plates of shapes other than circle and rectangle are used in some situations. A rigorous solution of the deflection for a plate with a more complicated shape is likely to be very difficult. Consider, for example, the bending of an isosceles triangular plate with simply supported edges under concentrated load P acting at an arbitrary point (Figure 2.36). A solution can be obtained for this plate by considering a mirror image of the plate as shown in the figure. The deflection of OBC of the square 1999 by CRC Press LLC
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FIGURE 2.36: Isosceles triangular plate. plate is identical with that of a simply supported triangular plate OBC. The deflection owing to the force P can be written as w1 =
∞ ∞ 4P a 2 X X sin(mπ x1 /a) sin(nπy1 /a) mπ x nπy sin sin a a π 4D (m2 + n2 )2
(2.82)
m=1 n=1
Upon substitution of −P for P , (a − y1 ) for x1 , and (a − x1) for y1 in Equation 2.82 we obtain the deflection due to the force −P at Ai : ∞ ∞ 4P a 2 X X mπ x sin(mπ x1 /a) sin(nπy1 /a) nπy (−1)m+n sin sin w2 = − 4 a a π D (m2 + n2 )2
(2.83)
m=1 n=1
The deflection surface of the triangular plate is then w = w1 + w2
(2.84)
Equilateral Triangular Plates
The deflection surface of a simply supported plate loaded by uniform moment Mo along its boundary and the surface of a uniformly loaded membrane, uniformly stretched over the same triangular boundary, are identical. The deflection surface for such a case can be obtained as 4 Mo x 3 − 3xy 2 − a(x 2 + y 2 ) + a 3 (2.85) w= 4aD 27 If the simply supported plate is subjected to uniform load po the deflection surface takes the form 4 3 4 2 po 3 2 2 2 2 2 x − 3xy − a(x + y ) + a a −x −y (2.86) w= 64aD 27 9 For the equilateral triangular plate (Figure 2.37) subjected to uniform load and supported at the corners approximate solutions based on the assumption that the total bending moment along each 1999 by CRC Press LLC
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FIGURE 2.37: Equilateral triangular plate with coordinate axes. side of the triangle vanishes were obtained by Vijakkhana et al. [58] who derived an equation for deflection surface as 8 qa 4 (7 + ν)(2 − ν) − (7 + ν)(1 − ν) w = 2 144(1 − ν )D 27 2 3 y2 xy 2 x x + − 3 − (5 − ν)(1 + ν) a2 a2 a3 a3 4 x x2y2 y4 9 + 2 + (2.87) + (1 − ν 2 ) 4 a4 a4 a4 The errors introduced by the approximate boundary condition, i.e., the total bending moment along each side of the triangle vanishes, are not significant because its influence on the maximum deflection and stress resultants is small for practical design purposes. The value of the twisting moment on the edge at the corner given by this solution is found to be exact. The details of the mathematical treatment may be found in [58]. Rectangular Plate Supported at Corners
Approximate solutions for rectangular plates supported at the corners and subjected to uniformly distributed load were obtained by Lee and Ballesteros [36]. The approximate deflection surface is given as b4 b2 qa 4 2 (10 + ν − ν ) 1 + − 2(7ν − 1) 2 w = 2 4 48(1 − ν )D a a 2 b x + 2 (1 + 5ν) 2 − (6 + ν − ν 2 ) a a 2 2 b y + 2 (1 + 5ν) − (6 + ν − ν 2 ) 2 a a2 4 4 x +y x2y2 +(2 + ν − ν 2 ) − 6(1 + ν) (2.88) a4 a4 The details of the mathematical treatment may be found in [36]. 1999 by CRC Press LLC
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2.5.7
Orthotropic Plates
Plates of anisotropic materials have important applications owing to their exceptionally high bending stiffness. A nonisotropic or anisotropic material displays directiondependent properties. Simplest among them are those in which the material properties differ in two mutually perpendicular directions. A material so described is orthotropic, e.g., wood. A number of manufactured materials are approximated as orthotropic. Examples include corrugated and rolled metal sheets, fillers in sandwich plate construction, plywood, fiber reinforced composites, reinforced concrete, and gridwork. The latter consists of two systems of equally spaced parallel ribs (beams), mutually perpendicular, and attached rigidly at the points of intersection. The governing equation for orthotropic plates similar to that of isotropic plates (Equation 2.42) takes the form δ4w δ4w δ4w (2.89) Dx 4 + 2H 2 2 + Dy 4 = q δx δx δy δy In which h3 Ey h3 Exy h3 Ex h3 G , Dy = , H = Dxy + 2Gxy , Dxy = , Gxy = 12 12 12 12 The expressions for Dx , Dy , Dxy , and Gxy represent the flexural rigidities and the torsional rigidity of an orthotropic plate, respectively. Ex , Ey , and G are the orthotropic plate moduli. Practical considerations often lead to assumptions, with regard to material properties, resulting in approximate expressions for elastic constants. The accuracy of these approximations is generally the most significant factor in the orthotropic plate problem. Approximate rigidities for some cases that are commonly encountered in practice are given in Figure 2.38. General solution procedures applicable to the case of isotropic plates are equally applicable to the orthotropic plates as well. Deflections and stressresultants can thus be obtained for orthotropic plates of different shapes with different support and loading conditions. These problems have been researched extensively and solutions concerning plates of various shapes under different boundary and loading conditions may be found in the references, namely [37, 52, 53, 56, 57]. Dx =
2.5.8
Buckling of Thin Plates
Rectangular Plates
Buckling of a plate involves bending in two planes and is therefore fairly complicated. From a mathematical point of view, the main difference between columns and plates is that quantities such as deflections and bending moments, which are functions of a single independent variable, in columns become functions of two independent variables in plates. Consequently, the behavior of plates is described by partial differential equations, whereas ordinary differential equations suffice for describing the behavior of columns. A significant difference between columns and plates is also apparent if one compares their buckling characteristics. For a column, buckling terminates the ability of the member to resist axial load, and the critical load is thus the failure load of the member. However, the same is not true for plates. These structural elements can, subsequently to reaching the critical load, continue to resist increasing axial force, and they do not fail until a load considerably in excess of the critical load is reached. The critical load of a plate is, therefore, not its failure load. Instead, one must determine the loadcarrying capacity of a plate by considering its postbuckling behavior. To determine the critical inplane loading of a plate by the concept of neutral equilibrium, a governing equation in terms of biaxial compressive forces Nx and Ny and constant shear force Nxy as shown in Figure 2.39 can be derived as 4 δ4w δ4w δ2 w δ2 w δ2 w δ w =0 + 2 + + N + 2N (2.90) + N D x y xy δxδy δx 4 δx 2 δy 2 δy 4 δx 2 δy 2 1999 by CRC Press LLC
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FIGURE 2.38: Various orthotropic plates.
The critical load for uniaxial compression can be determined from the differential equation 4 δ4w δ4w δ2 w δ w + 2 + (2.91) + Nx 2 = 0 D 4 2 2 4 δx δx δy δy δx which is obtained by setting Ny = Nxy = 0 in Equation 2.90. For example, in the case of a simply supported plate Equation 2.91 can be solved to give 2 π 2 a 2 D m2 n2 + 2 Nx = m2 a2 b 1999 by CRC Press LLC
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(2.92)
FIGURE 2.39: Plate subjected to inplane forces. The critical value of Nx , i.e., the smallest value, can be obtained by taking n equal to 1. The physical meaning of this is that a plate buckles in such a way that there can be several halfwaves in the direction of compression but only one halfwave in the perpendicular direction. Thus, the expression for the critical value of the compressive force becomes 2 π 2D 1 a2 (2.93) m+ (Nx )cr = 2 m b2 a The first factor in this expression represents the Euler load for a strip of unit width and of length a. The second factor indicates in what proportion the stability of the continuous plate is greater than the stability of an isolated strip. The magnitude of this factor depends on the magnitude of the ratio a/b and also on the number m, which gives the number of halfwaves into which the plate buckles. If ‘a’ is smaller than ‘b’, the second term in the parenthesis of Equation 2.93 is always smaller than the first and the minimum value of the expression is obtained by taking m = 1, i.e., by assuming that the plate buckles in one halfwave. The critical value of Nx can be expressed as Ncr =
kπ 2 D b2
(2.94)
The factor k depends on the aspect ratio a/b of the plate and m, the number of halfwaves into which the plate buckles in the x direction. The variation of k with a/b for different values of m can be plotted, as shown in Figure 2.40. The critical value of Nx is the smallest value that is obtained for m = 1 and the corresponding value of k is equal to 4.0. This formula is analogous to Euler’s formula for buckling of a column. In the more general case in which normal forces Nx and Ny and the shearing forces Nxy are acting on the boundary of the plate, the same general method can be used. The critical stress for the case of a uniaxially compressed simply supported plate can be written as 2 h π 2E (2.95) σcr = 4 12(1 − ν 2 ) b The critical stress values for different loading and support conditions can be expressed in the form 2 h π 2E (2.96) fcr = k 12(1 − ν 2 ) b in which fcr is the critical value of different loading cases. Values of k for plates with several different boundary and loading conditions are given in Figure 2.41. 1999 by CRC Press LLC
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FIGURE 2.40: Buckling stress coefficients for unaxially compressed plate.
Circular Plates
The critical value of the compressive forces Nr uniformly distributed around the edge of a circular plate of radius ro , clamped along the edge (Figure 2.42) can be determined by using the governing equation Qr 2 dφ d 2φ −φ =− (2.97) r2 2 + r dr D dr in which φ is the angle between the axis of revolution of the plate surface and any normal to the plate, r is the distance of any point measured from the center of the plate, and Q is the shearing force per unit of length. When there are no lateral forces acting on the plate, the solution of Equation 2.97 involves a Bessel function of the first order of the first and second kind and the resulting critical value of Nr is obtained as 14.68D (2.98) (Nr )cr = r02 The critical value of Nr for the plate when the edge is simply supported can be obtained in the same way as 4.20D (2.99) (Nr )cr = r02
2.6 2.6.1
Shell Stress Resultants in Shell Element
A thin shell is defined as a shell with a thickness that is relatively small compared to its other dimensions. Also, deformations should not be large compared to the thickness. The primary difference between a shell structure and a plate structure is that the former has a curvature in the unstressed state, whereas the latter is assumed to be initially flat. The presence of initial curvature is of little consequence as far as flexural behavior is concerned. The membrane behavior, however, is affected significantly by the curvature. Membrane action in a surface is caused by inplane forces. These forces may be primary forces caused by applied edge loads or edge deformations, or they may be secondary forces resulting from flexural deformations. 1999 by CRC Press LLC
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FIGURE 2.41: Values of K for plate with different boundary and loading conditions.
In the case of the flat plates, secondary inplane forces do not give rise to appreciable membrane action unless the bending deformations are large. Membrane action due to secondary forces is, therefore, neglected in small deflection theory. If the surface, as in the case of shell structures, has an initial curvature, membrane action caused by secondary inplane forces will be significant regardless of the magnitude of the bending deformations. A plate is likened to a twodimensional beam and resists transverse loads by two dimensional bending and shear. A membrane is likened to a twodimensional equivalent of the cable and resists loads through tensile stresses. Imagine a membrane with large deflections (Figure 2.43a), reverse the load and the membrane and we have the structural shell (Figure 2.43b) provided that the shell is stable for the type of load shown. The membrane resists the load through tensile stresses but the ideal thin shell must be capable of developing both tension and compression. Consider an infinitely small shell element formed by two pairs of adjacent planes which are normal to the middle surface of the shell and which contain its principal curvatures as shown in Figure 2.44a. The thickness of the shell is denoted as h. Coordinate axes x and y are taken tangent at ‘O’ to the lines of principal curvature and the axis z normal to the middle surface. rx and ry are the principal 1999 by CRC Press LLC
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FIGURE 2.42: Circular plate under compressive loading.
FIGURE 2.43: Membrane with large deflections. radii of curvature lying in the xz and yz planes, respectively. The resultant forces per unit length of the normal sections are given as Z =
Nx
h/2
−h/2 h/2
σx
z 1− ry
dz,
z τxy 1 − dz, ry −h/2 Z h/2 z = τxz 1 − dz, ry −h/2 Z
=
Nxy Qx
z = σy 1 − dz rx −h/2 Z h/2 z = τyx 1 − dz rx −h/2 Z h/2 z = τyz 1 − dz rx −h/2 Z
Ny Nyx Qy
h/2
(2.100)
The bending and twisting moments per unit length of the normal sections are given by z σx z 1 − dz, My ry −h/2 Z h/2 z =− τxy z 1 − dz, Myx ry −h/2 Z
Mx Mxy
=
h/2
z σy z 1 − dz rx −h/2 Z h/2 z = τyx z 1 − dz rx −h/2 Z
=
h/2
(2.101)
It is assumed, in bending of the shell, that linear elements as AD and BC (Figure 2.44), which are normal to the middle surface of the shell, remain straight and become normal to the deformed middle surface of the shell. If the conditions of a shell are such that bending can be neglected, the 1999 by CRC Press LLC
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FIGURE 2.44: A shell element. problem of stress analysis is greatly simplified because the resultant moments (Equation 2.101) vanish along with shearing forces Qx and Qy in Equation 2.100. Thus, the only unknowns are Nx , Ny , and Nxy = Nyx and these are called membrane forces.
2.6.2
Membrane Theory of Shells of Revolution
Shells having the form of surfaces of revolution find extensive application in various kinds of containers, tanks, and domes. Consider an element of a shell cut by two adjacent meridians and two parallel circles as shown in Figure 2.45. There will be no shearing forces on the sides of the element because of the symmetry of loading. By considering the equilibrium in the direction of the tangent to the meridian and z, two equations of equilibrium are written, respectively, as d (Nφ r0 ) − Nθ r1 cos φ + Y r1 r0 = 0 dφ Nφ r0 + Nθ r1 sin φ + Zr1 r0 = 0
(2.102)
The force Nθ and Nφ can be calculated from Equation 2.102 if the radii r0 and r1 and the components Y and Z of the intensity of the external load are given.
2.6.3
Spherical Dome
The spherical shell shown in Figure 2.46 is assumed to be subjected to its own weight; the intensity of the self weight is assumed as a constant value qo per unit area. Considering an element of the shell at an angle φ, the self weight of the portion of the shell above this element is obtained as Z φ a 2 qo sin φdφ r = 2π 0
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FIGURE 2.45: An element from shells of revolution—symmetrical loading.
FIGURE 2.46: Spherical dome. =
2π a 2 qo (1 − cos φ)
Considering the equilibrium of the portion of the shell above the parallel circle defined by the angle φ, we can write (2.103) 2π r0 Nφ sin φ + R = 0 Therefore, Nφ = −
aq(1 − cos φ) sin φ 2
=−
aq 1 + cos φ
We can write from Equation 2.102
Nφ Nθ + = −Z r1 r2 Substituting for Nφ and R into Equation 2.104 1 − cos φ Nθ = −aq 1 + cos φ 1999 by CRC Press LLC
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(2.104)
It is seen that the forces Nφ are always negative. Thus, there is a compression along the meridians that increases as the angle φ increases. The forces Nθ are also negative for small angles φ. The stresses as calculated above will represent the actual stresses in the shell with great accuracy if the supports are of such a type that the reactions are tangent to meridians as shown in the figure.
2.6.4
Conical Shells
If a force P is applied in the direction of the axis of the cone as shown in Figure 2.47, the stress distribution is symmetrical and we obtain Nφ = −
P 2π r0 cos α
By Equation 2.104, one obtains Nθ = 0.
FIGURE 2.47: Conical shell. In the case of a conical surface in which the lateral forces are symmetrically distributed, the membrane stresses can be obtained by using Equations 2.103 and 2.104. The curvature of the meridian in the case of a cone is zero and hence r1 = ∞; Equations 2.103 and 2.104 can, therefore, be written as R Nφ = − 2π r0 sin φ and Nθ = −r2 Z = −
Zr0 sin φ
If the load distribution is given, Nφ and Nθ can be calculated independently. For example, a conical tank filled with a liquid of specific weight γ is considered as shown in Figure 2.48. The pressure at any parallel circle mn is p = −Z = γ (d − y) For the tank, φ = α + Therefore,
π 2
and r0 = y tan α. Nθ =
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γ (d − y)y tan α cos α
FIGURE 2.48: Inverted conical tank. Nθ is maximum when y =
d 2
and hence (Nθ )max =
γ d 2 tan α 4 cos α
The term R in the expression for Nφ is equal to the weight of the liquid in the conical part mno and the cylindrical part must be as shown in Figure 2.47. Therefore, 1 3 2 2 2 R = − πy tan α + πy (d − y) tan α γ 3 2 2 = −πγ y d − y tan2 α 3 Hence, Nφ = Nφ is maximum when y =
3 4d
γ y d − 23 y tan α 2 cos α
and (Nφ )max =
3 d 2 γ tan α 16 cos α
The horizontal component of Nφ is taken by the reinforcing ring provided along the upper edge of the tank. The vertical components constitute the reactions supporting the tank.
2.6.5
Shells of Revolution Subjected to Unsymmetrical Loading
Consider an element cut from a shell by two adjacent meridians and two parallel circles (Figure 2.49). In the general case, shear forces Nϕθ = Nθ ϕ in addition to normal forces Nϕ and Nθ will act on the sides of the element. Projecting the forces on the element in the y direction we obtain the equation ∂Nθ ϕ ∂ (Nϕ r0 ) + r1 − Nθ r1 cos ϕ + Y r1 r0 = 0 ∂ϕ ∂θ
(2.105)
Similarly the forces in the x direction can be summed up to give ∂Nθ ∂ (r0 Nϕθ ) + r1 + Nθ ϕ r1 cos ϕ + Xr0 r1 = 0 ∂ϕ ∂θ 1999 by CRC Press LLC
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(2.106)
FIGURE 2.49: An element from shells of revolution—unsymmetrical loading. Since the projection of shearing forces on the z axis vanishes, the third equation is the same as Equation 2.104. The problem of determining membrane stresses under unsymmetrical loading reduces to the solution of Equations 2.104, 2.105, and 2.106 for given values of the components X, Y , and Z of the intensity of the external load.
2.6.6
Membrane Theory of Cylindrical Shells
It is assumed that the generator of the shell is horizontal and parallel to the x axis. An element is cut from the shell by two adjacent generators and two crosssections perpendicular to the x axis, and its position is defined by the coordinate x and the angle ϕ. The forces acting on the sides of the element are shown in Figure 2.50b. The components of the distributed load over the surface of the element are denoted as X, Y, and Z. Considering the equilibrium of the element and summing up the forces in the x direction, we obtain ∂Nϕx ∂Nx rdϕdx + dϕdx + Xrdϕdx = 0 ∂x ∂ϕ The corresponding equations of equilibrium in the y and z directions are given, respectively, as ∂Nϕ ∂Nxϕ rdϕdx + dϕdx + Y rdϕdx = 0 ∂x ∂ϕ Nϕ dϕdx + Zrdϕdx = 0 The three equations of equilibrium can be simplified and represented in the following form: 1 ∂Nxϕ ∂Nx + = −X ∂x r ∂ϕ ∂Nxϕ 1 ∂Nϕ + = −Y ∂x r ∂ϕ Nϕ = −Zr
(2.107)
In each particular case we readily find the value of Nϕ . Substituting this value in the second of the equations, we then obtain Nxϕ by integration. Using the value of Nxϕ thus obtained we find Nx by integrating the first equation. 1999 by CRC Press LLC
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FIGURE 2.50: Membrane forces on a cylindrical shell element.
2.6.7
Symmetrically Loaded Circular Cylindrical Shells
In practical applications problems in which a circular shell is subjected to the action of forces distributed symmetrically with respect to the axis of the cylinder are common. To establish the equations required for the solution of these problems, we consider an element, as shown in Figures 2.50a and 2.51, and consider the equations of equilibrium. From symmetry, the membrane shearing forces
FIGURE 2.51: Stress resultants in a cylindrical shell element. Nxϕ = Nϕx vanish in this case; forces Nϕ are constant along the circumference. From symmetry, only the forces Qz do not vanish. Considering the moments acting on the element in Figure 2.51, 1999 by CRC Press LLC
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from symmetry it can be concluded that the twisting moments Mxϕ = Mϕx vanish and the bending moments Mϕ are constant along the circumference. Under such conditions of symmetry three of the six equations of equilibrium of the element are identically satisfied. We have to consider only the equations obtained by projecting the forces on the x and z axes and by taking the moment of the forces about the y axis. For example, consider a case in which external forces consist only of a pressure normal to the surface. The three equations of equilibrium are dN adxdϕ = 0 dx
dQx adxdϕ + Nϕ dxdϕ + Zadxdϕ = 0 dx dMx adxdϕ − Qx adxdϕ = 0 dx
(2.108)
The first one indicates that the forces Nx are constant, and they are taken equal to zero in the further discussion. If they are different from zero, the deformation and stress corresponding to such constant forces can be easily calculated and superposed on stresses and deformations produced by lateral load. The remaining two equations are written in the simplified form: 1 dQx + Nϕ = −Z dx a dMx − Qx = 0 dx
(2.109)
These two equations contain three unknown quantities: Nϕ , Qx , and Mx . We need, therefore, to consider the displacements of points in the middle surface of the shell. The component v of the displacement in the circumferential direction vanishes because of symmetry. Only the components u and w in the x and z directions, respectively, are to be considered. The expressions for the strain components then become εx =
du dx
εϕ = −
w a
(2.110)
By Hooke’s law, we obtain Nx Nϕ
w Eh Eh du − ν =0 (ε + νε ) = x ϕ a 1 − ν2 1 − ν 2 dx Eh Eh du w = (ε + νε ) = + ν =0 − ϕ x a dx 1 − ν2 1 − ν2 =
(2.111)
From the first of these equation it follows that w du =ν dx a and the second equation gives
Ehw (2.112) a Considering the bending moments, we conclude from symmetry that there is no change in curvature in the circumferential direction. The curvature in the x direction is equal to −d 2 w/dx 2 . Using the same equations as for plates, we then obtain Nϕ = −
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Mϕ
=
νMx
Mx
=
−D
d 2w dx 2
(2.113)
where D=
Eh3 12(1 − ν 2 )
is the flexural rigidity per unit length of the shell. Eliminating Qx from Equation 2.109, we obtain 1 d 2 Mx + Nϕ = −Z 2 a dx from which, by using Equations 2.112 and 2.113, we obtain d 2w Eh d2 D + 2w=Z dx 2 dx 2 a
(2.114)
All problems of symmetrical deformation of circular cylindrical shells thus reduce to the integration of Equation 2.114. The simplest application of this equation is obtained when the thickness of the shell is constant. Under such conditions, Equation 2.114 becomes D
d 4 w Eh + 2w=Z dx 4 a
Using the notation 3(1 − ν 2 ) Eh = 4a 2 D a 2 h2 Equation 2.115 can be represented in the simplified form β4 =
Z d 4w + 4β 4 w = 4 D dx
(2.115)
(2.116)
The general solution of this equation is w
=
eβx (C1 cos βx + C2 sin βx) + e−βx (C3 cos βx + C4 sin βx) + f (x)
(2.117)
Detailed treatment of shell theory can be obtained from Timoshenko and WoinowskyKrieger [56].
2.6.8
Buckling of Shells
If a circular cylindrical shell is uniformly compressed in the axial direction, buckling symmetrical with respect to the axis of the cylinder (Figure 2.52) may occur at a certain value of the compressive load. The critical value of the compressive force Ncr per unit length of the edge of the shell can be obtained by solving the differential equation D
d 2w w d 4w + N + Eh 2 = 0 4 2 dx dx a
(2.118)
in which a is the radius of the cylinder and h is the wall thickness. Alternatively, the critical force per unit length may also be obtained by using the energy method. For a cylinder of length L simply supported at both ends one obtains 2 2 m π EhL2 + (2.119) Ncr = D L2 Da 2 m2 π 2 1999 by CRC Press LLC
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FIGURE 2.52: Buckling of a cylindrical shell. For each value of m there is a unique buckling mode shape and a unique buckling load. The lowest value is of greatest interest and is thus found by setting the derivative of Ncr with respect to L equal to zero for m = 1. With Poisson’s Ratio, = 0.3, the buckling load is obtained as Ncr = 0.605
Eh2 a
(2.120)
It is possible for a cylindrical shell be subjected to uniform external pressure or to the combined action of axial and uniform lateral pressure. In such cases the mathematical treatment is more involved and it requires special considerations. More detailed treatment of such cases may be found in Timoshenko and Gere [55].
2.7
Influence Lines
Structures such as bridges, industrial buildings with travelling cranes, and frames supporting conveyor belts are subjected to moving loads. Each member of these structures must be designed for the most severe conditions that can possibly be developed in that member. Live loads should be placed at the position where they will produce these severe conditions. The critical positions for placing live loads will not be the same for every member. On some occasions it is possible to determine by inspection where to place the loads to give the most critical forces, but on many other occasions it is necessary to resort to certain criteria to find the locations. The most useful of these methods is influence lines. An influence line for a particular response such as reaction, shear force, bending moment, and axial force is defined as a diagram the ordinate to which at any point equals the value of that response attributable to a unit load acting at that point on the structure. Influence lines provide a systematic procedure for determining how the force in a given part of a structure varies as the applied load moves about on the structure. Influence lines of responses of statically determinate structures consist only of straight lines whereas they are curves for statically indeterminate structures. They are primarily used to determine where to place live loads to cause maximum force and to compute the magnitude of those forces. The knowledge of influence lines helps to study the structural response under different moving load conditions.
2.7.1
Influence Lines for Shear in Simple Beams
Figure 2.53 shows influence lines for shear at two sections of a simply supported beam. It is assumed that positive shear occurs when the sum of the transverse forces to the left of a section is in the upward direction or when the sum of the forces to the right of the section is downward. A unit force is placed at various locations and the shear force at sections 11 and 22 are obtained for each position of the 1999 by CRC Press LLC
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FIGURE 2.53: Influence line for shear force.
unit load. These values give the ordinate of influence line with which the influence line diagrams for shear force at sections 11 and 22 can be constructed. Note that the slope of the influence line for shear on the left of the section is equal to the slope of the influence line on the right of the section. This information is useful in drawing shear force influence line in other cases.
2.7.2 Influence Lines for Bending Moment in Simple Beams Influence lines for bending moment at the same sections, 11 and 22 of the simple beam considered in Figure 2.53, are plotted as shown in Figure 2.54. For a section, when the sum of the moments of
FIGURE 2.54: Influence line for bending moment. 1999 by CRC Press LLC
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all the forces to the left is clockwise or when the sum to the right is counterclockwise, the moment is taken as positive. The values of bending moment at sections 11 and 22 are obtained for various positions of unit load and plotted as shown in the figure. It should be understood that a shear or bending moment diagram shows the variation of shear or moment across an entire structure for loads fixed in one position. On the other hand, an influence line for shear or moment shows the variation of that response at one particular section in the structure caused by the movement of a unit load from one end of the structure to the other. Influence lines can be used to obtain the value of a particular response for which it is drawn when the beam is subjected to any particular type of loading. If, for example, a uniform load of intensity qo per unit length is acting over the entire length of the simple beam shown in Figure 2.53, then the shear force at section 11 is given by the product of the load intensity, qo , and the net area under the influence line diagram. The net area is equal to 0.3l and the shear force at section 11 is, therefore, equal to 0.3qo l. In the same way, the bending moment at the section can be found as the area of the corresponding influence line diagram times the intensity of loading, qo . The bending moment at the section is, therefore, (0.08l 2 × qo =)0.08qo l 2 .
2.7.3
Influence Lines for Trusses
Influence lines for support reactions and member forces may be constructed in the same manner as those for various beam functions. They are useful to determine the maximum load that can be applied to the truss. The unit load moves across the truss, and the ordinates for the responses under consideration may be computed for the load at each panel point. Member force, in most cases, need not be calculated for every panel point because certain portions of influence lines can readily be seen to consist of straight lines for several panels. One method used for calculating the forces in a chord member of a truss is by the Method of Sections discussed earlier. The truss shown in Figure 2.55 is considered for illustrating the construction of influence lines for trusses. The member forces in U1 U2 , L1 L2 , and U1 L2 are determined by passing a section 11 and considering the equilibrium of the free body diagram of one of the truss segments. Unit load is placed at L1 first and the force in U1 U2 is obtained by taking moment about L2 of all the forces acting on the righthand segment of the truss and dividing the resulting moment by the lever arm (the perpendicular distance of the force in U1 U2 from L2 ). The value thus obtained gives the ordinate of the influence diagram at L1 in the truss. The ordinate at L2 obtained similarly represents the force in U1 U2 for unit load placed at L2 . The influence line can be completed with two other points, one at each of the supports. The force in the member L1 L2 due to unit load placed at L1 and L2 can be obtained in the same manner and the corresponding influence line diagram can be completed. By considering the horizontal component of force in the diagonal of the panel, the influence line for force in U1 L2 can be constructed. Figure 2.55 shows the respective influence diagram for member forces in U1 U2 , L1 L2 , and U1 L2 . Influence line ordinates for the force in a chord member of a “curvedchord” truss may be determined by passing a vertical section through the panel and taking moments at the intersection of the diagonal and the other chord.
2.7.4
Qualitative Influence Lines
One of the most effective methods of obtaining influence lines is by the use of M¨ullerBreslau’s principle, which states that “the ordinates of the influence line for any response in a structure are equal to those of the deflection curve obtained by releasing the restraint corresponding to this response and introducing a corresponding unit displacement in the remaining structure”. In this way, the shape of the influence lines for both statically determinate and indeterminate structures can be easily obtained especially for beams. 1999 by CRC Press LLC
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FIGURE 2.55: Influence line for truss. To draw the influence lines of 1. Support reaction: Remove the support and introduce a unit displacement in the direction of the corresponding reaction to the remaining structure as shown in Figure 2.56 for a symmetrical overhang beam.
FIGURE 2.56: Influence line for support reaction.
2. Shear: Make a cut at the section and introduce a unit relative translation (in the direction of positive shear) without relative rotation of the two ends at the section as shown in Figure 2.57. 3. Bending moment: Introduce a hinge at the section (releasing the bending moment) and apply bending (in the direction corresponding to positive moment) to produce a unit relative rotation of the two beam ends at the hinged section as shown in Figure 2.58. 1999 by CRC Press LLC
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FIGURE 2.57: Influence line for midspan shear force.
FIGURE 2.58: Influence line for midspan bending moment.
2.7.5
Influence Lines for Continuous Beams
Using M¨ullerBreslau’s principle, the shape of the influence line of any response of a continuous beam can be sketched easily. One of the methods for beam deflection can then be used for determining the ordinates of the influence line at critical points. Figures 2.59 to 2.61 show the influence lines of bending moment at various points of two, three, and four span continuous beams.
FIGURE 2.59: Influence lines for bending moments—two span beam.
2.8
Energy Methods in Structural Analysis
Energy methods are a powerful tool in obtaining numerical solutions of statically indeterminate problems. The basic quantity required is the strain energy, or work stored due to deformations, of the structure. 1999 by CRC Press LLC
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FIGURE 2.60: Influence lines for bending moments—three span beam.
FIGURE 2.61: Influence lines for bending moments—four span beam.
2.8.1
Strain Energy Due to Uniaxial Stress
In an axially loaded bar with constant crosssection, the applied load causes normal stress σy as shown in Figure 2.62. The tensile stress σy increases from zero to a value σy as the load is gradually applied. The original, unstrained position of any section such as C − C will be displaced by an amount v. A section D − D located a differential length below C − C will have been displaced by an amount v + ∂v ∂y dy. As σy varies with the applied load, from zero to σy , the work done by the forces external to the element can be shown to be dV =
1 1 2 σy Ady = σy εy Ady 2E 2
in which A is the area of crosssection of the bar and εy is the strain in the direction of σy .
1999 by CRC Press LLC
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(2.121)
FIGURE 2.62: Axially loaded bar.
2.8.2
Strain Energy in Bending
It can be shown that the strain energy of a differential volume dxdydz stressed in tension or compression in the x direction only by a normal stress σx will be dV =
1 1 2 σ dxdydz = σx εx dxdydz 2E x 2
(2.122)
1 M y When σx is the bending stress given by σx = My I (see Figure 2.63), then dV = 2E I 2 dxdydz, where I is the moment of inertia of the crosssectional area about the neutral axis. 2 2
FIGURE 2.63: Beam under arbitrary bending load.
The total strain energy of bending of a beam is obtained as Z Z Z V = volume
where
1 M2 2 y dzdydx 2E I 2
Z Z y 2 dzdy
I= area
1999 by CRC Press LLC
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Therefore,
Z V = length
2.8.3
M2 dx 2EI
(2.123)
Strain Energy in Shear
Figure 2.64 shows an element of volume dxdydz subjected to shear stress τxy and τyx .
FIGURE 2.64: Shear loading.
For static equilibrium, it can readily be shown that τxy = τyx The shear strain, γ is defined as AB/AC. For small deformations, it follows that γxy =
AB AC
Hence, the angle of deformation γxy is a measure of the shear strain. The strain energy for this differential volume is obtained as dV =
1 1 τxy dzdx γxy dy = τxy γxy dxdydz 2 2
(2.124)
Hooke’s Law for shear stress and strain is γxy =
τxy G
(2.125)
where G is the shear modulus of elasticity of the material. The expression for strain energy in shear reduces to 1 2 τ dxdydz (2.126) dV = 2G xy 1999 by CRC Press LLC
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2.8.4
The Energy Relations in Structural Analysis
The energy relations or laws such as (1) Law of Conservation of Energy, (2) Theorem of Virtual Work, (3) Theorem of Minimum Potential Energy, and (4) Theorem of Complementary Energy are of fundamental importance in structural engineering and are used in various ways in structural analysis. The Law of Conservation of Energy
There are many ways of stating this law. For the purpose of structural analysis it will be sufficient to state it in the following way: If a structure and the external loads acting on it are isolated so that these neither receive nor give out energy, then the total energy of this system remains constant. A typical application of the Law of Conservation of Energy can be made by referring to Figure 2.65 which shows a cantilever beam of constant crosssections subjected to a concentrated load at its end. If only bending strain energy is considered, External work Pδ 2
= =
Internal work Z L 2 M dx 2EI 0
Substituting M = −P x and integrating along the length gives δ=
P L3 3EI
(2.127)
FIGURE 2.65: Cantilever beam.
The Theorem of Virtual Work
The Theorem of Virtual Work can be derived by considering the beam shown in Figure 2.66. The full curved line represents the equilibrium position of the beam under the given loads. Assume the beam to be given an additional small deformation consistent with the boundary conditions. This is called a virtual deformation and corresponds to increments of deflection 1y1 , 1y2 , ..., 1yn at loads P1 , P2 , ..., Pn as shown by the broken line. The change in potential energy of the loads is given by 1(P .E.) =
n X
Pi 1yi
(2.128)
i=1
By the Law of Conservation of Energy this must be equal to the internal strain energy stored in the beam. Hence, we may state the Theorem of Virtual Work in the following form: 1999 by CRC Press LLC
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FIGURE 2.66: Equilibrium of a simply supported beam under loading.
If a body in equilibrium under the action of a system of external loads is given any small (virtual) deformation, then the work done by the external loads during this deformation is equal to the increase in internal strain energy stored in the body. The Theorem of Minimum Potential Energy
Let us consider the beam shown in Figure 2.67. The beam is in equilibrium under the action
FIGURE 2.67: Simply supported beam under point loading. of loads, P1 , P2 , P3 , ..., Pi , ..., Pn . The curve ACB defines the equilibrium positions of the loads and reactions. Now apply by some means an additional small displacement to the curve so that it is defined by AC 0 B. Let yi be the original equilibrium displacement of the curve beneath a particular load Pi . The additional small displacement is called δyi . The potential energy of the system while it is in the equilibrium configuration is found by comparing the potential energy of the beam and loads in equilibrium and in the undeflected position. If the change in potential energy of the loads is W and the strain energy of the beam is V , the total energy of the system is U =W +V
(2.129)
δU = δ(W + V ) = 0
(2.130)
If we neglect the secondorder terms, then
The above is expressed as the Principle or Theorem of Minimum Potential Energy which can be stated as Of all displacements satisfying given boundary conditions, those that satisfy the equilibrium conditions make the potential energy a minimum. 1999 by CRC Press LLC
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Castigliano’s Theorem
An example of application of energy methods to the field of structural engineering is Castigliano’s Theorem. The theorem applies only to structures stressed within the elastic limit. Also, all deformations must be linear homogeneous functions of the loads. Castigliano’s Theorem can be derived using the expression for total potential energy as follows: For a beam in equilibrium loaded as in Figure 2.66, the total energy is U = −[P1 y1 + P2 y2 + ...Pj yj + ...Pn yn ] + V
(2.131)
For an elastic system, the strain energy, V , turns out to be one half the change in the potential energy of the loads. i=n 1X Pi yi (2.132) V = 2 i=1
Castigliano’s Theorem results from studying the variation in the strain energy, V , produced by a differential change in one of the loads, say Pj . If the load Pj is changed by a differential amount δPj and if the deflections y are linear functions of the loads, then i=n 1 X ∂yi 1 ∂V = Pi + yj = yj (2.133) ∂Pj 2 ∂Pj 2 i=1
Castigliano’s Theorem is stated as follows: The partial derivatives of the total strain energy of any structure with respect to any one of the applied forces is equal to the displacement of the point of application of the force in the direction of the force. To find the deflection of a point in a beam that is not the point of application of a concentrated load, one should apply a load P = 0 at that point and carry the term P into the strain energy equation. Finally, introduce the true value of P = 0 into the expression for the answer.
EXAMPLE 2.6:
For example, it is required to determine the bending deflection at the free end of a cantilever loaded as shown in Figure 2.68. Solution Z L M2 V = dx 0 2EI Z L M ∂M ∂V = dx 1 = ∂W1 EI ∂W1 0 L M = W1 x 0<x 2 L ` <x Cc 23(Kl/r)2 1999 by CRC Press LLC
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FIGURE 3.6: Definition of widththickness ratio of selected crosssections.
1999 by CRC Press LLC
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TABLE 3.4 Limiting WidthThickness Ratios for Compression Elements Under Pure Compression Widththickness ratio
Component element Flanges of Ishaped sections; plates projecting from compression elements; outstanding legs of pairs of angles in continuous contact; flanges of channels. Flanges of square and rectangular box and hollow structural sections of uniform thickness; flange cover plates and diaphragm plates between lines of fasteners or welds. Unsupported width of cover plates perforated with a succession of access holes. Legs of single angle struts; legs of double angle struts with separators; unstiffened elements (i.e., elements supported along one edge). Flanges projecting from builtup members. Stems of tees. All other uniformly compressed elements (i.e., elements supported along two edges). Circular hollow sections.
ak c Fy
= =
b/t
Limiting value, λr p 95/ fy
b/t
p 238/ fy
b/t
p 317/ fy
b/t
p 76/ fy
b/t d/t b/t h/tw D/t D = outside diameter t = wall thickness
q 109/ (Fy /kca ) p 127/pFy 253/ Fy
3,300/Fy
√ 4/ (h/tw ), and 0.35 ≤ kc ≤ 0.763 for Ishaped sections, kc = 0.763 for other sections. specified minimum yield stress, in ksi.
where Kl/r is the slenderness ratio, K is the effective length factor of the compression member (see Section 3.4.3), l is the unbraced memberqlength, r is the radius of gyration of the crosssection,
E is the modulus of elasticity, and Cc = (2π 2 E/Fy ) is the slenderness ratio that demarcates between inelastic member buckling from elastic member buckling. Kl/r should be evaluated for both buckling axes and the larger value used in Equation 3.16 to compute Fa . The first of Equation 3.16 is the allowable stress for inelastic buckling, and the second of Equation 3.16 is the allowable stress for elastic buckling. In ASD, no distinction is made between flexural, torsional, and flexuraltorsional buckling.
3.4.2
Load and Resistance Factor Design
Compression members are to be designed so that the design compressive strength φc Pn will exceed the required compressive strength Pu . φc Pn is to be calculated as follows for the different types of overall buckling modes. Flexural Buckling (with widththickness ratio < λr ): h i 2 0.85 Ag (0.658λc )Fy , if λc ≤ 1.5 (3.17) φ c Pn = i h 0.85 Ag 0.877 Fy , > 1.5 if λ c 2 λ c
where λc = Ag = Fy = E = K = l = r =
p (KL/rπ) (Fy /E) is the slenderness parameter gross crosssectional area specified minimum yield stress modulus of elasticity effective length factor unbraced member length radius of gyration of the crosssection
1999 by CRC Press LLC
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The first of Equation 3.17 is the design strength for inelastic buckling and the second of Equation 3.17 is the design strength for elastic buckling. The slenderness parameter λc = 1.5 is therefore the value that demarcates between inelastic and elastic behavior. Torsional Buckling (with widththickness ratio < λr ): φc Pn is to be calculated from Equation 3.17, but with λc replaced by λe given by λe = where
q (Fy /Fe )
(3.18)
π 2 ECw 1 + GJ Fe = 2 Ix + Iy (Kz L)
(3.19)
in which = warping constant Cw G = shear modulus = 11,200 ksi (77,200 MPa) Ix , Iy = moment of inertia about the major and minor principal axes, respectively J = torsional constant = effective length factor for torsional buckling Kz The warping constant Cw and the torsional constant J are tabulated for various steel shapes in the AISCLRFD Manual [22]. Equations for calculating approximate values for these constants for some commonly used steel shapes are shown in Table 3.5. TABLE 3.5
Approximate Equations for Cw and J
Structural shape
Warping constant, Cw
I
h02 Ic It /(Ic + It )
C
(b0 − 3Eo )h02 b02 tf /6 + Eo2 Ix where Eo = b02 tf /(2b0 tf + h0 tw /3)
b0 h0 h00 l1 , l2 t1 , t2 bf tf tw Ic It Ix
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T
3 )/36 (bf3 tf3 /4 + h003 tw (≈ 0 for small t )
L
(l13 t13 + l23 t23 )/36 (≈ 0 for small t )
= = = = = = = = = = =
Torsional constant, J P Ci (bi ti3 /3)
where bi = width of component element i ti = thickness of component element i Ci = correction factor for component element i (see values below) bi /ti 1.00 1.20 1.50 1.75 2.00 2.50 3.00 4.00 5.00 6.00 8.00 10.00 ∞
Ci 0.423 0.500 0.588 0.642 0.687 0.747 0.789 0.843 0.873 0.894 0.921 0.936 1.000
distance measured from toe of flange to center line of web distance between centerline lines of flanges distance from centerline of flange to tip of stem length of the legs of the angle thickness of the legs of the angle flange width average thickness of flange thickness of web moment of inertia of compression flange taken about the axis of the web moment of inertia of tension flange taken about the axis of the web moment of inertia of the crosssection taken about the major principal axis
FlexuralTorsional Buckling (with widththickness ratio ≤ λr ): Same as for torsional buckling except Fe is now given by For singly symmetric sections: s " # Fes + Fez 4Fes Fez H 1− 1− Fe = 2H (Fes + Fez )2
(3.20)
where Fes = Fex if the xaxis is the axis of symmetry of the crosssection, or Fey if the yaxis is the axis of symmetry of the crosssection Fex = π 2 E/(Kl/r)2x Fey = π 2 E/(Kl/r)2x H = 1 − (xo2 + yo2 )/ro2 in which Kx , Ky = effective length factors for buckling about the x and y axes, respectively l = unbraced member length = radii of gyration about the x and y axes, respectively rx , ry xo , yo = the shear center coordinates with respect to the centroid Figure 3.7 = xo2 + yo2 + rx2 + ry2 ro2 Numerical values for ro and H are given for hotrolled W, channel, tee, and single and doubleangle sections in the AISCLRFD Manual [22]. For unsymmetric sections: Fe is to be solved from the cubic equation (Fe − Fex )(Fe − Fey )(Fe − Fez ) − Fe2 (Fe
xo − Fey ) ro
2 − Fe2 (Fe
yo − Fex ) ro
2 =0
(3.21)
The terms in the above equations are defined the same as in Equation 3.20. Local Buckling (with widththickness ratio ≥ λr ): Local buckling in a component element of the crosssection is accounted for in design by introducing a reduction factor Q in Equation 3.17 as follows: h i √ 2 0.85 Ag Q 0.658Qλ Fy , if λ Q ≤ 1.5 (3.22) φ c Pn = i h √ 0.85 Ag 0.877 Fy , Q > 1.5 if λ λ2 where λ = λc for flexural buckling, and λ = λe for flexuraltorsional buckling. The Q factor is given by Q = Qs Qa
(3.23)
where Qs is the reduction factor for unstiffened compression elements of the crosssection (see Table 3.6); and Qa is the reduction factor for stiffened compression elements of the crosssection (see Table 3.7)
3.4.3
BuiltUp Compression Members
Builtup members are members made by bolting and/or welding together two or more standard structural shapes. For a builtup member to be fully effective (i.e., if all component structural shapes are to act as one unit rather than as individual units), the following conditions must be satisfied: 1999 by CRC Press LLC
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FIGURE 3.7: Location of shear center for selected crosssections. 1. The ends of the builtup member must be prevented from slippage during buckling. 2. Adequate fasteners must be provided along the length of the member. 3. The fasteners must be able to provide sufficient gripping force on all the component shapes being connected. Condition 1 is satisfied if all component shapes in contact at the ends of the member are connected by a weld having a length not less than the maximum width of the member or by fully tightened bolts spaced longitudinally not more than four diameters apart for a distance equal to 11/2 times the maximum width of the member. Condition 2 is satisfied if continuous welds are used throughout the length of the builtup compression member. Condition 3 is satisfied if either welds or fully tightened bolts are used as the fasteners. While condition 1 is mandatory, conditions 2 and 3 can be violated in design. If condition 2 or 3 is violated, the builtup member is not fully effective and slight slippage among component shapes 1999 by CRC Press LLC
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TABLE 3.6
Formulas for Qs
Range of b/t
Qs
p 76.0/ Fy < b/t < 155/ Fy
p 1.340 − 0.00447(b/t) fy
Structural element p
Single angles
p b/t ≥ 155/ fy p p 95.0/ Fy < b/t < 176/ fy
p 1.415 − 0.00437(b/t) fy
p b/t ≥ 176/ Fy
20, 000/[Fy (b/t)2 ]
q p 109/ (Fy /kca ) < b/t < 200/ (Fy /kc )
p 1.415 − 0.00381(b/t) (Fy /kc )
p b/t ≥ 200/ (Fy /kc )
26, 200kc/[Fy (b/t)2 ]
p p 127/ Fy < b/t < 176/ Fy
p 1.908 − 0.00715(b/t) Fy
p b/t ≥ 176/ fy
20, 000/[Fy (b/t)2 ]
Flanges, angles, and plates projecting from columns or other compression members
Flanges, angles, and plates projecting from builtup columns or other compression members
Stems of tees
15, 500/[Fy (b/t)2 ]
a see footnote a in Table 3.4
Fy b t
TABLE 3.7
= = =
specified minimum yield stress, in ksi width of the component element thickness of the component element
Formula for Qa Qs = effective area actual area
The effective area is equal to the summation of the effective areas of the stiffened elements of the crosssection. The effective area of a stiffened element is equal to the product of its thickness t and its effective width be given by: a √ For flanges of square and rectangular sections of uniform thickness: when b/t ≥ 238 f
√ be = 326t f
h
√ 1 − 64.9 (b/t) f
i
≤b
a √ For other uniformly compressed elements: when b/t ≥ 253 f
h √ 1− be = 326t f
57.2 √ (b/t) f
i
≤b
where b = actual width of the stiffened element f = computed elastic compressive stress in the stiffened elements, in ksi ab e
=
b otherwise.
may occur. To account for the decrease in capacity due to slippage, a modified slenderness ratio is used for the computation of the design compressive strength when buckling of the builtup member is about an axis coincide or parallel to at least one plane of contact for the component shapes. The modified slenderness ratio (KL/r)m is given as follows: If condition 2 is violated:
1999 by CRC Press LLC
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KL r
s
m
=
KL r
2
0.82α 2 + (1 + α 2 ) o
a rib
2 (3.24)
If conditions 2 and 3 are violated:
KL r
s
m
=
KL r
2 a + r i o
2
(3.25)
In the above equations, (KL/r)o = (KL/r)x if the buckling axis is the xaxis and at least one plane of contact between component shapes is parallel to that axis; (KL/r)o = (KL/r)y if the buckling axis is the y axis and at least one plane of contact is parallel to that axis. a is the longitudinal spacing of the fasteners, ri is the minimum radius of gyration of any component element of the builtup crosssection, rib is the radius of gyration of an individual component relative to its centroidal axis parallel to the axis of buckling of the member, h is the distance between centroids of component elements measured perpendicularly to the buckling axis of the builtup member. No modification to (KL/r) is necessary if the buckling axis is perpendicular to the planes of contact of the component shapes. Modifications to both (KL/r)x and (KL/r)y are required if the builtup member is so constructed that planes of contact exist in both the x and y directions of the crosssection. Once the modified slenderness ratio is computed, it is to be used in the appropriate equation to calculate Fa in allowable stress design, or φc Pn in load and resistance factor design. An additional requirement for the design of builtup members is that the effective slenderness ratio, Ka/ri , of each component shape, where K is the effective length factor of the component shape between adjacent fasteners, does not exceed 3/4 of the governing slenderness ratio of the builtup member. This provision is provided to prevent component shape buckling between adjacent fasteners from occurring prior to overall buckling of the builtup member.
EXAMPLE 3.2:
Using LRFD, determine the size of a pair of cover plates to be bolted, using snugtight bolts, to the flanges of a W24x229 section as shown in Figure 3.8 so that its design strength, φc Pn , will be increased by 15%. Also, determine the spacing of the bolts in the longitudinal direction of the builtup column.
FIGURE 3.8: Design of cover plates for a compression member. 1999 by CRC Press LLC
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The effective lengths of the section about the major (KL)x and minor (KL)y axes are both equal to 20 ft. A36 steel is to be used. Determine design strength for the W24x229 section: Since (KL)x = (KL)y and rx > ry , (KL/r)y will be greater than (KL/r)x and the design strength will be controlled by flexural buckling about the minor axis. Using section properties, ry = 3.11 in. and A = 67.2 in.2 , obtained from the AISCLRFD Manual [22], the slenderness parameter λc about the minor axis can be calculated as follows: 1 (λc )y = π
KL r
r y
Fy 1 = E 3.142
20 × 12 3.11
s
36 = 0.865 29, 000
Substituting λc = 0.865 into Equation 3.17, the design strength of the section is h i 2 φc Pn = 0.85 67.2 0.6580.865 36 = 1503 kips Alternatively, the above value of φc Pn can be obtained directly from the column tables contained in the AISCLRFD Manual. Determine design strength for the builtup section: The builtup section is expected to possess a design strength which is 15% in excess of the design strength of the W24x229 section, so (φc Pn )req 0 d = (1.15)(1503) = 1728 kips Determine size of the cover plates: After cover plates are added, the resulting section is still doubly symmetric. Therefore, the overall failure mode is still flexural buckling. For flexural buckling about the minor axis (yy), no modification to (KL/r) is required because the buckling axis is perpendicular to the plane of contact of the component shapes and no relative movement between the adjoining parts is expected. However, for flexural buckling about the major (xx) axis, modification to (KL/r) is required because the buckling axis is parallel to the plane of contact of the adjoining structural shapes and slippage between the component pieces will occur. We shall design the cover plates assuming flexural buckling about the minor axis will control and check for flexural buckling about the major axis later. A W24x229 section has a flange width of 13.11 in.; so, as a trial, use cover plates with widths of 13 in. as shown in Figure 3.8a. Denoting t as the thickness of the plates, we have s (ry )builtup = and (λc )y,builtup
1 = π
(Iy )Wshape + (Iy )plates = AWshape + Aplates
KL r
y,builtup
r
r
651 + 183.1t 67.2 + 26t
r Fy 67.2 + 26t = 2.69 E 651 + 183.1t
Assuming (λ)y,built−up is less than 1.5, one can substitute the above expression for λc in Equation 3.17. With φc Pn equals 1728, we can solve for t. The result is t = 1/2 in. Backsubstituting t = 1/2 into the above expression, we obtain (λ)c,built−up = 0.884 which is indeed 0.6Fy Afg = 0.6(36)(7.005)(0.505) = 76.4 kips
Cover Plates
0.5Fu Af n = 0.5(58)(7 − 2 × 1/2)(1/2) = 87 kips
> 0.6Fy Af g = 0.6(36)(7)(1/2) = 75.6 kips so the use of the gross crosssectional area to compute section properties is justified. In the event that the condition is violated, crosssectional properties should be evaluated using an effective tension flange area Af e given by 5 Fu Af n Af e = 6 Fy Use 1/2” diameter A325N bolts spaced 4.5” apart longitudinally in two lines 4” apart to connect the cover plates to the beam flanges. 1999 by CRC Press LLC
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3.5.2
Load and Resistance Factor Design
Flexural Strength Criterion
Flexural members must be designed to satisfy the flexural strength criterion of φb Mn ≥ Mu
(3.36)
where φb Mn is the design flexural strength and Mu is the required strength. The design flexural strength is determined as follows: Compact Section Members Bent About Their Major Axes For Lb ≤ Lp , (Plastic hinge formation) φb Mn = 0.90Mp
(3.37)
For Lp < Lb ≤ Lr , (Inelastic lateral torsional buckling) Lb − Lp ≤ 0.90Mp φb Mn = 0.90Cb Mp − (Mp − Mr ) Lr − Lp
(3.38)
For Lb > Lr , (Elastic lateral torsional buckling) For Ishaped members and channels: π φb Mn = 0.90Cb Lb
s
EIy GJ +
πE Lb
2
Iy Cw ≤ 0.90Mp
(3.39)
For solid rectangular bars and symmetric box sections: √ 57, 000 J A ≤ 0.90Mp φb Mn = 0.90Cb Lb /ry The variables used in the above equations are defined in the following. Lb
= lateral unsupported length of the member
Lp , Lr = limiting lateral unsupported lengths given in the following table 1999 by CRC Press LLC
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(3.40)
Lp
Structural shape Ishaped chanels
p 300ry / Fyf
sections,
Solid rectangular bars, symmetric box sections
Lr ry X1 /FL
(s 1+
r
1 + X2 FL2
)
where
where
ry = radius of gyration about minor axis, in. Fyf = flange yield stress, ksi
ry = radius of√gyration about minor axis, in. X1 = (π/Sx ) (EGJ A/2) X2 = (4Cw /Iy )(Sx /GJ )2 FL = smaller of (Fyf − Fr ) or Fyw Fyf = flange yield stress, ksi Fyw = web yield stress, ksi Fr = 10 ksi for rolled shapes, 16.5 ksi for welded shapes Sx = elastic section modulus about the major axis, in.3 (use Sxc , the elastic section modulus about the major axis with respect to the compression flange if the compression flange is larger than the tension flange) Iy = moment of inertia about the minor axis, in.4 J = torsional constant, in.4 Cw = warping constant, in.6 E = modulus of elasticity, ksi G = shear modulus, ksi
√ 3, 750ry (J A) /Mp
√ 57, 000ry (J A) /Mr
where
where
ry = radius of gyration about minor axis, in. J = torsional constant, in.4 A = crosssectional area, in.2 Mp = plastic moment capacity = Fy Zx Fy = yield stress, ksi Zx = plastic section modulus about the major axis, in.3
ry = radius of gyration about minor axis, in. J = torsional constant, in.4 A = crosssectional area, in.2 Mr = Fy Sx for solid rectangular bar, Fyf Seff for box sections Fy = yield stress, ksi Fyf = flange yield stress, ksi Sx = plastic section modulus about the major axis, in.3
Note: Lp given in this table are valid only if the bending coefficient Cb is equal to unity. If Cb > 1, the value of Lp can be increased. However, using the Lp expressions given above for Cb > 1 will give a conservative value for the flexural design strength.
and Mp = Fy Zx Mr = FL Sx for Ishaped sections and channels, Fy Sx for solid rectangular bars, Fyf Seff for box sections FL = smaller of (Fyf − Fr ) or Fyw Fyf = flange yield stress, ksi Fyw = web yield stress Fr = 10 ksi for rolled sections, 16.5 ksi for welded sections Fy = specified minimum yield stress Sx = elastic section modulus about the major axis Seff = effective section modular, calculated using effective width be , in Table 3.7 Zx = plastic section modulus about the major axis Iy = moment of inertia about the minor axis J = torsional constant Cw = warping constant E = modulus of elasticity G = shear modulus Cb = 12.5Mmax /(2.5Mmax + 3MA + 4MB + 3MC ) 1999 by CRC Press LLC
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Mmax , MA , MB , MC
= maximum moment, quarterpoint moment, midpoint moment, and threequarter point moment along the unbraced length of the member, respectively.
Cb is a factor that accounts for the effect of moment gradient on the lateral torsional buckling strength of the beam. Lateral torsional buckling strength increases for a steep moment gradient. The worst loading case as far as lateral torsional buckling is concerned is when the beam is subjected to a uniform moment resulting in single curvature bending. For this case Cb =1. Therefore, the use of Cb =1 is conservative for the design of beams. Compact Section Members Bent About Their Minor Axes Regardless of Lb , the limit state will be a plastic hinge formation φb Mn = 0.90Mpy = 0.90Fy Zy
(3.41)
Noncompact Section Members Bent About Their Major Axes For Lb ≤ L0p , (Flange or web local buckling) φb Mn =
φb Mn0
where L0p
λ − λp = 0.90 Mp − (Mp − Mr ) λr − λp
Mp − Mn0 = Lp + (Lr − Lp ) Mp − Mr
(3.42)
(3.43)
Lp , Lr , Mp , Mr are defined as before for compact section members, and For flange local buckling: λ = bf /2t p f for Ishaped members, bf /tf for channels λp = 65/ Fy p λr = 141/ (Fy − 10) For web local buckling: λ = hc /twp λp = 640/ Fy p λr = 970/ Fy in which bf = flange width tf = flange thickness hc = twice the distance from the neutral axis to the inside face of the compression flange less the fillet or corner radius tw = web thickness For L0p < Lb ≤ Lr , (Inelastic lateral torsional buckling), φb Mn is given by Equation 3.38 except that the limit 0.90Mp is to be replaced by the limit 0.90Mn0 . For Lb > Lr , (Elastic lateral torsional buckling), φb Mn is the same as for compact section members as given in Equation 3.39 or Equation 3.40. Noncompact Section Members Bent About Their Minor Axes Regardless of the value of Lb , the limit state will be either flange or web local buckling, and φb Mn is given by Equation 3.42. 1999 by CRC Press LLC
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Slender Element Sections Refer to the section on Plate Girder. Tees and Double Angle Bent About Their Major Axes The design flexural strength for tees and doubleangle beams with flange and web slenderness ratios less than the corresponding limiting slenderness ratios λr shown in Table 3.8 is given by " p # p π EIy GJ 2 (B + 1 + B ) ≤ 0.90(CMy ) φb Mn = 0.90 Lb where
B = ±2.3
d Lb
r
Iy J
(3.44)
(3.45)
C = 1.5 for stems in tension, and 1.0 for stems in compression. Use the plus sign for B if the entire length of the stem along the unbraced length of the member is in tension. Otherwise, use the minus sign. The other variables in Equation 3.44 are defined as before in Equation 3.39. Shear Strength Criterion
For a satisfactory design, the design shear strength of the webs must exceed the factored shear acting on the crosssection, i.e., (3.46) φv Vn ≥ Vu Depending on the slenderness ratios of the webs, three limit states can be identified: shear yielding, inelastic shear buckling, and elastic shear buckling. The design shear strength that corresponds to each of these limit states is given as follows: p For h/tw ≤ 418/ Fyw , (Shear yielding of web)
p For 418/ Fyw
φv Vn = 0.90[0.60Fyw Aw ] p < h/tw ≤ 523/ Fyw , (Inelastic shear buckling of web) # p 418/ Fyw φv Vn = 0.90 0.60Fyw Aw h/tw
(3.47)
"
(3.48)
p For 523/ Fyw < h/tw ≤ 260, (Elastic shear buckling of web) φv Vn = 0.90
132,000Aw (h/tw )2
The variables used in the above equations are defined in the following: h tw Fyw Aw d
= = = = =
clear distance between flanges less the fillet or corner radius, in. web thickness, in. yield stress of web, ksi dtw , in.2 overall depth of section, in.
1999 by CRC Press LLC
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(3.49)
Criteria for Concentrated Loads
When concentrated loads are applied normal to the flanges in planes parallel to the webs of flexural members, the flange(s) and web(s) must be checked to ensure that they have sufficient strengths φRn to withstand the concentrated forces Ru , i.e., φRn ≥ Ru
(3.50)
The design strength for a variety of limit states are given below: Local Flange Bending The design strength for local flange bending is given by φRn ≥ 0.90[6.25tf2 Fyf ]
(3.51)
where = flange thickness of the loaded flange, in. tf Fyf = flange yield stress, ksi Local Web Yielding The design strength for yielding of a beam web at the toe of the fillet under tensile or compressive loads acting on one or both flanges are: If the load acts at a distance from the beam end which exceeds the depth of the member φRn = 1.00[(5k + N )Fyw tw ]
(3.52)
If the load acts at a distance from the beam end which does not exceed the depth of the member φRn = 1.00[(2.5k + N )Fyw tw ] where k = N = Fyw = tw =
(3.53)
distance from outer face of flange to web toe of fillet length of bearing on the beam flange web yield stress web thickness
Web Crippling The design strength for crippling of a beam web under compressive loads acting on one or both flanges are: If the load acts at a distance from the beam end which exceeds half the depth of the beam ( φRn = 0.75
"
N 1+3 d
135tw2
tw tf
1.5 # s
Fyw tf tw
) (3.54)
If the load acts at a distance from the beam end which does not exceed half the depth of the beam and if N/d ≤ 0.2 ( φRn = 0.75 1999 by CRC Press LLC
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" 68tw2
N 1+3 d
tw tf
1.5 # s
Fyw tf tw
) (3.55)
If the load acts at a distance from the beam end which does not exceed half the depth of the beam and if N/d>0.2 ( " ) 1.5 # s Fyw tf tw 4N 2 − 0.2 (3.56) φRn = 0.75 68tw 1 + d tf tw where d = overall depth of the section, in. = flange thickness, in. tf The other variables are the same as those defined in Equations 3.52 and 3.53. Sidesway Web Buckling Sidesway web buckling may occur in the web of a member if a compressive concentrated load is applied to a flange which is not restrained against relative movement by stiffeners or lateral bracings. The sidesway web buckling design strength for the member is: If the loaded flange is restrained against rotation about the longitudinal member axis and (hc /tw )(l/bf ) ≤ 2.3 ( " #) Cr tw3 tf h/tw 3 (3.57) 1 + 0.4 φRn = 0.85 l/bf h2 If the loaded flange is not restrained against rotation about the longitudinal member axis and (hc /tw )(l/bf ) ≤ 1.7 " ( #) Cr tw3 tf h/tw 3 0.4 (3.58) φRn = 0.85 l/bf h2 where = flange thickness, in. tf tw = web thickness, in. h = clear distance between flanges less the fillet or corner radius for rolled shapes; distance between adjacent lines of fasteners or clear distance between flanges when welds are used for builtup shapes, in. bf = flange width, in. l = largest laterally unbraced length along either flange at the point of load, in. Cr = 960,000 if Mu /My 234
r
Cv r
kv Fyw
kv Fyw
√
187 kv /Fyw h/tw 44,000kv (h/tw )2 Fyw
FlexureShear Interaction
Plate girders designed for tension field action must satisfy the flexureshear interaction criterion in regions where 0.60φVn ≤ Vu ≤ φVn and 0.75φMn ≤ Mu ≤ φMn Vu Mu + 0.625 ≤ 1.375 φMn φVn
(3.89)
where φ = 0.90. Bearing Stiffeners
Bearing stiffeners must be provided for a plate girder at unframed girder ends and at points of concentrated loads where the web yielding or the web crippling criterion is violated (see section on Concentrated Load Criteria). Bearing stiffeners shall be provided in pairs and extended from the upper flange to the lower flange of the girder. Denoting bst as the width of one stiffener and tst as its thickness, bearing stiffeners shall be portioned to satisfy the following limit states: For the limit state of local buckling
95 bst ≤p tst Fy
(3.90)
For the limit state of compression The design compressive strength, φc Pn , must exceed the required compressive force acting on the stiffeners. φc Pn is to be determined based on an effective length factor K of 0.75 and an effective area, Aeff , equal to the area of the bearing stiffeners plus a portion of the web. For end bearing, this effective area is equal to 2(bst tst ) + 12tw2 ; and for interior bearing, this effective area is equal to 2(bst tst ) + 25tw2 . tw is the p web thickness. The slenderness parameter, λc , is to be calculated using a radius of gyration, r = (Ist /Aeff ), where Ist = tst (2bst + tw )3 /12. For the limit state of bearing The bearing strength, φRn , must exceed the required compression force acting on the stiffeners. φRn is given by (3.91) φRn ≥ 0.75[1.8Fy Apb ] where Fy is the yield stress and Apb is the bearing area. 1999 by CRC Press LLC
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Intermediate Stiffeners
Intermediate stiffeners shall be provided if (1) the shear strength capacity is calculated based on tension field action, (2) the shearpcriterion is violated (i.e., when the Vu exceeds φv Vn ), or (3) the web slenderness h/tw exceeds 418/ Fyw . Intermediate stiffeners can be provided in pairs or on one side of the web only in the form of plates or angles. They should be welded to the compression flange and the web but they may be stopped short of the tension flange. The following requirements apply to the design of intermediate stiffeners: Local Buckling The widththickness ratio of the stiffener must be proportioned so that Equation 3.90 is satisfied to prevent failure by local buckling. Stiffener Area The crosssection area of the stiffener must satisfy the following criterion: Fyw Vu 2 − 18tw ≥ 0 0.15Dhtw (1 − Cv ) Ast ≥ Fy φv Vn
(3.92)
where Fy = yield stress of stiffeners D = 1.0 for stiffeners in pairs, 1.8 for single angle stiffeners, and 2.4 for single plate stiffeners The other terms in Equation 3.92 are defined as before in Equation 3.87 and Equation 3.88. Stiffener Moment of Inertia The moment of inertia for stiffener pairs taken about an axis in the web center or for single stiffeners taken in the face of contact with the web plate must satisfy the following criterion: Ist ≥
atw3
2.5 − 2 ≥ 0.5atw3 (a/ h)2
(3.93)
Stiffener Length The length of the stiffeners, lst , should fall within the range h − 6tw < lst < h − 4tw
(3.94)
where h is the clear distance between the flanges less the widths of the flangetoweb welds and tw is the web thickness. If intermittent welds are used to connect the stiffeners to the girder web, the clear distance between welds shall not exceed 16tw , or 10 in. If bolts are used, their spacing shall not exceed 12 in. Stiffener Spacing The spacing of the stiffeners, a, shall be determined from the shear criterion φv Vn ≥ Vu . This spacing shall not exceed the smaller of 3h and [260/(h/tw )]2 h.
EXAMPLE 3.7:
Using LRFD, design the crosssection of an Ishaped plate girder shown in Figure 3.12a to support a factored moment Mu of 4600 kipft (6240 kNm), dead weight of the girder is included. The girder is a 60ft (18.3m) long simply supported girder. It is laterally supported at every 20ft (6.1m) interval. Use A36 steel. 1999 by CRC Press LLC
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FIGURE 3.12: Design of a plate girder crosssection.
Proportion of the girder web Ordinarily, the overall depthtospan ratio d/L of a building girder is in the prange 1/12 to 1/10. p So, let us try h =70 in. Also, knowing h/tw of a plate girder is in the range 970/ Fyf and 2,000/ Fyf , let us try tw = 5/16 in. Proportion of the girder flanges For a preliminary design, the required area of the flange can be determined using the flange area method 4600 kipft x12 in./ft Mu = = 21.7 in.2 Af ≈ Fy h (36 ksi )(70 in.) So, let bf = 20 in. and tf = 11/8 in. giving Af = 22.5 in.2 1999 by CRC Press LLC
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Determine the design flexural strength φb Mn of the girder: Calculate Ix : X [Ii + Ai yi2 ] Ix = = [8932 + (21.88)(0)2 ] + 2[2.37 + (22.5)(35.56)2 ] = 65840 in.4 Calculate Sxt , Sxc : Sxt = Sxc =
Ix Ix 65840 = 1823 in.3 = = ct cc 35 + 1.125
Calculate rT : Refer to Figure 3.12b, s s IT (1.125)(20)3 /12 + (11.667)(5/16)3 /12 = 5.36 in. = rT = 1 Af + 6 Aw 22.5 + 16 (21.88) Calculate Fcr : For Flange Local Buckling (FLB),
# " bf 65 20 65 = 8.89 < p = = √ = 10.8 so, Fcr = Fyf = 36 ksi 2tf 2(1.125) Fyf 36
For Lateral Torsional Buckling (LTB), # " 300 20 × 12 300 Lb = 44.8 < p = = √ = 50 so, Fcr = Fyf = 36 ksi rT 5.36 Fyf 36 Calculate RP G : RP G
√ √ 0.972[70/(5/16) − 970/ 36] ar (hc /tw − 970/ Fcr ) =1− = 0.96 =1− (1,200 + 300ar ) [1,200 + 300(0.972)]
Calculate φb Mn : φb Mn
=
0.90 Sxt Re Fy t = (0.90)(1823)(1)(36) = 59,065 kipin. 0.90 Sxc RP G Re Fcr = (0.90)(1823)(0.96)(1)(36) = 56,700 kipin. 56,700 kipin.
=
4725 kipft.
=
smaller of
Since [φb Mn = 4725 kipft ] > [Mu = 4600 kipft ], the crosssection is acceptable. Use web plate 5/16”x70” and two flange plates 11/8”x20” for the girder crosssection.
EXAMPLE 3.8:
Design bearing stiffeners for the plate girder of the preceding example for a factored end reaction of 260 kips. Since the girder end is unframed, bearing stiffeners are required at the supports. The size of the stiffeners must be selected to ensure that the limit states of local buckling, compression, and bearing are not violated. 1999 by CRC Press LLC
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Limit state of local buckling Refer to p Figure 3.13, try bst = 8 in. To avoid problems with local buckling, bst /2tst must not exceed 95/ Fy = 15.8. Therefore, try tst = 1/2 in. So, bst /2tst = 8 which is less than 15.8.
FIGURE 3.13: Design of bearing stiffeners.
Limit state of compression Aeff
=
2(bst tst ) + 12tw2 = 2(8)(0.5) + 12(5/16)2 = 9.17 in.2
Ist
=
rst
=
tst (2bst + tw )3 /12 = 0.5[2(8) + 5/16]3 /12 = 181 in.4 q p (Ist /Aeff ) = (181/9.17) = 4.44 in.
Kh/rst
=
λc
=
0.75(70)/4.44 = 11.8 q p (Kh/πrst ) (Fy /E) = (11.8/3.142) (36/29,000) = 0.132
and from Equation 3.17 φc Pn = 0.85(0.658λc )Fy Ast = 0.85(0.658)0.132 (36)(9.17) = 279 kips 2
2
Since φc Pn > 260 kips, the design is satisfactory for compression. Limit state of bearing Assuming there is a 1/4in. weld cutout at the corners of the bearing stiffeners at the junction of the stiffeners and the girder flanges, the bearing area for the stiffener pairs is Apb = (8 − 0.25)(0.5)(2) = 7.75 in.2 . Substitute this into Equation 3.91, we have φRn = 0.75(1.8)(36)(7.75) = 377 kips, which exceeds the factored reaction of 260 kips. So, bearing is not a problem. Use two 1/2”x 8” plates for bearing stiffeners.
3.11
Connections
Connections are structural elements used for joining different members of a framework. Connections can be classified according to: 1999 by CRC Press LLC
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• the type of connecting medium used: bolted connections, welded connections, boltedwelded connections, riveted connections • the type of internal forces the connections are expected to transmit: shear (semirigid, simple) connections, moment (rigid) connections • the type of structural elements that made up the connections: single plate angle connections, double web angle connections, top and seated angle connections, seated beam connections, etc. • the type of members the connections are joining: beamtobeam connections (beam splices), columntocolumn connections (column splices), beamtocolumn connections, hanger connections, etc. To properly design a connection, a designer must have a thorough understanding of the behavior of the joint under loads. Different modes of failure can occur depending on the geometry of the connection and the relative strengths and stiffnesses of the various components of the connection. To ensure that the connection can carry the applied loads, a designer must check for all perceivable modes of failure pertinent to each component of the connection and the connection as a whole.
3.11.1
Bolted Connections
Bolted connections are connections whose components are fastened together primarily by bolts. The four basic types of bolts commonly used for steel construction are discussed in the section on Structural Fasteners. Depending on the direction and line of action of the loads relative to the orientation and location of the bolts, the bolts may be loaded in tension, shear, or a combination of tension and shear. For bolts subjected to shear forces, the design shear strength of the bolts also depends on whether or not the threads of the bolts are excluded from the shear planes. A letter X or N is placed at the end of the ASTM designation of the bolts to indicate whether the threads are excluded or not excluded from the shear planes, respectively. Thus, A325X denotes A325 bolts whose threads are excluded from the shear planes and A490N denotes A490 bolts whose threads are not excluded from the shear planes. Because of the reduced shear areas for bolts whose threads are not excluded from the shear planes, these bolts have lower design shear strengths than their counterparts whose threads are excluded from the shear planes. Bolts can be used in both bearingtype connections and slipcritical connections. Bearingtype connections rely on bearing between the bolt shanks and the connecting parts to transmit forces. Some slippage between the connected parts is expected to occur for this type of connection. Slipcritical connections rely on the frictional force developing between the connecting parts to transmit forces. No slippage between connecting elements is expected for this type of connection. Slipcritical connections are used for structures designed for vibratory or dynamic loads such as bridges, industrial buildings, and buildings in regions of high seismicity. Bolts used in slipcritical connections are denoted by the letter F after their ASTM designation, e.g., A325F, A490F. Bolt Holes
Holes made in the connected parts for bolts may be standard size, oversized, short slotted, or long slotted. Table 3.10 gives the maximum hole dimension for ordinary construction usage. Standard holes can be used for both bearingtype and slipcritical connections. Oversized holes shall be used only for slipcritical connections. Short and longslotted holes can be used for both bearingtype and slipcritical connections provided that when such holes are used for bearing, the direction of slot is transverse to the direction of loading. 1999 by CRC Press LLC
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TABLE 3.10
Nominal Hole Dimensions
Bolt
Hole dimensions
diameter, d (in.)
Standard (dia.)
Oversize (dia.)
Shortslot (width × length)
Longslot (width × length)
1/2 5/8 3/4 7/8 1 ≥ 11/8
9/16 11/16 13/16 15/16 11/16 d+1/16
5/8 13/16 15/16 11/16 11/4 d+5/16
9/16×11/16 11/16×7/8 13/16×1 15/16×11/8 11/16×15/16 (d+1/16)×(d+3/8)
9/16×11/4 11/16×19/16 13/16×17/8 15/16×23/16 11/16×21/2 (d+1/16)×(2.5d)
Note: 1 in. = 25.4 mm.
Bolts Loaded in Tension
If a tensile force is applied to the connection such that the direction of the load is parallel to the longitudinal axes of the bolts, the bolts will be subjected to tension. The following condition must be satisfied for bolts under tensile stresses. Allowable Stress Design: ft ≤ Ft
(3.95)
where ft = computed tensile stress in the bolt, ksi Ft = allowable tensile stress in bolt (see Table 3.11) Load and Resistance Factor Design: φt Ft ≥ ft
(3.96)
where φt = 0.75 ft = tensile stress produced by factored loads, ksi Ft = nominal tensile strength given in Table 3.11 TABLE 3.11
Ft of Bolts, ksi ASD
Bolt type
Ft , ksi (static loading)
A307 A325
20 44.0
Ft , ksi (fatigue loading)
Not allowed If N ≤ 20,000: Ft = same as for static loading
LRFD Ft , ksi (static loading)
45.0 90.0
If 20,000 < N ≤ 500,000: Ft = 40 (A325) = 49 (A490)
Ft , ksi
(fatigue loading) Not allowed If N ≤ 20,000: Ft = same as for static loading If 20,000 < N ≤ 500,000: Ft = 0.30Fu (at service loads)
If N > 500,000: A490
54.0
Ft = 31(A325) = 38 (A490)
where N = number of cycles Fu = minimum specified tensile strength, ksi Note: 1 ksi = 6.895 MPa.
1999 by CRC Press LLC
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113
If N > 500,000: Ft = 0.25Fu (at service loads) where N = number of cycles Fu = minimum specified tensile strength, ksi
Bolts Loaded in Shear
When the direction of load is perpendicular to the longitudinal axes of the bolts, the bolts will be subjected to shear. The condition that needs to be satisfied for bolts under shear stresses is as follows. Allowable Stress Design: fv ≤ Fv
(3.97)
where fv = computed shear stress in the bolt, ksi Fv = allowable shear stress in bolt (see Table 3.12) Load and Resistance Factor Design: φv Fv ≥ fv
(3.98)
where φv = 0.75 (for bearingtype connections), 1.00 (for slipcritical connections when standard, oversized, shortslotted, or longslotted holes with load perpendicular to the slots are used), 0.85 (for slipcritical connections when longslotted holes with load in the direction of the slots are used) fv = shear stress produced by factored loads (for bearingtype connections), or by service loads (for slipcritical connections), ksi Fv = nominal shear strength given in Table 3.12 TABLE 3.12
Fv of Bolts, ksi
Bolt type A307 A325N A325X A325Fb
A490N A490X A490Fb
Fv , ksi
ASD
LRFD
10.0a (regardless of whether or not threads
24.0a (regardless of whether or not threads
are excluded from shear planes) 21.0a 30.0a 17.0 (for standard size holes) 15.0 (for oversized and shortslotted holes) 12.0 (for longslotted holes when direction of load is transverse to the slots) 10.0 (for longslotted holes when direction of load is parallel to the slots) 28.0a 40.0a 21.0 (for standard size holes) 18.0 (for oversized and shortslotted holes) 15.0 (for longslotted holes when direction of load is transverse to the slots) 13.0 (for longslotted holes when direction of load is parallel to the slots)
are excluded from shear planes) 48.0a 60.0a 17.0 (for standard size holes) 15.0 (for oversized and shortslotted holes) 12.0 (for longslotted holes)
60.0a 75.0a 21.0 (for standard size holes) 18.0 (for oversized and shortslotted holes) 15.0 (for longslotted holes)
a tabulated values shall be reduced by 20% if the bolts are used to splice tension members having a fastener pattern whose length,
measured parallel to the line of action of the force, exceeds 50 in.
b tabulated values are applicable only to class A surface, i.e., clean mill surface and blast cleaned surface with class A coatings (with
slip coefficient = 0.33). For design strengths with other coatings, see RCSC “Load and Resistance Factor Design Specification to Structural Joints Using ASTM A325 or A490 Bolts” [28] Note: 1 ksi = 6.895 MPa.
Bolts Loaded in Combined Tension and Shear
If a tensile force is applied to a connection such that its line of action is at an angle with the longitudinal axes of the bolts, the bolts will be subjected to combined tension and shear. The conditions that need to be satisfied are given as follows. Allowable Stress Design: fv ≤ Fv and ft ≤ Ft 1999 by CRC Press LLC
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(3.99)
where fv , Fv ft Ft
= as defined in Equation 3.97 = computed tensile stress in the bolt, ksi = allowable tensile stress given in Table 3.13
Load and Resistance Factor Design: φv Fv ≥ fv and φt Ft ≥ ft
(3.100)
where φv , Fv , fv = as defined in Equation 3.98 = 1.0 φt = tensile stress due to factored loads (for bearingtype connection), or due to service ft loads (for slipcritical connections), ksi = nominal tension stress limit for combined tension and shear given in Table 3.13 Ft TABLE 3.13
Ft for Bolts Under Combined Tension and Shear, ksi Bearingtype connections ASD
Threads not excluded from the shear plane
Bolt type
LRFD Threads excluded from the shear plane
261.8fv ≤ 20 q q (442 − 4.39fv2 ) (442 − 2.15fv2 ) q q (542 − 3.75fv2 ) (542 − 1.82fv2 )
A307 A325 A490
Threads not excluded from the shear plane
Threads excluded from the shear plane
591.9fv ≤ 45 117 − 1.9fv ≤ 90
117 − 1.5fv ≤ 90
147 − 1.9fv ≤ 113
147 − 1.5fv ≤ 113
Slipcritical connections For ASD: Ft = Fv =
values given above [1 − (ft Ab /Tb )]× (values of Fv given in Table 3.12)
where ft Tb Fu Ab
computed tensile stress in the bolt, ksi pretension load = 0.70Fu Ab , kips minimum specified tensile strength, ksi nominal crosssectional area of bolt, in.2
= = = =
For LRFD: Ft = Fv =
values given above [1 − (T /Tb )]× (values of Fv given in Table 3.12)
where T Tb Fu Ab
service tensile force, kips pretension load = 0.70Fu Ab , kips minimum specified tensile strength, ksi nominal crosssectional area of bolt, in.2
= = = =
Note: 1 ksi = 6.895 MPa.
Bearing Strength at Fastener Holes
Connections designed on the basis of bearing rely on the bearing force developed between the fasteners and the holes to transmit forces and moments. The limit state for bearing must therefore be checked to ensure that bearing failure will not occur. Bearing strength is independent of the type of fastener. This is because the bearing stress is more critical on the parts being connected than on the fastener itself. The AISC specification provisions for bearing strength are based on preventing 1999 by CRC Press LLC
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excessive hole deformation. As a result, bearing capacity is expressed as a function of the type of holes (standard, oversized, slotted), bearing area (bolt diameter times the thickness of the connected parts), bolt spacing, edge distance (Le ), strength of the connected parts (Fu ) and the number of fasteners in the direction of the bearing force. Table 3.14 summarizes the expressions used in ASD and LRFD for calculating the bearing strength and the conditions under which each expression is valid. TABLE 3.14
Bearing Capacity
Conditions 1. For standard or shortslotted holes with Le ≥ 1.5d, s ≥ 3d and number of fasteners in the direction of bearing ≥ 2 2. For longslotted holes with direction of slot transverse to the direction of bearing and Le ≥ 1.5d, s ≥ 3d and the number of fasteners in the direction of bearing ≥ 2 3. If neither condition 1 nor 2 above is satisfied
ASD
LRFD
Allowable bearing stress, Fp , ksi
Design bearing strength, φRn , ksi
1.2Fu
0.75[2.4dtFu ]
1.0Fu
0.75[2.0dtFu ]
Le Fu /2d ≤ 1.2Fu
For the bolt hole nearest the edge: 0.75[Le tFu ] ≤ 0.75[2.4dtFu ]a For the remaining bolt holes: 0.75[(s − d/2)tFu ] ≤ 0.75[2.4dtFu ]a
1.5Fu
For the bolt hole nearest the edge: 0.75[Le tFu ] ≤ 0.75[3.0dtFu ] For the remaining bolt holes: 0.75[(s − d/2)tFu ] ≤ 0.75[3.0dtFu ]
4. If hole deformation is not a design consideration and adequate spacing and edge distance is provided (see sections on Minimum Fastener Spacing and Minimum Edge Distance)
a For longslotted bolt holes with direction of slot transverse to the direction of bearing, this limit is
0.75[2.0dtFu ] = edge distance (i.e., distance measured from the edge of the connected part to the center of Le a standard hole or the center of a short and longslotted hole perpendicular to the line of force. For oversized holes and short and longslotted holes parallel to the line of force, Le shall be increased by the edge distance increment C2 given in Table 3.16) s = fastener spacing (i.e., center to center distance between adjacent fasteners measured in the direction of bearing. For oversized holes and short and longslotted holes parallel to the line of force, s shall be increased by the spacing increment C1 given in Table 3.15) d = nominal bolt diameter, in. t = thickness of the connected part, in. Fu = specified minimum tensile strength of the connected part, ksi
TABLE 3.15
Values of Spacing Increment, C1 , in. Slotted Holes
Nominal
Parallel to line of force
diameter of fastener (in.)
Standard holes
Oversized holes
Transverse to line of force
Shortslots
Longslotsa
≤ 7/8 1 ≥ 11/8
0 0 0
1/8 3/16 1/4
0 0 0
3/16 1/4 5/16
3d /21/16 23/16 3d /21/16
a When length of slot is less than the value shown in Table 3.10, C may be reduced by the 1
difference between the value shown and the actual slot length. Note: 1 in. = 25.4 mm.
1999 by CRC Press LLC
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Minimum Fastener Spacing
To ensure safety, efficiency, and to maintain clearances between bolt nuts as well as to provide room for wrench sockets, the fastener spacing, s, should not be less than 3d where d is the nominal fastener diameter. TABLE 3.16
Values of Edge Distance Increment, C2 , in.
Nominal diameter
Slotted holes
of fastener
Slot transverse to edge
Slot parallel to
(in.)
Oversized holes
Shortslot
Longslota
edge
≤ 7/8 1 ≤ 11/8
1/16 1/8 1/8
1/8 1/8 3/16
3d/4 3d/4 3d/4
0
a If the length of the slot is less than the maximum shown in Table 3.10, the value shown may
be reduced by onehalf the difference between the maximum and the actual slot lengths. Note: 1 in. = 25.4 mm.
Minimum Edge Distance
To prevent excessive deformation and shear rupture at the edge of the connected part, a minimum edge distance Le must be provided in accordance with the values given in Table 3.17 for standard holes. For oversized and slotted holes, the values shown must be incremented by C2 given in Table 3.16. TABLE 3.17
Minimum Edge Distance for Standard Holes, in.
Nominal fastener diameter (in.)
At sheared edges
At rolled edges of plates, shapes, and bars or gas cut edges
1/2 5/8 3/4 7/8 1 11/8 11/4 over 11/4
7/8 11/8 11/4 11/2 13/4 2 21/4 13/4 x diameter
3/4 7/8 1 11/8 11/4 11/2 15/8 11/4 x diameter
Note: 1 in. = 25.4 mm.
Maximum Fastener Spacing
A limit is placed on the maximum value for the spacing between adjacent fasteners to prevent the possibility of gaps forming or buckling from occurring in between fasteners when the load to be transmitted by the connection is compressive. The maximum fastener spacing measured in the direction of the force is given as follows. For painted members or unpainted members not subject to corrosion: smaller of 24t where t is the thickness of the thinner plate and 12 in. For unpainted members of weathering steel subject to atmospheric corrosion: smaller of 14t where t is the thickness of the thinner plate and 7 in. 1999 by CRC Press LLC
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Maximum Edge Distance
A limit is placed on the maximum value for edge distance to prevent prying action from occurring. The maximum edge distance shall not exceed the smaller of 12t where t is the thickness of the connected part and 6 in.
EXAMPLE 3.9:
Check the adequacy of the connection shown in Figure 3.4a. The bolts are 1in. diameter A325N bolts in standard holes. Check bolt capacity All bolts are subjected to double shear. Therefore, the design shear strength of the bolts will be twice that shown in Table 3.12. Assuming each bolt carries an equal share of the factored applied load, we have from Equation 3.98 208 2 = 44.1 ksi [φv Fv = 0.75(2 × 48) = 72 ksi] > fv = (6) π41 The shear capacity of the bolt is therefore adequate. Check bearing capacity of the connected parts With reference to Table 3.14, it can be seen that condition 1 applies for the present problem. Therefore, we have 3 208 (58) = 39.2 kips] > Ru = = 34.7 kips [φRn = 0.75(2.4)(1) 8 6 and so bearing is not a problem. Note that bearing on the gusset plate is more critical than bearing on the webs of the channels because the thickness of the gusset plate is less than the combined thickness of the double channels. Check bolt spacing The minimum bolt spacing is 3d = 3(1) = 3 in. The maximum bolt spacing is the smaller of 14t = 14(.303) = 4.24 in. or 7 in. The actual spacing is 3 in. which falls within the range of 3 to 4.24 in., so bolt spacing is adequate. Check edge distance From Table 3.17, it can be determined that the minimum edge distance is 1.25 in. The maximum edge distance allowed is the smaller of 12t = 12(0.303) = 3.64 in. or 6 in. The actual edge distance is 3 in. which falls within the range of 1.25 to 3.64 in., so edge distance is adequate. The connection is adequate. Bolted Hanger Type Connections
A typical hanger connection is shown in Figure 3.14. In the design of such connections, the designer must take into account the effect of prying action. Prying action results when flexural deformation occurs in the tee flange or angle leg of the connection (Figure 3.15). Prying action tends to increase the tensile force, called prying force, in the bolts. To minimize the effect of prying, the fasteners should be placed as close to the tee stem or outstanding angle leg as the wrench clearance will permit (see Tables on Entering and Tightening Clearances in Volume IIConnections of the AISCLRFD Manual [22]). In addition, the flange and angle thickness should be proportioned so that the full tensile capacities of the bolts can be developed. 1999 by CRC Press LLC
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FIGURE 3.14: Hanger connections.
Two failure modes can be identified for hanger type connections: formation of plastic hinges in the tee flange or angle leg at crosssections 1 and 2, and tensile failure of the bolts when the tensile force including prying action Bc (= T + Q) exceeds the tensile capacity of the bolt B. Since the determination of the actual prying force is rather complex, the design equation for the required thickness for the tee flange or angle leg is semiempirical in nature. It is given by the following. If ASD is used:
s treq 0 d =
8T b0 pFy (1 + δα 0 )
(3.101)
where T = tensile force per bolt due to service load exclusive of initial tightening and prying force, kips The other variables are as defined in Equation 3.102 except that B in the equation for α 0 is defined as the allowable tensile force per bolt. A design is considered satisfactory if the thickness of the tee flange or angle leg tf exceeds treq 0 d and B > T . If LRFD is used:
s treq 0 d =
4Tu b0 φb pFy (1 + δα 0 )
(3.102)
where φb = 0.90 Tu = factored tensile force per bolt exclusive of initial tightening and prying force, kips p = length of flange tributary to each bolt measured along the longitudinal axis of the tee or double angle section, in. δ = ratio of net area at bolt line to gross area at angle leg or stem face = (p − d 0 )/p d 0 = diameter of bolt hole = bolt diameter +1/800 , in. α 0 = [(B/Tu − 1)(a 0 /b0 )]/{δ[1 − (B/Tu − 1)(a 0 /b0 )]}, but not larger than 1 (if α 0 is less than zero, use α 0 = 1) B = design tensile strength of one bolt = φFt Ab , kips (φFt is given in Table 3.11 and Ab is the nominal diameter of the bolt) a 0 = a + d/2 b0 = b − d/2 1999 by CRC Press LLC
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FIGURE 3.15: Prying action in hanger connections. a b
= distance from bolt centerline to edge of tee flange or angle leg but not more than 1.25b, in. = distance from bolt centerline to face of tee stem or outstanding leg, in.
A design is considered satisfactory if the thickness of the tee flange or angle leg tf exceeds treg 0 d and B > Tu . Note that if tf is much larger than treg 0 d , the design will be too conservative. In this case α 0 should be recomputed using the equation # " 1 4Tu b0 0 −1 (3.103) α = δ φb ptf2 Fy As before, the value of α 0 should be limited to the range 0 ≤ α 0 ≤ 1. This new value of α 0 is to be used in Equation 3.102 to recalculate treg 0 d . Bolted Bracket Type Connections
Figure 3.16 shows three commonly used bracket type connections. The bracing connection shown in Figure 3.16a should be designed so that the line of action the force passes through is the centroid of the bolt group. It is apparent that the bolts connecting the bracket to the column flange are subjected to combined tension and shear. As a result, the capacity of the connection is limited 1999 by CRC Press LLC
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FIGURE 3.16: Bolted brackettype connections.
1999 by CRC Press LLC
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to the combined tensileshear capacities of the bolts in accordance with Equation 3.99 in ASD and Equation 3.100 in LRFD. For simplicity, fv and ft are to be computed assuming that both the tensile and shear components of the force are distributed evenly to all bolts. In addition to checking for the bolt capacities, the bearing capacities of the column flange and the bracket should also be checked. If the axial component of the force is significant, the effect of prying should also be considered. In the design of the eccentrically loaded connections shown in Figure 3.16b, it is assumed that the neutral axis of the connection lies at the center of gravity of the bolt group. As a result, the bolts above the neutral axis will be subjected to combined tension and shear and so Equation 3.99 or Equation 3.100 needs to be checked. The bolts below the neutral axis are subjected to shear only and so Equation 3.97 or Equation 3.98 applies. In calculating fv , one can assume that all bolts in the bolt group carry an equal share of the shear force. In calculating ft , one can assume that the tensile force varies linearly from a value of zero at the neutral axis to a maximum value at the bolt farthest away from the neutral axis. Using this assumption, ft can be calculated from the equation P ey/I wherePy is the distance from the neutral axis to the location of the bolt above the neutral axis and I = Ab y 2 is the moment of inertia of the bolt areas with Ab equal to the crosssectional area of each bolt. The capacity of the connection is determined by the capacities of the bolts and the bearing capacity of the connected parts. For the eccentrically loaded bracket connection shown in Figure 3.16c, the bolts are subjected to shear. The shear force in each bolt can be obtained by adding vectorally the shear caused by the applied load P and the moment P χo . The design of this type of connection is facilitated by the use of tables contained in the AISC Manuals for Allowable Stress Design and Load and Resistance Factor Design [21, 22]. In addition to checking for bolt shear capacity, one needs to check the bearing and shear rupture capacities of the bracket plate to ensure that failure will not occur in the plate. Bolted Shear Connections
Shear connections are connections designed to resist shear force only. These connections are not expected to provide appreciable moment restraint to the connection members. Examples of these connections are shown in Figure 3.17. The framed beam connection shown in Figure 3.17a consists of two web angles which are often shopbolted to the beam web and then fieldbolted to the column flange. The seated beam connection shown in Figure 3.17b consists of two flange angles often shopbolted to the beam flange and fieldbolted to the column flange. To enhance the strength and stiffness of the seated beam connection, a stiffened seated beam connection shown in Figure 3.17c is sometimes used to resist large shear force. Shear connections must be designed to sustain appreciable deformation and yielding of the connections is expected. The need for ductility often limits the thickness of the angles that can be used. Most of these connections are designed with angle thickness not exceeding 5/8 in. The design of the connections shown in Figure 3.17 is facilitated by the use of design tables contained in the AISCASD and AISCLRFD Manuals. These tables give design loads for the connections with specific dimensions based on the limit states of bolt shear, bearing strength of the connection, bolt bearing with different edge distances, and block shear (for coped beams). Bolted MomentResisting Connections
Momentresisting connections are connections designed to resist both moment and shear. These connections are often referred to as rigid or fully restrained connections as they provide full continuity between the connected members and are designed to carry the full factored moments. Figure 3.18 shows some examples of momentresisting connections. Additional examples can be found in the AISCASD and AISCLRFD Manuals and Chapter 4 of the AISC Manual on Connections [20]. 1999 by CRC Press LLC
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FIGURE 3.17: Bolted shear connections. (a) Bolted frame beam connection. (b) Bolted seated beam connection. (c) Bolted stiffened seated beam connection.
1999 by CRC Press LLC
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FIGURE 3.18: Bolted moment connections.
1999 by CRC Press LLC
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Design of MomentResisting Connections
An assumption used quite often in the design of moment connections is that the moment is carried solely by the flanges of the beam. The moment is converted to a couple Ff given by Ff = M/(d − tf ) acting on the beam flanges as shown in Figure 3.19.
FIGURE 3.19: Flange forces in moment connections.
The design of the connection for moment is considered satisfactory if the capacities of the bolts and connecting plates or structural elements are adequate to carry the flange force Ff . Depending on the geometry of the bolted connection, this may involve checking: (a) the shear and/or tensile capacities of the bolts, (b) the yield and/or fracture strength of the moment plate, (c) the bearing strength of the connected parts, and (d) bolt spacing and edge distance as discussed in the foregoing sections. As for shear, it is common practice to assume that all the shear resistance is provided by the shear plates or angles. The design of the shear plates or angles is governed by the limit states of bolt shear, bearing of the connected parts, and shear rupture. If the moment to be resisted is large, the flange force may cause bending of the column flange, or local yielding, crippling, or buckling of the column web. To prevent failure due to bending of the column flange or local yielding of the column web (for a tensile Ff ) as well as local yielding, crippling or buckling of the column web (for a compressive Ff ), column stiffeners should be provided if any one of the conditions discussed in the section on Criteria on Concentrated Loads is violated. Following is a set of guidelines for the design of column web stiffeners [21, 22]: 1. If local web yielding controls, the area of the stiffeners (provided in pairs) shall be determined based on any excess force beyond that which can be resisted by the web alone. The stiffeners need not extend more than onehalf the depth of the column web if the concentrated beam flange force Ff is applied at only one column flange. 1999 by CRC Press LLC
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2. If web crippling or compression buckling of the web controls, the stiffeners shall be designed as axially loaded compression members (see section on Compression Members). The stiffeners shall extend the entire depth of the column web. 3. The welds that connect the stiffeners to the column shall be designed to develop the full strength of the stiffeners. In addition, the following recommendations are given: 1. The width of the stiffener plus onehalf of the column web thickness should not be less than onehalf the width of the beam flange nor the moment connection plate which applies the force. 2. The stiffener thickness should not be less than onehalf the thickness of the beam flange. 3. If only one flange of the column is connected by a moment connection, the length of the stiffener plate does not have to exceed onehalf the column depth. 4. If both flanges of the column are connected by moment connections, the stiffener plate should extend through the depth of the column web and welds should be used to connect the stiffener plate to the column web with sufficient strength to carry the unbalanced moment on opposite sides of the column. 5. If column stiffeners are required on both the tension and compression sides of the beam, the size of the stiffeners on the tension side of the beam should be equal to that on the compression size for ease of construction. In lieu of stiffener plates, a stronger column section could be used to preclude failure in the column flange and web. For a more thorough discussion of bolted connections, the readers are referred to the book by Kulak et al. [16]. Examples on the design of a variety of bolted connections can be found in the AISCLRFD Manual [22] and the AISC Manual on Connections [20]
3.11.2
Welded Connections
Welded connections are connections whose components are joined together primarily by welds. The four most commonly used welding processes are discussed in the section on Structural Fasteners. Welds can be classified according to: • types of welds: groove, fillet, plug, and slot welds. • positions of the welds: horizontal, vertical, overhead, and flat welds. • types of joints: butt, lap, corner, edge, and tee. Although fillet welds are generally weaker than groove welds, they are used more often because they allow for larger tolerances during erection than groove welds. Plug and slot welds are expensive to make and they do not provide much reliability in transmitting tensile forces perpendicular to the faying surfaces. Furthermore, quality control of such welds is difficult because inspection of the welds is rather arduous. As a result, plug and slot welds are normally used just for stitching different parts of the members together. Welding Symbols
A shorthand notation giving important information on the location, size, length, etc. for the various types of welds was developed by the American Welding Society [6] to facilitate the detailing of welds. This system of notation is reproduced in Figure 3.20. 1999 by CRC Press LLC
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FIGURE 3.20: Basic weld symbols.
1999 by CRC Press LLC
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Strength of Welds
In ASD, the strength of welds is expressed in terms of allowable stress. In LRFD, the design strength of welds is taken as the smaller of the design strength of the base material φFBM and the design strength of the weld electrode φFW . These allowable stresses and design strengths are summarized in Table 3.18 [18, 21]. When a design uses ASD, the computed stress in the weld shall not exceed its allowable value. When a design uses LRFD, the design strength of welds should exceed the required strength obtained by dividing the load to be transmitted by the effective area of the welds. TABLE 3.18
Strength of Welds
Types of weld and stressa
ASD allowable stress
Material
LRFD φFBM or φFW
Required weld strength levelb,c
Full penetration groove weld Tension normal to effective area Compression normal to effective area Tension of compression parallel to axis of weld Shear on effective area
Base
Same as base metal
0.90Fy
Base
Same as base metal
0.90Fy
Base
Same as base metal
0.90Fy
Base weld electrode
0.30× nominal tensile strength of weld metal
0.90[0.60Fy ] 0.80[0.60FEXX ]
“Matching” weld must be used Weld metal with a strength level equal to or less than “matching” must be used
Partial penetration groove welds Compression normal to effective area Tension or compression parallel to axis of weldd Shear parallel to axis of weld
Base
Same as base metal
0.90Fy
Base weld electrode
0.75[0.60FEXX ]
Tension normal to effective area
Base weld electrode
0.30× nominal tensile strength of weld metal 0.30× nominal tensile strength of weld metal ≤ 0.18× yield stress of base metal
Weld metal with a strength level equal to or less than “matching” weld metal may be used
0.90Fy 0.80[0.60FEXX ]
Fillet welds Stress on effective area
Tension or compression parallel to axis of weldd
Base weld electrode
0.30× nominal tensile strength of weld metal
0.75[0.60FEXX ] 0.90Fy
Base
Same as base metal
0.90Fy
Weld metal with a strength level equal to or less than “matching” weld metal may be used
Plug or slot welds Shear parallel to faying surfaces (on effective area)
Base weld electrode
0.30×nominal tensile strength of weld metal
0.75[0.60FEXX ]
Weld metal with a strength level equal to or less than “matching” weld metal may be used
a see below for effective area b see AWS D1.1 for “matching”weld material c weld metal one strength level stronger than “matching” weld metal will be permitted
d fillet welds partialpenetration groove welds joining component elements of builtup members such as flangetoweb con
nections may be designed without regard to the tensile or compressive stress in these elements parallel to the axis of the welds
1999 by CRC Press LLC
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Effective Area of Welds
The effective area of groove welds is equal to the product of the width of the part joined and the effective throat thickness. The effective throat thickness of a fullpenetration groove weld is taken as the thickness of the thinner part joined. The effective throat thickness of a partialpenetration groove weld is taken as the depth of the chamfer for J, U, bevel, or V (with bevel ≥ 60◦ ) joints and it is taken as the depth of the chamfer minus 1/8 in. for bevel or V joints if the bevel is between 45◦ and 60◦ . For flare bevel groove welds the effective throat thickness is taken as 5R/16 and for flare Vgroove the effective throat thickness is taken as R/2 (or 3R/8 for GMAW process when R ≥ 1 in.). R is the radius of the bar or bend. The effective area of fillet welds is equal to the product of length of the fillets including returns and the effective throat thickness. The effective throat thickness of a fillet weld is the shortest distance from the root of the joint to the face of the diagrammatic weld as shown in Figure 3.21. Thus, for
FIGURE 3.21: Effective throat of fillet welds. an equal leg fillet weld, the effective throat is given by 0.707 times the leg dimension. For fillet weld made by the submerged arc welding process (SAW), the effective throat thickness is taken as the leg size (for 3/8in. and smaller fillet welds) or as the theoretical throat plus 0.11in. (for fillet weld over 3/8in.). A larger value for the effective throat thickness is permitted for welds made by the SAW process to account for the inherently superior quality of such welds. The effective area of plug and slot welds is taken as the nominal crosssectional area of the hole or slot in the plane of the faying surface. 1999 by CRC Press LLC
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Size and Length Limitations of Welds
To ensure effectiveness, certain size and length limitations are imposed for welds. For partialpenetration groove welds, minimum values for the effective throat thickness are given in Table 3.19. TABLE 3.19 Minimum Effective Throat Thickness of PartialPenetration Groove Welds Thickness of the thicker part joined, t (in.)
Minimum effective throat thickness (in.)
t ≤ 1/4 1/4 < t ≤ 1/2 1/2 < t ≤ 3/4 3/4 < t ≤ 11/2 11/2 < t ≤ 21/4 21/4 < t ≤ 6 >6
1/8 3/16 1/4 5/16 3/8 1/2 5/8
Note: 1 in. = 25.4 mm.
For fillet welds, the following size and length limitations apply: Minimum Size of Leg—The minimum leg size is given in Table 3.20. TABLE 3.20
Minimum Leg Size of Fillet Welds
Thickness of thicker part joined, t (in.)
Minimum leg size (in.)
≤ 1/4 1/4 < t ≤ 1/2 1/2 < t ≤ 3/4 > 3/4
1/8 3/16 1/4 5/16
Note: 1 in. = 25.4 mm.
Maximum Size of Leg—Along the edge of a connected part less than 1/4 thick, the maximum leg size is equal to the thickness of the connected part. For thicker parts, the maximum leg size is t minus 1/16 in. where t is the thickness of the part. Minimum effective length of weld—The minimum effective length of a fillet weld is four times its nominal size. If a shorter length is used, the leg size of the weld shall be taken as 1/4 its effective length for purpose of stress computation. The length of fillet welds used for flat bar tension members shall not be less than the width of the bar if the welds are provided in the longitudinal direction only. The transverse distance between longitudinal welds should not exceed 8 in. unless the effect of shear lag is accounted for by the use of an effective net area. Maximum effective length of weld—The maximum effective length of a fillet weld loaded by forces parallel to the weld shall not exceed 70 times the size of the fillet weld leg. End returns—End returns must be continued around the corner and must have a length of at least two times the size of the weld leg. Welded Connections for Tension Members
Figure 3.22 shows a tension angle member connected to a gusset plate by fillet welds. The applied tensile force P is assumed to act along the center of gravity of the angle. To avoid eccentricity, the lengths of the two fillet welds must be proportioned so that their resultant will also act along the 1999 by CRC Press LLC
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FIGURE 3.22: An eccentrically loaded welded tension connection.
center of gravity of the angle. For example, if LRFD is used, the following equilibrium equations can be written: Summing force along the axis of the angle (φFM )teff L1 + (φFm )teff L2 = Pu
(3.104)
Summing moment about the center of gravity of the angle (φFM )teff L1 d1 = (φFM )teff L2 d2
(3.105)
where Pu is the factored axial force, φFM is the design strength of the welds as given in Table 3.18, teff is the effective throat thickness, L1 , L2 are the lengths of the welds, and d1 , d2 are the transverse distances from the center of gravity of the angle to the welds. The two equations can be used to solve for L1 and L2 . If end returns are used, the added strength of the end returns should also be included in the calculations. Welded Bracket Type Connections
A typical welded bracket connection is shown in Figure 3.23. Because the load is eccentric with respect to the center of gravity of the weld group, the connection is subjected to both moment and shear. The welds must be designed to resist the combined effect of direct shear for the applied load and any additional shear from the induced moment. The design of the welded bracket connection is facilitated by the use of design tables in the AISCASD and AISCLRFD Manuals. In both ASD and LRFD, the load capacity for the connection is given by P = CC1 Dl where P = l = D = C1 = C =
allowable load (in ASD), or factored load, Pu (in LRFD), kips length of the vertical weld, in. number of sixteenths of an inch in fillet weld size coefficients for electrode used (see table below) coefficients tabulated in the AISCASD and AISCLRFD Manuals. In the tables, values of C for a variety of weld geometries and dimensions are given
1999 by CRC Press LLC
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(3.106)
FIGURE 3.23: An eccentrically loaded welded bracket connection.
Electrode ASD LRFD
Fv (ksi) C1 FEXX (ksi) C1
E60
E70
E80
E90
E100
E110
18 0.857 60 0.857
21 1.0 70 1.0
24 1.14 80 1.03
27 1.29 90 1.16
30 1.43 100 1.21
33 1.57 110 1.34
Welded Connections with Welds Subjected to Combined Shear and Flexure
Figure 3.24 shows a welded framed connection and a welded seated connection. The welds for these connections are subjected to combined shear and flexure. For purpose of design, it is common practice to assume that the shear force per unit length, RS , acting on the welds is a constant and is given by P (3.107) RS = 2l where P is the allowable load (in ASD), or factored load, Pu (in LRFD), and l is the length of the vertical weld. In addition to shear, the welds are subjected to flexure as a result of load eccentricity. There is no general agreement on how the flexure stress should be distributed on the welds. One approach is to assume that the stress distribution is linear with half the weld subjected to tensile flexure stress and half is subjected to compressive flexure stress. Based on this stress distribution and ignoring the returns, the flexure tension force per unit length of weld, RF , acting at the top of the weld can be written as Pe (l/2) 3Pe Mc = 3 = 2 (3.108) RF = I 2l /12 l where e is the load eccentricity. The resultant force per unit length acting on the weld, R, is then q R = RS2 + RF2 1999 by CRC Press LLC
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(3.109)
FIGURE 3.24: Welds subjected to combined shear and flexure.
1999 by CRC Press LLC
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For a satisfactory design, the value R/teff where teff is the effective throat thickness of the weld should not exceed the allowable values or design strengths given in Table 3.18. Welded Shear Connections
Figure 3.25 shows three commonly used welded shear connections: a framed beam connection, a seated beam connection, and a stiffened seated beam connection. These connections can be designed by using the information presented in the earlier sections on welds subjected to eccentric shear and welds subjected to combined tension and flexure. For example, the welds that connect the angles to the beam web in the framed beam connection can be considered as eccentrically loaded welds and so Equation 3.106 can be used for their design. The welds that connect the angles to the column flange can be considered as welds subjected to combined tension and flexure and so Equation 3.109 can be used for their design. Like bolted shear connections, welded shear connections are expected to exhibit appreciable ductility and so the use of angles with thickness in excess of 5/8 in. should be avoided. To prevent shear rupture failure, the shear rupture strength of the critically loaded connected parts should be checked. To facilitate the design of these connections, the AISCASD and AISCLRFD Manuals provide design tables by which the weld capacities and shear rupture strengths for different connection dimensions can be checked readily. Welded MomentResisting Connections
Welded momentresisting connections (Figure 3.26), like bolted momentresisting connections, must be designed to carry both moment and shear. To simplify the design procedure, it is customary to assume that the moment, to be represented by a couple Ff as shown in Figure 3.19, is to be carried by the beam flanges and that the shear is to be carried by the beam web. The connected parts (e.g., the moment plates, welds, etc.) are then designed to resist the forces Ff and shear. Depending on the geometry of the welded connection, this may include checking: (a) the yield and/or fracture strength of the moment plate, (b) the shear and/or tensile capacity of the welds, and (c) the shear rupture strength of the shear plate. If the column to which the connection is attached is weak, the designer should consider the use of column stiffeners to prevent failure of the column flange and web due to bending, yielding, crippling, or buckling (see section on Design of MomentResisting Connections). Examples on the design of a variety of welded shear and momentresisting connections can be found in the AISC Manual on Connections [20] and the AISCLRFD Manual [22].
3.11.3
Shop WeldedField Bolted Connections
A large percentage of connections used for construction are shop welded and field bolted types. These connections are usually more cost effective than fully welded connections and their strength and ductility characteristics often rival those of fully welded connections. Figure 3.27 shows some of these connections. The design of shop welded–field bolted connections is also covered in the AISC Manual on Connections and the AISCLRFD Manual. In general, the following should be checked: (a) Shear/tensile capacities of the bolts and/or welds, (b) bearing strength of the connected parts, (c) yield and/or fracture strength of the moment plate, and (d) shear rupture strength of the shear plate. Also, as for any other types of moment connections, column stiffeners shall be provided if any one of the following criteria is violated: column flange bending, local web yielding, crippling, and compression buckling of the column web. 1999 by CRC Press LLC
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FIGURE 3.25: Welded shear connections. (a) Framed beam connection, (b) seated beam connection, (c) stiffened beam connection.
1999 by CRC Press LLC
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FIGURE 3.26: Welded moment connections.
3.11.4
Beam and Column Splices
Beam and column splices (Figure 3.28) are used to connect beam or column sections of different sizes. They are also used to connect beams or columns of the same size if the design calls for an extraordinarily long span. Splices should be designed for both moment and shear unless it is the intention of the designer to utilize the splices as internal hinges. If splices are used for internal hinges, provisions must be made to ensure that the connections possess adequate ductility to allow for large hinge rotation. Splice plates are designed according to their intended functions. Moment splices should be designed to resist the flange force Ff = M/(d − tf ) (Figure 3.19) at the splice location. In particular, the following limit states need to be checked: yielding of gross area of the plate, fracture of net area of the plate (for bolted splices), bearing strengths of connected parts (for bolted splices), shear capacity of bolts (for bolted splices), and weld capacity (for welded splices). Shear splices should be designed to resist the shear forces acting at the locations of the splices. The limit states that need to be checked include: shear rupture of the splice plates, shear capacity of bolts under an eccentric load (for bolted splices), bearing capacity of the connected parts (for bolted splices), shear capacity of bolts (for bolted splices), and weld capacity under an eccentric load (for welded splices). Design examples of beam and column splices can be found in the AISC Manual of Connections [20] and the AISCLRFD Manuals [22].
1999 by CRC Press LLC
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FIGURE 3.27: Shopwelded fieldbolted connections.
1999 by CRC Press LLC
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FIGURE 3.28: Bolted and welded beam and column splices.
3.12
Column Base Plates and Beam Bearing Plates (LRFD Approach)
3.12.1
Column Base Plates
Column base plates are steel plates placed at the bottom of columns whose function is to transmit column loads to the concrete pedestal. The design of column base plates involves two major steps: (1) determining the size N × B of the plate, and (2) determining the thickness tp of the plate. Generally, the size of the plate is determined based on the limit state of bearing on concrete and the thickness of the plate is determined based on the limit state of plastic bending of critical sections in the plate. Depending on the types of forces (axial force, bending moment, shear force) the plate will be subjected to, the design procedures differ slightly. In all cases, a layer of grout should be placed between the base plate and its support for the purpose of leveling and anchor bolts should be provided to stabilize the column during erection or to prevent uplift for cases involving large bending moment. 1999 by CRC Press LLC
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Axially Loaded Base Plates
Base plates supporting concentrically loaded columns in frames in which the column bases are assumed pinned are designed with the assumption that the column factored load Pu is distributed uniformly to the area of concrete under the base plate. The size of the base plate is determined from the limit state of bearing on concrete. The design bearing strength of concrete is given by the equation s # " A2 0 (3.110) φc Pp = 0.60 0.85fc A1 A1 where fc0 = compressive strength of concrete A1 = area of base plate A2 = area of concrete pedestal that is geometrically similar to and concentric with the loaded area, A1 ≤ A2 ≤ 4A1 From Equation 3.110, it can be seen that the bearing capacity increases when the concrete area is greater than the plate area. This accounts for the beneficial effect of confinement. The upper limit of the bearing strength is obtained when A2 = 4A1 . Presumably, the concrete area in excess of 4A1 is not effective in resisting the load transferred through the base plate. Setting the column factored load, Pu , equal to the bearing capacity of the concrete pedestal, φc Pp , and solving for A1 from Equation 3.110, we have A1 =
1 A2
Pu 0.6(0.85fc0 )
2 (3.111)
The length, N, and width, B, of the plate should be established so that N × B > A1 . For an efficient design, the length can be determined from the equation p (3.112) N ≈ A1 + 0.50(0.95d − 0.80bf ) where 0.95d and 0.80bf define the socalled effective load bearing area shown crosshatched in Figure 3.29a. Once N is obtained, B can be solved from the equation B=
A1 N
(3.113)
Both N and B should be rounded up to the nearest full inches. The required plate thickness, treg 0 d , is to be determined from the limit state of yield line formation along the most severely stressed sections. A yield line develops when the crosssection moment capacity is equal to its plastic moment capacity. Depending on the size of the column relative to the plate and the magnitude of the factored axial load, yield lines can form in various patterns on the plate. Figure 3.29 shows three models of plate failure in axially loaded plates. If the plate is large compared to the column, yield lines are assumed to form around the perimeter of the effective load bearing area (the crosshatched area) as shown in Figure 3.29a. If the plate is small and the column factored load is light, yield lines are assumed to form around the inner perimeter of the Ishaped area as shown in Figure 3.29b. If the plate is small and the column factored load is heavy, yield lines are assumed to form around the inner edge of the column flanges and both sides of the column web as shown in Figure 3.29c. The following equation can be used to calculate the required plate thickness s 2Pu (3.114) treq 0 d = l 0.90Fy BN 1999 by CRC Press LLC
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FIGURE 3.29: Failure models for centrally loaded column base plates.
where l is the larger of m, n, and λn0 given by
m
=
n = n0 1999 by CRC Press LLC
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=
(N − 0.95d) 2 (B − 0.80bf ) 2 p dbf 4
and
√ 2 X ≤1 λ= √ 1+ 1−X
in which
X=
4dbf (d + bf )2
Pu φc Pp
Base Plates for Tubular and Pipe Columns
The design concept for base plates discussed above for Ishaped sections can be applied to the design of base plates for rectangular tubes and circular pipes. The critical section used to determine the plate thickness should be based on 0.95 times the outside column dimension for rectangular tubes and 0.80 times the outside dimension for circular pipes [11]. Base Plates with Moments
For columns in frames designed to carry moments at the base, base plates must be designed to support both axial forces and bending moments. If the moment is small compared to the axial force, the base plate can be designed without consideration of the tensile force which may develop in the anchor bolts. However, if the moment is large, this effect should be considered. To quantify the relative magnitude of this moment, an eccentricity e = Mu /Pu is used. The general procedures for the design of base plates for different values of e will be given in the following [11]. Small eccentricity, e ≤ N/6 If e is small, the bearing stress is assumed to distribute linearly over the entire area of the base plate (Figure 3.30). The maximum bearing stress is given by fmax =
Pu Mu c + BN I
(3.115)
where c = N/2 and I = BN 3 /12.
FIGURE 3.30: Eccentrically loaded column base plate (small load eccentricity).
The size of the plate is to be determined by a trial and error process. The size of the base plate should be such that the bearing stress calculated using Equation 3.115 does not exceed φc Pp /A1 , 1999 by CRC Press LLC
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given by
s
" 0.60
0.85fc0
A2 A1
# ≤ 0.60[1.7fc0 ]
The thickness of the plate is to be determined from s 4Mplu tp = 0.90Fy
(3.116)
(3.117)
where Mplu is the moment per unit width of critical section in the plate. Mplu is to be determined by assuming that the portion of the plate projecting beyond the critical section acts as an inverted cantilever loaded by the bearing pressure. The moment calculated at the critical section divided by the length of the critical section (i.e., B) gives Mplu . Moderate eccentricity, N/6 < e ≤ N/2 For plates subjected to moderate moments, only portions of the plate will be subjected to bearing stress (Figure 3.31). Ignoring the tensile force in the anchor bolt in the region of the plate where no
FIGURE 3.31: Eccentrically loaded column base plate (moderate load eccentricity). bearing occurs and denoting A as the length of the plate in bearing, the maximum bearing stress can be calculated from force equilibrium consideration as fmax =
2Pu AB
(3.118)
where A = 3(N/2 − e) is determined from moment equilibrium. The plate should be portioned such that fmax does not exceed the value calculated using Equation 3.116. tp is to be determined from Equation 3.117. Large eccentricity, e > N/2 For plates subjected to large bending moments so that e > N/2, one needs to take into consideration the tensile force developing in the anchor bolts (Figure 3.32). Denoting T as the resultant force in the anchor bolts, force equilibrium requires that T + Pu = 1999 by CRC Press LLC
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fmax AB 2
(3.119)
FIGURE 3.32: Eccentrically loaded column base plate (large load eccentricity). and moment equilibrium requires that fmax AB N A +M = N0 − Pu N 0 − 2 2 3
(3.120)
The above equations can be used to solve for A and T . The size of the plate is to be determined using a trialanderror process. The size should be chosen such that fmax does not exceed the value calculated using Equation 3.116, A should be smaller than N 0 and T should not exceed the tensile capacity of the bolts. Once the size of the plate is determined, the plate thickness tp is to be calculated using Equation 3.117. Note that there are two critical sections on the plate, one on the compression side of the plate and the other on the tension side of the plate. Two values of Mplu are to be calculated and the larger value should be used to calculate tp . Base Plates with Shear
Under normal circumstances, the factored column base shear is adequately resisted by the frictional force developed between the plate and its support. Additional shear capacity is also provided by the anchor bolts. For cases in which exceptionally high shear force is expected, such as in a bracing connection or in which uplift occurs which reduces the frictional resistance, the use of shear lugs may be necessary. Shear lugs can be designed based on the limit states of bearing on concrete and bending of the lugs. The size of the lug should be proportioned such that the bearing stress on concrete does not exceed 0.60(0.85fc0 ). The thickness of the lug can be determined from Equation 3.117. Mplu is the moment per unit width at the critical section of the lug. The critical section is taken to be at the junction of the lug and the plate (Figure 3.33).
3.12.2
Anchor Bolts
Anchor bolts are provided to stabilize the column during erection and to prevent uplift for cases involving large moments. Anchor bolts can be castinplace bolts or drilledin bolts. The latter are placed after the concrete is set and are not too often used. Their design is governed by the manufacturer’s specifications. Castinplace bolts are hooked bars, bolts, or threaded rods with nuts (Figure 3.34) placed before the concrete is set. Of the three types of castinplace anchors shown in the figure, the hooked bars are recommended for use only in axially loaded base plates. They are not normally relied upon to carry significant tensile force. Bolts and threaded rods with nuts can be used 1999 by CRC Press LLC
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FIGURE 3.33: Column base plate subjected to shear.
FIGURE 3.34: Base plate anchors.
for both axially loaded base plates or base plates with moments. Threaded rods with nuts are used when the length and size required for the specific design exceed those of standard size bolts. Failure of bolts or threaded rods with nuts occur when their tensile capacities are reached. Failure is also considered to occur when a cone of concrete is pulled out from the pedestal. This cone pullout type of failure is depicted schematically in Figure 3.35. The failure cone is assumed to radiate out from the bolt head or nut at p an angle of 45◦ with tensile failure occurring along the surface of the cone at an average stress of 4 fc0 where fc0 is the compressive strength of concrete in psi. The load that will cause this cone pullout failure is given by the product of this average stress and the projected area the cone Ap [23, 24]. The design of anchor bolts is thus governed by the limit states of tensile fracture of the anchors and cone pullout. 1999 by CRC Press LLC
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FIGURE 3.35: Cone pullout failure.
Limit State of Tensile Fracture The area of the anchor should be such that Ag ≥
Tu φt 0.75Fu
(3.121)
where Ag is the required gross area of the anchor, Fu is the minimum specified tensile strength, and φt is the resistance factor for tensile fracture which is equal to 0.75. Limit State of Cone PullOut From Figure 3.35, it is clear that the size of the cone is a function of the length of the anchor. Provided that there is sufficient edge distance and spacing between adjacent anchors, the amount of tensile force required to cause cone pullout failure increases with the embedded length of the anchor. This concept can be used to determine the required embedded length of the anchor. Assuming that the failure cone does not intersect with another failure cone nor the edge of the pedestal, the required embedded length can be calculated from the equation r L≥
s Ap = π
p (Tu /φt 4 fc0 ) π
(3.122)
where Ap is the projected area of the failure cone, Tu is the required bolt force in pounds, fc0 is the compressive strength of concrete in psi and φt is the resistance factor assumed to be equal to 0.75. If failure cones from adjacent anchors overlap one another or intersect with the pedestal edge, the projected area Ap must be adjusted according (see, for example [23, 24]). The length calculated using the above equation should not be less than the recommended values given by [29]. These values are reproduced in the following table. Also shown in the table are the recommended minimum edge distances for the anchors. 1999 by CRC Press LLC
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Bolt type (material)
Minimum embedded length
Minimum edge distance
A307 (A36)
12d
5d > 4 in.
A325 (A449)
17d
7d > 4 in.
d = nominal diameter of the anchor
3.12.3
Beam Bearing Plates
Beam bearing plates are provided between main girders and concrete pedestals to distribute the girder reactions to the concrete supports (Figure 3.36). Beam bearing plates may also be provided between cross beams and girders if the cross beams are designed to sit on the girders.
FIGURE 3.36: Beam bearing plate.
Beam bearing plates are designed based on the limit states of web yielding, web crippling, bearing on concrete, and plastic bending of the plate. The dimension of the plate along the beam axis, i.e., N, is determined from the web yielding or web crippling criterion (see section on Concentrated Load Criteria), whichever is more critical. The dimension B of the plate is determined from Equation 3.113 with A1 calculated using Equation 3.111. Pu in Equation 3.111 is to be replaced by Ru , the factored reaction at the girder support. 1999 by CRC Press LLC
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Once the size B × N is determined, the plate thickness tp can be calculated using the equation s tp =
2Ru n2 0.90Fy BN
(3.123)
where Ru is the factored girder reaction, Fy is the yield stress of the plate and n = (B − 2k)/2 in which k is the distance from the web toe of the fillet to the outer surface of the flange. The above equation was developed based on the assumption that the critical sections for plastic bending in the plate occur at a distance k from the centerline of the web.
3.13
Composite Members (LRFD Approach)
Composite members are structural members made from two or more materials. The majority of composite sections used for building constructions are made from steel and concrete. Steel provides strength and concrete provides rigidity. The combination of the two materials often results in efficient loadcarrying members. Composite members may be concreteencased or concretefilled. For concreteencased members (Figure 3.37a), concrete is casted around steel shapes. In addition to enhancing strength and providing rigidity to the steel shapes, the concrete acts as a fireproofing material to the steel shapes. It also serves as a corrosion barrier shielding the steel from corroding under adverse environmental conditions. For concretefilled members (Figure 3.37b), structural steel tubes are filled with concrete. In both concreteencased and concretefilled sections, the rigidity of the concrete often eliminates the problem of local buckling experienced by some slender elements of the steel sections. Some disadvantages associated with composite sections are that concrete creeps and shrinks. Furthermore, uncertainties with regard to the mechanical bond developed between the steel shape and the concrete often complicate the design of beamcolumn joints.
3.13.1
Composite Columns
According to the LRFD Specification [18], a compression member is regarded as a composite column if (1) the crosssectional area of the steel shape is at least 4% of the total composite area. If this condition is not satisfied, the member should be designed as a reinforced concrete column. (2) Longitudinal reinforcements and lateral ties are provided for concreteencased members. The crosssectional area of the reinforcing bars shall be 0.007 in.2 per inch of bar spacing. To avoid spalling, lateral ties shall be placed at a spacing not greater than 2/3 the least dimension of the composite crosssection. For fire and corrosion resistance, a minimum clear cover of 1.5 in. shall be provided. (3) The compressive strength of concrete fc0 used for the composite section falls within the range 3 to 8 ksi for normal weight concrete and not less than 4 ksi for light weight concrete. These limits are set because they represent the range of test data available for the development of the design equations. (4) The specified minimum yield stress for the steel shapes and reinforcing bars used in calculating the strength of the composite column does not exceed 55 ksi. This limit is set because this stress corresponds to a strain below which the concrete remains unspalled andp stable. (5) The minimum wall thickness of the steel shapes for concrete filled members is equal to b (Fy /3E) for rectangular sections of width b and p D (Fy /8E) for circular sections of outside diameter D. Design Compressive Strength
The design compressive strength, φc Pn , shall exceed the factored compressive force, Pu . The design compressive strength is given as follows: 1999 by CRC Press LLC
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FIGURE 3.37: Composite columns. For λc ≤ 1.5
h i 0.85 0.658λ2c As Fmy , i h φc Pn = F A , 0.85 0.877 s my 2 λ c
where λc =
KL rm π
Fmy = Fy + c1 Fyr
q Ar As
Fmy Em
Em = E + c3 Ec Ac Ar As E Ec Fy Fyr
= = = = = = =
if λc > 1.5
Ac As
Ac As
area of concrete, in.2 area of longitudinal reinforcing bars, in.2 area of steel shape, in.2 modulus of elasticity of steel, ksi modulus of elasticity of concrete, ksi specified minimum yield stress of steel shape, ksi specified minimum yield stress of longitudinal reinforcing bars, ksi
1999 by CRC Press LLC
c
+ c2 fc0
if λc ≤ 1.5
(3.124)
(3.125) (3.126) (3.127)
fc0 = specified compressive strength of concrete, ksi c1 , c2 , c3 = coefficients given in table below Type of composite section Concrete encased shapes Concretefilled pipes and tubings
c1
c2
c3
0.7
0.6
0.2
1.0
0.85
0.4
In addition to satisfying the condition φc Pn ≥ Pu , the bearing condition for concrete must also be satisfied. Denoting φc Pnc (= φc Pn,composite section −φc Pn,steel shape alone ) as the portion of compressive strength resisted by the concrete and AB as the loaded area (the condition), then if the supporting concrete area is larger than the loaded area, the bearing condition that needs to be satisfied is φc Pnc ≤ 0.60[1.7fc0 AB ]
3.13.2
(3.128)
Composite Beams
For steel beams fully encased in concrete, no additional anchorage for shear transfer is required if (1) at least 1.5 in. concrete cover is provided on top of the beam and at least 2 in. cover is provided over the sides and at the bottom of the beam, and (2) spalling of concrete is prevented by adequate mesh or other reinforcing steel. The design flexural strength φb Mn can be computed using either an elastic or plastic analysis. If an elastic analysis is used, φb shall be taken as 0.90. A linear strain distribution is assumed for the crosssection with zero strain at the neutral axis and maximum strains at the extreme fibers. The stresses are then computed by multiplying the strains by E (for steel) or Ec (for concrete). Maximum stress in steel shall be limited to Fy , and maximum stress in concrete shall be limited to 0.85fc0 . Tensile strength of concrete shall be neglected. Mn is to be calculated by integrating the resulting stress block about the neutral axis. If a plastic analysis is used, φc shall be taken as 0.90, and Mn shall be assumed to be equal to Mp , the plastic moment capacity of the steel section alone.
3.13.3 Composite BeamColumns Composite beamcolumns shall be designed to satisfy the interaction equation of Equation 3.68 or Equation 3.69, whichever is applicable, with φc Pn calculated based on Equations 3.124 to 3.127, Pe calculated using the equation Pe = As Fmy /λ2c , and φb Mn calculated using the following equation [14]: Aw Fy 1 h2 − (3.129) Aw Fy φb Mn = 0.90 ZFy + (h2 − 2cr )Ar Fyr + 3 2 1.7fc0 h1 where Z = plastic section modulus of the steel section, in.3 cr = average of the distance measured from the compression face to the longitudinal reinforcement in that face and the distance measured from the tension face to the longitudinal reinforcement in that face, in. h1 = width of the composite section perpendicular to the plane of bending, in. h2 = width of the composite section parallel to the plane of bending, in. Ar = crosssectional area of longitudinal reinforcing bars, in.2 Aw = web area of the encased steel shape (= 0 for concretefilled tubes) 1999 by CRC Press LLC
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If 0 < (Pu /φc Pn ) ≤ 0.3, a linear interpolation of φb Mn calculated using the above equation assuming Pu /φc Pn = 0.3 and that for beams with Pu /φc Pn = 0 (see section on Composite Beams) should be used.
3.13.4
Composite Floor Slabs
Composite floor slabs (Figure 3.38) can be designed as shored or unshored. In shored construction,
FIGURE 3.38: Composite floor slabs. 1999 by CRC Press LLC
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temporary shores are used during construction to support the dead and accidental live loads until the concrete cures. The supporting beams are designed on the basis of their ability to develop composite action to support all factored loads after the concrete cures. In unshored construction, temporary shores are not used. As a result, the steel beams alone must be designed to support the dead and accidental live loads before the concrete has attained 75% of its specified strength. After the concrete is cured, the composite section should have adequate strength to support all factored loads. Composite action for the composite floor slabs shown in Figure 3.38 is developed as a result of the presence of shear connectors. If sufficient shear connectors are provided so that the maximum flexural strength of the composite section can be developed, the section is referred to as fully composite. Otherwise, the section is referred to as partially composite. The flexural strength of a partially composite section is governed by the shear strength of the shear connectors. The horizontal shear force Vh , which should be designed for at the interface of the steel beam and the concrete slab, is given by: In regions of positive moment Vh = min(0.85fc0 Ac , As Fy , In regions of negative moment Vh = min(Ar Fyr , where fc0 Ac tc beff
X
X
Qn )
Qn )
(3.130)
(3.131)
= = = = = =
compressive strength of concrete, ksi effective area of the concrete slab = tc beff , in.2 thickness of the concrete slab, in. effective width of the concrete slab, in. min(L/4, s), for an interior beam min(L/8+ distance from beam centerline to edge of slab, s/2+ distance from beam centerline to edge of slab), for an exterior beam L = beam span measured from centertocenter of supports, in. s = spacing between centerline of adjacent beams, in. = crosssectional area of the steel beam, in.2 As Fy = yield stress of the steel beam, ksi = area of reinforcing steel within the effective area of the concrete slab, in.2 Ar Fyr = yield stress of the reinforcing steel, ksi 6Qn = sum of nominal shear strengths of the shear connectors, kips The nominal shear strength of a shear connector (used without a formed steel deck) is given by: For a stud shear connector
p Qn = 0.5Asc fc0 Ec ≤ Asc Fu
(3.132)
p Qn = 0.3(tf + 0.5tw )Lc fc0 Ec
(3.133)
For a channel shear connector
where Asc = crosssectional area of the shear stud, in.2 fc0 = compressive strength of concrete, ksi Ec = modulus of elasticity of concrete, ksi 1999 by CRC Press LLC
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Fu = minimum specified tensile strength of the shear stud, ksi tf = flange thickness of the channel, in. tw = web thickness of the channel, in. Lc = length of the channel, in. If a formed steel deck is used, Qn must be reduced by a reduction factor. The reduction factor depends on whether the deck ribs are perpendicular or parallel to the steel beam. Expressions for the reduction factor are given in the AISCLRFD Specification [18]. For full composite action, the number of connectors required between the maximum moment point and the zero moment point of the beam is given by N=
Vh Qn
(3.134)
For partial composite action, the number of connectors required is governed by the condition φb Mn ≥ Mu , where φb Mn is governed by the shear strength of the connectors. The placement and spacing of the shear connectors should comply with the following guidelines: 1. The shear connectors shall be uniformly spaced with the region of maximum moment and zero moment. However, the number of shear connectors placed between a concentrated load point and the nearest zero moment point must be sufficient to resist the factored moment Mu . 2. Except for connectors installed in the ribs of formed steel decks, shear connectors shall have at least 1 in. of lateral concrete cover. 3. Unless located over the web, diameter of shear studs must not exceed 2.5 times the thickness of the beam flange. 4. The longitudinal spacing of the studs should fall in the range 6 times the stud diameter to 8 times the slab thickness if a solid slab is used or 4 times the stud diameter to 8 times the slab thickness if a formed steel deck is used. The design flexural strength φb Mn of the composite beam with shear connectors is determined as follows: In regions of positive moments p For hc /tw ≤ 640/ Fyf , φb = 0.85, Mn = moment capacity determined using a plastic stress distribution assuming concrete crushes at a stress of 0.85fc0 and steel yields at a stress of Fy . If a portion of the concrete slab is in tension, the strength contribution of that portion of concrete is ignored. The determination of Mn using this method is very similar to the technique used for computing the moment capacity of a reinforced concrete beam according to the ultimate strength method. p For hc /tw > 640/ Fyf , φb = 0.90, Mn = moment capacity determined using superposition of elastic stress, considering the effect of shoring. The determination of Mn using this method is quite similar to the technique used for computing the moment capacity of a reinforced concrete beam according to the working stress method. In regions of negative moments φb Mn is to be determined for the steel section alone in accordance with the requirements discussed in the section on Flexural Members. To facilitate design, numerical values of φb Mn for composite beams with shear studs in solid slabs are given in tabulated form by the AISCLRFD Manual. Values of φb Mn for composite beams with formed steel decks are given in a publication by the Steel Deck Institute [19]. 1999 by CRC Press LLC
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3.14
Plastic Design
Plastic analysis and design is permitted only for steels with yield stress not exceeding 65 ksi. The reason for this is that steels with high yield stress lack the ductility required for inelastic rotation at hinge locations. Without adequate inelastic rotation, moment redistribution (which is an important characteristic for plastic design) cannot take place. In plastic design, the predominant limit state is the formation of plastic hinges. Failure occurs when sufficient plastic hinges have formed for a collapse mechanism to develop. To ensure that plastic hinges can form and can undergo large inelastic rotation, the following conditions must be satisfied: 1. Sections must be compact. That is, the widththickness ratios of flanges in compression and webs must not exceed λp in Table 3.8. 2. For columns, the slenderness parameter λc (see section on Compression Members) shall not exceed 1.5K where K is the effective length factor, and Pu from gravity and horizontal loads shall not exceed 0.75Ag Fy . 3. For beams, the lateral unbraced length Lb shall not exceed Lpd where For doubly and singly symmetric Ishaped members loaded in the plane of the web Lpd =
3,600 + 2,200(M1 /M2 ) ry Fy
(3.135)
and for solid rectangular bars and symmetric box beams Lpd =
3,000ry 5,000 + 3,000(M1 /M2 ) ry ≥ Fy Fy
(3.136)
In the above equations, M1 is the smaller end moment within the unbraced length of the beam. M2 = Mp is the plastic moment (= Zx Fy ) of the crosssection. ry is the radius of gyration about the minor axis, in inches, and Fy is the specified minimum yield stress, in ksi. Lpd is not defined for beams bent about their minor axes nor for beams with circular and square crosssections because these beams do not experience lateral torsional bucking when loaded.
3.14.1
Plastic Design of Columns and Beams
Provided that the above limitations are satisfied, the design of columns shall meet the condition 1.7Fa A ≥ Pu where Fa is the allowable compressive stress given in Equation 3.16, A is the gross crosssectional area, and Pu is the factored axial load. The design of beams shall satisfy the conditions Mp ≥ Mu and 0.55Fy tw d ≥ Vu where Mu and Vu are the factored moment and shear, respectively. Mp is the plastic moment capacity Fy is the minimum specified yield stress, tw is the beam web thickness, and d is the beam depth. For beams subjected to concentrated loads, all failure modes associated with concentrated loads (see section on Concentrated Load Criteria) should also be prevented. Except at the location where the last hinge forms, a beam bending about its major axis must be braced to resist lateral and torsional displacements at plastic hinge locations. The distance between adjacent braced points should not exceed lcr given by 1375 + 25 ry , if − 0.5 < MMp < 1.0 F y (3.137) lcr = 1375 ry , if − 1.0 < MMp ≤ −0.5 Fy
1999 by CRC Press LLC
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where ry M Mp M/Mp
3.14.2
= = = =
radius of gyration about the weak axis smaller of the two end moments of the unbraced segment plastic moment capacity is taken as positive if the unbraced segment bends in reverse curvature, and it is taken as negative if the unbraced segment bends in single curvature
Plastic Design of BeamColumns
Beamcolumns designed on the basis of plastic analysis shall satisfy the following interaction equations for stability (Equation 3.138) and for strength (Equation 3.139). Pu Pcr
+
Pu Py
Cm Mu 1− PPue Mm
+
Mu 1.18Mp
≤ 1.0
≤ 1.0
(3.138) (3.139)
where Pu = Pcr = Py = Pe = Cm = Mu = Mp = Mm = = = l = ry = Mpx = Fy =
factored axial load 1.7Fa A, Fa is defined in Equation 3.16 and A is the crosssectional area yield load = AFy Euler buckling load = π 2 EI /(Kl)2 coefficient defined in the section on Compression Members factored moment plastic moment = ZFy maximum moment that can be resisted by the member in the absence of axial load Mpx if the member p is braced in the weak direction {1.07 − [(l/ry ) Fy ]/3160}Mpx ≤ Mpx if the member is unbraced in the weak direction unbraced length of the member radius of gyration about the minor axis plastic moment about the major axis = Zx Fy minimum specified yield stress
3.15
Defining Terms
ASD: Acronym for Allowable Stress Design. Beamxcolumns: Structural members whose primary function is to carry loads both along and transverse to their longitudinal axes. Biaxial bending: Simultaneous bending of a member about two orthogonal axes of the crosssection. Builtxup members: Structural members made of structural elements jointed together by bolts, welds, or rivets. Composite members: Structural members made of both steel and concrete. Compression members: Structural members whose primary function is to carry loads along their longitudinal axes Design strength: Resistance provided by the structural member obtained by multiplying the nominal strength of the member by a resistance factor. Drift: Lateral deflection of a building. Factored load: The product of the nominal load and a load factor. 1999 by CRC Press LLC
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Flexural members: Structural members whose primary function is to carry loads transverse to their longitudinal axes. Limit state: A condition in which a structural or structural component becomes unsafe (strength limit state) or unfit for its intended function (serviceability limit state). Load factor: A factor to account for the unavoidable deviations of the actual load from its nominal value and uncertainties in structural analysis in transforming the applied load into a load effect (axial force, shear, moment, etc.) LRFD: Acronym for Load and Resistance Factor Design. PD: Acronym for Plastic Design. Plastic hinge: A yielded zone of a structural member in which the internal moment is equal to the plastic moment of the crosssection. Resistance factor: A factor to account for the unavoidable deviations of the actual resistance of a member from its nominal value. Service load: Nominal load expected to be supported by the structure or structural component under normal usage. Sidesway inhibited frames: Frames in which lateral deflections are prevented by a system of bracing. Sidesway uninhibited frames: Frames in which lateral deflections are not prevented by a system of bracing. Shear lag: The phenomenon in which the stiffer (or more rigid) regions of a structure or structural component attract more stresses than the more flexible regions of the structure or structural component. Shear lag causes stresses to be unevenly distributed over the crosssection of the structure or structural component. Tension field action: Postbuckling shear strength developed in the web of a plate girder. Tension field action can develop only if sufficient transverse stiffeners are provided to allow the girder to carry the applied load using trusstype action after the web has buckled.
References [1] AASHTO. 1992. Standard Specification for Highway Bridges. 15th ed., American Association of State Highway and Transportation Officials, Washington D.C. [2] ASTM. 1988. Specification for Carbon Steel Bolts and Studs, 60000 psi Tensile Strength (A30788a). American Society for Testing and Materials, Philadelphia, PA. [3] ASTM. 1986. Specification for High Strength Bolts for Structural Steel Joints (A32586). American Society for Testing and Materials, Philadelphia, PA. [4] ASTM. 1985. Specification for HeatTreated Steel Structural Bolts, 150 ksi Minimum Tensile Strength (A49085). American Society for Testing and Materials, Philadelphia, PA. [5] ASTM. 1986. Specification for Quenched and Tempered Steel Bolts and Studs (A44986). American Society for Testing and Materials, Philadelphia, PA. [6] AWS. 1987. Welding Handbook. 8th ed., 1, Welding Technology, American Welding Society, Miami, FL. [7] AWS. 1996. Structural Welding CodeSteel. American Welding Society, Miami, FL. [8] Blodgett, O.W. Distortion... How to Minimize it with Sound Design Practices and Controlled Welding Procedures Plus Proven Methods for Straightening Distorted Members. Bulletin G261, The Lincoln Electric Company, Cleveland, OH. [9] Chen, W.F. and Lui, E.M. 1991. Stability Design of Steel Frames, CRC Press, Boca Raton, FL. 1999 by CRC Press LLC
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[10] CSA. 1994. Limit States Design of Steel Structures. CSA Standard CAN/CSA S16.194, Canadian Standards Association, Rexdale, Ontantio. [11] Dewolf, J.T. and Ricker, D.T. 1990. Column Base Plates. Steel Design Guide Series 1, American Institute of Steel Construction, Chicago, IL. [12] Disque, R.O. 1973. Inelastic Kfactor in column design. AISC Eng. J., 10(2):3335. [13] Galambos, T.V., Ed. 1988. Guide to Stability Design Criteria for Metal Structures. 4th ed., John Wiley & Sons, New York. [14] Galambos, T.V. and Chapuis, J. 1980. LRFD Criteria for Composite Columns and Beam Columns. Washington University, Department of Civil Engineering, St. Louis, MO. [15] Gaylord, E.H., Gaylord, C.N., and Stallmeyer, J.E. 1992. Design of Steel Structures, 3rd ed., McGrawHill, New York. [16] Kulak, G.L., Fisher, J.W., and Struik, J.H.A. 1987. Guide to Design Criteria for Bolted and Riveted Joints, 2nd ed., John Wiley & Sons, New York. [17] Lee, G.C., Morrel, M.L., and Ketter, R.L. 1972. Design of Tapered Members. WRC Bulletin No.
173. [18] Load and Resistance Factor Design Specification for Structural Steel Buildings. 1993. American Institute of Steel Construction, Chicago, IL. [19] LRFD Design Manual for Composite Beams and Girders with Steel Deck. 1989. Steel Deck Institute, Canton, OH. [20] Manual of Steel ConstructionVolume II Connections. 1992. ASD 1st ed./LRFD 1st ed., American Institute of Steel Construction, Chicago, IL. [21] Manual of Steel ConstructionAllowable Stress Design. 1989. 9th ed., American Institute of Steel Construction, Chicago, IL. [22] Manual of Steel ConstructionLoad and Resistance Factor Design. 1994. Vol. I and II, 2nd ed., American Institute of Steel Construction, Chicago, IL. [23] Marsh, M.L. and Burdette, E.G. 1985. Multiple bolt anchorages: Method for determining the effective projected area of overlapping stress cones. AISC Eng. J., 22(1):2932. [24] Marsh, M.L. and Burdette, E.G. 1985. Anchorage of steel building components to concrete. AISC Eng. J., 22(1):3339. [25] Munse, W.H. and Chesson E., Jr. 1963. Riveted and Bolted Joints: Net Section Design. ASCE J. Struct. Div., 89(1):107126. [26] Rains, W.A. 1976. A new era in fire protective coatings for steel. Civil Eng., ASCE, September:8083. [27] RCSC. 1985. Allowable Stress Design Specification for Structural Joints Using ASTM A325 or A490 Bolts. American Institute of Steel Construction, Chicago, IL. [28] RCSC. 1988. Load and Resistance Factor Design Specification for Structural Joints Using ASTM A325 or A490 Bolts. American Institute of Steel Construction, Chicago, IL. [29] Shipp, J.G. and Haninge, E.R. 1983. Design of headed anchor bolts. AISC Eng. J., 20(2):5869. [30] SSRC. 1993. Is Your Structure Suitably Braced? Structural Stability Research Council, Bethlehem, PA.
Further Reading The following publications provide additional sources of information for the design of steel structures:
General Information [1] Chen, W.F. and Lui, E.M. 1987. Structural Stability—Theory and Implementation, Elsevier, New York. 1999 by CRC Press LLC
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[2] Englekirk, R. 1994. Steel Structures—Controlling Behavior Through Design, John Wiley & Sons, New York. [3] Stability of Metal Structures—A World View. 1991. 2nd ed., Lynn S. Beedle (editorinchief), Structural Stability Research Council, Lehigh University, Bethlehem, PA. [4] Trahair, N.S. 1993. FlexuralTorsional Buckling of Structures, CRC Press, Boca Raton, FL.
Allowable Stress Design [5] Adeli, H. 1988. Interactive MicrocomputerAided Structural Steel Design, PrenticeHall, Englewood Cliffs, NJ. [6] Cooper S.E. and Chen A.C. 1985. Designing Steel Structures—Methods and Cases, PrenticeHall, Englewood Cliffs, NJ. [7] Crawley S.W. and Dillon, R.M. 1984. Steel Buildings Analysis and Design, 3rd ed., John Wiley & Sons, New York. [8] Fanella, D.A., Amon, R., Knobloch, B., and Mazumder, A. 1992. Steel Design for Engineers and Architects, 2nd ed., Van Nostrand Reinhold, New York. [9] Kuzmanovic, B.O. and Willems, N. 1983. Steel Design for Structural Engineers, 2nd ed., PrenticeHall, Englewood Cliffs, NJ. [10] McCormac, J.C. 1981. Structural Steel Design, 3rd ed., Harper & Row, New York. [11] Segui, W.T. 1989. Fundamentals of Structural Steel Design, PWSKENT, Boston, MA. [12] Spiegel, L. and Limbrunner, G.F. 1986. Applied Structural Steel Design, PrenticeHall, Englewood Cliffs, NJ.
Plastic Design [13] Horne, M.R. and Morris, L.J. 1981. Plastic Design of LowRise Frames, Constrado Monographs, Collins, London, England. [14] Plastic Design in SteelA Guide and Commentary. 1971. 2nd ed., ASCE Manual No. 41, ASCEWRC, New York. [15] Chen, W.F. and Sohal, I.S. 1995. Plastic Design and SecondOrder Analysis of Steel Frames, SpringerVerlag, New York.
Load and Resistance Factor Design [16] Geschwindner, L.F., Disque, R.O., and Bjorhovde, R. 1994. Load and Resistance Factor Design of Steel Structures, PrenticeHall, Englewood Cliffs, NJ. [17] McCormac, J.C. 1995. Structural Steel Design—LRFD Method, 2nd ed., Harper & Row, New York. [18] Salmon C.G. and Johnson, J.E. 1990. Steel Structures—Design and Behavior, 3rd ed., Harper & Row, New York. [19] Segui, W.T. 1994. LRFD Steel Design, PWS, Boston, MA. [20] Smith, J.C. 1996. Structural Steel Design—LRFD Approach, 2nd ed., John Wiley & Sons, New York. [21] Chen, W.F. and Kim, S.E. 1997. LRFD Steel Design Using Advanced Analysis, CRC Press, Boca Raton, FL. [22] Chen, W.F., Goto, Y., and Liew, J.Y.R. 1996. Stability Design of SemiRigid Frames, John Wiley & Sons, New York.
1999 by CRC Press LLC
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Grider, A.; Ramirez, J.A. and Yun, Y.M. “Structural Concrete Design” Structural Engineering Handbook Ed. Chen WaiFah Boca Raton: CRC Press LLC, 1999
Structural Concrete Design
1
4.1 4.2 4.3 4.4
4.5
Properties of Concrete and Reinforcing Steel
Properties of Concrete • Lightweight Concrete • Heavyweight Concrete • HighStrength Concrete • Reinforcing Steel
Proportioning and Mixing Concrete
Proportioning Concrete Mix • Admixtures • Mixing
Flexural Design of Beams and OneWay Slabs
Reinforced Concrete Strength Design • Prestressed Concrete Strength Design
Columns under Bending and Axial Load
Short Columns under Minimum Eccentricity • Short Columns under Axial Load and Bending • Slenderness Effects • Columns under Axial Load and Biaxial Bending
Shear and Torsion Reinforced Concrete Beams and OneWay Slabs Strength Design • Prestressed Concrete Beams and OneWay Slabs Strength Design
4.6
4.7
4.8
Development of Reinforcement
Development of Bars in Tension • Development of Bars in Compression • Development of Hooks in Tension • Splices, Bundled Bars, and Web Reinforcement
TwoWay Systems
Definition • Design Procedures • Minimum Slab Thickness and Reinforcement • Direct Design Method • Equivalent Frame Method • Detailing
Frames
Analysis of Frames • Design for Seismic Loading
4.9 Brackets and Corbels 4.10 Footings
Amy Grider and Julio A. Ramirez School of Civil Engineering, Purdue University, West Lafayette, IN
Young Mook Yun Department of Civil Engineering, National University, Taegu, South Korea
Types of Footings • Design Considerations • Wall Footings • SingleColumn Spread Footings • Combined Footings • TwoColumn Footings • Strip, Grid, and Mat Foundations • Footings on Piles
4.11 Walls
Panel, Curtain, and Bearing Walls • Basement Walls • Partition Walls • Shears Walls
4.12 Defining Terms References Further Reading
1 The material in this chapter was previously published by CRC Press in The Civil Engineering Handbook, W.F. Chen, Ed.,
1995. 1999 by CRC Press LLC
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At this point in the history of development of reinforced and prestressed concrete it is necessary to reexamine the fundamental approaches to design of these composite materials. Structural engineering is a worldwide industry. Designers from one nation or a continent are faced with designing a project in another nation or continent. The decades of efforts dedicated to harmonizing concrete design approaches worldwide have resulted in some successes but in large part have led to further differences and numerous different design procedures. It is this abundance of different design approaches, techniques, and code regulations that justifies and calls for the need for a unification of design approaches throughout the entire range of structural concrete, from plain to fully prestressed [5]. The effort must begin at all levels: university courses, textbooks, handbooks, and standards of practice. Students and practitioners must be encouraged to think of a single continuum of structural concrete. Based on this premise, this chapter on concrete design is organized to promote such unification. In addition, effort will be directed at dispelling the present unjustified preoccupation with complex analysis procedures and often highly empirical and incomplete sectional mechanics approaches that tend to both distract the designers from fundamental behavior and impart a false sense of accuracy to beginning designers. Instead, designers will be directed to give careful consideration to overall structure behavior, remarking the adequate flow of forces throughout the entire structure.
4.1
Properties of Concrete and Reinforcing Steel
The designer needs to be knowledgeable about the properties of concrete, reinforcing steel, and prestressing steel. This part of the chapter summarizes the material properties of particular importance to the designer.
4.1.1
Properties of Concrete
Workability is the ease with which the ingredients can be mixed and the resulting mix handled, transported, and placed with little loss in homogeneity. Unfortunately, workability cannot be measured directly. Engineers therefore try to measure the consistency of the concrete by performing a slump test. The slump test is useful in detecting variations in the uniformity of a mix. In the slump test, a mold shaped as the frustum of a cone, 12 in. (305 mm) high with an 8 in. (203 mm) diameter base and 4 in. (102 mm) diameter top, is filled with concrete (ASTM Specification C143). Immediately after filling, the mold is removed and the change in height of the specimen is measured. The change in height of the specimen is taken as the slump when the test is done according to the ASTM Specification. A wellproportioned workable mix settles slowly, retaining its original shape. A poor mix crumbles, segregates, and falls apart. The slump may be increased by adding water, increasing the percentage of fines (cement or aggregate), entraining air, or by using an admixture that reduces water requirements; however, these changes may adversely affect other properties of the concrete. In general, the slump specified should yield the desired consistency with the least amount of water and cement. Concrete should withstand the weathering, chemical action, and wear to which it will be subjected in service over a period of years; thus, durability is an important property of concrete. Concrete resistance to freezing and thawing damage can be improved by increasing the watertightness, entraining 2 to 6% air, using an airentraining agent, or applying a protective coating to the surface. Chemical agents damage or disintegrate concrete; therefore, concrete should be protected with a resistant coating. Resistance to wear can be obtained by use of a highstrength, dense concrete made with hard aggregates. 1999 by CRC Press LLC
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Excess water leaves voids and cavities after evaporation, and water can penetrate or pass through the concrete if the voids are interconnected. Watertightness can be improved by entraining air or reducing water in the mix, or it can be prolonged through curing. Volume change of concrete should be considered, since expansion of the concrete may cause buckling and drying shrinkage may cause cracking. Expansion due to alkaliaggregate reaction can be avoided by using nonreactive aggregates. If reactive aggregates must be used, expansion may be reduced by adding pozzolanic material (e.g., fly ash) to the mix. Expansion caused by heat of hydration of the cement can be reduced by keeping cement content as low as possible; using Type IV cement; and chilling the aggregates, water, and concrete in the forms. Expansion from temperature increases can be reduced by using coarse aggregate with a lower coefficient of thermal expansion. Drying shrinkage can be reduced by using less water in the mix, using less cement, or allowing adequate moist curing. The addition of pozzolans, unless allowing a reduction in water, will increase drying shrinkage. Whether volume change causes damage usually depends on the restraint present; consideration should be given to eliminating restraints or resisting the stresses they may cause [8]. Strength of concrete is usually considered its most important property. The compressive strength at 28 d is often used as a measure of strength because the strength of concrete usually increases with time. The compressive strength of concrete is determined by testing specimens in the form of standard cylinders as specified in ASTM Specification C192 for research testing or C31 for field testing. The test procedure is given in ASTM C39. If drilled cores are used, ASTM C42 should be followed. The suitability of a mix is often desired before the results of the 28d test are available. A formula proposed by W. A. Slater estimates the 28d compressive strength of concrete from its 7d strength: p (4.1) S28 = S7 + 30 S7 where S28 = 28d compressive strength, psi S7 = 7d compressive strength, psi Strength can be increased by decreasing watercement ratio, using higher strength aggregate, using a pozzolan such as fly ash, grading the aggregates to produce a smaller percentage of voids in the concrete, moist curing the concrete after it has set, and vibrating the concrete in the forms. The shorttime strength can be increased by using Type III portland cement, accelerating admixtures, and by increasing the curing temperature. The stressstrain curve for concrete is a curved line. Maximum stress is reached at a strain of 0.002 in./in., after which the curve descends. The modulus of elasticity, Ec , as given in ACI 31889 (Revised 92), Building Code Requirements for Reinforced Concrete [1], is: Ec
=
Ec
=
p wc1.5 33 fc0 lb/ft3 and psi p wc1.5 0.043 fc0 kg/m3 and MPa
(4.2a) (4.2b)
where wc = unit weight of concrete fc0 = compressive strength at 28 d p Tensile strength of concrete is much lower than the compressive strength—about 7 fc0 for the p higherstrength concretes and 10 fc0 for the lowerstrength concretes. Creep is the increase in strain with time under a constant load. Creep increases with increasing watercement ratio and decreases with an increase in relative humidity. Creep is accounted for in design by using a reduced modulus of elasticity of the concrete. 1999 by CRC Press LLC
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4.1.2 Lightweight Concrete Structural lightweight concrete is usually made from aggregates conforming to ASTM C330 that are usually produced in a kiln, such as expanded clays and shales. Structural lightweight concrete has a density between 90 and 120 lb/ft3 (1440 to 1920 kg/m3 ). Production of lightweight concrete is more difficult than normalweight concrete because the aggregates vary in absorption of water, specific gravity, moisture content, and amount of grading of undersize. Slump and unit weight tests should be performed often to ensure uniformity of the mix. During placing and finishing of the concrete, the aggregates may float to the surface. Workability can be improved by increasing the percentage of fines or by using an airentraining admixture to incorporate 4 to 6% air. Dry aggregate should not be put into the mix because it will continue to absorb moisture and cause the concrete to harden before placement is completed. Continuous water curing is important with lightweight concrete. Nofines concrete is obtained by using pea gravel as the coarse aggregate and 20 to 30% entrained air instead of sand. It is used for low dead weight and insulation when strength is not important. This concrete weighs from 105 to 118 lb/ft3 (1680 to 1890 kg/m3 ) and has a compressive strength from 200 to 1000 psi (1 to 7 MPa). A porous concrete made by gap grading or singlesize aggregate grading is used for low conductivity or where drainage is needed. Lightweight concrete can also be made with gasforming of foaming agents which are used as admixtures. Foam concretes range in weight from 20 to 110 lb/ft3 (320 to 1760 kg/m3 ). The modulus of elasticity of lightweight concrete can be computed using the same formula as normal concrete. The shrinkage of lightweight concrete is similar to or slightly greater than for normal concrete.
4.1.3
Heavyweight Concrete
Heavyweight concretes are used primarily for shielding purposes against gamma and xradiation in nuclear reactors and other structures. Barite, limonite and magnetite, steel punchings, and steel shot are typically used as aggregates. Heavyweight concretes weigh from 200 to 350 lb/ft3 (3200 to 5600 kg/m3 ) with strengths from 3200 to 6000 psi (22 to 41 MPa). Gradings and mix proportions are similar to those for normal weight concrete. Heavyweight concretes usually do not have good resistance to weathering or abrasion.
4.1.4
HighStrength Concrete
Concretes with strengths in excess of 6000 psi (41 MPa) are referred to as highstrength concretes. Strengths up to 18,000 psi (124 MPa) have been used in buildings. Admixtures such as superplasticizers, silica fume, and supplementary cementing materials such as fly ash improve the dispersion of cement in the mix and produce workable concretes with lower watercement ratios, lower void ratios, and higher strength. Coarse aggregates should be strong finegrained gravel with rough surfaces. For concrete strengths in excess of 6000 psi (41 MPa), the modulus of elasticity should be taken as p Ec = 40,000 fc0 + 1.0 × 106 where fc0 = compressive strength at 28 d, psi [4] The shrinkage of highstrength concrete is about the same as that for normal concrete. 1999 by CRC Press LLC
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(4.3)
4.1.5
Reinforcing Steel
Concrete can be reinforced with welded wire fabric, deformed reinforcing bars, and prestressing tendons. Welded wire fabric is used in thin slabs, thin shells, and other locations where space does not allow the placement of deformed bars. Welded wire fabric consists of cold drawn wire in orthogonal patterns—square or rectangular and resistancewelded at all intersections. The wire may be smooth (ASTM A185 and A82) or deformed (ASTM A497 and A496). The wire is specified by the symbol W for smooth wires or D for deformed wires followed by a number representing the crosssectional area in hundredths of a square inch. On design drawings it is indicated by the symbol WWF followed by spacings of the wires in the two 90◦ directions. Properties for welded wire fabric are given in Table 4.1. TABLE 4.1
Wire and Welded Wire Fabric Steels Wire size AST designation
Minimum yield stress,a fy
Minimum tensile strength
designation
ksi
MPa
ksi
MPa
A8279 (colddrawn wire) (properties apply when material is to be used for fabric)
W1.2 and largerb Smaller than W1.2
65 56
450 385
75 70
520 480
A18579 (welded wire fabric)
Same as A82; this is A82 material fabricated into sheet (socalled “mesh”) by the process of electric welding
A49678 (deformed steel wire) (properties apply when material is to be used for fabric)
D1D31c
A49779
Same as A82 or A496; this specification applies for fabric made from A496, or from a combination of A496 and A82 wires
70
480
80
550
a The term “yield stress” refers to either yield point, the welldefined deviation from perfect elasticity, or yield strength,
the value obtained by a specified offset strain for material having no welldefined yield point.
b The W number represents the nominal crosssectional area in square inches multiplied by 100, for smooth wires. c The D number represents the nominal crosssectional area in square inches multiplied by 100, for deformed wires.
The deformations on a deformed reinforcing bar inhibit longitudinal movement of the bar relative to the concrete around it. Table 4.2 gives dimensions and weights of these bars. Reinforcing bar steel can be made of billet steel of grades 40 and 60 having minimum specific yield stresses of 40,000 and 60,000 psi, respectively (276 and 414 MPa) (ASTM A615) or lowalloy steel of grade 60, which is intended for applications where welding and/or bending is important (ASTM A706). Presently, grade 60 billet steel is the most predominantly used for construction. Prestressing tendons are commonly in the form of individual wires or groups of wires. Wires of different strengths and properties are available with the most prevalent being the 7wire lowrelaxation strand conforming to ASTM A416. ASTM A416 also covers a stressrelieved strand, which is seldom used in construction nowadays. Properties of standard prestressing strands are given in Table 4.3. Prestressing tendons could also be bars; however, this is not very common. Prestressing bars meeting ASTM A722 have been used in connections between members. The modulus of elasticity for nonprestressed steel is 29,000,000 psi (200,000 MPa). For prestressing steel, it is lower and also variable, so it should be obtained from the manufacturer. For 7wires strands conforming to ASTM A416, the modulus of elasticity is usually taken as 27,000,000 psi (186,000 MPa). 1999 by CRC Press LLC
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TABLE 4.2
Reinforcing Bar Dimensions and Weights Nominal dimensions
Bar number 3 4 5 6 7 8 9 10 11 14 18
Diameter (in.) 0.375 0.500 0.625 0.750 0.875 1.000 1.128 1.270 1.410 1.693 2.257
TABLE 4.3
(mm) 9.5 12.7 15.9 19.1 22.2 25.4 28.7 32.3 35.8 43.0 57.3
(cm2 )
0.11 0.20 0.31 0.44 0.60 0.79 1.00 1.27 1.56 2.25 4.00
0.71 1.29 2.00 2.84 3.87 5.10 6.45 8.19 10.06 14.52 25.81
(lb/ft) 0.376 0.668 1.043 1.502 2.044 2.670 3.400 4.303 5.313 7.65 13.60
(kg/m) 0.559 0.994 1.552 2.235 3.041 3.973 5.059 6.403 7.906 11.38 20.24
Standard Prestressing Strands, Wires, and Bars Nominal dimension
fpu ksi
Diameter in.
Area in.2
Weight plf
Sevenwire strand
250 270 250 270 250 270 250
1/4 3/8 3/8 1/2 1/2 0.6 0.6
0.036 0.085 0.080 0.153 0.144 0.215 0.216
0.12 0.29 0.27 0.53 0.49 0.74 0.74
Prestressing wire
250 240 235
0.196 0.250 0.276
0.0302 0.0491 0.0598
0.10 0.17 0.20
Deformed prestressing bars
157 150 150 150
5/8 1 1 1/4 1 3/8
0.28 0.85 1.25 1.58
0.98 3.01 4.39 5.56
Tendon type
4.2.1
. Weight
(in.2 )
Grade
4.2
. Area
Proportioning and Mixing Concrete Proportioning Concrete Mix
A concrete mix is specified by the weight of water, sand, coarse aggregate, and admixture to be used per 94pound bag of cement. The type of cement (Table 4.4), modulus of the aggregates, and maximum size of the aggregates (Table 4.5) should also be given. A mix can be specified by the weight ratio of cement to sand to coarse aggregate with the minimum amount of cement per cubic yard of concrete. In proportioning a concrete mix, it is advisable to make and test trial batches because of the many variables involved. Several trial batches should be made with a constant watercement ratio but varying ratios of aggregates to obtain the desired workability with the least cement. To obtain results similar to those in the field, the trial batches should be mixed by machine. When time or other conditions do not allow proportioning by the trial batch method, Table 4.6 may be used. Start with mix B corresponding to the appropriate maximum size of aggregate. Add just enough water for the desired workability. If the mix is undersanded, change to mix A; if oversanded, change to mix C. Weights are given for dry sand. For damp sand, increase the weight of sand 10 lb, and for very wet sand, 20 lb, per bag of cement. 1999 by CRC Press LLC
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TABLE 4.4
Types of Portland Cementa
Type I
Usage Ordinary construction where special properties are not required Ordinary construction when moderate sulfate resistance or moderate heat of hydration is desired When high early strength is desired When low heat of hydration is desired When high sulfate resistance is desired
II III IV V
a According to ASTM C150.
TABLE 4.5
Recommended Maximum Sizes of Aggregatea Maximum size, in., of aggregate for:
Minimum dimension of section, in. 5 or less 6–11 12–29 30 or more
Reinforcedconcrete beams, columns, walls
Heavily reinforced slabs
Lightly reinforced or unreinforced slabs
··· 3/4 – 1 1/2 1 1/2 – 3 1 1/2 – 3
3/4 – 1 1/2 1 1/2 3 3
3/4 – 1 1/2 1 1/2 – 3 3–6 6
a Concrete Manual. U.S. Bureau of Reclamation.
TABLE 4.6
Typical Concrete Mixesa Aggregate, lb per bag of cement
Maximum size of
Bags of cement
Sand
aggregate, in.
Mix designation
per yd3 of concrete
Airentrained concrete
Concrete without air
Gravel or crushed stone
1/2
A B C
7.0 6.9 6.8
235 225 225
245 235 235
170 190 205
3/4
A B C
6.6 6.4 6.3
225 225 215
235 235 225
225 245 265
1
A B C
6.4 6.2 6.1
225 215 205
235 225 215
245 275 290
1 1/2
A B C
6.0 5.8 5.7
225 215 205
235 225 215
290 320 345
2
A B C
5.7 5.6 5.4
225 215 205
235 225 215
330 360 380
a Concrete Manual. U.S. Bureau of Reclamation.
4.2.2
Admixtures
Admixtures may be used to modify the properties of concrete. Some types of admixtures are set accelerators, water reducers, airentraining agents, and waterproofers. Admixtures are generally helpful in improving quality of the concrete. However, if admixtures are not properly used, they could have undesirable effects; it is therefore necessary to know the advantages and limitations of the proposed admixture. The ASTM Specifications cover many of the admixtures. Set accelerators are used (1) when it takes too long for concrete to set naturally; such as in cold weather, or (2) to accelerate the rate of strength development. Calcium chloride is widely used as a set accelerator. If not used in the right quantities, it could have harmful effects on the concrete and reinforcement. Water reducers lubricate the mix and permit easier placement of the concrete. Since the workability of a mix can be improved by a chemical agent, less water is needed. With less water but the same 1999 by CRC Press LLC
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cement content, the strength is increased. Since less water is needed, the cement content could also be decreased, which results in less shrinkage of the hardened concrete. Some water reducers also slow down the concrete set, which is useful in hot weather and integrating consecutive pours of the concrete. Airentraining agents are probably the most widely used type of admixture. Minute bubbles of air are entrained in the concrete, which increases the resistance of the concrete to freezethaw cycles and the use of iceremoval salts. Waterproofing chemicals are often applied as surface treatments, but they can be added to the concrete mix. If applied properly and uniformly, they can prevent water from penetrating the concrete surface. Epoxies can also be used for waterproofing. They are more durable than silicone coatings, but they may be more costly. Epoxies can also be used for protection of wearing surfaces, patching cavities and cracks, and glue for connecting pieces of hardened concrete.
4.2.3
Mixing
Materials used in making concrete are stored in batch plants that have weighing and control equipment and bins for storing the cement and aggregates. Proportions are controlled by automatic or manually operated scales. The water is measured out either from measuring tanks or by using water meters. Machine mixing is used whenever possible to achieve uniform consistency. The revolving drumtype mixer and the countercurrent mixer, which has mixing blades rotating in the opposite direction of the drum, are commonly used. Mixing time, which is measured from the time all ingredients are in the drum, “should be at least 1.5 minutes for a 1yd3 mixer, plus 0.5 min for each cubic yard of capacity over 1 yd3 ” [ACI 30473, 1973]. It also is recommended to set a maximum on mixing time since overmixing may remove entrained air and increase fines, thus requiring more water for workability; three times the minimum mixing time can be used as a guide. Readymixed concrete is made in plants and delivered to job sites in mixers mounted on trucks. The concrete can be mixed en route or upon arrival at the site. Concrete can be kept plastic and workable for as long as 1.5 hours by slow revolving of the mixer. Mixing time can be better controlled if water is added and mixing started upon arrival at the job site, where the operation can be inspected.
4.3 4.3.1
Flexural Design of Beams and OneWay Slabs Reinforced Concrete Strength Design
The basic assumptions made in flexural design are: 1. Sections perpendicular to the axis of bending that are plane before bending remain plane after bending. 2. A perfect bond exists between the reinforcement and the concrete such that the strain in the reinforcement is equal to the strain in the concrete at the same level. 3. The strains in both the concrete and reinforcement are assumed to be directly proportional to the distance from the neutral axis (ACI 10.2.2) [1]. 4. Concrete is assumed to fail when the compressive strain reaches 0.003 (ACI 10.2.3). 5. The tensile strength of concrete is neglected (ACI 10.2.5). 6. The stresses in the concrete and reinforcement can be computed from the strains using stressstrain curves for concrete and steel, respectively. 1999 by CRC Press LLC
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7. The compressive stressstrain relationship for concrete may be assumed to be rectangular, trapezoidal, parabolic, or any other shape that results in prediction of strength in substantial agreement with the results of comprehensive tests (ACI 10.2.6). ACI 10.2.7 outlines the use of a rectangular compressive stress distribution which is known as the Whitney rectangular stress block. For other stress distributions see Reinforced Concrete Mechanics and Design by James G. MacGregor [8]. Analysis of Rectangular Beams with Tension Reinforcement Only Equations for Mn and φMn : Tension Steel Yielding Consider the beam shown in Figure 4.1. The compressive force, C, in the concrete is (4.4) C = 0.85fc0 ba
The tension force, T , in the steel is
T = As fy
(4.5)
For equilibrium, C = T , so the depth of the equivalent rectangular stress block, a, is a=
As fy 0.85fc0 b
(4.6)
Noting that the internal forces C and T form an equivalent forcecouple system, the internal moment is (4.7) Mn = T (d − a/2) or Mn = C(d − a/2) φMn is then
φMn = φT (d − a/2)
or φMn = φC(d − a/2) where φ =0.90.
FIGURE 4.1: Stresses and forces in a rectangular beam.
1999 by CRC Press LLC
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(4.8)
Equation for Mn and φMn : Tension Steel Elastic
The internal forces and equilibrium are
given by: C
= T = As fs = ρbdEs εs
0.85fc0 ba 0.85fc0 ba From strain compatibility (see Figure 4.1),
εs = εcu
d −c c
(4.9)
(4.10)
Substituting εs into the equilibrium equation, noting that a = β1 c, and simplifying gives 0.85fc0 a 2 + (d)a − β1 d 2 = 0 ρEs εcu
(4.11)
which can be solved for a. Equations 4.7 and 4.8 can then be used to obtain Mn and φMn . Reinforcement Ratios The reinforcement ratio, ρ, is used to represent the relative amount of tension reinforcement in a beam and is given by ρ=
As bd
(4.12)
At the balanced strain condition the maximum strain, εcu , at the extreme concrete compression fiber reaches 0.003 just as the tension steel reaches the strain εy = fy /Es . The reinforcement ratio in the balanced strain condition, ρb , can be obtained by applying equilibrium and compatibility conditions. From the linear strain condition, Figure 4.1, εcu cb = = d εcu + εy
0.003 fy 0.003 + 29,000,000
87,000 87,000 + fy
=
(4.13)
The compressive and tensile forces are: Cb Tb
0.85fc0 bβ1 cb fy Asb = ρb bdfy
= =
(4.14)
Equating Cb to Tb and solving for ρb gives 0.85fc0 β1 cb fy d
ρb = which on substitution of Equation 4.13 gives
0.85fc0 β1 ρb = fy
87,000 87,000 + fy
(4.15)
(4.16)
ACI 10.3.3 limits the amount of reinforcement in order to prevent nonductile behavior: maxρ = 0.75ρb
(4.17)
ACI 10.5 requires a minimum amount of flexural reinforcement: ρmin =
1999 by CRC Press LLC
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200 fy
(4.18)
Analysis of Beams with Tension and Compression Reinforcement
For the analysis of doubly reinforced beams, the crosssection will be divided into two beams. Beam 1 consists of the compression reinforcement at the top and sufficient steel at the bottom so that T1 = Cs ; beam 2 consists of the concrete web and the remaining tensile reinforcement, as shown in Figure 4.2
FIGURE 4.2: Strains, stresses, and forces in beam with compression reinforcement.
Equation for Mn : Compression Steel Yields The area of tension steel in beam 1 is obtained by setting T1 = Cs , which gives As1 = A0s . The nominal moment capacity of beam 1 is then (4.19) Mn1 = A0s fy d − d 0
Beam 2 consists of the concrete and the remaining steel, As2 = As −As1 = As −A0s . The compression force in the concrete is (4.20) C = 0.85fc0 ba and the tension force in the steel for beam 2 is
T = As − A0s fy
(4.21)
The depth of the compression stress block is then
As − A0s fy a= 0.85fc0 b
Therefore, the nominal moment capacity for beam 2 is Mn2 = As − A0s fy (d − a/2)
(4.22)
(4.23)
The total moment capacity for a doubly reinforced beam with compression steel yielding is the summation of the moment capacity for beam 1 and beam 2; therefore, (4.24) Mn = A0s fy d − d 0 + As − A0s fy (d − a/2) 1999 by CRC Press LLC
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Equation for Mn : Compression Steel Does Not Yield the internal forces in the beam are
T Cc Cs where
Assuming that the tension steel yields,
= As fy = 0.85fc0 ba = A0s Es εs0
(4.25)
β1 d 0 (0.003) εs0 = 1 − a
(4.26)
β1 d 0 (0.003) = As fy 0.85fc0 ba + A0s Es 1 − a
(4.27)
From equilibrium, Cs + Cc = T or
This can be rewritten in quadratic form as 0.85fc0 b a 2 + 0.003A0s Es − As Fy a − 0.003A0s Es β1 d 0 = 0
(4.28)
where a can be calculated by means of the quadratic equation. Therefore, the nominal moment capacity in a doubly reinforced concrete beam where the compression steel does not yield is a + Cs d − d 0 (4.29) Mn = Cc d − 2 Reinforcement Ratios The reinforcement ratio at the balanced strain condition can be obtained in a similar manner as that for beams with tension steel only. For compression steel yielding, the balanced ratio is 0.85fc0 β1 87,000 0 (4.30) ρ−ρ b = fy 87,000 + fy
For compression steel not yielding, the balanced ratio is 0.85fc0 β1 87,000 ρ 0 fs0 = ρ− fy b fy 87,000 + fy
(4.31)
The maximum and minimum reinforcement ratios as given in ACI 10.3.3 and 10.5 are ρmax
=
ρmin
=
0.75ρb 200 fy
(4.32)
4.3.2 Prestressed Concrete Strength Design Elastic Flexural Analysis
In developing elastic equations for prestress, the effects of prestress force, dead load moment, and live load moment are calculated separately, and then the separate stresses are superimposed, giving f =− 1999 by CRC Press LLC
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F ey My F ± ± A I I
(4.33)
where (−) indicates compression and (+) indicates tension. It is necessary to check that the stresses in the extreme fibers remain within the ACIspecified limits under any combination of loadings that many occur. The stress limits for the concrete and prestressing tendons are specified in ACI 18.4 and 18.5 [1]. ACI 18.2.6 states that the loss of area due to open ducts shall be considered when computing section properties. It is noted in the commentary that section properties may be based on total area if the effect of the open duct area is considered negligible. In pretensioned members and in posttensioned members after grouting, section properties can be based on gross sections, net sections, or effective sections using the transformed areas of bonded tendons and nonprestressed reinforcement. Flexural Strength
The strength of a prestressed beam can be calculated using the methods developed for ordinary reinforced concrete beams, with modifications to account for the differing nature of the stressstrain relationship of prestressing steel compared with ordinary reinforcing steel. A prestressed beam will fail when the steel reaches a stress fps , generally less than the tensile strength fpu . For rectangular crosssections the nominal flexural strength is Mn = Aps fps d − where a=
a 2
Aps fps 0.85fc0 b
(4.34)
(4.35)
The steel stress fps can be found based on strain compatibility or by using approximate equations such as those given in ACI 18.7.2. The equations in ACI are applicable only if the effective prestress in the steel, fse , which equals Pe /Aps , is not less than 0.5 fpu . The ACI equations are as follows. (a) For members with bonded tendons: γp fpu d 0 (4.36) ω−ω ρ 0 + fps = fpu 1 − β1 fc dp If any compression reinforcement is taken into account when calculating fps with Equation 4.36, the following applies: fpu d 0 (4.37) ω−ω ≥ 0.17 ρp 0 + fc dp and
d 0 ≤ 0.15dp
(b) For members with unbonded tendons and with a spantodepth ratio of 35 or less: fc0 fpy fps = fse + 10,000 + ≤ fse + 60,000 100ρp
(4.38)
(c) For members with unbonded tendons and with a spantodepth ratio greater than 35: fc0 fpy ≤ (4.39) fps = fse + 10,000 + fse + 30,000 300ρp The flexural strength is then calculated from Equation 4.34. The design strength is equal to φMn , where φ = 0.90 for flexure. 1999 by CRC Press LLC
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Reinforcement Ratios
ACI requires that the total amount of prestressed and nonprestressed reinforcement be adequate to develop a factored load at least 1.2 times the cracking load calculated on the basis of a modulus of p rupture of 7.5 fc0 . To control cracking in members with unbonded tendons, some bonded reinforcement should be uniformly distributed over the tension zone near the extreme tension fiber. ACI specifies the minimum amount of bonded reinforcement as As = 0.004A
(4.40)
where A is the area of the crosssection between the flexural tension face and the center of gravity of the gross crosssection. ACI 19.9.4 gives the minimum length of the bonded reinforcement. To ensure adequate ductility, ACI 18.8.1 provides the following requirement: ω p d 0 ωp + ω−ω dp ≤ 0.36β1 (4.41) d 0 ωw − ωw ωpw + dp ACI allows each of the terms on the left side to be set equal to 0.85 a/dp in order to simplify the equation. When a reinforcement ratio greater than 0.36 β1 is used, ACI 18.8.2 states that the design moment strength shall not be greater than the moment strength based on the compression portion of the moment couple.
4.4
Columns under Bending and Axial Load
4.4.1
Short Columns under Minimum Eccentricity
When a symmetrical column is subjected to a concentric axial load, P , longitudinal strains develop uniformly across the section. Because the steel and concrete are bonded together, the strains in the concrete and steel are equal. For any given strain it is possible to compute the stresses in the concrete and steel using the stressstrain curves for the two materials. The forces in the concrete and steel are equal to the stresses multiplied by the corresponding areas. The total load on the column is the sum of the forces in the concrete and steel: (4.42) Po = 0.85fc0 Ag − Ast + fy Ast To account for the effect of incidental moments, ACI 10.3.5 specifies that the maximum design axial load on a column be, for spiral columns, (4.43) φPn(max) = 0.85φ .85fc0 Ag − Ast + fy Ast and for tied columns, φPn(max) = 0.80φ .85fc0 Ag − Ast + fy Ast
(4.44)
For high values of axial load, φ values of 0.7 and 0.75 are specified for tied and spiral columns, respectively (ACI 9.3.2.2b) [1]. Short columns are sufficiently stocky such that slenderness effects can be ignored. 1999 by CRC Press LLC
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4.4.2
Short Columns under Axial Load and Bending
Almost all compression members in concrete structures are subjected to moments in addition to axial loads. Although it is possible to derive equations to evaluate the strength of columns subjected to combined bending and axial loads, the equations are tedious to use. For this reason, interaction diagrams for columns are generally computed by assuming a series of strain distributions, each corresponding to a particular point on the interaction diagram, and computing the corresponding values of P and M. Once enough such points have been computed, the results are summarized in an interaction diagram. For examples on determining the interaction diagram, see Reinforced Concrete Mechanics and Design by James G. MacGregor [8] or Reinforced Concrete Design by ChuKia Wang and Charles G. Salmon [11]. Figure 4.3 illustrates a series of strain distributions and the resulting points on the interaction diagram. Point A represents pure axial compression. Point B corresponds to crushing at one face and zero tension at the other. If the tensile strength of concrete is ignored, this represents the onset of cracking on the bottom face of the section. All points lower than this in the interaction diagram represent cases in which the section is partially cracked. Point C, the farthest right point, corresponds to the balanced strain condition and represents the change from compression failures for higher loads and tension failures for lower loads. Point D represents a strain distribution where the reinforcement has been strained to several times the yield strain before the concrete reaches its crushing strain. The horizontal axis of the interaction diagram corresponds to pure bending where φ = 0.9. A transition is required from φ = 0.7 or 0.75 for high axial loads to φ = 0.9 for pure bending. The change in φ begins at a capacity φPa , which equals the smaller of the balanced load, φPb , or 0.1 fc0 Ag . Generally, φPb exceeds 0.1 fc0 Ag except for a few nonrectangular columns. ACI publication SP17A(85), A Design Handbook for Columns, contains nondimensional interaction diagrams as well as other design aids for columns [2].
4.4.3
Slenderness Effects
ACI 10.11 describes an approximate slendernesseffect design procedure based on the moment magnifier concept. The moments are computed by ordinary frame analysis and multiplied by a moment magnifier that is a function of the factored axial load and the critical buckling load of the column. The following gives a summary of the moment magnifier design procedure for slender columns in frames. 1. Length of Column. The unsupported length, lu , is defined in ACI 10.11.1 as the clear distance between floor slabs, beams, or other members capable of giving lateral support to the column. 2. Effective length. The effective length factors, k, used in calculating δb shall be between 0.5 and 1.0 (ACI 10.11.2.1). The effective length factors used to compute δs shall be greater than 1 (ACI 10.11.2.2). The effective length factors can be estimated using ACI Fig. R10.11.2 or using ACI Equations (A)–(E) given in ACI R10.11.2. These two procedures require that the ratio, ψ, of the columns and beams be known: P (Ec Ic / lc ) ψ=P (Eb Ib / lb )
(4.45)
In computing ψ it is acceptable to take the EI of the column as the uncracked gross Ec Ig of the columns and the EI of the beam as 0.5 Ec Ig . 3. Definition of braced and unbraced frames. The ACI Commentary suggests that a frame is braced if either of the following are satisfied: 1999 by CRC Press LLC
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FIGURE 4.3: Strain distributions corresponding to points on interaction diagram. (a) If the stability index, Q, for a story is less than 0.04, where P Q=
Pu 1u ≤ 0.04 Hu hs
(4.46)
(b) If the sum of the lateral stiffness of the bracing elements in a story exceeds six times the lateral stiffness of all the columns in the story. 4. Radius of gyration. For a rectangular crosssection √ r equals 0.3 h, and for a circular crosssection r equals 0.25 h. For other sections, r equals I /A. 5. Consideration of slenderness effects. ACI 10.11.4.1 allows slenderness effects to be neglected for columns in braced frames when M1b klu < 34 − 12 r M2b
(4.47)
ACI 10.11.4.2 allows slenderness effects to be neglected for columns in unbraced frames when klu < 22 r
(4.48)
If klu /r exceeds 100, ACI 10.11.4.3 states that design shall be based on secondorder analysis. 1999 by CRC Press LLC
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6. Minimum moments. For columns in a braced frame, M2b shall be not less than the value given in ACI 10.11.5.4. In an unbraced frame ACI 10.11.5.5 applies for M2s . 7. Moment magnifier equation. ACI 10.11.5.1 states that columns shall be designed for the factored axial load, Pu , and a magnified factored moment, Mc , defined by Mc = δb M2b + δs M2s
(4.49)
where M2b is the larger factored end moment acting on the column due to loads causing no appreciable sidesway (lateral deflections less than l/1500) and M2s is the larger factored end moment due to loads that result in an appreciable sidesway. The moments are computed from a conventional firstorder elastic frame analysis. For the above equation, the following apply:
δb
=
δs
=
Cm ≥ 1.0 1 − Pu /φPc 1 P P ≥ 1.0 1 − Pu /φ Pc
(4.50)
For members braced against sidesway, ACI 10.11.5.1 gives δs = 1.0. Cm = 0.6 + 0.4
M1b ≥ 0.4 M2b
(4.51)
The ratio M1b /M2b is taken as positive if the member is bent in single curvature and negative if the member is bent in double curvature. Equation 4.51 applies only to columns in braced frames. In all other cases, ACI 10.11.5.3 states that Cm = 1.0.
Pc =
π 2 EI (klu )2
(4.52)
where EI =
Ec Ig /5 + Es Ise 1 + βd
(4.53)
Ec Ig /2.5 1 + βd
(4.54)
or, approximately
EI =
1999 by CRC Press LLC
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When computing δb , Axial load due to factored dead load Total factored axial load
(4.55)
Factored sustained lateral shear in the story Total factored lateral shear in the story
(4.56)
βd = When computing δs , βd =
If δb or δs is found to be negative, the column should be enlarged. If either δb or δs exceeds 2.0, consideration should be given to enlarging the column.
4.4.4
Columns under Axial Load and Biaxial Bending
The nominal ultimate strength of a section under biaxial bending and compression is a function of three variables, Pn , Mnx , and Mny , which may also be expressed as Pn acting at eccentricities ey = Mnx /Pn and ex = Mny /Pn with respect to the x and y axes. Three types of failure surfaces can be defined. In the first type, S1 , the three orthogonal axes are defined by Pn , ex , and ey ; in the second type, S2 , the variables defining the axes are 1/Pn , ex , and ey ; and in the third type, S3 the axes are Pn , Mnx , and Mny . In the presentation that follows, the Bresler reciprocal load method makes use of the reciprocal failure surface S2 , and the Bresler load contour method and the PCA load contour method both use the failure surface S3 . Bresler Reciprocal Load Method
Using a failure surface of type S2 , Bresler proposed the following equation as a means of approximating a point on the failure surface corresponding to prespecified eccentricities ex and ey : 1 1 1 1 = + − Pni Pnx Pny P0
(4.57)
where Pni = nominal axial load strength at given eccentricity along both axes Pnx = nominal axial load strength at given eccentricity along x axis Pny = nominal axial load strength at given eccentricity along y axis P0 = nominal axial load strength for pure compression (zero eccentricity) Test results indicate that Equation 4.57 may be inappropriate when small values of axial load are involved, such as when Pn /P0 is in the range of 0.06 or less. For such cases the member should be designed for flexure only. Bresler Load Contour Method The failure surface S3 can be thought of as a family of curves (load contours) each corresponding to a constant value of Pn . The general nondimensional equation for the load contour at constant Pn may be expressed in the following form: Mny α2 Mnx α1 + = 1.0 (4.58) Mox Moy
where Mnx = Pn ey ; Mny = Pn ex 1999 by CRC Press LLC
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Mox = Mnx capacity at axial load Pn when Mny (or ex ) is zero Moy = Mny capacity at axial load Pn when Mnx (or ey ) is zero The exponents α1 and α2 depend on the column dimensions, amount and arrangement of the reinforcement, and material strengths. Bresler suggests taking α1 = α2 = α. Calculated values of α vary from 1.15 to 1.55. For practical purposes, α can be taken as 1.5 for rectangular sections and between 1.5 and 2.0 for square sections. PCA (ParmeGowens) Load Contour Method
This method has been developed as an extension of the Bresler load contour method in which the Bresler interaction Equation 4.58 is taken as the basic strength criterion. In this approach, a point on the load contour is defined in such a way that the biaxial moment strengths Mnx and Mny are in the same ratio as the uniaxial moment strengths Mox and Moy , Moy Mny = =β Mnx Mox
(4.59)
The actual value of β depends on the ratio of Pn to P0 as well as the material and crosssectional properties, with the usual range of values between 0.55 and 0.70. Charts for determining β can be found in ACI Publication SP17A(85), A Design Handbook for Columns [2]. Substituting Equation 4.59 into Equation 4.58, βMoy α βMox α + = 1 Mox Moy 2β α = 1 (4.60) β α = 1/2 log 0.5 α = log β thus,
Mnx Mox
log0.5/logβ
+
Mny Moy
log0.5/logβ
=1
(4.61)
For more information on columns subjected to biaxial bending, see Reinforced Concrete Design by ChuKia Wang and Charles G. Salmon [11].
4.5 4.5.1
Shear and Torsion Reinforced Concrete Beams and OneWay Slabs Strength Design
The cracks that form in a reinforced concrete beam can be due to flexure or a combination of flexure and shear. Flexural cracks start at the bottom of the beam, where the flexural stresses are the largest. Inclined cracks, also called shear cracks or diagonal tension cracks, are due to a combination of flexure and shear. Inclined cracks must exist before a shear failure can occur. Inclined cracks form in two different ways. In thinwalled Ibeams in which the shear stresses in the web are high while the flexural stresses are low, a webshear crack occurs. The inclined cracking shear can be calculated as the shear necessary to cause a principal tensile stress equal to the tensile strength of the concrete at the centroid of the beam. In most reinforced concrete beams, however, flexural cracks occur first and extend vertically in the beam. These alter the state of stress in the beam and cause a stress concentration near the tip of the crack. In time, the flexural cracks extend to become flexureshear cracks. Empirical equations have 1999 by CRC Press LLC
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been developed to calculate the flexureshear cracking load, since this cracking cannot be predicted by calculating the principal stresses. In the ACI Code, the basic design equation for the shear capacity of concrete beams is as follows: Vu ≤ φVn
(4.62)
where Vu is the shear force due to the factored loads, φ is the strength reduction factor equal to 0.85 for shear, and Vn is the nominal shear resistance, which is given by Vn = Vc + Vs
(4.63)
where Vc is the shear carried by the concrete and Vs is the shear carried by the shear reinforcement. The torsional capacity of a beam as given in ACI 11.6.5 is as follows: Tu ≤ φTn
(4.64)
where Tu is the torsional moment due to factored loads, φ is the strength reduction factor equal to 0.85 for torsion, and Tn is the nominal torsional moment strength given by Tn = Tc + Tc
(4.65)
where Tc is the torsional moment strength provided by the concrete and Ts is the torsional moment strength provided by the torsion reinforcement. Design of Beams and OneWay Slabs Without Shear Reinforcement: for Shear
The critical section for shear in reinforced concrete beams is taken at a distance d from the face of the support. Sections located at a distance less than d from the support are designed for the shear computed at d. Shear Strength Provided by Concrete Beams without web reinforcement will fail when inclined cracking occurs or shortly afterwards. For this reason the shear capacity is taken equal to the inclined cracking shear. ACI gives the following equations for calculating the shear strength provided by the concrete for beams without web reinforcement subject to shear and flexure: p (4.66) Vc = 2 fc0 bw d or, with a more detailed equation: p p Vu d 0 bw d ≤ 3.5 fc0 bw d Vc = 1.9 fc + 2500ρw Mu
(4.67)
The quantity Vu d/Mu is not to be taken greater than 1.0 in computing Vc where Mu is the factored moment occurring simultaneously with Vu at the section considered. Combined Shear, Moment, and Axial Load For members that are also subject to axial compression, ACI modifies Equation 4.66 as follows (ACI 11.3.1.2): p Nu fc0 bw d (4.68) Vc = 2 1 + 2000Ak where Nu is positive in compression. ACI 11.3.2.2 contains a more detailed calculation for the shear strength of members subject to axial compression. For members subject to axial tension, ACI 11.3.1.3 states that shear reinforcement shall be designed to carry total shear. As an alternative, ACI 11.3.2.3 gives the following for the shear strength of members subject to axial tension: p Nu fc0 bw d (4.69) Vc = 2 1 + 500Ag 1999 by CRC Press LLC
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p where Nu is negative in tension. In Equation 4.68 and 4.69 the terms fc0 , Nu /Ag , 2000, and 500 all have units of psi. Combined Shear, Moment, and Torsion For members subject to torsion, ACI 11.3.1.4 gives the equation for the shear strength of the concrete as the following: p 2 fc0 bw d (4.70) Vc = q 1 + (2.5Ct Tu /Vu )2 where
p X x2y Tu ≥ φ 0.5 fc0
Design of Beams and OneWay Slabs Without Shear Reinforcements: for Torsion
ACI 11.6.1 requires that torsional moments be considered in design if p X x2y Tu ≥ φ 0.5 fc0
(4.71)
Otherwise, torsion effects may be neglected. The critical section for torsion is taken at a distance d from the face of support, and sections located at a distance less than d are designed for the torsion at d. If a concentrated torque occurs within this distance, the critical section is taken at the face of the support. Torsional Strength Provided by Concrete Torsion seldom occurs by itself; bending moments and shearing forces are typically present also. In an uncracked member, shear forces as well as torques produce shear stresses. Flexural shear forces and torques interact in a way that reduces the strength of the member compared with what it would be if shear or torsion were acting alone. The interaction between shear and torsion is taken into account by the use of a circular interaction equation. For more information, refer to Reinforced Concrete Mechanics and Design by James G. MacGregor [8]. The torsional moment strength provided by the concrete is given in ACI 11.6.6.1 as p 0.8 fc0 x 2 y (4.72) Tc = q 1 + (0.4Vu /Ct Tu )2 Combined Torsion and Axial Load For members subject to significant axial tension, ACI 11.6.6.2 states that the torsion reinforcement must be designed to carry the total torsional moment, or as an alternative modify Tc as follows: p 0.8 fc0 x 2 y Nu (4.73) 1+ Tc = q 500Ag 1 + (0.4Vu /Ct Tu )2
where Nu is negative for tension. Design of Beams and OneWay Slabs without Shear Reinforcement: Minimum Reinforcement ACI 11.5.5.1 requires a minimum amount of web reinforcement to be provided for shear and torsion if the factored shear force Vu exceeds one half the shear strength provided by the concrete (Vu ≥ 0.5φVc ) except in the following:
(a) Slabs and footings (b) Concrete joist construction 1999 by CRC Press LLC
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(c) Beams with total depth not greater than 10 inches, 2 1/2 times the thickness of the flange, or 1/2 the width of the web, whichever is greatest The minimum area of shear reinforcement shall be at least p X 50bw s x2y for Tu < φ 0.5 fc0 Av(min) = fy
(4.74)
When torsion is to be considered in design, the sum of the closed stirrups for shear and torsion must satisfy the following: 50bw s (4.75) Av + 2At ≥ fy where Av is the area of two legs of a closed stirrup and At is the area of only one leg of a closed stirrup. Design of Stirrup Reinforcement for Shear and Torsion Shear Reinforcement
Shear reinforcement is to be provided when Vu ≥ φVc , such that Vs ≥
Vu − Vc φ
(4.76)
The design yield strength of the shear reinforcement is not to exceed 60,000 psi. When the shear reinforcement is perpendicular to the axis of the member, the shear resisted by the stirrups is Av fy d (4.77) Vs = s If the shear reinforcement is inclined at an angle α, the shear resisted by the stirrups is Vs =
Av fy (sin α + cos α) d s
(4.78)
Tu − Tc φ
(4.79)
p The maximum shear strength of the shear reinforcement is not to exceed 8 fc0 bw d as stated in ACI 11.5.6.8. Spacing Limitations for Shear Reinforcement ACI 11.5.4.1 sets the maximum spacing of vertical stirrups as the smaller of d/2 or 24 inches. The maximum spacing of inclined stirrups is such that a 45◦ line extending from midheight of the member to the tension reinforcement will intercept at least one stirrup.p If Vs exceeds 4 fc0 bw d, the maximum allowable spacings are reduced to one half of those just described. Torsion Reinforcement Torsion reinforcement is to be provided when Tu ≥ φTc , such that Ts ≥
The design yield strength of the torsional reinforcement is not to exceed 60,000 psi. The torsional moment strength of the reinforcement is computed by Ts =
At αt x1 y1 fy s
(4.80)
where αt = [0.66 + 0.33 (yt /xt )] ≥ 1.50 1999 by CRC Press LLC
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(4.81)
where At is the area of one leg of a closed stirrup resisting torsion within a distance s. The torsional moment strength is not to exceed 4 Tc as given in ACI 11.6.9.4. Longitudinal reinforcement is to be provided to resist axial tension that develops as a result of the torsional moment (ACI 11.6.9.3). The required area of longitudinal bars distributed around the perimeter of the closed stirrups that are provided as torsion reinforcement is to be Al
≥
Al
≥
(x1 + y1 ) 2At s " ! # 400xs Tu x1 + y1 = 2A t fy s Tu + Vu
(4.82)
3Ct
Spacing Limitations for Torsion Reinforcement ACI 11.6.8.1 gives the maximum spacing of closed stirrups as the smaller of (x1 + y1 )/4 or 12 inches. The longitudinal bars are to spaced around the circumference of the closed stirrups at not more than 12 inches apart. At least one longitudinal bar is to be placed in each corner of the closed stirrups (ACI 11.6.8.2). Design of Deep Beams
ACI 11.8 covers the shear design of deep beams. This section applies to members with ln /d 60,000 psi
(4.106) (4.107)
but not less than 12 inches. For fc0 less than 3000 psi, the lap length must be increased by onethird. When different size bars are lap spliced in compression, the splice length is to be the larger of: 1. Compression splice length of the smaller bar, or 2. Compression development length of larger bar. Compression lap splices are allowed for no. 14 and no. 18 bars to no. 11 or smaller bars (ACI 12.16.2). EndBearing Splices Endbearing splices are allowed for compression only where the compressive stress is transmitted by bearing of square cut ends held in concentric contact by a suitable device. According to ACI 12.16.4.2 bar ends must terminate in flat surfaces within 1 1/2◦ of right angles to the axis of the bars and be fitted within 3◦ of full bearing after assembly. Endbearing splices are only allowed in members containing closed ties, closed stirrups, or spirals. Welded Splices or Mechanical Connections Bars stressed in tension or compression may be spliced by welding or by various mechanical connections. ACI 12.14.3, 12.15.3, 12.15.4, and 12.16.3 govern the use of such splices. For further information see Reinforced Concrete Design, by ChuKia Wang and Charles G. Salmon [11]. Bundled Bars
The requirements of ACI 12.4.1 specify that the development length for bundled bars be based on that for the individual bar in the bundle, increased by 20% for a threebar bundle and 33% for a fourbar bundle. ACI 12.4.2 states that “a unit of bundled bars shall be treated as a single bar of a diameter derived from the equivalent total area” when determining the appropriate modification factors in ACI 12.2.3 and 12.2.4.3. Web Reinforcement
ACI 12.13.1 requires that the web reinforcement be as close to the compression and tension faces as cover and barspacing requirements permit. The ACI Code requirements for stirrup anchorage are illustrated in Figure 4.4. (a) ACI 12.13.3 requires that each bend away from the ends of a stirrup enclose a longitudinal bar, as seen in Figure 4.4(a). 1999 by CRC Press LLC
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FIGURE 4.4: Stirrup detailing requirements.
(b) For no. 5 or D31 wire stirrups and smaller with any yield strength and for no. 6, 7, and 8 bars with a yield strength of 40,000 psi or less, ACI 12.13.2.1 allows the use of a standard hook around longitudinal reinforcement, as shown in Figure 4.4(b). (c) For no. 6, 7, and 8 stirrups with fy greater than 40,000 psi, ACI 12.13.2.2 requires a standard hook around a longitudinal bar plus an embedment p between midheight of the member and the outside end of the hook of at least 0.014 db fy / fc0 . (d) Requirements for welded wire fabric forming U stirrups are given in ACI 12.13.2.3. 1999 by CRC Press LLC
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(e) Pairs of U stirrups that form a closed unit shall have a lap length of 1.3ld as shown in Figure 4.4(c). This type of stirrup has proven unsuitable in seismic areas. (f) Requirements for longitudinal bars bent to act as shear reinforcement are given in ACI 12.13.4.
4.7 4.7.1
TwoWay Systems Definition
When the ratio of the longer to the shorter spans of a floor panel drops below 2, the contribution of the longer span in carrying the floor load becomes substantial. Since the floor transmits loads in two directions, it is defined as a twoway system, and flexural reinforcement is designed for both directions. Twoway systems include flat plates, flat slabs, twoway slabs, and waffle slabs (see Figure 4.5). The choice between these different types of twoway systems is largely a matter of the architectural layout, magnitude of the design loads, and span lengths. A flat plate is simply a slab of uniform thickness supported directly on columns, generally suitable for relatively light loads. For larger loads and spans, a flat slab becomes more suitable with the column capitals and drop panels providing higher shear and flexural strength. A slab supported on beams on all sides of each floor panel is generally referred to as a twoway slab. A waffle slab is equivalent to a twoway joist system or may be visualized as a solid slab with recesses in order to decrease the weight of the slab.
FIGURE 4.5: Twoway systems.
4.7.2
Design Procedures
The ACI code [1] states that a twoway slab system “may be designed by any procedure satisfying conditions of equilibrium and geometric compatibility if shown that the design strength at every section is at least equal to the required strength . . . and that all serviceability conditions, including 1999 by CRC Press LLC
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specified limits on deflections, are met” (p.204). There are a number of possible approaches to the analysis and design of twoway systems based on elastic theory, limit analysis, finite element analysis, or combination of elastic theory and limit analysis. The designer is permitted by the ACI Code to adopt any of these approaches provided that all safety and serviceability criteria are satisfied. In general, only for cases of a complex twoway system or unusual loading would a finite element analysis be chosen as the design approach. Otherwise, more practical design approaches are preferred. The ACI Code details two procedures—the direct design method and the equivalent frame method—for the design of floor systems with or without beams. These procedures were derived from analytical studies based on elastic theory in conjunction with aspects of limit analysis and results of experimental tests. The primary difference between the direct design method and equivalent frame method is in the way moments are computed for twoway systems. The yieldline theory is a limit analysis method devised for slab design. Compared to elastic theory, the yieldline theory gives a more realistic representation of the behavior of slabs at the ultimate limit state, and its application is particularly advantageous for irregular column spacing. While the yieldline method is an upperbound limit design procedure, strip method is considered to give a lowerbound design solution. The strip method offers a wide latitude of design choices and it is easy to use; these are often cited as the appealing features of the method. Some of the earlier design methods based on moment coefficients from elastic analysis are still favored by many designers. These methods are easy to apply and give valuable insight into slab behavior; their use is especially justified for many irregular slab cases where the preconditions of the direct design method are not met or when column interaction is not significant. Table 4.7 lists the moment coefficients taken from method 2 of the 1963 ACI Code. TABLE 4.7
Elastic Moment Coefficients for TwoWay Slabs Short span
Long span, all
Span ratio, short/long 1.0
0.9
0.8
0.7
0.6
0.5 and less
span ratios
Case 1—Interior panels Negative moment at: Continuous edge Discontinuous edge Positive moment at midspan
0.033 — 0.025
0.040 — 0.030
0.048 — 0.036
0.055 — 0.041
0.063 — 0.047
0.083 — 0.062
0.033 — 0.025
Case 2—One edge discontinuous Negative moment at: Continuous edge Discontinuous edge Positive moment at midspan
0.041 0.021 0.031
0.048 0.024 0.036
0.055 0.027 0.041
0.062 0.031 0.047
0.069 0.035 0.052
0.085 0.042 0.064
0.041 0.021 0.031
Case 3—Two edges discontinuous Negative moment at: Continuous edge Discontinuous edge Positive moment at midspan:
0.049 0.025 0.037
0.057 0.028 0.043
0.064 0.032 0.048
0.071 0.036 0.054
0.078 0.039 0.059
0.090 0.045 0.068
0.049 0.025 0.037
Case 4—Three edges discontinuous Negative moment at: Continuous edge Discontinuous edge Positive moment at midspan:
0.058 0.029 0.044
0.066 0.033 0.050
0.074 0.037 0.056
0.082 0.041 0.062
0.090 0.045 0.068
0.098 0.049 0.074
0.058 0.029 0.044
Case 5—Four edges discontinuous Negative moment at: Continuous edge Discontinuous edge Positive moment at midspan
— 0.033 0.050
— 0.038 0.057
— 0.043 0.064
— 0.047 0.072
— 0.053 0.080
— 0.055 0.083
— 0.033 0.050
Moments
As in the 1989 code, twoway slabs are divided into column strips and middle strips as indicated by Figure 4.6, where l1 and l2 are the centertocenter span lengths of the floor panel. A column strip is 1999 by CRC Press LLC
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FIGURE 4.6: Definitions of equivalent frame, column strip, and middle strip. (From ACI Committee 318. 1992. Building Code Requirements for Reinforced Concrete and Commentary, ACI 31889 (Revised 92) and ACI 318R89 (Revised 92), Detroit, MI. With permission.)
a design strip with a width on each side of a column centerline equal to 0.25l2 or 0.25l1 , whichever is less. A middle strip is a design strip bounded by two column strips. Taking the moment coefficients from Table 4.7, bending moments per unit width M for the middle strips are computed from the formula M = (Coef.)wls2
(4.108)
where w is the total uniform load per unit area and ls is the shorter span length of l1 and l2 . The average moments per unit width in the column strip is taken as twothirds of the corresponding moments in the middle strip.
4.7.3
Minimum Slab Thickness and Reinforcement
ACI Code Section 9.5.3 contains requirements to determine minimum slab thickness of a twoway system for deflection control. For slabs without beams, the thickness limits are summarized by Table 4.8, but thickness must not be less than 5 in. for slabs without drop panels or 4 in. for slabs with drop panels. In Table 4.8 ln is the length of clear span in the long direction and α is the ratio of flexural stiffness of beam section to flexural stiffness of a width of slab bounded laterally by centerline of adjacent panel on each side of beam. For slabs with beams, it is necessary to compute the minimum thickness h from fy ln 0.8 + 200, 000 h= 1 36 + 5β αm − 0.12 1 + β 1999 by CRC Press LLC
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(4.109)
but not less than
h=
fy 0.8 + 200, 000 36 + 9β
ln
and need not be more than
h=
fy 0.8 + 200, 000 36
ln
(4.110)
(4.111)
where β is the ratio of clear spans in longtoshort direction and αm is the average value of α for all beams on edges of a panel. In no case should the slab thickness be less than 5 in. for αm < 2.0 or less than 3 1/2 in. for αm ≥ 2.0. Minimum reinforcement in twoway slabs is governed by shrinkage and temperature controls to minimize cracking. The minimum reinforcement area stipulated by the ACI Code shall not be less than 0.0018 times the gross concrete area when grade 60 steel is used (0.0020 when grade 40 or grade 50 is used). The spacing of reinforcement in twoway slabs shall exceed neither two times the slab thickness nor 18 in. TABLE 4.8 Minimum Thickness of TwoWay Slabs without Beams Yield stress
Exterior panels
fy , psia
Without edge beams
40,000
ln /33
ln /36
ln /36
60,000
ln /30
ln /33
ln /33
With edge beamsb
Interior panels
Without drop panels
With drop panels 40,000
ln /36
ln /40
ln /40
ln /33 ln /36 ln /36 60,000 a For values of reinforcement yield stress between 40,000
and 60,000 psi minimum thickness shall be obtained by linear interpolation.
b Slabs with beams between columns along exterior
edges. The value of α for the edge beam shall not be less than 0.8. From ACI Committee 318. 1992. Building Code Requirements for Reinforced Concrete and Commentary, ACI 31889 (Revised 92) and ACI 318R89 (Revised 92), Detroit, MI. With permission.
4.7.4
Direct Design Method
The direct design method consists of a set of rules for the design of twoways slabs with or without beams. Since the method was developed assuming simple designs and construction, its application is restricted by the code to twoway systems with a minimum of three continuous spans, successive span lengths that do not differ by more than onethird, columns with offset not more than 10% of the span, and all loads are due to gravity only and uniformly distributed with live load not exceeding three times dead load. The direct design method involves three fundamental steps: (1) determine the total factored static moment; (2) distribute the static moment to negative and positive sections; 1999 by CRC Press LLC
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and (3) distribute moments to column and middle strips and to beams, if any. The total factored static moment Mo for a span bounded laterally by the centerlines of adjacent panels (see Figure 4.6) is given by wu l2 ln2 (4.112) Mo = 8 In an interior span, 0.65 Mo is assigned to each negative section and 0.35 Mo is assigned to the positive section. In an end span, Mo is distributed according to Table 4.9. If the ratio of dead load to live load is less than 2, the effect of pattern loading is accounted for by increasing the positive moment following provisions in ACI Section 13.6.10. Negative and positive moments are then proportioned to the column strip following the percentages in Table 4.10, where βt is the ratio of the torsional stiffness of edge beam section to flexural stiffness of a width of slab equal to span length of beam. The remaining moment not resisted by the column strip is proportionately assigned to the corresponding half middle strip. If beams are present, they are proportioned to resist 85% of column strip moments. When (αl2 / l1 ) is less than 1.0, the proportion of column strip moments resisted by beams is obtained by linear interpolation between 85% and zero. The shear in beams is determined from loads acting on tributary areas projected from the panel corners at 45 degrees. TABLE 4.9
Direct Design Method—Distribution of Moment in End Span (1)
(2)
(3) (4) Slab without beams between interior supports
Slab with
Interior negativefactored moment Positivefactored moment Exterior negativefactored moment
(5)
Exterior edge unrestrained
beams between all supports
Without edge beam
With edge beam
Exterior edge fully restrained
0.75
0.70
0.70
0.70
0.65
0.63
0.57
0.52
0.50
0.35
0
0.16
0.26
0.30
0.65
From ACI Committee 318. 1992. Building Code Requirements for Reinforced Concrete and Commentary, ACI 31889 (Revised 92) and ACI 318R89 (Revised 92), Detroit, MI. With permission.
TABLE 4.10 Proportion of Moment to Column Strip in Percent Interior negativefactored moment `2 /`1 (α1 `2 /`1 ) = 0 (α1 `2 /`1 ) ≥ 1.0
0.5 75 90
1.0 75 75
2.0 75 45
100 75 100 75
100 75 100 45
Positivefactored moment (α1 `2 /`1 ) = 0 (α1 `2 /`1 ) ≥ 1.0
Bt Bt Bt Bt
=0 ≥ 2.5 =0 = 2.5
100 75 100 90
Exterior negativefactored moment (α1 `2 /`1 ) = 0 (α1 `2 /`1 ) ≥ 1.0
60 90
60 75
60 45
From ACI Committee 318. 1992. Building Code Requirements for Reinforced Concrete and Commentary, ACI 31889 (Revised 92) and ACI 318R89 (Revised 92), Detroit, MI. With permission.
1999 by CRC Press LLC
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4.7.5
Equivalent Frame Method
For twoway systems not meeting the geometric or loading preconditions of the direct design method, design moments may be computed by the equivalent frame method. This is a more general method and involves the representation of the threedimensional slab system by dividing it into a series of twodimensional “equivalent” frames (Figure 4.6). The complete analysis of a twoway system consists of analyzing the series of equivalent interior and exterior frames that span longitudinally and transversely through the system. Each equivalent frame, which is centered on a column line and bounded by the center lines of the adjacent panels, comprises a horizontal slabbeam strip and equivalent columns extending above and below the slab beam (Figure 4.7). This structure is analyzed
FIGURE 4.7: Equivalent column (columns plus torsional members).
as a frame for loads acting in the plane of the frame, and the moments obtained at critical sections across the slabbeam strip are distributed to the column strip, middle strip, and beam in the same manner as the direct design method (see Table 4.10). In its original development, the equivalent frame method assumed that analysis would be done by moment distribution. Presently, frame analysis is more easily accomplished in design practice with computers using general purpose programs based on the direct stiffness method. Consequently, the equivalent frame method is now often used as a method for modeling a twoway system for computer analysis. For the different types of twoway systems, the moment of inertias for modeling the slabbeam element of the equivalent frame are indicated in Figure 4.8. Moments of inertia of slab beams are based on the gross area of concrete; the variation in moment of inertia along the axis is taken into account, which in practice would mean that a node would be located on the computer model where a change of moment of inertia occurs. To account for the increased stiffness between the center of the column and the face of column, beam, or capital, the moment of inertia is divided by the quantity (1 − c2 / l2 )2 , where c2 and l2 are measured transverse to the direction of the span. For 1999 by CRC Press LLC
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FIGURE 4.8: Slabbeam stiffness by equivalent frame method. (From ACI Committee 318. 1992. Building Code Requirements for Reinforced Concrete and Commentary, ACI 31889 (Revised 92) and ACI 318R89 (Revised 92), Detroit, MI. With permission.)
column modeling, the moment of inertia at any crosssection outside of joints or column capitals may be based on the gross area of concrete, and the moment of inertia from the top to bottom of the slabbeam joint is assumed infinite. Torsion members (Figure 4.7) are elements in the equivalent frame that provide moment transfer between the horizontal slab beam and vertical columns. The crosssection of torsional members are assumed to consist of the portion of slab and beam having a width according to the conditions depicted 1999 by CRC Press LLC
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FIGURE 4.9: Torsional members. (From ACI Committee 318. 1992. Building Code Requirements for Reinforced Concrete and Commentary, ACI 31889 (Revised 92) and ACI 318R89 (Revised 92), Detroit, MI. With permission.) in Figure 4.9. The stiffness Kt of the torsional member is calculated by the following expression: Kt =
X
9Ecs C 3 l2 1 − cl22
(4.113)
where Ecs is the modulus of elasticity of the slab concrete and torsional constant C may be evaluated by dividing the crosssection into separate rectangular parts and carrying out the following summation: C=
X
1 − 0.63
x y
x3y 3
(4.114)
where x and y are the shorter and longer dimension, respectively, of each rectangular part. Where beams frame into columns in the direction of the span, the increased torsional stiffness Kta is obtained by multiplying the value Kt obtained from Equation 4.113 by the ratio of (a) moment inertia of slab with such beam, to (b) moment of inertia of slab without such beam. Various ways have been suggested for incorporating torsional members into a computer model of an equivalent frame. The model implied by the ACI Code is one that has the slab beam connected to the torsional members, which are projected out of the plane of the columns. Others have suggested that the torsional 1999 by CRC Press LLC
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members be replaced by rotational springs at column ends or, alternatively, at the slabbeam ends. Or, instead of rotational springs, columns may be modeled with an equivalent value of the moment of inertia modified by the equivalent column stiffness Kec given in the commentary of the code. Using Figure 4.7, Kec is computed as Kct + Kcb (4.115) Kec = Kct + Kcb 1+ Kta + Kta where Kct and Kcb are the top and bottom flexural stiffnesses of the column.
4.7.6
Detailing
The ACI Code specifies that reinforcement in twoway slabs without beams have minimum extensions as prescribed in Figure 4.10. Where adjacent spans are unequal, extensions of negative moment
FIGURE 4.10: Minimum extensions for reinforcement in twoway slabs without beams. (From ACI Committee 318. 1992. Building Code Requirements for Reinforced Concrete and Commentary, ACI 31889 (Revised 92) and ACI 318R89 (Revised 92), Detroit, MI. With permission.)
reinforcement shall be based on the longer span. Bent bars may be used only when the depthspan ratio permits use of bends 45 degrees or less. And at least two of the column strip bottom bars in each direction shall be continuous or spliced at the support with Class A splices or anchored within support. These bars must pass through the column and be placed within the column core. The purpose of this “integrity steel” is to give the slab some residual capacity following a single punching shear failure. The ACI Code requires drop panels to extend in each direction from centerline of support a distance not less than onesixth the span length, and the drop panel must project below the slab at 1999 by CRC Press LLC
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least onequarter of the slab thickness. The effective support area of a column capital is defined by the intersection of the bottom surface of the slab with the largest right circular cone whose surfaces are located within the column and capital and are oriented no greater than 45 degrees to the axis of the column.
4.8
Frames
A structural frame is a threedimensional structural system consisting of straight members that are built monolithically and have rigid joints. The frame may be one bay long and one story high—such as portal frames and gable frames—or it may consist of multiple bays and stories. All members of frame are considered continuous in the three directions, and the columns participate with the beams in resisting external loads. Consideration of the behavior of reinforced concrete frames at and near the ultimate load is necessary to determine the possible distributions of bending moment, shear force, and axial force that could be used in design. It is possible to use a distribution of moments and forces different from that given by linear elastic structural analysis if the critical sections have sufficient ductility to allow redistribution of actions to occur as the ultimate load is approached. Also, in countries that experience earthquakes, a further important design is the ductility of the structure when subjected to seismictype loading, since present seismic design philosophy relies on energy dissipation by inelastic deformations in the event of major earthquakes.
4.8.1
Analysis of Frames
A number of methods have been developed over the years for the analysis of continuous beams and frames. The socalled classical methods—such as application of the theorem of three moments, the method of least work, and the general method of consistent deformation—have proved useful mainly in the analysis of continuous beams having few spans or of very simple frames. For the more complicated cases usually met in practice, such methods prove to be exceedingly tedious, and alternative approaches are preferred. For many years the closely related methods of slope deflection and moment distribution provided the basic analytical tools for the analysis of indeterminate concrete beams and frames. In offices with access to highspeed digital computers, these have been supplanted largely by matrix methods of analysis. Where computer facilities are not available, moment distribution is still the most common method. Approximate methods of analysis, based either on an assumed shape of the deformed structure or on moment coefficients, provide a means for rapid estimation of internal forces and moments. Such estimates are useful in preliminary design and in checking more exact solutions, and in structures of minor importance may serve as the basis for final design. Slope Deflection
The method of slope deflection entails writing two equations for each member of a continuous frame, one at each end, expressing the end moment as the sum of four contributions: (1) the restraining moment associated with an assumed fixedend condition for the loaded span, (2) the moment associated with rotation of the tangent to the elastic curve at the near end of the member, (3) the moment associated with rotation of the tangent at the far end of the member, and (4) the moment associated with translation of one end of the member with respect to the other. These equations are related through application of requirements of equilibrium and compatibility at the joints. A set of simultaneous, linear algebraic equations results for the entire structure, in which the structural displacements are unknowns. Solution for these displacements permits the calculation of all internal forces and moments. This method is well suited to solving continuous beams, provided there are not very many spans. 1999 by CRC Press LLC
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Its usefulness is extended through modifications that take advantage of symmetry and antisymmetry, and of hingeend support conditions where they exist. However, for multistory and multibay frames in which there are a large number of members and joints, and which will, in general, involve translation as well as rotation of these joints, the effort required to solve the correspondingly large number of simultaneous equations is prohibitive. Other methods of analysis are more attractive. Moment Distribution
The method of moment distribution was developed to solve problems in frame analysis that involve many unknown joint displacements. This method can be regarded as an iterative solution of the slopedeflection equations. Starting with fixedend moments for each member, these are modified in a series of cycles, each converging on the precise final result, to account for rotation and translation of the joints. The resulting series can be terminated whenever one reaches the degree of accuracy required. After obtaining memberend moments, all member stress resultants can be obtained by use of the laws of statics. Matrix Analysis
Use of matrix theory makes it possible to reduce the detailed numerical operations required in the analysis of an indeterminate structure to systematic processes of matrix manipulation which can be performed automatically and rapidly by computer. Such methods permit the rapid solution of problems involving large numbers of unknowns. As a consequence, less reliance is placed on special techniques limited to certain types of problems; powerful methods of general applicability have emerged, such as the matrix displacement method. Account can be taken of such factors as rotational restraint provided by members perpendicular to the plane of a frame. A large number of alternative loadings may be considered. Provided that computer facilities are available, highly precise analyses are possible at lower cost than for approximate analyses previously employed. Approximate Analysis In spite of the development of refined methods for the analysis of beams and frames, increasing attention is being paid to various approximate methods of analysis. There are several reasons for this. Prior to performing a complete analysis of an indeterminate structure, it is necessary to estimate the proportions of its members in order to know their relative stiffness upon which the analysis depends. These dimensions can be obtained using approximate analysis. Also, even with the availability of computers, most engineers find it desirable to make a rough check of results—using approximate means—to detect gross errors. Further, for structures of minor importance, it is often satisfactory to design on the basis of results obtained by rough calculation. Provided that points of inflection (locations in members at which the bending moment is zero and there is a reversal of curvature of the elastic curve) can be located accurately, the stress resultants for a framed structure can usually be found on the basis of static equilibrium alone. Each portion of the structure must be in equilibrium under the application of its external loads and the internal stress resultants. The use of approximate analysis in determining stress resultants in frames is illustrated using a simple rigid frame in Figure 4.11. ACI Moment Coefficients
The ACI Code [1] includes moment and shear coefficients that can be used for the analysis of buildings of usual types of construction, span, and story heights. They are given in ACI Code Sec. 8.3.3. The ACI coefficients were derived with due consideration of several factors: a maximum allowable ratio of live to dead load (3:1); a maximum allowable span difference (the larger of two adjacent spans not exceed the shorter by more than 20%); the fact that reinforced concrete beams are never simply supported but either rest on supports of considerable width, such as walls, or are 1999 by CRC Press LLC
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FIGURE 4.11: Approximate analysis of rigid frame.
built monolithically like columns; and other factors. Since all these influences are considered, the ACI coefficients are necessarily quite conservative, so that actual moments in any particular design are likely to be considerably smaller than indicated. Consequently, in many reinforced concrete structures, significant economy can be effected by making a more precise analysis. Limit Analysis
Limit analysis in reinforced concrete refers to the redistribution of moments that occurs throughout a structure as the steel reinforcement at a critical section reaches its yield strength. Under working loads, the distribution of moments in a statically indeterminate structure is based on elastic theory, and the whole structure remains in the elastic range. In limit design, where factored loads are used, the distribution of moments at failure when a mechanism is reached is different from that distribution based on elastic theory. The ultimate strength of the structure can be increased as more sections reach their ultimate capacity. Although the yielding of the reinforcement introduces large deflections, which should be avoided under service, a statically indeterminate structure does not collapse when the reinforcement of the first section yields. Furthermore, a large reserve of strength is present between the initial yielding and the collapse of the structure. In steel design the term plastic design is used to indicate the change in the distribution of moments in the structure as the steel fibers, at a critical section, are stressed to their yield strength. Limit analysis of reinforced concrete developed as a result of earlier research on steel structures. Several studies had been performed on the principles of limit design and the rotation capacity of reinforced concrete plastic hinges. Full utilization of the plastic capacity of reinforced concrete beams and frames requires an extensive analysis of all possible mechanisms and an investigation of rotation requirements and capacities at 1999 by CRC Press LLC
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all proposed hinge locations. The increase of design time may not be justified by the limited gains obtained. On the other hand, a restricted amount of redistribution of elastic moments can safely be made without complete analysis and may be sufficient to obtain most of the advantages of limit analysis. A limited amount of redistribution is permitted under the ACI Code, depending upon a rough measure of available ductility; without explicit calculation of rotation requirements and capacities. The ratio ρ/ρb —or in the case of doubly reinforced members, (ρ − ρ 0 )/ρb —is used as an indicator of rotation capacity, where ρb is the balanced steel ratio. For singly reinforced members with ρ = ρb , experiments indicate almost no rotation capacity, since the concrete strain is nearly equal to εcu when steel yielding is initiated. Similarly, in a doubly reinforced member, when ρ − ρ 0 = ρb , very little rotation will occur after yielding before the concrete crushes. However, when ρ or ρ − ρ 0 is low, extensive rotation is usually possible. Accordingly, ACI Code Sec. 8.3 provides as follows: Except where approximate values for moments are used, it is permitted to increase or decrease negative moments calculated by elastic theory at supports of continuous flexural members for any assumed loading arrangement by not more than 20 [1 − (ρ − ρ 0 )/ρb ] percent. The modified negative moments shall be used for calculating moments at sections within the spans. Redistribution of negative moments shall be made only when the section at which moment is reduced is so designed that ρ or ρ − ρ 0 is not greater than 0.5 ρb [1992].
4.8.2
Design for Seismic Loading
The ACI Code contains provisions that are currently considered to be the minimum requirements for producing a monolithic concrete structure with adequate proportions and details to enable the structure to sustain a series of oscillations into the inelastic range of response without critical decay in strength. The provisions are intended to apply to reinforced concrete structures located in a seismic zone where major damage to construction has a high possibility of occurrence, and are designed with a substantial reduction in total lateral seismic forces due to the use of lateral loadresisting systems consisting of ductile momentresisting frames. The provisions for frames are divided into sections on flexural members, columns, and joints of frames. Some of the important points stated are summarized below. Flexural Members
Members having a factored axial force not exceeding Ag fc0 /10, where Ag is gross section of area (in.2 ), are regarded as flexural members. An upper limit is placed on the flexural steel ratio ρ. The maximum value of ρ should not exceed 0.025. Provision is also made to ensure that a minimum quantity of top and bottom reinforcement is always present. Both the top and the bottom steel are to have a steel ratio of a least 200/fy , with the steel yield strength fy in psi throughout the length of the member. Recommendations are also made to ensure that sufficient steel is present to allow for unforeseen shifts in the points of contraflexure. At column connections, the positive moment capacity should be at least 50% of the negative moment capacity, and the reinforcement should be continuous through columns where possible. At external columns, beam reinforcement should be terminated in the far face of the column using a hook plus any additional extension necessary for anchorage. The design shear force Ve should be determined from consideration of the static forces on the portion of the member between faces of the joints. It should be assumed that moments of opposite sign corresponding to probable strength Mpr act at the joint faces and that the member is loaded with the factored tributary gravity load along its span. Figure 4.12 illustrates the calculation. Minimum web reinforcement is provided throughout the length of the member, and spacing should not exceed 1999 by CRC Press LLC
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FIGURE 4.12: Design shears for girders and columns. (From ACI Committee 318. 1992. Building Code Requirements for Reinforced Concrete and Commentary, ACI 31889 (Revised 92) and ACI 318R89 (Revised 92), Detroit, MI. With permission.) d/4 in plastic hinge zones and d/2 elsewhere, where d is effective depth of member. The stirrups should be closed around bars required to act as compression reinforcement and in plastic hinge regions, and the spacing should not exceed specified values. Columns
Members having a factored axial force exceeding Ag fc0 /10 are regarded as columns of frames serving to resist earthquake forces. These members should satisfy the conditions that the shortest crosssectional dimension—measured on a straight line passing through the geometric centroid— should not be less than 12 in. and that the ratio of the shortest crosssectional dimension to the perpendicular dimension should not be less than 0.4. The flexural strengths of the columns should satisfy X X Mg (4.116) Me ≥ (6/5) P where Me is sum of moments, at the center of the joint, P corresponding to the design flexural strength of the columns framing into that joint and where Mg is sum of moments, at the center of the joint, corresponding to the design flexural strengths of the girders framing into that joint. Flexural strengths should be summed such that the column moments oppose the beam moments. 1999 by CRC Press LLC
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Equation 4.116 should be satisfied for beam moments acting in both directions in the vertical plane of the frame considered. The requirement is intended to ensure that plastic hinges form in the girders rather than the columns. The longitudinal reinforcement ratio is limited to the range of 0.01 to 0.06. The lower bound to the reinforcement ratio refers to the traditional concern for the effects of timedependent deformations of the concrete and the desire to have a sizable difference between the cracking and yielding moments. The upper bound reflects concern for steel congestion, load transfer from floor elements to column in lowrise construction, and the development of large shear stresses. Lap splices are permitted only within the center half of the member length and should be proportioned as tension splices. Welded splices and mechanical connections are allowed for splicing the reinforcement at any section, provided not more than alternate longitudinal bars are spliced at a section and the distance between splices is 24 in. or more along the longitudinal axis of the reinforcement. If Equation 4.116 is not satisfied at a joint, columns supporting reactions from that joint should be provided with transverse reinforcement over their full height to confine the concrete and provide lateral support to the reinforcement. Where a spiral is used, the ratio of volume of spiral reinforcement to the core volume confined by the spiral reinforcement, ρs , should be at least that given by f 0 Ag −1 (4.117) ρs = 0.45 c fy Ac but not less than 0.12 fc0 /fyh , where Ac is the area of core of spirally reinforced compression member measured to outside diameter of spiral in in.2 and fyh is the specified yield strength of transverse reinforcement in psi. When rectangular reinforcement hoop is used, the total crosssectional area of rectangular hoop reinforcement should not be less than that given by (4.118) Ash = 0.3 shc fc0 /fyh Ag /Ach − 1 0 Ash = 0.09shc fc /fyh (4.119) where s is the spacing of transverse reinforcement measured along the longitudinal axis of column, hc is the crosssectional dimension of column core measured centertocenter of confining reinforcement, and Ash is the total crosssectional area of transverse reinforcement (including crossties) within spacing s and perpendicular to dimension hc . Supplementary crossties, if used, should be of the same diameter as the hoop bar and should engage the hoop with a hook. Special transverse confining steel is required for the full height of columns that support discontinuous shear walls. The design shear force Ve should be determined from consideration of the maximum forces that can be generated at the faces of the joints at each end of the column. These joint forces should be determined using the maximum probable moment strength Mpr of the column associated with the range of factored axial loads on the column. The column shears need not exceed those determined from joint strengths based on the probable moment strength Mpr , of the transverse members framing into the joint. In no case should Ve be less than the factored shear determined by analysis of the structure Figure 4.12. Joints of Frames
Development of inelastic rotations at the faces of joints of reinforced concrete frames is associated with strains in the flexural reinforcement well in excess of the yield strain. Consequently, joint shear force generated by the flexural reinforcement is calculated for a stress of 1.25 fy in the reinforcement. Within the depth of the shallowed framing member, transverse reinforcement equal to at least onehalf the amount required for the column reinforcement should be provided where members frame into all four sides of the joint and where each member width is at least threefourths the column width. Transverse reinforcement as required for the column reinforcement should be provided through the 1999 by CRC Press LLC
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joint to provide confinement for longitudinal beam reinforcement outside the column core if such confinement is not provided by a beam framing into the joint. The nominal shear strength of the joint should not be taken greater than the forces specified below for normal weight aggregate concrete: p 20 fc0 Aj
for joints confined on all four faces
p 15 fc0 Aj
for joints confined on three faces or on two opposite faces
p 12 fc0 Aj
for others
where Aj is the effective crosssectional area within a joint in a plane parallel to plane of reinforcement generating shear in the joint (Figure 4.13). A member that frames into a face is considered to provide
FIGURE 4.13: Effective area of joint. (From ACI Committee 318. 1992. Building Code Requirements for Reinforced Concrete and Commentary, ACI 31889 (Revised 92) and ACI 318R89 (Revised 92), Detroit, MI. With permission.)
confinement to the joint if at least threequarters of the face of the joint is covered by the framing member. A joint is considered to be confined if such confining members frame into all faces of the joint. For lightweightaggregate concrete, the nominal shear strength of the joint should not exceed threequarters of the limits given above. Details of minimum development length for deformed bars with standard hooks embedded in normal and lightweight concrete and for straight bars are contained in ACI Code Sec. 21.6.4.
1999 by CRC Press LLC
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4.9
Brackets and Corbels
Brackets and corbels are cantilevers having shear span depth ratio, a/d, not greater than unity. The shear span a is the distance from the point of load to the face of support, and the distance d shall be measured at face of support (see Figure 4.14).
FIGURE 4.14: Structural action of a corbel. (From ACI Committee 318. 1992. Building Code Requirements for Reinforced Concrete and Commentary, ACI 31889 (Revised 92) and ACI 318R89 (Revised 92), Detroit, MI. With permission.)
The corbel shown in Figure 4.14 may fail by shearing along the interface between the column and the corbel by yielding of the tension tie, by crushing or splitting of the compression strut, or by localized bearing or shearing failure under the loading plate. The depth of a bracket or corbel at its outer edge should be less than onehalf of the required depth d at the support. Reinforcement should consist of main tension bars with area As and shear reinforcement with area Ah (see Figure 4.15 for notation). The area of primary tension reinforcement
FIGURE 4.15: Notation used. (From ACI Committee 318. 1992. Building Code Requirements for Reinforced Concrete and Commentary, ACI 31889 (Revised 92) and ACI 318R89 (Revised 92), Detroit, MI. With permission.) 1999 by CRC Press LLC
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As should be made equal to the greater of (Af + An ) or (2Avf /3 − An ), where Af is the flexural reinforcement required to resist moment [Vu a + Nuc (h − d)], An is the reinforcement required to resist tensile force Nuc , and Avf is the shearfriction reinforcement required to resist shear Vu : Af
=
An
=
Avf
=
Mu Vu a + Nuc (h − d) = φfy j d φfy j d Nuc φfy Vu φfy µ
(4.120) (4.121) (4.122)
In the above equations, fy is the reinforcement yield strength; φ is 0.9 for Equation 4.120 and 0.85 for Equations 4.121 and 4.122. In Equation 4.120, the lever arm j d can be approximated for all practical purposes in most cases as 0.85d. Tensile force Nuc in Equation 4.121 should not be taken less than 0.2 Vu unless special provisions are made to avoid tensile forces. Tensile force Nuc should be regarded as a live load even when tension results from creep, shrinkage, or temperature change. In Equation 4.122, Vu /φ(= Vn ) should not be taken greater than 0.2 fc0 bw d nor 800bw d in pounds in normalweight concrete. For “alllightweight” or “sandlightweight” concrete, shear strength Vn should not be taken greater than (0.2 − 0.07a/d)fc0 bw d nor (800 − 280a/d)bw d in pounds. The coefficient of friction µ in Equation 4.122 should be 1.4λ for concrete placed monolithically, 1.0λ for concrete placed against hardened concrete with surface intentionally roughened, 0.6λ for concrete placed against hardened concrete not intentionally roughened, and 0.7λ for concrete anchored to asrolled structural steel by headed studs or by reinforcing bars, where λ is 1.0 for normal weight concrete, 0.85 for “sandlightweight” concrete, and 0.75 for “alllightweight” concrete. Linear interpolation of λ is permitted when partial sand replacement is used. The total area of closed stirrups or ties Ah parallel to As should not be less than 0.5(As − An ) and should be uniformly distributed within twothirds of the depth of the bracket adjacent to As . At front face of bracket or corbel, primary tension reinforcement As should be anchored in one of the following ways: (a) by a structural weld to a transverse bar of at least equal size; weld to be designed to develop specified yield strength fy of As bars; (b) by bending primary tension bars As back to form a horizontal loop, or (c) by some other means of positive anchorage. Also, to ensure development of the yield strength of the reinforcement As near the load, bearing area of load on bracket or corbel should not project beyond straight portion of primary tension bars As , nor project beyond interior face of transverse anchor bar (if one is provided). When corbels are designed to resist horizontal forces, the bearing plate should be welded to the tension reinforcement As .
4.10
Footings
Footings are structural members used to support columns and walls and to transmit and distribute their loads to the soil in such a way that (a) the load bearing capacity of the soil is not exceeded, (b) excessive settlement, differential settlement, and rotations are prevented, and (c) adequate safety against overturning or sliding is maintained. When a column load is transmitted to the soil by the footing, the soil becomes compressed. The amount of settlement depends on many factors, such as the type of soil, the load intensity, the depth below ground level, and the type of footing. If different footings of the same structure have different settlements, new stresses develop in the structure. Excessive differential settlement may lead to the damage of nonstructural members in the buildings, even failure of the affected parts. Vertical loads are usually applied at the centroid of the footing. If the resultant of the applied loads does not coincide with the centroid of the bearing area, a bending moment develops. In this case, 1999 by CRC Press LLC
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the pressure on one side of the footing will be greater than the pressure on the other side, causing higher settlement on one side and a possible rotation of the footing. If the bearing soil capacity is different under different footings—for example, if the footings of a building are partly on soil and partly on rock—a differential settlement will occur. It is customary in such cases to provide a joint between the two parts to separate them, allowing for independent settlement.
4.10.1
Types of Footings
Different types of footings may be used to support building columns or walls. The most commonly used ones are illustrated in Figure 4.16(a–g). A simple file footing is shown in Figure 4.16(h).
FIGURE 4.16: Common types of footings for walls and columns. (From ACI Committee 340. 1990. Design Handbook in Accordance with the Strength Design Method of ACI 31889. Volume 2, SP17. With permission.)
For walls, a spread footing is a slab wider than the wall and extending the length of the wall [Figure 4.16(a)]. Square or rectangular slabs are used under single columns [Figure 4.16(b–d)]. When two columns are so close that their footings would merge or nearly touch, a combined footing [Figure 4.16(e)] extending under the two should be constructed. When a column footing cannot project in one direction, perhaps because of the proximity of a property line, the footing may be helped out by an adjacent footing with more space; either a combined footing or a strap (cantilever) footing [Figure 4.16(f)] may be used under the two. For structures with heavy loads relative to soil capacity, a mat or raft foundation [Figure 4.16(g)] may prove economical. A simple form is a thick, twowayreinforcedconcrete slab extending under 1999 by CRC Press LLC
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the entire structure. In effect, it enables the structure to float on the soil, and because of its rigidity it permits negligible differential settlement. Even greater rigidity can be obtained by building the raft foundation as an inverted beamandgirder floor, with the girders supporting the columns. Sometimes, also, inverted flat slabs are used as mat foundations.
4.10.2
Design Considerations
Footings must be designed to carry the column loads and transmit them to the soil safely while satisfying code limitations. The design procedure must take the following strength requirements into consideration: • The area of the footing based on the allowable bearing soil capacity • Twoway shear or punching shear • Oneway shear • Bending moment and steel reinforcement required • Dowel requirements • Development length of bars • Differential settlement These strength requirements will be explained in the following sections. Size of Footings The required area of concentrically loaded footings is determined from
Areq =
D+L qa
(4.123)
where qa is allowable bearing pressure and D and L are, respectively, unfactored dead and live loads. Allowable bearing pressures are established from principles of soil mechanics on the basis of load tests and other experimental determinations. Allowable bearing pressures qa under service loads are usually based on a safety factor of 2.5 to 3.0 against exceeding the ultimate bearing capacity of the particular soil and to keep settlements within tolerable limits. The required area of footings under the effects of wind W or earthquake E is determined from the following: Areq =
D+L+W 1.33qa
or
D+L+E 1.33qa
(4.124)
It should be noted that footing sizes are determined for unfactored service loads and soil pressures, in contrast to the strength design of reinforced concrete members, which utilizes factored loads and factored nominal strengths. A footing is eccentrically loaded if the supported column is not concentric with the footing area or if the column transmits—at its juncture with the footing—not only a vertical load but also a bending moment. In either case, the load effects at the footing base can be represented by the vertical load P and a bending moment M. The resulting bearing pressures are again assumed to be linearly distributed. As long as the resulting eccentricity e = M/P does not exceed the kern distance k of the footing area, the usual flexure formula qmax, min = 1999 by CRC Press LLC
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Mc P + A I
(4.125)
FIGURE 4.17: Assumed bearing pressures under eccentric footings. permits the determination of the bearing pressures at the two extreme edges, as shown in Figure 4.17(a). The footing area is found by trial and error from the condition qmax ≤ qa . If the eccentricity falls outside the kern, Equation 4.125 gives a negative value for q along one edge of the footing. Because no tension can be transmitted at the contact area between soil and footing, Equation 4.125 is no longer valid and bearing pressures are distributed as in Figure 4.17(b). Once the required footing area has been determined, the footing must then be designed to develop the necessary strength to resist all moments, shears, and other internal actions caused by the applied loads. For this purpose, the load factors of the ACI Code apply to footings as to all other structural components. Depth of footing above bottom reinforcement should not be less than 6 in. for footings on soil, nor less than 12 in. for footings on piles. TwoWay Shear (Punching Shear) ACI Code Sec. 11.12.2 allows a shear strength Vc of footings without shear reinforcement for twoway shear action as follows: p 4 p 0 fc bo d ≤ 4 fc0 bo d (4.126) Vc = 2 + βc
where βc is the ratio of long side to short side of rectangular area, bo is the perimeter of the critical section taken at d/2 from the loaded area (column section), and d is the effective depth of footing. This shear is a measure of the diagonal tension caused by the effect of the column load on the footing. Inclined cracks may occur in the footing at a distance d/2 from the face of the column on all sides. The footing will fail as the column tries to punch out part of the footing, as shown in Figure 4.18. OneWay Shear
For footings with bending action in one direction, the critical section is located at a distance d from the face of the column. The diagonal tension at section mm in Figure 4.19 can be checked as is done in beams. The allowable shear in this case is equal to p (4.127) φVc = 2φ fc0 bd where b is the width of section mm. The ultimate shearing force at section mm can be calculated as follows: L c − −d (4.128) Vu = qu b 2 2 where b is the side of footing parallel to section mm. 1999 by CRC Press LLC
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FIGURE 4.18: Punching shear (twoway).
FIGURE 4.19: Oneway shear. Flexural Reinforcement and Footing Reinforcement
The theoretical sections for moment occur at face of the column (section nn, Figure 4.20). The bending moment in each direction of the footing must be checked and the appropriate reinforcement must be provided. In square footings the bending moments in both directions are equal. To determine the reinforcement required, the depth of the footing in each direction may be used. As the bars in one direction rest on top of the bars in the other direction, the effective depth d varies with the diameter of the bars used. The value of dmin may be adopted. The depth of footing is often controlled by the shear, which requires a depth greater than that required by the bending moment. The steel reinforcement in each direction can be calculated in the case of flexural members as follows: As =
Mu φfy (d − a/2)
(4.129)
The minimum steel percentage requirement in flexural member is equal to 200/fy . However, ACI Code Sec. 10.5.3 indicates that for structural slabs of uniform thickness, the minimum area 1999 by CRC Press LLC
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FIGURE 4.20: Critical section of bending moment. and maximum spacing of steel in the direction of bending should be as required for shrinkage and temperature reinforcement. This last minimum steel reinforcement is very small and a higher minimum reinforcement ratio is recommended, but not greater than 200/fy . The reinforcement in oneway footings and twoway footings must be distributed across the entire width of the footing. In the case of twoway rectangular footings, ACI Code Sec 15.4.4 specifies that in the long direction the total reinforcement must be placed uniformly within a band width equal to the length of the short side of the footing according to 2 Reinforcement band width = Total reinforcement in short direction β +1
(4.130)
where β is the ratio of the long side to the short side of the footing. The band width must be centered on the centerline of the column (Figure 4.21). The remaining reinforcement in the short direction must be uniformly distributed outside the band width. This remaining reinforcement percentage should not be less than required for shrinkage and temperature.
FIGURE 4.21: Band width for reinforcement distribution. When structural steel columns or masonry walls are used, the critical sections for moments in footing are taken at halfway between the middle and the edge of masonry walls, and halfway between the face of the column and the edge of the steel base place (ACI Code Sec. 15.4.2). 1999 by CRC Press LLC
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Bending Capacity of Column at Base
The loads from the column act on the footing at the base of the column, on an area equal to the area of the column crosssection. Compressive forces are transferred to the footing directly by bearing on the concrete. Tensile forces must be resisted by reinforcement, neglecting any contribution by concrete. Forces acting on the concrete at the base of the column must not exceed the bearing strength of concrete as specified by the ACI Code Sec.10.15: (4.131) N = φ 0.85fc0 A1 where φ is 0.7 and A1 is the bearing area of the column. The value of the bearing strength given √ in Equation 4.131 may be multiplied by a factor A2 /A1 ≤ 2.0 for bearing on footings when the supporting surface is wider on all sides other than the loaded area. Here A2 is the area of the part of the supporting footing that is geometrically similar to and concentric with the load area √ (Figure 4.22). Since A2 > A1 , the factor A2 /A1 is greater than unity, indicating that the allowable
FIGURE 4.22: Bearing areas on footings. A1 = c2 , A2 = b2 .
bearing strength is increased because of the lateral support from the footing area surrounding the √ column base. If the calculated bearing force is greater than N or the modified one with r A2 /A1 , reinforcement must be provided to transfer the excess force. This is achieved by providing dowels or extending the column bars √ into the footing. If the calculated bearing force is less than either N or the modified one with r A2 /A1 , then minimum reinforcement must be provided. ACI Code Sec. 15.8.2 indicates that the minimum area of the dowel reinforcement is at least 0.005Ag but not less than 4 bars, where Ag is the gross area of the column section of the supported member. The minimum reinforcement requirements apply √ to the case in which the calculated bearing forces are greater than N or the modified one with r A2 /A1 . Dowels on Footings
It was explained earlier that dowels are required in any case, even if the bearing strength is adequate. The ACI Code specifies a minimum steel ratio ρ = 0.005 of the column section as compared to ρ = 0.01 as minimum reinforcement for the column itself. The minimum number of dowel bars needed is four; these may be placed at the four corners of the column. The dowel bars are usually extended into the footing, bent at their ends, and tied to the main footing reinforcement. ACI Code Sec. 15.8.2 indicates that #14 and #18 longitudinal bars, in compression only, may be lapspliced with dowels. Dowels should not be larger than #11 bar and should extend (1) into supported member a distance not less than the development length of #14 or 18” bars or the splice length of the dowels—whichever is greater, and (2) into the footing a distance not less than the development length of the dowels. 1999 by CRC Press LLC
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Development Length of the Reinforcing Bars
The critical sections for checking the development length of reinforcing bars are the same as those for bending moments. Calculated tension or compression in reinforcement at each section should be developed on each side of that section by embedment length, hook (tension only) or mechanical device, or a combination thereof. The development length for compression bar is p (4.132) ld = 0.02fy db fc0 but not less than 0.0003fy db ≥ 8 in. For other values, refer to ACI Code, Chapter 12. Dowel bars must also be checked for proper development length. Differential Settlement
Footings usually support the following loads: • Dead loads from the substructure and superstructure • Live loads resulting from materials or occupancy • Weight of materials used in backfilling • Wind loads Each footing in a building is designed to support the maximum load that may occur on any column due to the critical combination of loadings, using the allowable soil pressure. The dead load, and maybe a small portion of the live load, may act continuously on the structure. The rest of the live load may occur at intervals and on some parts of the structure only, causing different loadings on columns. Consequently, the pressure on the soil under different loadings will vary according to the loads on the different columns, and differential settlement will occur under the various footings of one structure. Since partial settlement is inevitable, the problem is defined by the amount of differential settlement that the structure can tolerate. The amount of differential settlement depends on the variation in the compressibility of the soils, the thickness of compressible material below foundation level, and the stiffness of the combined footing and superstructure. Excessive differential settlement results in cracking of concrete and damage to claddings, partitions, ceilings, and finishes. For practical purposes it can be assumed that the soil pressure under the effect of sustained loadings is the same for all footings, thus causing equal settlements. The sustained load (or the usual load) can be assumed equal to the dead load plus a percentage of the live load, which occurs very frequently on the structure. Footings then are proportioned for these sustained loads to produce the same soil pressure under all footings. In no case is the allowable soil bearing capacity to be exceeded under the dead load plus the maximum live load for each footing.
4.10.3
Wall Footings
The spread footing under a wall [Figure 4.16(a)] distributes the wall load horizontally to preclude excessive settlement. The wall should be so located on the footings as to produce uniform bearing pressure on the soil (Figure 4.23), ignoring the variation due to bending of the footing. The pressure is determined by dividing the load per foot by the footing width. The footing acts as a cantilever on opposite sides of the wall under downward wall loads and upward soil pressure. For footings supporting concrete walls, the critical section for bending moment is at the face of the wall; for footings under masonry walls, halfway between the middle and edge of the 1999 by CRC Press LLC
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FIGURE 4.23: Reinforcedconcrete wall footing. wall. Hence, for a onefootlong strip of symmetrical concretewall footing, symmetrically loaded, the maximum moment, ftlb, is 1 (4.133) Mu = qu (L − a)2 8 where qu is the uniform pressure on soil (lb/ft2 ), L is the width of footing (ft), and a is wall thickness (ft). For determining shear stresses, the vertical shear force is computed on the section located at a distance d from the face of the wall. Thus, L−a (4.134) −L Vu = qu 2 The calculation of development length is based on the section of maximum moment.
4.10.4
SingleColumn Spread Footings
The spread footing under a column [Figure 4.16(b–d)] distributes the column load horizontally to prevent excessive total and differential settlement. The column should be located on the footing so as to produce uniform bearing pressure on the soil, ignoring the variation due to bending of the footing. The pressure equals the load divided by the footing area. In plan, singlecolumn footings are usually square. Rectangular footings are used if space restrictions dictate this choice or if the supported columns are of strongly elongated rectangular crosssection. In the simplest form, they consist of a single slab [Figure 4.16(b)]. Another type is that of Figure 4.16(c), where a pedestal or cap is interposed between the column and the footing slab; the pedestal provides for a more favorable transfer of load and in many cases is required in order to provide the necessary development length for dowels. This form is also known as a stepped footing. All parts of a stepped footing must be poured in a single pour in order to provide monolithic action. Sometimes sloped footings like those in Figure 4.16(d) are used. They requires less concrete than stepped footings, but the additional labor necessary to produce the sloping surfaces (formwork, etc.) usually makes stepped footings more economical. In general, singleslab footings [Figure 4.16(b)] are most economical for thicknesses up to 3 ft. The required bearing area is obtained by dividing the total load, including the weight of the footing, by the selected bearing pressure. Weights of footings, at this stage, must be estimated and usually amount to 4 to 8% of the column load, the former value applying to the stronger types of soils. Once the required footing area has been established, the thickness h of the footing must be determined. In single footings the effective depth d is mostly governed by shear. Two different types of 1999 by CRC Press LLC
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shear strength are distinguished in single footings: twoway (or punching) shear and oneway (or beam) shear. Based on the Equations 4.126 and 4.127 for punching and oneway shear strength, the required effective depth of footing d is calculated. Singlecolumn footings represent, as it were, cantilevers projecting out from the column in both directions and loaded upward by the soil pressure. Corresponding tension stresses are caused in both these directions at the bottom surface. Such footings are therefore reinforced by two layers of steel, perpendicular to each other and parallel to the edge. The steel reinforcement in each direction can be calculated using Equation 4.129. The critical sections for development length of footing bars are the same as those for bending. Development length may also have to be checked at all vertical planes in which changes of section or of reinforcement occur, as at the edges of pedestals or where part of the reinforcement may be terminated. When a column rests on a footing or pedestal, it transfers its load to only a part of the total area of the supporting member. The adjacent footing concrete provides lateral support to the directly loaded part of the concrete. This causes triaxial compression stresses that increase the strength of the concrete, which is loaded directly under the column. The design bearing strength of concrete must not exceed the one given √ in Equation 4.131 for forces acting on the concrete at the base of column and the modified one with r A2 /A1 for supporting area wider than the loaded area. If the calculated bearing force is greater than the design bearing strength, reinforcement must be provided to transfer the excess force. This is done either by extending the column bars into the footing or by providing dowels, which are embedded in the footing and project above it.
4.10.5
Combined Footings
Spread footings that support more than one column or wall are known as combined footings. They can be divided into two categories: those that support two columns, and those that support more than two (generally large numbers of) columns. In buildings where the allowable soil pressure is large enough for single footings to be adequate for most columns, twocolumn footings are seen to become necessary in two situations: (1) if columns are so close to the property line that singlecolumn footings cannot be made without projecting beyond that line, and (2) if some adjacent columns are so close to each other that their footings would merge. When the bearing capacity of the subsoil is low so that large bearing areas become necessary, individual footings are replaced by continuous strip footings, which support more than two columns and usually all columns in a row. Mostly, such strips are arranged in both directions, in which case a grid foundation is obtained, as shown in Figure 4.24. Such a grid foundation can be done by single footings because the individual strips of the grid foundation represent continuous beams whose moments are much smaller than the cantilever moments in large single footings that project far out from the column in all four directions. For still lower bearing capacities, the strips are made to merge, resulting in a mat foundation, as shown in Figure 4.25. That is, the foundation consists of a solid reinforced concrete slab under the entire building. In structural action such a mat is very similar to a flat slab or a flat plate, upside down—that is, loaded upward by the bearing pressure and downward by the concentrated column reactions. The mat foundation evidently develops the maximum available bearing area under the building. If the soil’s capacity is so low that even this large bearing area is insufficient, some form of deep foundation, such as piles or caissons, must be used. Grid and mat foundations may be designed with the column pedestals—as shown in Figures 4.24 and 4.25—or without them, depending on whether or not they are necessary for shear strength and the development length of dowels. Apart from developing large bearing areas, another advantage of grid and mat foundations is that their continuity and rigidity help in reducing differential settlements of individual columns relative to each other, which may otherwise be caused by local variations in the 1999 by CRC Press LLC
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FIGURE 4.24: Grid foundation.
FIGURE 4.25: Mat foundation.
quality of subsoil, or other causes. For this purpose, continuous spread foundations are frequently used in situations where the superstructure or the type of occupancy provides unusual sensitivity to differential settlement.
4.10.6
TwoColumn Footings
The ACI Codes does not provide a detailed approach for the design of combined footings. The design, in general, is based on an empirical approach. It is desirable to design combined footings so that the centroid of the footing area coincides with the resultant of the two column loads. This produces uniform bearing pressure over the entire area and forestalls a tendency for the footings to tilt. In plan, such footings are rectangular, trapezoidal, or T shaped, the details of the shape being arranged to produce coincidence of centroid and resultant. The simple relationships of Figure 4.26 facilitate the determination of the shapes of the bearing area [7]. In general, the distances m and n are given, the former being the distance from the center of the exterior column to the property line and the latter the distance from that column to the resultant of both column loads. Another expedient, which is used if a single footing cannot be centered under an exterior column, is to place the exterior column footing eccentrically and to connect it with the nearest interior column 1999 by CRC Press LLC
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FIGURE 4.26: Twocolumn footings. (From Fintel, M. 1985. Handbook of Concrete Engineering, 2nd ed., Van Nostrand Reinhold, New York. With permission.)
by a beam or strap. This strap, being counterweighted by the interior column load, resists the tilting tendency of the eccentric exterior footings and equalizes the pressure under it. Such foundations are known as strap, cantilever, or connected footings. The strap may be designed as a rectangular beam spacing between the columns. The loads on it include its own weight (when it does not rest on the soil) and the upward pressure from the footings. Width of the strap usually is selected arbitrarily as equal to that of the largest column plus 4 to 8 inches so that column forms can be supported on top of the strap. Depth is determined by the maximum bending moment. The main reinforcing in the strap is placed near the top. Some of the steel can be cut off where not needed. For diagonal tension, stirrups normally will be needed near the columns (Figure 4.27). In addition, longitudinal placement steel is set near the bottom of the strap, plus reinforcement to guard against settlement stresses. The footing under the exterior column may be designed as a wall footing. The portions on opposite sides of the strap act as cantilevers under the constant upward pressure of the soil. The interior footing should be designed as a singlecolumn footing. The critical section for punching shear, however, differs from that for a conventional footing. This shear should be computed on a section at a distance d/2 from the sides and extending around the column at a distance d/2 from its faces, where d is the effective depth of the footing.
1999 by CRC Press LLC
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FIGURE 4.27: Strap (cantilever) footing. (From Fintel, M. 1985. Handbook of Concrete Engineering, 2nd ed., Van Nostrand Reinhold, New York. With permission.)
4.10.7
Strip, Grid, and Mat Foundations
In the case of heavily loaded columns, particularly if they are to be supported on relatively weak or uneven soils, continuous foundations may be necessary. They may consist of a continuous strip footing supporting all columns in a given row or, more often, of two sets of such strip footings intersecting at right angles so that they form one continuous grid foundation (Figure 4.24). For even larger loads or weaker soils the strips are made to merge, resulting in a mat foundation (Figure 4.25). For the design of such continuous foundations it is essential that reasonably realistic assumptions be made regarding the distribution of bearing pressures, which act as upward loads on the foundation. For compressible soils it can be assumed in first approximation that the deformation or settlement of the soil at a given location and the bearing pressure at that location are proportional to each other. If columns are spaced at moderate distances and if the strip, grid, or mat foundation is very rigid, the settlements in all portions of the foundation will be substantially the same. This means that the bearing pressure, also known as subgrade reaction, will be the same provided that the centroid of the foundation coincides with the resultant of the loads. If they do not coincide, then for such rigid foundations the subgrade reaction can be assumed as linear and determined from statics in the same manner as discussed for single footings. In this case, all loads—the downward column loads as well as the upwardbearing pressures—are known. Hence, moments and shear forces in the foundation can be found by statics alone. Once these are determined, the design of strip and grid foundations is similar to that of inverted continuous beams, and design of mat foundations is similar to that of inverted flat slabs or plates. 1999 by CRC Press LLC
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On the other hand, if the foundation is relatively flexible and the column spacing large, settlements will no longer be uniform or linear. For one thing, the more heavily loaded columns will cause larger settlements, and thereby larger subgrade reactions, than the lighter ones. Also, since the continuous strip or slab midway between columns will deflect upward relative to the nearby columns, soil settlement—and thereby the subgrade reaction—will be smaller midway between columns than directly at the columns. This is shown schematically in Figure 4.28. In this case the subgrade reaction can no longer be assumed as uniform. A reasonably accurate but fairly complex analysis can then be made using the theory of beams on elastic foundations.
FIGURE 4.28: Strip footing. (From Fintel, M. 1985. Handbook of Concrete Engineering, 2nd ed., Van Nostrand Reinhold, New York. With permission.)
A simplified approach has been developed that covers the most frequent situations of strip and grid foundations [4]. The method first defines the conditions under which a foundation can be regarded as rigid so that uniform or overall linear distribution of subgrade reactions can be assumed. This is the case when the average of two adjacent span lengths in a continuous strip does not exceed 1.75/λ, provided also that the adjacent span and column loads do not differ by more than 20% of the larger value. Here, s ks b (4.135) λ=4 3Ec I where ks = Sks0 ks0 = coefficient of subgrade reaction as defined in soils mechanics, basically force per unit area required to produce unit settlement, kips/ft3 b = width of footing, ft Ec = modulus of elasticity of concrete, kips/ft2 I = moment of inertia of footing, ft4 S = shape factor, being [(b + 1)/2b]2 for granular soils and (n + 0.5)/1.5n for cohesive soils, where n is the ratio of longer to shorter side of strip If the average of two adjacent spans exceeds 1.75/λ, the foundation is regarded as flexible. Provided that adjacent spans and column loads differ by no more than 20%, the complex curvelinear distribution of subgrade reaction can be replaced by a set of equivalent trapezoidal reactions, which are also shown in Figure 4.28. The report of ACI Committee 436 contains fairly simple equations 1999 by CRC Press LLC
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for determining the intensities of the equivalent pressures under the columns and at the middle of the spans and also gives equations for the positive and negative moments caused by these equivalent subgrade reactions. With this information, the design of continuous strip and grid footings proceeds similarly to that of footings under two columns. Mat foundations likewise require different approaches, depending on whether they can be classified as rigid or flexible. As in strip footings, if the column spacing is less than 1/λ, the structure may be regarded as rigid, soil pressure can be assumed as uniformly or linearly distributed, and the design is based on statics. On the other hand, when the foundation is considered flexible as defined above, and if the variation of adjacent column loads and spans is not greater than 20%, the same simplified procedure as for strip and grid foundations can be applied to mat foundations. The mat is divided into two sets of mutually perpendicular strip footings of width equal to the distance between midspans, and the distribution of bearing pressures and bending moments is carried out for each strip. Once moments are determined, the mat is in essence treated the same as a flat slab or plate, with the reinforcement allocated between column and middle strips as in these slab structures. This approach is feasible only when columns are located in a regular rectangular grid pattern. When a mat that can be regarded as rigid supports columns at random locations, the subgrade reactions can still be taken as uniform or as linearly distributed and the mat analyzed by statics. If it is a flexible mat that supports such randomly located columns, the design is based on the theory of plates on elastic foundation.
4.10.8
Footings on Piles
If the bearing capacity of the upper soil layers is insufficient for a spread foundation, but firmer strata are available at greater depth, piles are used to transfer the loads to these deeper strata. Piles are generally arranged in groups or clusters, one under each column. The group is capped by a spread footing or cap that distributes the column load to all piles in the group. Reactions on caps act as concentrated loads at the individual piles, rather than as distributed pressures. If the total of all pile reactions in a cluster is divided by area of the footing to obtain an equivalent uniform pressure, it is found that this equivalent pressure is considerably higher in pile caps than for spread footings. Thus, it is in any event advisable to provide ample rigidity—that is, depth for pile caps—in order to spread the load evenly to all piles. As in singlecolumn spread footings, the effective portion of allowable bearing capacities of piles, Ra , available to resist the unfactored column loads is the allowable pile reaction less the weight of footing, backfill, and surcharge per pile. That is, Re = Ra − Wf
(4.136)
where Wf is the total weight of footing, fill, and surcharge divided by the number of piles. Once the available or effective pile reaction Re is determined, the number of piles in a concentrically loaded cluster is the integer next larger than n=
D+L Re
(4.137)
The effects of wind and earthquake moments at the foot of the columns generally produce an eccentrically loaded pile cluster in which different piles carry different loads. The number and location of piles in such a cluster is determined by successive approximation from the condition that the load on the most heavily loaded pile should not exceed the allowable pile reaction Ra . Assuming a linear distribution of pile loads due to bending, the maximum pile reaction is Rmax = 1999 by CRC Press LLC
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P M + n Ipg /c
(4.138)
where P is the maximum load (including weight of cap, backfill, etc.), M is the moment to be resisted by the pile group, both referred to the bottom of the cap, Ipg is the moment of inertia of the entire pile group about the centroidal axis about which bendings occurs, and c is the distance from that axis to the extreme pile. Piles are generally arranged in tight patterns, which minimizes the cost of the caps, but they cannot be placed closer than conditions of deriving and of undisturbed carrying capacity will permit. AASHTO requires that piles be spaced at least 2 ft 6 in. center to center and that the distance from the side of a pile to the nearest edge of the footing be 9 in. or more. The design of footings on piles is similar to that of singlecolumn spread footings. One approach is to design the cap for the pile reactions calculated for the factored column loads. For a concentrically loaded cluster this would give Ru = (1.4D + 1.7L)/n. However, since the number of piles was taken as the next larger integer according to Equation 4.138, determining Ru in this manner can lead to a design where the strength of the cap is less than the capacity of the pile group. It is therefore recommended that the pile reaction for strength design be taken as Ru = Re × Average load factor
(4.139)
where the average load factor is (1.4D + 1.7L)/(D + L). In this manner the cap is designed to be capable of developing the full allowable capacity of the pile group. As in singlecolumn spread footings, the depth of the pile cap is usually governed by shear. In this regard both punching and oneway shear need to be considered. The critical sections are the same as explained earlier under “TwoWay Shear (Punching Shear)” and “OneWay Shear.” The difference is that shears on caps are caused by concentrated pile reactions rather than by distributed bearing pressures. This poses the question of how to calculate shear if the critical section intersects the circumference of one or more piles. For this case the ACI Code accounts for the fact that pile reaction is not really a point load, but is distributed over the pilebearing area. Correspondingly, for piles with diameters dp , it stipulates as follows: Computation of shear on any section through a footing on piles shall be in accordance with the following: (a) The entire reaction from any pile whose center is located dp /2 or more outside this section shall be considered as producing shear on that section. (b) The reaction from any pile whose center is located dp /2 or more inside the section shall be considered as producing no shear on that section. (c) For intermediate portions of the pile center, the portion of the pile reaction to be considered as producing shear on the section shall be based on straightline interpolation between the full value at dp /2 outside the section and zero at dp /2 inside the section [1]. In addition to checking punching and oneway shear, punching shear must be investigated for the individual pile. Particularly in caps on a small number of heavily loaded piles, it is this possibility of a pile punching upward through the cap which may govern the required depth. The critical perimeter for this action, again, is located at a distance d/2 outside the upper edge of the pile. However, for relatively deep caps and closely spaced piles, critical perimeters around adjacent piles may overlap. In this case, fracture, if any, would undoubtedly occur along an outwardslanting surface around both adjacent piles. For such situations the critical perimeter is so located that its length is a minimum, as shown for two adjacent piles in Figure 4.29. 1999 by CRC Press LLC
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FIGURE 4.29: Modified critical section for shear with overlapping critical perimeters.
4.11
Walls
4.11.1
Panel, Curtain, and Bearing Walls
As a general rule, the exterior walls of a reinforced concrete building are supported at each floor by the skeleton framework, their only function being to enclose the building. Such walls are called panel walls. They may be made of concrete (often precast), cinder concrete block, brick, tile blocks, or insulated metal panels. The thickness of each of these types of panel walls will vary according to the material, type of construction, climatological conditions, and the building requirements governing the particular locality in which the construction takes place. The pressure of the wind is usually the only load that is considered in determining the structural thickness of a wall panel, although in some cases exterior walls are used as diaphragms to transmit forces caused by horizontal loads down to the building foundations. Curtain walls are similar to panel walls except that they are not supported at each story by the frame of the building; rather, they are self supporting. However, they are often anchored to the building frame at each floor to provide lateral support. A bearing wall may be defined as one that carries any vertical load in addition to its own weight. Such walls may be constructed of stone masonry, brick, concrete block, or reinforced concrete. Occasional projections or pilasters add to the strength of the wall and are often used at points of load concentration. Bearing walls may be of either single or double thickness, the advantage of the latter type being that the air space between the walls renders the interior of the building less liable to temperature variation and makes the wall itself more nearly moistureproof. On account of the greater gross thickness of the double wall, such construction reduces the available floor space. According to ACI Code Sec. 14.5.2 the load capacity of a wall is given by " # klc 2 0 (4.140) φPnw = 0.55φfc Ag 1 − 32h where φPnw = design axial load strength = gross area of section, in.2 Ag lc = vertical distance between supports, in. h = thickness of wall, in. φ = 0.7 and where the effective length factor k is taken as 0.8 for walls restrained against rotation at top or bottom or both, 1.0 for walls unrestrained against rotation at both ends, and 2.0 for walls not braced against lateral translation. 1999 by CRC Press LLC
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In the case of concentrated loads, the length of the wall to be considered as effective for each should not exceed the centertocenter distance between loads; nor should it exceed the width of the bearing plus 4 times the wall thickness. Reinforced concrete bearing walls should have a thickness of a least 1/25 times the unsupported height or width, whichever is shorter. Reinforced concrete bearing walls of buildings should be not less than 4 in. thick. Minimum ratio of horizontal reinforcement area to gross concrete area should be 0.0020 for deformed bars not larger than #5—with specified yield strength not less than 60,000 psi or 0.0025 for other deformed bars—or 0.0025 for welded wire fabric not larger than W31 or D31. Minimum ratio of vertical reinforcement area to gross concrete area should be 0.0012 for deformed bars not larger than #5—with specified yield strength not less than 60,000 psi or 0.0015 for other deformed bars—or 0.0012 for welded wire fabric not larger than W31 or D31. In addition to the minimum reinforcement, not less than two #5 bars shall be provided around all window and door openings. Such bars shall be extended to develop the bar beyond the corners of the openings but not less than 24 in. Walls more than 10 in. thick should have reinforcement for each direction placed in two layers parallel with faces of wall. Vertical and horizontal reinforcement should not be spaced further apart than three times the wall thickness, or 18 in. Vertical reinforcement need not be enclosed by lateral ties if vertical reinforcement area is not greater than 0.01 times gross concrete area, or where vertical reinforcement is not required as compression reinforcement. Quantity of reinforcement and limits of thickness mentioned above are waived where structural analysis shows adequate strength and stability. Walls should be anchored to intersecting elements such as floors, roofs, or to columns, pilasters, buttresses, and intersecting walls, and footings.
4.11.2
Basement Walls
In determining the thickness of basement walls, the lateral pressure of the earth, if any, must be considered in addition to other structural features. If it is part of a bearing wall, the lower portion may be designed either as a slab supported by the basement and floors or as a retaining wall, depending upon the type of construction. If columns and wall beams are available for support, each basement wall panel of reinforced concrete may be designed to resist the earth pressure as a simple slab reinforced in either one or two directions. A minimum thickness of 7.5 in. is specified for reinforced concrete basement walls. In wet ground a minimum thickness of 12 in. should be used. In any case, the thickness cannot be less than that of the wall above. Care should be taken to brace a basement wall thoroughly from the inside (1) if the earth is backfilled before the wall has obtained sufficient strength to resist the lateral pressure without such assistance, or (2) if it is placed before the firstfloor slab is in position.
4.11.3
Partition Walls
Interior walls used for the purpose of subdividing the floor area may be made of cinder block, brick, precast concrete, metal lath and plaster, clay tile, or metal. The type of wall selected will depend upon the fire resistance required; flexibility of rearrangement; ease with which electrical conduits, plumbing, etc. can be accommodated; and architectural requirements.
4.11.4
Shears Walls
Horizontal forces acting on buildings—for example, those due to wind or seismic action—can be resisted by a variety of means. Rigidframe resistance of the structure, augmented by the contribution of ordinary masonry walls and partitions, can provide for wind loads in many cases. However, when heavy horizontal loading is likely—such as would result from an earthquake—reinforced concrete 1999 by CRC Press LLC
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shear walls are used. These may be added solely to resist horizontal forces; alternatively, concrete walls enclosing stairways or elevator shafts may also serve as shear walls. Figure 4.30 shows a building with wind or seismic forces represented by arrows acting on the edge of each floor or roof. The horizontal surfaces act as deep beams to transmit loads to vertical resisting
FIGURE 4.30: Building with shear walls subject to horizontal loads: (a) typical floor; (b) front elevation; (c) end elevation.
elements A and B. These shear walls, in turn, act as cantilever beams fixed at their base to carry loads down to the foundation. They are subjected to (1) a variable shear, which reaches maximum at the base, (2) a bending moment, which tends to cause vertical tension near the loaded edge and compression at the far edge, and (3) a vertical compression due to ordinary gravity loading from the structure. For the building shown, additional shear walls C and D are provided to resist loads acting in the log direction of the structure. The design basis for shear walls, according to the ACI Code, is of the same general form as that used for ordinary beams: Vu Vn
≤ φVn = Vc + Vs
(4.141) (4.142)
Shear p strength Vn at any horizontal section for shear in plane of wall should not be taken greater than 10 fc0 hd. In this and all other equations pertaining to the design of shear walls, the distance of d may be taken equal to 0.8lw . A larger value of d, equal to the distance from the extreme compression face to the center of force of all reinforcement in tension, may be used when determined by a strain compatibility analysis. The value of Vc , the nominal shear strength provided by the concrete, may be based on the usual equations for beams, according to ACI Code. For walls subjected to vertical compression, p (4.143) Vc = 2 fc0 hd and for walls subjected to vertical tension Nu , Vc = 2 1 + 1999 by CRC Press LLC
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Nu 500Ag
p fc0 hd
(4.144)
where Nu is the factored axial load in pounds, taken negative for tension, and Ag is the gross area of horizontal concrete section in square inches. Alternatively, the value of Vc may be based on a more detailed calculation, as the lesser of p Nu d Vc = 3.3 fc0 hd + 4lw or
p # 0 + 0.2N / l h p f l 1.25 w u w c hd Vc = 0.6 fc0 + Mu /Vu − lw /2
(4.145)
"
(4.146)
p Equation 4.145 corresponds to the occurrence of a principal tensile stress of approximately 4 fc0 at the centroid of the shearwallpsection. Equation 4.146 corresponds approximately to the occurrence of a flexural tensile stress of 6 fc0 at a section lw /2 above the section being investigated. Thus the two equations predict, respectively, webshear cracking and flexureshear cracking. When the quantity Mu /Vu − lw /2 is negative, Equation 4.146 is inapplicable. According to the ACI Code, horizontal sections located closer to the wall base than a distance lw /2 or hw /2, whichever less, may be designed for the same Vc as that computed at a distance lw /2 or hw /2. When the factored shear force Vu does not exceed φVc /2, a wall may be reinforced according to the minimum requirements given in Sec. 12.1. When Vu exceeds φVc /2, reinforcement for shear is to be provided according to the following requirements. The nominal shear strength Vs provided by the horizontal wall steel is determined on the same basis as for ordinary beams: Av fy d (4.147) Vs = s2 where Av is the area of horizontal shear reinforcement within vertical distance s2 , (in.2 ), s2 is the vertical distance between horizontal reinforcement, (in.), and fy is the yield strength of reinforcement, psi. Substituting Equation 4.147 into Equation 4.142, then combining with Equation 4.141, one obtains the equation for the required area of horizontal shear reinforcement within a distance s2 : Av =
(Vu − φVc ) s2 φfy d
(4.148)
The minimum permitted ratio of horizontal shear steel to gross concrete area of vertical section, ρn , is 0.0025 and the maximum spacing s2 is not exceed lw /5, 3h, or 18 in. Test results indicate that for low shear walls, vertical distributed reinforcement is needed as well as horizontal reinforcement. Code provisions require vertical steel of area Ah within a spacing s1 , such that the ratio of vertical steel to gross concrete area of horizontal section will not be less than hw (4.149) ρn = 0.0025 + 0.5 2.5 − (ρh − 0.0025) lw nor less than 0.0025. However, the vertical steel ratio need not be greater than the required horizontal steel ratio. The spacing of the vertical bars is not to exceed lw /3, 3h, or 18 in. Walls may be subjected to flexural tension due to overturning moment, even when the vertical compression from gravity loads is superimposed. In many but not all cases, vertical steel is provided, concentrated near the wall edges, as in Figure 4.31. The required steel area can be found by the usual methods for beams. The ACI Code contains requirements for the dimensions and details of structural walls serving as part of the earthquakeforce resisting systems. The reinforcement ratio, ρv (= Asv /Acv ; where Acv is the net area of concrete section bounded by web thickness and length of section in the direction of 1999 by CRC Press LLC
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FIGURE 4.31: Geometry and reinforcement of typical shear wall: (a) cross section; (b) elevation. shear force considered, and Asv is the projection on Acv of area of distributed shear reinforcement crossing the plane of Acv ), for structural walls should not be less than 0.0025 along the longitudinal and transverse axes. Reinforcement provided for shear strength should be continuous p and should be distributed across the shear plane. If the design shear force does not exceed Acv fc0 , the shear reinforcement may conform to the reinforcement ratio given in Sec. 12.1. At least two curtains of reinforcement p should be used in a wall if the inplane factored shear force assigned to the wall exceeds 2Acv fc0 . All continuous reinforcement in structural walls should be anchored or spliced in accordance with the provisions for reinforcement in tension for seismic design. Proportioning and details of structural walls that resist shear forces caused by earthquake motion is contained in the ACI Code Sec. 21.7.3.
4.12
Defining Terms
The terms common in concrete engineering as defined in and selected from the Cement and Concrete Terminology Report of ACI Committee 116 are given below [1, Further Reading]. Allowable stress: Maximum permissible stress used in design of members of a structure and based on a factor of safety against yielding or failure of any type. Allowable stress design (ASD): Design principle according to which stresses resulting from service or working loads are not allowed to exceed specified allowable values. Balanced load: Combination of axial force and bending moment that causes simultaneous crushing of concrete and yielding of tension steel. Balanced reinforcement: An amount and distribution of flexural reinforcement such that the tensile reinforcement reaches its specified yield strength simultaneously with the concrete in compression reaching its assumed ultimate strain of 0.003. Beam: A structural member subjected primarily to flexure; depthtospan ratio is limited to 2/5 for continuous spans, or 4/5 for simple spans, otherwise the member is to be treated as a deep beam. 1999 by CRC Press LLC
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Beamcolumn: A structural member that is subjected simultaneously to bending and substantial axial forces. Bond: Adhesion and grip of concrete or mortar to reinforcement or to other surfaces against which it is placed; to enhance bond strength, ribs or other deformations are added to reinforcing bars. Camber: A deflection that is intentionally built into a structural element or form to improve appearance or to offset the deflection of the element under the effects of loads, shrinkage, and creep. Castinplace concrete: Concrete poured in its final or permanent location; also called in situ concrete; opposite of precast concrete. Column: A member used to support primarily axial compression loads with a height of at least three times its least lateral dimensions; the capacity of short columns is controlled by strength; the capacity of long columns is limited by buckling. Column strip: The portion of a flat slab over a row of columns consisting of the two adjacent quarter panels on each side of the column centerline. Composition construction: A type of construction using members made of different materials (e.g., concrete and structural steel), or combining members made of castinplace concrete and precast concrete such that the combined components act together as a single member; strictly speaking, reinforced concrete is also composite construction. Compression member: A member subjected primarily to longitudinal compression; often synonymous with “column”. Compressive strength: Strength typically measured on a standard 6 × 12 in. cylinder of concrete in an axial compression test, 28 d after casting. Concrete: A composite material that consists essentially of a binding medium within which are embedded particles or fragments of aggregate; in portland cement concrete, the binder is a mixture of portland cement and water. Confined concrete: Concrete enclosed by closely spaced transverse reinforcement, which restrains the concrete expansion in directions perpendicular to the applied stresses. Construction joint: The surface where two successive placements of concrete meet, across which it may be desirable to achieve bond, and through which reinforcement may be continuous. Continuous beam or slab: A beam or slab that extends as a unit over three or more supports in a given direction and is provided with the necessary reinforcement to develop the negative moments over the interior supports; a redundant structure that requires a statically indeterminant analysis (opposite of simple supported beam or slab). Cover: In reinforced concrete, the shortest distance between the surface of the reinforcement and the outer surface of the concrete; minimum values are specified to protect the reinforcement against corrosion and to assure sufficient bond strength. Cracks: Results of stresses exceeding concrete’s tensile strength capacity; cracks are ubiquitous in reinforced concrete and needed to develop the strength of the reinforcement, but a design goal is to keep their widths small (hairline cracks). 1999 by CRC Press LLC
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Cracked section: A section designed or analyzed on the assumption that concrete has no resistance to tensile stress. Cracking load: The load that causes tensile stress in a member to be equal to the tensile strength of the concrete. Deformed bar: Reinforcing bar with a manufactured pattern of surface ridges intended to prevent slip when the bar is embedded in concrete. Design strength: Ultimate loadbearing capacity of a member multiplied by a strength reduction factor. Development length: The length of embedded reinforcement to develop the design strength of the reinforcement; a function of bond strength. Diagonal crack: An inclined crack caused by a diagonal tension, usually at about 45 degrees to the neutral axis of a concrete member. Diagonal tension: The principal tensile stress resulting from the combination of normal and shear stresses acting upon a structural element. Drop panel: The portion of a flat slab in the area surrounding a column, column capital, or bracket which is thickened in order to reduce the intensity of stresses. Ductility: Capability of a material or structural member to undergo large inelastic deformations without distress; opposite of brittleness; very important material property, especially for earthquakeresistant design; steel is naturally ductile, concrete is brittle but it can be made ductile if well confined. Durability: The ability of concrete to maintain its qualities over long time spans while exposed to weather, freezethaw cycles, chemical attack, abrasion, and other service load conditions. Effective depth: Depth of a beam or slab section measured from the compression face to the centroid of the tensile reinforcement. Effective flange width: Width of slab adjoining a beam stem assumed to function as the flange of a Tsection. Effective prestress: The stress remaining in the prestressing steel or in the concrete due to prestressing after all losses have occurred. Effective span: The lesser of the distance between centers of supports and the clear distance between supports plus the effective depth of the beam or slab. Flat slab: A concrete slab reinforced in two or more directions, generally without beams or girders to transfer the loads to supporting members, but with drop panels or column capitals or both. Highearly strength cement: Cement producing strength in mortar or concrete earlier than regular cement. Hoop: A onepiece closed reinforcing tie or continuously wound tie that encloses the longitudinal reinforcement. Interaction diagram: Failure curve for a member subjected to both axial force and the bending moment, indicating the moment capacity for a given axial load and vice versa; used to develop design charts for reinforced concrete compression members. 1999 by CRC Press LLC
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Lightweight concrete: Concrete of substantially lower unit weight than that made using normalweight gravel or crushed stone aggregate. Limit analysis: See Plastic analysis. Limit design: A method of proportioning structural members based on satisfying certain strength and serviceability limit states. Load and resistance factor design (LRFD): See Ultimate strength design. Load factor: A factor by which a service load is multiplied to determine the factored load used in ultimate strength design. Modulus of elasticity: The ratio of normal stress to corresponding strain for tensile of compressive stresses below the proportional limit of the material; for steel, Es = 29,000 ksi; for concrete 0 it is highly variable with stress level and the strength p fc ; for normalweight concrete and low 0 stresses, a common approximation is Ec = 57,000 fc . Modulus of rupture: The tensile strength of concrete as measured in a flexural test of a small prismatic specimen of plain concrete. Mortar: A mixture of cement paste and fine aggregate; in fresh concrete, the material occupying the interstices among particles of coarse aggregate. Nominal strength: The strength of a structural member based on its assumed material properties and sectional dimensions, before application of any strength reduction factor. Plastic analysis: A method of structural analysis to determine the intensity of a specified load distribution at which the structure forms a collapse mechanism. Plastic hinge: Region of flexural member where the ultimate moment capacity can be developed and maintained with corresponding significant inelastic rotation, as main tensile steel is stressed beyond the yield point. Posttensioning: A method of prestressing reinforced concrete in which the tendons are tensioned after the concrete has hardened (opposite of pretensioning). Precast concrete: Concrete cast elsewhere than its final position, usually in factories or factorylike shop sites near the final site (opposite of castinplace concrete). Prestressed concrete: Concrete in which internal stresses of such magnitude and distribution are introduced that the tensile stresses resulting from the service loads are counteracted to a desired degree; in reinforced concrete the prestress is commonly introduced by tensioning embedded tendons. Prestressing steel: High strength steel used to prestress concrete, commonly sevenwire strands, single wires, bars, rods, or groups of wires or strands. Pretensioning: A method of prestressing reinforced concrete in which the tendons are tensioned before the concrete has hardened (opposite of posttensioning). Readymixed concrete: Concrete manufactured for delivery to a purchaser in a plastic and unhardened state; usually delivered by truck. Rebar: Short for reinforcing bar; see Reinforcement. 1999 by CRC Press LLC
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Reinforced concrete: Concrete containing adequate reinforcement (prestressed or not) and designed on the assumption that the two materials act together in resisting forces. Reinforcement: Bars, wires, strands, and other slender members that are embedded in concrete in such a manner that the reinforcement and the concrete act together in resisting forces. Safety factor: The ratio of a load producing an undesirable state (such as collapse) and an expected or service load. Service loads: Loads on a structure with high probability of occurrence, such as dead weight supported by a member or the live loads specified in building codes and bridge specifications. Shear key: A recess or groove in a joint between successive lifts or placements of concrete, which is filled with concrete of the adjacent lift, giving shear strength to the joint. Shear span: The distance from a support of a simply supported beam to the nearest concentrated load. Shear wall: See Structural wall. Shotcrete: Mortar or concrete pneumatically projected at high velocity onto a surface. Silica fume: Very fine noncrystalline silica produced in electric arc furnaces as a byproduct of the production of metallic silicon and various silicon alloys (also know as condensed silica fume); used as a mineral admixture in concrete. Slab: A flat, horizontal (or neatly so) molded layer of plain or reinforced concrete, usually of uniform thickness, either on the ground or supported by beams, columns, walls, or other frame work. See also Flat slab. Slump: A measure of consistency of freshly mixed concrete equal to the subsidence of the molded specimen immediately after removal of the slump cone, expressed in inches. Splice: Connection of one reinforcing bar to another by lapping, welding, mechanical couplers, or other means. Split cylinder test: Test for tensile strength of concrete in which a standard cylinder is loaded to failure in diametral compression applied along the entire length (also called Brazilian test). Standard cylinder: Cylindric specimen of 12in. height and 6in. diameter, used to determine standard compressive strength and splitting tensile strength of concrete. Stiffness coefficient: The coefficient kij of stiffness matrix K for a multidegree of freedom structure is the force needed to hold the ith degree of freedom in place, if the jth degree of freedom undergoes a unit of displacement, while all others are locked in place. Stirrup: A reinforcement used to resist shear and diagonal tension stresses in a structural member; typically a steel bar bent into a U or rectangular shape and installed perpendicular to or at an angle to the longitudinal reinforcement, and properly anchored; the term “stirrup” is usually applied to lateral reinforcement in flexural members and the term “tie” to lateral reinforcement in compression members. See Tie. Strength design: See Ultimate strength design. 1999 by CRC Press LLC
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Strength reduction factor: Capacity reduction factor (typically designated as φ) by which the nominal strength of a member is to be multiplied to obtain the design strength; specified by the ACI Code for different types of members. Structural concrete: Concrete used to carry load or to form an integral part of a structure (opposite of, for example, insulating concrete). Tbeam: A beam composed of a stem and a flange in the form of a “T”, with the flange usually provided by a slab. Tension stiffening effect: The added stiffness of a single reinforcing bar due to the surrounding uncracked concrete between bond cracks. Tie: Reinforcing bar bent into a loop to enclose the longitudinal steel in columns; tensile bar to hold a form in place while resisting the lateral pressure of unhardened concrete. Ultimate strength design (USD): Design principle such that the actual (ultimate) strength of a member or structure, multiplied by a strength factor, is no less than the effects of all service load combinations, multiplied by respective overload factors. Unbonded tendon: A tendon that is not bonded to the concrete. Underreinforced beam: A beam with less than balanced reinforcement such that the reinforcement yields before the concrete crushes in compression. Watercement ratio: Ratio by weight of water to cement in a mixture; inversely proportional to concrete strength. Waterreducing admixture: An admixture capable of lowering the mix viscosity, thereby allowing a reduction of water (and increase in strength) without lowering the workability (also called superplasticizer). Whitney stress block: A rectangular area of uniform stress intensity 0.85fc0 , whose area and centroid are similar to that of the actual stress distribution in a flexural member near failure. Workability: General property of freshly mixed concrete that defines the ease with which it can be placed into forms without honeycombs; closely related to slump. Working stress design: See Allowable stress design. Yieldline theory: Method of structural analysis of plate structures at the verge of collapse under factored loads.
References [1] ACI Committee 318. 1992. Building Code Requirements for Reinforced Concrete and Commentary, ACI 31889 (Revised 92) and ACI 318R89 (Revised 92) (347pp.). Detroit, MI. [2] ACI Committee 340. 1990. Design Handbook in Accordance with the Strength Design Method of ACI 31889. Volume 2, SP17 (222 pp.). [3] ACI Committee 363. 1984. Stateoftheart report on high strength concrete. ACI J. Proc. 81(4):364411. [4] ACI Committee 436. 1996. Suggested design procedures for combined footings and mats. J. ACI. 63:10411057. [5] Breen, J.E. 1991. Why structural concrete? IASE Colloq. Struct. Concr. Stuttgart, pp.1526. 1999 by CRC Press LLC
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[6] Collins, M.P. and Mitchell, D. 1991. Prestressed Concrete Structures, 1st ed., Prentice Hall, Englewood Cliffs, N.J. [7] Fintel, M. 1985. Handbook of Concrete Engineering. 2nd ed., Van Nostrand Reinhold, New York. [8] MacGregor, J.G. 1992. Reinforced Concrete Mechanics and Design, 2nd ed., Prentice Hall, Englewood Cliffs, N.J. [9] Nilson, A.H. and Winter, G. 1992. Design of Concrete Structures, 11th ed., McGrawHill, New York. [10] Standard Handbook for Civil Engineers, 2nd ed., McGrawHill, New York. [11] Wang, C.K. and Salmon, C. G. 1985. Reinforced Concrete Design, 4th ed., Harper Row, New York.
Further Reading [1] ACI Committee 116. 1990. Cement and Concrete Terminology, Report 116R90, American Concrete Institute, Detroit, MI. [2] Ferguson, P.M., Breen, J.E., and Jirsa, J.O. 1988. Reinforced Concrete Fundamentals, 5th ed., John Wiley & Sons, New York. [3] Lin, TY. and Burns, N.H. 1981. Design of Prestressed Concrete Structures, 3rd ed., John Wiley & Sons, New York. [4] Meyer, C. 1996. Design of Concrete Structures, PrenticeHall, Upper Saddle River, NJ.
1999 by CRC Press LLC
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Scawthorn, C. “Earthquake Engineering” Structural Engineering Handbook Ed. Chen WaiFah Boca Raton: CRC Press LLC, 1999
Earthquake Engineering 5.1 5.2
5.3 5.4
Charles Scawthorn EQE International, San Francisco, California and Tokyo, Japan
5.1
Introduction Earthquakes
Causes of Earthquakes and Faulting • Distribution of Seismicity • Measurement of Earthquakes • Strong Motion Attenuation and Duration • Seismic Hazard and Design Earthquake • Effect of Soils on Ground Motion • Liquefaction and LiquefactionRelated Permanent Ground Displacement
Seismic Design Codes
Purpose of Codes • Historical Development of Seismic Codes • Selected Seismic Codes
Earthquake Effects and Design of Structures
Buildings • NonBuilding Structures
5.5 Defining Terms References Further Reading
Introduction
Earthquakes are naturally occurring broadbanded vibratory ground motions, caused by a number of phenomena including tectonic ground motions, volcanism, landslides, rockbursts, and humanmade explosions. Of these various causes, tectonicrelated earthquakes are the largest and most important. These are caused by the fracture and sliding of rock along faults within the Earth’s crust. A fault is a zone of the earth’s crust within which the two sides have moved — faults may be hundreds of miles long, from 1 to over 100 miles deep, and not readily apparent on the ground surface. Earthquakes initiate a number of phenomena or agents, termed seismic hazards, which can cause significant damage to the built environment — these include fault rupture, vibratory ground motion (i.e., shaking), inundation (e.g., tsunami, seiche, dam failure), various kinds of permanent ground failure (e.g., liquefaction), fire or hazardous materials release. For a given earthquake, any particular hazard can dominate, and historically each has caused major damage and great loss of life in specific earthquakes. The expected damage given a specified value of a hazard parameter is termed vulnerability, and the product of the hazard and the vulnerability (i.e., the expected damage) is termed the seismic risk. This is often formulated as Z E(D  H )p(H )dH ψ (5.1) E(D) = H
where Hψ p(·)ψ Dψ
= hazard = refers to probability = damage
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E(DH ) = vulnerability E(·) = the expected value operator Note that damage can refer to various parameters of interest, such as casualties, economic loss, or temporal duration of disruption. It is the goal of the earthquake specialist to reduce seismic risk. The probability of having a specific level of damage (i.e., p(D) = d) is termed the fragility. For most earthquakes, shaking is the dominant and most widespread agent of damage. Shaking near the actual earthquake rupture lasts only during the time when the fault ruptures, a process that takes seconds or at most a few minutes. The seismic waves generated by the rupture propagate long after the movement on the fault has stopped, however, spanning the globe in about 20 minutes. Typically earthquake ground motions are powerful enough to cause damage only in the near field (i.e., within a few tens of kilometers from the causative fault). However, in a few instances, long period motions have caused significant damage at great distances to selected lightly damped structures. A prime example of this was the 1985 Mexico City earthquake, where numerous collapses of mid and highrise buildings were due to a Magnitude 8.1 earthquake occurring at a distance of approximately 400 km from Mexico City. Ground motions due to an earthquake will vibrate the base of a structure such as a building. These motions are, in general, threedimensional, both lateral and vertical. The structure’s mass has inertia which tends to remain at rest as the structure’s base is vibrated, resulting in deformation of the structure. The structure’s load carrying members will try to restore the structure to its initial, undeformed, configuration. As the structure rapidly deforms, energy is absorbed in the process of material deformation. These characteristics can be effectively modeled for a single degree of freedom (SDOF) mass as shown in Figure 5.1 where m represents the mass of the structure, the elastic spring (of stiffness k = force / displacement) represents the restorative force of the structure, and the dashpot damping device (damping coefficient c = force/velocity) represents the force or energy lost in the process of material deformation. From the equilibrium of forces on mass m due to the spring and
FIGURE 5.1: Single degree of freedom (SDOF) system. dashpot damper and an applied load p(t), we find: mu¨ + cu˙ + ku = p(t)
(5.2)
the solution of which [32] provides relations between circular frequency of vibration ω, the natural frequency f , and the natural period T : $2 = f = 1999 by CRC Press LLC
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1 T
=
$ 2π
k m
=
1 2π
q
(5.3) k m
(5.4)
Damping tends to reduce the amplitude of vibrations. Critical damping refers to the value of damping such that free vibration of a structure will cease after one cycle (ccrit = 2mω). Damping is conventionally expressed as a percent of critical damping and, for most buildings and engineering structures, ranges from 0.5 to 10 or 20% of critical damping (increasing with displacement amplitude). Note that damping in this range will not appreciably affect the natural period or frequency of vibration, but does affect the amplitude of motion experienced.
5.2 5.2.1
Earthquakes Causes of Earthquakes and Faulting
In a global sense, tectonic earthquakes result from motion between a number of large plates comprising the earth’s crust or lithosphere (about 15 in total), (see Figure 5.2). These plates are driven by the convective motion of the material in the earth’s mantle, which in turn is driven by heat generated at the earth’s core. Relative plate motion at the fault interface is constrained by friction and/or asperities (areas of interlocking due to protrusions in the fault surfaces). However, strain energy accumulates in the plates, eventually overcomes any resistance, and causes slip between the two sides of the fault. This sudden slip, termed elastic rebound by Reid [101] based on his studies of regional deformation following the 1906 San Francisco earthquake, releases large amounts of energy, which constitutes the earthquake. The location of initial radiation of seismic waves (i.e., the first location of dynamic rupture) is termed the hypocenter, while the projection on the surface of the earth directly above the hypocenter is termed the epicenter. Other terminology includes nearfield (within one source dimension of the epicenter, where source dimension refers to the length or width of faulting, whichever is less), farfield (beyond nearfield), and meizoseismal (the area of strong shaking and damage). Energy is radiated over a broad spectrum of frequencies through the earth, in body waves and surface waves [16]. Body waves are of two types: P waves (transmitting energy via pushpull motion), and slower S waves (transmitting energy via shear action at right angles to the direction of motion). Surface waves are also of two types: horizontally oscillating Love waves (analogous to S body waves) and vertically oscillating Rayleigh waves. While the accumulation of strain energy within the plate can cause motion (and consequent release of energy) at faults at any location, earthquakes occur with greatest frequency at the boundaries of the tectonic plates. The boundary of the Pacific plate is the source of nearly half of the world’s great earthquakes. Stretching 40,000 km (24,000 miles) around the circumference of the Pacific Ocean, it includes Japan, the west coast of North America, and other highly populated areas, and is aptly termed the Ring of Fire. The interiors of plates, such as ocean basins and continental shields, are areas of low seismicity but are not inactive — the largest earthquakes known to have occurred in North America, for example, occurred in the New Madrid area, far from a plate boundary. Tectonic plates move very slowly and irregularly, with occasional earthquakes. Forces may build up for decades or centuries at plate interfaces until a large movement occurs all at once. These sudden, violent motions produce the shaking that is felt as an earthquake. The shaking can cause direct damage to buildings, roads, bridges, and other humanmade structures as well as triggering fires, landslides, tidal waves (tsunamis), and other damaging phenomena. Faults are the physical expression of the boundaries between adjacent tectonic plates and thus may be hundreds of miles long. In addition, there may be thousands of shorter faults parallel to or branching out from a main fault zone. Generally, the longer a fault the larger the earthquake it can generate. Beyond the main tectonic plates, there are many smaller subplates (“platelets”) and simple blocks of crust that occasionally move and shift due to the “jostling” of their neighbors and/or the major plates. The existence of these many subplates means that smaller but still damaging earthquakes are possible almost anywhere, although often with less likelihood. 1999 by CRC Press LLC
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FIGURE 5.2: Global seismicity and major tectonic plate boundaries.
Faults are typically classified according to their sense of motion (Figure 5.3). Basic terms include
FIGURE 5.3: Fault types.
transform or strike slip (relative fault motion occurs in the horizontal plane, parallel to the strike of the fault), dipslip (motion at right angles to the strike, up or downslip), normal (dipslip motion, two sides in tension, move away from each other), reverse (dipslip, two sides in compression, move towards each other), and thrust (lowangle reverse faulting). Generally, earthquakes will be concentrated in the vicinity of faults. Faults that are moving more rapidly than others will tend to have higher rates of seismicity, and larger faults are more likely than others to produce a large event. Many faults are identified on regional geological maps, and useful information on fault location and displacement history is available from local and national geological surveys in areas of high seismicity. Considering this information, areas of an expected large earthquake in the near future (usually measured in years or decades) can be and have been identified. However, earthquakes continue to occur on “unknown” or “inactive” faults. An important development has been the growing recognition of blind thrust faults, which emerged as a result of several earthquakes in the 1980s, none of which were accompanied by surface faulting [120]. Blind thrust faults are faults at depth occurring under anticlinal folds — since they have only subtle surface expression, their seismogenic potential can be evaluated by indirect means only [46]. Blind thrust faults are particularly worrisome because they are hidden, are associated with folded topography in general, including areas of lower and infrequent seismicity, and therefore result in a situation where the potential for an earthquake exists in any area of anticlinal geology, even if there are few or no earthquakes in the historic record. Recent major earthquakes of this type have included the 1980 Mw 7.3 El Asnam (Algeria), 1988 Mw 6.8 Spitak (Armenia), and 1994 Mw 6.7 Northridge (California) events. Probabilistic methods can be usefully employed to quantify the likelihood of an earthquake’s occurrence, and typically form the basis for determining the design basis earthquake. However, the earthquake generating process is not understood well enough to reliably predict the times, sizes, and 1999 by CRC Press LLC
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locations of earthquakes with precision. In general, therefore, communities must be prepared for an earthquake to occur at any time.
5.2.2
Distribution of Seismicity
This section discusses and characterizes the distribution of seismicity for the U.S. and selected areas. Global
It is evident from Figure 5.2 that some parts of the globe experience more and larger earthquakes than others. The two major regions of seismicity are the circumPacific Ring of Fire and the TransAlpide belt, extending from the western Mediterranean through the Middle East and the northern India subcontinent to Indonesia. The Pacific plate is created at its South Pacific extensional boundary — its motion is generally northwestward, resulting in relative strikeslip motion in California and New Zealand (with, however, a compressive component), and major compression and subduction in Alaska, the Aleutians, Kuriles, and northern Japan. Subduction refers to the plunging of one plate (i.e., the Pacific) beneath another, into the mantle, due to convergent motion, as shown in Figure 5.4.
FIGURE 5.4: Schematic diagram of subduction zone, typical of west coast of South America, Pacific Northwest of U.S., or Japan.
Subduction zones are typically characterized by volcanism, as a portion of the plate (melting in the lower mantle) reemerges as volcanic lava. Subduction also occurs along the west coast of South America at the boundary of the Nazca and South American plate, in Central America (boundary of the Cocos and Caribbean plates), in Taiwan and Japan (boundary of the Philippine and Eurasian plates), and in the North American Pacific Northwest (boundary of the Juan de Fuca and North American 1999 by CRC Press LLC
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plates). The TransAlpide seismic belt is basically due to the relative motions of the African and Australian plates colliding and subducting with the Eurasian plate. U.S.
Table 5.1 provides a list of selected U.S. earthquakes. The San Andreas fault system in California and the Aleutian Trench off the coast of Alaska are part of the boundary between the North American and Pacific tectonic plates, and are associated with the majority of U.S. seismicity (Figure 5.5 and Table 5.1). There are many other smaller fault zones throughout the western U.S. that are also helping to release the stress that is built up as the tectonic plates move past one another, (Figure 5.6). While California has had numerous destructive earthquakes, there is also clear evidence that the potential exists for great earthquakes in the Pacific Northwest [11].
FIGURE 5.5: U.S. seismicity. (From Algermissen, S. T., An Introduction to the Seismicity of the United States, Earthquake Engineering Research Institute, Oakland, CA, 1983. With permission. Also after Coffman, J. L., von Hake, C. A., and Stover, C. W., Earthquake History of the United States, U.S. Department of Commerce, NOAA, Pub. 411, Washington, 1980.)
On the east coast of the U.S., the cause of earthquakes is less well understood. There is no plate boundary and very few locations of active faults are known so that it is more difficult to assess where earthquakes are most likely to occur. Several significant historical earthquakes have occurred, such as in Charleston, South Carolina, in 1886, and New Madrid, Missouri, in 1811 and 1812, indicating that there is potential for very large and destructive earthquakes [56, 131]. However, most earthquakes in the eastern U.S. are smaller magnitude events. Because of regional geologic differences, eastern and central U.S. earthquakes are felt at much greater distances than those in the western U.S., sometimes up to a thousand miles away [58]. 1999 by CRC Press LLC
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TABLE 5.1
Selected U.S. Earthquakes
Yr
M
D
Lat.
Long.
M
MMI
1755
11
18
8
1774 1791
2 5
21 16
7 8
1811 1812 1812 1817 1836 1838 1857 1865 1868 1868 1872 1886 1892 1892 1892 1897
12 1 2 10 6 6 1 10 4 10 3 9 2 4 5 5
16 23 7 5 10 0 9 8 3 21 26 1 24 19 16 31
36 36.6 36.6
N N N
90 89.6 89.6
W W W
8.6 8.4 8.7
38 37.5 35 37 19 37.5 36.5 32.9 31.5 38.5 14
N N N N N N N N N N N
122 123 119 122 156 122 118 80 117 123 143
W W W W W W W W W W E
8.3 6.8 8.5 7.7 5.8
12 12 8 10 10 7 9 10 10 10 9 10 9 8
1899 1906
9 4
4 18
60 38
N N
142 123
W W
8.3 8.3
11
1915 1925 1927 1933 1934 1935 1940 1944 1949 1951 1952 1954 1957 1958 1959 1962 1964 1965 1971 1975 1975 1975 1980 1980 1980 1980 1983 1983 1983 1984 1986 1987 1987 1989 1989 1990 1992 1992 1992 1992 1992 1993 1993 1994 1994 1994 1995
10 6 11 3 12 10 5 9 4 8 7 12 3 7 8 8 3 4 2 3 8 11 1 5 7 11 5 10 11 4 7 10 11 6 10 2 4 4 6 6 6 3 9 1 1 2 10
3 29 4 11 31 19 19 5 13 21 21 16 9 10 18 30 28 29 9 28 1 29 24 25 27 8 2 28 16 24 8 1 24 26 18 28 23 25 28 28 29 25 21 16 17 3 6
40.5 34.3 34.5 33.6 31.8 46.6 32.7 44.7 47.1 19.7 35 39.3 51.3 58.6 44.8 41.8 61 47.4 34.4 42.1 39.4 19.3 37.8 37.6 38.2 41.2 36.2 43.9 19.5 37.3 34 34.1 33.2 19.4 37.1 34.1 34 40.4 34.2 34.2 36.7 45 42.3 40.3 34.2 42.8 65.2
N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N
118 120 121 118 116 112 116 74.7 123 156 119 118 176 137 111 112 148 122 118 113 122 155 122 119 83.9 124 120 114 155 122 117 118 116 155 122 118 116 124 117 116 116 123 122 76 119 111 149
W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W
7.8 6.2 7.5 6.3 7.1 6.2 7.1 5.6 7 6.9 7.7 7 8.6 7.9 7.7 5.8 8.3 6.5 6.7 6.2 6.1 7.2 5.9 6.4 5.2 7 6.5 7.3 6.6 6.2 6.1 6 6.3 6.1 7.1 5.5 6.3 7.1 6.7 7.6 5.6 5.6 5.9 4.6 6.8 6 6.4
9 10 10 8 11 10 7 11 8 9 7 7 7 8 8 7 7 8 6 6 9 7 7 8 8 9 7 7 5 9 7 
Fat.
USD mills
81 3 50 60 
5
700?
400
13
8
115
40
2 9 8
19 6 2 25
13
60 3
5 131 7 65 2 1 5 2 8 2
2 540 13 553 1 6 4 4 2 1 3 31 13 7 8 5 358 
62 
6,000 13
3
92
2

57
30,000
66
Locale Nr Cape Ann, MA (MMI from STA) Eastern VA (MMI from STA) E. Haddam, CT (MMI from STA) New Madrid, MO New Madrid, MO New Madrid, MO Woburn, MA (MMI from STA) California California San Francisco, CA San Jose, Santa Cruz, CA Hawaii Hayward, CA Owens Valley, CA Charleston, SC, Ms from STA San Diego County, CA Vacaville, Winters, CA Agana, Guam Giles County, VA (mb from STA) Cape Yakataga, AK San Francisco, CA (deaths more?) Pleasant Valley, NV Santa Barbara, CA Lompoc, Port San Luis, CA Long Beach, CA Baja, Imperial Valley, CA Helena, MT SE of Elcentro, CA Massena, NY Olympia, WA Hawaii Central, Kern County, CA Dixie Valley, NV Alaska Lituyabay, AK—Landslide Hebgen Lake, MT Utah Alaska Seattle, WA San Fernando, CA Pocatello Valley, ID Oroville Reservoir, CA Hawaii Livermore, CA Mammoth Lakes, CA Maysville, KY N Coast, CA Central, Coalinga, CA Borah Peak, ID Kapapala, HI Central Morgan Hill, CA Palm Springs, CA Whittier, CA Superstition Hills, CA Hawaii Loma Prieta, CA Claremont, Covina, CA Joshua Tree, CA Humboldt, Ferndale, CA Big Bear Lake, Big Bear, CA Landers, Yucca, CA Border of NV and CA WashingtonOregon Klamath Falls, OR PA, Felt, Canada Northridge, CA Afton, WY AK (Oil pipeline damaged)
Note: STA refers to [3]. From NEIC, Database of Significant Earthquakes Contained in Seismicity Catalogs, National Earthquake Information Center, Goldon, CO, 1996. With permission.
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FIGURE 5.6: Seismicity for California and Nevada, 1980 to 1986. M >1.5 (Courtesy of Jennings, C. W., Fault Activity Map of California and Adjacent Areas, Department of Conservation, Division of Mines and Geology, Sacramento, CA, 1994.)
Other Areas
Table 5.2 provides a list of selected 20thcentury earthquakes with fatalities of approximately 10,000 or more. All the earthquakes are in the TransAlpide belt or the circumPacific ring of fire, and the great loss of life is almost invariably due to lowstrength masonry buildings and dwellings. Exceptions to this rule are the 1923 Kanto (Japan) earthquake, where most of the approximately 140,000 fatalities were due to fire; the 1970 Peru earthquake, where large landslides destroyed whole towns; and the 1988 Armenian earthquake, where large numbers were killed in Spitak and Leninakan due to poor quality precast concrete construction. The TransAlpide belt includes the Mediterranean, which has very significant seismicity in North Africa, Italy, Greece, and Turkey due to the Africa plate’s motion relative to the Eurasian plate; the Caucasus (e.g., Armenia) and the Middle East (Iran, Afghanistan), due to the Arabian plate being forced northeastward into the Eurasian plate by the African plate; and the Indian subcontinent (Pakistan, northern India), and the subduction boundary along the southwestern side of Sumatra and Java, which are all part of the IndianAustralian 1999 by CRC Press LLC
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plate. Seismicity also extends northward through Burma and into western China. The Philippines, Taiwan, and Japan are all on the western boundary of the Philippines sea plate, which is part of the circumPacific ring of fire. Japan is an island archipelago with a long history of damaging earthquakes [128] due to the interaction of four tectonic plates (Pacific, Eurasian, North American, and Philippine) which all converge near Tokyo. Figure 5.7 indicates the pattern of Japanese seismicity, which is seen to be higher in the north of Japan. However, central Japan is still an area of major seismic risk, particularly Tokyo,
FIGURE 5.7: Japanese seismicity (1960 to 1965).
which has sustained a number of damaging earthquakes in history. The Great Kanto earthquake of 1923 (M7.9, about 140,000 fatalities) was a great subduction earthquake, and the 1855 event (M6.9) had its epicenter in the center of presentday Tokyo. Most recently, the 1995 MW 6.9 Hanshin (Kobe) earthquake caused approximately 6,000 fatalities and severely damaged some modern structures as well as many structures built prior to the last major updating of the Japanese seismic codes (ca. 1981). 1999 by CRC Press LLC
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The predominant seismicity in the Kuriles, Kamchatka, the Aleutians, and Alaska is due to subduction of the Pacific Plate beneath the North American plate (which includes the Aleutians and extends down through northern Japan to Tokyo). The predominant seismicity along the western boundary of North American is due to transform faults (i.e., strikeslip) as the Pacific Plate displaces northwestward relative to the North American plate, although the smaller Juan de Fuca plate offshore Washington and Oregon, and the still smaller Gorda plate offshore northern California, are driven into subduction beneath North American by the Pacific Plate. Further south, the Cocos plate is similarly subducting beneath Mexico and Central America due to the Pacific Plate, while the Nazca Plate lies offshore South America. Lesser but still significant seismicity occurs in the Caribbean, primarily along a series of trenches north of Puerto Rico and the Windward islands. However, the southern boundary of the Caribbean plate passes through Venezuela, and was the source of a major earthquake in Caracas in 1967. New Zealand’s seismicity is due to a major plate boundary (Pacific with IndianAustralian plates), which transitions from thrust to transform from the South to the North Island [108]. Lesser but still significant seismicity exists in Iceland where it is accompanied by volcanism due to a spreading boundary between the North American and Eurasian plates, and through FennoScandia, due to tectonics as well as glacial rebound. This very brief tour of the major seismic belts of the globe is not meant to indicate that damaging earthquakes cannot occur elsewhere — earthquakes can and have occurred far from major plate boundaries (e.g., the 18111812 New Madrid intraplate events, with several being greater than magnitude 8), and their potential should always be a consideration in the design of a structure. TABLE 5.2
Selected 20th Century Earthquakes with Fatalities Greater than 10,000
Yr
M
D
Lat.
Long.
M
MMI
Deaths
1976 1920 1923 1908 1932 1970 1990 1927 1915 1935 1939 1939 1978 1988 1976 1974 1948 1905 1917 1968 1962 1907 1960 1980 1934 1918 1933 1975
7 12 9 2 12 5 6 5 1 5 12 1 9 12 2 5 10 4 1 8 9 10 2 10 1 2 8 2
27 16 1 0 25 31 20 22 13 30 26 25 16 7 4 10 5 4 21 31 1 21 29 10 15 13 25 4
39.5 N 36.5 N 35.3 N 38.2 N 39.2 N 9.1 S 37 N 37.6 N 41.9 N 29.5 N 39.5 N 36.2 S 33.4 N 41 N 15.3 N 28.2 N 37.9 N 33 N 8S 33.9 N 35.6 N 38.5 N 30.4 N 36.1 N 26.5 N 23.5 N 32 N 40.6 N
118 E 106 E 140 E 15.6 E 96.5 E 78.8 W 49.4 E 103 E 13.6 E 66.8 E 38.5 E 72.2 W 57.5 E 44.2 E 89.2 W 104 E 58.6 E 76 E 115 E 59 E 49.9 E 67.9 E 9.6 W 1.4 E 86.5 E 117 E 104 E 123 E
8 8.5 8.2 7.5 7.6 7.8 7.7 8 7 7.5 7.9 8.3 7.4 6.8 7.5 6.8 7.2 8.6 — 7.3 7.3 7.8 5.9 7.7 8.4 7.3 7.4 7.4
10 — — — — 9 7 — 11 10 12 — — 10 9 — — — — — — 9 — — — — — 10
655,237 200,000 142,807 75,000 70,000 67,000 50,000 40,912 35,000 30,000 30,000 28,000 25,000 25,000 22,400 20,000 19,800 19,000 15,000 15,000 12,225 12,000 12,000 11,000 10,700 10,000 10,000 10,000
Damage USD millions $2,000 $2,800 $500
$100 $11 $16,200 $6,000
Locale China: NE: Tangshan China: Gansu and Shanxi Japan: Toyko, Yokohama, Tsunami Italy: Sicily China: Gansu Province Peru Iran: Manjil China: Gansu Province Italy: Abruzzi, Avezzano Pakistan: Quetta Turkey: Erzincan Chile: Chillan Iran: Tabas CIS: Armenia Guatemala: Tsunami China: Yunnan and Sichuan CIS: Turkmenistan: Aschabad India: Kangra Indonesia: Bali, Tsunami Iran Iran: NW CIS: Uzbekistan: SE Morocco: Agadir Algeria: Elasnam NepalIndia China: Guangdong Province China: Sichuan Province China: NE: Yingtao
From NEIC, Database of Significant Earthquakes Contained in Seismicity Catalogs, National Earthquake Information Center, Goldon, CO, 1996.
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5.2.3
Measurement of Earthquakes
Earthquakes are complex multidimensional phenomena, the scientific analysis of which requires measurement. Prior to the invention of modern scientific instruments, earthquakes were qualitatively measured by their effect or intensity, which differed from pointtopoint. With the deployment of seismometers, an instrumental quantification of the entire earthquake event — the unique magnitude of the event — became possible. These are still the two most widely used measures of an earthquake, and a number of different scales for each have been developed, which are sometimes confused.1 Engineering design, however, requires measurement of earthquake phenomena in units such as force or displacement. This section defines and discusses each of these measures. Magnitude
An individual earthquake is a unique release of strain energy. Quantification of this energy has formed the basis for measuring the earthquake event. Richter [103] was the first to define earthquake magnitude as (5.5) ML = log A − log Ao where ML is local magnitude (which Richter only defined for Southern California), A is the maximum trace amplitude in microns recorded on a standard WoodAnderson shortperiod torsion seismometer,2 at a site 100 km from the epicenter, log Ao is a standard value as a function of distance, for instruments located at distances other than 100 km and less than 600 km. Subsequently, a number of other magnitudes have been defined, the most important of which are surface wave magnitude MS , body wave magnitude mb , and moment magnitude MW . Due to the fact that ML was only locally defined for California (i.e., for events within about 600 km of the observing stations), surface wave magnitude MS was defined analogously to ML using teleseismic observations of surface waves of 20s period [103]. Magnitude, which is defined on the basis of the amplitude of ground displacements, can be related to the total energy in the expanding wave front generated by an earthquake, and thus to the total energy release. An empirical relation by Richter is log10 Es = 11.8 + 1.5Ms
(5.6)
where Es is the total energy in ergs.3 Note that 101.5 = 31.6, so that an increase of one magnitude unit is equivalent to 31.6 times more energy release, two magnitude units increase is equivalent to 1000 times more energy, etc. Subsequently, due to the observation that deepfocus earthquakes commonly do not register measurable surface waves with periods near 20 s, a body wave magnitude mb was defined [49], which can be related to Ms [38]: mb = 2.5 + 0.63Ms
(5.7)
Body wave magnitudes are more commonly used in eastern North America, due to the deeper earthquakes there. A number of other magnitude scales have been developed, most of which tend to saturate — that is, asymptote to an upper bound due to larger earthquakes radiating significant amounts of energy at periods longer than used for determining the magnitude (e.g., for Ms , defined by
1 Earthquake magnitude and intensity are analogous to a lightbulb and the light it emits. A particular lightbulb has only one energy level, or wattage (e.g., 100 watts, analogous to an earthquake’s magnitude). Near the lightbulb, the light intensity is very bright (perhaps 100 ftcandles, analogous to MMI IX), while farther away the intensity decreases (e.g., 10 ftcandles, MMI V). A particular earthquake has only one magnitude value, whereas it has many intensity values. 2 The instrument has a natural period of 0.8 s, critical damping ration 0.8, magnification 2,800. 3 Richter [104] gives 11.4 for the constant term, rather than 11.8, which is based on subsequent work. The uncertainty in the data make this difference, equivalent to an energy factor = 2.5 or 0.27 magnitude units, inconsequential.
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measuring 20 s surface waves, saturation occurs at about Ms > 7.5). More recently, seismic moment has been employed to define a moment magnitude Mw ( [53]; also denoted as boldface M) which is finding increased and widespread use: log Mo = 1.5Mw + 16.0
(5.8)
where seismic moment Mo (dynecm) is defined as [74] Mo = µAu¯
(5.9)
where µ is the material shear modulus, A is the area of fault plane rupture, and u¯ is the mean relative displacement between the two sides of the fault (the averaged fault slip). Comparatively, Mw and Ms are numerically almost identical up to magnitude 7.5. Figure 5.8 indicates the relationship between moment magnitude and various magnitude scales.
FIGURE 5.8: Relationship between moment magnitude and various magnitude scales. (From Campbell, K. W., Strong Ground Motion Attenuation Relations: A TenYear Perspective, Earthquake Spectra, 1(4), 759804, 1985. With permission.)
For lay communications, it is sometimes customary to speak of great earthquakes, large earthquakes, etc. There is no standard definition for these, but the following is an approximate categorization: Earthquake Magnitude∗
Micro Not felt
Small 8
∗ Not specifically defined.
From the foregoing discussion, it can be seen that magnitude and energy are related to fault rupture length and slip. Slemmons [114] and Bonilla et al. [17] have determined statistical relations between these parameters, for worldwide and regional data sets, segregated by type of faulting (normal, reverse, strikeslip). The worldwide results of Bonilla et al. for all types of faults are Ms = 6.04 + 0.708 log10 L s = .306 log10 L = −2.77 + 0.619Ms s = .286 1999 by CRC Press LLC
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(5.10) (5.11)
Ms = log10 d =
6.95 + 0.723 log10 d −3.58 + 0.550Ms
s = .323 s = .282
(5.12) (5.13)
which indicates, for example that, for Ms = 7, the average fault rupture length is about 36 km (and the average displacement is about 1.86 m). Conversely, a fault of 100 km length is capable of about a Ms = 7.54 event. More recently, Wells and Coppersmith [130] have performed an extensive analysis of a dataset of 421 earthquakes. Their results are presented in Table 5.3a and b. Intensity
In general, seismic intensity is a measure of the effect, or the strength, of an earthquake hazard at a specific location. While the term can be applied generically to engineering measures such as peak ground acceleration, it is usually reserved for qualitative measures of locationspecific earthquake effects, based on observed human behavior and structural damage. Numerous intensity scales were developed in preinstrumental times. The most common in use today are the Modified Mercalli Intensity (MMI) [134], RossiForel (RF), MedvedevSponheurKarnik (MSK) [80], and the Japan Meteorological Agency (JMA) [69] scales. MMI is a subjective scale defining the level of shaking at specific sites on a scale of I to XII. (MMI is expressed in Roman numerals to connote its approximate nature). For example, moderate shaking that causes few instances of fallen plaster or cracks in chimneys constitutes MMI VI. It is difficult to find a reliable relationship between magnitude, which is a description of the earthquake’s total energy level, and intensity, which is a subjective description of the level of shaking of the earthquake at specific sites, because shaking severity can vary with building type, design and construction practices, soil type, and distance from the event. Note that MMI X is the maximum considered physically possible due to “mere” shaking, and that MMI XI and XII are considered due more to permanent ground deformations and other geologic effects than to shaking. Other intensity scales are defined analogously (see Table 5.5, which also contains an approximate conversion from MMI to acceleration a [PGA, in cm/s2 , or gals]). The conversion is due to Richter [103] (other conversions are also available [84]. log a = MMI/3 − 1/2
(5.14)
Intensity maps are produced as a result of detailed investigation of the type of effects tabulated in Table 5.4, as shown in Figure 5.9 for the 1994 MW 6.7 Northridge earthquake. Correlations have been developed between the area of various MMIs and earthquake magnitude, which are of value for seismological and planning purposes. Figure 10 correlates Af elt vs. MW . For preinstrumental historical earthquakes, Af elt can be estimated from newspapers and other reports, which then can be used to estimate the event magnitude, thus supplementing the seismicity catalog. This technique has been especially useful in regions with a long historical record [4, 133]. Time History
Sensitive strong motion seismometers have been available since the 1930s, and they record actual ground motions specific to their location (Figure 5.11). Typically, the ground motion records, termed seismographs or time histories, have recorded acceleration (these records are termed accelerograms),
4 Note that L = g(M ) should not be inverted to solve for M = f (L), as a regression for y = f (x) is different than a s s
regression for x = g(y).
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1999 by CRC Press LLC
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Table 5.3a Regressions of Rupture Length, Rupture Width, Rupture Area and Moment Magnitude Equationa M = a + b ∗ log(SRL)
log(SRL) = a + b∗ M
M = a + b ∗ log(RLD)
log(RLD) = a + b∗ M
M = a + b ∗ log(RW )
log(RW ) = a + b∗ M
M = a + b ∗ log(RA)
log(RA) = a + b∗ M
Slip typeb
Number of events
SS R N All SS R N All SS R N All SS R N All SS R N All SS R N All SS R N All SS R N All
43 19 15 77 43 19 15 77 93 50 24 167 93 50 24 167 87 43 23 153 87 43 23 153 83 43 22 148 83 43 22 148
Coefficients and standard errors a (sa) b(sb) 5.16(0.13) 5.00(0.22) 4.86(0.34) 5.08(0.10) −3.55(0.37) −2.86(0.55) −2.01(0.65) −3.22(0.27) 4.33(0.06) 4.49(0.11) 4.34(0.23) 4.38(0.06) −2.57(0.12) −2.42(0.21) −1.88(0.37) −2.44(0.11) 3.80(0.17) 4.37(0.16) 4.04(0.29) 4.06(0.11) −0.76(0.12) −1.61(0.20) −1.14(0.28) −1.01(0.10) 3.98(0.07) 4.33(0.12) 3.93(0.23) 4.07(0.06) −3.42(0.18) −3.99(0.36) −2.87(0.50) −3.49(0.16)
1.12(0.08) 1.22(0.16) 1.32(0.26) 1.16(0.07) 0.74(0.05) 0.63(0.08) 0.50(0.10) 0.69(0.04) 1.49(0.05) 1.49(0.09) 1.54(0.18) 1.49(0.04) 0.62(0.02) 0.58(0.03) 0.50(0.06) 0.59(0.02) 2.59(0.18) 1.95(0.15) 2.11(0.28) 2.25(0.12) 0.27(0.02) 0.41(0.03) 0.35(0.05) 0.32(0.02) 1.02(0.03) 0.90(0.05) 1.02(0.10) 0.98(0.03) 0.90(0.03) 0.98(0.06) 0.82(0.08) 0.91(0.03)
Standard deviation s
Correlation coefficient r
Magnitude range
0.28 0.28 0.34 0.28 0.23 0.20 0.21 0.22 0.24 0.26 0.31 0.26 0.15 0.16 0.17 0.16 0.45 0.32 0.31 0.41 0.14 0.15 0.12 0.15 0.23 0.25 0.25 0.24 0.22 0.26 0.22 0.24
0.91 0.88 0.81 0.89 0.91 0.88 0.81 0.89 0.96 0.93 0.88 0.94 0.96 0.93 0.88 0.94 0.84 0.90 0.86 0.84 0.84 0.90 0.86 0.84 0.96 0.94 0.92 0.95 0.96 0.94 0.92 0.95
5.6 to 8.1 5.4 to 7.4 5.2 to 7.3 5.2 to 8.1 5.6 to 8.1 5.4 to 7.4 5.2 to 7.3 5.2 to 8.1 4.8 to 8.1 4.8 to 7.6 5.2 to 7.3 4.8 to 8.1 4.8 to 8.1 4.8 to 7.6 5.2 to 7.3 4.8 to 8.1 4.8 to 8.1 4.8 to 7.6 5.2 to 7.3 4.8 to 8.1 4.8 to 8.1 4.8 to 7.6 5.2 to 7.3 4.8 to 8.1 4.8 to 7.9 4.8 to 7.6 5.2 to 7.3 4.8 to 7.9 4.8 to 7.9 4.8 to 7.6 5.2 to 7.3 4.8 to 7.9
Length/width range (km) 1.3 to 432 3.3 to 85 2.5 to 41 1.3 to 432 1.3 to 432 3.3 to 85 2.5 to 41 1.3 to 432 1.5 to 350 1.1 to 80 3.8 to 63 1.1 to 350 1.5 to 350 1.1 to 80 3.8 to 63 1.1 to 350 1.5 to 350 1.1 to 80 3.8 to 63 1.1 to 350 1.5 to 350 1.1 to 80 3.8 to 63 1.1 to 350 3 to 5,184 2.2 to 2,400 19 to 900 2.2 to 5,184 3 to 5,184 2.2 to 2,400 19 to 900 2.2 to 5,184
a SRL—surface rupture length (km); RLD —subsurface rupture length (km); RW —downdip rupture width (km); RA—rupture area (km2 ). b SS—strike slip; R—reverse; N—normal.
From Wells, D. L. and Coopersmith, K. J., Empirical Relationships Among Magnitude, Rupture Length, Rupture Width, Rupture Area and Surface Displacements, Bull. Seis. Soc. Am., 84(4), 9741002, 1994. With permission.
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Table 5.3b Regressions of Displacement and Moment Magnitude Equationa M = a + b ∗ log(MD)
log(MD) = a + b∗ M
M = a + b ∗ log(AD)
log(AD) = a + b∗ M
Slip typeb
Number of events
SS { Rc N All SS {R N All SS {R N All SS {R N All
43 21 16 80 43 21 16 80 29 15 12 56 29 15 12 56
Coefficients and standard errors a (sa) b(sb) 6.81(0.05) 6.52(0.11) 6.61(0.09) 6.69(0.04) −7.03(0.55) − 1.84(1.14) −5.90(1.18) −5.46(0.51) 7.04(0.05) 6.64(0.16) 6.78(0.12) 6.93(0.05) −6.32(0.61) − 0.74(1.40) −4.45(1.59) −4.80(0.57)
0.78(0.06) 0.44(0.26) 0.71(0.15) 0.74(0.07) 1.03(0.08) 0.29(0.17) 0.89(0.18) 0.82(0.08) 0.89(0.09) 0.13(0.36) 0.65(0.25) 0.82(0.10) 0.90(0.09) 0.08(0.21) 0.63(0.24) 0.69(0.08)
Standard deviation s
Correlation coefficient r
Magnitude range
0.29 0.52 0.34 0.40 0.34 0.42 0.38 0.42 0.28 0.50 0.33 0.39 0.28 0.38 0.33 0.36
0.90 0.36 0.80 0.78 0.90 0.36 0.80 0.78 0.89 0.10 0.64 0.75 0.89 0.10 0.64 0.75
5.6 to 8.1 5.4 to 7.4 5.2 to 7.3 5.2 to 8.1 5.6 to 8.1 5.4 to 7.4 5.2 to 7.3 5.2 to 8.1 5.6 to 8.1 5.8 to 7.4 6.0 to 7.3 5.6 to 8.1 5.6 to 8.1 5.8 to 7.4 6.0 to 7.3 5.6 to 8.1
Displacement range (km) 0.01 to 14.6 0.11 to 6.5 } 0.06 to 6.1 0.01 to 14.6 0.01 to 14.6 0.11 to 6.5 } 0.06 to 6.1 0.01 to 14.6 0.05 to 8.0 0.06 to 1.5 } 0.08 to 2.1 0.05 to 8.0 0.05 to 8.0 0.06 to 1.5 } 0.08 to 2.1 0.05 to 8.0
a MD —maximum displacement (m); AD —average displacement (M).
b SS—strike slip; R—reverse; N—normal. c Regressions for reverseslip relationships shown in italics and brackets are not significant at a 95% probability level.
From Wells, D. L. and Coopersmith, K. J., Empirical Relationships Among Magnitude, Rupture Length, Rupture Width, Rupture Area and Surface Displacements, Bull. Seis. Soc. Am., 84(4), 9741002, 1994. With permission.
TABLE 5.4 I II III IV V VI VII VIII
IX X XI XII
Modified Mercalli Intensity Scale of 1931
Not felt except by a very few under especially favorable circumstances. Felt only by a few persons at rest, especially on upper floors of buildings. Delicately suspended objects may swing. Felt quite noticeably indoors, especially on upper floors of buildings, but many people do not recognize it as an earthquake. Standing motor cars may rock slightly. Vibration like passing truck. Duration estimated. During the day felt indoors by many, outdoors by few. At night some awakened. Dishes, windows, and doors disturbed; walls make creaking sound. Sensation like heavy truck striking building. Standing motor cars rock noticeably. Felt by nearly everyone; many awakened. Some dishes, windows, etc. broken; a few instances of cracked plaster; unstable objects overturned. Disturbance of trees, poles, and other tall objects sometimes noticed. Pendulum clocks may stop. Felt by all; many frightened and run outdoors. Some heavy furniture moved; a few instances of fallen plaster or damaged chimneys. Damage slight. Everybody runs outdoors. Damage negligible in buildings of good design and construction slight to moderate in well built ordinary structures; considerable in poorly built or badly designed structures. Some chimneys broken. Noticed by persons driving motor cars. Damage slight in specially designed structures; considerable in ordinary substantial buildings, with partial collapse; great in poorly built structures. Panel walls thrown out of frame structures. Fall of chimneys, factory stacks, columns, monuments, walls. Heavy furniture overturned. Sand and mud ejected in small amounts. Changes in well water. Persons driving motor cars disturbed. Damage considerable in specially designed structures; welldesigned frame structures thrown out of plumb; great in substantial buildings, with partial collapse. Buildings shifted off foundations. Ground cracked conspicuously. Underground pipes broken. Some wellbuilt wooden structures destroyed; most masonry and frame structures destroyed with foundations; ground badly cracked. Rails bent. Landslides considerable from river banks and steep slopes. Shifted sand and mud. Water splashed over banks. Few, if any (masonry), structures remain standing. Bridges destroyed. Broad fissures in ground. Underground pipelines completely out of service. Earth slumps and land slips in soft ground. Rails bent greatly. Damage total. Waves seen on ground surfaces. Lines of sight and level distorted. Objects thrown upward into the air.
After Wood, H. O. and Neumann, Fr., Modified Mercalli Intensity Scale of 1931, Bull. Seis. Soc. Am., 21, 277283, 1931.
TABLE 5.5 Comparison of Modified Mercalli (MMI) and Other Intensity Scales aa
MMIb
RFc
MSKd
JMAe
0.7 1.5 3 7 15 32 68 147 316 681 (1468)f (3162)f
I II III IV V VI VII VIII IX X XI XII
I I to II III IV to V V to VI VI to VII VIIIVIII+ to IX− IX+ X — —
I II III IV V VI VII VIII IX X XI XII
0 I II II to III III IV IV to V V V to VI VI VII
a gals b Modified Mercalli Intensity c RossiForel d MedvedevSponheurKarnik e Japan Meteorological Agency
f a values provided for reference only. MMI > X are due more
to geologic effects.
for many years in analog form on photographic film and, more recently, digitally. Analog records required considerable effort for correction due to instrumental drift, before they could be used. Time histories theoretically contain complete information about the motion at the instrumental location, recording three traces or orthogonal records (two horizontal and one vertical). Time histories (i.e., the earthquake motion at the site) can differ dramatically in duration, frequency content, and amplitude. The maximum amplitude of recorded acceleration is termed the peak ground acceleration, PGA (also termed the ZPA, or zero period acceleration). Peak ground velocity (PGV) and peak ground displacement (PGD) are the maximum respective amplitudes of velocity and displacement. Acceleration is normally recorded, with velocity and displacement being determined by numerical integration; however, velocity and displacement meters are also deployed, to a lesser extent. Accel1999 by CRC Press LLC
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FIGURE 5.9: MMI maps, 1994 MW 6.7 Northridge Earthquake. (1) Farfield isoseismal map. Roman numerals give average MMI for the regions between isoseismals; arabic numerals represent intensities in individual communities. Squares denote towns labeled in the figure. Box labeled “FIG. 2” identifies boundaries of that figure. (2) Distribution of MMI in the epicentral region. (Courtesy of Dewey, J.W. et al., Spacial Variations of Intensity in the Northridge Earthquake, in Woods, M.C. and Seiple, W.R., Eds., The Northridge California Earthquake of 17 January 1994, California Department of Conservation, Division of Mines and Geology, Special Publication 116, 3946, 1995.) eration can be expressed in units of cm/s2 (termed gals), but is often also expressed in terms of the fraction or percent of the acceleration of gravity (980.66 gals, termed 1g). Velocity is expressed in cm/s (termed kine). Recent earthquakes (1994 Northridge, Mw 6.7 and 1995 Hanshin [Kobe] Mw 6.9) have recorded PGA’s of about 0.8g and PGV’s of about 100 kine — almost 2g was recorded in the 1992 Cape Mendocino earthquake. Elastic Response Spectra
If the SDOF mass in Figure 5.1 is subjected to a time history of ground (i.e., base) motion similar to that shown in Figure 5.11, the elastic structural response can be readily calculated as a function of time, generating a structural response time history, as shown in Figure 5.12 for several oscillators with differing natural periods. The response time history can be calculated by direct integration of Equation 5.1 in the time domain, or by solution of the Duhamel integral [32]. However, this is timeconsuming, and the elastic response is more typically calculated in the frequency domain 1 v(t) = 2π
Z
∞
$ =−∞
H ($ )c($ ) exp(i$ t)d$
(5.15)
where v(t) = the elastic structural displacement response time history $ = frequency 1 is the complex frequency response function H ($ ) = −$ 2 m+ic+k R∞ c($ ) = $ =−∞ p(t) exp(−i$ t)dt is the Fourier transform of the input motion (i.e., the Fourier transform of the ground motion time history) which takes advantage of computational efficiency using the Fast Fourier Transform. 1999 by CRC Press LLC
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FIGURE 5.10: log Afelt (km2 ) vs. MW . Solid circles denote ENA events and open squares denote California earthquakes. The dashed curve is the MW − Afelt relationship of an earlier study, whereas the solid line is the fit determined by Hanks and Johnston, for California data. (Courtesy of Hanks J. W. and Johnston A. C., Common Features of the Excitation and Propagation of Strong Ground Motion for North American Earthquakes, Bull. Seis. Soc. Am., 82(1), 123, 1992.)
FIGURE 5.11: Typical earthquake accelerograms. (Courtesy of Darragh, R. B., Huang, M. J., and Shakal, A. F., Earthquake Engineering Aspects of Strong Motion Data from Recent California Earthquakes, Proc. Fifth U.S. Natl. Conf. Earthquake Eng., 3, 99108, 1994, Earthquake Engineering Research Institute. Oakland, CA.)
For design purposes, it is often sufficient to know only the maximum amplitude of the response time history. If the natural period of the SDOF is varied across a spectrum of engineering interest (typically, for natural periods from .03 to 3 or more seconds, or frequencies of 0.3 to 30+ Hz), then the plot of these maximum amplitudes is termed a response spectrum. Figure 5.12 illustrates this process, resulting in Sd , the displacement response spectrum, while Figure 5.13 shows (a) the Sd , 1999 by CRC Press LLC
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FIGURE 5.12: Computation of deformation (or displacement) response spectrum. (From Chopra, A. K., Dynamics of Structures, A Primer, Earthquake Engineering Research Institute, Oakland, CA, 1981. With permission.) displacement response spectrum, (b) Sv , the velocity response spectrum (also denoted PSV, the pseudo spectral velocity, pseudo to emphasize that this spectrum is not exactly the same as the relative velocity response spectrum [63], and (c) Sa , the acceleration response spectrum. Note that Sv =
2π Sd = $ Sd T
and Sa =
2π Sv = $ Sv = T
2π T
(5.16)
2 Sd = $ 2 Sd
(5.17)
Response spectra form the basis for much modern earthquake engineering structural analysis and design. They are readily calculated if the ground motion is known. For design purposes, however, response spectra must be estimated. This process is discussed below. Response spectra may be plotted in any of several ways, as shown in Figure 5.13 with arithmetic axes, and in Figure 5.14 where the 1999 by CRC Press LLC
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FIGURE 5.13: Response spectra spectrum. (From Chopra, A. K., Dynamics of Structures, A Primer, Earthquake Engineering Research Institute, Oakland, CA, 1981. With permission.)
velocity response spectrum is plotted on tripartite logarithmic axes, which equally enables reading of displacement and acceleration response. Response spectra are most normally presented for 5% of critical damping. While actual response spectra are irregular in shape, they generally have a concavedown arch or trapezoidal shape, when plotted on tripartite log paper. Newmark observed that response spectra tend to be characterized by three regions: (1) a region of constant acceleration, in the high frequency portion of the spectra; (2) constant displacement, at low frequencies; and (3) constant velocity, at intermediate frequencies, as shown in Figure 5.15. If a spectrum amplification factor is defined as the ratio of the spectral parameter to the ground motion parameter (where parameter indicates acceleration, velocity or displacement), then response spectra can be estimated from the data in Table 5.6, provided estimates of the ground motion parameters are available. An example spectra using these data is given in Figure 5.15. A standardized response spectra is provided in the Uniform Building Code [126] for three soil types. The spectra is a smoothed average of normalized 5% damped spectra obtained from actual ground 1999 by CRC Press LLC
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FIGURE 5.14: Response spectra, tripartite plot (El Centro S 0◦ E component). (From Chopra, A. K., Dynamics of Structures, A Primer, Earthquake Engineering Research Institute, Oakland, CA, 1981. With permission.)
motion records grouped by subsurface soil conditions at the location of the recording instrument, and are applicable for earthquakes characteristic of those that occur in California [111]. If an estimate of ZPA is available, these normalized shapes may be employed to determine a response spectra, appropriate for the soil conditions. Note that the maximum amplification factor is 2.5, over a period range approximately 0.15 s to 0.4  0.9 s, depending on the soil conditions. Other methods for estimation of response spectra are discussed below. 1999 by CRC Press LLC
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FIGURE 5.15: Idealized elastic design spectrum, horizontal motion (ZPA = 0.5g, 5% damping, one sigma cumulative probability. (From Newmark, N. M. and Hall, W. J., Earthquake Spectra and Design, Earthquake Engineering Research Institute, Oakland, CA, 1982. With permission.) TABLE 5.6 Spectrum Amplification Factors for Horizontal Elastic Response Damping,
One sigma (84.1%)
Median (50%)
% Critical
A
V
D
A
V
D
0.5 1 2 3 5 7 10 20
5.10 4.38 3.66 3.24 2.71 2.36 1.99 1.26
3.84 3.38 2.92 2.64 2.30 2.08 1.84 1.37
3.04 2.73 2.42 2.24 2.01 1.85 1.69 1.38
3.68 3.21 2.74 2.46 2.12 1.89 1.64 1.17
2.59 2.31 2.03 1.86 1.65 1.51 1.37 1.08
2.01 1.82 1.63 1.52 1.39 1.29 1.20 1.01
From Newmark, N. M. and Hall, W. J., Earthquake Spectra and Design, Earthquake Engineering Research Institute, Oakland, CA, 1982. With permission.
Inelastic Response Spectra
While the foregoing discussion has been for elastic response spectra, most structures are not expected, or even designed, to remain elastic under strong ground motions. Rather, structures are expected to enter the inelastic region — the extent to which they behave inelastically can be defined by the ductility factor, µ µ= 1999 by CRC Press LLC
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um uy
(5.18)
FIGURE 5.16: Normalized response spectra shapes. (From Uniform Building Code, Structural Engineering Design Provisions, vol. 2, Intl. Conf. Building Officials, Whittier, 1994. With permission.)
where um is the maximum displacement of the mass under actual ground motions, and uy is the displacement at yield (i.e., that displacement which defines the extreme of elastic behavior). Inelastic response spectra can be calculated in the time domain by direct integration, analogous to elastic response spectra but with the structural stiffness as a nonlinear function of displacement, k = k(u). If elastoplastic behavior is assumed, then elastic response spectra can be readily modified to reflect inelastic behavior [90] on the basis that (a) at low frequencies (0.3 Hz 33 Hz), accelerations are equal; and (c) at intermediate frequencies, the absorbed energy is preserved. Actual construction of inelastic response spectra on this basis is shown in Figure 5.17, where DV AAo is the elastic spectrum, which is reduced to D 0 and V 0 by the ratio of 1/µ for frequencies less than 2 Hz, and by the ratio of 1/(2µ − 1)1/2 between 2 and 8 Hz. Above 33 Hz there is no reduction. The result is the inelastic acceleration spectrum (D 0 V 0 A0 Ao ), while A00 A0o is the inelastic displacement spectrum. A specific example, for ZPA = 0.16g, damping = 5% of critical, and µ = 3 is shown in Figure 5.18. Response Spectrum Intensity and Other Measures
While the elastic response spectrum cannot directly define damage to a structure (which is essentially inelastic deformation), it captures in one curve the amount of elastic deformation for a wide variety of structural periods, and therefore may be a good overall measure of ground motion intensity. On this basis, Housner defined a response spectrum intensity as the integral of the elastic response spectrum velocity over the period range 0.1 to 2.5 s. Z SI (h) = 1999 by CRC Press LLC
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2.5
T =0.1
Sv(h, T )dT
(5.19)
FIGURE 5.17: Inelastic response spectra for earthquakes. (After Newmark, N. M. and Hall, W. J., Earthquake Spectra and Design, Earthquake Engineering Research Institute, Oakland, CA, 1982.)
where h = damping (as a percentage of ccrit ). A number of other measures exist, including Fourier amplitude spectrum [32] and Arias Intensity [8]: π IA = g
Z
t
a 2 (t)dt
(5.20)
0
Engineering Intensity Scale
Lastly, Blume [14] defined a measure of earthquake intensity, the Engineering Intensity Scale (EIS), which has been relatively underutilized but is worth noting as it attempts to combine the engineering benefits of response spectra with the simplicity of qualitative intensity scales, such as MMI. The EIS is simply a 10x9 matrix which characterizes a 5% damped elastic response spectra (Figure 5.19). Nine period bands (0.01.1, .2, .4, .6, 1.0, 2.0,  4.0, 7.0, 10,0 s), and ten Sv levels (0.010.1, 1.0, 4.0, 10.0, 30.0, 60.0, 100., 300., 1000. kine) are defined. As can be seen, since the response spectrum for the example ground motion in period band II (0.10.2 s) is predominantly in Sv level 5 (1030 kine), it is assigned EIS 5 (X is assigned where the response spectra does not cross a period band). In this manner, a ninedigit EIS can be assigned to a ground motion (in the example, it is X56,777,76X), which can be reduced to three digits (5,7,6) by averaging, or even to one digit (6, for this example). Numerically, single digit EIS values tend to be a unit or so lower than the equivalent MMI intensity value. 1999 by CRC Press LLC
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FIGURE 5.18: Example inelastic response spectra. (From Newmark, N. M. and Hall, W. J., Earthquake Spectra and Design, Earthquake Engineering Research Institute, Oakland, CA, 1982. With permission.)
5.2.4
Strong Motion Attenuation and Duration
The rate at which earthquake ground motion decreases with distance, termed attenuation, is a function of the regional geology and inherent characteristics of the earthquake and its source. Three major factors affect the severity of ground shaking at a site: (1) source — the size and type of the earthquake, (2) path — the distance from the source of the earthquake to the site and the geologic characteristics of the media earthquake waves pass through, and (3) sitespecific effects — type of soil at the site. In the simplest of models, if the seismogenic source is regarded as a point, then from considering the relation of energy and earthquake magnitude and the fact that the volume of a hemisphere is proportion to R 3 (where R represents radius), it can be seen that energy per unit volume is proportional to C10aM R −3 , where C is a constant or constants dependent on the earth’s crustal properties. The constant C will vary regionally — for example, it has long been observed that attenuation in eastern North America (ENA) varies significantly from that in western North America (WNA) — earthquakes in ENA are felt at far greater distances. Therefore, attenuation relations are regionally dependent. Another regional aspect of attenuation is the definition of terms, especially magnitude, where various relations are developed using magnitudes defined by local observatories. A very important aspect of attenuation is the definition of the distance parameter; because attenuation is the change of ground motion with location, this is clearly important. Many investigators use differing definitions; as study has progressed, several definitions have emerged: (1) hypocentral distance (i.e., straight line distance from point of interest to hypocenter, where hypocentral distance 1999 by CRC Press LLC
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FIGURE 5.19: Engineering intensity scale (EIS) matrix with example. (From Blume, J. A., An Engineering Intensity Scale for Earthquakes and Other Ground Motions, Bull. Seis. Soc. Am., 60(1), 217229, 1970. With permission.)
may be arbitrary or based on regression rather than observation), (2) epicentral distance, (3) closest distance to the causative fault, and (4) closest horizontal distance from the station to the point on the earth’s surface that lies directly above the seismogenic source. In using attenuation relations, it is critical that the correct definition of distance is consistently employed. Methods for estimating ground motion may be grouped into two major categories: empirical and methods based on seismological models. Empirical methods are more mature than methods based on seismological models, but the latter are advantageous in explicitly accounting for source and path, therefore having explanatory value. They are also flexible, they can be extrapolated with more confidence, and they can be easily modified for additional factors. Most seismological modelbased methods are stochastic in nature — Hanks and McGuire’s [54] seminal paper has formed the basis 1999 by CRC Press LLC
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for many of these models, which “assume that ground acceleration is a finiteduration segment of a stationary random process, completely characterized by the assumption that acceleration follows Brune’s [23] source spectrum (for California data, typically about 100 bars), and that the duration of strong shaking is equal to reciprocal of the source corner frequency” fo (the frequency above which earthquake radiation spectra vary with $ −3  below fo , the spectra are proportional to seismic moment [108]). Since there is substantial ground motion data in WNA, seismological modelbased relations have had more value in ENA, where few records exist. The HanksMcGuire method has, therefore, been usefully applied in ENA [123] where Boore and Atkinson [18] found, for hardrock sites, the relation: (5.21) log y = c0 + c1 r − log r where y = a ground motion parameter (PSV, unless ci coefficients for amax are used) r = hypocentral distance (km) P = ξoi + ξni (MW − 6)n I = 0, 1 summation for n = 1, 2, 3 (see Table 5.7) ci TABLE 5.7
Eastern North America HardRock Attenuation Coefficientsa
Response frequency (Hz)
ξ0
ξ1
ξ2
ξ3 −5.364E − 02
0.2
c0 : c1 :
1.743E + 00 −3.130E − 04
1.064E + 00 1.415E − 03
−4.293E − 02 −1.028E − 03
0.5
c0 : c1 :
2.141E + 00 −2.504E − 04
8.521E − 01
−1.670E − 01 −2.612E − 04
1.0
c0 : c1 :
2.300E + 00 −1.024E − 03
6.655E − 01 −1.144E − 04
−1.538E − 01 1.109E − 04
2.0
c0 : c1 :
2.317E + 00 −1.683E − 03
5.070E − 01 1.492E − 04
−9.317E − 02 1.203E − 04
5.0
c0 : c1 :
2.239E + 00 −2.537E − 03
3.976E − 01 5.468E − 04
−4.564E − 02 7.091E − 05
10.0
c0 : c1 :
2.144E + 00 −3.094E − 03
3.617E − 01 7.640E − 04
−3.163E − 02
20.0
c0 : c1 :
2.032E + 00 −3.672E − 03
3.438E − 01 8.956E − 04
−2.559E − 02 −4.219E − 05
amax
c0 : c1 :
3.763E + 00 −3.885E − 03
3.354E − 01 1.042E − 03
−2.473E − 02 −9.169E − 05
a See Equation 5.21. From Boore, D.M. and Atkinson, G.M., Stochastic Prediction of Ground Motion and Spectral Response Parameters at HardRock Sites in Eastern North America, Bull. Seis. Soc. Am., 77, 440487, 1987. With permission.
Similarly, Toro and McGuire [123] furnish the following relation for rock sites in ENA: ln Y = c0 + c1 M + c2 ln(R) + c3 R
(5.22)
where the c0  c3 coefficients are provided in Table 5.8, M represents mLg , and R is the closest distance between the site and the causative fault at a minimum depth of 5 km. These results are valid for hypocentral distances of 10 to 100 km, and mLg 4 to 7. More recently, Boore and Joyner [19] have extended their hardrock relations to deep soil sites in ENA: (5.23) log y = a 00 + b(m − 6) + c(m − 6)2 + d(m − 6)3 − log r + kr where a 00 and other coefficients are given in Table 5.9, m is moment magnitude (MW ), and r is hypocentral distance (km) although the authors suggest that, close to long faults, the distance should 1999 by CRC Press LLC
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TABLE 5.8
ENA Rock Attenuation Coefficientsa
Y
c0
c1
c2
c3
PSRV (1 Hz) PSRV (5 Hz) PSRV (10 Hz) PGA (cm/s2 )
−9.283 −2.757 −1.717 2.424
2.289 1.265 1.069 0.982
−1.000 −1.000 −1.000 −1.004
−.00183 −.00310 −.00391 −.00468
a See Equation 5.22. Spectral velocities are given in cm/s; peak acceleration is given in cm/s2 From Toro, G.R. and McGuire, R.K., An Investigation Into Earthquake Ground Motion Characteristics in Eastern North America, Bull. Seis. Soc. Am., 77, 468489, 1987. With permission.
be the nearest distance to seismogenic rupture. The coefficients in Table 5.9 should not be used outside the ranges 10 < r < 400 km, and 5.0 < MW < 8.5. TABLE 5.9 Coefficients for GroundMotion Estimation at DeepSoil Sites a in Eastern North America in Terms of MW a0
a 00
b
c
d
k
M at maxb
SV 0.05 0.10 0.15 0.20 0.30 0.40 0.50 0.75 1.00 1.50 2.00 3.00 4.00
0.020 0.040 0.015 0.015 0.010 0.015 0.010 0.000 0.000 0.000 0.000 0.000 0.000
1.946 2.267 2.377 2.461 2.543 2.575 2.588 2.586 2.567 2.511 2.432 2.258 2.059
0.431 0.429 0.437 0.447 0.472 0.499 0.526 0.592 0.655 0.763 0.851 0.973 1.039
− 0.028 − 0.026 − 0.031 − 0.037 − 0.051 − 0.066 − 0.080 − 0.111 − 0.135 − 0.165 − 0.180 − 0.176 − 0.145
− 0.018 − 0.018 − 0.017 − 0.016 − 0.012 − 0.009 − 0.007 − 0.001 0.002 0.004 0.002 − 0.008 − 0.022
− 0.00350 − 0.00240 − 0.00190 − 0.00168 − 0.00140 − 0.00110 − 0.00095 − 0.00072 − 0.00058 − 0.00050 − 0.00039 − 0.00027 − 0.00020
8.35 8.38 8.38 8.38 8.47 8.50 8.48 8.58 8.57 8.55 8.47 8.38 8.34
amax SV max SA max
0.030 0.020 0.040
3.663 2.596 4.042
0.448 0.608 0.433
− 0.037 − 0.038 − 0.029
− 0.016 − 0.022 − 0.017
− 0.00220 − 0.00055 − 0.00180
8.38 8.51 8.40
T (sec)
a The distance used is generally the hypocentral distance; we suggest that, close to long faults,
the distance should be the nearest distance to seismogenic rupture. The response spectra are for random horizontal components and 5% damping. The units of amax and SA are cm/s2 ; the units of SV are cm/s. The coefficients in this table should not be used outside the ranges < < < < 10= r = 400 km and 5.0 = M = 8.5. See also Equation 5.23. b “M at max” is the magnitude at which the cubic equation attains its maximum value; for larger magnitudes, we recommend that the motions be equated to those for “M at max”. From Boore, D.M. and Joyner, W.B., Estimation of Ground Motion at DeepSoil Sites in Eastern North America, Bull. Seis. Soc. Am., 81(6), 21672185, 1991. With permission.
In WNA, due to more data, empirical methods based on regression of the ground motion parameter vs. magnitude and distance have been more widely employed, and Campbell [28] offers an excellent review of North American relations up to 1985. Initial relationships were for PGA, but regression of the amplitudes of response spectra at various periods is now common, including consideration of fault type and effects of soil. Some current favored relationships are: Campbell and Bozorgnia [29] (PGA  Worldwide Data) ln(P GA)
=
−3.512 + 0.904M − 1.328 ln
q {Rs2 + [0.149 exp(0.647M)]2 }
+ [1.125 − 0.112 ln(Rs ) − 0.0957M]F 1999 by CRC Press LLC
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+ [0.440 − 0.171 ln(Rs )]Ssr + [0.405 − 0.222 ln(Rs )]Shr + ε where P GA M Rs F Ssr Shr Ssr = Shr ε
(5.24)
= the geometric mean of the two horizontal components of peak ground acceleration (g) = moment magnitude (MW ) = the closest distance to seismogenic rupture on the fault (km) = 0 for strikeslip and normal faulting earthquakes, and 1 for reverse, reverseoblique, and thrust faulting earthquakes = 1 for softrock sites = 1 for hardrock sites = 0 for alluvium sites = a random error term with zero mean and standard deviation equal to σln (P GA), the standard error of estimate of ln(P GA)
This relation is intended for meizoseismal applications, and should not be used to estimate PGA at distances greater than about 60 km (the limit of the data employed for the regression). The relation is based on 645 nearsource recordings from 47 worldwide earthquakes (33 of the 47 are California records — among the other 14 are the 1985 MW 8.0 Chile, 1988 MW 6.8 Armenia, and 1990 MW 7.4 Manjil Iran events). Rs should not be assigned a value less than the depth of the top of the seismogenic crust, or 3 km. Regarding the uncertainty, ε was estimated as: 0.55 if P GA < 0.068 if 0.068 ≤ P GA ≤ 0.21 σln (P GA) = 0.173 − 0.140 ln(P GA) 0.39 if P GA > 0.21 Figure 5.20 indicates, for alluvium, median values of the attenuation of peak horizontal acceleration with magnitude and style of faulting. Joyner and Boore (PSV  WNA Data) [20, 67] Similar to the above but using a twostep regression technique in which the ground motion parameter is first regressed against distance and then amplitudes regressed against magnitude, Boore, Joyner, and Fumal [20] have used WNA data to develop relations for PGA and PSV of the form: log Y where Y M r
= b1 + b2 (M − 6) + b3 (M − 6)2 + b4 r + b5 log10 r + b6 GB + b7 GC + εr + εe
(5.25)
= the ground motion parameter (in cm/s for PSV, and g for PGA) = moment magnitude (MW ) = (d 2 + h2 )(1/2) = distance (km), where h is a fictitious depth determined by regression, and d is the closest horizontal distance from the station to the point on the earth’s surface that lies directly above the rupture GB , GC = site classification indices (GB = 1 for class B site, GC =1 for class C site, both zero otherwise), where Site Class A has shear wave velocities (averaged over the upper 30 m) > 750 m/s, Site Class B is 360 to 750 m/s, and Site Class C is 180 to 360 m/s (class D sites, < 180 m/s, were not included). In effect, class A are rock, B are firm soil sites, C are deep alluvium/soft soils, and D would be very soft sites εr + εe = independent random variable measures of uncertainty, where εr takes on a specific value for each record, and εe for each earthquake = coefficients (see Table 5.10 and Table 5.11) bi , h The relation is valid for magnitudes between 5 and 7.7, and for distances (d) ≤ 100 km. The coefficients in Equation 5.25 are for 5% damped response spectra — Boore et al. [20] also provide similar coefficients for 2%, 10%, and 20% damped spectra, as well as for the random horizontal 1999 by CRC Press LLC
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FIGURE 5.20: Campbell and Bozorgnia worldwide attenuation relationship showing (for alluvium) the scaling of peak horizontal acceleration with magnitude and style of faulting. (From Campbell, K.W. and Bozorgnia, Y., NearSource Attenuation of Peak Horizontal Acceleration from Worldwide Accelerograms Recorded from 1957 to 1993, Proc. Fifth U.S. National Conference on Earthquake Engineering, Earthquake Engineering Research Institute, Oakland, CA, 1994. With permission.)
coefficient (i.e., both horizontal coefficients, not just the larger, are considered). Figure 5.21 presents curves of attenuation of PGA and PSV for Site Class C, using these relations, while Figure 5.22 presents a comparison of this, the Campbell and Bozorgnia [29] and Sadigh et al. [105] attenuation relations, for two magnitude events on alluvium. The foregoing has presented attenuation relations for PGA (Worldwide) and response spectra (ENA and WNA). While there is some evidence [136] that meizoseismal strong ground motion may not differ as much regionally as previously believed, regional attenuation in the farfield differs significantly (e.g., ENA vs. WNA). One regime that has been treated in a special class has been large subduction zone events, such as those that occur in the North American Pacific Northwest (PNW), in Alaska, off the west coast of Central and South America, offshore Japan, etc. This is due to the very large earthquakes that are generated in these zones, with long duration and a significantly different path. A number of relations have been developed for these events [10, 37, 81, 115, 138] which should be employed in those regions. A number of other investigators have developed attenuation relations for other regions, such as China, Japan, New Zealand, the TransAlpide areas, etc., which should be reviewed when working in those areas (see the References). In addition to the seismologically based and empirical models, there is another method for attenuation or ground motion modeling, which may be termed semiempirical methods (Figure 5.23) [129]. The approach discretizes the earthquake fault into a number of subfault elements, finite rupture on each of which is modeled with radiation therefrom modeled via Green’s functions. The resulting wavetrains are combined with empirical modeling of scattering and other factors to generate timehistories of ground motions for a specific site. The approach utilizes a rational framework with powerful explanatory features, and offers useful application in the very nearfield of large earthquakes, where it is increasingly being employed. The foregoing has also dealt exclusively with horizontal ground motions, yet vertical ground motions can be very significant. The common practice for many years has been to take the ratio (V /H ) 1999 by CRC Press LLC
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TABLE 5.10 Coefficients for 5% Damped PSV, for the Larger Horizontal Component T(s)
B1
B2
B3
B4
B5
B6
B7
H
.10 .11 .12 .13 .14 .15 .16 .17 .18 .19 .20 .22 .24 .26 .28 .30 .32 .34 .36 .38 .40 .42 .44 .46 .48 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00
1.700 1.777 1.837 1.886 1.925 1.956 1.982 2.002 2.019 2.032 2.042 2.056 2.064 2.067 2.066 2.063 2.058 2.052 2.045 2.038 2.029 2.021 2.013 2.004 1.996 1.988 1.968 1.949 1.932 1.917 1.903 1.891 1.881 1.872 1.864 1.858 1.849 1.844 1.842 1.844 1.849 1.857 1.866 1.878 1.891 1.905
.321 .320 .320 .321 .322 .323 .325 .326 .328 .330 .332 .336 .341 .345 .349 .354 .358 .362 .366 .369 .373 .377 .380 .383 .386 .390 .397 .404 .410 .416 .422 .427 .432 .436 .440 .444 .452 .458 .464 .469 .474 .478 .482 .485 .488 .491
−.104 −.110 −.113 −.116 −.117 −.117 −.117 −.117 −.115 −.114 −.112 −.109 −.105 −.101 −.096 −.092 −.088 −.083 −.079 −.076 −.072 −.068 −.065 −.061 −.058 −.055 −.048 −.042 −.037 −.033 −.029 −.025 −.022 −.020 −.018 −.016 −.014 −.013 −.012 −.013 −.014 −.016 −.019 −.022 −.025 −.028
.00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000
−.921 −.929 −.934 −.938 −.939 −.939 −.939 −.938 −.936 −.934 −.931 −.926 −.920 −.914 −.908 −.902 −.897 −.891 −.886 −.881 −.876 −.871 −.867 −.863 −.859 −.856 −.848 −.842 −.837 −.833 −.830 −.827 −.826 −.825 −.825 −.825 −.828 −.832 −.837 −.843 −.851 −.859 −.868 −.878 −.888 −.898
.039 .065 .087 .106 .123 .137 .149 .159 .169 .177 .185 .198 .208 .217 .224 .231 .236 .241 .245 .249 .252 .255 .258 .261 .263 .265 .270 .275 .279 .283 .287 .290 .294 .297 .301 .305 .312 .319 .326 .334 .341 .349 .357 .365 .373 .381
.128 .150 .169 .187 .203 .217 .230 .242 .254 .264 .274 .291 .306 .320 .333 .344 .354 .363 .372 .380 .388 .395 .401 .407 .413 .418 .430 .441 .451 .459 .467 .474 .481 .486 .492 .497 .506 .514 .521 .527 .533 .538 .543 .547 .551 .554
6.18 6.57 6.82 6.99 7.09 7.13 7.13 7.10 7.05 6.98 6.90 6.70 6.48 6.25 6.02 5.79 5.57 5.35 5.14 4.94 4.75 4.58 4.41 4.26 4.16 3.97 3.67 3.43 3.23 3.08 2.97 2.89 2.85 2.83 2.84 2.87 3.00 3.19 3.44 3.74 4.08 4.46 4.86 5.29 5.74 6.21
The equations are to be used for 5.0 1 are peculiar to shells in that they produce undulating deformations around the crosssection with no net translation. The relatively large Fourier coefficients associated with n = 2,3,4,5 indicate that a significant portion of the loading will cause shell deformations in these modes. In turn, the corresponding local forces are significantly higher than a beamlike response would produce. To account for the internal conditions in the tower during operation, it is common practice to add an axisymmetric internal suction coefficient H = 0.5 to the external pressure coefficients H (θ ). In terms of the Fourier series representation, this would increase A0 to −0.8922. The dynamic amplification of the effective velocity pressure is represented by the parameter g in Equation 14.5. This parameter reflects the resonant part of the response of the structure and may be as much as 0.2 depending on the dynamic characteristics of the structure. However, when the basis 1999 by CRC Press LLC
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FIGURE 14.11: Surface roughness k/a and maximum sidesuction. of q(z) includes some dynamic portion, such as the fastestmileofwind, (1 + g) is commonly taken as 1.0. Cooling towers are often constructed in groups and close to other structures, such as chimneys or boiler houses, which may be higher than the tower itself. When the spacing of towers is closer than 1.5 times the base diameter or 2 times the throat diameter, or when other tall structures are nearby, the wind pressure on any single tower may be altered in shape and intensity. Such effects should be studied carefully in boundarylayer wind tunnels in order not to overlook dramatic increases in the wind loading. Earthquake loading on hyperbolic cooling towers is produced by ground motions transmitted from the foundation through the supporting columns and the lintel into the shell. If the base motion is assumed to be uniform vertically and horizontally, the circumferential effects are axisymmetrical (n = 0) and antisymmetrical (n = 1), respectively (see Figure 14.12). In the meridional direction, the magnitude and distribution of the earthquakeinduced forces is a function of the mass of the tower and the dynamic properties of the structure (natural frequencies and damping) as well as the acceleration produced by the earthquake at the base of the structure. The most appropriate technique for determining the loads applied by a design earthquake to the shell and components is the response spectrum method which, in turn, requires a free vibration analysis to evaluate the natural frequencies [2, 3, 4]. It is common to use elastic spectra with 5% of critical damping. The supporting columns and foundation are critical for this loading condition and should be modeled in appropriate detail [3, 4]. Temperature variations on cooling towers arise from two sources: operating conditions and sunshine on one side. Typical operating conditions are an external temperature of −15◦ C and internal temperature of +30◦ C. This is an axisymmetrical effect, n = 0 on Figure 14.12. For sunshine, a 1999 by CRC Press LLC
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FIGURE 14.12: Harmonic components of the radial displacement. temperature gradient of 25◦ C constant over the height and distributed as a halfwave around one half of the circumference is appropriate. This loading would require a Fourier expansion in the form of Equation 14.6 and higher harmonic components, n >1, to be considered. Construction loads are generally caused by the fixing devices of climbing formwork, by tower crane anchors, and by attachments for material transport equipment as shown in Figure 14.13. These loads must be considered on the portion of the shell extant at the phase of construction. Nonuniform settlement due to varying subsoil stiffness may be a consideration. Such effects should be modeled considering the interaction of the foundation and the soil.
14.6
Methods of Analysis
Thin shells may resist external loading through forces acting parallel to the shell surface, forces acting perpendicular to the shell surface, and moments. While the analysis of such shells may be formulated within the threedimensional theory of elasticity, there are reduced theories which are twodimensional and are expressed in terms of force and moment intensities. These intensities are traditionally based on a reference surface, generally the middle surface, and are forces and moments per unit length of the middle surface element upon which they act. They are called stress resultants and stress couples, respectively, and are associated with the three directions: circumferential, θ 1 ; meridional, θ 2 ; and normal, θ 3 . In Figure 14.14, the extensional stress resultants, n11 and n22 , the inplane shearing stress resultants, n12 = n21 , and the transverse shear stress resultants, q12 = q21 , are shown in the left diagram along with the components of the applied loading in the circumferential, meridional, and normal directions, p1 , p2 , and p3 , respectively. The bending stress couples, m11 and m22 , and the twisting stress couples, m12 = m21 , are shown in the right diagram along with the displacements v1 , v2 , and v3 in the respective directions. Historically, doubly curved thin shells have been designed to resist applied loading primarily through the extensional and shearing forces in the “plane” of the shell surface, as opposed to the 1999 by CRC Press LLC
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FIGURE 14.13: Attachments on shell wall.
transverse shears and bending and twisting moments which predominate in flat plates loaded normally to their surface. This is known as membrane action, as opposed to bending action, and is consistent with an accompanying theory and calculation methodology which has the advantage of being statically determinate. This methodology was wellsuited for the precomputer age and enabled many large thin shells, including cooling towers, to be rationally designed and economically constructed [9]. Because the conditions that must be provided at the shell boundaries in order to insure membrane action are not always achievable, shell bending should be taken into account even for shells designed by membrane theory. Remarkably, the accompanying bending often is confined to narrow regions in the vicinity of the boundaries and other discontinuities and may have only a minor effect on the shell design, such as local thickening and/or additional reinforcement. Many clever and insightful techniques have been developed over the years to approximate the effects of local bending in shells designed by the membrane theory. As we have passed into and advanced in the computer age, it is no longer appropriate to use the membrane theory to analyze such extraordinary thin shells, except perhaps for preliminary design purposes. The finite element method is widely accepted as the standard contemporary technique and the attention shifts to the level of sophistication to be used in the finite element model. As is often the case, the greater the level of sophistication specified, the more data required. Consequently, 1999 by CRC Press LLC
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FIGURE 14.14: Surface loads, stress resultants, stress couples, and displacements.
a model may evolve through several stages, starting with a relatively simple version that enables the structure to be sized, to the most sophisticated version that may depict such phenomena as the sequence of progressive collapse of the asbuilt shell under various static and dynamic loading scenarios, the incremental effects of the progressive stages of construction, the influence of the operating environment, aging and deterioration on the structure, etc. The techniques described in the following paragraphs form a hierarchical progression from the relatively simple to the very complex, depending on the objective of the analysis. In modeling cooling tower shells using the finite element method, there are a number of options. For the shell wall, ring elements, triangular elements, or quadrilateral elements have been used. Earlier, flat elements adapted from the twodimensional elasticity and plate formulations were used to approximate the doubly curved surface. Such elements present a number of theoretical and computational problems and are not recommended for the analysis of shells. Currently, shell elements degenerated from threedimensional solid elements are very popular. These elements have been utilized in both the ring and quadrilateral form. The column region at the base of the shell presents a special modeling challenge. For static analysis, the lower boundary is often idealized as a uniform support at the lintel level. Then, a portion of the lower shell and the columns is considered in a subsequent analysis to account for the concentrated actions of the columns, which may penetrate only a relatively short distance into the shell wall. For dynamic analysis, it is important to include the column region along with the veil in the model. An equivalent shell element has proved useful in this regard if ring elements are used to model the shell [3, 4]. It may also be desirable to include some of the foundation elements, such as a ring beam at the base and even the supporting piles in a dynamic or settlement model. The linear static analysis method is based on the classical bending theory of thin shells. While this theory has been formulated for many years, solutions for doubly curved shells have not been readily achievable until the development of computerbased numerical methods, most notably the finite element method. The outputs of such an analysis are the stress resultants and couples, defined on Figure 14.14, over the entire shell surface and the accompanying displacements. The analysis is based on the initial geometry, linear elastic material behavior, and a linear kinematic law. Some representative results of such analyses for a large cooling tower (Figure 14.15) are shown in Figures 14.16 through 14.24 for some of the important loading conditions discussed in the preceding section. The finite element model used considers the shell to be fixed at the top of the columns and, thus, does not account for the effect of the concentrated column reactions. Also, in considering the analyses under the individual loading conditions, it should be remembered that the effects are to be factored and combined to produce design values. 1999 by CRC Press LLC
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FIGURE 14.15: Design project for a 200m high cooling tower: geometry.
FIGURE 14.16: Circumferential forces n11G under deadweight. 1999 by CRC Press LLC
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The dead load analysis results in Figures 14.16 and 14.17 indicate that the shell is always under compression in both directions, except for a small circumferential tension near the top. This is a very desirable feature of this geometrical form. In Figures 14.18 through 14.20, the results of an analysis for a quasistatic wind load using the K1.0 distribution from Figure 14.10 are shown. Large tensions in both the meridional and circumferential directions are present. The regions of tension may extend a considerable distance along the circumference from the windward meridian, and the magnitude of the forces is strongly dependent on the distribution selected. In contrast to bluff bodies, where the magnitude of the extensional force along the meridian would be essentially a function of the overturning moment, the cylindricaltype body is also strongly influenced by the circumferential distribution of the applied pressure, a function of the surface roughness. The major effect of the shearing forces is at the level of the lintel where they are transferred into the columns. The internal suction effects (Figures 14.21 and 14.22) are significant only in the circumferential direction. For the service temperature case shown in Figures 14.23 and 14.24, the main effects are bending in the lower region of the shell wall. The analysis of hyperbolic cooling towers for instability or buckling is a subject that has been investigated for several decades [1]. Shell buckling is a complex topic to treat analytically in any case due to the influence of imperfections; for reinforced concrete, it is even more difficult. While the governing equations may be generalized to treat instability by using nonlinear straindisplacement relations and thereby introducing the geometric stiffness matrix, the correlation between the resulting analytical solutions and the possible failure of a reinforced concrete cooling tower is questionable. Nevertheless, it has been common to analyze cooling tower shells under an unfactored combination of dead load plus wind load plus internal suction. The corresponding buckling pattern is shown in Figure 14.25. Interaction diagrams calibrated from experimental studies based on bifurcation buckling are also available [9, 12, 13]. Additionally, there are empirical methods based on wind tunnel tests that consider a snapthrough buckle at the upper edge at each stage of construction [13]. These formulas are proportional to h/R and are convenient for establishing an appropriate shell thickness. If buckling safety is evaluated based on such a linear buckling analysis or an experimental investigation, the buckling safety factor for realistic material parameters should exceed 5.0. Presently, however, the use of bifurcation buckling analyses should be confined to preliminary proportioning since more rational procedures based on nonlinear analysis have been developed to predict the collapse of reinforced concrete shells, as discussed in the following paragraphs. Advances in the analyses of reinforced concrete have produced the capability to analyze shells taking into account the layered composition of the crosssection as shown in Figure 14.26. Using realistic material properties for steel and for concrete, including tension stiffening in the form shown in Figures 14.27 through 14.29, loaddeflection relationships may be constructed for appropriate load combinations. These relationships progress from the linear elastic phase to initial cracking of the concrete through spreading of the cracks until collapse. Results from a nonlinear study are presented in Figures 14.30 through 14.33. The geometry of the shell is given in Figure 14.30, the wind load factor λ is plotted against the maximum lateral displacement at the top of the shell in Figure 14.31, and the deformed shape for the collapse load is shown in Figure 14.32. Also, the pattern of cracking corresponding to the initial yielding of the reinforcement is indicated in Figure 14.33. For reinforced concrete shells, this type of analysis represents the stateoftheart and provides a realistic evaluation of the capacity of such shells against extreme loading [8]. Also durability assessments can be performed by this concept, from which particularly weak and crackendangered regions of the shell can be identified and further reinforced [10]. It is possible to obtain an estimate of the wind load factor, λ, from the results of a linear elastic analysis, even from a calculation based on membrane theory. This estimate is computed as the cracking load for the shell under a combination of D + λW and is predicated on the notion that the 1999 by CRC Press LLC
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FIGURE 14.17: Meridional forces n22G under deadweight.
FIGURE 14.18: Circumferential forces n11W under wind load.
1999 by CRC Press LLC
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FIGURE 14.19: Meridional forces n22W under wind load.
FIGURE 14.20: Shear forces n12W under wind load.
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FIGURE 14.21: Circumferential forces n11S under internal suction. reinforcement may add only a modest amount of capacity to the tower beyond the cracking load [6]. The amount of reinforcement in the wall is often controlled by a specified minimum percentage augmented by that required to resist the net tension due to the factored load combinations. The steel provided is often less than the capacity of the concrete in tension, which is presumed to be lost when the concrete cracks. Therefore, the cracking load represents most of the ultimate capacity of the tower. The maximum meridional tension location under the wind loading is identified, for example, as the value of n22 = 863 kN/m in Figure 14.19. The dead load at this location is obtained from Figure 14.17 as −701 kN/m. Taking the concrete tensile capacity as 2,400 kN/m2 and the wall thickness as 16 cm, the tensile strength is 384 kN/m. Therefore, we have − 701 + λ863 = 384
(14.7)
giving λ = 1.26 as the lower bound on the ultimate strength of the tower. Note that the tower used for the linear elastic analysis is much taller than the one shown in Figure 14.30. The dynamic analysis of cooling towers is usually associated with design for earthquakeinduced forces. The most efficient approach is the response spectrum method, but a time history analysis may be appropriate if nonlinearities are to be included [2, 7]. For large shells the dynamic response due to wind is often investigated, at least to determine the positions of the nodal lines and areas of particularly intensive vibrations. In any case the first step is to carry out a free vibration analysis. This analysis represents the modes of free vibration associated with each natural frequency, f , or its inverse the natural period T, as the product of a circumferential mode proportional to sin nθ or cos nθ and a longitudinal mode along the z axis [3, 4]. Some representative results are shown on Figures 14.34 and 14.35, as discussed below. As an illustration, the cooling tower from Figure 14.4 is again considered. Some key circumferential and longitudinal modes for a fixedbase boundary condition are shown in Figure 14.35. Also, the effects of different cornice stiffnesses are demonstrated. This model may be regarded as preliminary 1999 by CRC Press LLC
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FIGURE 14.22: Meridional forces n22S under internal suction. in that the relatively soft column supports are not properly represented, but it illustrates the salient characteristics of the modes of vibration. Most interesting are the frequency curves on Figure 14.34 for the first 10 harmonics, also demonstrating the influence of different cornice stiffnesses. Note that the natural frequencies decrease with increasing n until a minimum is reached whereupon they increase, a very typical behavior for cylindricaltype shells. Also, the stiffening of the cornice tends to raise the minimum frequency, which is desirable for resistance to dynamic wind. Longitudinally, the cornice stiffness effect is significant for odd modes only. Specifically for earthquake effects, only the first mode participates in a linear analysis for uniform horizontal base motion and the respective values for n = 1 should be entered into the design response spectrum. Results from a seismic analysis of a cooling tower are presented in Figures 14.36 to 14.39. The cooling tower of Figure 14.4 is subjected to a horizontal base excitation based on Figure 14.36, leading to a first circumferential mode (n = 1) participation. A response spectrum analysis provides the lateral displacements w of the tower axis, the meridional forces n22 , and the shear forces n12 as shown on the indicated figures. In general, cooling tower shells have proven to be reasonably resistant against seismic excitations, but obviously the most critical region is the connection between the columns and the lintel as portrayed in Figure 14.40.
14.7
Design and Detailing of Components
The structural elements of the tower should be constructed with a suitable grade of concrete following the provisions of applicable codes and standards. The design of the mixture should reflect the conditions for placement of the concrete and the external and internal environment of the tower. The shell wall should be of a thickness which will permit two layers of reinforcement in two perpendicular directions to be covered by a minimum of 3 cm of concrete, and should be no less than 16 cm thick [7, 13]. The buckling considerations mentioned in the previous section have proven to 1999 by CRC Press LLC
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FIGURE 14.23: Circumferential bending moments m11T under service temperature.
FIGURE 14.24: Meridional bending moments m22T under service temperature.
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FIGURE 14.25: Buckling pattern of tower shell with upper ring beam: D + W + S.
FIGURE 14.26: Layered model for reinforced concrete shell.
be a convenient and evidently acceptable criteria for setting the minimum wall thickness, subject to a nonlinear analysis. The formula qc = 0.052E(h/R)2.3
(14.8)
where E = modulus of elasticity, has been used to estimate the critical shell buckling pressure qc [1, 13]. Then, h(z) is selected to provide a factor of safety of at least 5.0 with respect to the maximum velocity pressure along the windward meridian, q(z)(1 + g). Also, the cornice should have a minimum stiffness of (14.9) Ix /dH = 0.0015m3 where Ix is the moment of inertia of the uncracked crosssection about the vertical axis [13]. Some typical forms of the cornice crosssection are shown in Figure 14.41. 1999 by CRC Press LLC
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FIGURE 14.27: Elastoplastic material law for steel.
FIGURE 14.28: Biaxial failure envelope of Kupfter/Hilsdorf/R¨usch.
The elements of the cooling tower should be reinforced with deformed steel bars so as to provide for the tensile forces and moments arising from the controlling combination of factored loading cases. The shell walls may be proportioned as rectangular crosssections subjected to axial forces and bending. As mentioned above, a mesh of two orthogonal layers of reinforcement should be provided in the shell walls, generally in the meridional and circumferential directions [2]. In each direction, the inner and outer layers should generally be the same, except near the edges where the bending may require an unsymmetrical mesh. It is preferable to locate the circumferential reinforcement outside of the meridional reinforcement except near the lintel, where the meridional reinforcement should be on the outside to stabilize the circumferential bars [13]. A typical heavily reinforced segment of the lintel, also showing the anchorage of the column reinforcement into the shell, is depicted in Figure 14.42. A summary of the most important minimum construction values for the shell wall is given in Figure 14.43 [13]. The bars should not be smaller than 8 mm diameter and, for meridional bars, not 1999 by CRC Press LLC
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FIGURE 14.29: Additional modulus of elasticity due to tension stiffening.
FIGURE 14.30: Shell geometry and wall thickness.
smaller than 10 mm. Further, a minimum of 0.35 to 0.45%, depending on the admissible cracking, should be used in each direction. The minimum cover, as mentioned above, should be 3 cm, the maximum spacing of the bars should be 20 cm, and the splices should be staggered as specified for the construction of walls in the applicable codes or standards. Particular attention should be given to splices in tensile zones. The supporting columns should ideally be proportioned for the forces and moments computed from an analysis in which they are represented as discrete members, using the appropriate factored loading combinations [3]. If the column region has not been modeled discretely, but rather by a 1999 by CRC Press LLC
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FIGURE 14.31: Loaddisplacement diagram for load combination D + λ • W .
FIGURE 14.32: Displacement plot of the shell for load combination D + λ • W : load factor = 1.83.
continuum approximation, the columns may be proportioned to resist the tributary factored forces and moments at the interface with the lintel, as computed from the shell analysis. The effective length may be taken as unity. Particular attention should be directed toward splices of the column bars when net tension is present. Since large bars will be involved, welded splices are recommended in such regions. It is possible to add discrete circumferential stiffeners to the shell to increase the stability or to restore capacity that may have been lost due to cracking or other deterioration [5] (see Figure 14.9). Such stiffeners can generally be included in a finite element model of the shell wall and should be proportioned for the forces computed from such an analysis. The eccentricity of the stiffeners with respect to the circumferential axis should be considered when the stiffeners are only on one side of the shell. The foundations should be proportioned for the factored forces induced by the column reactions, or from the computed forces if the foundation is included in the model with the shell and columns. 1999 by CRC Press LLC
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FIGURE 14.33: Crack pattern of the outer face of the shell for load combination D + λ • W : load factor = 1.54.
FIGURE 14.34: Natural frequencies for different cornice stiffnesses for the cooling tower on Figure 14.4. Reinforcement detailing and cover should be in accordance with the applicable codes or standards. Several improved forms for cooling tower foundations have been suggested. Figure 14.44 shows a flat ring footing suitable for uniform soil conditions, while Figure 14.45 portrays a stiff ring beam foundation appropriate for soil conditions that are nonuniform around the circumference. An example of an individual pier on bedrock is given in Figure 14.46.
1999 by CRC Press LLC
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FIGURE 14.35: Normalized natural vibration modes for the cooling tower on Figure 14.4.
FIGURE 14.36: Seismic response spectrum.
14.8
Construction
Tolerances for tall concrete cooling tower shells have been debated for many years and reasonable values should take into consideration what is achievable and what is measurable. It should be noted that stateoftheart finite element models are capable of analyzing the asbuilt shell as well as the design configuration, so that the effects of those irregularities arising during construction, or even those discovered later, may be quantitatively studied and sometimes corrected. It is recommended that the actual wall thickness be no less than the design thickness and exceed this thickness by not more than 10%. The imperfections of the shell wall middle surface should not exceed onehalf of the wall thickness or 10 cm. Deviations from the design geometry occurring 1999 by CRC Press LLC
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FIGURE 14.37: First axial mode seismic response.
FIGURE 14.38: Second axial mode seismic response.
during the construction should be corrected gradually, limiting the angular change in either direction to 1.5%. The column heads should be within 0.005 times the column height or ±6.0 cm of the design position, and foundation structures should also be within ±6.0 cm of the design location [13]. Formwork and scaffolding systems are generally proprietary and are provided by the constructor. Nevertheless, their influence on the shell quality is of utmost importance and diligent attention of the engineer is required. In general, the system should be designed to provide safety to operating personnel and to produce a sound structure. The working platforms should be designed for realistic loading, and scaffolding systems used for continuous material transport should be designed and built taking into account the resulting loads. The connections and joints between individual scaffolding units should be designed and built to act independently in case of collapse, so that the loss of one unit would not affect the adjacent units. Furthermore, at least two independent safety devices should be in place to prevent collapse. 1999 by CRC Press LLC
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FIGURE 14.39: SRSS superposition of first and second axial modes.
FIGURE 14.40: Column to lintel connection. 1999 by CRC Press LLC
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FIGURE 14.41: Suitable forms of the cornice.
FIGURE 14.42: Lintel reinforcement.
1999 by CRC Press LLCFIGURE 14.43:
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Important minimum construction values.
FIGURE 14.44: Flat ring foundation.
FIGURE 14.45: Ring beam foundation.
FIGURE 14.46: Individual reinforced concrete foundation on concrete base. 1999 by CRC Press LLC
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The shell wall should be designed to resist the anchor loads of the scaffolding, based on the strength of the concrete which is expected to be available when the anchors are loaded. Continuous monitoring of the concrete strength during the climbing process is essential. Cooling tower shells are subjected to a relatively severe environment over their lifetime, which may span several decades, and special care must be taken in order to provide a durable structure. The tower is subjected to the physical loads produced by wind, temperature, and moisture acting on concrete which may still be drying and hardening. Over the lifetime of the structure, it may be exposed to severe frost action in a saturated state, chemical attacks due to noxious substances in the atmosphere and in the water and water vapor, biological attacks due to microorganisms, and possibly additional chemical attacks due to reintroduced cleaned flue gases. The concrete should be of highquality approved materials including flyash. It should have the following properties: • • • •
High resistance against chemical attacks High early strength High structural density High resistance against frost
The surface finish should be of high quality and the surface should be smooth and essentially free of shrink holes. Air bubbles deeper than 4 mm and unintended surface irregularities at joints should be avoided. The shell should be coated with a curing agent providing a high blocking effect and long durability. Several single component (acrylate or polyurethenebased) or double component (epoxy resinbased) coating systems are approved worldwide and are in a process of continual improvement. Of utmost importance for any coating is the homogeneity of the applied film between ≥ 200µm for single and ≥ 300µ for double component systems, since the durability of the complete coating is determined by the thinnest film spots.
References [1] Abel, J.F. and Gould, P.L. 1981. Buckling of Concrete Cooling Towers Shells, ACI SP67, American Concrete Institute, Detroit, Michigan, pp. 135160. [2] ACIASCE Committee 334. 1977. Recommended Practice for the Design and Construction of Reinforced Concrete Cooling Towers, ACI J., 74(1), 2231. [3] Gould, P.L. 1985. Finite Element Analysis of Shells of Revolution, Pitman. [4] Gould, P.L., Suryoutomo, H., and Sen, S.K. 1974. Dynamic analysis of columnsupported hyperboloidal shells, Earth. Eng. Struct. Dyn., 2, 269279. [5] Gould, P.L. and Guedelhoefer, O.C. 1988. Repair and Completion of Damaged Cooling Tower, J. Struct. Eng., 115(3), 576593. [6] Hayashi, K. and Gould, P.L. 1983. Cracking load for a windloaded reinforced concrete cooling tower, ACI J., 80(4), 318325. [7] IASSRecommendations for the Design of Hyperbolic or Other Similarly Shaped Cooling Towers. 1977. Intern. Assoc. for Shell and Space Structures, Working Group No. 3, Brussels. [8] Kr¨atzig, W.B. and Zhuang, Y. 1992. Collapse simulation of reinforced natural draught cooling towers, Eng. Struct., 14(5), 291299. [9] Kr¨atzig, W.B. and Meskouris, K. 1993. Natural draught cooling towers: An increasing need for structural research, Bull. IASS, 34(1), 3751. [10] Kr¨atzig, W.B. and Gruber, K.P. 1996. LifeCycle Damage Simulations of Natural Draught Cooling Towers in Natural Draught Cooling Towers, Wittek, U. and Kr¨atzig, W., Eds., A.A. Balkema, Rotterdam, 151158. 1999 by CRC Press LLC
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[11] Minimum Design Loads for Buildings and Other Structures. 1994. ASCE Standard 793, ASCE, New York. [12] Mungan, I. 1976. Buckling stress states of hyperboloidal shells, J. Struct. Div., ASCE, 102, 20052020. [13] VGB Guideline. 1990. Structural Design of Cooling Towers, VGBTechnical Committee, “Civil Engineering Problems of Cooling Towers”, Essen, Germany.
Further Reading [1] Proceedings (First) International Symposium on Very Tall Reinforced Concrete Cooling Towers. 1978. I.A.S.S., E.D.F., Paris, France, November. [2] Gould, P.L., Kr¨atzig, W.B., Mungan, I., and Wittek, U., Eds. 1984. Natural Draught Cooling Towers. Proceedings of the 2nd International Symposium on Natural Draught Cooling Towers, SpringerVerlag, Heidelberg. [3] Proceedings Third International Symposium on Natural Draught Cooling Towers. 1989. I.A.S.S., E.D.F., Paris, France, April. [4] Wittek, U. and Kr¨atzig, W.B., Eds. 1996. Natural Draught Cooling Towers. Proceedings of the 4th International Symposium on Natural Draught Cooling Towers, A.A. Balkema, Rotterdam. [5] British Standard Institution. 1996. BS 4485, Part 4: British Standard for Water Cooling Towers. Document 96/17117 DC 22. [6] Syndicat National du B´eton Arm´e et des Techniques Industrialis´ees. 1996. Regles de conception et de realisation des refrigerants atmospheriques en beton arm´e, Paris.
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Fang, S.J.; Roy, S. and Kramer, J. “Transmission Structures” Structural Engineering Handbook Ed. Chen WaiFah Boca Raton: CRC Press LLC, 1999
Transmission Structures 15.1 Introduction and Application
Application • Structure Configuration and Material • Constructibility • Maintenance Considerations • Structure Families • State of the Art Review
15.2 Loads on Transmission Structures
General • Calculation of Loads Using NESC Code • Calculation of Loads Using the ASCE Guide • Special Loads • Security Loads • Construction and Maintenance Loads • Loads on Structure • Vertical Loads • Transverse Loads • Longitudinal Loading
15.3 Design of Steel Lattice Tower
Tower Geometry • Analysis and Design Methodology • Allowable Stresses • Connections • Detailing Considerations • Tower Testing
15.4 Transmission Poles
General • Stress Analysis • Tubular Steel Poles • Wood Poles • Concrete Poles • Guyed Poles
15.5 Transmission Tower Foundations
Shujin Fang, Subir Roy, and Jacob Kramer Sargent & Lundy, Chicago, IL
15.1
Geotechnical Parameters • Foundation Types—Selection and Design • Anchorage • Construction and Other Considerations • Safety Margins for Foundation Design • Foundation Movements • Foundation Testing • Design Examples
15.6 Defining Terms References
Introduction and Application
Transmission structures support the phase conductors and shield wires of a transmission line. The structures commonly used on transmission lines are either lattice type or pole type and are shown in Figure 15.1. Lattice structures are usually composed of steel angle sections. Poles can be wood, steel, or concrete. Each structure type can also be selfsupporting or guyed. Structures may have one of the three basic configurations: horizontal, vertical, or delta, depending on the arrangement of the phase conductors.
15.1.1 Application Pole type structures are generally used for voltages of 345kV or less, while lattice steel structures can be used for the highest of voltage levels. Wood pole structures can be economically used for relatively shorter spans and lower voltages. In areas with severe climatic loads and/or on higher voltage lines with multiple subconductors per phase, designing wood or concrete structures to meet the large 1999 by CRC Press LLC
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FIGURE 15.1: Transmission line structures.
loads can be uneconomical. In such cases, steel structures become the costeffective option. Also, if greater longitudinal loads are included in the design criteria to cover various unbalanced loading contingencies, Hframe structures are less efficient at withstanding these loads. Steel lattice towers can be designed efficiently for any magnitude or orientation of load. The greater complexity of these towers typically requires that fullscale load tests be performed on new tower types and at least the 1999 by CRC Press LLC
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tangent tower to ensure that all members and connections have been properly designed and detailed. For guyed structures, it may be necessary to prooftest all anchors during construction to ensure that they meet the required holding capacity.
15.1.2
Structure Configuration and Material
Structure cost usually accounts for 30 to 40% of the total cost of a transmission line. Therefore, selecting an optimum structure becomes an integral part of a costeffective transmission line design. A structure study usually is performed to determine the most suitable structure configuration and material based on cost, construction, and maintenance considerations and electric and magnetic field effects. Some key factors to consider when evaluating the structure configuration are: • A horizontal phase configuration usually results in the lowest structure cost. • If rightofway costs are high, or the width of the rightofway is restricted or the line closely parallels other lines, a vertical configuration may be lower in total cost. • In addition to a wider rightofway, horizontal configurations generally require more tree clearing than vertical configurations. • Although vertical configurations are narrower than horizontal configurations, they are also taller, which may be objectionable from an aesthetic point of view. • Where electric and magnetic field strength is a concern, the phase configuration is considered as a means of reducing these fields. In general, vertical configurations will have lower field strengths at the edge of the rightofway than horizontal configurations, and delta configurations will have the lowest singlecircuit field strengths and a doublecircuit with reverse or lowreactance phasing will have the lowest possible field strength. Selection of the structure type and material depends on the design loads. For a single circuit 230kV line, costs were estimated for singlepole and Hframe structures in wood, steel, and concrete over a range of design span lengths. For this example, wood Hframes were found to have the lowest installed cost, and a design span of 1000 ft resulted in the lowest cost per mile. As design loads and other parameters change, the relative costs of the various structure types and materials change.
15.1.3
Constructibility
Accessibility for construction of the line should be considered when evaluating structure types. Mountainous terrain or swampy conditions can make access difficult and use of helicopter may become necessary. If permanent access roads are to be built to all structure locations for future maintenance purposes, all sites will be accessible for construction. To minimize environmental impacts, some lines are constructed without building permanent access roads. Most construction equipment can traverse moderately swampy terrain by use of widetrack vehicles or temporary mats. Transporting concrete for foundations to remote sites, however, increases construction costs. Steel lattice towers, which are typically set on concrete shaft foundations, would require the most concrete at each tower site. Grillage foundations can also be used for these towers. However, the cost of excavation, backfill and compaction for these foundations is often higher than the cost of a drilled shaft. Unless subsurface conditions are poor, most pole structures can be directly embedded. However, if unguyed pole structures are used at medium to large line angles, it may be necessary to use drilled shaft foundations. Guyed structures can also create construction difficulties in that a wider area must be accessed at each structure site to install the guys and anchors. Also, careful coordination is required to ensure that all guys are tensioned equally and that the structure is plumb. 1999 by CRC Press LLC
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Hauling the structure materials to the site must also be considered in evaluating constructibility. Transporting concrete structures, which weigh at least five times as much as other types of structures, will be difficult and will increase the construction cost of the line. Heavier equipment, more trips to transport materials, and more matting or temporary roadwork will be required to handle these heavy poles.
15.1.4
Maintenance Considerations
Maintenance of the line is generally a function of the structure material. Steel and concrete structures should require very little maintenance, although the maintenance requirements for steel structures depends on the type of finish applied. Tubular steel structures are usually galvanized or made of weathering steel. Lattice structures are galvanized. Galvanized or painted structures require periodic inspection and touchup or reapplication of the finish while weathering steel structures should have relatively low maintenance. Wood structures, however, require more frequent and thorough inspections to evaluate the condition of the poles. Wood structures would also generally require more frequent repair and/or replacement than steel or concrete structures. If the line is in a remote location and lacks permanent access roads, this can be an important consideration in selecting structure material.
15.1.5
Structure Families
Once the basic structure type has been established, a family of structures is designed, based on the line route and the type of terrain it crosses, to accommodate the various loading conditions as economically as possible. The structures consist of tangent, angle, and deadend structures. Tangent structures are used when the line is straight or has a very small line angle, usually not exceeding 3◦ . The line angle is defined as the deflection angle of the line into adjacent spans. Usually one tangent type design is sufficient where terrain is flat and the span lengths are approximately equal. However, in rolling and mountainous terrain, spans can vary greatly. Some spans, for example, across a long valley, may be considerably larger than the normal span. In such cases, a second tangent design for long spans may prove to be more economical. Tangent structures usually comprise 80 to 90% of the structures in a transmission line. Angle towers are used where the line changes direction. The point at which the direction change occurs is generally referred to as the point of intersection (P.I.) location. Angle towers are placed at the P.I. locations such that the transverse axis of the cross arm bisects the angle formed by the conductor, thus equalizing the longitudinal pulls of the conductors in the adjacent spans. On lines where large numbers of P.I. locations occur with varying degrees of line angles, it may prove economical to have more than one angle structure design: one for smaller angles and the other for larger angles. When the line angle exceeds 30◦ , the usual practice is to use a deadend type design. Deadend structures are designed to resist wire pulls on one side. In addition to their use for large angles, the deadend structures are used as terminal structures or for sectionalizing a long line consisting of tangent structures. Sectionalizing provides a longitudinal strength to the line and is generally recommended every 10 miles. Deadend structures may also be used for resisting uplift loads. Alternately, a separate strain structure design with deadend insulator assemblies may prove to be more economical when there is a large number of structures with small line angle subjected to uplift. These structures are not required to resist the deadend wire pull on one side.
15.1.6
State of the Art Review
A major development in the last 20 years has been in the area of new analysis and design tools. These include software packages and design guidelines [12, 6, 3, 21, 17, 14, 9, 8], which have greatly 1999 by CRC Press LLC
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improved design efficiency and have resulted in more economical structures. A number of these tools have been developed based on test results, and many new tests are ongoing in an effort to refine the current procedures. Another area is the development of the reliability based design concept [6]. This methodology offers a uniform procedure in the industry for calculation of structure loads and strength, and provides a quantified measure of reliability for the design of various transmission line components. Aside from continued refinements in design and analysis, significant progress has been made in the manufacturing technology in the last two decades. The advance in this area has led to the increasing usage of cold formed shapes, structures with mixed construction such as steel poles with lattice arms or steel towers with FRP components, and prestressed concrete poles [7].
15.2
Loads on Transmission Structures
15.2.1
General
Prevailing practice and most state laws require that transmission lines be designed, as a minimum, to meet the requirements of the current edition of the National Electrical Safety Code (NESC) [5]. NESC’s rules for the selection of loads and overload capacity factors are specified to establish a minimum acceptable level of safety. The ASCE Guide for Electrical Transmission Line Structural Loading (ASCE Guide) [6] provides loading guidelines for extreme ice and wind loads as well as security and safety loads. These guidelines use reliability based procedures and allow the design of transmission line structures to incorporate specified levels of reliability depending on the importance of the structure.
15.2.2
Calculation of Loads Using NESC Code
NESC code [5] recognizes three loading districts for ice and wind loads which are designated as heavy, medium, and light loading. The radial thickness of ice and the wind pressures specified for the loading districts are shown in Table 15.1. Ice buildup is considered only on conductors and shield wires, and is usually ignored on the structure. Ice is assumed to weigh 57 lb/ft3 . The wind pressure applies to cylindrical surfaces such as conductors. On the flat surface of a lattice tower member, the wind pressure values are multiplied by a force coefficient of 1.6. Wind force is applied on both the windward and leeward faces of a lattice tower. TABLE 15.1
Ice, Wind, and Temperature Loading districts
Radial thickness of ice (in.) Horizontal wind pressure (lb/ft2 ) Temperature (◦ F)
Heavy
Medium
Light
0.50
0.25
0
4
4
9
0
+15
+30
NESC also requires structures to be designed for extreme wind loading corresponding to 50 year fastest mile wind speed with no ice loads considered. This provision applies to all structures without conductors, and structures over 60 ft supporting conductors. The extreme wind speed varies from a basic speed of 70 mph to 110 mph in the coastal areas. In addition, NESC requires that the basic loads be multiplied by overload capacity factors to 1999 by CRC Press LLC
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determine the design loads on structures. Overload capacity factors make it possible to assign relative importance to the loads instead of using various allowable stresses for different load conditions. Overload capacity factors specified in NESC have a larger value for wood structures than those for steel and prestressed concrete structures. This is due to the wide variation found in wood strengths and the aging effect of wood caused by decay and insect damage. In the 1990 edition, NESC introduced an alternative method, where the same overload factors are used for all the materials but a strength reduction factor is used for wood.
15.2.3
Calculation of Loads Using the ASCE Guide
The ASCE Guide [6] specifies extreme ice and extreme wind loads, based on a 50year return period, which are assigned a reliability factor of 1. These loads can be increased if an engineer wants to use a higher reliability factor for an important line, for example a long line, or a line which provides the only source of load. The load factors used to increase the ASCE loads for different reliability factors are given in Table 15.2. TABLE 15.2
Load Factor to Adjust Line Reliability
Line reliability factor, LRF Load return period, RP Corresponding load factor, a˜
1 50 1.0
2 100 1.15
4 200 1.3
8 400 1.4
In calculating wind loads, the effects of terrain, structure height, wind gust, and structure shape are included. These effects are explained in detail in the ASCE Guide. ASCE also recommends that the ice loads be combined with a wind load equal to 40% of the extreme wind load.
15.2.4
Special Loads
In addition to the weather related loads, transmission line structures are designed for special loads that consider security and safety aspects of the line. These include security loads for preventing cascading type failures of the structures and construction and maintenance loads that are related to personnel safety.
15.2.5
Security Loads
Longitudinal loads may occur on the structures due to accidental events such as broken conductors, broken insulators, or collapse of an adjacent structure in the line due to an environmental event such as a tornado. Regardless of the triggering event, it is important that a line support structure be designed for a suitable longitudinal loading condition to provide adequate resistance against cascading type failures in which a larger number of structures fail sequentially in the longitudinal direction or parallel to the line. For this reason, longitudinal loadings are sometimes referred to as “anticascading”, “failure containment”, or “security loads”. There are two basic methods for reducing the risk of cascading failures, depending on the type of structure, and on local conditions and practices. These methods are: (1) design all structures for broken wire loads and (2) install stop structures or guys at specified intervals. Design for Broken Conductors
Certain types of structures such as squarebased lattice towers, 4guyed structures, and single shaft steel poles have inherent longitudinal strength. For lines using these types of structures, the 1999 by CRC Press LLC
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recommended practice is to design every structure for one broken conductor. This provides the additional longitudinal strength for preventing cascading failures at a relatively low cost. Anchor Structures
When single pole wood structures or Hframe structures having low longitudinal strength are used on a line, designing every structure for longitudinal strength can be very expensive. In such cases, stop or anchor structures with adequate longitudinal strength are provided at specific intervals to limit the cascading effect. The Rural Electrification Administration [19] recommends a maximum interval of 5 to 10 miles between structures with adequate longitudinal capacity.
15.2.6
Construction and Maintenance Loads
Construction and maintenance (C&M) loads are, to a large extent, controllable and are directly related to construction and maintenance methods. A detailed discussion on these types of loads is included in the ASCE Loading Guide, and Occupation Safety and Health Act (OSHA) documents. It should be emphasized, however, that workers can be seriously injured as a result of structure overstress during C&M operations; therefore, personnel safety should be a paramount factor when establishing C&M loads. Accordingly, the ASCE Loading Guide recommends that the specified C&M loads be multiplied by a minimum load factor of 1.5 in cases where the loads are “static” and well defined; and by a load factor of 2.0 when the loads are “dynamic”, such as those associated with moving wires during stringing operations.
15.2.7
Loads on Structure
Loads are calculated on the structures in three directions: vertical, transverse, and longitudinal. The transverse load is perpendicular to the line and the longitudinal loads act parallel to the line.
15.2.8
Vertical Loads
The vertical load on supporting structures consists of the weight of the structure plus the superimposed weight, including all wires, ice coated where specified. Vertical load of wire Vw in. (lb/ft) is given by the following equations: Vw = wt. of bare wire (lb/f t) + 1.24(d + I )I
(15.1)
where d = diameter of wire (in.) I = ice thickness (in.) Vertical wire load on structure (lb) = V w × vertical design span × load factor
(15.2)
Vertical design span is the distance between low points of adjacent spans and is indicated in Figure 15.2.
15.2.9
Transverse Loads
Transverse loads are caused by wind pressure on wires and structure, and the transverse component of the line tension at angles. 1999 by CRC Press LLC
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FIGURE 15.2: Vertical and horizontal design spans. Wind Load on Wires
The transverse load due to wind on the wire is given by the following equations: Wh
= =
p × d/12 × Horizontal Span × OCF (without ice) p × (d + 2I )/12 × Horizontal Span × OCF (with ice)
(15.3) (15.4)
where = transverse wind load on wire in lb Wh p = wind pressure in lb/ft2 d = diameter of wire in in. I = radial thickness of ice in in. OCF = Overload Capacity Factor Horizontal span is the distance between midpoints of adjacent spans and is shown in Figure 15.2. Transverse Load Due to Line Angle
Where a line changes direction, the total transverse load on the structure is the sum of the transverse wind load and the transverse component of the wire tension. The transverse component of the tension may be of significant magnitude, especially for large angle structures. To calculate the total load, a wind direction should be used which will give the maximum resultant load considering the effects on the wires and structure. The transverse component of wire tension on the structure is given by the following equation: H = 2T sin θ/2
(15.5)
where H = transverse load due to wire tension in pounds T = wire tension in pounds θ = Line angle in degrees Wind Load on Structures
In addition to the wire load, structures are subjected to wind loads acting on the exposed areas of the structure. The wind force coefficients on lattice towers depend on shapes of member sections, solidity ratio, angle of incidence of wind (faceon wind or diagonal wind), and shielding. Methods 1999 by CRC Press LLC
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for calculating wind loads on transmission structures are given in the ASCE Guide as well the NESC code.
15.2.10
Longitudinal Loading
There are several conditions under which a structure is subjected to longitudinal loading: Deadend Structures—These structures are capable of withstanding the full tension of the conductors and shield wires or combinations thereof, on one side of the structure. Stringing— Longitudinal load may occur at any one phase or shield wire due to a hangup in the blocks during stringing. The longitudinal load is taken as the stringing tension for the complete phase (i.e., all subconductors strung simultaneously) or a shield wire. In order to avoid any prestressing of the conductors, stringing tension is typically limited to the minimum tension required to keep the conductor from touching the ground or any obstructions. Based on common practice and according to the IEEE “Guide to the Installation of Overhead Transmission Line Conductors” [4], stringing tension is generally about onehalf of the sagging tension. Therefore, the longitudinal stringing load is equal to 50% of the initial, unloaded tension at 60◦ F. Longitudinal Unbalanced Load—Longitudinal unbalanced forces can develop at the structures due to various conditions on the line. In rugged terrain, large differentials in adjacent span lengths, combined with inclined spans, could result in significant longitudinal unbalanced load under ice and wind conditions. Nonuniform loading of adjacent spans can also produce longitudinal unbalanced loads. This load is based on an ice shedding condition where ice is dropped from one span and not the adjacent spans. Reference [12] includes a software that is commonly used for calculating unbalanced loads on the structure.
EXAMPLE 15.1: Problem
Determine the wire loads on a small angle structure in accordance with the data given below. Use NESC medium district loading and assume all intact conditions. Given Data: Conductor: 954 kcm 45/7 ACSR Diameter = 1.165 in. Weight = 1.075 lb/ft Wire tension for NESC medium loading = 8020 lb Shield Wire: 3 No.6 Alumoweld Diameter = 0.349 in. Weight = 0.1781 lb/ft Wire tension for NESC medium loading = 2400 lb Wind Span = 1500 ft Weight Span = 1800 ft Line angle = 5◦ Insulator weight = 170 lb 1999 by CRC Press LLC
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Solution NESC Medium District Loading
4 psf wind, 1/4in. ice Ground Wire Iced Diameter = 0.349 + 2 × 0.25 = 0.849 in. Conductor Ice Diameter = 1.165 + 2 × 0.25 = 1.665 in. Overload Capacity Factors for Steel Transverse Wind = 2.5 Wire Tension = 1.65 Vertical = 1.5 Conductor Loads On Tower Transverse Wind = 4 psf × 1.665"/12 × 1500 × 2.5 = 2080 lb Line Angle = 2 × 8020 × sin 2.5◦ × 1.65 = 1150 lb Total = 3230 lb Vertical Bare Wire = 1.075 × 1800 × 1.5 = 2910 lb Ice = {1.24(d + I )I }1800 × 1.5 = 1.24(1.165 + .25).25 × 1800 × 1.5 = 1185 lb Insulator = 170 × 1.5 = 255 lb Total = 4350 lb Ground Wire Loads on Tower Transverse Wind = 4 psf × 0.849/12 × 1500 × 2.5 = 1060 lb Line Angle = 2 × 2400 × sin 2.5 × 1.65 = 350 lb Total = 1410 lb
15.3
Design of Steel Lattice Tower
15.3.1
Tower Geometry
A typical single circuit, horizontal configuration, selfsupported lattice tower is shown in Figure 15.3. The design of a steel lattice tower begins with the development of a conceptual design, which establishes the geometry of the structure. In developing the geometry, structure dimensions are established for the tower window, crossarms and bridge, shield wire peak, bracing panels, and the slope of the 1999 by CRC Press LLC
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FIGURE 15.3: Single circuit lattice tower.
tower leg below the waist. The most important criteria for determining structure geometry are the minimum phase to phase and phase to steel clearance requirements, which are functions of the line voltage. Spacing of phase conductors may sometimes be dictated by conductor galloping considerations. Height of the tower peak above the crossarm is based on shielding considerations for lightning protection. The width of the tower base depends on the slope of the tower leg below the waist . The overall structure height is governed by the span length of the conductors between structures. The lattice tower is made up of a basic body, body extension, and leg extensions. Standard designs are developed for these components for a given tower type. The basic body is used for all the towers regardless of the height. Body and leg extensions are added to the basic body to achieve the desired tower height. The primary members of a tower are the leg and the bracing members which carry the vertical and shear loads on the tower and transfer them to the foundation. Secondary or redundant bracing members are used to provide intermediate support to the primary members to reduce their unbraced length and increase their load carrying capacity. The slope of the tower leg from the waist down has a significant influence on the tower weight and should be optimized to achieve an economical tower 1999 by CRC Press LLC
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design. A flatter slope results in a wider tower base which reduces the leg size and the foundation size, but will increase the size of the bracing. Typical leg slopes used for towers range from 3/4 in. 12 for light tangent towers to 2 1/2 in. 12 for heavy deadend towers. The minimum included angle ∞ between two intersecting members is an important factor for proper force distribution. Reference [3] recommends a minimum included angle of 15◦ , intended to develop a truss action for load transfer and to minimize moment in the member. However, as the tower loads increase, the preferred practice is to increase the included angle to 20◦ for angle towers and 25◦ for deadend towers [23]. Bracing members below the waist can be designed as a tension only or tension compression system as shown in Figure 15.4. In a tension only system shown in (a), the bracing members are designed
FIGURE 15.4: Bracing systems.
to carry tension forces only, the compression forces being carried by the horizontal strut. In a tension/compression system shown in (b) and (c), the braces are designed to carry both tension and compression. A tension only system may prove to be economical for lighter tangent towers. But for heavier towers, a tension/compression system is recommended as it distributes the load equally to the tower legs. A staggered bracing pattern is sometimes used on the adjacent faces of a tower for ease of connections and to reduce the number of bolt holes at a section. Tests [23] have shown that staggering of main bracing members may produce significant moment in the members especially for heavily loaded towers. For heavily loaded towers, the preferred method is to stagger redundant bracing members and connect the main bracing members on the adjacent faces at a common panel point.
15.3.2
Analysis and Design Methodology
The ASCE Guide for Design of Steel Transmission Towers [3] is the industry document governing the analysis and design of lattice steel towers. A lattice tower is analyzed as a space truss. Each member of the tower is assumed pinconnected at its joints carrying only axial load and no moment. Today, finite element computer programs [12, 21, 17] are the typical tools for the analysis of towers for ultimate design loads. In the analytical model the tower geometry is broken down into a discrete number of joints (nodes) and members (elements). User input consists of nodal coordinates, member end incidences and properties, and the tower loads. For symmetric towers, most programs can generate the complete geometry from a part of the input. Loads applied on the tower are ultimate loads which include overload capacity factors discussed in Section 15.2. Tower members are then designed to 1999 by CRC Press LLC
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the yield strength or the buckling strength of the member. Tower members typically consist of steel angle sections, which allow ease of connection. Both single and doubleangle sections are used. Aluminum towers are seldom used today due to the high cost of aluminum. Steel types commonly used on towers are ASTM A36 (Fy = 36 ksi) or A572 (F y = 50 ksi). The most common finish for steel towers is hotdipped galvanizing. Selfweathering steel is no longer used for towers due to the “packout” problems experienced in the past resulting in damaged connections. Tower members are designed to carry axial compressive and tensile forces. Allowable stress in compression is usually governed by buckling, which causes the member to fail at a stress well below the yield strength of the material. Buckling of a member occurs about its weakest axis, which for a single angle section is at an inclination to the geometric axes. As the unsupported length of the member increases, the allowable stress in buckling is reduced. Allowable stress in a tension member is the full yield stress of the material and does not depend on the member length. The stress is resisted by a net crosssection, the area of which is the gross area minus the area of the bolt holes at a given section. Tension capacity of an angle member may be affected by the type of end connection [3]. For example, when one leg of the angle is connected, the tension capacity is reduced by 10%. A further reduction takes place when only the short leg of an unequal angle is connected.
15.3.3
Allowable Stresses
Compression Member
The allowable compressive stress in buckling on the gross crosssectional area of axially loaded compression members is given by the following equations [3]: i 1 − (KL/R)2 /(2Cc2 ) F y
h
if KL/R = Cc or less
Fa
=
Fa Cc
= 286000/(kl/r)2 if KL/R > Cc 1/2 = (3.14)(2E/Fy)
(15.6) (15.7) (15.8)
where Fa = allowable compressive stress (ksi) Fy = yield strength (ksi) E = modulus of elasticity (ksi) L/R = maximum slenderness ratio = unbraced length /radius of gyration K = effective length coefficient The angle member must also be checked for local buckling considerations. If the ratio of the angle effective width to angle thickness (w/t) exceeds 80/(F y)1/2 , the value of F a will be reduced in accordance with the provisions of Reference [3]. The above formulas indicate that the allowable buckling stress is largely dependent on the effective slenderness ratio (kl/r) and the material yield strength (F y). It may be noted, however, that Fy influences the buckling capacity for short members only (kl/r < Cc). For long members (kl/r > Cc), the allowable buckling stress is unaffected by the material strength. The slenderness ratio is calculated for different axes of buckling and the maximum value is used for the calculation of allowable buckling stress. In some cases, a compression member may have an intermediate lateral support in one plane only. This support prevents weak axis and inplane buckling but not the outofplane buckling. In such cases, the slenderness ratio in the member geometric axis will be greater than in the member weak axis, and will control the design of the member. The effective length coefficient K adjusts the member slenderness ratio for different conditions of framing eccentricity and the restraint against rotation provided at the connection. Values of K 1999 by CRC Press LLC
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for six different end conditions, curves one through six, have been defined in Reference [3]. This reference also specifies maximum slenderness ratios of tower members, which are as follows: Type of Member Leg Bracing Redundant
Maximum KL/R 150 200 250
Tests have shown that members with very low L/R are subjected to substantial bending moment in addition to axial load. This is especially true for heavily loaded towers where members are relatively stiff and multiple bolted rigid joints are used [22]. A minimum L/R of 50 is recommended for compression members. Tension Members
The allowable tensile force on the net crosssectional area of a member is given by the following equation [3]: Pt = F y · An · K
(15.9)
where Pt = allowable tensile force (kips) F y = yield strength of the material (ksi) An = net crosssectional area of the angle after deducting for bolt holes (in.2 ). For unequal angles, if the short leg is connected, An is calculated by considering the unconnected leg to be the same size as the connected leg K = 1.0 if both legs of the angle connected = 0.9 if one leg connected The allowable tensile force must also meet the block shear criteria at the connection in accordance with the provisions of Reference [3]. Although the allowable force in a tension member does not depend on the member length, Reference [3] specifies a maximum L/R of 375 for these members. This limit minimizes member vibration under everyday steady state wind, and reduces the risk of fatigue in the connection.
15.3.4
Connections
Transmission towers typically use bearing type bolted connections. Commonly used bolt sizes are 5/8", 3/4", and 7/8" in diameter. Bolts are tightened to a snug tight condition with torque values ranging from 80 to 120 ftlb. These torques are much smaller than the torque used in friction type connections in steel buildings. The snug tight torque ensures that the bolts will not slip back and forth under everyday wind loads thus minimizing the risk of fatigue in the connection. Under full design loads, the bolts would slip adding flexibility to the joint, which is consistent with the truss assumption. Load carrying capacity of the bolted connections depends on the shear strength of the bolt and the bearing strength of the connected plate. The most commonly used bolt for transmission towers is A394, Type 0 bolt with an allowable shear stress of 55.2 ksi across the threaded part. The maximum allowable stress in bearing is 1.5 times the minimum tensile strength of the connected part or the bolt. Use of the maximum bearing stress requires that the edge distance from the center of the bolt hole to the edge of the connected part be checked in accordance with the provisions of Reference [3]. 1999 by CRC Press LLC
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15.3.5
Detailing Considerations
Bolted connections are detailed to minimize eccentricity as much as possible. Eccentric connections give rise to a bending moment causing additional shear force in the bolts. Sometimes small eccentricities may be unavoidable and should be accounted for in the design. The detailing specification should clearly specify the acceptable conditions of eccentricity. Figure 15.5 shows two connections, one with no eccentricity and the second with a small eccentricity. In the first case the lines of force passing through the center of gravity (c.g.) of the members
FIGURE 15.5: Brace details. intersect at a common point. This is the most desired condition producing no eccentricity. In the second case, the lines of force of the two bracing members do not intersect with that of the leg member thus producing an eccentricity in the connection. It is common practice to accept a small eccentricity as long as the intersection of the lines of force of the bracing members does not fall outside the width of the leg member. In some cases it may be necessary to add gusset plates to avoid large eccentricities. In detailing double angle members, care should be taken to avoid a large gap between the angles that are typically attached together by stitch bolts at specified intervals. Tests [23] have shown that a double angle member with a large gap between the angles does not act as a composite member. This results in one of the two angles carrying significantly more load than the other angle. It is recommended that the gap between the two angles of a double angle member be limited to 1/2 in. The minimum size of a member is sometimes dictated by the size of the bolt on the connected leg. The minimum width of members that can accommodate a single row of bolts is as follows: Bolt diameter
Minimum width of member
5/8" 3/4" 7/8"
1 3/4" 2" 2 1/2"
Tension members are detailed with draw to facilitate erection. Members 15 ft in length, or less, are detailed 1/8 in. short, plus 1/16 in. for each additional 10 ft. Tension members should have at least two bolts on one end to facilitate the draw.
15.3.6
Tower Testing
Full scale load tests are conducted on new tower designs and at least the tangent tower to verify the adequacy of the tower members and connections to withstand the design loads specified for that structure. Towers are required to pass the tests at 100% of the ultimate design loads. Tower tests 1999 by CRC Press LLC
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also provide insight into actual stress distribution in members, fitup verification and action of the structure in deflected positions. Detailed procedures of tower testing are given in Reference [3].
EXAMPLE 15.2:
Description Check the adequacy of the following tower components shown in Figure 15.3. Member 1 (compressive leg of the leg extension) Member force = 132 kips (compression) Angle size = L5 × 5 × 3/8" Fy = 50 ksi Member 2 (tension member) Tensile force = 22 kips Angle size = L2 1/2 × 2 × 3/16 (long leg connected) Fy = 36 ksi Bolts at the splice connection of Member 1 Number of 5/8" bolts = 6 (Butt Splice) Type of bolt = A394, Type O Solution Member 1
Member force = 132 kips (compression) Angle size = L5 × 5 × 3/8" Fy = 50 ksi Find maximum L/R Properties of L 5 × 5 × 3/8" Area = 3.61 in.2 rx = ry = 1.56 in. rz = 0.99 in. Member 1 has the same bracing pattern in adjacent planes. Thus, the unsupported length is the same in the weak (z − z) axis and the geometric axes (x − x and y − y). lz = lx = ly = 61" Maximum L/R = 61/0.99 = 61.6 Allowable Compressive Stress: Using Curve 1 for leg member (no framing eccentricity), per Reference [3], k = 1.0 KL/R 1999 by CRC Press LLC
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=
L/R = 61.6
Cc
Fa
= (3.14)(2E/Fy)1/2 = (3.14)(2 × 29000/50)1/2 = 107.0 which is > KL/R i h = 1 − (KL/R)2 /(2Cc2 ) F y i h = 1 − (61.6)2 /(2 × 107.02 ) 50.0 =
41.7 ksi
Allowable compressive load = 41.7 ksi × 3.61 in. = 150.6 kips > 132 kips → O.K. Check local buckling: w/t 80/(Fy)1/2
= (5.0 − 7/8)/(3/8) = 11.0 = 80/(50)1/2 = 11.3 > 11.0 O.K.
Member 2 Tensile force = 22 kips Angle size = L 2 − 1/2 × 2 × 3/16 Area = 0.81 in.2 Fy = 36 ksi Find tension capacity Pt = Fy · An · K Diameter of bolt hole = 5/8" + 1/16" = 11/16" Assuming one bolt hole deduction in 2 − 1/2" leg width, Area of bolt hole
= =
angle th. × hole diam. (3/16)(11/16) = 0.128 in.2
An
=
gross area − bolt hole area
K Pt
= = =
0.81 − 0.128 = 0.68 in.2 0.9, since member end is connected by one leg (36)(0.68)(0.9) = 22.1 kips > 22.0 kips, O.K.
Bolts for Member 1 Number of 5/8" bolts = 6 (Butt Splice) Type of bolt = A394, Type O Shear Strength F v = 55.2 ksi Root area thru threads = 0.202 in.2 1999 by CRC Press LLC
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Shear capacity of bolts: Bolts act in double shear at butt splice Shear capacity of 6 bolts in double shear = 2 × (Root area) × 55.2 ksi × 6 = 133.8 kips > 132 kips ⇒ O.K. Bearing capacity of connected part: Thickness of connected angle = 3/8" Fy of angle = 50 ksi Capacity of bolt in bearing = 1.5 × F u × th. of angle × dia. of bolt F u of 50 ksi material = 65 ksi Capacity of 6 bolts in bearing = 1.5 × 65 × 3/8 × 5/8 × 6 = 137.1 kips > 132 kips, O.K.
15.4
Transmission Poles
15.4.1
General
Transmission poles made of wood, steel, or concrete are used on transmission lines at voltages up to 345kv. Wood poles can be economically used for relatively shorter spans and lower voltages whereas steel poles and concrete poles have greater strength and are used for higher voltages. For areas where severe climatic loads are encountered, steel poles are often the most costeffective choice. Pole structures have two basic configurations: single pole and Hframe (Figure 15.1). Single pole structures are used for lower voltages and shorter spans. Hframe structures consist of two poles connected by a framing comprised of the cross arm, the Vbraces, and the Xbraces. The use of Xbraces significantly increases the load carrying capacity of Hframe structures. At line angles or deadend conditions, guying is used to decrease pole deflections and to increase their transverse or longitudinal structural strength. Guys also help prevent uplift on Hframe structures. Large deflections would be a hindrance in stringing operations.
15.4.2
Stress Analysis
Transmission poles are flexible structures and may undergo relatively large lateral deflections under design loads. A secondary moment (or P − 1 effect) will develop in the poles due to the lateral deflections at the load points. This secondary moment can be a significant percent of the total moment. In addition, large deflections of poles can affect the magnitude and direction of loads caused by the line tension and stringing operations. Therefore, the effects of pole deflections should be included in the analysis and design of single and multipole transmission structures. To properly analyze and design transmission structures, the standard industry practice today is to use nonlinear finite element computer programs. These computer programs allow efficient evaluation of pole structures considering geometric and/or material nonlinearities. For wood poles, there are several popular computer software programs available from EPRI [15]. They are specially developed for design and analysis of wood pole structures. Other general purpose commercial programs auch as SAP90 and STAAD [20, 10] are available for performing small displacement P − 1 analysis. 1999 by CRC Press LLC
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15.4.3
Tubular Steel Poles
Steel transmission poles are fabricated from uniformly tapered hollow steel sections. The crosssections of the poles vary from round to 16sided polygonal with the 12sided dodecagonal as the most common shape. The poles are formed into design crosssections by braking, rolling, or stretch bending. For these structures the usual industry practice is that the analysis, design, and detailing are performed by the steel pole supplier. This facilitates the design to be more compatible with fabrication practice and available equipment. Design of tubular steel poles is governed by the ASCE Manual # 72 [9]. The Manual provides detailed design criteria including allowable stresses for pole masts and connections and stability considerations for global and local buckling. It also defines the requirements for fabrication, erection, load testing, and quality assurance. It should be noted that steel transmission pole structures have several unique design features as compared to other tubular steel structures. First, they are designed for ultimate, or maximum anticipated loads. Thus, stress limits of the Manual #72 are not established for working loads but for ultimate loads. Second, Manual #72 requires that stability be provided for the structure as a whole and for each structural element. In other words, the effects of deflected structural shape on structural stability should be considered in the evaluation of the whole structure as well as the individual element. It relies on the use of the large displacement nonlinear computer analysis to account for the P −1 effect and check for stability. To prevent excessive deflection effects, the lateral deflection under factored loads is usually limited to 5 to 10% of the pole height. Precambering of poles may be used to help meet the imposed deflection limitation on angle structures. Lastly, due to its polygonal crosssections combined with thin material, special considerations must be given to calculation of member section properties and assessment of local buckling. To ensure a polygonal tubular member can reach yielding on its extreme fibers under combined axial and bending compression, local buckling must be prevented. This can be met by limiting the width to thickness ratio, w/t, to 240/(F y)1/2 for tubes with 12 or fewer sides and 215/(F y)1/2 for hexdecagonal tubes. If the axial stress is 1 ksi or less, the w/t limit may be increased to 260/(F y)1/2 for tubes with 8 or fewer sides [9]. Special considerations should be given in the selection of the pole materials where poles are to be subjected to subzero temperatures. To mitigate potential brittle fracture, use of steel with good impact toughness in the longitudinal direction of the pole is necessary. Since the majority of pole structures are manufactured from steels of a yield strength of 50 to 65 ksi (i.e., ASTM A871 and A572), it is advantageous to specify a minimum CharpyVnotch impact energy of 15 ftlb at 0◦ F for plate thickness of 1/2 in. or less and 15 ftlb at −20◦ F for thicker plates. Likewise, high strength anchor bolts made of ASTM A61587 Gr.75 steel should have a minimum Charpy Vnotch of 15 ftlbs at −20◦ F. Corrosion protection must be considered for steel poles. Selection of a specific coating or use of weathering steel depends on weather exposure, past experience, appearance, and economics. Weathering steel is best suited for environments involving proper wetting and drying cycles. Surfaces that are wet for prolonged periods will corrode at a rapid rate. A protective coating is required when such conditions exist. When weathering steel is used, poles should also be detailed to provide good drainage and avoid water retention. Also, poles should either be sealed or well ventilated to assure the proper protection of the interior surface of the pole. Hotdip galvanizing is an excellent alternate means for corrosion protection of steel poles above grade. Galvanized coating should comply with ASTM A123 for its overall quality and for weight/thickness requirements. Pole sections are normally joined by telescoping or slip splices to transfer shears and moments. They are detailed to have a lap length no less than 1.5 times the largest inside diameter. It is important 1999 by CRC Press LLC
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to have a tight fit in slip joint to allow load transfer by friction between sections. Locking devices or flanged joints will be needed if the splice is subjected to uplift forces.
15.4.4
Wood Poles
Wood poles are available in different species. Most commonly used are Douglas Fir and Southern Yellow Pine, with a rupture bending stress of 8000 psi, and Western Red Cedar with a rupture bending stress of 6000 psi. The poles are usually treated with a preservative (pentachlorophenol or creosote). Framing materials for crossarm and braces are usually made of Douglas Fir or Southern Yellow Pine. Crossarms are typically designed for a rupture bending stress of 7400 psi. Wood poles are grouped into a wide range of classes and heights. The classification is based on minimum circumference requirements specified by the American National Standard (ANSI) specification 05.1 for each species, each class, and each height [2]. The most commonly used pole classes are class 1, 2, 3, and H1. Table 15.3 lists the moment capacities at groundline for these common classes of wood poles. Poles of the same class and length have approximately the same capacity regardless of the species. TABLE 15.3 Moment Capacity at Ground Line for 8000 psi Douglas Fir and Southern Pine Poles Class Minimum circumference at top (in.) Length of pole (ft)
Ground line distance from butt (ft)
50 55 60 65 70 75 80 85 90 95 100 105 110
7 7.5 8 8.5 9 9.5 10 10.5 11 11 11 12 12
H1 29
1 27
2 25
3 23
Ultimate moment capacity, ftlb 220.3 246.4 266.8 288.4 311.2 335.3 360.6 387.2 405.2 438.0 461.5 461.5 514.2
187.2 204.2 222.3 241.5 261.9 283.4 306.2 321.5 337.5 357.3 387.3 387.3 424.1
152.1 167.1 183.0 200.0 218.1 230.3 250.2 263.7 285.5 303.2 321.5 321.5 354.1
121.7 134.7 148.7 163.5 179.4 190.2 201.5 213.3 225.5 —
The basic design principle for wood poles, as in steel poles, is to assure that the applied loads with appropriate overload capacity factors do not exceed the specified stress limits. In the design of a single unguyed wood pole structure, the governing criteria is to keep the applied moments below the moment capacity of wood poles, which are assumed to have round solid sections. Theoretically the maximum stress for single unguyed poles under lateral load does not always occur at the ground line. Because all data have been adjusted to the ground line per ANSI 05.1 pole dimensions, only the stress or moment at the ground line need to be checked against the moment capacity. The total ground line moment is the sum of the moment due to transverse wire loads, the moment due to wind on pole, and the secondary moment. The moment due to the eccentric vertical load should also be included if the conductors are not symmetrically arranged. Design guidelines for wood pole structures are given in the REA (Rural Electrification Administration) Bulletin 621 [18] and IEEE Wood Transmission Structural Design Guide [15]. Because of the use of high overload factors, the REA and NESC do not require the consideration of secondary moments in the design of wood poles unless the pole is very flexible. It also permits the use of rupture stress. In contrast, IEEE requires the secondary moments be included in the design and recommends 1999 by CRC Press LLC
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lower overload factors and use of reduction factors for computing allowable stresses. Designers can use either of the two standards to evaluate the allowable horizontal span for a given wood pole. Conversely, a wood pole can be selected for a given span and pole configuration. For Hframes with Xbraces, maximum moments may not occur at ground line. Sections at braced location of poles should also be checked for combined moments and axial loads.
15.4.5
Concrete Poles
Prestressed concrete poles are more durable than wood or steel poles and they are aesthetically pleasing. The reinforcing of poles consists of a spiral wire cage to prevent longitudinal cracks and high strength longitudinal strands for prestressing. The pole is spinned to achieve adequate concrete compaction and a dense smooth finish. The concrete pole typically utilizes a high strength concrete (around 12000 psi) and 270 ksi prestressing strands. Concrete poles are normally designed by pole manufacturers. The guideline for design of concrete poles is given in Reference [8]. Standard concrete poles are limited by their ground line moment capacity. Concrete poles are, however, much heavier than steel or wood poles. Their greater weight increases transportation and handling costs. Thus, concrete poles are used most costeffectively when there is a manufacturing plant near the project site.
15.4.6
Guyed Poles
At line angles and deadends, single poles and Hframes are guyed in order to carry large transverse loads or longitudinal loads. It is a common practice to use bisector guys for line angles up to 30◦ and inline guys for structures at deadends or larger angles. The large guy tension and weight of conductors and insulators can exert significant vertical compression force on poles. Stability is therefore a main design consideration for guyed pole structures. Structural Stability
The overall stability of guyed poles under combined axial compression and bending can be assessed by either a large displacement nonlinear finite element stress analysis or by the use of simplified approximate methods. The rigorous stability analysis is commonly used by steel and concrete pole designers. The computer programs used are capable of assessing the structural stability of the guyed poles considering the effects of the stressdependent structural stiffness and large displacements. But, in most cases, guys are modeled as tensiononly truss elements instead of geometrically nonlinear cable elements. The effect of initial tension in guys is neglected in the analysis. The simplified stability method is typically used in the design of guyed wood poles. The pole is treated as a strut carrying axial loads only and guys are to carry the lateral loads. The critical buckling load for a tapered guyed pole may be estimated by the Gere and Carter method [13]. P cr = P (Dg/Da)e
(15.10)
where P is the Euler buckling load for a pole with a constant diameter of Da at guy attachment and is equal to 9.87 EI /(kl)2 ; Dg is the pole diameter at groundline; kl is the effective column length depending on end condition; e is an exponent constant equal to 2.7 for fixedfree ends and 2.0 for other end conditions. It should be noted that the exact end condition at the guyed attachment is difficult to evaluate. Common practice is to assume a hingedhinged condition with k equal to 1.0. A higher k value should be chosen when there is only a single back guy. For a pole guyed at multiple levels, the column stability may be checked as follows by comparing the maximum axial compression against the critical buckling load, P cr, at the lowest braced location 1999 by CRC Press LLC
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of the pole [15]: [P 1 + P 2 + P 3 + · · ·] /P cr < 1/OCF
(15.11)
where OCF is the overload capacity factor and P 1, P 2, and P 3 are axial loads at various guy levels. Design of Guys
Guys are made of strands of cable attached to the pole and anchor by shackles, thimbles, clips, or other fittings. In the tall microwave towers, initial tension in the guys is normally set between 8 to 15% of the rated breaking strength (RBS) of the cable. However, there is no standard initial tension specified for guyed transmission poles. Guys are installed before conductors and ground wires are strung and should be tightened to remove slack without causing noticeable pole deflections. Initial tension in guys are normally in the range of 5 to 10% of RBS. For design of guys, the maximum tension under factored loads per NESC shall not exceed 90% of the cable breaking strength. Note that for failure containment (broken conductors) the guy tension may be limited to 0.85 RBS. A lower allowable of 65% of RBS would be needed if a linear loaddeformation behavior of guyed poles is desired for extreme wind and ice conditions per ASCE Manual #72. Considerations should be given to the range of ambient temperatures at the site. A large temperature drop may induce a significant increase of guy tension. Guys with an initial tension greater than 15% of RBS of the guy strand may be subjected to aeolian vibrations.
EXAMPLE 15.3:
Description Select a Douglas Fir pole unguyed tangent structure shown below to withstand the NESC heavy district loads. Use an OCF of 2.5 for wind and 1.5 for vertical loads and a strength reduction factor of 0.65. Horizontal load span is 400 ft and vertical load span is 500 ft. Examine both cases with and without the P − 1 effect. The NESC heavy loading is 0.5 in. ice, 4 psf wind, and 0◦ F.
1999 by CRC Press LLC
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Ground Wire Loads H 1 = 0.453#/ft V 1 = 0.807#/ft Conductor Loads H 2 = H 3 = H 4 = 0.732#/ft V 2 = V 3 = V 4 = 2.284#/ft Horizontal Span = 400 ft Vertical Span = 500 ft Line Angle = 0◦ Solution A 75ft class 1 pole is selected as the first trial. The pole will have a length of 9.5 ft buried below the groundline. The diameter of the pole is 9.59 in. at the top (Dt) and 16.3 in. at the groudline (Dg). Moment at groundline due to transverse wind on wire loads is
Mh = (0.732)(2.5)(400)(58 + 53.5 + 49) + (0.453)(2.5)(400)(65) = 146930 ftlbs Moment at groundline due to vertical wire loads Mv = (2.284)(1.5)(500)(8 + 7 − 7) = 13700 ftlbs Moment due to 4 psf wind on pole Mw
= (wind pressure) (OCF )H 2 (Dg + 2Dt)/72 = (4)(2.5)(65.5)2 (16.3 + 9.59 × 2)/72 = 21140 ftlbs
The total moment at groundline Mt = 146930 + 13700 + 21140 = 181770 ftlbs or 181.7 ftkips This moment is less than the moment capacity of the 75ft class 1 pole, 184.2 ftkips ( i.e., 0.65 × 283.4, refer to Table 15.3). Thus, the 75ft class 1 pole is adequate if the P − 1 effect is ignored. To include the effect of the pole displacement, the same pole was modeled on the SAP90 computer program using a modulus of elasticity of 1920 ksi. Under the factored NESC loading, the maximum displacement at the top of the pole is 67.9 in. The associated secondary moment at the groundline is 28.5 ftkips, which is approximately 15.7% of the primary moment. As a result, a 75ft class H1 Douglas Fir pole with an allowable moment of 217.9 ftkips is needed when the P − 1 effect is considered.
15.5
Transmission Tower Foundations
Tower foundation design requires competent engineering judgement. Soil data interpretation is critical as soil and rock properties can vary significantly along a transmission line. In addition, construction procedures and backfill compaction greatly influence foundation performance. Foundations can be designed for site specific loads or for a standard maximum load design. The best approach is to use both a site specific and standardized design. The selection should be based on the number of sites that will have a geotechnical investigation, inspection, and verification of soil conditions. 1999 by CRC Press LLC
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15.5.1
Geotechnical Parameters
To select and design the most economical type of foundation for a specific location, soil conditions at the site should be known through existing site knowledge or new explorations. Inspection should also be considered to verify that the selected soil parameters are within the design limits. The subsurface investigation program should be consistent with foundation loads, experience in the rightofway conditions, variability of soil conditions, and the desired level of reliability. In designing transmission structure foundations, considerations must be given to frost penetration, expansive or shrinking soils, collapsing soils, black shales, sinkholes, and permafrost. Soil investigation should consider the unit weight, angle of internal friction, cohesion, blow counts, and modulus of deformation. The blow count values are correlated empirically to the soil value. Lab tests can measure the soil properties more accurately especially in clays.
15.5.2
Foundation Types—Selection and Design
There are many suitable types of tower foundations such as steel grillages, pressed plates, concrete footings, precast concrete, rock foundations, drilled shafts with or without bells, direct embedment, pile foundations, and anchors. These foundations are commonly used as support for lattice, poles, and guyed towers. The selected type depends on the cost and availability [14, 24]. Steel Grillages
These foundations consist entirely of steel members and should be designed in accordance with Reference [3]. The surrounding soil should not be considered as bracing the leg. There are pyramid arrangements that transfer the horizontal shear to the base through truss action. Other types transfer the shear through shear members that engage the lateral resistance of the compacted backfill. The steel can be purchased with the tower steel and concrete is not required at the site. Cast in Place Concrete
Cast in place concrete foundation consists of a base mat and a square of cylindrical pier. Most piers are kept in vertical position. However, the pier may be battered to allow the axial loads in the tower legs to intersect the mat centroid. Thus, the horizontal shear loads are greatly reduced for deadends and large line angles. Either stub angles or anchor bolts are embedded in the top of the pier so that the upper tower section can be spliced directly to the foundation. Bolted clip angles, welded stud shear connectors, or bottom plates are added to the stub angle. This type can also be precast elsewhere and delivered to the site. The design is accomplished by Reference [1]. Drilled Concrete Shafts
The drilled concrete shaft is the most common type of foundation now being used to support transmission structures. The shafts are constructed by power auguring a circular excavation, placing the reinforcing steel and anchor, and pouring concrete. Tubular steel poles are attached to the shafts using base plates welded to the pole with anchor bolts embedded in the foundation (Figure 15.6a). Lattice towers are attached through the use of stub angles or base plates with anchor bolts. Loose granular soil may require a casing or a slurry. If there is a water level, tremi concrete is required. The casing, if used, should be pulled as the concrete is poured to allow friction along the sides. A minimum 4" slump should allow good concrete flow. Belled shafts should not be attempted in granular soil. If conditions are right, this foundation type is the fastest and most economical to install as there is no backfilling required with dependency on compaction. Lateral procedures for design of drilled shafts under lateral and uplift loads are given in References [14] and [25]. 1999 by CRC Press LLC
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FIGURE 15.6: Direct embedment.
Rock Foundations
If bedrock is close to the surface, a rock foundation can be installed. The rock quality designation (RQD) is useful in evaluating rock. Uplift capacity can be increased with drilled anchor rods or by shaping the rock. Blasting may cause shatter or fracture to rock. Drilling or power hammers are therefore preferred. It is also helpful to wet the hole before placing concrete to ensure a good bond. Direct Embedment
Direct embedment of structures is the oldest form of foundation as it has been used on wood pole transmission lines since early times. Direct embedment consists of digging a hole in the ground, inserting the structure into the hole, and backfilling. Thus, the structure acts as its own foundation 1999 by CRC Press LLC
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transferring loads to the in situ soil via the backfill. The backfill can be a stone mix, stonecement mix, excavated material, polyurethane foam, or concrete (see Figure 15.6b and c). The disadvantage of direct embedment is the dependency on the quality of backfill material. To accurately get deflection and rotation of direct embedded structures, the stiffness of the embedment must be considered. Rigid caisson analysis will not give accurate results. The performance criteria for deflection should be for the combined pole and foundation. Instability of the augured hole and the presence of water may require a liner or double liners (see Figure 15.6d). The design procedure for direct embedment is similar to drilled shafts [14, 25, 16]. Vibratory Shells
Steel shells are installed by using a vibratory hammer. The top 6 or 8 ft (similar to slip joint requirements) of soil inside the shell is excavated and the pole is inserted. The annulus is then filled with a high strength nonshrink grout. The pole can also be attached through a flange connection which eliminates excavating and grouting. The shell design is similar to drilled shafts. Piles
Piles are used to transmit loads through soft soil layers to stiffer soils or rock. The piles can be of wood, prestressed concrete, cast in place concrete, concrete filled shells, steel H piles, steel pipes filled with concrete, and prestressed concrete cylinder piles. The pipe selection depends on the loads, materials, and cost. Pile foundations are normally used more often for lattice towers than for Hframed structures or poles because piles have high axial load capacity and relatively low shear and bending capacity. Besides the external loading, piles can be subjected to the handling, drying, and soil stresses. If piles are not tested, the design should be conservative. Reference [14] should be consulted for bearing, uplift, lateral capacity, and settlement. Driving formulas can be used to estimate dynamic capacity of the pile or group. Timber piles are susceptible to deterioration and should be treated with a preservative. Anchors
Anchors are usually used to support guyed structures. The uplift capacity of rock anchors depends on the quality of the rock, the bond of the grout and rock with steel, and the steel strength. The uplift capacity of soil anchors depends on the resistance between grout and soil and end bearing if applicable. Multibelled anchors in cohesive soil depend on the number of bells. The capacity of Helix anchors can be determined by the installation torque developed by the manufacturer. Spread anchor plate anchors depend on the soil weight plus the soil resistance. Anchors provide resistance to upward forces. They may be prestressed or deadman anchors. Deadmen anchors are not loaded until the structure is loaded, while prestressed anchors are loaded when installed or proof loaded. Helix soil anchors have deformed plates installed by rotating the anchor into the ground with a truckmounted power auger. The capacity of the anchor is correlated to the amount of torque. Anchors are typically designed in accordance with the procedure given in Reference [14].
15.5.3
Anchorage
Anchorage of the transmission tower can consist of anchor bolts, stub angles with clip angles, or shear connectors and designed by Reference [3]. The anchor bolts can be smooth bars with a nut or head at the bottom, or deformed reinforcing bars with the embedment determined by Reference [1]. If the anchor bolt base plate is in contact with the foundation, the lateral or shear load is transferred to the foundation by shear friction. If there is no contact between the base plate and the concrete (anchor 1999 by CRC Press LLC
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bolts with leveling nuts), the lateral load is transferred to the concrete by the side bearing of the anchor bolt. Thus, anchor bolts should be designed for a combination of tension (or compression), shear, and bending by linear interaction.
15.5.4
Construction and Other Considerations
Backfill
Excavated foundations require a high level of compaction that should be inspected and tested. During the original design the degree of compaction that may actually be obtained should be considered. This construction procedure of excavation and compaction increases the foundation costs. Corrosion
The type of soil, moisture, and stray electric currents could cause corrosion of metals placed below the ground. Obtaining resistivity measurements would determine if a problem exists. Consideration could then be given to increasing the steel thickness, a heavier galvanizing coat, a bituminous coat, or in extreme cases a cathodic protection system. Hard epoxy coatings can be applied to steel piles. In addition, concrete can deteriorate in acidic or high sulfate soils.
15.5.5
Safety Margins for Foundation Design
The NESC requires the foundation design loads to be taken the same as NESC load cases used for design of the transmission structures. The engineer must use judgement in determining safety factors depending on the soil conditions, importance of the structures, and reliability of the transmission line. Unlike structural steel or concrete, soil does not have welldefined properties. Large variations exist in the geotechnical parameters and construction techniques. Larger safety margins should be provided where soil conditions are less uniform and less defined. Although foundation design is based on ultimate strength design, there is no industry standard on strength reduction factors at present. The latest research [11] shows that uplift test results differed significantly from analytical predictions and uplift capacity. Based on a statistical analysis of 48 uplift tests on drilled piers and 37 tests on grillages and plates, the coefficients of variation were found to be approximately 30%. To achieve a 95% reliability, which is a 5% exclusion limit, an uplift strength reduction factor of 0.8 to 0.9 is recommended for drilled shafts and 0.7 to 0.8 for backfilled types of foundations.
15.5.6
Foundation Movements
Foundation movements may change the structural configuration and cause load redistribution in lattice structures and framed structures. For pole structures a small foundation movement can induce a large displacement at the top of the pole which will reduce ground clearance or cause problems in wire stringing. The amount of tolerable foundation settlements depends on the structure type and load conditions. However, there is no industry standard at the present time. For lattice structures, it is suggested that the maximum vertical foundation movement be limited to 0.004 times the base dimensions. If larger movements are expected, foundations can be designed to limit their movements or the structures can be designed to withstand the specified foundation movements.
15.5.7
Foundation Testing
Transmission line foundations are load tested to verify the foundation design for specific soils, adequacy of the foundation, research investigation, and to determine strength reduction factors. The 1999 by CRC Press LLC
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load tests will refine foundation selection and verify the soil conditions and construction techniques. The load tests may be in uplift, download, lateral loads, overturning moment, or any necessary combination. There should also be a geotechnical investigation at the test site to correlate the soil data with other locations. There are various test setups, depending on what type of loading is to be applied and what type of foundation is to be tested. The results should compare the analytical methods used to actual behaviors. The load vs. the foundation movements should be plotted in order to evaluate the foundation performance.
15.5.8
Design Examples
EXAMPLE 15.4: Spread Footing
Problem—Determine the size of a square spread footing for a combined moment (175 ftk) and axial load (74 kips) using two alternate methods. In the first method, the minimum factor of safety against overturning is 1.7 and the maximum soil pressure is kept below an allowable soil bearing of 4000 psf. In the second method, no factor of safety against overturning is specified. Instead, the spread footing is designed so that the resultant reaction is within the middle third. This example shows that keeping the resultant in the middle third is a conservative design. Solution Method 1
Try a 8 ft x 8 ft footing
1999 by CRC Press LLC
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P
=
74 kips
Mo
=
175 kipft
P increase for footing size increase = 0.3 kips/ft2 e = 175 kft/74 kips = 2.4 ft > 8 ft/6 = 1.33 ft Therefore, resultant is outside the middle third of the mat. (40 − 2.40 ) × 3 = 4.8 ft S.P. = (74 k)(2)/(4.8 ft)(8 ft) = 3850 psf < 4000 psf MR = (74 k)(4 ft) = 296 kft MR /Mo = 296/175 = 1.7 FOS against overturning, O.K. Method 2 (increase mat size to keep the resultant in the middle third) Try a 11.3 ft x 11.3 ft mat
h i P increase = (11.3 ft)2 − (8 ft)2 × 0.3 k/ft2 = 19.1 kips e = 175 kft/(74 + 19.1) kips = 1.88 ft = 11.3 ft/6 Resultant is within middle third. S.P. = (93.1 k)(2)/(11.3)2 = 1460 lbs/ft2 < 4000 lbs/ft2 Therefore, O.K. Increase in mat size = (11.3/8)2 = 1.99 Therefore, mat size has doubled, assuming that the mat thickness remains the same. 1999 by CRC Press LLC
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EXAMPLE 15.5: Design of a Drilled Shaft
Problem—Determine the depth of a 5ft diameter drilled shaft in cohesive soil with a cohesion of 1.25 ksf by both Broms and modified Broms methods. The foundation is subjected to a combined moment of 2000 ftk and a shear of 20 kips under extreme wind loading. Manual calculation by Broms method is shown herein while the modified Broms method is made by the use of a computer program (CADPRO) [25], which determines the depth required, lateral displacement, and rotation of the foundation top. Calculations are made for various factors of safety (or strength reduction factor). The equations used in this example are based on Reference [25]. Foundation in Cohesive Soil: M
=
2000 ftkips
V
=
20 kips
Cohesion: C D
Solution 1. Use Broms Method [14] M H
=
2000 + 20 × 1
=
2020 ftk
=
M/V = 2020/20 = 101
1999 by CRC Press LLC
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= 1.25 ksf = 5"
q
=
V /(9 CD) = 20/(9)(1.25)(5) = 0.356 i h L = 1.5D + q 1 + (2 + (4H + 6D)/q)0.5 i h = (1.5)(5) + .356 1 + (2 + (4)(101) + (6)(5))/0.356)0.5 = 20.3 ft 2. Comparison of Results of Broms Method and Modified Broms Method. Depth from
where F OS 8 1 θ
= = = =
Modified Brom
F OS
8
C used
broms (ft)
D (ft)
1
θ
1.0 1.33 1.5 1.75 2.0
1.0 0.75 0.667 0.575 0.5
1.25 .9375 .833 .714 .625
20.3 22.3 23.2 24.6 25.8
19 19.5 20.5 23.0 24.0
.935 .89 .81 .653 .603
.457 .474 .366 .262 .23
factor of safety strength reduction factor displacement, in. rotation, degrees
3. Conclusions This example demonstrates that the modified Broms method provides a more economical design than the Broms method. It also shows that as the depth increases by 26%, the factor of safety increased from 1.0 to 2.0. The cost will also increase proportionally.
15.6
Defining Terms
Bearing connection: Shear resistance is provided by bearing of bolt against the connected part. Block shear: A combination of shear and tensile failure through the end connection of a member. Buckling: Mode of failure of a member under compression at stresses below the material yield stress. Cascading effect: Progressive failure of structures due to an accident event. Circuit: A system of usually three phase conductors. Eccentric connection: Lines of force in intersecting members do not pass through a common work point, thus producing moment in the connection. Galloping: High amplitude, low frequency oscillation of snow covered conductors due to wind on uneven snow formation. Horizontal span: The horizontal distance between the midspan points of adjacent spans. Leg and bracing members: Tension or compression members which carry the loads on the structure to the foundation. Line angle: Denotes the change in the direction of a transmission line. Line tension: The longitudinal tension in a conductor or shield wire. Longitudinal load: Load on the supporting structure in a direction parallel to the line. Overload capacity factor: A multiplier used with the unfactored load to establish the design factored load. Phase conductors: Wires or cables intended to carry electric currents, extending along the route of the transmission line, supported by transmission structures. 1999 by CRC Press LLC
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Redundant member: Members that reduce the enbraced length of leg or brace members by providing intermediate support. Sag: The distance measured vertically from a conductor to the straight line joining its two points of support. Self supported structure: Unguyed structure supported on its own foundation. Shear friction: A mechanism to transfer the shear force at anchor bolts to the concrete through wedge action and tension of anchor. Shield wires: Wires installed on transmission structures intended to protect phase conductors against lightning strokes. Slenderness ratio: Ratio of the member unsupported length to its least radius of gyration. Span length: The horizontal distance between two adjacent supporting structures. Staggered bracing: Brace members on adjoining faces of a lattice tower are not connected to a common point on the leg. Stringing: Installation of conductor or shield wire on the structure. Transverse load: Load on the supporting structure in a direction perpendicular to the line. Uplift load: Vertically upward load at the wire attachment to the structure. Vertical span: The horizontal distance between the maximum sag points of adjacent spans. Voltage: The effective potential difference between any two conductors or between a conductor and ground.
References [1] ACI Committee 318, 1995, Building Code Requirements for Reinforced Concrete with Commentary, American Concrete Institute (ACI), Detroit, MI. [2] ANSI, 1979, Specification and Dimensions for Wood Poles, ANSI 05.1, American National Standard Institute, New York. [3] ANSI/ASCE, 1991, Design of Steel Latticed Transmission Structures, Standard 1090, American National Standard Institute and American Society of Civil Engineers, New York. (Former ASCE Manual No. 52). [4] ANSI/IEEE, 1992, IEEE Guide to the Installation of Overhead Transmission Line Conductors, Standard 524, American National Structure Institute and Institute of Electrical and Electronic Engineers, New York. [5] ANSI/IEEE, 1993, National Electrical Safety Code, Standard C2, American National Standard Institute and Institute of Electrical and Electronic Engineers, New York. [6] ASCE, 1984, Guideline for Transmission Line Structural Loading, Committee on Electrical Transmission Structures, American Society of Civil Engineers, New York. [7] ASCE, 1986, Innovations in the Design of Electrical Transmission Structures, Proc. Conf. Struct. Div. Am. Soc. Civil Eng., New York. [8] ASCE, 1987, Guide for the Design and Use of Concrete Pole, American Society of Civil Engineers, New York. [9] ASCE, 1990, Design of Steel Transmission Pole Structures, ASCE Manual No. 72, Second ed., American Society of Civil Engineers, New York. [10] CSI, 1992, SAP90—A Series of Computer Programs for the Finite Element Analysis of Structures—Structural Analysis User’s Manual, Computer and Structures, Berkeley, CA. [11] EPRI, 1983, Transmission Line Structure Foundations for UpliftCompression Loading: Load Test Summaries, EPRI Report EL3160, Electric Power Research Institute, Palo Alto, CA. 1999 by CRC Press LLC
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[12] EPRI, 1990, T.L. Workstation Code, EPRI (Electric Power Research Institute), Report EL6420, Vol. 123, Palo Alto, CA. [13] Gere, J.M. and Carter, W.O., 1962, Critical Buckling Loads for Tapered Columns, J. Struct. Div., ASCE, 88(ST1), 112. [14] IEEE, 1985, IEEE TrialUse Guide for Transmission Structure Foundation Design, Standard 891, Institute of Electrical and Electronics Engineers, New York. [15] IEEE, 1991, IEEE TrialUse Guide for Wood Transmission Structures, IEEE Standard 751, Institute of Electrical and Electronic Engineers, New York. [16] Kramer, J. M., 1978, Direct Embedment of Transmission Structures, Sargent & Lundy Transmission and Substation Conference, Chicago, IL. [17] Peyrot, A.H., 1985, Microcomputer Based Nonlinear Structural Analysis of Transmission Line Systems, IEEE Trans. Power Apparatus and Systems, PAS104 (11). [18] REA, 1980, Design Manual for High Voltage Transmission Lines, Rural Electrification Administration (REA) Bulletin 621. [19] REA, 1992, Design Manual for High Voltage Transmission Lines, Rural Electrification Administration (REA), Bulletin 1724E200. [20] REI, 1993, Program STAADIII—Structural Analysis and Design—User’s Manual, Research Engineers, Orange, CA. [21] Rossow, E.C., Lo, D., and Chu, S.L, 1975, Efficient DesignAnalysis of Physically Nonlinear Trusses, J. Struct. Div., 839853, ASCE, New York. [22] Roy, S., Fang, S., and Rossow, E.C., 1982, Secondary Effects of Large Defection in Transmission Tower Structures, J. Energy Eng., ASCE, 1102, 157172. [23] Roy, S. and Fang, S., 1993, Designing and Testing Heavy DeadEnd Towers, Proc. Am. Power Conf., 55I, 839853, ASCE, New York. [24] Simpson, K.D. and Yanaga, C.Y., 1982, Foundation Design Considerations for Transmission Structure, Sargent & Lundy Transmission and Distribution Conference, Chicago, IL. [25] Simpson, K.D., Strains, T.R., et. al., 1992, Transmission Line Computer Software: The New Generation of Design Tool, Sargent & Lundy Transmission and Distribution Conference, Chicago, IL.
1999 by CRC Press LLC
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Duan, L. and Reno, M. “PerformanceBased Seismic Design Criteria For Bridges” Structural Engineering Handbook Ed. Chen WaiFah Boca Raton: CRC Press LLC, 1999
PerformanceBased Seismic Design Criteria For Bridges Notations 16.2 Introduction
Damage to Bridges in Recent Earthquakes • NoCollapseBased Design Criteria • PerformanceBased Design Criteria • Background of Criteria Development
16.3 Performance Requirements
General • Safety Evaluation Earthquake • Functionality Evaluation Earthquake • Objectives of Seismic Design
16.4 Loads and Load Combinations
Load Factors and Combinations • Earthquake Load Load • Buoyancy and Hydrodynamic Mass
•
Wind
16.5 Structural Materials
Existing Materials • New Materials
16.6 Determination of Demands
Analysis Methods • Modeling Considerations
16.7 Determination of Capacities
Limit States and Resistance Factors • Effective Length of Compression Members • Nominal Strength of Steel Structures • Nominal Strength of Concrete Structures • Structural Deformation Capacity • Seismic Response Modification Devices
16.8 Performance Acceptance Criteria
General • Structural Component Classifications • Steel Structures • Concrete Structures • Seismic Response Modification Devices
Lian Duan and Mark Reno Division of Structures, California Department of Transportation, Sacramento, CA
Defining Terms Acknowledgments References Further Reading Appendix A 16.A.1 Section Properties for Latticed Members 16.A.2 Buckling Mode Interaction For Compression Builtup members 16.A.3 Acceptable Force D/C Ratios and Limiting Values 16.A.4 Inelastic Analysis Considerations
Notations The following symbols are used in this chapter. The section number in parentheses after definition of a symbol refers to the section where the symbol first appears or is defined. Aψ = crosssectional area (Figure 16.9) 1999 by CRC Press LLC
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Ab Aclose Ad Ae Aequiv
= = = = =
Af Ag Agt Agv Ai Ant Anv Ap Ar As Aw A∗i A∗equiv
= = = = = = = = = = = = =
B C Cb Cw D1
= = = = =
DCaccept E Ec Es Et (EI )eff FL Fr
= = = = = = = =
Fu Fumax Fy Fyf Fymax Fyw G Ib If Ii Is Ix−x Iy−y
= = = = = = = = = = = = =
crosssectional area of batten plate (Section 17.A.1) area enclosed within mean dimension for a box (Section 17.A.1) crosssectional area of all diagonal lacings in one panel (Section 17.A.1) effective net area (Figure 16.9) crosssectional area of a thinwalled plate equivalent to lacing bars considering shear transferring capacity (Section 17.A.1) flange area (Section 17.A.1) gross section area (Section 16.7.3) gross area subject to tension (Figure 16.9) gross area subject to shear (Figure 16.9) crosssectional area of individual component i (Section 17.A.1) net area subject to tension (Figure 16.9) net area subject to shear (Figure 16.9) crosssectional area of pipe (Section 16.7.3) nominal area of rivet (Section 16.7.3) crosssectional area of steel members (Figure 16.8) crosssectional area of web (Figure 16.12) crosssectional area above or below plastic neutral axis (Section 17.A.1) crosssectional area of a thinwalled plate equivalent to lacing bars or battens assuming full section integrity (Section 17.A.1) ratio of width to depth of steel box section with respect to bending axis (Section 17.A.4) distance from elastic neutral axis to extreme fiber (Section 17.A.1) bending coefficient dependent on moment gradient (Figure 16.10) warping constant, in.6 (Table 16.2) damage index defined as ratio of elastic displacement demand to ultimate displacement (Section 17.A.3) Acceptable force demand/capacity ratio (Section 16.8.1) modulus of elasticity of steel (Figure 16.8) modulus of elasticity of concrete (Section 16.5.2) modulus of elasticity of reinforcement (Section 16.5.2) tangent modulus (Section 17.A.4) effective flexural stiffness (Section 17.A.4) smaller of (Fyf − Fr ) or Fyw , ksi (Figure 16.10) compressive residual stress in flange; 10 ksi for rolled shapes, 16.5 ksi for welded shapes (Figure 16.10) specified minimum tensile strength of steel, ksi (Section 16.5.2) specified maximum tensile strength of steel, ksi (Section 16.5.2) specified minimum yield stress of steel, ksi (Section 16.5.2) specified minimum yield stress of the flange, ksi (Figure 16.10) specified maximum yield stress of steel, ksi (Section 16.5.2) specified minimum yield stress of the web, ksi (Figure 16.10) shear modulus of elasticity of steel (Table 16.2) moment of inertia of a batten plate (Section 17.A.1) moment of inertia of one solid flange about weak axis (Section 17.A.1) moment of inertia of individual component i (Section 17.A.1) moment of inertia of the stiffener about its own centroid (Section 16.7.3) moment of inertia of a section about xx axis (Section 17.A.1) moment of inertia of a section about yy axis considering shear transferring capacity (Section 17.A.1)
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Iy J Ka K L Lg M M1 M2 Mn MnFLB MnLTB MnWLB Mp Mr Mu My Mp−batten Mεc
= = = = = = = = = = = = = = = = = = =
Ns P Pcr
= = =
PL PG Pn Pu Py Pn∗ PnLG Pnb f Pn Pns y Pn comp Pn Pnten Q Qi Re Rn S Seff Sx Tn Vc Vn Vp Vs Vt
= = = = = = = = = = = = = = = = = = = = = = = = = =
moment of inertia about minor axis, in.4 (Table 16.2) torsional constant, in.4 (Figure 16.10) effective length factor of individual components between connectors (Figure 16.8) effective length factor of a compression member (Section 16.7.2) unsupported length of a member (Figure 16.8) free edge length of gusset plate (Section 16.7.3) bending moment (Figure 16.26) larger moment at end of unbraced length of beam (Table 16.2) smaller moment at end of unbraced length of beam (Table 16.2) nominal flexural strength (Figure 16.10) nominal flexural strength considering flange local buckling (Figure 16.10) nominal flexural strength considering lateral torsional buckling (Figure 16.10) nominal flexural strength considering web local buckling (Figure 16.10) plastic bending moment (Figure 16.10) elastic limiting buckling moment (Figure 16.10) factored bending moment demand (Section 16.7.3) yield moment (Figure 16.10) plastic moment of a batten plate about strong axis (Figure 16.12) moment at which compressive strain of concrete at extreme fiber equal to 0.003 (Section 16.7.4) number of shear planes per rivet (Section 16.7.3) axial force (Section 17.A.4) elastic buckling load of a builtup member considering buckling mode interaction (Section 17.A.2) elastic buckling load of an individual component (Section 17.A.2) elastic buckling load of a global member (Section 17.A.2) nominal axial strength (Figure 16.8) factored axial load demands (Figure 16.13) yield axial strength (Section 16.7.3) nominal compressive strength of column (Figure 16.8) nominal compressive strength considering buckling mode interaction (Figure 16.8) nominal tensile strength considering block shear rupture (Figure 16.9) nominal tensile strength considering fracture in net section (Figure 16.9) nominal compressive strength of a solid web member (Figure 16.8) nominal tensile strength considering yielding in gross section (Figure 16.9) nominal compressive strength of lacing bar (Figure 16.12) nominal tensile strength of lacing bar (Figure 16.12) full reduction factor for slender compression elements (Figure 16.8) force effect (Section 16.4.1) hybrid girder factor (Figure 16.10) nominal shear strength (Section 16.7.3) elastic section modulus (Figure 16.10) effective section modulus (Figure 16.10) elastic section modulus about major axis, in.3 (Figure 16.10) nominal tensile strength of a rivet (Section 16.7.3) nominal shear strength of concrete (Section 16.7.4) nominal shear strength (Figure 16.12) plastic shear strength (Section 16.7.3) nominal shear strength of transverse reinforcement (Section 16.7.4) shear strength carried bt truss mechanism (Section 16.7.4)
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Vu X1 X2 Z a b bi d fc0 fcmin fr fyt h k kv l
= = = = = = = = = = = = = = = =
m
=
mbatten mlacing n nr
= = = =
r ri ry t ti tequiv tw vc x xi xi∗
= = = = = = = = = = =
y yi∗
= =
1ed 1u α αx αy β βm βt
= = = = = = = =
βx βy
= =
factored shear demand (Section 16.7.3) beam buckling factor defined by AISCLRFD [4] (Figure 16.11) beam buckling factor defined by AISCLRFD [4] (Figure 16.11) plastic section modulus (Figure 16.10) distance between two connectors along member axis (Figure 16.8) width of compression element (Figure 16.8) length of particular segment of (Section 17.A.1) effective depth of (Section 16.7.4) specified compressive strength of concrete (Section 16.7.5) specified minimum compressive strength of concrete (Section 16.5.2) modulus of rupture of concrete (Section 16.5.2) probable yield strength of transverse steel (Section 16.7.4) depth of web (Figure 16.8) or depth of member in lacing plane (Section 17.A.1) buckling coefficient (Table 16.3) web plate buckling coefficient (Figure 16.12) length from the last rivet (or bolt) line on a member to first rivet (or bolt) line on a member measured along the centerline of member (Section 16.7.3) number of panels between point of maximum moment to point of zero moment to either side [as an approximation, half of member length (L/2) may be used] (Section 17.A.1) number of batten planes (Figure 16.12) number of lacing planes (Figure 16.12) number of equally spaced longitudinal compression flange stiffeners (Table 16.3) number of rivets connecting lacing bar and main component at one joint (Figure 16.12) radius of gyration, in. (Figure 16.8) radius of gyration of local member, in. (Figure 16.8) radius of gyration about minor axis, in. (Figure 16.10) thickness of unstiffened element (Figure 16.8) average thickness of segment bi (Section 17.A.1) thickness of equivalent thinwalled plate (Section 17.A.1) thickness of the web (Figure 16.10) permissible shear stress carried by concrete (Section 16.7.4) subscript relating symbol to strong axis or xx axis (Figure 16.13) distance between yy axis and center of individual component i (Section 17.A.1) distance between center of gravity of a section A∗i and plastic neutral yy axis (Section 17.A.1) subscript relating symbol to strong axis or yy axis (Figure 16.13) distance between center of gravity of a section A∗i and plastic neutral xx axis (Section 17.A.1) elastic displacement demand (Section 17.A.3) ultimate displacement (Section 17.A.3) separation ratio (Section 17.A.2) parameter related to biaxial loading behavior for xx axis (Section 17.A.4) parameter related to biaxial loading behavior for yy axis (Section 17.A.4) 0.8, reduction factor for connection (Section 16.7.3) reduction factor for moment of inertia specified by Equation 16.28 (Section 17.A.1) reduction factor for torsion constant may be determined Equation 16.38 (Section 17.A.1) parameter related to uniaxial loading behavior for xx axis (Section 17.A.4) parameter related to uniaxial loading behavior for yy axis (Section 17.A.4)
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δo γLG λ λb λbp λbr λbpr λc λcp λcpr λcr λp λpr λr λp−Seismic µ1 µφ ρ 00 φ φ φb φbs φc φt φtf φty comp σc σcten σs τu εs εsh comp εc γi η
= imperfection (outofstraightness) of individual component (Section 17.A.2) = buckling mode interaction factor to account for buckling model interaction (Figure 16.8) = widththickness ratio of compression element (Figure 16.8) = rLy (slenderness parameter of flexural moment dominant members) (Figure 16.10) = limiting beam slenderness parameter for plastic moment for seismic design (Figure 16.10) = limiting beam slenderness parameter for elastic lateral torsional buckling (Figure 16.10) = limiting q beam slenderness parameter determined by Equation 16.25 (Table 16.2) Fy = KL rπ E (slenderness parameter of axial load dominant members) (Figure 16.8) = 0.5 (limiting column slenderness parameter for 90% of the axial yield load based on AISCLRFD [4] column curve) (Table 16.2) = limiting column slenderness parameter determined by Equation 16.24 (Table 16.2) = limiting column slenderness parameter for elastic buckling (Table 16.2) = limiting widththickness ratio for plasticity development specified in Table 16.3 (Figure 16.10) = limiting widththickness ratio determined by Equation 16.23 (Table 16.2) = limiting widththickness ratio (Figure 16.8) = limiting widththickness ratio for seismic design (Table 16.2) = displacement ductility, ratio of ultimate displacement to yield displacement (Section 16.7.4) = curvature ductility, ratio of ultimate curvature to yield curvature (Section 17.A.3) = ratio of transverse reinforcement volume to volume of confined core (Section 16.7.4) = resistance factor (Section 16.7.1) = angle between diagonal lacing bar and the axis perpendicular to the member axis (Figure 16.12) = resistance factor for flexure (Figure 16.13) = resistance factor for block shear (Section 16.7.1) = resistance factor for compression (Figure 16.13) = resistance factor for tension (Figure 16.9) = resistance factor for tension fracture in net (section 16.7.1) = resistance factor for tension yield (Figure 16.9) = maximum concrete stress under uniaxial compression (Section 16.7.5) = maximum concrete stress under uniaxial tension (Section 16.7.5) = maximum steel stress under uniaxial tension (Section 16.7.5) = shear strength of a rivet (Section 16.7.3) = maximum steel strain under uniaxial tension (Section 16.7.5) = strain hardening strain of steel (Section 16.5.2) = maximum concrete strain under uniaxial compression (Section 16.7.5) = load factor corresponding to Qi (Section 16.4.1) = a factor relating to ductility, redundancy, and operational importance (Section 16.4.1)
16.2
Introduction
16.2.1
Damage to Bridges in Recent Earthquakes
Since the beginning of civilization, earthquake disasters have caused both death and destruction — the structural collapse of homes, buildings, and bridges. About 20 years ago, the 1976 Tangshan earthquake in China resulted in the tragic death of 242,000 people, while 164,000 people were severely 1999 by CRC Press LLC
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injured, not to mention the entire collapse of the industrial city of Tangshan [39]. More recently, the 1989 Loma Prieta and the 1994 Northridge earthquakes in California [27, 28] and the 1995 Kobe earthquake in Japan [29] have exacted their tolls in the terms of deaths, injuries, and the collapse of the infrastructure systems which can in turn have detrimental effects on the economies. The damage and collapse of bridge structures tend to have a more lasting image on the public. Figure 16.1 shows the collapsed elevated steel conveyor at Lujiatuo Mine following the 1976 Tangshan earthquake in China. Figures 16.2 and 16.3 show damage from the 1989 Loma Prieta earthquake: the San FranciscoOakland Bay Bridge east span drop off and the collapsed double deck portion of the Cypress freeway, respectively. Figure 16.4 shows a portion of the R14/I5 interchange following the 1994 Northridge earthquake, which also collapsed following the 1971 San Fernando earthquake in California while it was under construction. Figure 16.5 shows a collapsed 500m section of the elevated Hanshin Expressway during the 1995 Kobe earthquake in Japan. These examples of bridge damage, though tragic, have served as fullscale laboratory tests and have forced bridge engineers to reconsider their design principles and philosophies. Since the 1971 San Fernando earthquake, it has been a continuing challenge for bridge engineers to develop a safe seismic design procedure so that the structures are able to withstand the sometimes unpredictable devastating earthquakes.
FIGURE 16.1: Collapsed elevated steel conveyor at Lujiatuo Mine following the 1976 Tangshan earthquake in China. (From California Institute of Technology, The Greater Tangshan Earthquake, California, 1996. With permission.)
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FIGURE 16.2: Aerial view of collapsed upper and lower decks of the San FranciscoOakland Bay Bridge (I80) following the 1989 Loma Prieta earthquake in California. (Photo by California Department of Transportation. With permission.)
16.2.2 NoCollapseBased Design Criteria For seismic design and retrofit of ordinary bridges, the primary philosophy is to prevent collapse during severe earthquakes [13, 24, 25]. The structural survival without collapse has been a basis of seismic design and retrofit for many years [13]. To prevent the collapse of bridges, two alternative design approaches are commonly in use. First is the conventional forcebased approach where the adjustment factor Z for ductility and risk assessment [12], or the response modification factor R [1], is applied to elastic member force levels obtained by acceleration spectra analysis. The second approach is the newer displacementbased design approach [13] where displacements are a major consideration in design. For more detailed information, reference is made to a comprehensive and stateoftheart book by Prietley et al. [35]. Much of the information in this book is backed by California Department of Transportation (Caltrans)supported research, directed at the seismic performance of bridge structures.
16.2.3
PerformanceBased Design Criteria
Following the 1989 Loma Prieta earthquake, bridge engineers recognized the need for sitespecific and projectspecific design criteria for important bridges. A bridge is defined as “important” when one of the following criteria is met: • The bridge is required to provide secondary life safety. • Time for restoration of functionality after closure creates a major economic impact. • The bridge is formally designated as critical by a local emergency plan. 1999 by CRC Press LLC
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FIGURE 16.3: Collapsed Cypress Viaduct (I880) following the 1989 Loma Prieta earthquake in California.
FIGURE 16.4: Collapsed SR14/I5 south connector overhead following the 1994 Northridge earthquake in California. (Photo by James MacIntyre. With permission.)
Caltrans, in cooperation with various emergency agencies, has designated and defined the various important routes throughout the state of California. For important bridges, such as I880 replacement [23] and R14/I5 interchange replacement projects, the design criteria [10, 11] including sitespecific Acceleration Response Spectrum (ARS) curves and specific design procedures to reflect the desired performance of these structures were developed. 1999 by CRC Press LLC
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FIGURE 16.5: Collapsed Hanshin Expressway following the 1995 Kobe earthquake in Japan. (Photo by Mark Yashinsky. With permission.)
In 1995, Caltrans, in cooperation with engineering consulting firms, began the task of seismic retrofit design for the seven major toll bridges including the San FranciscoOakland Bay Bridge (SFOBB) in California. Since the traditional seismic design procedures could not be directly applied to these toll bridges, various analysis and design concepts and strategies have been developed [7]. These differences can be attributed to the different postearthquake performance requirements. As shown in Figure 16.6, the performance requirements for a specific project or bridge must be the first item to be established. Loads, materials, analysis methods and approaches, and detailed acceptance criteria are then developed to achieve the expected performance. The nocollapsebased design criteria shall be used unless performancebased design criteria is required.
16.2.4
Background of Criteria Development
It is the purpose of this chapter to present performancebased criteria that may be used as a guideline for seismic design and retrofit of important bridges. More importantly, this chapter provides concepts for the general development of performancebased criteria. The appendices, as an integral part of the criteria, are provided for background and information of criteria development. However, it must be recognized that the desired performance of the structure during various earthquakes ultimately defines the design procedures. Much of this chapter was primarily based on the Seismic Retrofit Design Criteria (Criteria) which was developed for the SFOBB West Spans [17]. The SFOBB Criteria was developed and based on past successful experience, various codes, specifications, and stateoftheart knowledge. The SFOBB, one of the national engineering wonders, provides the only direct highway link between San Francisco and the East Bay Communities. SFOBB (Figure 16.7) carries Interstate Highway 80 approximately 81/4 miles across San Francisco Bay since it first opened to traffic in 1936. The west spans of SFOBB, consisting of twin, endtoend doubledeck suspension bridges and a threespan doubledeck continuous truss, crosses the San Francisco Bay from the city of San Francisco to Yerba Buena Island. The seismic retrofit design of SFOBB West Spans, as the top priority project of the California Department of Transportation, is a challenge to bridge engineers. A performancebased design Criteria [17] was, therefore, developed for SFOBB West Spans. 1999 by CRC Press LLC
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FIGURE 16.6: Development procedure of performancebased seismic design criteria for important bridges.
16.3
Performance Requirements
16.3.1
General
The seismic design and retrofit of important bridges shall be performed by considering both the higher level Safety Evaluation Earthquake (SEE), which has a mean return period in the range of 1000 to 2000 years, and the lower level Functionality Evaluation Earthquake (FEE), which has a mean return period of 300 years with a 40% probability of exceedance during the expected life of the bridge. It is important to note that the return periods of both the SEE and FEE are dictated by the engineers and seismologists.
16.3.2
Safety Evaluation Earthquake
The bridge shall remain serviceable after a SEE. Serviceable is defined as sustaining repairable damage with minimum impact to functionality of the bridge structure. In addition, the bridge will be open to emergency vehicles immediately following the event, provided bridge management personnel can provide access. 1999 by CRC Press LLC
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(a) West crossing spans.
(b) East crossing spans. FIGURE 16.7: San FranciscoOakland Bay Bridge. (Photo by California Department of Transportation. With permission.)
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16.3.3
Functionality Evaluation Earthquake
The bridge shall remain fully operational after a FEE. Fully operational is defined as full accessibility to the bridge by current normal daily traffic. The structure may suffer repairable damage, but repair operations may not impede traffic in excess of what is currently required for normal daily maintenance.
16.3.4
Objectives of Seismic Design
The objectives of seismic design are as follows: 1. To keep the Critical structural components in the essentially elastic range during the SEE. 2. To achieve safety, reliability, serviceability, constructibility, and maintainability when the Seismic Response Modification Devices (SRMDs), i.e., energy dissipation and isolation devices, are installed in bridges. 3. To devise expansion joint assemblies between bridge frames that either retain traffic support or, with the installation of deck plates, are able to carry the designated traffic after being subjected to SEE displacements. 4. To provide ductile load paths and detailing to ensure bridge safety in the event that future demands might exceed those demands resulting from current SEE ground motions.
16.4
Loads and Load Combinations
16.4.1
Load Factors and Combinations
New and retrofitted bridge components shall be designed for the applicable load combinations in accordance with the requirements of AASHTOLRFD [1]. The load effect shall be obtained by Load effect = η
X
γi Qi
(16.1)
where Qi = force effect η = a factor relating to ductility, redundancy, and operational importance = load factor corresponding to Qi γi . The AASHTOLRFD load factors or load factors η = 1.0 and γi = 1.0 may be used for seismic design. The live load on the bridge shall be determined by ADTT (Average Daily Truck Traffic) value for the project. The bridge shall be analyzed for the worst case with or without live load. The mass of the live loads shall not be included in the dynamic calculations. The intent of the live load combination is to include the weight effect of the vehicles only.
16.4.2
Earthquake Load
The earthquake load – ground motions and response spectra shall be considered at two levels: SEE and FEE. The ground motions and response spectra may be generated in accordance with Caltrans Guidelines [14, 15]. 1999 by CRC Press LLC
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16.4.3
Wind Load
1. Wind load on structures — Wind loads shall be applied as a static equivalent load in accordance with AASHTOLRFD [1] . 2. Wind load on live load — Wind pressure on vehicles shall be represented by a uniform load of 0.100 kips/ft (1.46 kN/m) applied at right angles to the longitudinal axis of the structure and 6.0 ft (1.85 m) above the deck to represent the wind load on vehicles. 3. Wind load dynamics — The expansion joints, SRMDs, and wind locks (tongues) shall be evaluated for the dynamic effects of wind loads.
16.4.4
Buoyancy and Hydrodynamic Mass
The buoyancy shall be considered to be an uplift force acting on all components below design water level. Hydrodynamic mass effects [26] shall be considered for bridges over water.
16.5
Structural Materials
16.5.1
Existing Materials
For seismic retrofit design, aged concrete with specified strength of 3250 psi (22.4 MPa) can be considered to have a compressive strength of 5000 psi (34.5 MPa). If possible, cores of existing concrete should be taken. Behavior of structural steel and reinforcement shall be based on mill certificate or tensile test results. If they are not available in bridge archives, a nominal strength of 1.1 times specified yield strength may be used [13].
16.5.2
New Materials
Structural Steel
New structural steel used shall be AASHTO designation M270 (ASTM designation A709) Grade 36 and Grade 50. Welds shall be as specified in the Bridge Welding Code ANSI/AASHTO/AWS D1.595 [8]. Partial penetration welds shall not be used in regions of structural components subjected to possible inelastic deformation. High strength bolts conforming to ASTM designation A325 shall be used for all new connections and for upgrading strengths of existing riveted connections. New bolted connections shall be designed as bearingtype for seismic loads and shall be slipcritical for all other load cases. All bolts with a required length under the head greater than 8 in. shall be designated as ASTM A449 threaded rods (requiring nuts at each end) unless a verified source of longer bolts can be identified. New anchor bolts shall be designated as ASTM A449 threaded rods. Structural Concrete
All concrete shall be normal weight concrete with the following properties: Specified compressive strength: Modulus of elasticity: Modulus of rupture:
fcmin = 4, 000 p Ec = 57,000 fc0 p fr = 5 fc0
psi (27.6MPa) psi psi
Reinforcement
All reinforcement shall use ASTM A706 (Grade 60) with the following specified properties: 1999 by CRC Press LLC
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Specified minimum yield stress: Specified minimum tensile strength: Specified maximum yield stress: Specified maximum tensile strength: Modulus of elasticity: Strain hardening strain: εsh =
16.6
Determination of Demands
16.6.1
Analysis Methods
Fy = 60 ksi Fu = 90 ksi Fymax = 78 ksi Fumax = 107 ksi Es = 29,000 ksi 0.0150 0.0125 0.0100 0.0075 0.0050
(414 MPa) (621 MPa) (538 MPa) (738 MPa) (200,000 MPa)
for #8 and smallers bars for #9 for #10 and #11 for #14 for #18
Static Linear Analysis
Static linear analysis shall be used to determine member forces due to self weight, wind, water currents, temperature, and live load. Dynamic Response Spectrum Analysis
1. Dynamic response spectrum analysis shall be used for the local and regional stand alone models and the simplified global model described in Section 16.6.2 to determine mode shapes, structure periods, and initial estimates of seismic force and displacement demands. 2. Dynamic response spectrum analysis may be used on global models prior to time history analysis to verify model behavior and eliminate modeling errors. 3. Dynamic response spectrum analysis may be used to identify initial regions or members of likely inelastic behavior which need further refined analysis using inelastic nonlinear elements. 4. Site specific ARS curves shall be used, with 5% damping. 5. Modal responses shall be combined using the Complete Quadratic Combination (CQC) method and the resulting orthogonal responses shall be combined using either the Square Root of the Sum of the Squares (SRSS) method or the “30%” rule, e.g., RH = Max(Rx + 0.3Ry , Ry + 0.3Rx ) [13]. 6. Due to the expected levels of inelastic structural response in some members and regions, dynamic response spectrum analysis shall not be used to determine final design demand values or to assess the performance of the retrofitted structures. Dynamic Time History Analysis
Site specific multisupport dynamic time histories shall be used in a dynamic time history analysis. All analyses incorporating significant nonlinear behavior shall be conducted using nonlinear inelastic dynamic time history procedures. 1. Linear elastic dynamic time history analysis — Linear elastic dynamic time history analysis is defined as dynamic time history analysis with considerations of geometrical linearity 1999 by CRC Press LLC
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(small displacement), linear boundary conditions, and elastic members. It shall only be used to check regional and global models. 2. Nonlinear elastic dynamic time history analysis — Nonlinear elastic time history analysis is defined as dynamic time history analysis with considerations of geometrical nonlinearity, linear boundary conditions, and elastic members. It shall be used to determine areas of inelastic behavior prior to incorporating inelasticity into the regional and global models. 3. Nonlinear inelastic dynamic time history analysis – Level I — Nonlinear inelastic dynamic time history analysis – Level I is defined as dynamic time history analysis with considerations of geometrical nonlinearity, nonlinear boundary conditions, other inelastic elements (for example, dampers) and elastic members. It shall be used for the final determination of force and displacement demands for existing structures in combination with static gravity, wind, thermal, water current, and live load as specified in Section 16.4. 4. Nonlinear inelastic dynamic time history analysis – Level II — Nonlinear inelastic dynamic time history analysis – Level II is defined as dynamic time history analysis with considerations of geometrical nonlinearity, nonlinear boundary conditions, other inelastic elements (for example, dampers) and inelastic members. It shall be used for the final evaluation of response of the structures. Reduced material and section properties, and the yield surface equation suggested in the Appendix may be used for inelastic considerations.
16.6.2
Modeling Considerations
Global, Regional, and Local Models
The global models focus on the overall behavior and may include simplifications of complex structural elements. Regional models concentrate on regional behavior. Local models emphasize the localized behavior, especially complex inelastic and nonlinear behavior. In regional and global models where more than one foundation location is included in the model, multisupport time history analysis shall be used. Boundary Conditions
Appropriate boundary conditions shall be included in the regional models to represent the interaction between the regional model and the adjacent portion of the structure not explicitly included. The adjacent portion not specifically included may be modeled using simplified structural combinations of springs, dashpots, and lumped masses. Appropriate nonlinear elements such as gap elements, nonlinear springs, SRMDs, or specialized nonlinear finite elements shall be included where the behavior and response of the structure is determined to be sensitive to such elements. SoilFoundationStructureInteraction
SoilFoundationStructureInteraction may be considered using nonlinear or hysteretic springs in the global and regional models. Foundation springs at the base of the structure which reflect the dynamic properties of the supporting soil shall be included in both regional and global models. Section Properties of Latticed Members
For latticed members, the procedure proposed in the Appendix may be used for member characterization. 1999 by CRC Press LLC
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Damping
When nonlinear member properties are incorporated in the model, Rayleigh damping shall be reduced, for example by 20%, compared with analysis with elastic member properties. Seismic Response Modification Devices
The SRMDs, i.e., energy dissipation and isolation devices, shall be modeled explicitly using their hysteretic characteristics as determined by tests.
16.7
Determination of Capacities
16.7.1 Limit States and Resistance Factors Limit States
The limit states are defined as those conditions of a structure at which it ceases to satisfy the provisions for which it was designed. Two kinds of limit states corresponding to SEE and FEE specified in Section 16.3 apply for seismic design and retrofit. Resistance Factors
To account for unavoidable inaccuracies in the theory, variation in the material properties, workmanship, and dimensions, nominal strength of structural components should be modified by a resistance factor φ to obtain the design capacity or strength (resistance). The following resistance factors shall be used for seismic design: • • • •
16.7.2
For tension fracture in net section For block shear For bolts and welds For all other cases
φtf φbs φ φ
= 0.8 = 0.8 = 0.8 = 1.0
Effective Length of Compression Members
The effective length factor K for compression members shall be determined in accordance with Chapter 17 of this Handbook.
16.7.3
Nominal Strength of Steel Structures
Members
1. General — Steel members include rolled members and builtup members, such as latticed, battened, and perforated members. The design strength of those members shall be according to applicable provisions of AISCLRFD [4]. Section properties of latticed members shall be determined in accordance with the Appendix. 2. Compression members — For compression members, the nominal strength shall be determined in accordance with Section E2 and Appendix B of AISCLRFD [4]. For builtup members, effects of interaction of buckling modes shall be considered in accordance with the Appendix. A detailed procedure in a flowchart format is shown in Figure 16.8. 3. Tension Members — For tension members, the design strength shall be determined in accordance with Sections D1 and J4 of AISCLRFD [4]. It is the smallest value obtained according to (i) yielding in gross section, (ii) fracture in net section, and (iii) block shear rupture. A detailed procedure is shown in Figure 16.9. 1999 by CRC Press LLC
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FIGURE 16.8: Evaluation procedure for nominal compressive strength of steel members.
4. Flexural members — For flexural members, the nominal flexural strength shall be determined in accordance with Section F1 and Appendices B, F, and G of AISCLRFD [4]. • For critical members, the nominal flexural strength is the smallest value according to (i) initial yielding, (ii) lateraltorsional buckling, (iii) flange local buckling, and (iv) web local buckling. • For other members, the nominal flexural strength is the smallest value according to (i) plastic moment, (ii) lateraltorsional buckling, (iii) flange local buckling, and (iv) web local buckling.
1999 by CRC Press LLC
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FIGURE 16.9: Evaluation procedure for tensile strength of steel members.
Detailed procedures for flexural strength of box and Ishaped members are shown in Figures 16.10 and 16.11, respectively. 5. Nominal shear strength — For solidweb steel members, the nominal shear strength shall be determined in accordance with Appendix F2 of AISCLRFD [4]. For latticed members, the shear strength shall be based on shearflow transfercapacity of lacing bar, battens, and connectors as discussed in the Appendix. A detailed procedure for shear strength is shown in Figure 16.12. 6. Members subjected to bending and axial force — For members subjected to bending and axial force, the evaluation shall be according to Section H1 of AISCLRFD [4], i.e., the bilinear interaction equation shall be used. The recent study on “Cyclic Testing of Latticed Members for San FranciscoOakland Bay Bridge” at UCSD [37] recommends that the AISCLRFD interaction equation can be used directly for seismic evaluation of latticed members. A detailed procedure for steel beamcolumns is shown in Figure 16.13. Gusset Plate Connections
1. General description — Gusset plates shall be evaluated for shear, bending, and axial forces according to Article 6.14.2.8 of AASHTOLRFD [1]. The internal stresses in the gusset plate shall be determined according to Whitmore’s method in which the effective area is defined as the width bound by two 30◦ lines drawn from the first row of the bolt or rivet 1999 by CRC Press LLC
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FIGURE 16.10: Evaluation procedure for nominal flexural strength of boxshaped steel members.
group to the last bolt or rivet line. The stresses in the gusset plate may be determined by more rational methods or refined computer models. 2. Tension strength — The tension capacity of the gusset plates shall be calculated according to Article 6.13.5.2 of AASHTOLRFD [1]. 3. Compressive strength — The compression capacity of the gusset plates shall be calculated according to Article 6.9.4.1 of AASHTOLRFD [1]. In using the AASHTOLRFD Equations (6.9.4.11) and (6.9.4.12), symbol l is the length from the last rivet (or bolt) line on a member to first rivet (or bolt) line on a chord measured along the centerline of the member; K is effective length factor = 0.65; As is average effective crosssection area defined by Whitmore’s method. 4. Limit of free edge to thickness ratio of gusset p plate — When the free edge length to thickness ratio of a gusset plate Lg /t > 1.6 E/Fy , the compression stress of a gusset plate shall be less than 0.8 Fy ; otherwise the plate shall be stiffened. The free edge length to thickness ratio of a gusset plate shall satisfy the following limit specified in Article 6.14.2.8 of AASHTOLRFD [1]. s Lg E ≤ 2.06 (16.2) t Fy When the free edge is stiffened, the following requirements shall be satisfied: • The stiffener plus a width of 10t of gusset plate shall have an l/r ratio less than or equal to 40. 1999 by CRC Press LLC
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FIGURE 16.11: Evaluation procedure for nominal flexural strength of Ishaped steel members.
• The stiffener shall have an l/r ratio less than or equal to 40 between fasteners. • The stiffener moment of inertia shall satisfy [38]: ( Is ≥
q 1.83t 4 (b/t)2 − 144 9.2t 4
(16.3)
where Is = the moment of inertia of the stiffener about its own centroid b
= the width of the gusset plate perpendicular to the edge
t
= the thickness of the gusset plate
5. Inplane moment strength of gusset plate (strong axis) — The nominal moment strength of a gusset plate shall be calculated by the following equation in Article 6.14.2.8 of AASHTOLRFD [1]: Mn = SFy
(16.4)
where S = elastic section modulus about the strong axis 6. Inplane shear strength for a gusset plate — The nominal shear strength of a gusset plate shall be calculated by the following equations: 1999 by CRC Press LLC
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FIGURE 16.12: Evaluation procedure for nominal shear strength of steel members.
Based on gross section: Vn = smaller
0.4Fy Agv 0.6Fy Agv
for flexural shear for uniform shear
(16.5)
0.4Fu Anv 0.6Fu Anv
for flexural shear for uniform shear
(16.6)
Based on net section: Vn = smaller
where Agv = gross area subject to shear Anv = net area subject to shear Fu = minimum tensile strength of the gusset plate 7. Initial yielding of gusset plate in combined inplane moment, shear, and axial load — The initial yielding strength of a gusset plate subjected to a combined inplane moment, shear, and axial load shall be determined by the following equations: Pu Mu + ≤1 Mn Py 1999 by CRC Press LLC
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(16.7)
FIGURE 16.13: Evaluation procedure for steel beamcolumns.
or
Vu Vn
2 +
Pu Py
2 ≤1
(16.8)
where Vu = factored shear Mu = factored moment Pu = factored axial load Mn = nominal moment strength determined by Equation 16.4 Vn = nominal shear strength determined by Equation 16.5 Py = yield axial strength (Ag Fy ) Ag = gross section area of gusset plate 8. Full yielding of gusset plate in combined inplane moment, shear, and axial load — Full yielding strength for a gusset plate subjected to combined inplane moment, shear, and axial load has the form [6]: 1999 by CRC Press LLC
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Mu + Mp
Pu Py
2
+ 1−
4
Vu Vp
Pu Py
2 = 1
(16.9)
where Mp = plastic moment of pure bending (ZFy ) Vp = shear capacity of gusset plate (0.6Ag Fy ) Z = plastic section modulus 9. Block shear capacity — The block shear capacity shall be calculated according to Article 6.13.4 of AASHTOLRFD [1]. 10. Outofplane moment and shear consideration — Moment will be resolved into a couple acting on the near and far side gusset plates. This will result in tension or compression on the respective plates. This force will produce weak axis bending of the gusset plate. Connections Splices
The splice section shall be evaluated for axial tension, flexure, and combined axial and flexural loading cases according to AISCLRFD [4]. The member splice capacity shall be equal to or greater than the capacity of the smaller of the two members being spliced. Eyebars
The tensile capacity of the eyebars shall be calculated according to Article D3 of AISCLRFD [4]. Anchor Bolts (Rods) and Anchorage Assemblies 1. Anchorage assemblies for nonrocking mechanisms shall be anchored with sufficient capacity to develop the lesser of the seismic force demand and plastic strength of the columns. Anchorage assemblies may be designed for rocking mechanisms where yield is permitted — at which point rocking commences. Shear keys shall be provided to prevent excess lateral movement. The nominal shear strength of pipe guided shear keys shall be calculated by:
Rn = 0.6Fy Ap
(16.10)
where Ap = crosssection area of pipe 2. Evaluation of anchorage assemblies shall be based on reinforced concrete structure behavior with bonded or unbonded anchor rods under combined axial load and bending moment. All anchor rods outside of the compressive region may be taken to full minimum tensile strength. 3. The nominal strength of anchor bolts (rods) for shear, tension, and combined shear and tension shall be calculated according to Article 6.13.2 of AASHTOLRFD [1]. 4. Embedment length of anchor rods shall be such that p a ductile failure occurs. Concrete failure surfaces shall be based on a shear stress of 2 fc0 and account for edge distances and overlapping shear zones. In no case should edge distances or embedments be less than those shown in Table 826 of the AISCLRFD Manual [3]. New anchor rods shall be threaded to assure development. 1999 by CRC Press LLC
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Rivets and Holes
1. The bearing capacity on rivet holes shall be calculated according to Article 6.13.2.9 of AASHTOLRFD [1]. 2. Nominal shear strength of a rivet shall be calculated by the following formula: Rn = 0.75βFu Ar Ns
(16.11)
where β = 0.8, reduction factor for connections with more than two rivets and to account for deformation of connected material which causes nonuniform rivet shear force (see Article C6.13.2.7 of AASHTOLRFD [1]) Fu = minimum tensile strength of the rivet Ar = the nominal area of the rivet (before driving) Ns = number of shear planes per rivet It should be pointed out that the 0.75 factor is the ratio of the shear strength τu to the tensile strength Fu of a rivet. The research work by Kulak et al. [31] found that this ratio is independent of the rivet grade, installation procedure, diameter, and grip length and is about 0.75. 3. Tension capacity of a rivet shall be calculated by the following formula: Tn = Ar Fu
(16.12)
4. Tensile capacity of a rivet subjected to combined tension and shear shall be calculated by the following formula: s Tn = Ar Fu 1 −
Vu Rn
(16.13)
where Vu = factored shear force Rn = nominal shear strength of a rivet determined by Equation 16.11
Bolts and Holes
1. The bearing capacity on bolt holes shall be calculated according to Article 6.13.2.9 of AASHTOLRFD [1]. 2. The nominal strength of a bolt for shear, tension, and combined shear and tension shall be calculated according to Article 6.13.2 of AASHTOLRFD [1].
Prying Action
Additional tension forces resulting from prying action must be accounted for in determining applied loads on rivets or bolts. The connected elements (primarily angles) must also be checked for adequate flexural strength. Prying action forces shall be determined from the equations presented in AISCLRFD Manual Volume 2, Part 11 [3].
1999 by CRC Press LLC
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16.7.4
Nominal Strength of Concrete Structures
Nominal Moment Strength
The nominal moment strength Mn shall be calculated by considering combined biaxial bending and axial loads. It is defined as: My (16.14) Mn = smaller Mεc where My = moment corresponding to first steel yield Mεc = moment at which compressive strain of concrete at extreme fiber equal to 0.003 Nominal Shear Strength
The nominal shear strength Vn shall be calculated by the following equations [12, 13]. Vn or Vn
where Ag = As = Vt = D0 = Pu = d = s = fyt = µ1 =
16.7.5
Vc
=
νc
=
Factor 1
=
ρ 00
=
Vs
=
= =
Vc + Vs Vc + Vt
0.8νc Ag p p Pu 2 1 + 2,000A fc0 ≤ 3 fc0 g p larger p Factor 1 × 1 + Pu fc0 ≤ 4 fc0 2,000Ag ρ 00 fyt + 3.67 − µ1 ≤ 3.0 150 volume of transverse reinforcement volume of confined core A f d/s for rectangular sections ν yt A s fyt D 0 for circular sections 2s
(16.15a) (16.15b)
(16.16) (16.17)
(16.18)
(16.19)
gross section area of concrete member crosssectional area of transverse reinforcement within space s shear strength carried by truss mechanism hoop or spiral diameter factored axial load associated with design shear Vu and Pu /Ag is in psi effective depth of section space of transverse reinforcement probable yield strength of transverse steel (psi) ductility demand ratio (1.0 will be used)
Structural Deformation Capacity
Steel Structures
Displacement capacity shall be evaluated by considering both material and geometrical nonlinearity. Proper boundary conditions for various structures shall be carefully adjusted. The ultimate 1999 by CRC Press LLC
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available displacement capacity is defined as the displacement corresponding to a load that drops a maximum of 20% from the peak load. Reinforced Concrete Structures
Displacement capacity shall be evaluated using standalone pushover analysis models. Both the geometrical and material nonlinearities, as well as the foundation (nonlinear soil springs) shall be taken into account. The ultimate available displacement capacity is defined as the displacement corresponding to a maximum of 20% load reduction from the peak load, or to a specified stressstrain failure limit (surface), whichever occurs first. The following parameters shall be used to define stressstrain failure limit (surface): comp
=
σcten
= = = =
σc
comp
εc
σs εs where comp σc σcten fc0 σs εs comp εc
= = = = = =
16.7.6
0.85fc0
p fr = 5 fc0 0.003 Fu 0.12
maximum concrete stress under uniaxial compression maximum concrete stress under uniaxial tension specified compressive concrete strength maximum steel stress under uniaxial tension maximum steel strain under uniaxial tension maximum concrete strain under uniaxial compression
Seismic Response Modification Devices
General
The SRMDs include the energy dissipation and seismic isolation devices. The basic purpose of energy dissipation devices is to increase the effective damping of the structure by adding dampers to the structure thereby reducing forces, deflections, and impact effects. The basic purpose of isolation devices is to change the fundamental mode of vibration so that the structure is subjected to lower earthquake forces. However, the reduction in force may be accompanied by an increase in displacement demand that shall be accommodated within the isolation system and any adjacent structures. Determination of SRMDs Properties
The properties of SRMDs shall be determined by the specified testing program. References are made to AASHTOGuide [2], Caltrans [18], and JMC [30]. The following items shall be addressed rigorously in the testing specification: • • • •
Scales of specimens; at least two fullscale tests are required Loading (including lateral and vertical) history and rate Durability — design life Expected levels of strength and stiffness deterioration
1999 by CRC Press LLC
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16.8
Performance Acceptance Criteria
16.8.1
General
To achieve the performance objectives stated in Section 16.3, the various structural components shall satisfy the acceptable demand/capacity ratios, DCaccept , specified in this section. The general design format is given by the formula: Demand (16.20) ≤ DCaccept Capacity where demand, in terms of various factored forces (moment, shear, axial force, etc.), and deformations (displacement, rotation, etc.) shall be obtained by the nonlinear inelastic dynamic time history analysis – Level I defined in Section 16.6; and capacity, in terms of factored strength and deformations, shall be obtained according to the provisions set forth in Section 16.7. For members subjected to combined loadings, the definition of force D/C ratio:] D/C ratios is given in the Appendix.
16.8.2
Structural Component Classifications
Structural components are classified into two categories: critical and other. It is the aim that other components may be permitted to function as “fuses” so that the critical components of the bridge system can be protected during FEE and SEE. As an example, Table 16.1 shows structural component classifications and their definition for SFOBB West Span components. TABLE 16.1
Structural Component Classification
Component classification
Critical
Other
Definition
Example (SFOBB West Spans)
Components on a critical path that carry bridge gravity load directly. The loss of capacity of these components would have serious consequences on the structural integrity of the bridge
Suspension cables Continuous trusses Floor beams and stringers Tower legs Central anchorage AFrame Piers W1 and W2 Bents A and B Caisson foundations Anchorage housings Cable bents
All components other than critical
All other components
Note: Structural components include members and connections.
16.8.3
Steel Structures
General Design Procedure
Seismic design of steel members shall be in accordance with the procedure shown in Figure 16.14. Seismic retrofit design of steel members shall be in accordance with the procedure shown in Figure 16.15. Connections
Connections shall be evaluated over the length of the seismic event. For connecting members with force D/C ratios larger than one, 25% greater than the nominal capacities of the connecting members shall be used for connection design. 1999 by CRC Press LLC
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FIGURE 16.14: Steel member seismic design procedure.
General Limiting Slenderness Parameters and WidthThickness Ratios
For all steel members regardless of their force D/C ratios, slenderness parameters λc for axial load dominant members, and λb for flexural dominant members shall not exceed the limiting values (0.9λcr or 0.9λbr for critical, λcr or λbr for other) shown in Table 16.2. Acceptable Force D/C Ratios and Limiting Values
Acceptable force D/C ratios, DCaccept and associated limiting slenderness parameters and widththickness ratios for various members are specified in Table 16.2. For all members with D/C ratios larger than one, slenderness parameters and width thickness ratios shall not exceed the limiting values specified in Table 16.2. For existing steel members with D/C ratios less than one, widththickness ratios may exceed λr specified in Table 16.3 and AISCLRFD [4]. The following symbols are used in Table 16.2: Mn = nominal moment strength of a member determined by Section 16.7 Pn = nominal axial strength of a member determined by Section 16.7 λ = widththickness (b/t or h/tw ) ratio of compressive elements p λc = (KL/rπ) Fy /E, slenderness parameter of axial load dominant members λb = L/ry , slenderness parameter of flexural moment dominant members 1999 by CRC Press LLC
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FIGURE 16.15: Steel member seismic retrofit design procedure.
TABLE 16.2 Acceptable Force Demand/Capacity Ratios and Limiting Slenderness Parameters and Width/Thickness Ratios Limiting ratios Slenderness parameter (λc and λb )
Width/thickness λ (b/t or h/tw )
force D/C ratio D/Caccept
Axial load dominant Pu /Pn ≥ Mu /Mn Flexural moment dominant Mu /Mn > Pu /Pn
0.9 λcr λcpr λcp 0.9 λbr λbpr λbp
λr λpr λp λr λpr λp
DCr = 1.0 1.0 ∼ 1.2 DCp = 1.2 DCr = 1.0 1.2 ∼ 1.5 DCp = 1.5
Axial load dominant Pu /Pn ≥ Mu /Mn Flexural moment dominant Mu /Mn > Pu /Pn
λcr λcpr λcp λbr λbpr λbp
λr λpr
DCr = 1.0 1.0 ∼ 2.0 DCp = 2 DCr = 1.0 1.0 ∼ 2.5 DCp = 2.5
Member classification
Critical
Other
1999 by CRC Press LLC
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Acceptable
λp−Seismic λr λpr λp−Seismic
TABLE 16.3
Limiting WidthThickness Ratio
1999 by CRC Press LLC
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λcp = 0.5, limiting column slenderness parameter for 90% of the axial yield load based on AISCLRFD [4] column curve λbp = limiting beam slenderness parameter for plastic moment for seismic design λcr = 1.5, limiting column slenderness parameter for elastic buckling based on AISCLRFD [4] column curve λbr = limiting beam slenderness parameter for elastic lateral torsional buckling
λbr
=
Mr
=
X1
=
where A = L = J = r = ry = Fyw = Fyf = E = G = Sx = Iy = Cw =
√ 57,000 J A rMr X1 FL
1+
q
for solid rectangular bars and box sections 1 + X2 FL2
for doubly symmetric Ishaped members and channels
FL Sx for Ishaped member Fyf Sx for solid rectangular and box section r π EGJ A 4Cw Sx 2 Fyw ; X2 = ; FL = smaller F Sx 2 Iy GJ yf − Fr
crosssectional area, in.2 unsupported length of a member torsional constant, in.4 radius of gyration, in. radius of gyration about minor axis, in. yield stress of web, ksi yield stress of flange, ksi modulus of elasticity of steel (29,000 ksi) shear modulus of elasticity of steel (11,200 ksi) section modulus about major axis, in.3 moment of inertia about minor axis, in.4 warping constant, in.6
For doubly symmetric and singly symmetric Ishaped members with compression flange equal to or larger than the tension flange, including hybrid members (strong axis bending): [3,600+2,200(M1 /M2 )] for other members Fy (16.21) λbp = for critical members √300 Fyf
in which = larger moment at end of unbraced length of beam M1 = smaller moment at end of unbraced length of beam M2 (M1 /M2 ) = positive when moments cause reverse curvature and negative for single curvature For solid rectangular bars and symmetric box beam (strong axis bending): ( λbp =
[5,000+3,000(M1 /M2 )] √ Fy 3,750 JA Mp
≥
in which Mp = plastic moment (Zx Fy ) Zx = plastic section modulus about major axis 1999 by CRC Press LLC
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3,000 Fy
for other members for critical members
(16.22)
FIGURE 16.16: Typical crosssections for steel members (SFOBB west spans). λr , λp , λp−Seismic are limiting width thickness ratios specified by Table 16.3 h DCp −DCaccept i λp + λr − λp for critical members DCp −DCr i h λpr = DC −DC p accept λ for other members p−Seismic + λr − λp−Seismic DCp −DCr For axial load dominant members (Pu /Pn ≥ Mu /Mn ) DCp −DCaccept λcp + 0.9λcr − λcp for critical members DCp −DCr λcpr = λ + λ − λ DCp −DCaccept for other members cp cr cp DCp −DCr For flexural moment dominant members (Mu /Mn > Pu /Pn ) DCp −DCaccept λbp + 0.9λbr − λbp for critical members DCp −DCr λbpr = DCp −DCaccept λ + λ −λ for other members bp br bp DCp −DCr 1999 by CRC Press LLC
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(16.23)
(16.24)
(16.25)
16.8.4
Concrete Structures
General
For all concrete compression members regardless of their force D/C ratios, slenderness parameters KL/r shall not exceed 60. For critical components, force DCaccept = 1.2 and deformation DCaccept = 0.4. For other components, force DCaccept = 2.0 and deformation DCaccept = 0.67. BeamColumn (Bent Cap) Joints
For concrete box girder bridges, the beamcolumn (bent cap) joints shall be evaluated and designed in accordance with the following guidelines [16, 40]: 1. Effective Superstructure Width — The effective width of a superstructure (box girder) on either side of a column to resist longitudinal seismic moment at bent (support) shall not be taken as larger than the superstructure depth. • The immediately adjacent girder on either side of a column within the effective superstructure width is considered effective. • Additional girders may be considered effective if refined bentcap torsional analysis indicates that the additional girders can be mobilized. 2. Minimum BentCap Width — Minimum cap width outside the column shall not be less than D/4 (D is column diameter or width in that direction) or 2 ft (0.61 m). 3. Acceptable Joint Shear Stress • For p existing unconfined joints, acceptable principal tensile stress shall be taken as p 3.5 fc0 psi (0.29 fc0 MPa). If the principal tensile stress demand exceeds this value, the joint shear reinforcement specified in (4) shall be provided. p • For new joints, acceptable principal tensile stress shall be taken as 12 fc0 psi (1.0 p fc0 MPa). • For existing and new joints, acceptable principal compressive stress shall be taken as 0.25 fc0 . 4. Joint Shear Reinforcement • Typical flexure and shear reinforcement (see Figures 16.17 and 16.18) in bent caps shall be supplemented in the vicinity of columns to resist joint shear. All joint shear reinforcement shall be well distributed and provided within D/2 from the face of column. • Vertical reinforcement including cap stirrups and added bars shall be 20% of the column reinforcement anchored into the joint. Added bars shall be hooked around main longitudinal cap bars. Transverse reinforcement in the joint region shall consist of hoops with a minimum reinforcement ratio of 0.4 (column steel area)/(embedment length of column bar into the bent cap)2 . • Horizontal reinforcement shall be stitched across the cap in two or more intermediate layers. The reinforcement shall be shaped as hairpins, spaced vertically at not more than 18 in. (457 mm). The hairpins shall be 10% of column reinforcement. Spacing shall be denser outside the column than that used within the column. • Horizontal side face reinforcement shall be 10% of the main cap reinforcement including top and bottom steel. 1999 by CRC Press LLC
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FIGURE 16.17: Example cap joint shear reinforcement — skews 0◦ to 20◦ . • For bent caps skewed greater than 20◦ , the vertical Jbars hooked around longitudinal deck and bent cap steel shall be 8% of the column steel (see Figure 16.18). The Jbars shall be alternatively 24 in. (600 mm) and 30 in. (750 mm) long and placed within a width of the column dimension on either side of the column centerline. • All vertical column bars shall be extended as high as practically possible without interfering with the main cap bars.
16.8.5
Seismic Response Modification Devices
General
Analysis methods specified in Section 16.6 shall apply for determining seismic design forces and displacements on SRMDs. Properties or capacities of SRMDs shall be determined by specified tests. Acceptance Criteria
SRMDs shall be able to perform their intended function and maintain their design parameters for the design life (for example, 40 years) and for an ambient temperature range (for example from 30◦ 1999 by CRC Press LLC
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FIGURE 16.18: Example cap joint shear reinforcement — skews > 20◦ .
to 125◦ F). The devices shall have accessibility for periodic inspections, maintenance, and exchange. In general, the SRMDs shall satisfy at least the following requirements:
• To remain stable and provide increasing resistance with the increasing displacement. Stiffness degradation under repeated cyclic load is unacceptable. • To dissipate energy within the design displacement limits. For example: provisions may be made to limit the maximum total displacement imposed on the device to prevent device displacement failure, or the device shall have a displacement capacity 50% greater than the design displacement. • To withstand or dissipate the heat buildup during reasonable seismic displacement time history. • To survive for the number of cycles of displacement expected under wind excitation during the life of the device and to function at maximum wind force and displacement levels for at least, for example, five hours.
1999 by CRC Press LLC
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Defining Terms
Bridge: A structure that crosses over a river, bay, or other obstruction, permitting the smooth and safe passage of vehicles, trains, and pedestrians. Buckling model interaction: A behavior phenomenon of compression builtup member; that is, interaction between the individual (or local) buckling mode and the global buckling mode. Builtup member: A member made of structural metal elements that are welded, bolted, and/or riveted together. Capacity: factored strength and deformation capacity obtained according to specified provisions. Critical components: Structural components on a critical path that carry bridge gravity load directly. The loss of capacity of these components would have serious consequences on the structural integrity of the bridge. Damage index: A ratio of elastic displacement demand to ultimate displacement. D/C ratio: A ratio of demand to capacity. Demands: In terms of various forces (moment, shear, axial force, etc.) and deformation (displacement, rotation, etc.) obtained by structural analysis. Ductility: A nondimensional factor, i.e., ratio of ultimate deformation to yield deformation. Effective length factor K: A factor that when multiplied by actual length of the endrestrained column gives the length of an equivalent pinended column whose elastic buckling load is the same as that of the endrestrained column Functionality evaluation earthquake (FEE): An earthquake that has a mean return period of 300 years with a 40% probability of exceedance during the expected life of the bridge. Latticed member: A member made of metal elements that are connected by lacing bars and batten plates. LRFD (Load and Resistance Factor Design): A method of proportioning structural components (members, connectors, connecting elements, and assemblages) such that no applicable limit state is exceeded when the structure is subjected to all appropriate load combinations. Limit states: Those conditions of a structure at which it ceases to satisfy the provision for which it was designed. Nocollapsebased design: Design that is based on survival limit state. The overall design concern is to prevent the bridge from catastrophic collapse and to save lives. Other components: All components other than critical. Performancebased seismic design: Design that is based on bridge performance requirements. The design philosophy is to accept some repairable earthquake damage and to keep bridge functional performance after earthquakes. Safety evaluation earthquake (SEE): An earthquake that has a mean return period in the range of 1000 to 2000 years. Seismic design: Design and analysis considering earthquake loads. Seismic response modification devices (SRMDs): Seismic isolation and energy dissipation devices including isolators, dampers, or isolation/dissipation (I/D) devices. Ultimate deformation: Deformation refers to a loading state at which structural system or a structural member can undergo change without losing significant loadcarrying capacity. 1999 by CRC Press LLC
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The ultimate deformation is usually defined as the deformation corresponding to a load that drops a maximum of 20% from the peak load. Yield deformation: Deformation corresponds to the points beyond which the structure starts to respond inelastically.
Acknowledgments First, we gratefully acknowledge the support from Professor WaiFah Chen, Purdue University. Without his encouragement, drive, and review, this chapter would not have been done in this timely manner. Much of the material presented in this chapter was taken from the San Francisco – Oakland Bay Bridge West Spans Seismic Retrofit Design Criteria (Criteria) which was developed by Caltrans engineers. Substantial contribution to the Criteria came from the following: Lian Duan, Mark Reno, Martin Pohll, Kevin Harper, Rod Simmons, Susan Hida, Mohamed Akkari, and Brian Sutliff. We would like to acknowledge the careful review of the SFOBB Criteria by Caltrans engineers: Abbas Abghari, Steve Altman, John Fujimoto, Don Fukushima, Richard Heninger, Kevin Keady, John Kung, Mike Keever, Rick Land, Ron Larsen, Brian Maroney, Steve Mitchell, Ramin Rashedi, Jim Roberts, Bob Tanaka, Vinacs Vinayagamoorthy, Ray Wolfe, Ray Zelinski, and Gus Zuniga. The SFOBB Criteria was also reviewed by the Caltrans Peer Review Committee for the Seismic Safety Review of the Toll Bridge Retrofit Designs: Chuck Seim (Chairman), T.Y. Lin International; Professor Frieder Seible, University of California at San Diego; Professor Izzat M. Idriss, University of California at Davis; and Gerard Fox, Structural Consultant in New York. We are thankful for their input. We are also appreciative of the review and suggestions of IHong Chen, Purdue University; Professor Ahmad Itani, University of Nevada at Reno; Professor Dennis Mertz, University of Delaware; and Professor ChiaMing Uang, University of California at San Diego. We express our sincere thanks to Enrico Montevirgen and Jerry Helm for their careful preparation of figures. Finally, we gratefully acknowledge the continuous support of the California Department of Transportation.
References [1] AASHTO. 1994. LRFD Bridge Design Specifications, 1st ed., American Association of State Highway and Transportation Officials, Washington, D.C. [2] AASHTO. 1997. Guide Specifications for Seismic Isolation Design, American Association of State Highway and Transportation Officials, Washington, D.C. [3] AISC. 1994. Manual of Steel Construction — Load and Resistance Factor Design, Vol. 12, 2nd ed., American Institute of Steel Construction, Chicago, IL. [4] AISC. 1993. Load and Resistance Factor Design Specification for Structural Steel Buildings, 2nd ed., American Institute of Steel Construction, Chicago, IL. [5] AISC. 1992. Seismic Provisions for Structural Steel Buildings, American Institute of Steel Construction, Chicago, IL. [6] ASCE. 1971. Plastic Design in Steel — A Guide and Commentary, 2nd ed., American Society of Civil Engineers, New York. [7] AstanehAsl, A. and Roberts, J. eds. 1996. Seismic Design, Evaluation and Retrofit of Steel Bridges, Proceedings of the Second U.S. Seminar, San Francisco, CA. 1999 by CRC Press LLC
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[8] AWS. 1995. Bridge Welding Code. (ANSI/AASHTO/AWS D1.595), American Welding Society, Miami, FL. [9] Bazant , Z. P. and Cedolin, L. 1991. Stability of Structures, Oxford University Press, New York. [10] Caltrans. 1993. Design Criteria for I880 Replacement, California Department of Transportation, Sacramento, CA. [11] Caltrans. 1994. Design Criteria for SR14/I5 Replacement, California Department of Transportation, Sacramento, CA. [12] Caltrans. 1995. Bridge Design Specifications, California Department of Transportation, Sacramento, CA. [13] Caltrans. 1995. Bridge Memo to Designers (204), California Department of Transportation, Sacramento, CA. [14] Caltrans. 1996. Guidelines for Generation of ResponseSpectrumCompatible Rock Motion Time History for Application to Caltrans Toll Bridge Seismic Retrofit Projects, Caltrans Seismic Advisory Board, California Department of Transportation, Sacramento, CA. [15] Caltrans. 1996. Guidelines for Performing Site Response Analysis to Develop Seismic Ground Motions for Application to Caltrans Toll Bridge Seismic Retrofit Projects, Caltrans Seismic Advisory Board, California Department of Transportation, Sacramento, CA. [16] Caltrans. 1996. Seismic Design Criteria for retrofit of the West Approach to the San FranciscoOakland Bay Bridge, Prepared by Keever, M., California Department of Transportation, Sacramento, CA. [17] Caltrans. 1997. San Francisco – Oakland Bay Bridge West Spans Seismic Retrofit Design Criteria, Prepared by Reno, M. and Duan, L., edited by Duan, L., California Department of Transportation, Sacramento, CA. [18] Caltrans. 1997. Full Scale Isolation Bearing Testing Document (Draft), Prepared by Mellon, D., California Department of Transportation, Sacramento, CA. [19] Duan, L. and Chen, W.F. 1990. “A Yield Surface Equation for Doubly Symmetrical Section,” Structural Eng., 12(2), 114119. CRC [20] Duan, L. and Cooper, T. R. 1995. “Displacement Ductility Capacity of Reinforced Concrete Columns,” ACI Concrete Int., 17(11). 6165. [21] Duan, L. and Reno, M. 1995. “Section Properties of Latticed Members,” Research Report, California Department of Transportation, Sacramento, CA. [22] Duan, L., Reno, M., and Uang, C.M. 1997. “Buckling Model Interaction for Compression Builtup Members,” AISC Eng. J. (in press). [23] ENR. 1997. Seismic Superstar Billion Dollar California Freeway — Cover story: Rising form the Rubble, New Freeway Soars and Swirls Near Quake, Engineering News Records, Jan. 20, McGrawHill, New York. [24] FHWA. 1987. Seismic Design and Retrofit Manual for Highway Bridges, Report No. FHWAIP876, Federal Highway Administration, Washington, D.C. [25] FHWA. 1995. Seismic Retrofitting Manual for Highway Bridges, Publication No. FHWARD94052, Federal Highway Administration, Washington, D.C. [26] Goyal, A. and Chopra, A.K. 1989. Earthquake Analysis & Response of IntakeOutlet Towers, EERC Report No. UCB/EERC89/04, University of California, Berkeley, CA. [27] Housner, G.W. 1990.Competing Against Time, Report to Governor George Deuknejian from The Governor’s Broad of Inquiry on the 1989 Loma Prieta Earthquake, Sacramento, CA. [28] Housner, G.W. 1994. The Continuing Challenge — The Northridge Earthquake of January 17, 1994, Report to Director, California Department of Transportation, Sacramento, CA. [29] Institute of Industrial Science (IIS). 1995. Incede Newsletter, Special Issue, International Center for DisasterMitigation Engineering, Institute of Industrial Science, The University of Tokyo, Japan. 1999 by CRC Press LLC
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[30] Japan Ministry of Construction (JMC). 1994. Manual of Menshin Design of Highway Bridges, English Version: EERC, Report 94/10, University of California, Berkeley, CA. [31] Kulak, G.L., Fisher, J.W., and Struik, J.H. 1987. Guide to the Design Criteria for Bolted and Riveted Joints, 2nd ed., John Wiley & Sons, New York. [32] Liew, J.Y.R. 1992. “Advanced Analysis for Frame Design,” Ph.D. Dissertation, Purdue University, West Lafayette, IN. [33] McCormac, J. C. 1989. Structural Steel Design, LRFD Method, Harper & Row, New York. [34] Park, R. and Paulay, T. 1975. Reinforced Concrete Structures, John Wiley & Sons, New York. [35] Priestley, M.J.N., Seible, F., and Calvi, G.M. 1996. Seismic Design and Retrofit of Bridges, John Wiley & Sons, New York. [36] Salmon, C. G. and Johnson, J. E. 1996. Steel Structures: Design and Behavior, Emphasizing Load and Resistance Factor Design, Fourth ed., HarperCollins College Publishers, New York. [37] Uang, C. M. and Kleiser, M. 1997. “Cyclic Testing of Latticed Members for San FranciscoOakland Bay Bridge,” Final Report, Division of Structural Engineering, University of California at San Diego, La Jolla, CA. [38] USS. 1968. Steel Design Manual, Brockenbrough, R.L. and Johnston, B.G., Eds., United States Steel Corporation, ADUSS 27340002, Pittsburgh, PA. [39] Xie, L. L. and Housner, G. W. 1996. The Greater Tangshan Earthquake, Vol. I and IV, California Institute of Technology, Pasadena, CA. [40] Zelinski, R. 1994. Seismic Design Momo Various Topics Preliminary Guidelines, California Department of Transportation, Sacramento, CA.
Further Reading [1] Chen, W.F. and Duan, L. 1998. Handbook of Bridge Engineering, (in press) CRC Press, Boca Raton, FL. [2] Clough, R.W. and Penzien, J. 1993. Dynamics of Structures, 2nd ed., McGrawHill, New York. [3] Fukumoto, Y. and Lee, G. C. 1992. Stability and Ductility of Steel Structures under Cyclic Loading, CRC Press, Boca Raton, FL. [4] Gupta, A.K. 1992. Response Spectrum Methods in Seismic Analysis and Design of Structures, CRC Press, Boca Raton, FL.
Appendix A 16.A.1 Section Properties for Latticed Members This section presents practical formulas proposed by Duan and Reno [21] for calculating section properties for latticed members. Concept
It is generally assumed that section properties can be computed based on crosssections of main components if the lacing bars and battens can assure integral action of the solid main components [33, 36]. To consider actual section integrity, reduction factors βm for moment of inertia, and βt for torsional constant are proposed depending on shearflow transferringcapacity of lacing bars and connections. For clarity and simplicity, typical latticed members as shown in Figure 16.19 are discussed. 1999 by CRC Press LLC
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FIGURE 16.19: Typical latticed members. Section Properties
1. Crosssectional area — The contribution of lacing bars is assumed negligible. The crosssectional area of latticed member is only based on main components. A=
X
Ai
(16.26)
where Ai is crosssectional area of individual component i. 2. Moment of inertia — For lacing bars or battens within web plane (bending about yy axis in Figure 16.19) Iy−y =
X
I(y−y)i + βm
X
Ai xi2
(16.27)
where Iy−y = moment of inertia of a section about yy axis considering shear transferring capacity = moment of inertia of individual component i Ii = distance between yy axis and center of individual component i xi βm = reduction factor for moment of inertia and may be determined by the following formula:
1999 by CRC Press LLC
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For laced member (Figure 16.19a): ( m sin φ × smaller of βm
comp
mlacing Pn
+ Pnten
mlacing nr Ar (0.6Fu ) Fyf Af
=
≤ 1.0
(16.28a)
For battened member (Figure 16.19b)
βm
=
mbatten Ab 0.6Fyw m × smaller of mbatten 2Mp−batten / h mbatten nr Ar (0.6Fu ) Fyf Af
≤ 1.0
(16.28b)
in which φ = the angle between the diagonal lacing bar and the axis perpendicular to the member axis (see Figure 16.19) = crosssectional area of batten plate Ab = flange area Af Fyf = yield strength of flange = yield strength of web member (battens or lacing bars) Fyw = ultimate strength of rivets Fu m = number of panels between point of maximum moment to point of zero moment to either side (as an approximation, half of member length L/2 may be used) = number of batten planes mbatten = number of lacing planes mlacing = number of rivets of connecting lacing bar and main component at one nr joint h = depth of member in lacing plane = nominal area of rivet Ar Mp−batten = plastic moment of a batten plate about strong axis comp = nominal compressive strength of lacing bar and can be determined by Pn AISCLRFD [4] column curve = nominal tensile strength of lacing bar and can be determined by AISCPnten LRFD [4] Since the section integrity mainly depends on the shear transference between various components, it is rational to introduce the βm factor in Equation 16.27. As seen in Equations 16.28a and 16.28b, βm is defined as the ratio of the shear capacity transferred by lacing bars/battens and connections to the shearflow (Fyf Af ) required by the plastic bending moment of a fully integral section. For laced members, the shear transferring capacity is controlled by either lacing bars or connecting rivets, the smaller of the two values should be used in Equation 16.28a. For battened members, the shear transferring capacity is controlled by either pure shear strength of battens (0.6 Fyw Ab ), or flexural strength of battens or connecting rivets, the smaller of the three values should be used in Equation 16.28b. It is important to point out that the limiting value unity for βm implies a fully integral section when shear can be transferred fully by lacings and connections. For lacing bars or battens within flange plane (bending about xx axis in Figure 16.19).
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The contribution of lacing bars is assumed negligible and only the main components are considered. X X I(x−x)i + Ai yi2 (16.29) Ix−x = 3. Elastic section modulus S=
I C
(16.30)
where S = elastic section modulus of a section C = distance from elastic neutral axis to extreme fiber 4. Plastic section modulus For lacing bars or battens within flange plane (bending about xx axis in Figure 16.19) Zx−x =
X
yi∗ A∗i
(16.31)
For lacing bars or battens within web plane (bending about yy axis in Figure 16.19) Zy−y = βm where Z = xi∗ = yi∗ = A∗i =
X
xi∗ A∗i
(16.32)
plastic section modulus of a section about plastic neutral axis distance between center of gravity of a section A∗i and plastic neutral yy axis distance between center of gravity of a section A∗i and plastic neutral xx axis crosssection area above or below plastic neutral axis
It should be pointed out that the plastic neutral axis is generally different from the elastic neutral axis. The plastic neutral axis is defined by equal plastic compression and tension forces for this section. 5. Torsional constant For a boxshaped section J =
4 (Aclose )2 P bi
(16.33)
ti
For an open thinwalled section J =
X bi t 3 i
3
(16.34)
where Aclose = area enclosed within mean dimension for a box = length of a particular segment of the section bi = average thickness of segment bi ti For determination of torsional constant of a latticed member, it is proposed that the lacing bars or batten plates be replaced by reduced equivalent thinwalled plates defined as:
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Aequiv = βt A∗equiv
(16.35)
For laced member (Figure 16.19a) A∗equiv
=
3.12Ad sin φ cos2 φ
(16.36a)
For battened member (Figure 16.19b) A∗equiv
=
tequiv =
74.88 2ah Ib
+
(16.36b)
a2 If
Aequiv h
(16.37)
where a = distance between two battens along member axis Aequiv = crosssection area of a thinwalled plate equivalent to lacing bars considering shear transferring capacity A∗equiv = crosssection area of a thinwalled plate equivalent to lacing bars or battens assuming full section integrity tequiv = thickness of equivalent thinwalled plate = crosssectional area of all diagonal lacings in one panel Ad = moment of inertia of a batten plate Ib If = moment of inertia of a side solid flange about the weak axis = reduction factor for torsion constant may be determined by the following βt formula: For laced member (Figure 16.19a) cos φ × smaller of βt
=
comp
Pn + Pnten nr Ar (0.6Fu )
0.6Fyw A∗equiv
≤ 1.0
(16.38a)
For battened member (Figure 16.19b)
βt
=
Ab 0.6Fyw h/a 2Mp−batten /a smaller of nr Ar (0.6Fu ) h/a 0.6Fyw A∗equiv
≤ 1.0
(16.38b)
The torsional integrity is from lacings and battens. A reduction factor βt , similar to that used for the moment of inertia, is introduced to consider section integrity when the lacing is weaker than the solid plate side of the section. βt factor is defined as the ratio of the shear capacity transferred by lacing bars and connections to the shearflow (0.6Fyw A∗equiv ) required by the equivalent thinwalled plate. It is seen that the limiting value of unity for βt implies a fully integral section when shear in the equivalent thinwalled plate can be transferred fully by lacings and connections. 1999 by CRC Press LLC
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Based on the equal lateral stiffness principle, an equivalent thinwalled plate for a lacing plane, Equation 16.36a and 16.36b can be obtained by considering E/G = 2.6 for steel material and shape factor for shear n = 1.2 for a rectangular section. 6. Warping constant For a boxshaped section Cw ≈ 0
(16.39)
For an Ishaped section Cw =
If h2 2
(16.40)
where If = moment of inertia of one solid flange about the weak axis (perpendicular to the flange) of the crosssection h = distance between center of gravity of two flanges
16.A.2 Buckling Mode Interaction For Compression Builtup members An important phenomenon of the behavior of these compressive builtup members is the interaction of buckling modes [9]; i.e., interaction between the individual (or local) component buckling (Figure 16.20a) and the global member buckling (Figure 16.20b). This section presents the practical approach proposed by Duan, Reno, and Uang [22] that may be used to determine the effects of interaction of buckling modes for capacity assessment of existing builtup members. Buckling Mode Interaction Factor
To consider buckling mode interaction between the individual components and the global member, it is proposed that the usual effective length factor K of a builtup member be multiplied by a buckling mode interaction factor, γLG , that is, KLG = γLG K
(16.41)
where K = usual effective length factor of a builtup member KLG = effective length factor considering buckling model interaction Limiting Effective Slenderness Ratios
For practical design, the following two limiting effective slenderness ratios are suggested for consideration of buckling mode interaction: p (16.42) Ka a/rf = 1.1 E/F Ka a/rf = 0.75(KL/r) (16.43) where (Ka a/rf ) is the largest effective slenderness ratio of individual components between connectors and (KL/r) is governing effective slenderness ratio of a builtup member. The first limit Equation 16.42 is based on the argument that if an individual component is very short, the failure mode of the component would be material yielding (say 95% of the yield load), not member buckling. This implies that no interaction of buckling modes occurs when an individual component is very short. 1999 by CRC Press LLC
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FIGURE 16.20: Buckling modes of builtup members.
The second limit Equation 16.43 is set forth by the current design specifications. Both the AISCLRFD [4] and AASHTOLRFD [1] imply that when the effective slenderness ratio (Ka a/ri ) of each individual component between the connectors does not exceed 75% of the governing effective slenderness ratio of the builtup member, no strength reduction due to interaction of buckling modes needs to be considered. The study reported by Duan, Reno, and Uang [22] has justified that the rule of (Ka a/ri ) < 0.75(KL/r) is consistent with the theory. Analytical Equation
The buckling mode interaction factor γLG is defined as: s s PG π 2 EI = γLG = Pcr (KL)2 Pcr where Pcr = elastic buckling load considering buckling mode interaction PG = elastic buckling load without considering buckling mode interaction L = unsupported member length I = moment of inertia of a builtup member γLG can be computed by the following equation: v u 1 + α2 u γLG = u u α2 u1 + u (δo /a)2 (Ka a/rf )2 u 1+ " #3 t (Ka a/r )2 2 1−
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f (γLG KL/r)2
(16.44)
(16.45)
where (δo /a) = imperfection (outofstraightness) parameter of individual component (see Figure 16.21) α = separation as defined as: h α= 2
s
Af h = If 2rf
(16.46)
in which If = moment inertia of one side individual components (see Figure 16.21) Af = crosssection area of one side individual components (see Figure 16.21) h = depth of latticed member, distance between center of gravity of two flanges in lacing plane (see Figure 16.21) r = radius of gyration of a builtin member rf = radius of gyration of individual component
FIGURE 16.21: Typical crosssection and local components. For widely separated builtup members with α ≥ 2, the buckling mode interaction factor γLG can be accurately estimated by the following equation on the conservative side:
γLG
v u u (δo /a)2 (Ka a/rj )2 =u i t1 + h (K a/rj )2 3 2 1 − (γ aKL/r) 2
(16.47)
LG
Graphical Solution
Although γLG can be obtained by solving Equation 16.45, an iteration procedure must be used. For design purposes, solutions in chart forms are more desirable. Figures 16.22 to 16.24 provide engineers with alternative graphic solutions for widely separated builtup members with α ≥ 2. In these figures, the outofstraightness ratios (δo /a) considered are 1/500, 1/1000, and 1/1500, and the effective slenderness ratios (KL/r) considered are 20, 40, 60, 100, and 140. In all these figures, the top line represents KL/r = 20 and the bottom line represents KL/r = 140. 1999 by CRC Press LLC
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FIGURE 16.22: Buckling mode interaction factor γLG for δ0 /a = 1/500.
FIGURE 16.23: Buckling mode interaction factor γLG for δ0 /a = 1/1000.
16.A.3 Acceptable Force D/C Ratios and Limiting Values Since it is uneconomical and impossible to design bridges to withstand seismic forces elastically, the nonlinear inelastic responses of the bridges are expected. The performancebased criteria accepts certain seismic damage in some other components so that the critical components and the bridges will be kept essentially elastic and functional after the SEE and FEE. This section presents the concept of acceptable force D/C ratios, limiting member slenderness parameters and limiting widththickness ratios, as well as expected ductility. 1999 by CRC Press LLC
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FIGURE 16.24: Buckling mode interaction factor γLG for δ0 /a = 1/1500.
FIGURE 16.25: Definition of force D/C ratios for combined loadings.
Definition of Force Demand/Capacity (D/C) Ratios
For members subjected to an individual load, force demand is defined as a factored individual force, such as factored moment, shear, axial force, etc. which shall be obtained by the nonlinear dynamic time history analysis – Level I specified in Section 16.6, and capacity is defined according to the provisions in Section 16.7. For members subjected to combined loads, force D/C ratio is based on the force interaction. For example, for a member subjected to combined axial load and bending moment (Figure 16.25), force demand D is defined as the distance from the origin point O(0, 0) to the factored force point d(Pu , Mu ), and capacity C is defined as the distance from the origin point O(0, 0) to the point c(P ∗ , M ∗ ) on the specified interaction surface or curve (failure surface or curve). 1999 by CRC Press LLC
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Ductility and LoadDeformation Curves
“Ductility” is usually defined as a nondimensional factor, i.e., ratio of ultimate deformation to yield deformation [20, 34]. It is normally expressed by two forms: 1. curvature ductility (µφ = φu /φy ) 2. displacement ductility (µ1 = 1u /1y ) Representing section flexural behavior, curvature ductility is dependent on the section shape and material properties. It is based on the momentcurvature curve. Indicating structural system or member behavior, displacement ductility is related to both the structural configuration and its section behavior. It is based on the loaddisplacement curve. A typical loaddeformation curve including both the ascending and descending branches is shown in Figure 16.26. The yield deformation (1y or φy ) corresponds to a loading state beyond which the structure starts to respond inelastically. The ultimate deformation (1u or φu ) refers to the a loading state at which a structural system or a structural member can undergo without losing significant loadcarrying capacity. It is proposed that the ultimate deformation (curvature or displacement) is defined as the deformation corresponding to a load that drops a maximum of 20% from the peak load.
FIGURE 16.26: Loaddeformation curves.
Force D/C Ratios and Ductility
The following discussion will give engineers a direct measure of seismic damage of structural components during an earthquake. Figure 16.27 shows a typical loadresponse curve for a single degree of freedom system. Displacement ductility is: µ1 =
1u 1y
(16.48)
A new term, Damage Index, is defined herein as the ratio of elastic displacement demand to ultimate 1999 by CRC Press LLC
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FIGURE 16.27: Response of a single degree of freedom of system. displacement: D1 =
1ed 1u
(16.49)
When damage index D1 < 1/µ1 (1ed < 1y ), it implies that no damage occurs and the structure responds elastically; 1/µ1 < D1 < 1.0 indicates certain damages occur and the structure responds inelastically; D1 > 1.0, however, means that a structural system collapses. Based on the equal displacement principle, the following relationship is obtained as: 1ed Force Demand = µ1 D1 = Force Capacity 1y
(16.50)
It is seen from Equation 16.50 that the force D/C ratio is related to both the structural characters in terms of ductility µ1 and the degree of damage in terms of damage index D1 . Table 16.4 shows detailed data for this relationship. TABLE 16.4 Index Force D/C ratio 1.0 1.2 1.5 2.0 2.5
Force D/C Ratio and Damage
Damage index D1 No damage 0.4 0.5 0.67 0.83
Expected system displacement ductility µ1 No requirement 3.0 3.0 3.0 3.0
General Limiting Values
To ensure the important bridges have ductile load paths, general limiting slenderness parameters and widththickness ratios are specified in Sections 16.8.3 and 16.8.4. For steel members, λcr is the limiting member slenderness parameter for column elastic buckling and is taken as 1.5 from AISCLRFD [4] and λbr is the limiting member slenderness parameter for 1999 by CRC Press LLC
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beam elastic torsional buckling and is calculated by AISCLRFD [4]. For a critical member, a more strict requirement, 90% of those elastic buckling limits is proposed. Regardless of the force D/C ratios, all steel members must not exceed these limiting values. For existing steel members with D/C ratios less than one, this limit may be relaxed. For concrete members, the general limiting parameter KL/r = 60 is proposed. Acceptable Force D/C Ratios DCaccept
Acceptable force D/C ratios (DCaccept ) depends on both the structural characteristics in terms of ductility µ1 and the degree of damage to the structure that can be accepted by practicing engineers in terms of damage index D1 . To ensure a steel member has enough inelastic deformation capacity during an earthquake, it is necessary to limit both the member slenderness parameters and the section widththickness ratios within the specified ranges so that the acceptable D/C ratios and the energy dissipation can be achieved. Upper Bound Acceptable D/C Ratio DCp 1. For other members, the large acceptable force D/C ratios (DCp = 2 to 2.5) are proposed in Table 16.2. This implies that the damage index equals 0.67 ∼ 0.83 and more damage will occur at other members and large member ductility will be expected. To achieve this, • the limiting widththickness ratio was taken as λp−Seismic from AISCSeismic Provisions [5], which can provide flexural ductility 8 to 10. • the limiting slenderness parameters were taken as λbp for flexural moment dominant members from AISCLRFD [4], which can provide flexural ductility 8 to 10. 2. For critical members, small acceptable force D/C ratios (DCp = 1.2 to 1.5) are proposed in Table 16.2, as the design purpose is to keep critical members essentially elastic and allow little damage (damage index equals to 0.4 ∼ 0.5). Thus, small member ductility is expected. To achieve this, • the limiting widththickness ratio was taken as λp from AISCLRFD [5], which can provide flexural ductility at least 4. • the limiting slenderness parameters were taken as λbp for flexural moment dominant members from AISCLRFD [4], which can provide flexural ductility at least 4. 3. For axial load dominant members, the limiting slenderness parameters were taken as λcp = 0.5, corresponding to 90% of the axial yield load by the AISCLRFD [4] column curve. This limit will provide the potential for axial load dominated members to develop expected inelastic deformation. Lower Bound Acceptable D/C Ratio DCr
The lower bound acceptable force D/C ratio DCrc = 1 is proposed in Table 16.2. For DCaccept =1, it is unnecessary to enforce more strict limiting values for members and sections. Therefore, the limiting slenderness parameters for elastic global buckling specified in Table 16.2 and the limiting widththickness ratios specified in Table 16.3 for elastic local buckling are proposed. Acceptable D/C Ratios Between Upper and Lower Bounds DCr < DCaccept < DCp
When acceptable force D/C ratios are between the upper and the lower bounds, DCr < DCaccept < DCp , a linear interpolation as shown in Figure 16.28 is proposed to determine the limiting slenderness 1999 by CRC Press LLC
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parameters and widththickness ratios. The following formulas can be used: h DCp −DCaccept i λp + λr − λp for critical members DCp −DCr i h λpr = DCp −DCaccept λ for other members p−Seismic + λr − λp−Seismic DCp −DCr
(16.51)
FIGURE 16.28: Acceptable D/C ratios and limiting slenderness parameters and widththickness ratios. For axial load dominant members (Pu /Pn ≥ Mu /Mn ) DCp −DCaccept λcp + 0.9λcr − λcp for critical members DCp −DCr λcpr = λ + λ − λ DCp −DCaccept for other members cp cr cp DCp −DCr For flexural moment dominant members (Mu /Mn > Pu /Pn ) DCp −DCaccept λbp + 0.9λbr − λbp for critical members DCp −DCr λbpr = λ + λ − λ DCp −DCaccept for other members bp br bp DCp −DCr
(16.52)
(16.53)
where λr , λp , λp−Seismic are limiting widththickness ratios specified by Table 16.3. Limiting WidthThickness Ratios
The basic limiting widththickness ratios λr , λp , λp−Seismic specified in Table 16.3 are proposed for important bridges.
16.A.4 Inelastic Analysis Considerations This section presents concepts and formulas of reduced material and section properties and yield surface for steel members for possible use in inelastic analysis. 1999 by CRC Press LLC
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Stiffness Reduction
Concepts of stiffness reduction — tangent modulus — have been used to calculate inelastic effective length factors by AISCLRFD [4], and to account for both the effects of residual stresses and geometrical imperfection by Liew [32]. To consider inelasticity of a material, the tangent modulus of the material Et may be used in analysis. For practical application, stiffness reduction factor (SRF) = (Et /E) can be taken as the ratio of the inelastic to elastic buckling load of the column: SRF =
(Pcr )inelastic Et ≈ E (Pcr )elastic
(16.54)
where (Pcr )inelastic and (Pcr )elastic can be calculated by AISCLRFD [4] column equations: (Pcr )inelastic
=
(Pcr )elastic
=
0.658λc As Fy 0.877 As Fy λ2c 2
(16.55) (16.56)
in which As is the gross section area of the member and λc is the slenderness parameter. By utilizing the calculated axial compression load P , the tangent modulus Et can be obtained as: E for P /Py ≤ 0.39 (16.57) Et = −3(P /Py ) ln(P /Py ) for P /Py > 0.39 Reduced Section Properties
In an initial structural analysis, the section properties based on a fully integral section of a latticed member may be used. If section forces obtained from this initial analysis are lower than the section strength controlled by the shearflow transferring capacity, assumed fully integral section properties used in the analysis are rational. Otherwise, section properties considering a partially integral section, as discussed in Section 17.A.1, Iequiv and Jequiv , may be used in the further analysis. This concept is similar to “cracked section” analysis for reinforced concrete structures [13]. 1. Moment of inertia — latticed members (a) For lacing bars or battens within web plane (bending about yy axis in Figure 16.19) The following assumptions are made: • Momentcurvature curve (Figure 16.29) behaves bilinearly until the section reaches its ultimate moment capacity. • For moments less than Md = βm Mu , the moment at first stiffness degradation, the section can be considered fully integral. • For moments larger than Md and less than Mu , the ultimate moment capacity of the section, the section is considered as a partially integral one; that is, bending stiffness should be based on a reduced moment of inertia defined by Equation 16.27. An equivalent moment of inertia, Iequiv , based on the secant stiffness can be obtained as: Iequiv =
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∗ I Iy−y y−y
∗ βm Iy−y + (1 − βm )Iy−y
(16.58)
FIGURE 16.29: Idealized momentcurvature curve of a latticed member section.
where Iy−y = moment of inertia of a section about the yy axis considering shear transferring capacity ∗ = moment of inertia of a section about the yy axis assuming full section integrity Iy−y βm = reduction factor for the moment of inertia and may be determined by Equation 16.28 (b) For lacing bars or battens within flange plane (bending about xx axis in Figure 16.19) Equation 16.29 is still valid for structural analysis. 2. Effective flexural stiffness — For steel members, when 8y < 8 < 8u , the further reduced section property, effective flexural stiffness (EI )eff may be used in the analysis. Mu Mu ≤ EIequiv ≤ (EI )eff = 8u 8
(16.59)
3. Torsional constant — latticed members Based on similar assumptions and principles used for moment of inertia, an equivalent torsional constant, Jequiv , is derived as follows: Jequiv =
J ∗J βt J + (1 − βt )J ∗
(16.60)
where J = torsional constant of a section considering shear transferring capacity (See Section 17.A.1) J ∗ = torsional constant of a section assuming full section integrity βt = reduction factor for torsional constant may be determined by Equation 16.38 1999 by CRC Press LLC
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Yield Surface Equation for Doubly Symmetrical Sections
The yield or failure surface concept has been conveniently used in inelastic analysis to describe the full plastification of steel sections under action of axial force combined with biaxial bending. A four parameter yield surface equation for doubly symmetrical steel sections (I, thinwalled circular tube, thinwalled box, solid rectangular and circular sections), developed by Duan and Chen [19], is presented in this section for possible use in a nonlinear analysis. The general shape of the yield surface for a doubly symmetrical steel section as shown in Figure 16.30 can be described approximately by the following general equation: My αy Mx αx + = 1.0 (16.61) Mpcx Mpcy where Mpcx and Mpcy are the moment capacities about the respective axes, reduced for the presence
FIGURE 16.30: Typical yield surface for doubly symmetrical sections.
of axial load; they can be obtained by the following formulas: " βx # P Mpcx = Mpx 1 − Py " βy # P Mpcy = Mpy 1 − Py
(16.62)
(16.63)
where P = axial force Mx = bending moment about the xx principal axis My = bending moment about the yy principal axis Mpx = plastic moment about xx principal axis Mpy = plastic moment about yy principal axis The four parameters αx , αy , βx , and βy are dependent on sectional shapes and area distribution. It is seen that αx and αy represent biaxial loading behavior, while βx and βy describe uniaxial loading behavior. They are listed in Table 16.5: Equation 16.61 represents a smooth and convex surface in the threedimensional stressresultant space. It meets all special conditions and is easy to implement in a computerbased structural analysis. 1999 by CRC Press LLC
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TABLE 16.5
Parameters for Doubly Symmetrical Sections αx
αy
βx
βy
Solid rectangular Solid circular Ishape
1.7 + 1.3 (P /Py ) 2.0 2.0
1.7 + 1.3 (P /Py ) 2.0 1.2 + 2 (P /Py )
2.0 2.1 1.3
2.0 2.1 2 + 1.2 (Aw /Af )
Thinwalled box Thinwalled circular
1.7 + 1.5 (P /Py ) 2.0
1.7 + 1.5 (P /Py ) 2.0
2 − 0.5 B ≥ 1.3 1.75
2 − 0.5 B ≥ 1.3 1.75
Section types
Note: B is the ratio of widthtodepth of the box section with respect to the bending axis.
1999 by CRC Press LLC
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Duan, L. and Chen, W.F. “Effective Length Factors of Compression Members” Structural Engineering Handbook Ed. Chen WaiFah Boca Raton: CRC Press LLC, 1999
Effective Length Factors of Compression Members 17.1 17.2 17.3 17.4
Introduction Basic Concept Isolated Columns Framed Columns—Alignment Chart Method
Alignment Chart Method • Requirements for Braced Frames • Simplified Equations to Alignment Charts
17.5 Modifications to Alignment Charts
Different Restraining Girder End Conditions • Different Restraining Column End Conditions • Column Restrained by Tapered Rectangular Girders • Unsymmetrical Frames • Effects of Axial Forces in Restraining Members in Braced Frames • Consideration of Partial Column Base Fixity • Inelastic Kfactor
17.6 Framed Columns—Alternative Methods
LeMessurier Method • Lui Method • Remarks
17.7 Unbraced Frames With Leaning Columns Rigid Columns • Leaning Columns • Remarks
17.8 Cross Bracing Systems 17.9 Latticed and BuiltUp Members
Lian Duan Division of Structures, California Department of Transportation, Sacramento, CA
W.F. Chen School of Civil Engineering, Purdue University, West Lafayette, IN
17.1
Laced Columns • Columns with Battens • LacedBattened Columns • Columns with Perforated Cover Plates • BuiltUp Members with Bolted and Welded Connectors
17.10Tapered Columns 17.11Crane Columns 17.12Columns in Gable Frames 17.13Summary 17.14Defining Terms References Further Reading .
Introduction
The concept of the effective length factors of columns has been well established and widely used by practicing engineers and plays an important role in compression member design. The most structural design codes and specifications have provisions concerning the effective length factor. The aim of this chapter is to present a stateoftheart engineering practice of the effective length factor for the design of columns in structures. In the first part of this chapter, the basic concept of the effective length 1999 by CRC Press LLC
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factor is discussed. And then, the design implementation for isolated columns, framed columns, crossing bracing systems, latticed members, tapered columns, crane columns, as well as columns in gable frames is presented. The determination of whether a frame is braced or unbraced is also addressed. Several detailed examples are given to illustrate the determination of effective length factors for different cases of engineering applications.
17.2
Basic Concept
Mathematically, the effective length factor or the elastic Kfactor is defined as: s K=
Pe = Pcr
s
π 2 EI L2 Pcr
(17.1)
where Pe is the Euler load, the elastic buckling load of a pinended column; Pcr is the elastic buckling load of an endrestrained framed column; E is the modulus of elasticity; I is the moment of inertia in the flexural buckling plane; and L is the unsupported length of column. Physically, the Kfactor is a factor that when multiplied by actual length of the endrestrained column (Figure 17.1a) gives the length of an equivalent pinended column (Figure 17.1b) whose buckling load is the same as that of the endrestrained column. It follows that effective length, KL,
FIGURE 17.1: Isolated columns. 1999 by CRC Press LLC
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of an endrestrained column is the length between adjacent inflection points of its pure flexural buckling shape. Specifications provide the resistance equations for pinended columns, while the resistance of framed columns can be estimated through the Kfactor to the pinended columns strength equation. Theoretical Kfactor is determined from an elastic eigenvalue analysis of the entire structural system, while practical methods for the Kfactor are based on an elastic eigenvalue analysis of selected subassemblages. The effective length concept is the only tool currently available for the design of compression members in engineering structures, and it is an essential part of analysis procedures.
17.3
Isolated Columns
From an eigenvalue analysis, the general Kfactor equation of an endrestrained column as shown in Figure 17.1 is obtained as: C + RkA L S −(C + S) EI kB L −(C + S) S C + REI (17.2) det =0 3 −(C + S) −(C + S) 2(C + S) − π 2 + Tk L K
EI
where the stability functions C and S are defined as: C
=
(π/K) sin (π/K) − (π/K)2 cos (π/K) 2 − 2 cos (π/K) − (π/K) sin (π/K)
(17.3)
S
=
(π/K)2 − (π/K) sin (π/K) 2 − 2 cos (π/K) − (π/K) sin (π/K)
(17.4)
The largest value of K that satisfies Equation 17.2 gives the elastic buckling load of an endrestrained column. Figure 17.2 [1, 3, 4] summarizes the theoretical Kfactors for columns with some idealized end conditions. The recommended Kfactors are also shown in Figure 17.2 for practical design applications. Since actual column conditions seldom comply fully with idealized conditions used in buckling analysis, the recommended Kfactors are always equal to or greater than their theoretical counterparts.
17.4
Framed Columns—Alignment Chart Method
In theory, the effective length factor K for any column in a framed structure can be determined from a stability analysis of the entire structural analysis—eigenvalue analysis. Methods available for stability analysis include the slopedeflection method [17, 35, 71], threemoment equation method [13], and energy methods [42]. In practice, however, such analysis is not practical, and simple models are often used to determine the effective length factors for framed columns [38, 47, 55, 72]. One such practical procedure that provides an approximate value of the elastic Kfactor is the alignment chart method [46]. This procedure has been adopted by the AISC [3, 4], ACI 31895 [2], and AASHTO [1] specifications, among others. At present, most engineers use the alignment chart method in lieu of an actual stability analysis.
17.4.1
Alignment Chart Method
The structural models employed for determination of Kfactor for framed columns in the alignment chart method are shown in Figure 17.3. The assumptions used in these models are [4, 17]: 1999 by CRC Press LLC
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FIGURE 17.2: Theoretical and recommended Kfactors for isolated columns with idealized end conditions. 1. 2. 3. 4.
All members have constant crosssection and behave elastically. Axial forces in the girders are negligible. All joints are rigid. For braced frames, the rotations at the near and far ends of the girders are equal in magnitude and opposite in direction (i.e., girders are bent in single curvature). 5. For unbraced frames, the rotations at the near and far ends of the girders are equal in magnitude and direction (i.e., girders are bent in double curvature). √ 6. The stiffness parameters, L P /EI , of all columns are equal. 7. All columns buckle simultaneously. Using the slopedeflection equation method and stability functions, the effective length factor equations of framed columns are obtained as follows. For columns in braced frames: π/K 2 tan(π/2K) GA + GB GA GB 1− + −1=0 (π/K)2 + 4 2 tan(π/K) π/K
(17.5)
For columns in unbraced frames: π/K GA GB (π/K)2 − 362 =0 − 6 (GA + GB ) tan (π/K)
(17.6)
where GA and GB are stiffness ratios of columns and girders at two end joints, A and B, of the column section being considered, respectively. They are defined by: P (Ec Ic /Lc ) (17.7) GA = P A A Eg Ig /Lg 1999 by CRC Press LLC
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FIGURE 17.3: Subassemblage models for Kfactors of framed columns.
GB
=
P (Ec Ic /Lc ) PB B Eg Ig /Lg
(17.8)
P where indicates a summation of all members rigidly connected to the joint and lying in the plane in which buckling of column is being considered; subscripts c and g represent columns and girders, respectively. Equations 17.5 and 17.6 can be expressed in the form of alignment charts, as shown in Figure 17.4. It is noted that for columns in braced frames, the range of K is 0.5 ≤ K ≤ 1.0; for columns in unbraced frames, the range is 1.0 ≤ K ≤ ∞. For column ends supported by but not rigidly connected to a footing or foundations, G is theoretically infinity, but, unless actually designed as a true friction free pin, may be taken as 10 for practical design. If the column end is rigidly attached to a properly designed footing, G may be taken as 1.0 [4].
EXAMPLE 17.1:
Given: A twostory steel frame is shown in Figure 17.5. Using the alignment chart, determine the Kfactor for the elastic column DE. E = 29,000 ksi (200 GPa) and F y = 36 ksi (248 MPa). Solution
1. For the given frame, section properties are 1999 by CRC Press LLC
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FIGURE 17.4: Alignment charts for effective length factors of framed columns.
Members
Section
Ix in.4 (mm4 × 108 )
L in. (mm)
Ix /L in.3 (mm3 )
AB and GH BC and HI DE EF BE EH CF FI
W 10x22 W10x22 W10x45 W10x45 W18x50 W18x86 W16x40 W16x67
118 (0.49) 118 (0.49) 248 (1.03) 248 (1.03) 800 (3.33) 1530 (6.37) 518 (2.16) 954 (3.97)
180 (4,572) 144 (3,658) 180 (4,572) 144 (3,658) 300 (7,620) 360 (9,144) 300 (7,620) 360 (9,144)
0.656(10,750) 0.819(13,412) 1.378(22,581) 1.722(28,219) 2.667(43,704) 4.250(69,645) 1.727(28,300) 2.650(43,426)
2. Calculate Gfactor for column DE: GE
=
GD
=
P 1.378 + 1.722 (Ec Ic /Lc ) = = 0.448 PE 2.667 + 4.250 I /L E g g g E 10 (AISCLRFD, 1993)
3. From the alignment chart in Figure 17.4b, K = 1.8 is obtained.
17.4.2
Requirements for Braced Frames
In stability design, one of the major decisions engineers have to make is the determination of whether a frame is braced or unbraced. The AISCLRFD [4] states that a frame is braced when “lateral stability is provided by diagonal bracing, shear walls or equivalent means”. However, there is no specific provision for the “amount of stiffness required to prevent sidesway buckling” in the AISC, 1999 by CRC Press LLC
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FIGURE 17.5: An unbraced twostory frame. AASHTO, and other specifications. In actual structures, a completely braced frame seldom exists. But in practice, some structures can be analyzed as braced frames as long as the lateral stiffness provided by the bracing system is large enough. The following brief discussion may provide engineers with the tools to make engineering decisions regarding the basic requirements for a braced frame. 1. Lateral Stiffness Requirement Galambos [34] presented a simple conservative procedure to evaluate minimum lateral stiffness provided by a bracing system so that the frame is considered braced. P Pn (17.9) Required lateral stiffness, Tk = Lc P where represents the summation of all columns in one story, Pn is the nominal axial compression strength of a column using the effective length factor K = 1, and Lc is the unsupported length of a column. 2. Bracing Size Requirement Galambos [34] applied Equation 17.9 to a diagonal bracing (Figure 17.6) and obtained minimum requirements of diagonal bracing for a braced frame as Ab =
1 + (Lb /Lc )2
3/2 P
(Lb /Lc ) E 2
Pn
(17.10)
where Ab is the crosssectional area of diagonal bracing and Lb is the span length of the beam. A recent study by AristizabalOchoa [8] indicates that the size of the diagonal bracing required for a totally braced frame is about 4.9 and 5.1% of the column crosssection for 1999 by CRC Press LLC
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FIGURE 17.6: Diagonal cross bracing system. a “rigid frame” and “simple framing”, respectively, and increases with the moment inertia of the column, the beam span, and the beamtocolumn span ratio, Lb /Lc .
17.4.3
Simplified Equations to Alignment Charts
1. ACI 31895 Equations The ACI Building Code [2] recommends the use of alignment charts as the primary design aid for estimating Kfactors, following two sets of simplified Kfactor equations as an alternative: For braced frames [19]: K K
= 0.7 + 0.05 (GA + GB ) ≤ 1.0 = 0.85 + 0.05Gmin ≤ 1.0
(17.11) (17.12)
The smaller of the above two expressions provides an upper bound to the effective length factor for braced compression members. For unbraced frames [32]: For Gm < 2 K=
20 − Gm p 1 + Gm 20
(17.13)
For Gm ≥ 2 p K = 0.9 1 + Gm For columns hinged at one end 1999 by CRC Press LLC
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(17.14)
K = 2.0 + 0.3G
(17.15)
where Gm is the average of G values at the two ends of the columns. 2. DuanKingChen Equations A graphical alignment chart determination of the Kfactor is easy to perform, while solving the chart Equations 17.5 and 17.6 always involves iteration. Although the ACI code provides simplified Kfactor equations, generally, they may not lead to an economical design [40]. To achieve both accuracy and simplicity for design purposes, the following alternative Kfactor equations were proposed by Duan, King and Chen [48]. For braced frames: 1 1 1 − − 5 + 9GA 5 + 9GB 10 + GA GB
(17.16)
1 1 1 − − 1 + 0.2GA 1 + 0.2GB 1 + 0.01GA GB
(17.17)
K =1− For unbraced frames: For K < 2 K =4− For K ≥ 2
K=
2π a √ 0.9 + 0.81 + 4ab
(17.18)
where
a
=
b
=
GA GB +3 GA + GB 36 +6 GA + GB
(17.19) (17.20)
3. French Equations For braced frames: K=
3GA GB + 1.4 (GA + GB ) + 0.64 3GA GB + 2.0 (GA + GB ) + 1.28
(17.21)
For unbraced frames: s K=
1.6GA GB + 4.0 (GA + GB ) + 7.5 GA + GB + 7.5
(17.22)
Equations 17.21 and 17.22 first appeared in the French Design Rules for Steel Structure [31] since 1966, and were later incorporated into the European Recommendation for Steel Construction [28]. They provide a good approximation to the alignment charts [26]. 1999 by CRC Press LLC
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17.5
Modifications to Alignment Charts
In using the alignment charts in Figure 17.4 and Equations 17.5 and 17.6, engineers must always be aware of the assumptions used in the development of these charts. When actual structural conditions differ from these assumptions, unrealistic design may result [4, 43, 53]. The SSRC (Structural Stability Research Council) guide [43] provides methods that enable engineers to make simple modifications of the charts for some special conditions, such as unsymmetrical frames, column base conditions, girder far end conditions, and flexible conditions. A procedure that can be used to account for far ends of restraining columns being hinged or fixed was proposed by Duan and Chen [21, 22]. Consideration of effects of material inelasticity on the Kfactor was developed originally by Yura [73] and expanded by Disque [20]. LeMessurier [52] presented an overview of unbraced frames with or without leaning columns. An approximate procedure is also suggested by AISCLRFD [4]. Special attention should also be paid to calculation of the proper G values [10, 49] when partially restrained (PR) connections are used in frames. Several commonly used modifications are summarized in this section.
17.5.1
Different Restraining Girder End Conditions
When the end conditions of restraining girders are not rigidly jointed to columns, the girder stiffness (Ig /Lg ) used in the calculation of GA and GB in Equations 17.7 and 17.8 should be multiplied by a modification factor, αk , as: P (Ec Ic /Lc ) (17.23) G= P αk Eg Ig /Lg where the modification factor, αk , for braced frames developed by Duan and Lu [25] and for unbraced frames proposed by Kishi, Chen, and Goto [49] are given in Table 17.1 and 17.2. In these tables, RkN and RkF are elastic spring constants at the near and far ends of a restraining girder, respectively. RkN and RkF are the tangent stiffness of a semirigid connection at buckling. TABLE 17.1 Modification Factor αk for Braced Frames with SemiRigid Connections End conditions of restraining girder Near end Far end Rigid Rigid
Rigid Hinged
Rigid
Semirigid
Rigid
Fixed
Semirigid
Rigid
Semirigid
Hinged
αk
1.0 1.5 6Eg Ig 4Eg Ig 1+ L R / 1+ L R g kF g kF 2.04E I 1/ 1 + L Rg g g kN
3E I 1.5/ 1 + L Rg g g kN
Semirigid Semirigid
Semirigid Fixed
6Eg Ig 1+ L R g kF
/R ∗
4E I 2/ 1 + L Rg g g kN
E I 2 4E I 4Eg Ig 4 1+ L R − Lg g Note: R ∗ = 1 + L Rg g RkN RkF g kN g kF g
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TABLE 17.2 Modification Factor, αk , for Unbraced Frames with SemiRigid Connections End conditions of restraining girder Near end Far end Rigid Rigid
Rigid Hinged
Rigid
Semirigid
Rigid
Fixed
Semirigid
Rigid
αk
1 0.5 2Eg Ig 4Eg Ig 1+ L R / 1+ L R g kF g kF
2/3 4E I 1/ 1 + L Rg g
Semirigid
Hinged
Semirigid
Semirigid
3E I 0.5/ 1 + L Rg g g kN 2Eg Ig 1+ L R /R ∗
Semirigid
g kN
g kF
4E I
(2/3)/ 1 + L Rg g g kN
Fixed
E I 2 4E I 4Eg Ig 4 Note:R ∗ = 1 + L Rg g 1+ L R − Lg g RkN RkF g kN g kF g
EXAMPLE 17.2:
Given: A steel frame is shown in Figure 17.5. Using the alignment chart with the necessary modifications, determine the Kfactor for elastic column EF . E = 29,000 ksi (200 GPa) and Fy = 36 ksi (248 MPa). Solution 1. Calculate Gfactor with modification for column EF . Since the far end of restraining girders are hinged, girder stiffness should be multiplied by 0.5 (see Table 17.2). Using section properties in Example 17.1, we obtain:
GF
P 1.722 (Ec Ic /Lc ) = = P = 0.787 0.5(1.727) + 0.5(2.650) αk Eg Ig /Lg = 0.448
GE
2. From the alignment chart in Figure 17.4b, K = 1.22 is obtained.
17.5.2
Different Restraining Column End Conditions
To consider different far end conditions of restraining columns, the general effective length factor equations for column C2 (Figure 17.3) were derived by Duan and Chen [21, 22, 23]. By assuming that the far ends of columns C1 and C3 are hinged and using the slopedeflection equation approach for the subassemblies shown in Figure 17.3, we obtain the following. 1. For a Braced Frame [21]: " C2
− S2
GAC1 + GBC3 + GAC2 GBC2 + + 2C
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1 GA
+
1 GB
2GBC3 2GAC1 GA + GB
+
C
4 GA GB
=0
2 − GAC1 GBC3 CS
#
(17.24)
where C and S are stability functions as defined by Equations 17.3 and 17.4; GA and GB are defined in Equations 17.7 and 17.8; GAC1 , GAC2 , GBC2 , and GBC3 are stiffness ratios of columns at Ath and Bth ends of the columns being considered, respectively. They are defined as: Eci Ici /Lci GCi = P (Eci Ici /Lci )
(17.25)
P where indicates a summation of all columns rigidly connected to the joint and lying in the plane in which buckling of the column is being considered. Although Equation 17.24 was derived for the special case in which the far ends of both columns C1 and C3 are hinged, this equation is also applicable if adjustment to GCi is made as follows: (1) if the far end of column Ci(C1 or C3) is fixed, then take GCi = 0 (except for GC2 ); (2) if the far end of the column Ci(C1 or C3) is rigidly connected, then take GCi = 0 and GC2 = 1.0. Therefore, Equation 17.24 can be specialized for the following conditions: (a) If the far ends of both columns C1 and C3 are fixed, we have GAC1 = GBC3 = 0 and Equation 17.24 reduces to C − S (GAC2 GBC2 ) + 2C 2
2
1 1 + GA GB
+
4 =0 GA GB
(17.26)
(b) If the far end of column C1 is rigidly connected and the far end of column C3 is fixed, we have GAC2 = 1.0 and GAC1 = GBC3 = 0, and Equation 17.24 reduces to C 2 − S 2 + GBC2 + 2C
1 1 + GA GB
+
4 =0 GA GB
(17.27)
(c) If the far end of column C1 is rigidly connected and the far end of column C3 is hinged, we have GAC1 = 0 and GAC2 = 1.0, and Equation 17.24 reduces to
C2
−
2GBC3 S 2 GBC3 + GBC2 + GA C 1 4 1 + =0 + + 2C GA GB GA GB
(17.28)
(d) If the far end of column C1 is hinged and the far end of column C3 is fixed, we have GBC3 = 0 and Equation 17.24 reduces to
C2
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−
2GAC1 S 2 GAC1 + GAC2 GBC2 + GB C 1 4 1 + =0 + + 2C GA GB GA GB
(17.29)
(e) If the far ends of both columns C1 and C3 are rigidly connected (i.e., assumptions used in developing the alignment chart), we have GC2 = 1.0 and GCi = 0, and Equation 17.24 reduces to C − S + 2C 2
2
1 1 + GA GB
+
4 =0 GA GB
(17.30)
which can be rewritten in the form of Equation 17.5. 2. For an Unbraced Frame [22, 23]: a11 det a21 a31
a12 a22 a32
a13 a23 a33
=0
(17.31)
or
a11 a22 a33 + a21 a32 a13 + a31 a23 a12 − a31 a22 a13 − a21 a12 a33 + a11 a23 a32 = 0
(17.32)
where
a11
=
a22
=
a33
=
a12 a21 a31
= = =
a13
=
a23
=
6 S2 − GAC1 GA C 6 S2 C+ − GBC3 G C B 1 π 2 −2 C + S − 2 K GAC2 S GBC2 S a32 = C + S S2 −(C + S) + GAC1 S + C S2 −(C + S) + GBC3 S + C C+
(17.33) (17.34) (17.35) (17.36) (17.37) (17.38) (17.39)
(17.40)
Although Equation 17.31 was derived for the special case in which the far ends of both columns C1 and C3 are hinged, it can be adjusted to account for the following cases: (1) if the far end of column Ci(C1 or C3) is fixed, then take GCi = 0 (except for GC2 ); (2) if the far end of column Ci(C1 or C3) is rigidly connected, then take GCi = 0 and GC2 = 1.0. Therefore, Equation 17.31 can be used for the following conditions: (a) If the far ends of both columns C1 and C3 are fixed, we take GC1 = GC3 = 0, and obtain from Equations 17.33, 17.34, 17.39, and 17.40, 1999 by CRC Press LLC
c
a11
=
a22
=
a13
=
6 GA 6 C+ GB a23 = −(C + S) C+
(17.41) (17.42) (17.43)
(b) If the far end of column C1 is rigidly connected and the far end of column C3 is fixed, we take GAC2 = 1.0 and GAC1 = GBC3 = 0, and obtain from Equations 17.33, 17.34, 17.36, 17.39, and 17.40, a11
=
a22
=
a12 a13
= =
6 GA 6 C+ GB S a23 = −(C + S) C+
(17.44) (17.45) (17.46) (17.47)
(c) If the far end of column C1 is rigidly connected and the far end of column C3 is hinged, we take GAC1 = 0 and GAC2 = 1.0, and obtain from Equations 17.33, 17.36, and 17.39, 6 GA
a11
=
C+
a12 a13
= =
S −(C + S)
(17.48) (17.49) (17.50)
(d) If the far end of column C1 is hinged and the far end of column C3 is fixed, we have GBC3 = 0.0, and obtain from Equations 17.34 and 17.40, a22
=
a23
=
6 GB −(C + S) C+
(17.51) (17.52)
(e) If the far ends of both columns C1 and C3 are rigidly connected (i.e., assumptions used in developing the alignment chart, that is θC = θB and θD = θA ), we take GC2 = 1.0 and GCi = 0, and obtain from Equations 17.33 to 17.40,
1999 by CRC Press LLC
c
a11
=
a22
=
a12 a13
= =
6 GA 6 C+ GB a21 = S a23 = −(C + S) C+
(17.53) (17.54) (17.55) (17.56)
Equation 17.31 is reduced to the form of Equation 17.6. The procedures to obtain the Kfactor directly from the alignment charts without resorting to solve Equations 17.24 and 17.31 were also proposed by Duan and Chen [21, 22].
17.5.3
Column Restrained by Tapered Rectangular Girders
A modification factor, αT , was developed by King et al. [48] for those framed columns restrained by tapered rectangular girders with different far end conditions. The following modified Gfactor is introduced in connection with the use of alignment charts: P (Ec Ic /Lc ) G= P αT Eg Ig /Lg
(17.57)
where Ig is the moment of inertia of the girder at near end. Both closedform and approximate solutions for modification factor αT were derived. It is found that the following twoparameter power function can describe the closedform solutions very well: αT = D (1 − r)β
(17.58)
in which the parameter D is a constant depending on the far end conditions, and β is a function of the far end conditions and tapering factor, α and r, as defined in Figure 17.7.
FIGURE 17.7: Tapered rectangular girders. (From King, W.S., Duan, L., et al., Eng. Struct., 15(5), 369, 1993. With kind permission from Elsevier Science, Ltd, The Boulevard, Langford Lane, Kidlington OX5 IGB, UK.) 1999 by CRC Press LLC
c
For a braced frame:
For an unbraced frame:
1.0 2.0 D= 1.5
rigid far end fixed far end hinged far end
1.0 2/3 D= 0.5
rigid far end fixed far end hinged far end
1. For a linearly tapered rectangular girder (Figure 17.7a) For a braced frame: rigid far end 0.02 + 0.4r 0.75 − 0.1r fixed far end β= 0.75 − 0.1r hinged far end
(17.59)
(17.60)
(17.61)
For an unbraced frame: 0.95 0.70 β= 0.70
rigid far end fixed far end hinged far end
2. For a symmetrically tapered rectangular girder (Figure 17.7b) For a braced frame: rigid far end 3 − 1.7a 2 − 2a β= fixed far end 3 + 2.5a 2 − 5.55a hinged far end 3 − a 2 − 2.7a
(17.62)
(17.63)
For an unbraced frame: 3 + 3.8a 2 − 6.5a β= 3 + 2.3a 2 − 5.45a 3 − 0.3a
rigid far end fixed far end hinged far end
(17.64)
EXAMPLE 17.3:
Given: A onestory frame with a symmetrically tapered rectangular girder is shown in Figure 17.8. Assuming r = 0.5, a = 0.2, and Ig = 2Ic = 2I , determine Kfactor for column AB. Solution
1. Using the alignment chart with modification For joint A, since the far end of the girder is rigid, use Equations 17.64 and 17.58, 1999 by CRC Press LLC
c
FIGURE 17.8: A simple frame with rectangular sections. (From King, W.S., Duan, L., et al., Eng. Struct., 15(5), 369, 1993. With kind permission from Elsevier Science, Ltd, The Boulevard, Langford Lane, Kidlington OX5 IGB, UK.)
β αT GA GB
= 3 + 3.8 (0.2)2 − 6.5 (0.2) = 1.852 = (1 − 0.5)1.852 = 0.277 P Ec Ic /Lc EI /L = 3.61 = P = 0.277E(2I )/2L αT Eg Ig /Lg = 1.0 (AISC – LRFD 1993)
From the alignment chart in Figure 17.4b, K = 1.59 is obtained. 2. Using the alignment chart without modification A direct use of Equations 17.7 and 17.8 with an average section (0.75 h) results in:
Ig
=
GA
=
0.753 (2I ) = 0.844I EI /L = 2.37 GB = 1.0 0.844EI /2L
From the alignment chart in Figure 17.4b K = 1.49, or (1.49−1.59)/1.59 = −6 % in error on the less conservative side.
17.5.4
Unsymmetrical Frames
When the column sizes or column loads are not identical, adjustments to the alignment charts are necessary to obtain a correct Kfactor. SSRC Guide [43] presents a set of curves as shown in Figure 17.9 for a modification factor, β, originally developed by Chu and Chow [18]. Kadjusted = βKalignment chart 1999 by CRC Press LLC
c
(17.65)
FIGURE 17.9: Chart for the modification factor β in an unsymmetrical frame. If the Kfactor of the column under the load λP is desired, further modifications to K are necessary. Denoting K 0 as the effective length factor of the column with Ic0 = αIc subjected to the axial load, P 0 = λP , as shown in Figure 17.9, then we have: r L α 0 (17.66) K = Kadjusted 0 L λ Equation 17.66 can be used to determine Kfactors for columns in adjacent stories with different heights, L0 .
17.5.5
Effects of Axial Forces in Restraining Members in Braced Frames
Bridge and Fraser [14] observed that Kfactors of a column in a braced frame may be greater than unity due to “negative” restraining effects. Figure 17.10 shows the solutions obtained by considering both the “positive” and “negative” values of Gfactors. The shaded portion of the graph corresponds to the alignment chart shown in Figure 17.4a when both GA and GB are positive. To account for the effect of axial forces in the restraining members, Bridge and Fraser [14] proposed a more general expression for Gfactor: G 1999 by CRC Press LLC
c
=
(I /L) (I n /L)n γn mn
P
FIGURE 17.10: Effective length chart considering both positive and negative effects in braced frame. (From Bridge, R.Q. and Fraser, D.J., ASCE, J. Struct. Eng., 113(6), 1341, 1987. With permission.) =
stiffness of member i under investigation stiffness of all rigidly connected members
(17.67)
where γ is a function of the stability functions S and C (Equations 17.3 and 17.4); m is a factor to account for the end conditions of the restraining member (see Figure 17.11); and subscript n represents the other members rigidly connected to member i. The summation in the denominator is for all members meeting at the joint. Using Figures 17.10 and 17.11 and Equation 17.67, the effective length factor Ki for the ith member can be determined by the following steps: 1. Sketch the buckled shape of the structure under consideration. 2. Assume a value of Ki for the member being investigated. 3. Calculate values of Kn for each of the other members that are rigidly connected to the ith member using the equation Li Kn = Ki Ln 4. 5. 6. 7.
Pi Pn
In Ii
(17.68)
Calculate γ and obtain m from Figure 17.11 for each member. Calculate Gi for the ith member using Equation 17.67. Obtain Ki from Figure 17.10 and compare with the assumed Ki at Step 2. Repeat the procedure by using the calculated Ki as the assumed Ki until Ki calculated at the end the cycle is approximately (say 10%) equal to the Ki at the beginning of the cycle.
1999 by CRC Press LLC
c
s
FIGURE 17.11: Values of γ and m to account for the effect of axial forces in the restraining members.
8. Repeat steps 2 to 7 for other members of the frame. 9. The largest set of K values obtained is then used for design. The above procedure has been illustrated [14] and verified [50] to provide a good elastic Kfactor of columns in braced frames.
EXAMPLE 17.4:
Given: A braced column is shown in Figure 17.12. Consider axial force effects to determine Kfactors for columns AB and BC. Solutions 1. Sketch the buckled shape as shown in Figure 17.12b 2. Assume KAB = 0.94. 1999 by CRC Press LLC
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FIGURE 17.12: Braced columns. 3. Calculate KBC by Equation 17.68.
KBC
=
LAB KAB + LBC
=
1.22
s
PAB PBC
IBC IAB
L = 0.94 L
s
2P I P (1.2I )
4. Calculate γ and obtain m from Figure 17.11 for member BC. Since KBC > 1.0 γBC = 1 −
1 1 =1− = 0.33 2 1.222 KBC
Far end is pinned, mBC = 1.5 5. Calculate Gfactor for the member AB using Equation 17.67. (I /L) (1.2I /L) = 2.42 = (I /L) γ m (I /L)(0.33)(1.5) n n n n
GB
=
P
GA
=
∞
6. From Figure 17.10, KAB = 0.93. Comparing with the assumed KAB = 0.94 O.K. 7. Repeat the above procedure for member BC. 1999 by CRC Press LLC
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Assume KBC = 1.2 Calculate KAB by Equation 17.68 s r PBC IAB L P (1.2I ) LBC = 0.93 = 1.2 KAB = KBC LAB PAB IBC L 2P I Calculate γ and obtain m from Figure 17.11 for member AB Since KAB < 1.0 γAB = 2 −
2 2 =2− = −0.312 2 0.932 KAB
Far end is pinned, mAB = 1.5 Calculate Gfactor for the member BC using Equation 17.67. (I /L) (I /L) = = −1.78 (I /L) γ m (1.2I /L)(−0.312)(1.5) n n n n
GB
=
P
GA
=
∞
Read Figure 17.10, KBC = 1.18 Comparing with the assumed KAB = 1.20 O.K. 8. It is seen that the largest set of Kfactors is KAB = 1.22 and KBC = 0.93
17.5.6
Consideration of Partial Column Base Fixity
In computing the effective length factor for monolithic connections, it is important to properly evaluate the degree of fixity in foundation. The following two approaches can be used to account for foundation fixity. 1. Fictitious Restraining Beam Approach Galambos [33] proposed that the effect of partial base fixity can be modelled as a fictitious beam. The approximate expression for the stiffness of the fictitious beam accounting for rotation of foundation in the soil has the form: qBH 3 Is = LB 72Esteel
(17.69)
where q is the modulus of subgrade reaction (varies from 50 to 400 lb/in.3 , 0.014 to 0.109 N/mm3 ); B and H are the width and length (in bending plane) of the foundation; and Esteel is the modulus of elasticity of steel. Based on studies by Salmon, Schenker, and Johnston [65], the approximate expression for the stiffness of the fictitious beam accounting for the rotations between column ends and footing due to deformation of base plate, anchor bolts, and concrete can be written as: bd 2 Is = LB 72Esteel /Econcrete 1999 by CRC Press LLC
c
(17.70)
where b and d are the width and length of the base plate, and subscripts concrete and steel represent concrete and steel, respectively. Galambos [33] suggested that the smaller of the stiffness calculated by Equations 17.69 and 17.70 be used in determining Kfactors. 2. AASHTOLRFD Approach The following values are suggested by AASHTOLRFD [1]: G = 1.5 G = 3.0 G = 5.0 G = 1.0
17.5.7
footing anchored on rock footing not anchored on rock footing on soil footing on multiple rows of end bearing piles
Inelastic Kfactor
The effect of material inelasticity and end restrain on the Kfactors has been studied during the last two decades [12, 15, 20, 44, 45, 58, 64, 67, 68, 69, 73] The inelastic Kfactor developed originally by Yura [73] and expanded by Disque [20] makes use of the alignment charts with simple modifications. To consider inelasticity of material, the G values as defined by Equations 17.7 and 17.8 are replaced by G∗ [20] as follows: Et (17.71) G G∗ = SRF (G) = E in which Et is the tangent modulus of the material. For practical application, stiffness reduction factor (SRF ) = (Et /E) can be taken as the ratio of the inelastic to elastic buckling stress of the column Pu /Ag Et (Fcr )inelastic ≈ ≈ (17.72) SRF = E (Fcr )elastic (Fcr )elastic where Pu is the factored axial load and Ag is the crosssectional area of the member. (Fcr )inelastic and (Fcr )elastic can be calculated by AISCLRFD [4] column equations: (Fcr )inelastic
=
(Fcr )elastic
=
λc
=
(0.658)λc Fy 0.877 Fy λ2c r KL Fy rπ E 2
(17.73) (17.74) (17.75)
in which K is the elastic effective length factor and r is the radius of gyration about the plane of buckling. Table 17.3 gives SRF values for different stress levels and slenderness parameters.
EXAMPLE 17.5:
Given: A twostory steel frame is shown in Figure 17.5. Use the alignment chart to determine Kfactor for inelastic column DE. E = 29,000 ksi (200 GPa) and Fy = 36 ksi (248 MPa). Solution 1. Calculate the axial stress ratio: 300 Pu = 0.63 = Ag Fy 13.3(36) 2. Obtain SRF = 0.793 from Table 17.3 1999 by CRC Press LLC
c
Stiffness Reduction Factor (SRF) for G
TABLE 17.3 values
Pu Ag Fy
KL r elastic
36 ksi (248 MPa)
50 ksi (345 MPa)
λc
SRF (Eq. 18.72)
0.0 31.2 44.7 55.6 65.1 73.9 82.3 90.5 98.5 106.6 114.7 123.2 131.9 133.7
0.0 26.5 38.0 47.1 55.2 62.7 69.8 76.8 83.6 90.4 97.4 104.5 111.9 113.5
0.155 0.350 0.502 0.623 0.730 0.829 0.923 1.015 1.105 1.195 1.287 1.381 1.480 1.500
0.000 0.133 0.258 0.376 0.486 0.588 0.680 0.763 0.835 0.896 0.944 0.979 0.998 1.000
1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.39
3. Calculate modified Gfactor. GE G∗E GD
= = =
0.448 (Example 17.1) SRF (GE ) = 0.794(0.448) = 0.355 10 (AISCLRFD 1993)
4. From the alignment chart in Figure 17.4b , we have (KDE )inelastic = 1.75
17.6
Framed Columns—Alternative Methods
17.6.1
LeMessurier Method
Considering that all columns in a story buckle simultaneously and strong columns will brace weak columns (Figure 17.13), a more accurate approach to calculate Kfactors for columns in a sidesway frame was developed by LeMessurier [52]. The Ki value for the ith column in a story can be obtained by the following expression: s P P P + CL P π 2 EIi P (17.76) Ki = PL L2i Pi where Pi is the axial compressive force for member i, subscript i represents the ith column, and P P is the sum of the axial force of all columns in a story. PL β CL 1999 by CRC Press LLC
c
βEI L2 6 (GA + GB ) + 36 = 2 (GA + GB ) + GA GB + 3 Ko2 = β 2 −1 π =
(17.77) (17.78) (17.79)
FIGURE 17.13: Subassemblage of the LeMessurier method. in which Ko is the effective length factor obtained by the alignment chart for unbraced frames, and PL is only for rigid columns which provide sidesway stiffness.
EXAMPLE 17.6:
Given: A sway frame with columns of unequal height is shown in Figure 17.14a. Determine elastic Kfactors for columns by using the LeMessurier method. Member properties are: Member AB BD CD
A in.2 21.5 21.5 7.65
(mm2 ) I in.4 (13,871) 620 (13,871) 620 (4,935) 310
(mm4 × 108 ) L in. (2.58) 240 (2.58) 240 (1.29) 120
Solution
The detailed calculations are listed in Table 17.4. Using Equation 17.76, we obtain: s KAB
= s
1999 by CRC Press LLC
c
P
P+ P
P
CL P
PL
π 2 E(620) 3P + 0.495P = 0.83 0.271E (240)2 (2P ) s P P P + CL P π 2 EICD P = PL L2CD PCD s π 2 E(310) 3P + 0.495P = 1.66 = 0.271E (120)2 (P )
= KCD
π 2 EIAB L2AB PAB
(mm) (6,096) (6,096) (3,048)
FIGURE 17.14: A frame with unequal columns.
17.6.2
Lui Method
A simple and straightforward approach for determining the effective length factors for framed columns without the use of alignment charts and other charts was proposed by Lui [57]. The formulas take into account both the member instability and frame instability effects explicitly. The Kfactor for the ith column in a story was obtained in a simple form: v ! u u π 2 EIi X P 1 11 t P +P Ki = L 5 η H Pi L2i
(17.80)
P P where (P /L) represents the sum of the axial forcetolength ratio of all members in a story, H is the story lateral load producing 11 , 11 is the firstorder interstory deflection, and η is the member 1999 by CRC Press LLC
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TABLE 17.4 Example 17.6—Detailed Calculation by the LeMessurier Method Members
AB
CD
Sum
I in.4 (mm4 × 108 ) L in. (mm) Gtop Gbottom β Kio CL PL P CL P
620 (2.58) 240 (6,096) 1.0 0.0 8.4 1.17 0.165 0.09E 2P 0.33P
310 (1.29) 120 (3,048) 1.0 0.0 8.4 1.17 0.165 0.181E P 0.165P
—
Notes
— — — — — — 0.271E 3P 0.495P
Eq. 18.7 Eq. 18.7 Eq. 18.78 Alignment Chart Eq. 18.79 Eq. 18.77
stiffness index and can be calculated by 3 + 4.8m + 4.2m2 EI η= L3
(17.81)
in which m is the ratio of the smaller to larger end moments of the member; it is taken as positive if the member bends in reverse curvatureP and negative for single curvature. It is important to note that the term H used in Equation 17.80 is not the actual applied lateral load. Rather, it is a small disturbing or fictitious force (taken as a fraction of the story gravity loads) to be applied to each story of the frame. This fictitious force is applied in a direction such that the deformed configuration of the frame will resemble its buckled shape.
EXAMPLE 17.7:
Given: Determine Kfactors by using the Lui method for the frame shown in Figure 17.14a. E = 29,000 ksi (200 GPa). Solution
Apply fictitious lateral forces at B and D (Figure 17.14b) and perform a firstorder analysis. Detailed calculation is shown in Table 17.5. Using Equation 17.80, we obtain: v ! u u π 2 EIAB X P 1 11 t P +P KAB = L 5 η H PAB L2AB s 1 π 2 (29,000)(620) P + 0.019 = 0.76 = 60 5(56.24) (2P )(240)2 v ! u u π 2 EICD X P 1 11 t P P + KCD = L 5 η H PCD L2CD s 1 π 2 (29,000)(310) P + 0.019 = 1.52 = 60 5(56.24) (P )(120)2
1999 by CRC Press LLC
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TABLE 17.5 Lui Method
17.6.3
Example 17.7—Detailed Calculation by the
Members
AB
CD
Sum
I in.4 (mm4 × 108 ) L in. (mm) H kips (kN) 11 in. (mm) P 11 / H in./kips (mm/kN) Mtop kin. (kNm) Mbottom kin. (kNm) m η kips/in. (kN/mm) P /L kips/in. (kN/mm)
620 (2.58) 240 (6096) 1.0 (4.448) 0.0286 (0.7264) —
310 (1.29) 120 (3048) 0.5 (2.224) 0.0283 (0.7188) —
—
−38.8 (−4.38) −46.2 (−5.22) 0.84 13.00 (2.28) P /120 P /3048
56.53 (6.39) 81.18 (9.17) 0.69 43.24 (7.57) P /120 P /3048
Notes
— 1.5 (6.672) — 0.019 (0.108) —
Average
— — 56.24 (9.85) P /60 P /1524
Eq. 18.1
Remarks
For comparison, Table 17.6 summarizes Kfactors for the frame shown in Figure 17.14a obtained from the alignment chart, the LeMessurier and Lui methods, as well as an eigenvalue analysis. It is seen that errors in alignment chart results are rather significant in this case. Although Kfactors predicted by Lui’s and LeMessurier’s formulas are identical in most cases, the simplicity and independence of any chart in the case of Lui’s formulas make it more desirable for design office use [66]. TABLE 17.6 Comparison of K Factors for the Frame in Figure 17.14a
17.7
Columns
Theoretical
Alignment chart
Lui Eq. 18.80
LeMessurier Eq. 18.76
AB CD
0.70 1.40
1.17 1.17
0.76 1.52
0.83 1.67
Unbraced Frames With Leaning Columns
A column framed with simple connections is often called a leaning column. It has no lateral stiffness or sidesway resistance. A column framed with rigid momentresisting connections is called a rigid column. It provides the lateral stiffness or sidesway resistance to the frame. When a frame system (Figure 17.15a) includes leaning columns, the effective length factors of rigid columns must be modified. Several approaches to account for the effect of “leaning columns” were reported in the literature [16, 52, 54, 73]. A detailed discussion about the leaning columns for practical applications was presented by Geschwindner [37].
17.7.1
Rigid Columns
1. Yura Method Yura [73] discussed frames with leaning columns and noted the behavior of stronger columns assisting weaker ones in resisting sidesway. He concluded that the alignment 1999 by CRC Press LLC
c
FIGURE 17.15: A frame with leaning columns.
chart gives valid sidesway buckling solutions if the columns are in the elastic range and all columns in a story reach their individual buckling loads simultaneously. For columns that do not satisfy these two conditions, the alignment chart is generally overly conservative. Yura states that (a) The maximum loadcarrying capacity of an individual column is limited to the load permitted on that column for braced case K = 1.0. (b) The total gravity loads that produce sidesway are distributed among the columns, which provides lateral stiffness in a story. 2. Lim and McNamara Method Based on the story buckling concept and using the stability functions, Lim and McNamara [54] presented the following formula to account for the leaning column effect. 1999 by CRC Press LLC
c
s
P Q Fo 1+ P P Fn
Kn = Ko
(17.82)
where Kn is the effective length factor accounting for the leaning columns; Ko is the effective length factor determined by P Pthe alignment chart (Figure 17.3b) not accounting for the leaning columns; P and Q are the loads on the restraining columns and on the leaning columns in a story, respectively; and Fo and Fn are the eigenvalue solutions for a frame without and with leaning columns, respectively. For normal column end conditions that fall somewhere between fixed and pinned, Fo /Fn = 1 provides a Kfactor on the conservative side by less than 2% [37]. Using Fo /Fn = 1, Equation 17.82 becomes: s P Q (17.83) Kn = Ko 1 + P P Equation 17.83 gives the same Kfactor as the modified Yura approach [37]. 3. LeMessurier and Lui Methods Equation 17.76 developed by LeMessurier [52] and Equation 17.80 proposed by Lui [57] can be used for frames with and without leaning columns. Since the Kfactor expressions Equations 17.76 and 17.80 were derived for an entire story of the frame, they are applicable to frames with and without leaning columns. 4. AISCLRFD Method The current AISCLRFD [4] commentary adopts the following modified effective length factor, Ki0 , for the ith rigid column: s Ki0
=
π 2 EIi L2i Pui
P Pu P Pe2
(17.84)
P where Pe2 is the Euler loads of all columns in a story providing lateral stiffness for the frame based on the effective length factor obtained from the alignment chart for an unbraced P frame; Pui is the required axial compressive strength for the ith rigid column; and Pu is the required axial compressive strength of all columns in a story. When E and L2 are constant for all columns in a story, AISC [4] suggested that: v uP u I P u u i × P I Ki0 = t i Pui 2
(17.85)
Kio
except r Ki0
≥
5 Kio 8
(17.86)
where Kio is the effective length factor of a rigid column based on the alignment chart for unbraced frames.
1999 by CRC Press LLC
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EXAMPLE 17.8:
Given: A frame with a leaning column is shown in Figure 17.16a [59]. Evaluate the Kfactor for
FIGURE 17.16: A leaning column frame. column AB using various methods. The bottom of column AB is assumed to be ideally pinended 1999 by CRC Press LLC
c
for comparison purposes. E = 29,000 ksi (200 GPa). Solution 1. Alignment Chart Method GA
=
GB
=
∞ P Ec Ic /Lc EI /L P = = 2.0 0.5EI /L αk Eg Ig /Lg
From Figure 17.3b, we have KAB = 2.6 2. Lim and McNamara P Method P For this frame, P = Q = P and Ko = 2.6. From Equation 17.83, we have s P √ Q KAB = Ko 1 + P = 2.6 1 + 1 = 3.68 P 3. LeMessurier Method For column AB, GA = ∞ and GB = 2.0; from the alignment chart, Ko = 2.6. According to Equations 17.76 to 17.79 we have, β GA=∞ X
PL CL
KAB
6(GA + GB ) + 36 6 6 = = G = 1.5 2(GA + GB ) + GA GB + 3 A=∞ 2 + GB 2+2 βEI EI = (PL )AB = 2 = 1.5 2 L L Ko2 2.62 = β 2 − 1 = (1.5) 2 − 1 = 0.0274 π π s P P s 2 P + CL P π 2 EIAB π EI 2P + 0.0274P P = = PL L2 P 1.5EI /L2 L2AB PAB √ 13.34 = 3.65 = =
4. AISCLRFD Method Using Equation 17.85 for column AB: v uP u √ u Pu IAB × P I = Kio 2 = 3.68 KAB = t PAB 2 Kio
5. Lui Method (a) Apply a small lateral force, H = 1 kip, as shown in Figure 17.16b. (b) Perform a firstorder analysis and find 11 = 0.687 in. (17.45 mm). (c) Calculate η factors from Equation 17.81. Since column CD buckles in a single curvature, m = −1, ηCD = 1999 by CRC Press LLC
c
(3 + 4.8m + 4.2m2 )EI (3 − 4.8 + 4.2)EI 2.4EI = = 3 3 L L L3
For column AB, m = 0,
ηAB X
η
(3 + 4.8m + 4.2m2 )EI 3EI = 3 3 L L 2.4EI 3EI 5.4(29,000)(100) + = = L3 L3 (144)3 = 5.245 kips/in. (0.918 kN/mm) =
(d) Calculate the Kfactor from Equation 17.80.
KAB
v ! u u π 2 EIAB X P 1 11 t P P + = L 5 η H PAB L2AB s 1 0.687 π 2 (29,000)(100) 2P + = 3.73 = 144 5(5.245) 1 P (144)2
From an eigenvalue analysis, KAB = 3.69 is obtained. It is seen that a direct use of the alignment chart leads to a significant error for this frame, and other approaches give good results. However, the LeMessurier approach requires the use of the alignment chart, and the Lui approach requires a firstorder analysis subjected to a fictitious lateral loading.
17.7.2
Leaning Columns
Recognizing that a leaning column is being braced by rigid columns, Lui [57] proposed a model for the leaning column, as shown in Figure 17.15b. Rigid columns provide lateral stability to the whole structure and are represented by a translation spring with a spring stiffness, SK . The Kfactor for a leaning column can be obtained as: ( K = larger of
q1
π 2 EI SK L3
(17.87)
For most commonly framed structures, the term (π 2 EI /SK L3 ) normally does not exceed unity and so K = 1 often governs. AISCLRFD [4] suggests that leaning columns with K = 1 may be used in unbraced frames provided that the lack of lateral stiffness from simple connections to the frame (K = ∞) is included in the design of moment frame columns.
17.7.3
Remarks
Numerical studies by Geschwindner [37] found that the Yura approach gives overly conservative results for some conditions; Lim and McNamara’s approach provides sufficiently accurate results for design, and the LeMessurier approach is the most accurate of the three. The Lim and McNamara approach could be appropriate for preliminary design while the LeMessurier and Lui approaches would be appropriate for final design. 1999 by CRC Press LLC
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17.8
Cross Bracing Systems
Diagonal bracing or Xbracing is commonly used in steel structures to resist horizontal loads. In the current practice, the design of this type of bracing system is based on the assumptions that the compression diagonal has negligible capacity and the tension diagonal resists the total load. The assumption that compression diagonal has a negligible capacity usually results in an overdesign [62, 63]. Picard and Beaulieu [62, 63] reported theoretical and experimental studies on double diagonal cross bracings (Figure 17.6) and found that 1. A general effective length factor equation (Figure 17.17) is given as s K=
0.523 −
0.428 ≥ 0.50 C/T
(17.88)
where C and T represent compression and tension forces obtained from an elastic analysis, respectively.
FIGURE 17.17: Effective length factor of compression diagonal. (From Picard, A. and Beaulieu, D., AISC Eng. J., 24(3), 122, 1987. With permission.)
1999 by CRC Press LLC
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2. When the double diagonals are continuous and attached at their intersection point, the effective length of the compression diagonal is 0.5 times the diagonal length, i.e., K = 0.5, because the C/T ratio is usually smaller than 1.6. ELTayem and Goel [27] reported a theoretical and experimental study about the Xbracing system made from single equalleg angles. They concluded that: 1. Design of an Xbracing system should be based on an exclusive consideration of one half diagonal only. 2. For Xbracing systems made from single equalleg angles, an effective length of 0.85 times the half diagonal length is reasonable, i.e., K = 0.425.
17.9
Latticed and BuiltUp Members
The main difference of behavior between solidwebbed members, latticed members, and builtup members is the effect of shear deformation on their buckling strength. For solidwebbed members, shear deformation has a negligible effect on buckling strength. Whereas for latticed structural members using lacing bars and batten plates, shear deformation has a significant effect on buckling strength. It is a common practice that when a buckling model involves relative deformation produced by shear forces in the connectors, such as lacing bars and batten plates, between individual components, a modified effective length factor, Km , is defined as follows: Km = αv K
(17.89)
in which K is the usual effective length factor of a latticed member acting as a unit obtained from a structural analysis, and αv is the shear factor to account for shear deformation on the buckling strength, or the modified effective slenderness ratio, (KL/r)m should be used in the determination of the compressive strength. Details of the development of the shear factor, αv , can be found in textbooks by Bleich [13] and Timoshenko and Gere [70]. The following section briefly summarizes αv formulas for various latticed members.
17.9.1
Laced Columns
For laced members as shown in Figure 17.18, by considering shear deformation due to the lengthening of diagonal lacing bars in each panel and assuming hinges at joints, the shear factor, αv , has the form: s αv =
1 π 2 EI (KL2 ) Ad Ed sin φ cos2 φ
1+
(17.90)
where Ed is the modulus of elasticity of materials for the lacing bars, Ad is the crosssectional area of all diagonals in one panel, and φ is the angle between the lacing diagonal and the axis that is perpendicular to the member axis. If the length of the lacing bars is given (Figure 17.18), Equation 17.90 can be rewritten as: s αv =
1+
d3 π 2 EI 2 (KL ) Ad Ed ab2
(17.91)
where a, b, and d are the height of panel, depth of member, and length of diagonal, respectively. 1999 by CRC Press LLC
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FIGURE 17.18: Typical configurations of laced members. The SSRC [36] suggested that a conservative estimate of the influence of 60 or 45◦ lacing, as generally specified in bridge design practice, can be made by modifying the overall effective length factor, K, by multiplying a factor, αv , originally developed by Bleich [13] as follows: For KL r > 40, q (17.92) αv = 1 + 300/ (KL/r)2 For
KL r
≤ 40, αv = 1.1
(17.93)
EXAMPLE 17.9:
Given: A laced column with angles and cover plates is shown in Figure 17.19. Ky = 1.25, L = 30 ft (9144 mm). Determine the modified effective length factor, (Ky )m , by considering the shear deformation effect. Section properties:
1999 by CRC Press LLC
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Iy
=
2259 in.4 (9.4 × 108 mm4 )
E a
Ad = 1.69 in.2 (1090 mm2 ) = Ed = 6 in. (152 mm) b = 11 in. (279 mm)
d
=
12.53 in. (318 mm)
FIGURE 17.19: A laced column. Solution
1. Calculate the shear factor, αv , by Equation 17.91. s αv
=
1+ s
=
1+
π 2 EI d3 (KL)2 Ad Ed ab2 π 2 E(2259)
12.533 = 1.09 (1.25 × 30 × 12)2 1.69E(6)(11)2
2. Calculate (Ky )m by Equation 17.89. Ky m = αv Ky = 1.09(1.25) = 1.36
1999 by CRC Press LLC
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17.9.2
Columns with Battens
The battened column has a greater shear flexibility than either the laced column or the column with perforated cover plates, hence the effect of shear distortion must be taken into account in calculating the effective length of a column [43]. For the battened members shown in Figure 17.20a, assuming that points of inflection in the battens are at the batten midpoints, and that points of inflection in the longitudinal element occur midway between the battens, the shear factor, αv , is obtained as: s π 2 EI a2 ab + (17.94) αv = 1 + 24EIf (KL)2 12Eb Ib where Eb is the modulus of elasticity of materials for the batten plates, Ib is the moment inertia of all the battens in one panel in the buckling plane, and If is the moment inertia of one side of the main components taken about the centroid axis of the flange in the buckling plane.
FIGURE 17.20: Typical configurations of members with battens and with perforated cover plates.
EXAMPLE 17.10:
Given: A battened column is shown in Figure 17.21. Ky = 0.8, L = 30 ft (9144 mm). Determine the modified effective length factor, (Ky )m , by considering the shear deformation effect. Section properties: Iy 1999 by CRC Press LLC
c
=
144 in.4 (6.0 × 107 mm4 )
FIGURE 17.21: A battened column.
E If a
= Eb = 1.98 in.4 (8.24 × 105 mm4 ) = 15 in. (381 mm)
b Ib
= 9 in. (229 mm) = 9 in.4 (3.75 × 106 mm4 )
Solution
1. Calculate the shear factor, αv , by Equation 17.94. 1999 by CRC Press LLC
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s
αv
a2 ab = + 12EIb 24EIf s 152 π 2 E(144) 15(9) + = 1.05 1+ = (0.8 × 30 × 12)2 12E(9) 24E(1.98) π 2 EI 1+ (KL)2
2. Calculate (Ky )m by Equation 17.89. Ky
17.9.3
m
= αv Ky = 1.05(0.8) = 0.84
LacedBattened Columns
For the lacedbattened columns, as shown in Figure 17.20b, considering the shortening of the battens and the lengthening of the diagonal lacing bars in each panel, the shear factor, αv , can be expressed as: s π 2 EI b d3 + (17.95) αv = 1 + aAb Eb (KL)2 Ad Ed ab2 where Eb is the modulus of elasticity of the materials for battens and Ab is the crosssectional area of all battens in one panel.
17.9.4
Columns with Perforated Cover Plates
For members with perforated cover plates, shown in Figure 17.20c, considering the horizontal cross member as infinitely rigid, the shear factor, αv , has the form: s αv =
1+
π 2 EI (KL)2
9c3 64aEIf
(17.96)
where c is the length of a perforation. It should be pointed out that the usual Kfactor based on a solid member analysis is included in Equations 17.90 to 17.96. However, since the latticed members studied previously have pinended conditions, the Kfactor of the member in the frame was not included in the second terms of the square root of the above equations in their original derivations [13, 70].
EXAMPLE 17.11:
Given: A column with perforated cover plates is shown in Figure 17.22. Ky = 1.3, L = 25 ft (7620 mm). Determine the modified effective length factor, (Ky )m , by considering the shear deformation effect. Section properties: Iy
=
If a
= 35.5 in.4 (1.48 × 106 mm4 ) = 30 in. (762 mm)
c 1999 by CRC Press LLC
c
=
2467 in.4 (1.03 × 108 mm4 )
14 in. (356 mm)
FIGURE 17.22: A column with perforate cover plates.
Solution
1. Calculate the shear factor, αv , by Equation 17.96. s αv
=
1+ s
=
π 2 EI (KL)2
9c3 64aEIf
π 2 E(2467) 1+ (1.3 × 25 × 12)2
9(14)3 64(30)E(35.5)
2. Calculate (Ky )m by Equation 17.89. Ky 1999 by CRC Press LLC
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m
= αv Ky = 1.03(1.3) = 1.34
= 1.03
17.9.5
BuiltUp Members with Bolted and Welded Connectors
AISCLRFD [4] specifies that if the buckling of a builtup member produces shear forces in the connectors between individual component members, the usual slenderness ratio, KL/r, for compression members must be replaced by the modified slenderness ratio, KL r m , in determining the compressive strength. 1. For snugtight bolted connectors:
KL r
s
m
=
KL r
2 o
+
2 a ri
(17.97)
2. For welded connectors and for fully tightened bolted connectors:
KL r
s
m
=
KL r
2 o
+ 0.82
α2 (1 + α 2 )
a rib
2 (17.98)
KL where KL r o is the slenderness ratio of the builtup member acting as a unit, r m is the modified slenderness ratio of the builtup member, rai is the largest slenderness ratio of the individual components, raib is the slenderness ratio of the individual components relative to its centroidal axis parallel to the axis of buckling, a is the distance between connectors, ri is the minimum radius of gyration of individual components, rib is the radius of gyration of individual components relative to its centroidal axis parallel to the member axis of buckling, α is the separation ratio = h/2rib , and h is the distance between centroids of individual components perpendicular to the member axis of buckling. Equation 17.97 is the same as that used in the current Italian code as well as other European specifications, based on test results [74]. In the equation, the bending effect is considered in the first term in square root, and shear force effect is taken into account in the second term. Equation 17.98 was derived from elastic stability theory and verified by test data [9]. In both cases the end connectors must be welded or slipcritical bolted [9].
EXAMPLE 17.12:
Given: A builtup member with two backtoback angles is shown in Figure 17.23. Determine the modified slenderness ratio, (KL/r)m , in accordance with AISCLRFD [4] and Equation 17.98. rib = a = h = (KL/r)o =
0.735 in. (19 mm) 48 in. (1219 mm) 1.603 in. (41 mm) 70
Solution
1. Calculate the separation factor α. α= 1999 by CRC Press LLC
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1.603 h = 1.09 = 2rib 2(0.735)
FIGURE 17.23: A builtup member with backtoback angles.
2. Calculate the modified slenderness ratio, (KL/r)m , by Equation 17.98.
KL r
s
m
= s = =
17.10
KL r
2 o
+ 0.82
α2 (1 + α 2 )
1.092 (70) + 0.82 (1 + 1.092 ) 82.5 2
a rib
2
48 0.735
2
Tapered Columns
The stateoftheart design for tapered structural members was provided in the SSRC guide [36]. The charts shown in Figures 17.24 and 17.25 can be used to evaluate the effective length factors for tapered columns restrained by prismatic beams [36]. In these figures, IT and IB are the moment of inertia of the top and bottom beam, respectively; b and L are the length of beam and column, respectively; and γ is the tapering factor as defined by: γ =
d1 − do do
where do and d1 are the section depth of column at the smaller and larger end, respectively. 1999 by CRC Press LLC
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(17.99)
FIGURE 17.24: Effective length factor for tapered columns in braced frames.
17.11
Crane Columns
The columns in mill buildings and warehouses are designed to support overhead crane loads. The crosssection of a crane column may be uniform or stepped (see Figure 17.26). Over the past two decades, a number of simplified procedures have been developed for evaluating the Kfactors for crane columns [5, 6, 7, 11, 29, 30, 41, 51, 61]. Those procedures have limitations in terms of column geometry, loading, and boundary conditions. Most importantly, most of these studies ignored the interaction effect between the left and right column of frames and were based on isolated member analyses [59]. Recently, a simple yet reasonably accurate procedure for calculating the Kfactors for crane columns with any value of relative shaft length, moment of inertia, loading, and boundary conditions was developed by Lui and Sun [59]. Based on the story stiffness concept and accounting for both member and frame instability effects in the formulation, Lui and Sun [59] proposed the following procedure [see Figure 17.27]: 1. Apply the fictitious lateral loads, αP (α is an arbitrary factor; 0.001 may be used), in such a direction as to create a deflected geometry for the frame that closely approximates its actual buckled configuration. 2. Perform a firstorder elastic analysis P on the frame subjected to the fictitious lateral loads (Figure 17.27b). Calculate 11 / H , where 11 is the average lateral P deflection at the intermediate load points (i.e., points B and F) of columns, and H is the sum of all fictitious lateral loads that act at and above the intermediate load points. 3. Calculate η using results obtained from a firstorder elastic analysis for lower shafts (i.e., segments AB and F G), according to Equation 17.81. 4. Calculate the Kfactor for the lower shafts using Equation 17.80. 5. Calculate the Kfactor for upper shafts using the following formula: 1999 by CRC Press LLC
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FIGURE 17.25: Effective length factor for tapered columns in unbraced frames.
FIGURE 17.26: Typical crane columns. (From Lui, E.M. and Sun, M.Q., AISC Eng. J., 32(2), 98, 1995. With permission.)
1999 by CRC Press LLC
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FIGURE 17.27: Crane column model for effective length factor computation. (From Lui, E.M. and Sun, M.Q., AISC Eng. J., 32(2), 98, 1995. With permission.)
KU = KL
LL LU
s
PL + PU PU
IU IL
(17.100)
where P is the applied load and subscripts U and L represent upper and lower shafts, respectively.
EXAMPLE 17.13:
Given: A stepped crane column is shown in Figure 17.28a. The example is the same frame as used by Fraser [30] and Lui and Sun [59]. Determine the effective length factors for all columns using the Lui approach. E = 29,000 ksi (200 GPa). IAB 1999 by CRC Press LLC
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=
IF G = IL = 30,000 in.4 (1.25 × 1010 mm4 )
FIGURE 17.28: A pinbased stepped crane column. (From Lui, E.M. and Sun, M.Q., AISC Eng. J., 32(2), 98, 1995. With permission.)
AAB IBC ABC
= = =
AF G = AL = 75 in.2 (48,387 mm2 ) IEF = ICE = IU = 5,420 in.4 (2.26 × 109 mm4 ) AEF = ACE = AU = 34.14 in.2 (22,026 mm2 )
Solution
1. Apply a set of fictitious lateral forces with α = 0.001 as shown in Figure 17.28b. 1999 by CRC Press LLC
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2. Perform a firstorder analysis and find (11 )B = 0.1086 in.(2.76 mm) and (11 )F = 0.1077 in. (2.74 mm) so, (0.1086 + 0.1077)/2 11 P = = 0.198 in./kips (1.131 mm/kN) 0.053 + 0.3 + 0.053 + 0.14 H 3. Calculate η factors from Equation 17.81. Since the bottom of column AB and F G is pinbased, m = 0,
ηAB
= =
X
η
=
(3 + 4.8m + 4.2m2 )EI 3EI = 3 3 L L (3)(29,000)(30,000) = 42.03 kips/in. (7.36 mm/kN) (396)3
ηF G =
42.03 + 42.03 = 84.06 kips/in. (14.72 mm/kN)
4. Calculate the Kfactors for columns AB and F G using Equation 17.80. s
KAB
KF G
1 π 2 (29,000)(30,000) 353 + 193 + 0.198 396 5(84.06) (353)(396)2 = 6.55 s 1 π 2 (29,000)(30,000) 353 + 193 + 0.198 = 396 5(84.06) (193)(396)2 = 8.85 =
5. Calculate the Kfactors for columns BC and EF using Equation 17.100. s LAB PAB + PBC IBC = KAB LBC PBC IAB s 5420 353 396 = 18.2 = 6.55 156 53 30,000 s LF G PF G + PEF IEF = KF G LEF PEF IF G s 5420 193 396 = 18.2 = 8.85 156 53 30,000
KBC
KEF
The Kfactors calculated above are in good agreement with the theoretical values reported by Lui and Sun [59]. 1999 by CRC Press LLC
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17.12
Columns in Gable Frames
For a pinbased gable frame subjected to a uniformly distributed load on the rafter, as shown in Figure 17.29a, Lu [56] presented a graph (Figure 17.29b) to determine the effective length factors of columns. For frames having different member sizes for rafter and columns with (L/ h) of (f/ h)
FIGURE 17.29: Effective length factors for columns in a pinbased gable frame. (From Lu, L.W., AISC Eng. J., 2(2), 6, 1965. With permission.) ratios not covered in Figure 17.29, an approximate method is available for determining Kfactors of columns [39]. The method is to find an equivalent portal frame whose span length is equal to 1999 by CRC Press LLC
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twice the rafter length, Lr (see Figure 17.29a). The Kfactors can be determined form the alignment c/ h charts using Gtop = IrI/2L and corresponding Gbottom . r
17.13
Summary
This chapter summarizes the stateoftheart practice of the effective length factors for isolated columns, framed columns, diagonal bracing systems, latticed and builtup members, tapered columns, crane columns, and columns in gable frames. Design implementation with formulas, charts, tables, various modification factors adopted in current codes and specifications, as well as those used in engineering practice are described. Several examples illustrate the steps of practical applications of various methods.
17.14
Defining Terms
Alignment chart: A monograph for determining the effective length factor K for some types of compression members. Braced frame: A frame in which the resistance to lateral load or frame instability is primarily provided by diagonal bracing, shear walls, or equivalent means. Buildup member: A member made of structural metal elements that are welded, bolted, and riveted together. Column: A structural member whose primary function is to carry loads parallel to its longitudinal axis. Crane column: A column that is designed to support overhead crane loads. Effective length factor K: A factor that when multiplied by actual length of the endrestrained column gives the length of an equivalent pinended column whose elastic buckling load is the same as that of the endrestrained column. Framed column: A column in a framed structure. Gable frame: A frame with a gabled roof. Latticed member: A member made of two or more rolledshapes that are connected to one another by means of lacing bars, batten plates, or perforated plates. Leaning column: A column that is connected to a frame with simple connections and does not provide lateral stiffness or sidesway resistance. LRFD (Load and Resistance Factor Design): A method of proportioning structural components (members, connectors, connecting elements, and assemblages) such that no applicable limit state is exceeded when the structure is subjected to all appropriate load combinations. Tapered column: A column that has a continuous reduction in section from top to bottom. Unbraced frame: A frame in which the resistance to lateral loads is provided by the bending stiffness of frame members and their connections.
References [1] American Association of State Highway and Transportation Officials. 1994. LRFD Bridge Design Specifications, 1st ed., AASHTO, Washington, D.C. [2] American Concrete Institute. 1995. Building Code Requirements for Structural Concrete (ACI 31895) and Commentary (ACI 318R95). ACI, Farmington Hills, MI. 1999 by CRC Press LLC
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[3] American Institute of Steel Construction. 1989. Allowable Stress Design Specification for Structural Steel Buildings, 9th ed., AISC, Chicago, IL. [4] American Institute of Steel Construction. 1993. Load and Resistance Factor Design Specification for Structural Steel Buildings, 2nd ed., AISC, Chicago, IL. [5] Association of Iron and Steel Engineers. 1991. Guide for the Design and Construction of Mill Buildings, AISE, Technical Report, No. 13, Pittsburgh, PA. [6] Anderson, J.P. and Woodward, J.H. 1972. Calculation of Effective Lengths and Effective Slenderness Ratios of Stepped Columns. AISC Eng. J., 7(4):157166. [7] Agarwal, K.M. and Stafiej, A.P. 1980. Calculation of Effective Lengths of Stepped Columns. AISC Eng. J., 15(4):96105. [8] AristizabalOchoa, J.D. 1994. K Factors for Columns in Any Type of Construction: Nonparadoxical Approach. J. Struct. Eng., 120(4):12721290. [9] Aslani, F. and Goel, S.C. 1991. An Analytical Criteria for Buckling Strength of BuiltUp Compression Members. AISC Eng. J., 28(4):159168. [10] Barakat, M. and Chen, W.F. 1991. Design Analysis of SemiRigid Frames: Evaluation and Implementation. AISC Eng. J., 28(2):5564. [11] Bendapudi, K.V. 1994. Practical Approaches in Mill Building Columns Subjected to Heavy Crane Loads. AISC Eng. J., 31(4):125140. [12] Bjorhovde, R. 1984. Effect of End Restraints on Column Strength—Practical Application, AISC Eng. J., 21(1):113. [13] Bleich, F. 1952. Buckling Strength of Metal Structures. McGrawHill, New York. [14] Bridge, R.Q. and Fraser, D.J. 1987. Improved GFactor Method for Evaluating Effective Length of Columns. J. Struct. Eng., 113(6):13411356. [15] Chapius, J. and Galambos, T.V. 1982. Restrained Crooked Aluminum Columns, J. Struct. Div., 108(ST12):511524. [16] CheongSiatMoy, F. 1986. K Factor Paradox, J. Struct. Eng., 112(8):16471760. [17] Chen, W.F. and Lui, E.M. 1991. Stability Design of Steel Frames, CRC Press, Boca Raton, FL. [18] Chu, K.H. and Chow, H.L. 1969. Effective Column Length in Unsymmetrical Frames, Publ. Intl. Assoc. Bridge Struct. Eng., 29(1). [19] Cranston, W.B. 1972. Analysis and Design of Reinforced Concrete Columns. Research Report No. 20, Paper 41.020, Cement and Concrete Association, London. [20] Disque, R.O. 1973. Inelastic K Factor in Design. AISC Eng. J., 10(2):3335. [21] Duan, L. and Chen, W.F. 1988. Effective Length Factor for Columns in Braced Frames. J. Struct. Eng., 114(10):23572370. [22] Duan, L. and Chen, W.F. 1989. Effective Length Factor for Columns in Unbraced Frames. J. Struct. Eng., 115(1):149165. [23] Duan, L. and Chen, W.F. 1996. Errata of Paper: Effective Length Factor for Columns in Unbraced Frames. J. Struct. Eng., 122(1):224225. [24] Duan, L., King, W.S., and Chen, W.F. 1993. K Factor Equation to Alignment Charts for Column Design. ACI Struct. J., 90(3):242248. [25] Duan, L. and Lu, Z.G. 1996. A Modified GFactor for Columns in SemiRigid Frames. Research Report, Division of Structures, California Department of Transportation, Sacramento, CA. [26] Dumonteil, P. 1992. Simple Equations for Effective Length Factors. AISC Eng. J., 29(3):111115. [27] ElTayem, A.A. and Goel, S.C. 1986. Effective Length Factor for the Design of XBracing Systems. AISC Eng. J., 23(4):4145. [28] European Convention for Constructional Steelwork. 1978. European Recommendations for Steel Construction, ECCS. [29] Fraser, D.J. 1989. Uniform PinBased Crane Columns, Effective Length, AISC Eng. J., 26(2):6165.
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[30] Fraser, D.J. 1990. The InPlane Stability of a Frame Containing PinBased Stepped Column. AISC Eng. J., 27(2):4953. [31] French. 1975. Regles de calcul des constructions en acier CM66, Eyrolles, Paris, France. [32] Furlong, R.W. 1971. Column Slenderness and Charts for Design. ACI Journal, Proceedings, 68(1):918. [33] Galambos, T.V. 1960. Influence of Partial Base Fixity on Frame Instability. J. Struct. Div., 86(ST5):85108. [34] Galambos, T.V. 1964. Lateral Support for Tier Building Frames. AISC Eng. J., 1(1):1619. [35] Galambos, T.V. 1968. Structural Members and Frames. PrenticeHall International, London, U.K. [36] Galambos, T.V., Ed. 1988. Structural Stability Research Council, Guide to Stability Design Criteria for Metal Structures, 4th ed., John Wiley & Sons, New York. [37] Geschwindner, L.F. 1995. A Practical Approach to the “Leaning” Column, AISC Eng. J., 32(2):6372. [38] Gurfinkel, G. and Robinson, A.R. 1965. Buckling of Elasticity Restrained Column. J. Struct. Div., 91(ST6):159183. [39] Hansell, W.C. 1964. SingleStory Rigid Frames, in Structural Steel Design, Chapt. 20, Ronald Press, New York. [40] Hu, X.Y., Zhou, R.G., King, W.S., Duan, L., and Chen, W.F. 1993. On Effective Length Factor of Framed Columns in ACI Code. ACI Struct. J., 90(2):135143. [41] Huang, H.C. 1968. Determination of Slenderness Ratios for Design of Heavy Mill Building Stepped Columns. Iron Steel Eng., 45(11):123. [42] Johnson, D.E. 1960. Lateral Stability of Frames by Energy Method. J. Eng. Mech., 95(4):2341. [43] Johnston, B.G., Ed. 1976. Structural Stability Research Council, Guide to Stability Design Criteria for Metal Structures, 3rd ed., John Wiley & Sons, New York. [44] Jones, S.W., Kirby, P.A., and Nethercot, D.A. 1980. Effect of SemiRigid Connections on Steel Column Strength, J. Constr. Steel Res., 1(1):3846. [45] Jones, S.W., Kirby, P.A., and Nethercot, D.A. 1982. Columns with SemiRigid Joints, J. Struct. Div., 108(ST2):361372 [46] Julian, O.G. and Lawrence, L.S. 1959. Notes on J and L Nomograms for Determination of Effective Lengths. Unpublished report. [47] Kavanagh, T.C. 1962. Effective Length of Framed Column. Trans. ASCE, 127(II):81101. [48] King, W.S., Duan, L., Zhou, R.G., Hu, Y.X., and Chen, W.F. 1993. K Factors of Framed Columns Restrained by Tapered Girders in US Codes. Eng. Struct., 15(5):369378. [49] Kishi, N., Chen, W.F., and Goto, Y. 1995. Effective Length Factor of Columns in SemiRigid and Unbraced Frames. Structural Engineering Report CESTR955, School of Civil Engineering, Purdue University, West Lafayette, IN. [50] Koo, B. 1988. Discussion of Paper “Improved GFactor Method for Evaluating Effective Length of Columns” by Bridge and Fraser. J. Struct. Eng., 114(12):28282830. [51] Lay, M.G. 1973. Effective Length of Crane Columns, Steel Const., 7(2):919. [52] LeMessurier, W.J. 1977. A Practical Method of Second Order Analysis. Part 2—Rigid Frames. AISC Eng. J., 14(2):4967. [53] Liew, J.Y.R., White, D.W., and Chen, W.F. 1991. BeamColumn Design in Steel Frameworks— Insight on Current Methods and Trends. J. Const. Steel. Res., 18:269308. [54] Lim, L.C. and McNamara, R.J. 1972. Stability of Novel Building System, Structural Design of Tall Steel Buildings, Vol. II16, Proceedings, ASCEIABSE International Conference on the Planning and Design of Tall Buildings, Bethlehem, PA, 499524. [55] Lu, L.W. 1962. A Survey of Literature on the Stability of Frames. Weld. Res. Conc. Bull., New York. [56] Lu, L.W. 1965. Effective Length of Columns in Gable Frame, AISC Eng. J., 2(2):67. 1999 by CRC Press LLC
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[57] Lui, E.M. 1992. A Novel Approach for K Factor Determination. AISC Eng. J., 29(4):150159. [58] Lui, E.M. and Chen, W.F. 1983. Strength of Columns with Small End Restraints, J. Inst. Struct. Eng., 61B(1):1726 [59] Lui, E.M. and Sun, M.Q. 1995. Effective Length of Uniform and Stepped Crane Columns, AISC Eng. J., 32(2):98106. [60] Maquoi, R. and Jaspart, J.P. 1989. Contribution to the Design of Braced Framed with SemiRigid Connections. Proc. 4th International Colloquium, Structural Stability Research Council, 209220. Lehigh University, Bethlehem, PA. [61] Moore, W.E. II. 1986. A Programmable Solution for Stepped Crane Columns. AISC Eng. J., 21(2):5558. [62] Picard, A. and Beaulieu, D. 1987. Design of Diagonal Cross Bracings. Part 1: Theoretical Study. AISC Eng. J., 24(3):122126. [63] Picard, A. and Beaulieu, D. 1988. Design of Diagonal Cross Bracings. Part 2: Experimental Study. AISC Eng. J., 25(4):156160. [64] Razzaq, Z. 1983. End Restraint Effect of Column Strength. J. Struct. Div., 109(ST2):314334. [65] Salmon, C.G., Schenker, L., and Johnston, B.G. 1957. MomentRotation Characteristics of Column Anchorage. Trans. ASCE, 122:132154. [66] Shanmugam, N.E. and Chen, W.F. 1995. An Assessment of K Factor Formulas. AISC Eng. J., 32(3):311. [67] Sohal, I.S., Yong, Y.K., and Balagura, P.N. 1995. K Factor in Plastic and SOIA for Design of Steel Frames, Proceeding International Conference on Stability of Structures, ICSS 95, June 79, Coimbatore, India, 411421. [68] Sugimoto, H. and Chen, W.F. 1982. Small End Restraint Effects on Strength of HColumns, J. Struct. Div., 108(ST3):661681. [69] Vinnakota, S. 1982. Planar Strength of Restrained Beam Columns. J. Struct. Div., 108(ST11):23492516. [70] Timoshenko, S.P. and Gere, J.M. 1961. Theory of Elastic Stability, 2nd ed., McGrawHill, New York. [71] Winter, G. et al. 1948. Buckling of Trusses and Rigid Frames, Cornell Univ. Bull. No. 36, Engineering Experimental Station, Cornell University, Ithaca, NY. [72] Wood, R.H. 1974. Effective Lengths of Columns in MultiStorey Buildings. Struct. Eng., 52(7,8,9):234244, 295302, 341346. [73] Yura, J.A. 1971. The Effective Length of Columns in Unbraced Frames. AISC Eng. J., 8(2):3742. [74] Zandonini, R. 1985. Stability of Compact BuiltUp Struts: Experimental Investigation and Numerical Simulation (in Italian). Construzioni Metalliche, No. 4.
Further Reading [1] Chen, W.F. and Lui, E.M. 1987. Structural Stability: Theory and Implementation, Elsevier, New York. [2] Chen, W.F., Goto, Y. and Liew, J.Y.R. 1996. Stability Design of SemiRigid Frames, John Wiley and Sons, New York. [3] Chen, W.F. and Kim, S.E. 1997. LRFD Steel Design Using Advanced Analysis, CRC Press, Boca Raton, FL.
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Bjorhovde, R. “Stub Girder Floor Systems” Structural Engineering Handbook Ed. Chen WaiFah Boca Raton: CRC Press LLC, 1999
Stub Girder Floor Systems 18.1 Introduction 18.2 Description of the Stub Girder Floor System 18.3 Methods of Analysis and Modeling
General Observations • Preliminary Design Procedure • Choice of Stub Girder Component Sizes • Modeling of the Stub Girder
18.4 Design Criteria For Stub Girders
Reidar Bjorhovde Department of Civil and Environmental Engineering, University of Pittsburgh, Pittsburgh, PA
18.1
General Observations • Governing Sections of the Stub Girder • Design Checks for the Bottom Chord • Design Checks for the Concrete Slab • Design Checks for the Shear Transfer Regions • Design of Stubs for Shear and Axial Load • Design of Stud Shear Connectors • Design of Welds between Stub and Bottom Chord • Floor Beam Connections to Slab and Bottom Chord • Connection of Bottom Chord to Supports • Use of Stub Girder for Lateral Load System • Deflection Checks
18.5 Influence of Method of Construction 18.6 Defining Terms References Further Reading
Introduction
The stub girder system was developed in response to a need for new and innovative construction techniques that could be applied to certain parts of all multistory steelframed buildings. Originated in the early 1970s, the design concept aimed at providing construction economies through the integration of the electrical and mechanical service ducts into the part of the building volume that is occupied by the floor framing system [11, 12]. It was noted that the overall height of the floor system at times could be large, leading to significant increases in the overall height of the structure, and hence the steel tonnage for the project. At other times the height could be reduced, but only at the expense of having sizeable web penetrations for the ductwork to pass through. This solution was often accompanied by having to reinforce the web openings by stiffeners, increasing the construction cost even further. The composite stub girder floor system subsequently was developed. Making extensive use of relatively simple shop fabrication techniques, basic elements with limited fabrication needs, simple connections between the main floor system elements and the structural columns, and composite action between the concrete floor slab and the steel loadcarrying members, a floor system of significant strength, stiffness, and ductility was devised. This led to a reduction in the amount of structural steel that traditionally had been needed for the floor framing. When coupled with the use of continuous, 1999 by CRC Press LLC
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composite transverse floor beams and the shorter erection time that was needed for the stub girder system, this yielded attractive cost savings. Since its introduction, the stub girder floor system has been used for a variety of steelframed buildings in the U.S., Canada, and Mexico, ranging in height from 2 to 72 stories. Despite this relatively widespread usage, the analysis techniques and design criteria remain unknown to many designers. This chapter will offer examples of practical uses of the system, together with recommendations for suitable design and performance criteria.
18.2
Description of the Stub Girder Floor System
The main element of the system is a special girder, fabricated from standard hotrolled wideflange shapes, that serves as the primary framing element of the floor. Hotrolled wideflange shapes are also used as transverse floor beams, running in a direction perpendicular to the main girders. The girder and the beams are usually designed for composite action, although the system does not rely on having composite floor beams, and the latter are normally analyzed as continuous beams. As a result, the transverse floor beams normally use a smaller dropin span within the positive moment region. This results in further economies for the floor beam design, since it takes advantage of continuous beam action. Allowable stress design (ASD) or load and resistance factor design (LRFD) criteria are equally applicable for the design of stub girders, although LRFD is preferable, since it gives lower steel weights and simple connections. The costs that are associated with an LRFDdesigned stub girder therefore tend to be lower. Figure 18.1 shows the elevation of a typical stub girder. It is noted that the girder that is shown
FIGURE 18.1: Elevation of a typical stub girder (one half of span is shown).
makes use of four stubs, oriented symmetrically with respect to the midspan of the member. The locations of the transverse floor beams are assumed to be the quarter points of the span, and the supports are simple. In practice many variations of this layout are used, to the extent that the girders may utilize any number of stubs. However, three to five stubs is the most common choice. The locations of the stubs may differ significantly from the symmetrical case, and the exterior ( = end) stubs may have been placed at the very ends of the bottom chord. However, this is not difficult to 1999 by CRC Press LLC
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address in the modeling of the girder, and the essential requirements are that the forces that develop as a result of the choice of girder geometry be accounted for in the design of the girder components and the adjacent structure. These actual forces are used in the design of the various elements, as distinguished from the simplified models that are currently used for many structural components. The choices of elements, etc., are at the discretion of the design team, and depend on the service requirements of the building as seen from the architectural, structural, mechanical, and electrical viewpoints. Unique design considerations must be made by the structural engineer, for example, if it is decided to eliminate the exterior openings and connect the stubs to the columns in addition to the chord and the slab. Figure 18.1 shows the main components of the stub girder, as follows: 1. 2. 3. 4. 5. 6. 7. 8.
Bottom chord Exterior and interior stubs Transverse floor beams Formed steel deck Concrete slab with longitudinal and transverse reinforcement Stud shear connectors Stub stiffeners Beamtocolumn connection
The bottom chord should preferably be a hotrolled wideflange shape of columntype proportions, most often in the W12 to W14 series of wideflange shapes. Other chord crosssections have been considered [19]; for example, T shapes and rectangular tubes have certain advantages as far as welded attachments and fire protection are concerned, respectively. However, these other shapes also have significant drawbacks. The rolled tube, for example, cannot accommodate the shear stresses that develop in certain regions of the bottom chord. Rather than using a T or a tube, therefore, a smaller W shape (in the W10 series, for example) is most likely the better choice under these conditions. The steel grade for the bottom chord, in particular, is important, since several of the governing regions of the girder are located within this member, and tension is the primary stress resultant. It is therefore possible to take advantage of higher strength steels, and 50ksiyield stress steel has typically been the choice, although 65ksi steel would be acceptable as well. The floor beams and the stubs are mostly of the same size W shape, and are normally selected from the W16 and W18 series of shapes. This is directly influenced by the size(s) of the HVAC ducts that are to be used, and input from the mechanical engineer is essential at this stage. Although it is not strictly necessary that the floor beams and the stubs use identical shapes, it avoids a number of problems if such a choice is made. At the very least, these two components of the floor system should have the same height. The concrete slab and the steel deck constitute the top chord of the stub girder. It is made either from lightweight or normal weight concrete, although if the former is available, even at a modest cost premium, it is preferred. The reason is the lower dead load of the floor, especially since the shores that will be used are strongly influenced by the concrete weight. Further, the shores must support several stories before they can be removed. In other words, the stub girders must be designed for shored construction, since the girder requires the slab to complete the system. In addition, the bending rigidity of the girder is substantial, and a major fraction is contributed by the bottom chord. The reduction in slab stiffness that is prompted by the lower value of the modulus of elasticity for the lightweight concrete is therefore not as important as it may be for other types of composite bending members. Concrete strengths of 3000 to 4000 psi are most common, although the choice also depends on the limit state of the stud shear connectors. Apart from certain longspan girders, some local regions in the 1999 by CRC Press LLC
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slab, and the desired mode of behavior of the slabtostub connection (which limits the maximum fc0 value that can be used), the strength of the stub girder is not controlled by the concrete. Consequently, there is little that can gained by using highstrength concrete. The steel deck should be of the composite type, and a number of manufacturers produce suitable types. Normal deck heights are 2 and 3 in., but most floors are designed for the 3in. deck. The deck ribs are run parallel to the longitudinal axis of the girder, since this gives better deck support on the transverse floor beams. It also increases the top chord area, which lends additional stiffness to a member that can span substantial distances. Finally, the parallel orientation provides a continuous rib trough directly above the girder centerline, improving the composite interaction of the slab and the girder. Due to fire protection requirements, the thickness of the concrete cover over the top of the deck ribs is either 43/16 in. (normal weight concrete) or 31/4 in. (lightweight concrete). This eliminates the need for applying fire protective material to the underside of the steel deck. Stud shear connectors are distributed uniformly along the length of the exterior and interior stubs, as well as on the floor beams. The number of connectors is determined on the basis of the computed shear forces that are developed between the slab and the stubs. This is in contrast to the current design practice for simple composite beams, which is based on the smaller of the ultimate axial loadcarrying capacity of the slab and the steel beam [2, 3]. However, the simplified approach of current specifications is not applicable to members where the crosssection varies significantly along the length (nonprismatic beams). The computed shear force design approach also promotes connector economy, in the sense that a much smaller number of shear connectors is required in the interior shear transfer regions of the girder [5, 7, 21]. The stubs are welded to the top flange of the bottom chord with fillet welds. In the original uses of the system, the design called for allaround welds [11, 12]; subsequent studies demonstrated that the forces that are developed between the stubs and the bottom chord are concentrated toward the end of the stubs [5, 6, 21]. The welds should therefore be located in these regions. The type and locations of the stub stiffeners that are indicated for the exterior stubs in Figure 18.1, as well as the lack of stiffeners for the interior stubs, represent one of the major improvements that were made to the original stub girder designs. Based on extensive research [5, 21], it was found that simple endplate stiffeners were as efficient as the traditional fitted ones, and in many cases the stiffeners could be eliminated at no loss in strength and stiffness to the overall girder. Figure 18.1 shows that a simple (shear) connection is used to attach the bottom chord of the stub girder to the adjacent structure (column, concrete building core, etc.). This is the most common solution, especially when a duct opening needs to be located at the exterior end of the girder. If the support is an exterior column, the slab will rest on an edge member; if it is an interior column, the slab will be continuous past the column and into the adjacent bay. This may or may not present problems in the form of slab cracking, depending on the reinforcement details that are used for the slab around the column. The stub girder has sometimes been used as part of the lateral loadresisting system of steelframed buildings [13, 17]. Although this has certain disadvantages insofar as column moments and the concrete slab reinforcement are concerned, the girder does provide significant lateral stiffness and ductility for the frame. As an example, the maintenance facility for Mexicana Airlines at the Mexico City International Airport, a structure utilizing stub girders in this fashion [17], survived the 1985 Mexico City earthquake with no structural damage. Expanding on the details that are shown in Figure 18.1, Figure 18.2 illustrates the crosssection of a typical stub girder, and Figure 18.3 shows a complete girder assembly with lights, ducts, and suspended ceiling. Of particular note are the longitudinal reinforcing bars. They add flexural strength as well as ductility and stiffness to the girder, by helping the slab to extend its service range. The longitudinal rebars are commonly placed in two layers, with the top one just below the heads of the stud shear connectors. The lower longitudinal rebars must be raised above the deck proper, 1999 by CRC Press LLC
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FIGURE 18.2: Crosssections of a typical stub girder (refer to Figure 18.1 for section location).
FIGURE 18.3: Elevation of a typical stub girder, complete with ductwork, lights, and suspended ceiling (duct sizes, etc., vary from system to system).
using high chairs or other means. This assures that the bars are adequately confined. The transverse rebars are important for adding shear strength to the slab, and they also help in the shear transfer from the connectors to the slab. The transverse bars also increase the overall ductility of the stub girder, and placing the bars in a herring bone pattern leads to a small improvement in the effective width of the slab. The common choices for stub girder floor systems have been 36 or 50ksiyield stress steel, with a preference for the latter, because of the smaller bottom chord size that can be used. Due to its function in the girder, there is no reason why steels such as ASTM A913 (65 ksi) cannot be used for the bottom chord. However, all detail materials (stiffeners, connection angles, etc.) are made from 36ksi steel. Welding is usually done with 70grade low hydrogen electrodes, using either the SMAW, 1999 by CRC Press LLC
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FCAW, or GMAW process, and the stud shear connectors are welded in the normal fashion. All of the work is done in the fabricating shop, except for the shear connectors, which are applied in the field, where they are welded directly through the steel deck. The completed stub girders are then shipped to the construction site.
18.3
Methods of Analysis and Modeling
18.3.1
General Observations
In general, any number of methods of analysis may be used to determine the bending moments, shear forces, and axial forces throughout the components of the stub girder. However, it is essential to bear in mind that the modeling of the girder, or, in other words, how the actual girder is transformed into an idealized structural system, should reflect the relative stiffness of the elements. This means that it is important to establish realistic trial sizes of the components, through an appropriate preliminary design procedure. The subsequent modeling will then lead to stress resultants that are close to the magnitudes that can be expected in actual stub girders. Based on this approach, the design that follows is likely to require relatively few changes, and those that are needed are often so small that they have no practical impact on the overall stiffness distribution and final member forces. The preliminary design procedure is therefore a very important step in the overall design. However, it will be shown that by using an LRFD approach, the process is simple, efficient, and accurate.
18.3.2
Preliminary Design Procedure
Using the LRFD approach for the preliminary design, it is not necessary to make any assumptions as regards the stress distribution over the depth of the girder, other than to adhere to the strength model that was developed for normal composite beams [3, 15]. The stress distribution will vary anyway along the span because of the openings. The strength model of Hansell et al. [15] assumes that when the ultimate moment is reached, all or a portion of the slab is failing in compression, with a uniformly distributed stress of 0.85fc0 . The steel shape is simultaneously yielding in tension. Equilibrium is therefore maintained, and the internal stress resultants are determined using first principles. Tests have demonstrated excellent agreement with theoretical analyses that utilize this approach [5, 7, 15, 21]. The LRFD procedure uses load and resistance factors in accordance with the American Institute of Steel Construction (AISC) LRFD specification [3]. The applicable resistance factor is given by the AISC LRFD specification, Section D1, for the case of gross crosssection yielding. This is because the preliminary design is primarily needed to find the bottom chord size, and this component is primarily loaded in tension [5, 7, 10, 21]. The load factors of the LRFD specification are those of the American Society of Civil Engineers (ASCE) load standard [4], for the combination of dead plus live load. The load computations follow the choice of the layout of the floor framing plan, whereby girder and floor beam spans are determined. This gives the tributary areas that are needed to calculate the dead and live loads. The load intensities are governed by local building code requirements or by the ASCE recommendations, in the absence of a local code. Reduced live loads should be used wherever possible. This is especially advantageous for stub girder floor systems, since spans and tributary areas tend to be large. The ASCE load standard [4] makes use of a live load reduction factor, RF , that is significantly simpler to use, and also less conservative than that of earlier codes. The standard places some restrictions on the value of RF , to the effect that the reduced live load cannot be less than 50% of the nominal value for structural members that 1999 by CRC Press LLC
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support only one floor. Similarly, it cannot be less than 40% of the nominal live load if two or more floors are involved. Proceeding with the preliminary design, the stub girder and its floor beam locations determine the magnitudes of the concentrated loads that are to be applied at each of the latter locations. The following illustrative example demonstrates the steps of the solution.
FIGURE 18.4: Stub girder layout used for preliminary design example.
EXAMPLE 18.1:
Given: Figure 18.4 shows the layout of the stub girder for which the preliminary sizes are needed. Other computations have already given the sizes of the floor beam, the slab, and the steel deck. The span of the girder is 40 ft, the distance between adjacent girders is 30 ft, and the floor beams are located at the quarter points. The steel grade remains to be chosen (36 and 50ksiyield stress steel are the most common); the concrete is lightweight, with wc = 120 pcf and a compressive strength of fc0 = 4000 psi. Solution Loads: Estimated dead load = 74 psf Nominal live load = 50 psf Live load reduction factor: p RF = 0.25 + 15/ [2 × (30 × 30)] = 0.60 Reduced live load: RLL = 0.60 × 50 = 30 psf Load factors (for D + L combination): For dead load: 1.2 For live load: 1.6 1999 by CRC Press LLC
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Factored distributed loads: Dead Load, DL = 74 × 1.2 = 88.8 psf Live Load, LL = 30 × 1.6 = 48.0 psf Total = 136.8 psf Concentrated factored load at each floor beam location: Due to the locations of the floor beams and the spacing of the stub girders, the magnitude of each load, P , is: P = 136.8 × 30 × 10 = 41.0 kips Maximum factored midspan moment: The girder is symmetric about midspan, and the maximum moment therefore occurs at this location: Mmax = 1.5 × P × 20 − P × 10 = 820 kft Estimated interior moment arm for full stub girder crosssection at midspan (refer to Figure 18.2 for typical details): The interior moment arm (i.e., the distance between the compressive stress resultant in the concrete slab and the tensile stress resultant in the bottom chord) is set equal to the distance between the slab centroid and the bottom chord (wideflange shape) centroid. This is simplified and conservative. In the example, the distance is estimated as Interior moment arm: d = 27.5 in. This is based on having a 14 series W shape for the bottom chord, W16 floor beams and stubs, a 3in.high steel deck, and 31/4 in. of lightweight concrete over the top of the steel deck ribs (this allows the deck to be used without having sprayedon fire protective material on the underside). These are common sizes of the components of a stub girder floor system. In general, the interior moment arm varies between 24.5 and 29.5 in., depending on the heights of the bottom chord, floor beams/stubs, steel deck, and concrete slab. Slab and bottom chord axial forces, F (these are the compressive and tensile stress resultants): F = Mmax /d = (820 × 12)/27.5 = 357.9 kips Required crosssectional area of bottom chord, As : The required crosssectional area of the bottom chord can now be found. Since the chord is loaded in tension, the φ value is 0.9. It is also important to note that in the vierendeel analysis that is commonly used in the final evaluation of the stub girder, the member forces will be somewhat larger than those determined through the simplified preliminary procedure. It is therefore recommended that an allowance of some magnitude be given for the vierendeel action. This is done most easily by increasing the area, As , by a certain percentage. Based on experience [7, 10], an increase of onethird is suitable, and such has been done in the computations that follow. On the basis of the data that have been developed, the required area of the bottom chord is: (Mmax /d) 4 4 F × = × As = φ × Fy 3 0.9 × Fy 3 1999 by CRC Press LLC
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which gives As values for 36ksi and 50ksi steel of As
=
As
=
357.9 4 × = 14.73 in.2 (Fy = 36 ksi) 0.9 × 36 3 4 357.9 × = 10.60 in.2 (Fy = 50 ksi) 0.9 × 50 3
Conclusions: If 36ksi steel is chosen for the bottom chord of the stub girder, the wideflange shapes W12x50 and W14x53 will be suitable. If 50ksi steel is the choice, the sections may be W12x40 or W14x38. Obviously the final decision is up to the structural engineer. However, in view of the fact that the W12 series shapes will save approximately 2 in. in net floor system height, per story of the building, this would mean significant savings if the overall structure is 10 to 15 stories or more. The differences in stub girder strength and stiffness are not likely to play a role [7, 10, 14].
18.3.3
Choice of Stub Girder Component Sizes
Some examples have been given in the preceding for the choices of chord and floor beam sizes, deck height, and slab configuration. These were made primarily on the basis of acceptable geometries, deck size, and fire protection requirements, to mention some examples. However, construction economy is critical, and the following guidelines will assist the user. The data that are given are based on actual construction projects. Economical span lengths for the stub girder range from 30 to 50 ft, although the preferable spans are 35 to 45 ft; 50ft span girders are erectable, but these are close to the limit where the dead load becomes excessive, which has the effect of making the slab govern the overall design. This is usually not an economical solution. Spans shorter than 30 ft are known to have been used successfully; however, this depends on the load level and the type of structure, to mention the key considerations. Depending on the type and configuration of steel deck that has been selected, the floor beam spacing should generally be maintained between 8 and 12 ft, although larger values have been used. The decisive factor is the ability of the deck to span the distance between the floor beams. The performance of the stub girder is not particularly sensitive to the stub lengths that are used, as long as these are kept within reasonable limits. In this context it is important to observe that it is usually the exterior stub that controls the behavior of the stub girder. As a practical guideline, the exterior stubs are normally 5 to 7 ft long; the interior stubs are considerably shorter, normally around 3 ft, but components up to 5 ft long are known to have been used. When the stub lengths are chosen, it is necessary to bear in mind the actual purpose of the stubs and how they carry the loads on the stub girder. That is, the stubs are loaded primarily in shear, which explains why the interior stubs can be kept so much shorter than the exterior ones. The shear connectors that are welded to the top flange of the stub, the stub web stiffeners, and the welds between the bottom flange of the stub and the top flange of the bottom chord are crucial to the function of the stub girder system. For example, the first application of stub girders utilized fitted stiffeners at the ends and sometimes at midlength of all of the stubs. Subsequent research demonstrated that the midlength stiffener did not perform any useful function, and that only the exterior stubs needed stiffeners in order to provide the requisite web stability and shear capacity [5, 21]. Regardless of the span of the girder, it was found that the interior stubs could be left unstiffened, even when they were made as short as 3 ft [7, 14]. Similar savings were realized for the welds and the shear connectors. In particular, in lieu of allaround fillet welds for the connection between the stub and the bottom chord, the studies showed 1999 by CRC Press LLC
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that a significantly smaller amount of welding was needed, and often only in the vicinity of the stub ends. However, specific weld details must be based on appropriate analyses of the stub, considering overturning, weld capacity at the tension end of the stub, and adequate ability to transfer shear from the slab to the bottom chord.
18.3.4
Modeling of the Stub Girder
The original work of Colaco [11, 12] utilized a vierendeel modeling scheme for the stub girder to arrive at a set of stress resultants, which in turn were used to size the various components. Elastic finite element analyses were performed for some of the girders that had been tested, mostly to examine local stress distributions and the correlation between test and theory. However, the finite element solution is not a practical design tool. Other studies have examined approaches such as nonprismatic beam analysis [6, 21] and variations of the finite element method [16]. The nonprismatic beam solution is relatively simple to apply. On the other hand, it is not as accurate as the vierendeel approach, since it tends to overlook some important local effects and overstates the service load deflections [5, 21]. On the whole, therefore, the vierendeel modeling of the stub girder has been found to give the most accurate and consistent results, and the correlation with test results is good [5, 6, 11, 14, 21]. Finally, it offers the best physical similarity with actual girders; many designers have found this to be an important advantage. There are no “simple” methods of analysis that can be used to find the bending moments, shear forces, and axial forces in vierendeel girders. Once the preliminary sizing has been accomplished, a computer solution is required for the girder. In general, all that is required for the vierendeel evaluation is a twodimensional plane frame program for elastic structural analysis. This gives moments, shears, and axial forces, as well as deflections, joint rotations, and other displacement characteristics. The stress resultants are used to size the girder and its elements and connections; the displacements reflect the serviceability of the stub girder. Once the stress resultants are known, the detailed design of the stub girder can proceed. A final runthrough of the girder model should then be done, using the components that were chosen, to ascertain that the performance and strength are sufficient in all respects. Under normal circumstances no alterations are necessary at this stage. As an illustration of the vierendeel modeling of a stub girder, the girder itself is shown in Figure 18.5a and the vierendeel model in Figure 18.5b. The girder is the same as the one used for the preliminary design example. It has four stubs and is symmetrical about midspan; therefore, only half is illustrated. The boundary conditions are shown in Figure 18.5b. The bottom chord of the model is assigned a moment of inertia equal to the major axis I value, Ix , of the wideflange shape that was chosen in the preliminary design. However, some analysts believe that since the stub is welded to the bottom chord, a portion of its flexural stiffness should be added to that of the moment of inertia of the wideflange shape [5, 7, 14, 21] This approach is identical to treating the bottom chord W shape as if it has a cover plate on its top flange. The area of this cover plate is the same as the area of the bottom flange of the stub. This should be done only in the areas where the stubs are placed. In the regions of the interior and exterior stubs it is therefore realistic to increase the moment of inertia of the bottom chord by the parallelaxis value of Af × df2 , where Af designates the area of the bottom flange of the stub and df is the distance between the centroids of the flange plate and the W shape. The contribution to the overall stub girder stiffness is generally small. The bending stiffness of the top vierendeel chord equals that of the effective width portion of the slab. This should include the contributions of the steel deck as well as the reinforcing steel bars that are located within this width. In particular, the influence of the deck is important. The effective width is determined from the criteria in the AISC LRFD specification, Section I3.1 [3]. It is noted 1999 by CRC Press LLC
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FIGURE 18.5: An actual stub girder and its vierendeel model (due to symmetry, only one half of the span is shown). that these were originally developed on the basis of analyses and tests of prismatic composite beams. The approach has been found to give conservative results [5, 21], but should continue to be used until more accurate criteria are available. In the computations for the slab, the crosssection is conveniently subdivided into simple geometrical shapes. The individual areas and moments of inertia are determined on the basis of the usual transformation from concrete to steel, using the modular ratio n = E/Ec , where E is the modulus of elasticity of the steel and Ec is that of concrete. The latter must reflect the density of the concrete that is used, and can be computed from [1]: p (18.1) Ec = 33 × wc1.5 × fc0 The shear connectors used for the stub are required to develop 100% interaction, since the design is based on the computed shear forces, rather than the axial capacity of the steel beam or the concrete slab, as is used for prismatic beams in the AISC Specifications [2, 3]. However, it is neither common nor proper to add the moment of inertia contribution of the top flange of the stub to that of the slab, contrary to what is done for the bottom chord. The reason for this is that dissimilar materials are joined, and some local concrete cracking and/or crushing can be expected to take place around the shear connectors. The discretization of the stubs into vertical vierendeel girder components is relatively straightforward. Considering the web of the stub and any stiffeners, if applicable (for exterior stubs, most commonly, since interior stubs usually can be left unstiffened), the moment of inertia about an axis that is perpendicular to the plane of the web is calculated. As an example, Figure 18.6 shows the stub and stiffener configuration for a typical case. The stub is a 5ft long W16x26 with 51/2x1/2in. endplate stiffeners. The computations give: Moment of inertia about the Z − Z axis: h i IZZ = 0.25 × (60)3 /12 + 2 × 5.5 × 0.5 × (30)2 = 1999 by CRC Press LLC
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9450 in.4
FIGURE 18.6: Horizontal crosssection of stub with stiffeners. Depending on the number of vierendeel truss members that will represent the stub in the model, the bending stiffness of each is taken as a fraction of the value of IZZ . For the girder shown in Figure 18.5, where the stub is discretized as three vertical members, the magnitude of Ivert is found as: Moment of inertia of vertical member: Ivert = IZZ /(no. of verticals ) = 9450/3 = 3150 in.4 The crosssectional area of the stub, including the stiffeners, is similarly divided between the verticals: Area of vertical member: Avert
= [Aweb + 2 × Ast ] /(no. of verticals) = [0.25 × (60 − 2 × 0.5) + 2 × 5.5 × 0.5] /3 = 6.75 in.2
Several studies have aimed at finding the optimum number of vertical members to use for each stub. However, the strength and stiffness of the stub girder are only insignificantly affected by this choice, and a number between 3 and 7 is usually chosen. As a rule of thumb, it is advisable to have one vertical per foot length of stub, but this should serve only as a guideline. The verticals are placed at uniform intervals along the length of the stub, usually with the outside members close to the stub ends. Figure 18.5 illustrates the approach. As for end conditions, these vertical members are assumed to be rigidly connected to the top and bottom chords of the vierendeel girder. One vertical member is placed at each of the locations of the floor beams. This member is assumed to be pinned to the top and bottom chords, as shown in Figure 18.5, and its stiffness is conservatively set equal to the moment of inertia of a plate with a thickness equal to that of the web of the floor beam and a length equal to the beam depth. In the example, tw = 0.25 in.; the beam depth is 15.69 in. This gives a moment of inertia of i h 15.69 × 0.253 /12 = 0.02 in.4 and the crosssectional area is (15.69 × 0.25) = 3.92 in.2 The vierendeel model shown in Figure 18.5b indicates that the portion of the slab that spans across the opening between the exterior end of the exterior stub and the support for the slab (a column, or a corbel of the core of the structural frame) has been neglected. This is a realistic simplification, considering the relatively low rigidity of the slab in negative bending. 1999 by CRC Press LLC
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Figure 18.5b also shows the support conditions that are used as input data for the computer analysis. In the example, the symmetrical layout of the girder and its loads make it necessary to analyze only onehalf of the span. This cannot be done if there is any kind of asymmetry, and the entire girder must then be analyzed. For the girder that is shown, it is known that only vertical displacements can take place at midspan; horizontal displacements and end rotations are prevented at this location. At the far ends of the bottom chord only horizontal displacements are permitted, and end rotations are free to occur. The reactions that are found are used to size the support elements, including the bottom chord connections and the column. The structural analysis results are shown in Figure 18.7, in terms of the overall bending moment, shear force, and axial force distributions of the vierendeel model given in Figure 18.5b. Figure 18.7d repeats the layout details of the stub girder, to help identify the locations of the key stress resultant magnitudes with the corresponding regions of the girder.
FIGURE 18.7: Distributions of bending moments, shear forces, and axial forces in a stub girder (see Figure 18.5) (dead load = 74 psf; nominal live load = 50 psf).
The design of the stub girder and its various components can now be done. This must also include deflection checks, even though research has demonstrated that the overall design will never be gov1999 by CRC Press LLC
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erned by deflection criteria [7, 14]. However, since the girder has to be built in the shored condition, the girder is often fabricated with a camber, approximately equal to the dead load deflection [7, 10].
18.4
Design Criteria For Stub Girders
18.4.1
General Observations
In general, the design of the stub girder and its components must consider overall member strength criteria as well as local checks. For most of these, the AISC Specifications [2, 3] give requirements that address the needs. Further, although LRFD and ASD are equally applicable in the design of the girder, it is recommended that LRFD be used exclusively. The more rational approach of this specification makes it the method of choice. In several important areas there are no standardized rules that can be used in the design of the stub girder, and the designer must rely on rational engineering judgment to arrive at satisfactory solutions. This applies to the parts of the girder that have to be designed on the basis of computed forces, such as shear connectors, stiffeners, stubtochord welds, and slab reinforcement. The modeling and evaluation of the capacity of the central portion of the concrete slab are also subject to interpretation. However, the design recommendations that are given in the following are based on a wide variety of practical and successful applications. It is again emphasized that the design throughout is based on the stress resultants that have been determined in the vierendeel or other analysis, rather than on idealized code criteria. However, the capacities of materials and fasteners, as well as the requirements for the stability and strength of tension and compression members, adhere strictly to the AISC Specifications. Any interpretations that have been made are always to the conservative side.
18.4.2
Governing Sections of the Stub Girder
Figures 18.5 and 18.7 show certain circled numbers at various locations throughout the span of the stub girder. These reflect the sections of the girder that are the most important, for one reason or another, and are the ones that must be examined to determine the required member size, etc. These are the governing sections of the stub girder and are itemized as follows: 1. Points 1, 2, and 3 indicate the critical sections for the bottom chord. 2. Points 4, 5, and 6 indicate the critical sections for the concrete slab. 3. Point 7, which is a region rather than a specific point, indicates the critical shear transfer region between the slab and the exterior stub. The design checks that must be made for each of these areas are discussed in the following.
18.4.3
Design Checks for the Bottom Chord
The size of the bottom chord is almost always governed by the stress resultants at midspan, or point 3 in Figures 18.5 and 18.7. This is also why the preliminary design procedure focused almost entirely on determining the required chord crosssection at this location. As the stress resultant distributions in Figure 18.7 show, the bottom chord is subjected to combined positive bending moment and tensile force at point 3, and the design check must consider the beamtension member behavior in this area. The design requirements are given in Section H1.1, Eqs. (H11a) and (H11b), of the AISC LRFD Specification [3]. Combined bending and tension must also be evaluated at point 2, the exterior end of the interior stub. The local bending moment in the chord is generally larger here than at midspan, but the axial 1999 by CRC Press LLC
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force is smaller. Only a computation can confirm whether point 2 will govern in lieu of point 3. Further, although the location at the interior end of the exterior stub (point 2a) is rarely critical, the combination of negative moment and tensile force should be evaluated. At point 1 of the bottom chord, which is located at the exterior end of the exterior stub, the axial force is equal to zero. At this location the bottom chord must therefore be checked for pure bending, as well as shear. The preceding applies only to a girder with simple end supports. When it is part of the lateral loadresisting system, axial forces will exist in all parts of the chord. These must be resisted by the adjacent structural members.
18.4.4
Design Checks for the Concrete Slab
The top chord carries varying amounts of bending moment and axial force, as illustrated in Figure 18.7, but the most important areas are indicated as points 4 to 6. The axial forces are always compressive in the concrete slab; the bending moments are positive at points 5 and 6, but negative at point 4. As a result, this location is normally the one that governs the performance of the slab, not the least because the reinforcement in the positive moment region includes the substantial crosssectional area of the steel deck. The full effective width of the slab must be analyzed for combined bending and axial force at all of points 4 through 6. Either the composite beamcolumn criteria of the AISC LRFD specification [3] or the criteria of the reinforced concrete structures code of the American Concrete Institute (ACI) [1] may be used for this purpose.
18.4.5
Design Checks for the Shear Transfer Regions
Region 7 is the shear transfer region between the concrete slab and the exterior stub, and the combined shear and longitudinal compressive capacity of the slab in this area must be determined. The shear transfer region between the slab and the interior stub always has a smaller shear force. Region 7 is critical, and several studies have shown that the slab in this area will fail in a combination of concrete crushing and shear [5, 6, 7, 21]. The shear failure zone usually extends from corner to corner of the steel deck, over the top of the shear connectors, as illustrated in Figure 18.8. This also
FIGURE 18.8: Shear and compression failure regions in the slab of the stub girder. 1999 by CRC Press LLC
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emphasizes why the placement of the longitudinal reinforcing steel bars in the central flute of the steel deck is important, as well as the location of the transverse bars: both groups should be placed just below the level of the top of the shear connectors (see Figure 18.2). The welded wire mesh reinforcement that is used as a matter of course, mostly to control shrinkage cracking in the slab, also assists in improving the strength and ductility of the slab in this region.
18.4.6
Design of Stubs for Shear and Axial Load
The shear and axial force distributions indicate the governing stress resultants for the stub members. It is important to note that since the vierendeel members are idealized from the real (i.e., continuous) stubs, bending is not a governing condition. Given the sizes and locations of the individual vertical members that make up the stubs, the design checks are easily made for axial load and shear. For example, referring to Figure 18.7, it is seen that the shear and axial forces in the exterior and interior stubs, and the axial forces in the verticals that represent the floor beams, are the following: Exterior stub verticals: Shear forces: Axial forces:
103 kips −18 kips
63 kips 0.4 kips
99 kips 3 kips
Shear forces: Axial forces:
38 kips −5 kips
19 kips 0.8 kips
20 kips 4 kips
Interior stub verticals:
Floor beam verticals: Exterior: Axial force = −39 kips Interior: Axial force = −12 kips Shear forces are zero in these members. The areas and moments of inertia of the verticals are known from the modeling of the stub girder. Figure 18.7 also shows the shear and axial forces in the bottom and top chords, but the design for these elements has been addressed earlier in this chapter. The design checks that are made for the stub verticals will also indicate whether there is a need for stiffeners for the stubs, since the evaluations for axial load capacity should always first be made on the assumption that there are no stiffeners. However, experience has shown that the exterior stubs always must be stiffened; the interior stubs, on the other hand, will almost always be satisfactory without stiffeners, although exceptions can occur. The axial forces that are shown for the stub verticals in the preceding are small, but typical, and it is clear that in all probability only the exterior end of the exterior stub really requires a stiffener. This was examined in one of the stub girder research studies, where it was found that a single stiffener would suffice, although the resulting lack of structural symmetry gave rise to a tensile failure in the unstiffened area of the stub [21]. Although this occurred at a very late stage in the test, the type of failure represents an undesirable mode of behavior, and the use of single stiffeners therefore was discarded. Further, by reason of ease of fabrication and erection, stiffeners should always be provided at both stub ends. It is essential to bear in mind that if stiffeners are required, the purpose of such elements is to add to the area and moment of inertia of the web, to resist the axial load that is applied. There is no need to provide bearing stiffeners, since the load is not transmitted in this fashion. The most economical solution is to make use of endplate stiffeners of the kind that is shown in Figure 18.1; extensive research evaluations showed that this was the most efficient and economical choice [5, 6, 21]. 1999 by CRC Press LLC
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The vertical stub members are designed as columns, using the criteria of Section E1 of the AISC Specification [3]. For a conservative solution, an effective length factor of 1.0 may be used. However, it is more realistic to utilize a K value of 0.8 for the verticals of the stubs, recognizing the end restraint that is provided by the connections between the chords and the stubs. The Kfactor for the floor beam verticals must be 1.0, due to the pinned ends that are assumed in the modeling of these components, as well as the flexibility of the floor beam itself in the direction of potential buckling of the vertical member.
18.4.7
Design of Stud Shear Connectors
The shear forces that must be transferred between the slab and the stubs are given by the vierendeel girder shear force diagram. These are the factored shear force values which are to be resisted by the connectors. The example shown in Figure 18.7 indicates the individual shear forces for the stub verticals, as listed in the preceding section. However, in the design of the overall shear connection, the total shear force that is to be transmitted to the stub is used, and the stud connectors are then distributed uniformly along the stub. The design strength of each connector is determined in accordance with Section I5.3 of the LRFD Specification [3], including any deck profile reduction factor (Section I3.5). Analyzing the girder whose data are given in Figure 18.7, the following is known: Exterior stub: Total shear force = Ves = 103 + 63 + 99 = 265 kips Interior stub: Total shear force = Vis = 38 + 19 + 20 = 77 kips The nominal strength, Qn , of the stud shear connectors is given by Eq. (I51) in Section I5.3 of the LRFD Specification, thus: p (18.2) Qn = 0.5 × Asc fc0 × Ec ≤ Asc × Fu where Asc is the crosssectional area of the stud shear connector, fc0 and Ec are the compressive strength and modulus of elasticity of the concrete, and Fu is the specified minimum tensile strength of the stud shear connector steel, or 60 ksi (ASTM A108). In the equation for Qn , the lefthand side reflects the ultimate limit state of shear yield failure of the connector; the righthand side gives the ultimate limit state of tension fracture of the stud. Although shear almost always governs and is the desirable mode of behavior, a check has to be made to ensure that tension fracture will not take place. This as achieved by the appropriate value of Ec , setting Fu = 60 ksi, and solving for fc0 from Equation 18.2. The requirement that must be satisfied in order for the stud shear limit state to govern is given by Equation 18.3: fc0 ≤
57,000 wc
(18.3)
This gives the limiting values for concrete strength as related to the density; data are given in Table 18.1. For concrete with wc = 120 pcf and fc0 = 4,000 psi, as used in the design example, Ec = 2,629,000 psi. Using 3/4in. diameter studs, the nominal shear capacity is: h ip h i Qn = 0.5 π(0.75)2 /4 (4 × 2,629) ≤ π(0.75)2 /4 60 which gives Qn = 22.7 kips < 26.5 kips 1999 by CRC Press LLC
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TABLE 18.1 Concrete Strength Limitations for Ductile Shear Connector Failure Concrete density, wc (pcf)
Maximum concrete strength, fc0 (psi)
145 (= N W ) 120 110 100 90
4000 4800 5200 5700 6400
Note: N W = normal weight.
The LRFD Specification [3] does not give a resistance factor for shear connectors, on the premise that the φ value of 0.85 for the overall design of the composite member incorporates the stud strength variability. This is not satisfactory for composite members such as stub girders and composite trusses. However, a study was carried out to determine the resistance factors for the two ultimate limit states for stud shear connectors [20]. Briefly, on the basis of extensive analyses of test data from a variety of sources, and using the Qn equation as the nominal strength expression, the values of the resistance factors that apply to the shear yield and tension fracture limit states, respectively, are: Stud shear connector resistance factors: Limit state of shear yielding: φconn = 0.90 Limit state of tension fracture: φconn = 0.75 The required number of shear connectors can now be found as follows, using the total stub shear forces, Ves and Vis , computed earlier in this section: Exterior stub: nes
= Ves /(0.9 × Qn ) = Ves /(φconn Qn ) = 265/(0.9 × 22.7) = 13.0
i.e., use nes = 143/4in. diameter stud shear connectors, placed in pairs and distributed uniformly along the length of the top flange of each of the exterior stubs. Interior stub: nis
= Vis /(0.9 × Qn ) = Vis /(φconn Qn ) = 77/(0.9 × 22.7) = 3.8
i.e., use nis = 43/4in. diameter stud shear connectors, placed singly and distributed uniformly along the length of the top flange of each of the interior stubs. Considering the shear forces for the stub girder of Figures 18.5 and 18.7, the number of connectors for the exterior stub is approximately three times that for the interior one, as expected. Depending on span, loading, etc., there are instances when it will be difficult to fit the required number of studs on the exterior stub, since typical usage entails a double row, spaced as closely as permitted (four diameters in any direction [Section I5.6, AISC LRFD Specification [3]]). Several avenues may be followed to remedy such a problem; the easiest one is most likely to use a higher strength concrete, as long as the limit state requirements for Qn and Table 18.1 are satisfied. This entails only minor reanalysis of the girder. 1999 by CRC Press LLC
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18.4.8
Design of Welds between Stub and Bottom Chord
The welds that are needed to fasten the stubs to the top flange of the bottom chord are primarily governed by the shear forces that are transferred between these components of the stub girder. The shear force distribution gives these stress resultants, which are equal to those that must be transferred between the slab and the stubs. Thus, the factored forces, Ves and Vis , that were developed in Section 18.4.7 are used to size the welds. Axial loads also act between the stubs and the chord; these may be compressive or tensile. In Figure 18.7 it is seen that the only axial force of note occurs in the exterior vertical of the exterior stub (load = 18 kips); the other loads are very small compressive or tensile forces. Unless a significant tensile force is found in the analysis, it will be a safe simplification to ignore the presence of the axial forces insofar as the weld design is concerned. The primary shear forces that have to be taken by the welds are developed in the outer regions of the stubs, although it is noted that in the case of Figure 18.5, the central vertical element in both stubs carries forces of some magnitude (63 and 19 kips, respectively). However, this distribution is a result of the modeling of the stubs; analyses of girders where many more verticals were used have confirmed that the major part of the shear is transferred at the ends [7, 10, 21]. The reason is that the stub is a full shear panel, where the internal moment is developed through stress resultants that act at points toward the ends, in a form of bending action. Tests have also verified this characteristic of the girder behavior [6, 21]. Finally, concentrating the welds at the stub ends will have significant economic impact [5, 7, 21]. In view of these observations, the most effective placement of the welds between the stubs and the bottom chord is to concentrate them across the ends of the stubs and along a short distance of both sides of the stub flanges. For ease of fabrication and structural symmetry, the same amount of welding should be placed at both ends, although the forces are always smaller at the interior ends of the stubs. Such Ushaped welds were used for a number of the fullsize girders that were tested [5, 6, 21], with only highly localized yielding occurring in the welds. A typical detail is shown in Figure 18.9; this reflects what is recommended for use in practice. Prior to the research that led to the change of the welded joint design, the stubs were welded with allaround fillet welds for the exterior as well as the interior elements. The improved, Ushaped detail provided for weld metal savings of approximately 75% for interior stubs and around 50% for exterior stubs. For the sample stub girder, W16x26 shapes are used for the stubs. The total forces to be taken by the welds are: Exterior stub: Ves = 265 kips Interior stub: Vis = 77 kips Using E70XX electrodes and 5/16in. fillet welds (the fillet weld size must be smaller than the thickness of the stub flange, which is 3/8 in. for the W16x26), the total weld length for each stub is Lw , given by (refer to Figure 18.9): Lw = 2(bf s + 2`) since Ushaped welds of length (bf s +2`) are placed at each stub end. The total weld lengths required for the stub girder in question are therefore: Exterior stub: (Lw )es
1999 by CRC Press LLC
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=
Ves /(0.707aφw Fw )
=
265/ [0.707(5/16) × 0.75(0.6 × 70)] = 38.1 in.
FIGURE 18.9: Placement of Ushaped fillet weld for attachment at each end of stub to bottom chord.
Interior stub: (Lw )is
= Vis /(0.707aφw Fw ) = 77/ [0.707(5/16) × 0.75(0.6 × 70)] = 11.1 in.
In the above expressions, a = 5/16 in. = fillet weld size, φw = 0.75, and Fw = 0.6FEXX = 0.6 × 70 = 42 ksi for E70XX electrodes (Table J2.3, AISC LRFD Specification [3]). The total Uweld lengths at each stub end are therefore: Exterior stub: LU es = 19.1 in. Interior stub: LU is = 5.6 in. With a flange width for the W16x26 of 5.50 in., the above lengths can be simplified as: LU es = 5.50 + 7.0 + 7.0 where `es is chosen as 7.0 in. For the interior stub: LU is = 5.50 + 2.0 + 2.0 where `is is chosen as 2.0 in. The details chosen are a matter of judgment. In the example, the interior stub for all practical purposes requires no weld other than the one across the flange, although at least a minimum weld return of 1/2 in. should be used. 1999 by CRC Press LLC
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18.4.9
Floor Beam Connections to Slab and Bottom Chord
In the vierendeel model, the floor beam is represented as a pinnedend compression member. It is designed using a Kfactor of 1.0, and the floor beam web by itself is almost always sufficient to take the axial load. However, the floor beam must be checked for web crippling and web buckling under shoring conditions. No shear is transferred from the beam to the slab or the bottom chord. In theory, therefore, any attachment device between the floor beam and the other components should not be needed. However, due to construction stability requirements, as well as the fact that the floor beam usually is designed for composite action normal to the girder, fasteners are needed. In practice, these are not actually designed; rather, one or two stud shear connectors are placed on the top flange of the beam, and two highstrength bolts attach the lower flange to the bottom chord.
18.4.10
Connection of Bottom Chord to Supports
In the traditional use of stub girders, the girder is supported as a simple beam, and the bottom chord end connections need to be able to transfer vertical reactions to the supports. The latter structural elements may be columns, or the girder may rest on corbels or other types of supports that are part of the concrete core of the building. For both of these cases the reactions that are to be carried to the adjacent structure are given by the analysis, and the response needs for the supports are clear. Any sheartype beam connections may be used to connect the bottom chord to a column or a corbel or similar bracket. It is important to ascertain that the chord web shear capacity is sufficient, including block shear (Section J5 of the AISC LRFD Specification [3]). Some designers prefer to use slotted holes for the connections, and to delay the final tightening of the bolts until after the shoring has been removed. This is done on the premise that the procedure will leave the slab essentially stress free from the construction loads, leading to less cracking in the slab during service. Other designers specify additional slab reinforcement to take care of any cracking problem. Experience has shown that both methods are suitable. The slab may be supported on an edge beam or similar element at the exterior side of the floor system. There is no force transfer ability required of this support. In the interior of the building the slab will be continuously cast across other girders and around columns; this will almost always lead to some cracking, both in the vicinity of the columns as well as along beams and girders. With suitable placement of floor slab joints, this can be minimized, and appropriate transverse reinforcement for the slab will reduce, if not eliminate, the longitudinal cracks. Data on the effects of various types of cracks in composite floor systems are scarce. Current opinion appears to be that the strength may not be influenced very much. In any case, the mechanics of the short and longterm service response of composite beams is not well understood. Recent studies have developed models for the cracking mechanism and the crack propagation [18]; the correlation with a wide variety of laboratory tests is good. However, a comprehensive study of concrete cracking and its implications for structural service and strength needs to be undertaken.
18.4.11
Use of Stub Girder for Lateral Load System
The stub girder was originally conceived only as being part of the vertical loadcarrying system of structural frames, and the use of simple connections, as discussed in Section 18.4.9, came from this development. However, because a deep, longspan member can be very effective as a part of the lateral loadresisting system for a structure, several attempts have been made to incorporate the stub girder into moment frames and similar systems. The projects of Colaco in Houston [13] and MartinezRomero [17] in Mexico City were successful, although the designers noted that the cost premium could be substantial. For the Colaco structure, his applications reduced drift, as expected, but gave much more complex 1999 by CRC Press LLC
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beamtocolumn connections and reinforcement details in the slab around the columns. Thus, the exterior stubs were moved to the far ends of the girders, and moment connections were designed for the full depth. For the Mexico City building, the added ductility was a prime factor in the survival of the structure during the 1985 earthquake. The advantages of using the stub girders in moment frames are obvious. Some of the disadvantages have been outlined; in addition, it must be recognized that the lack of room for perimeter HVAC ducts may be undesirable. This can only be addressed by the mechanical engineering consultant. As a general rule, a designer who wishes to use stub girders as part of the lateral loadresisting system should examine all structural effects, but also incorporate nonstructural considerations such as are prompted by HVAC and electronic communication needs.
18.4.12
Deflection Checks
The service load deflections of the stub girder are needed for several purposes. First, the overall dead load deflection is used to assess the camber requirements. Due to the long spans of typical stub girders, as well as the flexibility of the framing members and the connections during construction, it is important to end up with a floor system that is as level as possible by the time the structure is ready to be occupied. Thus, the girders must be built in the shored condition, and the camber should be approximately equal to the dead load deflection. Second, it is essential to bear in mind that each girder will be shored against a similar member at the level below the current construction floor. This member, in turn, is similarly shored, albeit against a girder whose stiffness is greater, due to the additional curing time of the concrete slab. This has a cumulative effect for the structure as a whole, and the dead load deflection computations must take this response into account. In other words, the support for the shores is a flexible one, and deflections therefore will occur in the girder as a result of floor system movements of the structure at levels in addition to the one under consideration. Although this is not unique to the stub girder system, the span lengths and the interaction with the frame accentuate the influence on the girder design. Depending on the structural system, it is also likely that the flexibility of the columns and the connections will add to the vertical displacements of the stub girders. The deflection calculations should incorporate these effects, preferably by utilizing realistic modified Ec values and determining displacements as they occur in the frame. Thus, the curing process for the concrete might be considered, since the strength development as a function of time is directly related to the value of Ec [1]. This is a subject that is open for study, although similar criteria have been incorporated in studies of the strength and behavior of composite frames [8, 9]. However, detailed evaluations of the influence of timedependent stiffness still need to be made for a wide variety of floor systems and frames. The cumulative deflection effects can be significant for the construction of the building, and consequently also must enter into the contractor’s planning. This subject is addressed briefly in Section 18.5. Third, the live load deflections must be determined to assess the serviceability of the floor system under normal operating conditions. However, several studies have demonstrated that such displacements will be significantly smaller than the L/360 requirement that is normally associated with live load deflections [6, 7, 10, 14, 21]. It is therefore rarely possible to design a girder that meets the strength and the deflection criteria simultaneously [14]. In other words, strength governs the overall design. Finally, although they rarely play a role in the overall response of the stub girder, the deflections and end rotations of the slab across the openings of the girder should also be checked. This is primarily done to assess the potential for local cracking, especially at the stub ends and at the floor beams. However, proper placement of the longitudinal girder reinforcement is usually sufficient to prevent problems of this kind, since the deformations tend to be small. 1999 by CRC Press LLC
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18.5
Influence of Method of Construction
A number of constructionrelated considerations have already been addressed in various sections of this chapter. The most important ones relate to the fact that the stub girders must be built in the shored condition. The placement and removal of the shores may have a significant impact on the performance of the member and the structure as a whole. In particular, too early shore removal may lead to excessive deflections in the girders at levels above the one where the shores were located. This is a direct result of the low stiffness of “green” concrete. It can also lead to “ponding” of the concrete slab, producing larger dead loads than accounted for in the original design. Finally, larger girder deflections can be translated into an “inward pulling” effect on the columns of the frame. However, this is clearly a function of the framing system. On the other hand, the use of high early strength cement and similar products can reduce this effect significantly. Further, since the concrete usually is able to reach about 75% of the 28day strength after 7 to 10 days, the problem is less severe than originally thought [5, 7, 10]. In any case, it is important for the structural engineer to interact with the general contractor, in order that the influence of the method of construction on the girders as well as the frame can be quantified, however simplistic the analysis procedure may be. Due to the larger loads that can be expected for the shores, the latter must either be designed as structural members or at least be evaluated by the structural engineer. The size of the shores is also influenced by the number of floors that are to have these supports left in place. As a general rule, when stub girders are used for multistory frames, the shores should be left in place for at least three floor levels. Some designers prefer a larger number; however, any choices of this kind should be based on computations for sizes and effects. Naturally, the more floors that are specified, the larger the shores will have to be.
18.6
Defining Terms
Composite: Steel and concrete acting in concert. Formed steel deck: A thin sheet of steel shaped into peaks and valleys called corrugations. Green concrete: concrete that has just been placed. HVAC: Heating, ventilating, and air conditioning. Lightweight: Refers to concrete with unit weights between 90 and 120 pcf. Normal weight: Refers to concrete with unit weights of 145 lb per cubic foot (pcf). Prismatic beam: A beam with a constant size crosssection over the full length. Rebar: An abbreviated name for reinforcing steel bars. Serviceability: The ability of a structure to function properly under normal operating condictions. Shoring: Temporary support. Vierendeel girder: A girder with top and bottom chords attached to each other through fully welded connections to vertical (generally) members.
References [1] American Concrete Institute. 1995. Building Code Requirements for Reinforced Concrete, ACI Standard No. 31895, ACI, Detroit, MI. [2] American Institute of Steel Construction. 1989. Specification for the Allowable Stress Design, Fabrication, and Erection of Structural Steel for Buildings, 9th ed., AISC, Chicago, IL. 1999 by CRC Press LLC
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[3] American Institute of Steel Construction. 1993. Specification for the Load and Resistance Factor Design, Fabrication, and Erection of Structural Steel for Buildings, 2nd ed., AISC, Chicago, IL. [4] American Society of Civil Engineers. 1995. Minimum Design Loads for Buildings and Other Structures, ASCE/ANSI Standard No. 795, ASCE, New York. [5] Bjorhovde, R., and Zimmerman, T.J. 1980. Some Aspects of Stub Girder Design, AISC Eng. J., 17(3), Third Quarter, September (pp. 5469). [6] Bjorhovde, R. 1981. FullScale Test of a Stub Girder, Report submitted to Dominion Bridge Company, Calgary, Alberta, Canada. Department of Civil Engineering, University of Alberta, Edmonton, Alberta, Canada, June. [7] Bjorhovde, R. 1985. Behavior and Strength of Stub Girder Floor Systems, in Composite and Mixed Construction, ASCE Special Publication, ASCE, New York. [8] Bjorhovde, R. 1987. Design Considerations for Composite Frames, Proceedings 2nd International and 5th Mexican National Symposium on Steel Structures, IMCA and SMIE, Morelia, Michoacan, Mexico, November 2324. [9] Bjorhovde, R. 1994. Concepts and Issues in Composite Frame Design, Steel Structures, Journal of the Singapore Society for Steel Structures, 5(1), December (pp. 314). [10] Chien, E.Y.L. and Ritchie, J.K. 1984. Design and Construction of Composite Floor Systems, Canadian Institute of Steel Construction (CISC), Willowdale (Toronto), Ontario, Canada. [11] Colaco, J.P. 1972. A Stub Girder System for HighRise Buildings, AISC Eng. J., 9(2), Second Quarter, July (pp. 8995). [12] Colaco, J. P. 1974. Partial Tube Concept for MidRise Structures, AISC Eng. J., 11(4), Fourth Quarter, December (pp. 8185). [13] Colaco, J.P. and Banavalkar, P.V. 1979. Recent Uses of the Stub Girder System, Proceedings 1979 National Engineering Conference, American Institute of Steel Construction, Chicago, IL, May. [14] Griffis, T.C. 1983. Stiffness Criteria for Stub Girder Floor Systems, M.S. thesis, University of Arizona, Tucson, AZ. [15] Hansell, W.C., Galambos, T.V., Ravindra, M.K., and Viest, I.M. 1978. Composite Beam Criteria in LRFD, J. Structural Div., ASCE, 104(ST9), September (pp. 14091426). [16] Hrabok, M.M. and Hosain, M.U. 1978. Analysis of Stub Girders Using SubStructuring, Intl. J. Computers and Structures, 8(5), 615619. [17] MartinezRomero, E. 1983. Continuous Stub Girder Structural System for Floor Decks, Technical report, EMRSA, Mexico City, Mexico, February. [18] Morcos, S.S. and Bjorhovde, R. 1995. Fracture Modeling of Concrete and Steel, J. Structural Eng., ASCE, 121(7), 11251133. [19] Wong, A.F. 1979. Conventional and Unconventional Composite Floor Systems, M.Eng. thesis, University of Alberta, Edmonton, Alberta, Canada. [20] Zeitoun, L.A. 1984. Development of Resistance Factors for Stud Shear Connectors, M.S. thesis, University of Arizona, Tucson, AZ. [21] Zimmerman, T.J. and Bjorhovde, R. 1981. Analysis and Design of Stub Girders, Structural Engineering Report No. 90, University of Alberta, Edmonton, Alberta, Canada, March.
Further Reading The references that accompany this chapter are allencompassing for the literature on stub girders. Primary references that should be studied in addition to this chapter are [5, 7, 10, 11], and [13].
1999 by CRC Press LLC
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Elgaaly, M. “Plate and Box Girders” Structural Engineering Handbook Ed. Chen WaiFah Boca Raton: CRC Press LLC, 1999
Plate and Box Girders 19.1 Introduction 19.2 Stability of the Compression Flange Vertical Buckling • Lateral Buckling Compression Flange of a Box Girder
Mohamed Elgaaly Department of Civil & Architectural Engineering, Drexel University, Philadelphia, PA
19.1
•
Torsional Buckling
•
19.3 Web Buckling Due to InPlane Bending 19.4 Nominal Moment Strength 19.5 Web Longitudinal Stiffeners for Bending Design 19.6 Ultimate Shear Capacity of the Web 19.7 Web Stiffeners for Shear Design 19.8 FlexureShear Interaction 19.9 Steel Plate Shear Walls 19.10InPlane Compressive Edge Loading 19.11Eccentric Edge Loading 19.12LoadBearing Stiffeners 19.13Web Openings 19.14Girders with Corrugated Webs 19.15Defining Terms References
Introduction
Plate and box girders are used mostly in bridges and industrial buildings, where large loads and/or long spans are frequently encountered. The high torsional strength of box girders makes them ideal for girders curved in plan. Recently, thin steel plate shear walls have been effectively used in buildings. Such walls behave as vertical plate girders with the building columns as flanges and the floor beams as intermediate stiffeners. Although traditionally simply supported plate and box girders are built up to 150 ft span, several threespan continuous girder bridges have been built in the U.S. with center spans exceeding 400 ft. In its simplest form a plate girder is made of two flange plates welded to a web plate to form an I section, and a box girder has two flanges and two webs for a singlecell box and more than two webs in multicell box girders (Figure 19.1). The designer has the freedom in proportioning the crosssection of the girder to achieve the most economical design and taking advantage of available highstrength steels. The larger dimensions of plate and box girders result in the use of slender webs and flanges, making buckling problems more relevant in design. Buckling of plates that are adequately supported along their boundaries is not synonymous with failure, and these plates exhibit postbuckling strength that can be several times their buckling strength, depending on the plate slenderness. Although plate 1999 by CRC Press LLC
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FIGURE 19.1: Plate and box girders.
buckling has not been the basis for design since the early 1960s, buckling strength is often required to calculate the postbuckling strength. The trend toward limit state format codes placed the emphasis on the development of new design approaches based on the ultimate strength of plate and box girders and their components. The postbuckling strength of plates subjected to shear is due to the diagonal tension field action. The postbuckling strength of plates subjected to uniaxial compression is due to the change in the stress distribution after buckling, higher near the supported edges. An effective width with a uniform stress, equal to the yield stress of the plate material, is used to calculate the postbuckling strength [40]. The flange in a box girder and the web in plate and box girders are often reinforced with stiffeners to allow for the use of thin plates. The designer has to find a combination of plate thickness and stiffener spacing that will optimize the weight and reduce the fabrication cost. The stiffeners in most cases are designed to divide the plate panel into subpanels, which are assumed to be supported along the stiffener lines. Recently, the use of corrugated webs resulted in employing thin webs without the need for stiffeners, thus reducing the fabrication cost and also improving the fatigue life of the girders. The web of a girder and loadbearing diaphragms can be subjected to inplane compressive patch loading. The ultimate capacity under this loading condition is controlled by web crippling, which can occur prior to or after local yielding. The presence of openings in plates subjected to inplane loads is unavoidable in some cases, and the presence of openings affects the stability and ultimate strength of plates.
19.2
Stability of the Compression Flange
The compression flange of a plate girder subjected to bending usually fails in lateral buckling, local torsional buckling, or yielding; if the web is slender the compression flange can fail by vertical buckling into the web (Figure 19.2). 1999 by CRC Press LLC
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FIGURE 19.2: Compression flange modes of failure.
19.2.1
Vertical Buckling
The following limiting value for the web slenderness ratio to preclude this mode of failure [4] can be used, q p Aw /Af (19.1) h/tw ≤ 0.68E/ Fyf Fyf + Fr where h and tw are the web height and thickness, respectively; Aw is the area of the web; Af is the area of the flange; E is Young’s modulus of elasticity; Fyf is the yield stress of the flange material; and Fr is the residual tension that must be overcome to achieve uniform yielding in compression. This limiting value may be too conservative since vertical buckling of the compression flange into the web occurs only after general yielding of the flange. This limiting value, however, can be helpful to avoid fatigue cracking under repeated loading due to outofplane flexing, and it also facilitates fabrication. The American Institute of Steel Construction (AISC) specification [32] uses Equation 19.1 when the spacing between the vertical stiffeners, a, is more than 1.5 times the web depth, h (a/ h > 1.5). In such a case the specification recommends that q (19.2) h/tw ≤ 14,000/ Fyf (Fyf + 16.5) where a minimum value of Aw /Af = 0.5 was assumed and the residual tension was taken to be 16.5 ksi. Furthermore, when a/ h is less than or equal to 1.5, higher web slenderness is permitted, namely p (19.3) h/tw ≤ 2000/ Fyf
19.2.2
Lateral Buckling
When a flange is not adequately supported in the lateral direction, elastic lateral buckling can occur. The compression flange, together with an effective area of the web equal to Aw /6, can be treated as a column and the buckling stress can be calculated from the Euler equation [2]: Fcr = π 2 E/(λ)2
(19.4)
where λ is the slenderness ratio, which is equal to Lb /rT ; Lb is the length of the unbraced segment of the beam; and rT is the radius of gyration of the compression flange plus onethird of the compression portion of the web. The AISC specification adopted Equation 19.4, rounding π 2 E to 286,000 and p assuming that elastic buckling will occur when the slenderness ratio, λ, is greater than λr (= 756/ Fyf ). Furthermore, Equation 19.4 is based on uniform compression; in most cases the bending is not uniform within the length of the unbraced segment of the beam. To account for nonuniform bending, 1999 by CRC Press LLC
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Equation 19.4 should be multiplied by a coefficient, Cb [25], where Cb = 12.5Mmax /(2.5Mmax + 3MA + 4MB + 3MC )
(19.5)
where Mmax = absolute value of maximum moment in the unbraced beam segment = absolute value of moment at quarter point of the unbraced beam segment MA = absolute value of moment at centerline of the unbraced beam segment MB = absolute value of moment at threequarter point of thepunbraced beam segment MC When the slenderness ratio, λ, is less than or equal to λp (= 300/ Fyf ), the flange will yield before it buckles, and Fcr = Fyf . When the flange slenderness ratio, λ, is greater than λp and smaller than or equal to λr , inelastic buckling will occur and a straight line equation must be adopted between yielding (λ ≤ λp ) and elastic buckling (λ > λr ) to calculate the inelastic buckling stress, namely (19.6) Fcr = Cb Fyf 1 − 0.5(λ − λp )/(λr − λp ) ≤ Fyf
19.2.3
Torsional Buckling
If the outstanding widthtothickness ratio of the flange is high, torsional buckling may occur. If one neglects any restraint provided by the web to the flange rotation, then the flange can be treated as a long plate, which is simply supported (hinged) at one edge and free at the other, subjected to uniaxial compression in the longitudinal direction. The elastic buckling stress under these conditions can be calculated from (19.7) Fcr = kc π 2 E/12(1 − µ2 )λ2 where kc is a buckling coefficient equal to 0.425 for a long plate simply supported and free at its longitudinal edges; λ is equal to bf /2tf ; bf and tf are the flange width and thickness, respectively; and E and µ are Young’s modulus of elasticity and the Poisson ratio, respectively. 2 2 The AISC √ specification adopted Equation 19.7, rounding π E/12(1−µ ) to 26,200 and assuming kc = 4 h/tw , where 0.35 ≤ kc ≥ 0.763. Furthermore, to allow for nonuniform bending, the buckling stress has to be multiplied by Cb , given by Equation p 19.5. Elastic torsional buckling of the compression flange will occur if λ is greater than λr (= 230/ Fyf /kc ). When λ is less than or equal p to λp (= 65/ Fyf ), the flange will yield before it buckles, and Fcr = Fyf . When λp < λ ≤ λr , inelastic buckling will occur and Equation 19.6 shall be used.
19.2.4
Compression Flange of a Box Girder
Lateraltorsional buckling does not govern the design of the compression flange in a box girder. Unstiffened flanges and flanges stiffened with longitudinal stiffeners can be treated as long plates supported along their longitudinal edges and subjected to uniaxial compression. In the AASHTO (American Association of State Highway and Transportation Officials ) specification [1], the nominal flexural stress,p Fn , for the compression flange is calculated as follows: If w/t ≤ 0.57 kE/Fyf , then the flange will yield before it buckles, and Fn = Fyf p If w/t > 1.23 kE/Fyf , then the flange will elastically buckle, and
(19.8)
Fn = kπ 2 E/12(1 − µ2 )(w/t)2 or Fn = 26,200 k(t/w)2 1999 by CRC Press LLC
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(19.9)
p p If 0.57 kE/Fyf < w/t ≤ 1.23 kE/Fyf , then the flange buckles inelastically, and Fn = 0.592Fyf [1 + 0.687 sin(cπ/2)]
(19.10)
In Equations 19.8 to 19.10, = the the longitudinal stiffeners, or the flange width for unstiffened flanges spacing between p = 1.23 − (w/t) Fyf /kE /0.66 1/3 ≤ 4.0, for n = 1 k = 8Is /wt 3 1/3 3 4 ≤ 4.0, for n = 2, 3, 4, or 5 k = 14.3Is /wt n n = number of equally spaced longitudinal stiffeners = the moment of inertia of the longitudinal stiffener about an axis parallel to the flange and Is taken at the base of the stiffener w c
The nominal stress, Fn , shall be reduced for hybrid girders to account for the nonlinear variation of stresses caused by yielding of the lower strength steel in the web of a hybrid girder. Furthermore, another reduction is made for slender webs to account for the nonlinear variation of stresses caused by local bend buckling of the web. The reduction factors for hybrid girders and slender webs will be given in Section 19.3. The longitudinal stiffeners shall be equally spaced across the compression flange width and shall satisfy the following requirements [1]. The projecting width, bs , of the stiffener shall satisfy: p (19.11) bs ≤ 0.48ts E/Fyc where = thickness of the stiffener ts = specified minimum yield strength of the compression flange Fyc The moment of inertia, Is , of each stiffener about an axis parallel to the flange and taken at the base of the stiffener shall satisfy: (19.12) Is ≥ 9wt 3 where 9 = = n = w =
0.125k 3 for n = 1 0.07k 3 n4 for n = 2, 3, 4, or 5 number of equally spaced longitudinal compression flange stiffeners larger of the width of compression flange between longitudinal stiffeners or the distance from a web to the nearest longitudinal stiffener t = compression flange thickness k = buckling coefficient as defined in connection with Equations 19.8 to 19.10 The presence of the inplane compression in the flange magnifies the deflection and stresses in the flange from local bending due to traffic loading. The amplification factor, 1/(1 − σa /σcr ), can be used to increase the deflections and stresses due to local bending; where σa and σcr are the inplane compressive and buckling stresses, respectively.
19.3
Web Buckling Due to InPlane Bending
Buckling of the web due to inplane bending does not exhaust its capacity; however, the distribution of the compressive bending stress changes in the postbuckling range and the web becomes less efficient. Only part of the compression portion of the web can be assumed effective after buckling. A reduction in the girder moment capacity to account for the web bend buckling can be used, and the following reduction factor [4] has been suggested: p (19.13) R = 1 − 0.0005(Aw /Af )(h/t − 5.7 E/Fyw ) 1999 by CRC Press LLC
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p It must be noted that when h/t = 5.7 E/Fyw , the web will yield before it buckles and there is no reduction in the moment capacity. This can be determined by equating the bend buckling stress to the web yield stress, i.e., h i (19.14) kπ 2 E/ 12(1 − µ2 )(h/t)2 = Fyw where k is the web bend buckling coefficient, which is equal to 23.9 if the flange simply supports the web and 39.6 if one assumes that the flange provides full fixity; the 5.7 in Equation 19.13 is based on a k value of 36. The AISC specification replaces the reduction factor given in Equation 19.13 by p (19.15) RP G = 1 − [ar /(1,200 + 300ar )] (h/t − 970/ Fcr ) √ where ar is equal to Aw /Af and 970 is equal to 5.7 29000; it must be noted that the yield stress in Equation 19.13 was replaced by the flange critical buckling stress, which can be equal to or less than the yield stress as discussed earlier. It must also be noted that in homogeneous girders the yield stresses of the web and flange materials are equal; in hybrid girders another reduction factor, Re , [39] shall be used: h i (19.16) Re = 12 + ar (3m − m3 ) /(12 + 2ar ) where ar is equal to the ratio of the web area to the compression flange area (≤ 10) and m is the ratio of the web yield stress to the flange yield or buckling stress.
19.4
Nominal Moment Strength
The nominal moment strength can be calculated as follows. Based on tension flange yielding: Mn = Sxt Re Fyt
(19.17a)
Mn = Sxc RP G Re Fcr
(19.17b)
or Based on compression flange buckling:
where Sxc and Sxt are the section moduli referred to the compression and tension flanges, respectively; Fyt is the tension flange yield stress; Fcr is the compression flange buckling stress calculated according to Section 19.2; RP G is the reduction factor calculated using Equation 19.15; and Re is a reduction factor to be used in the case of hybrid girders and can be calculated using Equation 19.16.
19.5
Web Longitudinal Stiffeners for Bending Design
Longitudinal stiffeners can increase the bending strength of plate girders. This increase is due to the control of the web lateral deflection, which increases its flexural stress capacity. The presence of the stiffener also improves the bending resistance of the flange due to a greater web restraint. If one longitudinal stiffener is used, its optimum location is 0.20 times the web depth from the compression flange. In this case the web plate elastic bend buckling stress increases more than five times that without the stiffener. Tests [8] showed that an adequately proportioned longitudinal stiffener at 0.2h from the compression flange eliminates bend buckling in girders with web slenderness, h/t, as large as 450. Girders with larger slenderness will require two or more longitudinal stiffeners to eliminate 1999 by CRC Press LLC
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web bend buckling. It must be noted that the increase in the bending strength of a longitudinally stiffened thinweb girder is usually small because the web contribution to the bending strength is small. However, longitudinal stiffeners can be important in a girder subjected to repeated loads because they reduce or eliminate the outofplane bending of the web, which increases resistance to fatigue cracking at the webtoflange juncture and allows more slender webs to be used [42]. The AISC specification does not address longitudinal stiffeners; on the other hand, the AASHTO specification states that longitudinal stiffeners should consist of either a plate welded longitudinally to one side of the web or a bolted angle, and shall be located at a distance of 0.4 Dc from the inner surface of the compression flange, where Dc is the depth of the web in compression at the section with the maximum compressive flexural stress. Continuous longitudinal stiffeners placed on the opposite side of the web from the transverse intermediate stiffeners, as shown in Figure 19.3, are preferred. If longitudinal and transverse stiffeners must be placed on the same side of the web, it is preferable
FIGURE 19.3: Longitudinal stiffener for flexure.
that the longitudinal stiffener not be interrupted for the transverse stiffener. Where the transverse stiffeners are interrupted, the interruptions must be carefully detailed with respect to fatigue. To prevent local buckling, the projecting width, bs of the stiffener shall satisfy the requirements of Equation 19.11. The section properties of the stiffener shall be based on an effective section consisting of the stiffener and a centrally located strip of the web not exceeding 18 times the web thickness. The moment of inertia of the longitudinal stiffener and the effective web strip about the edge in contact with the web, Is , and the corresponding radius of gyration, rs , shall satisfy the following requirements: i h (19.18) Is ≥ htw3 2.4(a/ h)2 − 0.13 and
p rs ≥ 0.234a Fyc /E
(19.19)
where a = spacing between transverse stiffeners
19.6
Ultimate Shear Capacity of the Web
As stated earlier, in most design codes buckling is not used as a basis for design. Minimum slenderness ratios, however, are specified to control outofplane deflection of the web. These ratios are derived to give a small factor of safety against buckling, which is conservative and in some cases extravagant. 1999 by CRC Press LLC
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Before the web reaches its theoretical buckling load the shear is taken by beam action and the shear stress can be resolved into diagonal tension and compression. After buckling, the diagonal compression ceases to increase and any additional loads will be carried by the diagonal tension. In very thin webs with stiff boundaries, the plate buckling load is very small and can be ignored and the shear is carried by a complete diagonal tension field action [41]. In welded plate and box girders the web is not very slender and the flanges are not very stiff; in such a case the shear is carried by beam action as well as incomplete tension field action. Based on test results, the analytical model shown in Figure 19.4 can be used to calculate the ultimate shear capacity of the web of a welded plate girder [5]. The flanges are assumed to be too flexible to
FIGURE 19.4: Tension field model by Basler.
support the vertical component from the tension field. The inclination and width of the tension field were defined by the angle 2, which is chosen to maximize the shear strength. The ultimate shear capacity of the web, Vu , can be calculated from Vu = τcr + 0.5σyw (1 − τcr /τyw ) sin 2d Aw (19.20) where = critical buckling stress in shear τcr = yield stress in shear τyw σyw = web yield stress 2d = angle of panel diagonal with flange = area of the web Aw In Equation 19.20, if τcr ≥ 0.8τyw , the buckling will be inelastic and p τcr = τcri = 0.8τcr τyw
(19.21)
It was shown later [23] that Equation 19.20 gives the shear strength for a complete tension field instead of the limited band shown in Figure 19.4. The results obtained from the formula, however, were in good agreement with the test results, and the formula was adopted in the AISC specification. Many variations of this incomplete tension field model have been developed; are view can be found in the SSRC Guide to Stability Design Criteria for Metal Structures [22]. The model shown in Figure 19.5 [36, 38] gives better results and has been adopted in codes in Europe. In the model shown in Figure 19.5, near failure the tensile membrane stress, together with the buckling stress, causes yielding, and failure occurs when hinges form in the flanges to produce a combined mechanism that includes the yield zone ABCD. The vertical component of the tension field is added to the shear at buckling and combined with the frame action shear to calculate the ultimate shear strength. The 1999 by CRC Press LLC
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FIGURE 19.5: Tension field model by Rockey et al.
ultimate shear strength is determined by adding the shear at buckling, the vertical component of the tension field, and the frame action shear, and is given by Vu = τcr Aw + σt Aw [(2c/ h) + cot 2 − cot 2d ] sin2 2 + 4Mp /c
(19.22)
where q 2 + (2.25 sin2 22 − 3)τ 2 σt = −1.5τcr sin 22 + σyw cr √ c = (2/ sin 2) Mp/(σt tw ) 0 ≤ c ≤ a = plastic moment capacity of the flange with an effective depth of the web, be , given by Mp = 30tw [1 − 2(τcr /τyw )] be where (τcr /τyw ) ≤ 0.5; reduction in Mp due to the effect of the flange axial compression shall be considered and when τcr > 0.8τyw , τcr = τcri = τyw [1 − 0.16(τyw /τcr )] The maximum value of Vu must be found by trial; 2 is the only independent variable in Equation 19.22, and the optimum is not difficult to determine by trial since it is between 2d /2 and 45 degrees, and Vu is not sensitive to small changes from the optimum 2. Recently [2, 33], it has been argued that the postbuckling strength arises not due to a diagonal tension field action, but by redistribution of shear stresses and local yielding in shear along the boundaries. A case in between is to model the web panel as a diagonal tension strip anchored by corner zones carrying shear stresses and act as gussets connecting the diagonal tension strip to the vertical stiffeners which are in compression [9]. On the basis of test results, it can be concluded that unstiffened webs possess a considerable reserve of postbuckling strength [16, 24]. The incomplete diagonal tension field approach, however, is only reasonably accurate up to a maximum aspect ratio (stiffeners spacing: web depth) equal to 6. Research is required to develop an appropriate method of predicting the postbuckling strength of unstiffened girders. In the AISC specification, the shear capacity of a plate girder web can be calculated, using the model shown in Figure p 19.4, as follows: For h/tw ≤ 187 kv /Fyw , the web yields before buckling, and Vn = 0.6Aw Fyw 1999 by CRC Press LLC
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(19.23)
p For h/tw > 187 kv /Fyw , the web will buckle and a tension field will develop, and q 2 Vn = 0.6Aw Fyw Cv + (1 − Cv )/1.15 1 + (a/ h)
(19.24)
where kv = buckling coefficient = 5 + 5/(a/ h)2 = 5, if (a/ h) > 3 or [260/(h/tw )]2 Cv = ratiopof the web buckling stresspto the shear yield stress of pthe web = 187 kv /Fyw /(h/tw ), for 187 kv /Fyw ≤ h/tw ≤ 234 kv /Fyw p = 44,000 kv /[(h/tw )2 Fyw ], for h/tw > 234 kv /Fyw p It must be noted that, in the above, the web buckling is elastic when h/tw > 234 kv /Fyw . The AISC specification does not permit the consideration of tension field action in end panels, hybrid and webtapered plate girders, and when a/ h exceeds 3.0 or [260/(h/tw)]2 . This is contrary to the fact that a tension field can develop in all these cases; however, little or no research was conducted. Furthermore, tension field can be considered for end panels if the end stiffener is designed for this purpose. When neglecting the tension field action, the nominal shear capacity can be calculated from Vn = 0.6Aw Fyw Cv
(19.25)
Care must be exercised in applying the tension field models developed primarily for welded plate girders to the webs of a box girder. The thin flange of a box girder can provide very little or no resistance against movements in the plane of the web. If the web of a box girder is transversely stiffened and if the model shown in Figure 19.4 is used, it may overpredict the web strength. Hence, it is advisable to use the model shown in Figure 19.5, assuming the plastic moment capacity of the flange to be negligible.
19.7
Web Stiffeners for Shear Design
Transverse stiffeners must be stiff enough to prevent outofplane displacement along the panel boundaries in computing shear buckling of plate girder webs. To provide the outofplane support an equation, developed for an infinitely long web with simply supported edges and equally spaced stiffeners, to calculate the required moment of inertia of the stiffeners, Is , namely for a ≤ h, Is = 2.5htw3 [(h/a) − 0.7(a/ h)]
(19.26)
The AASHTO formula for loadfactor design is Is = J atw3
(19.27)
where J = [2.5(h/a)2 − 2] ≥ 0.5 Equation 19.27 is the same as Equation 19.26 except that the coefficient of (a/ h) in the second term between brackets is 0.8 instead of 0.7. Equation 19.27 was adopted by the AISC specification as well. The moment of inertia of the transverse stiffener shall be taken about the edge in contact with the web for singlesided stiffeners and about the midthickness of the web for doublesided stiffeners. To prevent local buckling of transverse stiffeners, the width, bs , of each projecting stiffener element shall satisfy the requirements of Equation 19.11 using the yield stress of the stiffener material rather than that of the flange, as in Equation 19.11. Furthermore, bs shall also satisfy the following requirements: (19.28) 16.0ts ≥ bs ≤ 0.25bf 1999 by CRC Press LLC
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where bf = the full width of the flange Transverse stiffeners shall consist of plates or angles welded or bolted to either one or both sides of the web. Stiffeners that are not used as connection plates shall be a tight fit at the compression flange, but need not be in bearing with the tension flange. The distance between the end of the webtostiffener weld and the near edge of the webtoflange fillet weld shall not be less than 4tw or more than 6tw . Stiffeners used as connecting plates for diaphragms or crossframes shall be connected by welding or bolting to both flanges. In girders with longitudinal stiffeners the transverse stiffener must also support the longitudinal stiffener as it forces a horizontal node in the bend buckling configuration of the web. In such a case it is recommended that the transverse stiffener section modulus, ST , be equal to SL (h/a), where SL is the section modulus of the longitudinal stiffener and h and a are the web depth and the spacing between the transverse stiffeners, respectively. In the AASHTO specification, the moment of inertia of transverse stiffeners used in conjunction with longitudinal stiffeners shall also satisfy It ≥ (bt /bl )(h/3a)Il
(19.29)
where bt and bl = projecting width of transverse and longitudinal stiffeners, respectively, and It and Il = moment of inertia of transverse and longitudinal stiffeners, respectively. Transverse stiffeners in girders that rely on a tension field must also be designed for their role in the development of the diagonal tension. In this situation they are compression members, and so must be checked for local buckling. Furthermore, they must have crosssectional area adequate for the axial force that develops. The axial force, Fs , can be calculated based on the analytical model [5] shown in Figure 19.4, and is given by (19.30) Fs = 0.5Fyw atw 1 − τcr /τyw (1 − cos 2d ) The AISC and AASHTO specifications assume that a width of the web equal to 18tw acts with the stiffener and give the following formula for the crosssectional area, As , of the stiffeners: h i (19.31) As ≥ 0.15Bhtw (1 − Cv )Vu /0.9Vn − 18tw2 (Fyw /Fys ) where the new notations are 0.9Vu = shear due to factored loads B = 1.0 for doublesided stiffeners = 1.8 for singlesided angle stiffeners = 2.4 for singlesided plate stiffeners If longitudinally stiffened girders are used, h in Equation 19.31 shall be taken as the depth of the web, since the tension field will occur between the flanges and the transverse stiffeners. The optimum location of a longitudinal stiffener that is used to increase resistance to shear buckling is at the web middepth. In this case the two subpanels buckle simultaneously and the increase in the critical stress is substantial. To obtain the tension field shear resistance one can assume that only one tension field is developed between the flanges and transverse stiffeners even if longitudinal stiffeners are used.
19.8
FlexureShear Interaction
The shear capacity of a girder is independent of bending as long as the applied moment is less than the moment that can be taken by the flanges alone, Mf = σyf Af h; any larger moment must be resisted in part by the web, which reduces the shear capacity of the girder. When the girder is subjected to 1999 by CRC Press LLC
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pure bending with no shear, the maximum moment capacity is equal to the plastic moment capacity, Mp , due to yielding of the girder’s entire crosssection. In view of the tension field action and based on test results [3], a simple conservative interaction equation is given by (V /Vu )2 + (M − Mf )/(Mp − Mf ) = 1
(19.32)
where M and V are the applied moment and shear, respectively. In the AISC specification, plate girders with webs designed for tension field action and when the ultimate shear, Vu , is between 0.54 and 0.9Vn , and the ultimate moment, Mu , is between 0.675 and 0.9Mn , where Vn and Mn are the nominal shear and moment capacities in absence of one another; the following interaction equation must be satisfied, Mu /0.9Mn + 0.625(Vu /0.9Vn ) ≤ 1.375
19.9
(19.33)
Steel Plate Shear Walls
Although the postbuckling behavior of plates under monotonic loads has been under investigation for more than half a century, postbuckling strength of plates under cyclic loading has not been investigated until recently [7]. The results of this investigation indicate that plates can be subjected to few reversed cycles of loading in the postbuckling domain, without damage. In steel plate shear walls, the boundary members are stiff and the plate is relatively thin; in such cases a complete tension field can be developed. The plate can be modeled as a series of tension bars inclined at an angle, φ [27]. The angle of inclination, φ, is a function of the panel length and height, the plate thickness, the crosssectional areas of the surrounding beams and columns, and the moment of inertia of the columns. It can be determined by applying the principle of least work and is given by h i (19.34) tan4 α = [(2/tw L) + (1/Ac )] / (2/tw L) + (2h/Ab L) + (h4 /180Ic L2 ) where α = angle of inclination of tension field with the vertical axis L = panel length h = panel height tw = wall thickness Ab = crosssectional area of beam Ac = crosssectional area of column = moment of inertia of column about axis perpendicular to the plane of the wall Ic Although this model can predict the ultimate capacity to a reasonable degree of accuracy it cannot depict the loaddeflection characteristics to the same degree of accuracy. Based on test results and finite element analysis [17, 18], the stresses in the inclined tensile plate strips are not uniform but are higher near the supporting boundaries than the center of the plate, and yielding of these strips starts near their ends and propagates toward the midlength. The following method can be used to calculate the ultimate capacity and determine the loaddeflection characteristics of a thinsteelplate shear wall. The plate in the shear wall is replaced by a series of truss elements in the diagonal tension direction, as shown in Figure 19.6. A minimum of four truss members shall be used to replace the plate panel in order to depict the panel behavior to a reasonable degree of accuracy; however, six members are recommended. The stressstrain relationship for the truss elements shall be assumed to be bilinearly elastic perfectly plastic, as shown in Figure 19.7, where E is Young’s modulus of elasticity and σy is the tensile yield stress of the plate material. In Figure 19.7 the first slope represents the elastic 1999 by CRC Press LLC
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FIGURE 19.6: Steel plate shear wall analytical model.
FIGURE 19.7: Stressstrain relationship for truss element. response and the second represents the reduced stiffness caused by partial yielding; σy1 and E2 can be determined using a semiempirical approach for welded as well as bolted shear walls, and can be calculated using the following equations:
where
σy1 = (0.423 + 0.816be /L)σy
(19.35)
h i0.5 be = 14.6π 2 E/12(1 − ν 2 )τy t
(19.36)
L = length of the strip t = thickness of the plate and E2 = (σy2 − σy1 )(3 − σy1 )/σy2 (2 + α) E 1999 by CRC Press LLC
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(19.37)
where α is the ratio between the plastic strain, εp , and the strain at the initiation of yielding, εy . This ratio is in the range of 5 to 20, depending on the stiffness of the columns relative to the thickness of the plate; a value of 10 can be used in design. In the derivation of Equations 19.35 and 19.36 the inclination angle of the equivalent truss elements was assumed to be 45 degrees. The angle of inclination is usually in the range of 38 to 43 degrees and the effect of this assumption on the overall behavior of the wall is negligible. In order to define the load displacement relationship of the truss elements in a bolted shear wall, the parameters Py1 , K1 , Py , and K2 shown in Figure 19.8 need to be calculated. For a bolted plate,
FIGURE 19.8: Loaddisplacement relationship for truss element.
the initial yielding can occur when the plate at the bolted connection starts slipping or when it locally yields near the boundaries as in the welded plate. The load due to slippage is controlled by the friction coefficient between the connected surfaces and the normal force applied by the bolts. In case the bolts were pretensioned to 70% of their ultimate tensile strength, slip will occur at a load equal to Py1 = n(0.7µFub Ab )
(19.38)
where n is the number of bolts at one end of the truss element; Fub and Ab are the ultimate tensile strength and crosssection area of the bolt, respectively; and µ is the friction coefficient between the connected surfaces. The load that causes initial yielding at the ends of the strip is the same as for the welded plate, and can be obtained from Equation 19.35 by multiplying the stress, σy1 , by the strip crosssectional area, Ap . The usable load is the smaller of the loads that causes slippage or initial yielding at the ends of the strip. The initial stiffness of the equivalent truss element can be calculated as follows: K1 = EAp /L
(19.39)
The ultimate load is controlled by the total yielding of the plate strip, tearing at the bolt holes, or shearing of the bolts. The smallest failure load is the controlling ultimate capacity of the truss element. The ultimate load due to the strip yielding along its entire length, Py0 = σy Ap The bolted shear wall should be designed such that plate yielding controls. 1999 by CRC Press LLC
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(19.40)
Tearing at the bolt holes will occur when the edge distance is small or the bolt spacing is large, which should be avoided in design because it is a brittle failure. The tearing load, denoted as Py00 , can be calculated using the following formula: Py00 = 1.4nσu (Le − D/2)t
(19.41)
where n is the number of bolts at the end of the truss element, σu is the ultimate tensile strength of the plate material, Le is the distance from the edge of the plate to the centroid of the bolt hole, D is the diameter of the bolt, and t is the thickness of the plate. Note that in the above formula the ultimate strength of the plate material in shear was assumed to be 0.7, its ultimate strength in tension. The shear failure of the bolts can occur if the shear strength of the bolts is small or the spacing between them is large. In such case the ultimate capacity of the truss element, which is denoted as Py000 , can be calculated from one of the following formulas: Py000
=
0.45nFub Ab
(19.42a)
Py000
=
0.60nFub Ab
(19.42b)
where n is the number of bolts and Fub and Ab are the ultimate strength and the gross crosssection area of the bolts, respectively. Equation 19.42a is used if the shear plane is within the threaded part of the bolt and Equation 19.42b is used if the shear plane is not within the threaded part of the bolt. The ultimate load of the truss element is the smallest value of the plate total yielding capacity, the tearing capacity, and the bolt shearing capacity, i.e., Py = min(Py0 , Py00 , Py000 )
(19.43)
As stated earlier, the plate yielding shall control and the designer must ensure that Py0 is the smallest of the three values. As can be seen in Figure 19.8, in order to define the stiffness, K2 , the displacement of the truss element when the load reaches the ultimate capacity, Py , needs to be determined. This ultimate displacement includes the stretching of the element as well as the slippage and the bearing deformation of the plate and the bolts at the connections. The plate strip represented by the truss element is stretched under load; the elongation includes both elastic and plastic deformations. As discussed earlier, due to the nonuniform strain distribution along the length of the strip, the plastic deformation will occur mostly near the ends. If one assumes that the strain distribution along the length of the strip is a seconddegree parabola, and using a plastic deformation factor a, the elongation of the truss element due to the plate elastic and plastic deformations can be calculated using the following equation: 1def = (σy /3E)(2 + α)L
(19.44)
The elongation of the truss element due to slippage can be approximated by two times the hole clearance, taking into consideration the slippage at both ends of the element [26]. The local deformation at the bolt holes includes the effects of shearing, bending, and bearing deformations of the fastener as well as local deformation of the connected plates, and can be taken as 0.2 times the bolt diameter [21]. Having defined the ultimate elongation of the truss element, its reduced stiffness after slippage and/or initial yielding can be obtained using the following equation: K2 = (Py − Py1 )/ 0.125 + 0.2D + σy (α − 1)L/(3E)
1999 by CRC Press LLC
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(19.45)
19.10
InPlane Compressive Edge Loading
Webs of plate and box girders and loadbearing diaphragms in box girders can be subjected to local inplane compressive loads. Vertical (transverse) stiffeners can be provided at the location of the load to prevent web crippling; however, this is not always possible, such as in the case of a moving load, and it involves higher cost. Failure of the web under this loading is always due to crippling [10], as shown in Figure 19.9; in thin webs crippling occurs before yielding of the web and in stocky webs after yielding. The formula in the AISC specification predicts the crippling load, Pcr , to a reasonable
FIGURE 19.9: Deformed shape under inplane edge loading (only half of the beam is shown).
degree of accuracy [37]; this formula is h 1.5 i 0.5 Fyw tf /tw Pcr = 135tw2 1 + 3(N/d) tw /tf
(19.46)
The formula given by Equation 19.46 is applicable if the load is applied at a distance not less than half the member’s depth from its end; if the load is at a distance less than half the member’s depth, the following formulae shall be used [14]: h 1.5 i 0.5 (19.47a) Fyw tf /tw For N/d ≤ 0.2, Pcr = 68tw2 1 + 3(N/d) tw /tf h i 0.5 For N/d > 0.2, Pcr = 68tw2 1 + (4N/d − 0.2)(tw /tf )1.5 Fyw tf /tw (19.47b)
In addition to web crippling, the AISC specification requires a web yielding check; furthermore, when the relative lateral movement between the loaded compression flange and the tension flange is not restrained at the point of load application, sidesway buckling must be checked. 1999 by CRC Press LLC
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19.11
Eccentric Edge Loading
Eccentricities in loading with respect to the plane of the web are unavoidable, and it was found that there is a reduction in the web capacity due to the presence of an eccentricity [13, 14]; for example, in one case, an eccentricity of 0.5 in. reduced the web ultimate capacity to about half its capacity under inplane load. Furthermore, it was found that the effect of the load eccentricity in reducing the ultimate capacity decreases as the ratio of the flangetoweb thickness increases. A deformed beam subjected to eccentric load near failure is shown in Figure 19.10. Web strength reduction factors for various eccentricities as a function of the flange width and for various flangetoweb thickness ratios are given in Figure 19.11.
FIGURE 19.10: Deformed shape under eccentric edge loading.
The failure mechanism in the case of eccentric loading is different from that for inplane loading. The flange twisting moment acting at the webflange intersection can cause failure due to bending rather than crippling of the web, if the eccentricity is large enough. In most cases, however, the failure mode is due to a combination of web bending and crippling. Failure mechanisms were developed, and formulas to calculate the ultimate capacity of the web under eccentric edge loading were derived [15]. Currently, the AISC specification does not address the effect of the eccentricity on the reduction of the web crippling load. Eccentricities can also arise due to moments applied to the top flange in addition to vertical loads. An example would be a beam resting on the top flange of another beam and the two flanges are welded together. Rotation of the supported beam will impose a twisting moment in the flange of the supporting beam and bending of its web, which will reduce its crippling load. 1999 by CRC Press LLC
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FIGURE 19.11: Eccentricity reduction factor.
19.12
LoadBearing Stiffeners
Webs of girders are often strengthened with transverse stiffeners at points of concentrated loads and over intermediate and end supports. The AISC specification requires that these stiffeners be double sided, extend at least onehalf the beam depth, and either bear on or be welded to the loaded flange. The specification, further, requires that they be designed as axially loaded members with an effective length equals to 0.75 times the web depth; and a strip of the web, with a width equal to 25 times its thickness for intermediate stiffeners and 12 times the thickness for end stiffeners, shall be considered in calculating the geometric properties of the stiffener. The failure, in cases where the stiffener depth is less than 75% of the depth of the web, can be due to crippling of the web below the stiffener [14, 15], as shown in Figure 19.12. The failure, otherwise,
FIGURE 19.12: Web crippling below stiffener. 1999 by CRC Press LLC
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is due to global buckling of the stiffener, provided that the thickness of the stiffener is adequate to prevent local buckling. The optimum depth of the stiffener is 0.75 times the web depth. The AISC specification does not account for factors such as the stiffener depth and load eccentricity. In box girders intermediate diaphragms are provided to limit crosssectional deformation and loadbearing diaphragms are used at the supports to transfer loads to the bridge bearings. Diaphragm design is treated in the BS 5400: Part 3 (1983) [6] and discussed in Chapter 7 of the SSRC guide [22].
19.13
Web Openings
Openings are frequently encountered in the webs of plate and box girders. Research on the buckling and ultimate strength of plates with rectangular and circular openings subjected to inplane loads has been performed by many investigators. The research has included reinforced and unreinforced openings. A theoretical method of predicting the ultimate capacity of slender webs containing circular and rectangular holes, and subjected to shear, has been developed [34, 35]. The solution is obtained by considering the equilibrium of two tension bands, one above and the other below the opening. These bands have been chosen to conform to the failure pattern observed in tested plate girders with holes. Experimental results showed that the method gives satisfactory and safe predictions. The calculated values were found to be between 5 and 30% below the test results. Solutions for transversely stiffened webs subjected to shear and bending with centrally located holes are available [28, 30] and are applicable for webs with depthtothickness ratios of 120 to 360, panel aspect ratio between 0.7 and 1.5, hole depth greater than 1/10th of the web depth, and for circular, elongated circular, and rectangular holes.
19.14
Girders with Corrugated Webs
Corrugated webs can be used in an effort to decrease the weight of steel girders and reduce its fabrication cost. Studies have been conducted in Europe and Japan and girders with corrugated webs have been used in these countries [12]. The results of the studies indicate that the fatigue strength of girders with corrugated webs can be 50% higher compared to girders with flat stiffened webs. In addition to the improved fatigue life, the weight of girders with corrugated webs can be as much as 30 to 60% less than the weight of girders with flat webs and have the same capacity. Due to the weight savings, larger clear spans can be achieved. Beams and girders with corrugated webs are economical to use and can improve the aesthetics of the structure. Beams manufactured and used in Germany for buildings have a web thickness that varies between 2 and 5 mm, and the corresponding web heighttothickness ratio is in the range of 150 to 260. The corrugated webs of two bridges built in France were 8 mm thick and the web heighttothickness ratio was in the range of 220 to 375. Failure in shear is usually due to buckling of the web and the failure in bending is due to yielding of the compression flange and its vertical buckling into the corrugated web, which buckles [19, 20]. The shear buckling mode is global for dense corrugation and local for course corrugation, as shown in Figure 19.13. The loadcarrying capacity of the specimens drops after buckling, with some residual loadcarrying capacity after failure. In the local buckling mode, the corrugated web acts as a series of flatplate subpanels that mutually support each other along their vertical (longer) edges and are supported by the flanges at their horizontal (shorter) edges. These flatplate subpanels are subjected to shear, and the elastic buckling stress is given by h i τcre = ks π 2 E/12(1 − µ2 )(w/t)2
1999 by CRC Press LLC
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(19.48)
FIGURE 19.13: Local and global buckling.
where ks = buckling coefficient, which is a function of the panel aspect ratio, h/w, and the boundary support conditions h = the web depth t = the web thickness w = the flatplate subpanel width — the horizontal or the inclined, whichever is bigger E = Young’s modulus of elasticity µ = the Poisson ratio The buckling coefficient, ks, is given by ks = 5.34 + 2.31(w/ h) − 3.44(w/ h)2 + 8.39(w/ h)3 , for the longer edges simply supported and the shorter edges clamped ks = 8.98 + 5.6(w/ h)2 , in the case where all edges are clamped An average local buckling stress, τav (= 0.5[τssf + τf x ]), is recommended, and in the case of τcre ≥ 0.8τy , inelastic buckling will occur and the inelastic buckling stress, τcri , can be calculated by τcri = (0.8 ∗ τcre ∗ τy )0.5 , where τcri ≤ τy . As stated earlier, the mode of failure is local and/or global buckling; when global buckling controls, the buckling stress can be calculated for the entire corrugated web panel, using orthotropicplate buckling theory. The global elastic buckling stress, τcre , can be calculated from h 0.75 i /th2 τcre = ks (Dx )0.25 Dy where Dx Dy Iy ks
= = = =
(19.49)
(q/s)Et 3 /12 EIy /q 2bt (hr /2)2 + {t (hr )3 /6 sin 2} Buckling coefficient, equal to 31.6 for simply supported boundaries and 59.2 for clamped boundaries t = corrugated plate thickness b, hr , q, s, and 2 are as shown in Figure 19.14. 1999 by CRC Press LLC
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In the aforementioned, when τcre ≥ 0.8τy , inelastic buckling will occur and the inelastic buckling stress, τcri , can be calculated by τcri = (0.8 ∗ τcre ∗ τy )0.5 , where τ cri ≤ τy . For design, it is recommended that the local and global buckling values be calculated and the smaller value controls. As stated earlier, the failure in bending is due to compression flange yielding and vertical buckling into the web, as shown in Figure 19.15. The failure is sudden, with no appreciable residual strength. The web offers negligible contribution to the moment carrying capacity of the beam, and for design,
FIGURE 19.14: Dimensions of corrugation profile.
FIGURE 19.15: Bending failure of a beam with corrugate web.
the ultimate moment capacity can be calculated based on the flange yielding, ignoring any contribution from the web. The stresses in the web are equal to zero except near the flanges. This is because the corrugated web has no stiffness perpendicular to the direction of the corrugation, except for a 1999 by CRC Press LLC
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very small distance which is adjacent to and restrained by the flanges, and the stresses are appreciable only within the horizontal folds of the corrugation. It must be noted that the common practice is to fillet weld the web to the flanges from one side only; under static loading this welding detail was found to be adequate and there is no need to weld from both sides. Finally, the bracing requirements of the compression flange in beams and girders with corrugated webs are less severe compared to conventional beams and girders with flat webs. Lateraltorsional buckling of beams and girders with corrugated webs has been investigated [20].
19.15
Defining Terms
AASHTO: American Association of State Highway and Transportation Officials. AISC: American Institute of Steel Construction. Buckling load: The load at which a compressed element or member assumes a deflected position. Effective width: Reduced flat width of a plate element due to buckling, the reduced width is termed the effective width. Corrugated web: A web made of corrugated steel plates, where the corrugations are parallel to the depth of the girder. Factored load: The nominal load multiplied by a load factor to account for unavoidable deviations of the actual load from the nominal load. Limit state: A condition at which a structure or component becomes unsafe (strength limit state) or no longer useful for its intended function (serviceability limit state). LRFD (Load and Resistance Factor Design): A method of proportioning structural components such that no applicable limit state is exceeded when the structure is subjected to all appropriate load combinations. Shear wall: A wall in a building to carry lateral loads from wind and earthquakes. Stress: Force per unit area. Web crippling: Local buckling of the web plate under local loads. Web slenderness ratio: The depthtothickness ratio of the web.
References [1] American Association of State Highway and Transportation Officials. 1994. AASHTO LRFD Bridge Design Specifications, Washington, D.C. [2] Ajam, W. and Marsh, C. 1991. Simple Model for Shear Capacity of Webs, ASCE Struct. J., 117(2). [3] Basler, K. 1961. Strength of Plate Girders Under Combined Bending and Shear, ASCE J. Struct. Div., October, vol. 87. [4] Basler K. and Th¨urlimann, B. 1963. Strength of Plate Girders in Bending, Trans. ASCE, 128. [5] Basler, K. 1963. Strength of Plate Girders in Shear, Trans. ASCE, Vol. 128, Part II, 683. [6] British Standards Institution. 1983. BS 5400: Part 3, Code of Practice for Design of Steel Bridges, BSI, London. [7] Caccese, V., Elgaaly, M., and Chen, R. 1993. Experimental Study of Thin SteelPlate Shear Walls Under Cyclic Load, ASCE J. Struct. Eng., February. [8] Cooper, P.B. 1967. Strength of Longitudinally Stiffened Plate Girders, ASCE J. Struct. Div., 93(ST2), 419452. 1999 by CRC Press LLC
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[9] Dubas, P. and Gehrin, E. 1986. Behavior and Design of Steel Plated Structures, ECCS Publ. No. 44, TWG 8.3, 110112. [10] Elgaaly, M. 1983. Web Design Under Compressive Edge Loads, AISC Eng. J., Fourth Quarter. [11] Elgaaly, M. and Nunan W. 1989. Behavior of Rolled Sections Webs Under Eccentric Edge Compressive Loads, ASCE J. Struct. Eng., 115(7). [12] Elgaaly, M. and Dagher, H. 1990. Beams and Girders with Corrugated Webs, Proceedings of the SSRC Annual Technical Session, St. Louis, MO. [13] Elgaaly, M. and Salkar, R. 1990. Behavior of Webs Under Eccentric Compressive Edge Loads, Proceedings of IUTAM Symposium, Prague, Czechoslovakia. [14] Elgaaly, M. and Salkar, R. 1991. Web Crippling Under Edge Loading, Proceedings of the AISC National Steel Construction Conference, Washington, D.C. [15] Elgaaly, M., Salkar, R., and Eash, M. 1992. Unstiffened and Stiffened Webs Under Compressive Edge Loads, Proceedings of the SSRC Annual Technical Session, Pittsburgh, PA. [16] Evans, H.R. and Mokhtari, A.R. 1992. Plate Girders with Unstiffened or Profiled Web Plates, J. Singapore Struct. Steel Soc., 3(1), December. [17] Elgaaly, M., Caccese, V., and Du, C. 1993. Postbuckling Behavior of SteelPlate Shear Walls Under Cyclic Loads, ASCE J. Struct. Eng., 119(2). [18] Elgaaly, M., Liu, Y., Caccese, V., Du, C., Chen, R., and Martin, D. 1994. NonLinear Behavior of Steel Plate Shear Walls, Computational Structural Engineering for Practice, edited by Papadrakakis and Topping, CivilComp Press, Edinburgh, UK. [19] Elgaaly, M., Hamilton, R., and Seshadri, A. 1996. Shear Strength of Beams with Corrugated Webs, J. Struct. Eng., 122(4). [20] Elgaaly, M., Seshadri, A., and Hamilton, R. 1996. Bending Strength of Beams with Corrugated Webs, J. Struct. Eng., 123(6). [21] Fisher, J.W. 1965. Behavior of Fasteners and Plates with Holes, J. Struct. Eng., ASCE, 91(6). [22] Galambos, T.V., Ed. 1988. Guide to Stability Design Criteria for Metal Structures, 4th ed., Wiley Interscience, New York. [23] Gaylord, E.H. 1963. Discussion of K. Basler Strength of Plate Girders in Shear, Trans. ASCE, 128, Part II, 712. [24] Hoglund, T. 1971. Behavior and Load Carrying Capacity of Thin Plate IGirders, Division of Building Statics and Structural Engineering, Royal Institute of Technology, Bulletin No. 93, Stockholm. [25] Kirby, P.A. and Nethercat, D.A. 1979. Design for Structural Stability, John Wiley & Sons, New York. [26] Kulak, G.L., Fisher, J.W., and Struik, J.H.A. 1987. Guide to Design Criteria for Bolted and Riveted Joints, 2nd ed., Wiley Interscience, New York. [27] Kulak, G.L. 1985. Behavior of Steel Plate Shear Walls, Proc. of the AISC Int. Eng. Symp. on Struct. Steel, American Institute of Steel Construction (AISC), Chicago, IL. [28] Lee, M.M.K., Kamtekar, A.G., and Little, G.H. 1989. An Experimental Study of Perforated Steel Web Plates, Struct. Engineer, 67(2/24). [29] Lee, M.M.K. 1990. Numerical Study of Plate Girder Webs with Holes, Proc. Inst. Civ. Eng., Part 2. [30] Lee, M.M.K. 1991. A Theoretical Model for Collapse of Plate Girders with Perforated Webs, Struct. Eng., 68(4). [31] Lindner, J. 1990. LateralTorsional Buckling of Beams with Trapezoidally Corrugated Webs, Proceedings of the 4th International Colloquium on Stability of Steel Structures, Budapest, Hungary. [32] American Institute of Steel Construction. 1993. Load and Resistance Factor Design Specification for Structural Steel Buildings, AISC, Chicago. [33] Marsh, C. 1985. Photoelastic Study of Postbuckled Shear Webs, Canadian J. Civ. Eng., 12(2). 1999 by CRC Press LLC
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[34] Narayanan, R. and Der Avanessian, N.G.V. 1983. Strength of Webs Containing CutOuts, IABSE Proceedings P64/83. [35] Narayanan, R. and Der Avanessian, N.G.V. 1983. Equilibrium Solution for Predicting the Strength of Webs with Rectangular Holes, Proc. ICE, Part 2. [36] Porter, D.M., Rockey, K.C., and Evans, H.R. 1975. The Collapse Behavior of Plate Girders Loaded in Shear, Struct. Eng., 53(8), 313325. [37] Roberts, T.M. 1981. Slender Plate Girders Subjected to Edge Loading, Proc. Inst. Civil Eng., Part 2, 71. [38] Rockey, K.C. and Skaloud, M. 1972. The Ultimate Load Behavior of Plate Girders Loaded in Shear, Struct. Eng., 50(1). [39] Schilling, C.G. and Frost, R.W. 1964. Behavior of Hybrid Beams Subjected to Static Loads, ASCE, J. Struct. Div., 90(ST3), 5588. [40] Von Karman, T., Sechler, E.F., and Donnell, L.H. 1932. The Strength of Thin Plates in Compression, Trans. ASME, 54(2). [41] Wagner, H. 1931. Flat Sheet Metal Girder with Very Thin Metal Web, NACA Tech. Memo. Nos. 604, 605, 606. [42] Yen, B.T. and Mueller, J.A. 1966. Fatigue Tests of LargeSize Welded Plate Girders, Weld. Res. Counc. Bull., No. 118, November.
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Durkee, J. “Steel Bridge Construction” Structural Engineering Handbook Ed. Chen WaiFah Boca Raton: CRC Press LLC, 1999
Steel Bridge Construction 20.1 Introduction 20.2 Construction Engineering in Relation to Design Engineering 20.3 Construction Engineering Can Be Critical 20.4 Premises and Objectives of Construction Engineering 20.5 Fabrication and Erection Information Shown on\break Design Plans 20.6 Erection Feasibility 20.7 Illustrations of Challenges in Construction\break Engineering 20.8 Obstacles to Effective Construction Engineering 20.9 Examples of Inadequate Construction Engineering Allowances and Effort 20.10Considerations Governing Construction Engineering Practices 20.11Two General Approaches to Fabrication and\break Erection of Bridge Steelwork 20.12Example of Arch Bridge Construction 20.13Which Construction Procedure Is To Be Preferred? 20.14Example of Suspension Bridge Cable Construction 20.15Example of CableStayed Bridge Construction 20.16Field Checking at Critical Erection Stages 20.17Determination of Erection Strength Adequacy 20.18Philosophy of the Erection Rating Factor 20.19Minimum Erection Rating Factors 20.20Deficiencies of Typical Construction Procedure\break Drawings and Instructions 20.21Shop and Field Liaison by Construction Engineers 20.22Construction Practices and Specifications— The Future 20.23Concluding Comments Jackson Durkee Consulting Structural Engineer, Bethlehem, 20.24Further Illustrations References PA
20.1
Introduction
This chapter addresses some of the principles and practices applicable to the construction of mediumand longspan steel bridges — structures of such size and complexity that construction engineering becomes an important or even the governing factor in the successful fabrication and erection of the superstructure steelwork. 1999 by CRC Press LLC
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We begin with an explanation of the fundamental nature of construction engineering, then go on to explain some of the challenges and obstacles involved. Two general approaches to the fabrication and erection of bridge steelwork are described, with examples from experience with arch bridges, suspension bridges, and cablestayed bridges. The problem of erectionstrength adequacy of trusswork under erection is considered, and a method of appraisal offered that is believed to be superior to the standard workingstress procedure. Typical problems in respect to construction procedure drawings, specifications, and practices are reviewed, and methods for improvement suggested. Finally, we take a view ahead, to the future prospects for effective construction engineering in the U.S. This chapter also contains a large number of illustrations showing a variety of erection methods for several types of steel bridges.
20.2
Construction Engineering in Relation to Design Engineering
With respect to bridge steelwork the differences between construction engineering and design engineering should be kept firmly in mind. Design engineering is of course a concept and process well known to structural engineers; it involves preparing a set of plans and specifications — known as the contract documents — that define the structure in its completed configuration, referred to as the geometric outline. Thus, the design drawings describe to the contractor the steel bridge superstructure that the owner wants to see in place when the project is completed. A considerable design engineering effort is required to prepare a good set of contract documents. Construction engineering, however, is not so well known. It involves governing and guiding the fabrication and erection operations needed to produce the structural steel members to the proper cambered or “noload” shape, and get them safely and efficiently “up in the air” in place in the structure, such that the completed structure under the deadload conditions and at normal temperature will meet the geometric and stress requirements stipulated on the design drawings. Four key considerations may be noted: (1) design engineering is widely practiced and reasonably well understood, and is the subject of a steady stream of technical papers; (2) construction engineering is practiced on only a limited basis, is not as well understood, and is hardly ever discussed; (3) for medium and longspan bridges, the construction engineering aspects are likely to be no less important than design engineering aspects; and (4) adequately staffed and experienced construction engineering offices are a rarity.
20.3
Construction Engineering Can Be Critical
The construction phase of the total life of a major steel bridge will probably be much more hazardous than the serviceuse phase. Experience shows that a large bridge is more likely to suffer failure during erection than after completion. Many decades ago, steel bridge design engineering had progressed to the stage where the chance of structural failure under service loadings became altogether remote. However, the erection phase for a large bridge is inherently less secure, primarily because of the prospect of inadequacies in construction engineering and its implementation at the job site. Indeed, the hazards associated with the erection of large steel bridges will be readily apparent from a review of the illustrations in this chapter. For significant steel bridges the key to construction integrity lies in the proper planning and engineering of steelwork fabrication and erection. Conversely, failure to attend properly to construction engineering constitutes an invitation to disaster. In fact, this thesis is so compelling that whenever a steel bridge failure occurs during construction (see for example Figure 20.1), it is reasonable to assume 1999 by CRC Press LLC
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that the construction engineering investigation was either inadequate, not properly implemented, or both.
FIGURE 20.1: Failure of a steel girder bridge during erection, 1995. Steel bridge failures such as this one invite suspicion that the construction engineering aspects were not properly attended to.
20.4
Premises and Objectives of Construction Engineering
Obviously, when the structure is in its completed configuration it is ready for the service loadings. However, during the erection sequences the various components of major steel bridges are subject to stresses that may be quite different from those provided for by the designer. For example, during construction there may be a derrick moving and working on the partially erected structure, and the structure may be cantilevered out some distance causing tensiondesigned members to be in compression and vice versa. Thus, the steelwork contractor needs to engineer the bridge members through their various construction loadings, and strengthen and stabilize them as may be necessary. Further, the contractor may need to provide temporary members to support and stabilize the structure as it passes through its successive erection configurations. In addition to strength problems there are also geometric considerations. The steelwork contractor must engineer the construction sequences step by step to ensure that the structure will fit properly together as erection progresses, and that the final or closing members can be moved into position and connected. Finally, of course, the steelwork contractor must carry out the engineering studies needed to ensure that the geometry and stressing of the completed structure under normal temperature will be in accordance with the requirements of the design plans and specifications.
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20.5
Fabrication and Erection Information Shown on Design Plans
Regrettably, the level of engineering effort required to accomplish safe and efficient fabrication and erection of steelwork superstructures is not widely understood or appreciated in bridge design offices, nor indeed by a good many steelwork contractors. It is only infrequently that we find a proper level of capability and effort in the engineering of construction. The design drawings for an important bridge will sometimes display an erection scheme, even though most designers are not experienced in the practice of erection engineering and usually expend only a minimum or even superficial effort on erection studies. The scheme portrayed may not be practical, or may not be suitable in respect to the bidder or contractor’s equipment and experience. Accordingly, the bidder or contractor may be making a serious mistake if he relies on an erection scheme portrayed on the design plans. As an example of misplaced erection effort on the part of the designer, there have been cases where the design plans show cantilever erection by deck travelers, with the permanent members strengthened correspondingly to accommodate the erection loadings; but the successful bidder elected to use waterborne erection derricks with long booms, thereby obviating the necessity for most or all of the erection strengthening provided on the design plans. Further, even in those cases where the contractor would decide to erect by cantilevering as anticipated on the plans, there is hardly any way for the design engineer to know what will be the weight and dimensions of the contractor’s erection travelers.
20.6
Erection Feasibility
Of course, the bridge designer does have a certain responsibility to his client and to the public in respect to the erection of the bridge steelwork. This responsibility includes (1) making certain, during the design stage, that there is a feasible and economical method to erect the steelwork; (2) setting forth in the contract documents any necessary erection guidelines and restrictions; and (3) reviewing the contractor’s erection scheme, including any strengthening that may be needed, to verify its suitability. It may be noted that this latter review does not relieve the contractor from responsibility for the adequacy and safety of the field operations. Bridge annals include a number of cases where the designing engineer failed to consider erection feasibility. In one notable instance the design plans showed the 1200ft (366m) main span for a long crossing over a wide river as an esthetically pleasing steel tiedarch. However, erection of such a span in the middle of the river was impractical; one bidder found that the tonnage of falsework required was about the same as the weight of the permanent steelwork. Following opening of the bids, the owner found the prices quoted to be well beyond the resources available, and the tiedarch main span was discarded in favor of a throughcantilever structure, for which erection falsework needs were minimal and practical. It may be noted that designing engineers can stand clear of serious mistakes such as this one, by the simple expedient of conferring with prospective bidders during the preliminary design stage of a major bridge.
20.7
Illustrations of Challenges in Construction Engineering
Space does not permit comprehensive coverage of the numerous and difficult technical challenges that can confront the construction engineer in the course of the erection of various types of major steel bridges. However, some conception of the kinds of steelwork erection problems, the methods available to resolve them, and hazards involved can be conveyed by views of bridges in various stages of erection; refer to the illustrations in the text. 1999 by CRC Press LLC
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20.8
Obstacles to Effective Construction Engineering
There is an unfortunate tendency among designing engineers to view construction engineering as relatively unimportant. This view may be augmented by the fact that few designers have had any significant experience in the engineering of construction. Further, managers in the construction industry must look critically at costs, and they can readily develop the attitude that their engineers are doing unnecessary theoretical studies and calculations, detached from the practical world. (And indeed, this may sometimes be the case.) Such management apprehension can constitute a serious obstacle to staff engineers who see the need to have enough money in the bridge tender to cover a proper construction engineering effort for the project. There is the tendency for steelwork construction company management to cut back the construction engineering allowance, partly because of this apprehension and partly because of the concern that other tenderers will not be allotting adequate money for construction engineering. This effort is often thought of by company management as “a necessary evil” at best — something they would prefer not to be bothered with or burdened with. Accordingly, construction engineering tends to be a difficult area of endeavor. The way for staff engineers to gain the confidence of management is obvious — they need to conduct their investigations to a level of technical proficiency that will command management respect and support, and they must keep management informed as to what they are doing and why it is necessary. As for management’s concern that other bridge tenderers will not be putting into their packages much money for construction engineering, this concern is no doubt usually justified, and it is difficult to see how responsible steelwork contractors can cope with this problem.
20.9
Examples of Inadequate Construction Engineering Allowances and Effort
Even with the best of intentions, the bidder’s allocation of money to construction engineering can be inadequate. A case in point involved a very heavy, longspan cantilever truss bridge crossing a major river. The bridge superstructure carried a contract price of some $30 million, including an allowance of $150,000, or about onehalf of 1%, for construction engineering of the permanent steelwork (i.e., not including such matters as design of erection equipment). As fabrication and erection progressed, many unanticipated technical problems came forward, including brittlefracture aspects of certain grades of the highstrength structural steel, and aerodynamic instability of Hshaped vertical and diagonal truss members. In the end the contractor’s construction engineering effort mounted to about $1.3 million, almost nine times the estimated cost. Another significant example — this one in the domain of buildings — involved a designandconstruct project for airplane maintenance hangars at a prominent airport. There were two large and complicated buildings, each 100 × 150 m (328 × 492 ft) in plan and 37 m (121 ft) high with a 10m (33ft) deep spaceframe roof. Each building contained about 2300 tons of structural steelwork. The designandconstruct steelwork contractor had submitted a bid of about $30 million, and included therein was the magnificent sum of $5000 for construction engineering, under the expectation that this work could be done on an incidental basis by the project engineer in his “spare time”. As the steelwork contract went forward it quickly became obvious that the construction engineering effort had been grossly underestimated. The contractor proceeded of staffup appropriately and carried out indepth studies, leading to a detailed erection procedure manual of some 270 pages showing such matters as erection equipment and its positioning and clearances; falsework requirements; lifting tackle and jacking facilities; stress, stability, and geometric studies for gravity and wind loads; stepbystep instructions for raising, entering, and connecting steelwork components; closing and swinging the roof structure and portal frame; and welding guidelines and procedures. This 1999 by CRC Press LLC
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erection procedure manual turned out to be a key factor in the success of the fieldwork. The cost of this construction engineering effort amounted to ten times the estimate, but still came to a mere onefifth of 1% of the total contract cost. In yet another example a major steelwork general contractor was induced to sublet the erection of a longspan cantilever truss bridge to a reputable erection contractor, whose quoted price for the work was less than the general contractor’s estimated cost. During the erection cycle the general contractor’s engineers made some visits to the job site to observe progress, and were surprised and disconcerted to observe how little erection engineering and planning had been accomplished. For example, the erector had made no provision for installing jacks in the bottomchord jacking points for closure of the main span; it was left up to the field forces to provide the jack bearing components inside the bottomchord joints and to find the required jacks in the local market. When the jobbuilt installations were tested it was discovered that they would not lift the cantilevered weight, and the job had to be shut down while the field engineer scouted around to find largercapacity jacks. Further, certain compression members did not appear to be properly braced to carry the erection loadings; the erector had not engineered those members, but just assumed they were adequate. It became obvious that the erector had not appraised the bridge members for erection adequacy and had done little or no planning and engineering of the critical evolutions to be carried out in the field. Many further examples of inadequate attention to construction engineering could be presented. Experience shows that the amounts of money and time allocated by steelwork contractors for the engineering of construction are frequently far less than desirable or necessary. Clearly, effort spent on construction engineering is worthwhile; it is obviously more efficient and cheaper, and certainly much safer, to plan and engineer steelwork construction in the office in advance of the work, rather than to leave these important matters for the field forces to work out. Just a few bad moves on site, with the corresponding waste of labor and equipment hours, will quickly use up sums of money much greater than those required for a proper construction engineering effort — not to mention the costs of any job accidents that might occur. The obvious question is “Why is construction engineering not properly attended to?” Do not contractors learn, after a bad experience or two, that it is both necessary and cost effective to do a thorough job of planning and engineering the construction of important bridge projects? Experience and observation would seem to indicate that some steelwork contractors learn this lesson, while many do not. There is always pressure to reduce bid prices to the absolute minimum, and to add even a modest sum for construction engineering must inevitably reduce the chance of being the low bidder.
20.10
Considerations Governing Construction Engineering Practices
There are no textbooks or manuals that define how to accomplish a proper job of construction engineering. In bridge construction (and no doubt in building construction as well), the engineering of construction tends to be a matter of each firm’s experience, expertise, policies and practices. Usually there is more than one way to build the structure, depending on the contractor’s ingenuity and engineering skill, his risk appraisal and inclination to assume risk, the experience of his fabrication and erection work forces, his available equipment, and his personal preferences. Experience shows that each project is different; and although there will be similarities from one bridge of a given type to another, the construction engineering must be accomplished on an individual project basis. Many aspects of the project at hand will turn out to be different from those of previous similar jobs, and also there may be new engineering considerations and requirements for a given project that did not come forward on previous similar work. During the estimating and bidding phase of the project the prudent, experienced bridge steelwork contractor will “start from scratch” and perform his own fabrication and erection studies, irrespective 1999 by CRC Press LLC
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of any erection schemes and information that may be shown on the design plans. These studies can involve a considerable expenditure of both time and money, and thereby place that contractor at a disadvantage in respect to those bidders who are willing to rely on hasty, superficial studies, or — where the design engineer has shown an erection scheme — to simply assume that it has been engineered correctly and proceed to use it. The responsible contractor, on the other hand, will appraise the feasible construction methods and evaluate their costs and risks, and then make his selection. After the contract has been executed the contractor will set forth how he intends to fabricate and erect, in detailed plans that could involve a large number of calculation sheets and drawings along with construction procedure documents. It is appropriate for the design engineer on behalf of his client to review the contractor’s plans carefully, perform a check of construction considerations, and raise appropriate questions. Where the contractor does not agree with the designer’s comments the two parties get together for review and discussion, and in the end they concur on essential factors such as fabrication and erection procedures and sequences, the weight and positioning of erection equipment, the design of falsework and other temporary components, erection stressing and strengthening of the permanent steelwork, erection stability and bracing of critical components, any erection check measurements that may be needed, and span closing and swinging operations. The designing engineer’s approval is needed for certain fabrication plans, such as the cambering of individual members; however, in most cases the designer should stand clear of actual approval of the contractor’s construction plans since he is not in a position to accept construction responsibility, and too many things can happen during the field evolutions over which the designer has no control. It should be emphasized that even though the designing engineer has usually had no significant experience in steelwork construction, the contractor should welcome his comments and evaluate them carefully and respectfully. In major bridge projects many matters can get out of control or can be improved upon, and the contractor should take advantage of every opportunity to improve his prospects and performance. The experienced contractor will make sure that he works constructively with the designing engineer, standing well clear of antagonistic or confrontational posturing.
20.11
Two General Approaches to Fabrication and Erection of Bridge Steelwork
As has been stated previously, the objective in steel bridge construction is to fabricate and erect the structure so that it will have the geometry and stressing designated on the design plans, under full dead load at normal temperature. This geometry is known as the geometric outline. In the case of steel bridges there have been, over the decades, two general procedures for achieving this objective: 1. The “field adjustment” procedure — Carry out a continuing program of field surveys and measurements, and perform certain adjustments of selected steelwork components in the field as erection progresses, in an attempt to discover fabrication and erection deficiencies and compensate for them. 2. The “shop control” procedure — Place total reliance on firstorder surveying of span baselines and pier elevations, and on accurate steelwork fabrication and erection augmented by meticulous construction engineering; and proceed with erection without any field adjustments, on the basis that the resulting bridge deadload geometry and stressing will be as good as can possibly be achieved. Bridge designers have a strong tendency to overestimate the capability of field forces to accomplish accurate measurements and effective adjustments of the partially erected structure, and at the same time they tend to underestimate the positive effects of precise steel bridgework fabrication and 1999 by CRC Press LLC
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erection. As a result, we continue to find contract drawings for major steel bridges that call for field evolutions such as the following: 1. Continuous trusses and girders — At the designated stages, measure or “weigh” the reactions on each pier, compare them with calculated theoretical values, and add or remove bearingshoe shims to bring measured values into agreement with calculated values. 2. Arch bridges — With the arch ribs erected to midspan and only the short, closing “crown sections” not yet in place, measure thrust and moment at the crown, compare them with calculated theoretical values, and then adjust the shape of the closing sections to correct for errors in spanlength measurements and in bearingsurface angles at skewback supports, along with accumulated fabrication and erection errors. 3. Suspension bridges — Following erection of the first cable wire or strand across the spans from anchorage to anchorage, survey its sag in each span and adjust these sags to comport with calculated theoretical values. 4. Arch bridges and suspension bridges — Carry out a deckprofile survey along each side of the bridge under the steelloadonly condition, compare survey results with the theoretical profile, and shim the suspender sockets so as to render the bridge floorbeams level in the completed structure. 5. Cablestayed bridges — At each decksteelwork erection stage, adjust tensions in the newly erected cable stays so as to bring the surveyed deck profile and measured stay tensions into agreement with calculated theoretical data. There are two prime obstacles to the success of “field adjustment” procedures of whatever type: (1) field determination of the actual geometric and stress conditions of the partially erected structure and its components will not necessarily be definitive, and (2) calculation of the corresponding “proper” or “target” theoretical geometric and stress conditions will most likely prove to be less than authoritative.
20.12
Example of Arch Bridge Construction
In the case of the arch bridge closing sections referred to heretofore, experience on the construction of two major fixedarch bridges crossing the Niagara River gorge from the U.S. to Canada—the Rainbow and the LewistonQueenston arch bridges (see Figures 20.2 through 20.5)—has demonstrated the difficulty, and indeed the futility, of attempts to make fieldmeasured geometric and stress conditions agree with calculated theoretical values. The broad intent for both structures was to make such adjustments in the shape of the archrib closing sections at the crown (which were nominally about 1 ft [0.3 m] long) as would bring the archrib actual crown moments and thrusts into agreement with the calculated theoretical values, thereby correcting for errors in spanlength measurements, errors in bearingsurface angles at the skewback supports, and errors in fabrication and erection of the archrib sections. Following extensive theoretical investigations and onsite measurements the steelwork contractor found, in the case of each Niagara arch bridge, that there were large percentage differences between the fieldmeasured and the calculated theoretical values of archrib thrust, moment, and lineofthrust position, and that the measurements could not be interpreted so as to indicate what corrections to the theoretical closing crown sections, if any, should be made. Accordingly, the contractor concluded that the best solution in each case was to abandon any attempts at correction and simply install the theoreticalshape closing crown sections. In each case, the contractor’s recommendation was accepted by the designing engineer. Points to be noted in respect to these fieldclosure evolutions for the two longspan arch bridges 1999 by CRC Press LLC
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FIGURE 20.2: Erection of arch ribs, Rainbow Bridge, Niagara Falls, New York, 1941. Bridge span is 950 ft (290 m), with rise of 150 ft (46 m); box ribs are 3 × 12 ft (0.91 × 3.66 m). Tiebacks were attached starting at the end of the third tier and jumped forward as erection progressed (see Figure 20.3). Much permanent steelwork was used in tieback bents. Derricks on approaches load steelwork on material cars that travel up arch ribs. Travelers are shown erecting last fulllength archrib sections, leaving only the short, closing crown sections to be erected. Canada is at right, the U.S. at left. (Courtesy of Bethlehem Steel Corporation.)
are that accurate jackload closure measurements at the crown are difficult to obtain under field conditions; and calculation of corresponding theoretical crown thrusts and moments are likely to be questionable because of uncertainties in the dead loading, in the weights of erection equipment, and in the steelwork temperature. Therefore, attempts to adjust the shape of the closing crown sections so as to bring the actual stress condition of the arch ribs closer to the theoretical condition are not likely to be either practical or successful. It was concluded that for long, flexible arch ribs, the best construction philosophy and practice is (1) to achieve overall geometric control of the structure by performing all field survey work and steelwork fabrication and erection operations to a meticulous degree of accuracy, and then (2) to rely on that overall geometric control to produce a finished structure having the desired stressing and geometry. For the Rainbow arch bridge, these practical construction considerations were set forth definitively by the contractor in [2]. The contractor’s experience for the LewistonQueenston arch bridge was similar to that on Rainbow, and was reported — although in considerably less detail — in [10].
20.13
Which Construction Procedure Is To Be Preferred?
The contractor’s experience on the construction of the two longspan fixedarch bridges is set forth at length since it illustrates a key construction theorem that is broadly applicable to the fabrication 1999 by CRC Press LLC
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FIGURE 20.3: Rainbow Bridge, Niagara Falls, New York, showing successive arch tieback positions. Archrib erection geometry and stressing were controlled by means of measured tieback tensions in combination with surveyed archrib elevations.
FIGURE 20.4: LewistonQueenston arch bridge, near Niagara Falls, New York, 1962. The world’s longest fixedarch span, at 1000 ft (305 m); rise is 159 ft (48 m). Box archrib sections are typically about 3 × 131/2 ft (0.9 × 4.1 m) in crosssection and about 441/2 ft (13.6 m) long. Job was estimated using erection tiebacks (same as shown in Figure 20.3), but subsequent studies showed the long, sloping falsework bents to be more economical (even if less secure looking). Much permanent steelwork was used in the falsework bents. Derricks on approaches load steelwork onto material cars that travel up arch ribs. The 115toncapacity travelers are shown erecting the last fulllength archrib sections, leaving only the short, closing crown sections to be erected. Canada is at left, the U.S. at right. (Courtesy of Bethlehem Steel Corporation.) and erection of steel bridges of all types. This theorem holds that the contractor’s best procedure for achieving, in the completed structure, the deadload geometry and stressing stipulated on the design plans, is generally as follows: 1. Determine deadload stress data for the structure, at its geometric outline and under normal temperature, based on accurately calculated weights for all components. 2. Determine the cambered (i.e., “noload”) dimensions of each component. This involves determining the change of shape of each component from the deadload geometry, as its deadload stressing is removed and its temperature is changed from normal to the “shoptape” temperature. 3. Fabricate, with all due precision, each structural component to its proper noload dimensions — except for certain flexible components such as wire rope and strand members, which may require special treatment. 4. Accomplish shop assembly of members and “reaming assembled” of holes in joints, as needed. 5. Carry out comprehensive engineering studies of the structure under erection at each key erection stage, determining corresponding stress and geometric data, and prepare a stepbystep erection procedure plan, incorporating any check measurements that may be necessary or desirable. 6. During the erection program, bring all members and joints to the designated alignment prior to bolting or welding. 1999 by CRC Press LLC
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1999 by CRC Press LLC
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FIGURE 20.5: LewistonQueenston arch bridge, near Niagara Falls, New York. Crawler cranes erect steelwork for spans 1 and 6 and erect material derricks thereon. These derricks erect traveler derricks, which move forward and erect supporting falsework and spans 2, 5, and 4. Traveler derricks erect archrib sections 1 and 2 and supporting falsework at each skewback, then set up creeper derricks, which erect arches to midspan.
7. Enter and connect the final or closing structural components, following the closing procedure plan, without attempting any field measurements thereof or adjustments thereto. In summary, the key to construction success is to accomplish the field surveys of critical baselines and support elevations with all due precision, perform construction engineering studies comprehensively and shop fabrication accurately, and then carry the erection evolutions through in the field without any second guessing and illadvised attempts at measurement and adjustment. It may be noted that no special treatment is accorded to statically indeterminate members; they are fabricated and erected under the same governing considerations applicable to statically determinate members, as set forth above. It may be noted further that this general steel bridge construction philosophy does not rule out check measurements altogether, as erection goes forward; under certain special conditions, measurements of stressing and/or geometry at critical erection stages may be necessary or desirable in order to confirm structural integrity. However, before the erector calls for any such measurements he should make certain that they will prove to be practical and meaningful.
20.14
Example of Suspension Bridge Cable Construction
In order to illustrate the “shop control” construction philosophy further, its application to the main cables of the first Wm. Preston Lane, Jr., Memorial Bridge, crossing the Chesapeake Bay in Maryland, completed in 1952 (Figure 20.6), will be described. Suspension bridge cables constitute one of the most difficult bridge erection challenges. Up until “first Chesapeake” the cables of major suspension bridges had been adjusted to the correct position in each span by means of a sag survey of the firsterected cable wires or strands, using surveying instruments and target rods. However, on first Chesapeake, with its 1600ft (488m) main span, 661ft (201m) side spans, and 450ft (l37m) back spans, the steelwork contractor recommended abandoning the standard cablesag survey and adopting the “settingtomark” procedure for positioning the guide strands — a significant new concept in suspension bridge cable construction.
FIGURE 20.6: Suspension spans of first Chesapeake Bay Bridge, Maryland, 1952. Deck steelwork is under erection and is about 50% complete. A typical fourpanel throughtruss deck section, weighing about 100 tons, is being picked in west side span, and also in east side span in distance. Main span is 1600 ft (488 m) and side spans are 661 ft (201 m); towers are 324 ft (99 m) high. Cables are 14 in. (356 mm) in diameter and are made up of 61 helical bridge strands each (see Figure 20.8).
The steelwork contractor’s rationale for “setting to marks” was spelled out in a letter to the designing engineer (see Figure 20.7). (The complete letter is reproduced because it spells out significant 1999 by CRC Press LLC
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construction philosophies.) This innovation was accepted by the designing engineer. It should be noted that the contractor’s major argument was that setting to marks would lead to a more accurate cable placement than would the sag survey. The minor arguments, alluded to in the letter, were the resulting savings in preparatory office engineering work and in the field engineering effort, and most likely in construction time as well. Each cable consisted of 61 standard helicaltype bridge strands, as shown in Figure 20.8. To implement the settingtomark procedure each of three lowerlayer “guide strands” of each cable (i.e., strands 1, 2, and 3) was accurately measured in the manufacturing shop under the simulated fulldeadload tension, and circumferential marks were placed at the four centerofsaddle positions of each strand. Then, in the field, the guide strands (each about 3955 ft [1205 m] long) were erected and positioned according to the following procedure: 1. Place the three guide strands for each cable “on the mark” at each of the four saddles and set normal shims at each of the two anchorages. 2. Under conditions of uniform temperature and no wind, measure the sag differences among the three guide strands of each cable, at the center of each of the five spans. 3. Calculate the “centerofgravity” position for each guidestrand group in each span. 4. Adjust the sag of each strand to bring it to the centerofgravity position in each span. This position was considered to represent the correct theoretical guidestrand sag in each span. The maximum “spread” from the highest to the lowest strand at the span center, prior to adjustment, was found to be 13/4 in. (44 mm) in the main span, 31/2 in. (89 mm) in the side spans, and 33/4 in. (95 mm) in the back spans. Further, the maximum change of perpendicular sag needed to bring the guide strands to the centerofgravity position in each span was found to be 15/16 in. (24 mm) for the main span, 21/16 in. (52 mm) for the side spans, and 21/16 in. (52 mm) for the back spans. These small adjustments testify to the accuracy of strand fabrication and to the validity of the settingtomark strand adjustment procedure, which was declared to be a success by all parties concerned. It seems doubtful that such accuracy in cable positioning could have been achieved using the standard sagsurvey procedure. With the firstlayer strands in proper position in each cable, the strands in the second and subsequent layers were positioned to hang correctly in relation to the first layer, as is customary and proper for suspension bridge cable construction. This example provides good illustration that the construction engineering philosophy referred to as the shopcontrol procedure can be applied advantageously not only to typical rigidtype steel structures, such as continuous trusses and arches, but also to flexibletype structures, such as suspension bridges. There is, however, an important caveat: the steelwork contractor must be a firm of suitable caliber and experience.
20.15
Example of CableStayed Bridge Construction
In the case cablestayed bridges, the first of which were built in the 1950s, it appears that the governing construction engineering philosophy calls for field measurement and adjustment as the means for control of staycable and deckstructure geometry and stressing. For example, we have seen specifications calling for the completed bridge to meet the following geometric and stress requirements: 1. The deck elevation at midspan shall be within 12 in. (305 mm) of theoretical. 2. The deck profile at each cable attachment point shall be within 2 in. (50 mm) of a parabola passing through the actual (i.e., fieldmeasured) midspan point. 1999 by CRC Press LLC
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1999 by CRC Press LLC
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FIGURE 20.7: Setting cable guide strands to marks.
FIGURE 20.7: (Continued) Setting cable guide strands to marks.
1999 by CRC Press LLC
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FIGURE 20.8: Main cable of first Chesapeake Bay suspension bridge, Maryland. Each cable consists of 61 helicaltype bridge strands, 55 of 111/16 in. (43 mm) and 6 of 29/32 in. (23 mm) diameter. Strands 1, 2, and 3 were designated “guide strands” and were set to mark at each saddle and to normal shims at anchorages. 3. Cablestay tensions shall be within 5% of the “corrected theoretical” values. Such specification requirements introduce a number of problems of interpretation, field measurement, calculation, and field correction procedure, such as the following: 1. Interpretation: • The specifications are silent with respect to transverse elevation differentials. Therefore, two deckprofile control parabolas are presumably needed, one for each side of the bridge. 2. Field measurement of actual deck profile: • The temperature will be neither constant nor uniform throughout the structure during the survey work. • The survey procedure itself will introduce some inherent error. 3. Field measurement of cablestay tensions: • Hydraulic jacks, if used, are not likely to be accurate within 2%, perhaps even 5%; further, the exact point of “lift off ” will be uncertain. • Other procedures for measuring cable tension, such as vibration or strain gaging, do not appear to define tensions within about 5%. • All cable tensions cannot be measured simultaneously; an extended period will be needed, during which conditions will vary and introduce additional errors. 1999 by CRC Press LLC
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4. Calculation of “actual” bridge profile and cable tensions: • Fieldmeasured data must be transformed by calculation into “corrected actual” bridge profiles and cable tensions, at normal temperature and without erection loads. • Actual dead weights of structural components can differ by perhaps 2% from nominal weights, while temporary erection loads probably cannot be known within about 5%. • The actual temperature of structural components will be uncertain and not uniform. • The mathematical model itself will introduce additional error. 5. “Target condition” of bridge: • The “target condition” to be achieved by field adjustment will differ from the geometric condition, because of the absence of the deck wearing surface and other such components; it must therefore be calculated, introducing additional error. 6. Determining field corrections to be carried out by erector, to transform “corrected actual” bridge into “target condition” bridge: • The bridge structure is highly redundant, and changing any one cable tension will send geometric and cabletension changes throughout the structure. Thus, an iterative correction procedure will be needed. It seems likely that the total effect of all these practical factors could easily be sufficient to render ineffective the contractor’s attempts to fine tune the geometry and stressing of the aserected structure in order to bring it into agreement with the calculated bridge target condition. Further, there can be no assurance that the specifications requirements for the deckprofile geometry and cablestay tensions are even compatible; it seems likely that either the deck geometry or the cable tensions may be achieved, but not both. Specifications clauses of the type cited seem clearly to constitute unwarranted and unnecessary fieldadjustment requirements. Such clauses are typically set forth by bridge designers who have great confidence in computergenerated calculations, but do not have a sufficient background in and understanding of the practical factors associated with steel bridge construction. Experience has shown that field procedures for major bridges developed unilaterally by design engineers should be reviewed carefully to determine whether they are pr