nazmy a.s. three dimensional nonlinear static analysis of cable stayed bridges 1990

Compurers & Smcrures Vol. 34, No. 2. pp. 257-271, 1990 Printed in Great Britain.

co45-7949/90 s3.M) + 0.00 0 1990 Pergamon Press plc

THREE-DIMENSIONAL NONLINEAR STATIC ANALYSIS OF CABLE-STAYED BRIDGES

ALY S. NAZMYt and AHMED M. ABDEL-GHAFFAR~ TCivil Engineering Department, Polytechnic University, 333 Jay Street, Brooklyn, NY 11201, U.S.A.

ICivil Engineering Department, University of Southern California, Los Angeles, CA 900894242, U.S.A.

(Received 6 February 1989)

Abstract-The nonlinear static analysis of three-dimensional long-span cable-stayed bridges under the effect of their own dead weight and a set of initial cable tensions is formulated. All sources of geometric nonlinearity, such as cable sag, axial force&ending moment interaction in the bridge deck and towers, and change of the bridge geometry due to large displacements, are considered in the analysis. A computer program that uses a tangent stiffness iterative-incremental procedure is described to perform the nonlinear static analysis. Problems in modeling this special type of structure are addressed. Examples of three-dimensional mathematical models representing the present and future trends in cable-stayed bridge construction are presented. The results show that these flexible structures are highly nonlinear under dead loads, especially for very long span bridges. The analysis is essential to start linear or nonlinear dynamic analyses from the dead load deformed state utilizing the tangent stiffness matrix of the bridge under gravity load conditions.

INTRODUCTION

Long-span contemporary cable-stayed bridges [covering the range of 500 ft (152.5 m) to 2000 ft (610 m)], are very appealing aesthetically and also are very important lifeline structures. These bridges, which were introduced in the mid 1950s and have been developed over the last three decades, are becoming very popular in the U.S., Japan, Europe and the Third-World countries. It is estimated that the total, world-wide number of steel and concrete cable-stayed bridges that will be completed by the end of 1990 is about 90 [l]: this estimate counts only for those having center (or effective) span lengths of 800 ft (245 cm) or longer. In the U.S., eight such bridges have been constructed, seven are under construction, and about eight others are still in the design and/or consideration process.

The increasing popularity of these contemporary bridges among bridge engineers can be attributed to:

1. The appealing aesthetics. 2. The full and efficient utilization of structural

materials. 3. The increased stiffness over suspension bridges. 4. The efficient and fast mode of construction. 5. The relatively small size of the substructure.

Rapid progress has been made over the past 20 years in the design techniques of cable-stayed bridges; this progress is largely due to the use of electronic com- puters, the development of box-girders with ortho- tropic plate decks, and the manufacturing of high strength wires that can be used for cables. This progress has also led to increased competition among the bridge engineers in Japan, Europe, and the United States.

Cable-stayed bridges, in which the deck is elastically supported at points along its length by inclined cable stays, are now entering a new era, reaching to medium and long span lengths [with a range of 1300 ft (400 m) to 5000 ft (1500 m) of center span]. In Japan, there are plans for constructing even longer cable-stayed bridges [l] of both prestressed concrete and steel. The increase in span length combined with the trend to more shallow or slender stiffening girders in cable-stayed bridges has raised concern about their behavior under both service and environmental dynamic loadings, such as traffiic, wind and earthquake loadings. Because of the fact that these long- span, cable-supported structures constitute complex structural with mainly geometric nonlinearity it is essential to understand and realistically predict their structural response to these loadings. Accordingly, it is highly desirable in bridge engineering to develop and validate accurate procedures that can lead to a thorough understanding of the static, dynamic, seismic and wind problems of cable-stayed bridges.

Geometrically nonlinear static analysis of cable- stayed bridges, in which large displacements occur under design or service loads while the strains in the structural elements remain small, is essential to start linear or nonlinear dynamic analyses from the dead- load deformed state utilizing the tangent stiffness matrix of the bridge under gravity load conditions. Furthermore, the need for nonlinear analysis is essential not only for evaluating stresses and deformations induced by gravitational loads, but also for assuring safety during construction.

Although several investigators studied the two- dimensional nonlinear static behavior of cable-stayed bridges [2-6] very few, if any, tackled the three-

251

258 ALY S. NAZMV and AHMED M. ABDEL-GHAFFAR

dimensional problem. Now with the increase in the span and the utilization of complex geometry for the towers and accordingly for the two planes of the cables, it is imperative to utilize three-dimensional nonlinear analysis. In this paper, the stiffness method is utilized to perform nonlinear static analysis of three-dimensional cable-stayed bridges. The stiffness method was particularly used since it can handle various types of structural elements (such as cables, and beam-column elements) and it can conveniently be adapted for computer use. Numerical examples were presented to illustrate the behavior of recent and future trends in cable-stayed bridge construction.

SOURCES OF NONLINEARITY IN CABLESTAYED BRIDGES

In spite of the fact that the behavior of the material of the structural elements in a cable-stayed bridge is linear elastic, the overall load-displacement relationship for the structure is nonlinear under normal design loads [3,7]. This overall nonlinear behavior originates in three primary sources, namely: (1) the nonlinear axial forceelongation relationship for the inclined cable stays due to the sag caused by their own weight; (2) the nonlinear axial force- and bending moment-deformation relationships for the towers and longitudinal girder elements under combined bending and axial forces; and (3) the geometry change caused by large displacements in this type of structure under normal as well as environmental design loads.

Nonlinear behavior of cables

When a cable is suspended from its ends and subjected to its own weight and an externally applied axial tensile force, it sags into the shape of catenary. The nonlinear behavior of the individual cables in a cable-stayed bridge results from this sag phenom- enon. The axial stiffness of the cable varies non- linearly as a function of end displacements, since part of the end movement occurs due to material deformation and another part occurs due to change in sag. As the axial tension increases in the cable stay, this latter part, i.e. the change in cable sag, becomes smaller and smaller, and the end movement occurs mainly due to material deformation. Accordingly, the apparent axial stiffness of the cable increases as its tensile stresses increase.

Nonlinear behavior of bending members

When assuming small deformations in any structural system, the axial and flexural stiffnesses of bending members are usually considered to be uncoupled. However, when deformations are no longer small, there is an interaction between axial and flexural deformations in such members, under the combined effect of axial force and bending moment. The additional bending moment developed in a laterally deflected, or bent, member when subjected to a simultaneously applied axial force either increases

or reduces the original bending moment in the member. The result of this axial force-bending deformation interaction is that the effective bending stiffness of the member decreases for a compressive axial force and increases for a tensile force. In a similar manner, the presence of bending moments will affect the axial stiffness of the member due to an apparent shortening of the member caused by the bending deformations. In most conventional linear structures, this interaction or coupling effect is negligible. However, due to the large deformations that may occur in a cable-stayed bridge, as a flexible structure, this interaction can be significant and should be considered in any nonlinear analysis.

Geometry change due to large displacement

In linear structural analysis, it is assumed that the joint displacements of the structure under the applied loads are negligible with respect to the original joint coordinates. Thus, the geometric changes in the structure can be ignored and the overall stiffness of the structure in the deformed shape can be assumed to equal the stiffness of the undeformed structure. However, in cable-stayed bridges, displacements of several feet can occur under normal design loads, and accordingly, signficant changes in the bridge geometry can occur. In such a case, the stiffness of the bridge in the deformed shape should be computed from the new geometry of the structure.

THE NONLINEAR ANALYSIS TECHNIQUES

For a linear structural system, the static displacements can be easily computed by solving the set of linear simultaneous stiffness equations,

[Kl{DI = {PI (1)

in which [K] is the global stiffness matrix of the structure, {D} is the vector of joint displacements, and {P} is the vector of applied joint loads. The global stiffness matrix [K] can be constructed from the stiffness matrices of the individual members of the structure by the genera1 assembly procedure [8]. The terms in [K] are constants which do not change as the linear structure deforms.

For a nonlinear structural system, the stiffness changes as the structure deforms; this complicates the analysis of such structures to some extent. The stiffness matrix [K] in eqn (1) is a function, in this case, of the joint displacements {D), which are as yet unknown. It is not straightforward to solve the set of nonlinear stiffness equations. Therefore, numerical solution techniques are usually used for solving such nonlinear equations for the displacement vector {D}.

Available procedures for nonlinear st@ness analysis of cable-stayed bridges

There are several techniques for solving the nonlinear equation (eqn l), when the stiffness [K] varies

Three-dimensional nonlinear static analysis of cable-stayed bridges

rrrpontr

Fig. 1. Incremental procedure for cabled-stayed bridges. Fig. 3. Mixed Procedure I for cable-stayed bridges.

as a function of nodal displacements and member forces. These techniques are discussed in detail in the literature(!S131. These basic techniques can be classified into: (1) incremental or stepwise procedure, (2) the iterative or Newton procedure, and (3) mixed procedure.

In the incremental procedure, the load is applied in several small steps, and the structure is assumed to respond linearly within each step with its stiffness recomputed based on the structural geometry and member end actions at the end of the previous load step. This is a simple procedure which requires no iterations, but errors are likely to accumulate after several steps unless very fine steps are used (Fig. 1).

In the iterative procedure, the stiffness matrix of the structure is either reformulated at the beginning of each iteration cycle (Newton-Raphson iteration) or formed and decomposed only once (modified Newton-Raphson iteration) (Fig. 2). The advantage of the latter technique is the saving of computational time; however, convergence is slower than in Newton-Raphson iteration. Furthermore, in cable- stayed bridges, modified Newton-Raphson may never converge in some cases, as can be seen in Fig. 2(c), particularly when the nonlinearity is strong, and the load increment is relatively large.

In the mixed procedure, a combination of the incremental and iterative schemes is utiiized. The load is applied incrementally, and iterations are performed either after each load increment as shown in Fig. 3 (Mixed Procedure I), or after all the load increments as shown in Fig. 4 (Mixed Procedure II). Either Newton-Raphson or modified Newton-Raphson

Out-of-bolonca (ore* connot be

may be used for the iterations. This procedure yields high accuracy at the price of more computational effort. However, it may be necessary to use this procedure instead of the Newton-Raphson iterative procedure in the analysis of cable-stayed bridges if the out-of-balance force in the first cycle is very large, as shown in Figs 3 and 4.

To simplify the nonlinear static analysis the following assumptions and approximations were made: (1) All stresses in the bridge components remain within the elastic limit. (2) All cables are fixed to the tower and to the bridge at their points of attachment. (3) All cable stays are of uniform section and are assumed to be straight chord links between the towers and the bridge girder. An equivalent axial stiffness is assigned to these links to compensate for the sag effect. (4) The

Fig. 4. Mixed Procedure II for cable-stayed bridges.

Fig. 2. Iterative procedure for cable-stayed bridges. (a) Newton-Raphson iterations. (b) Modified Newton-Raphson iterations (converging case). (c) Modified Newton-Raphson iterations (diverging case).

260 ALY S. NAZMY and AHMED M. ABDEL-GHAFFAR

Unbalanced load

2W/3

Dfinal

Fig. 5. Mixed procedure used for the nonlinear static analysis.

cables are assumed to be perfectly flexible; that is, the flexural stiffness of the cables can be neglected.

The Mixed Procedure II mentioned previously is used in the present investigation for the nonlinear static analysis of cable-stayed bridges. It was found that this technique results in faster convergence. A graphical representation of this technique using three equal load increments in the first cycle is shown in Fig. 5.

The first load increment is applied using the tangent stiffness matrix of the undeformed structure [K,,], and the joint displacements {O\‘)} are then computed using eqn (1) with {P} = f{ W}. The tangent stiffness matrix [K,(D\‘))] that corresponds to the displaced shape of the structure is then computed, and used to compute the incremental joint displacements due to the second load increment. These incremental displacements are then added to the previously computed joint displacements {O\‘)} to obtain {O$‘)}, which correspond to point B in the Figure. The process is repeated for the last load increment until point C is reached, which represents the end of the first cycle. The computed displacements at the end of the cycle {D\‘)} (or simply {D”)}) actually correspond to loads on the true load- displacement curve, namely point E. The unbalanced loads at the end of the first cycle, {IV(‘)}, represented by line CE, are then computed and applied as a new set of joint loads during the second cycle of iteration, using the tangent stiffness [Kr(D(‘))], represented by line EF. The incremental displacements are then computed from

[K,(D”‘)]{AD’“} = {W”‘}, (2)

where (Wcn} is the out-of-balance force vector computed from:

{IV’“} = {W} - [K,(D’“)]{D’“}, (3)

in which [K,(D(‘)] is the secant stiffness matrix of the structure when the joint displacements are {DC’]}. The new displacements are then computed from:

{D(i+l)} = {D’“} + {AD’“}. (4)

The iteration process continues and the unbalanced loads at the end of each iteration cycle are computed. If the unbalanced loads, measured in some vector norm, where the maximum norm is usually used, are not less than a certain acceptable tolerance, another iteration cycle is required. This process continues until convergence to the correct displacements, represented by point H, occurs.

In eqns (2) and (3) the global tangent stiffness or secant stiffness of the structure is obtained by the standard assembly procedure [8], from the individual element stiffness matrices. The formulation of both tangent and secant stiffness matrices for cable, as well as bending, elements will be described later in this paper, where the element stiffness matrix [k,,,] will be given in the member local coordinates. A transformation of the element stiffness matrix from local to global coordinates is performed, before assembling the global stiffness matrix, by using the standard transformation formula [8-l 11:

PI = [~,lr[kJ~ml~ (5)

in which [k,] is the member stiffness matrix in global coordinates and [r,,,] is its rotation matrix given by:

[PI 0 0 0 1 0 0

km1 0

[r] = 0 0 [r] 0 ’ (6)

10 0 0 Hj

where [r] is a submatrix of order three, which gives the direction cosines of the member local axes with respect to the global axes [7]. Due to large displacements that occur in cable-stayed bridges under normal design loads, the joint coordinates used in computing these direction cosines are based on the new geometry of the bridge in its deformed state after applying the external loads.

The algorithm for the above described procedure is given later in this paper and a flow chart of the computer program is also listed. This algorithm is efficient for performing a nonlinear static analysis of three-dimensional cable-stayed bridges subjected to their own dead loads and a set of initial cable tensions. These initial cable tensions are specified such that the deformations in the bridge deck are minimized under dead loads. Several techniques for determining the initial tensions in the cables of a cable-stayed bridge are described in the literature [6, 14, 151. The adjustment of cable-length during erection to provide the predetermined initial stress in cables is discussed in Ref. [16].

NONLINEAR STIFFNESS FORMULATION OF CABLE ELEMENTS

Equivalent cable modulus of elasticity

As mentioned earlier in this paper the apparent axial stiffness of the cable stay in a cable-stayed

Three-dimensional nonlinear static analysis of cable-stayed bridges 261

CABLE _ STAYED BRIDGE

Fig. 6. Inclined cable member in a cable-stayed bridge.

bridge is affected by the cable sag which is greatly influenced by the amount of tension in the cable. When the cable tension increases, the sag decreases, and the apparent axial stiffness of the inclined cable increases. A convenient method to account for this variation in the cable axial stiffness is to consider an equivalent straight chord member, as shown in Fig. 6, with an equivalent modulus of elasticity which com- bines both the effects of material and geometric deformations such that the axial stiffness of the equivalent chord member becomes equal to the apparent axial stiffness of the actual curved cable. This concept was first introduced by Ernst [17], and has been verified by several additional investigators [ 1 g-221. This equivalent cable modulus of elasticity is given by

E Eeq = 1+ (wL)2AE ’ [ 1

(7)

12T3

in which Eq = equivalent modulus: E = cable material effective modulus; L = horizontal projected length of the cable; w = weight per unit length of the cable; A = cross-sectional area of the cable; and T = cable tension.

Equation (7) gives the tangential (or instan- taneous) value of the equivalent modulus when the tension in the cable equals T (or when the cable tensile stress equals Q as shown in Fig. 7). If the

ar t // ____________________--

b z W

E orctq recant Ecq

Q qr Qi __--- --------

orctq tonqential Ecq.

w STRAIN E

Fig. 7. Nonlinear stress-strain relationship for a cable stay.

’ /* Lc ,/ 4 /’

/’ /

/’ /’ y x

I/’ Y 2

Fig. 8. Degrees of freedom of a cable element in local coordinates.

tension in the cable changes from T, to T, during the application of a certain load increment (which is equivalent to a change in cable stress from ui to a/ as shown in Fig. 7), then the secant value of the equivalent modulus of elasticity over the load increment is given by:

Secant stlfness matrix of cable element

The secant stiffness matrix of a cable stay of chord length L, and cross-sectional area A, during a tension change in the cable from T, to T,, is simply equal to the elastic stiffness matrix of a truss element of length L, and cross-sectional area A, and of elastic modulus equal to that given by eqn (8). Thus, the elastic stiffness matrix in local coordinates for the cable element shown in Fig. 8 is given by

(9)

in which c stands for cable.

Tangent st@ness matrix of cable element

The tangent stiffness of a cable stay of chord length LC and cross-sectional area A, when it is subjected to an axial tension T, is equal to that of a truss element of length L, and cross-sectional area A, and of elastic modulus equal to that given by eqn (7), and subjected to axial tension T. There are different approaches for computing the tangent stiffness matrix of that truss element using the large deflection theory [lo, 23,241. All of these approaches lead to the same result, which is simply given by

WTIC = kElc + kYlc 1 (10)

where [k,], is the element tangent stiffness matrix in its local coordinates, [kEIC is the elastic stiffness matrix, as given by eqn (9), with Eq obtained from eqn (7), and [k,], is the geometric stiffness matrix of the truss element, and is given by:

(11)

262 ALY S. NAZMY and A~t.mo M. ABDEL-G-AR

Fig. 9. Degrees of freedom of a beam-column element in local coordinates.

in which the submatrix [G], is given by

NONLINEAR STIFFNESS FORMULATION OF BRIDGE GIRDER ELEMENTS

(12)

TOWER AND

As mentioned earlier the large deformations that occur in the tower and girder elements of a cable- stayed bridge under the combined effect of large bending moments and high axial forces produce a strong coupling between axial and Aexural stiffness in these members. This coupling can be considered in the refined nonlinear analysis by introducing the concept of stability function [8]. These functions are multiplication factors used to modify both the bending and axial stiffnesses of the member. The deri- vations of these stability functions are given in detail in Ref. [8] for the two-dimensjonal beam element. The extension to the three-dimensional beam-column element shown in Fig. 9, with 6 degrees of freedom at each end, is straightforward and the resulting 12 x 12 element stiffness matrix, in the local coordinate system, is given by:

k(2,6) = k(6,2) = k(2, 12) = k(12,2)

= -k(6,8) = -k(8,6) = -k(8, 12)

= -k(l2,8) = (6~Z~/~z)~2~ (14d)

k(3,5)=k(5.,3)=k(3, ll)=k(ll,3)= -k(5,9)

= -k(9,5)= -k(9, ll)= -k(ll,9)

= (-6Ez~~~*~~z.~,

k(4,4) = k(10, 10) = -k(4, 10)

(14e)

= -k(10,4)=GJ,/L,

k&5) = k(l1, 11) = (4EZJL)S3,,

k(6,6) = k(12, 12) = (4EI,/L)S3,,

(14f)

(14g)

U4h)

k(5, ll)= k(ll,5)=(2EI,/L)S4,, (14i)

k(6, 12) = k(I2,6) = (2EZJL)S4,, (14j)

where E is the member material modulus of elasticity, A is the cross-sectional area, L is the member length, Z, and I, are the moments of inertia of the cross- section about the local principal y and z axes, respec- tively, Z, is the torsional moment of inertia of the cross-section, G is the member material shear modulus, and the S’s are the stability functions. S1,y through S4, modify the bending stiffness of the member about the local y axis, while S 1, through S4, modify the bending stiffness about the local z axis, and S5 modifies the axial stiffness. If the axial force in the bending member is zero, all these S’s take the value one. The stability functions can be expressed in terms of the member axial force P, and the member end moments Ml and M2, at both ends about the member local y and z axes, as defined in Fig. 10.

in which

k(l,l) k(l,2) . . . . . . k(l,l2) k(2, 1) k(2,2) . . . . . k(2, 12)

[k&,= . . . . . . .q. . . . . . .

I

, (13)

k&l) k(ii;2) ::: ::: “* k( 12,12)

@Y k(1, 1) = k(7,7) = -k(l, 7) = -k(7, 1)

= (E.4 /L)S5,

k(2,2) = k(8,8) = -k(2,8) = -k(8,2)

= (12EZJL3)Sl,

(144

(14b)

k(3,3) = k(9,9) = -k(3,9) = -k(9,3)

= (12EZJL4S1,

p- 1 M’z

(14~) Fig. 10. Axial forces and end moments for a beam element.


For a tension member (P is positive), the stability functions Sl: through S4, are

Slz=cz3sinhw/12R,,

S2, = w2(cosh w - 1)/6R,,

S3, = w(w cash o - sinh w)/4R,,

S4, = w(sinh w - ~)~2R~,

where

and

w =I& and p2= P}Eiz,

R,=2-2coshwfosinhw.

(WI

tl5b)

(15c)

Wd)

(16)

(17)

While for a compression member (P is negative),

S1: = o3 sinh w/l2R,,

S2, = w=( 1 - cash o)/6R,,

S3, = w(sin w - 0 cos w)/4R,,

(lW

(l8b)

(18c)

S4, = w(w - sin w)/2R,, (lgd)

where

w = pL and p2 = P/E&, (19)

and

R,=2-2cosw -wsinw. (20)

The stability functions S 1, through S4, can be determined in the same way by replacing Zz by f,. in eqns (15) through (20).

The stability function S5 can be obtained as follows:

For a tension member (P is positive),

S5 = l/[l - EA(R,,,,J + R,)/4P3L21, (21)

where

R ,m~, =w,(MI~+ M2_~)(cothw,+w,cosecb’w,)

- 2(Ml,, f M2,$ + (Ml.,M5)

x (1 + O+ coth wY)(2wY cosech wJ), (22)

in which

(23)

CAS ,412-F

and

R,; = MMl; i- M2f)(coth w, + co1 cosech= w,)

- 2(Ml, + M2J2 + (Ml$42,)

x (1 + w: coth w,)(2wz cosech wz), (24)

in which

w,= p3L and ~3 = PIEI,. (25)

While for a compression member (P is negative),

S5 = l/[l - EA(R, + R,)/4P3LZf, (26)

where

R,,,,, = w,(Mlf + M2;)(cot w_? + wy cosec’ wy)

- 2(A4 I? + M5.)2 + (M l,“M2y)

x (1 + wY cot 0~~) (20, cosec w,), (27)

my=pyL and p:= PIEI,, (28)

and

R,,, = w,(M 12 + MZS)(cot w, + o,cosec2 0,)

- 2(Ml, + M2,)2 + (~1?~2~)

x (1 + 0, cot w,)(2w, cosec oz),

in which

(29)

(30)

The stiffness matrix of a beam-column element as given by eqn (13) is in fact the secant stiffness matrix, The tangent stiffness matrix of such an element can be obtained using the large deflection theory and nonlinear strain-displacement relationships 123-271 to give:

[Ml = Mb + [&lb? (31)

where [kflb is the beam element tangent stiffness matrix in its local coordinates, [k& is the elastic stiffness matrix of the beam element as given by eqn (13), and [kF],, is the geometric stiffness matrix of the beam element. Reference [23] gives this geometric stiffness matrix for the 2-D beam element. It can be extended to the 3-D beam element to give [28]


M7=;

'0

0 6

J

0 0 6

5

00 0 0

0 0 -$ 0 ;L2

0; 0 0 0

00 0 0 0

o-g 0 0 0

0 0 -; 0 L

3

00 0 0 0

0 0 _$ 0 _g

0; 0 0 0

$L2

0 0

L -- 10

0

0 0

0 0

0 0

L2 0 -- 30

Symmetric

6

‘J

6 0

s

0 0 0

0 L

i5 0 $L2

L --

10 0 00 hL2

(32)

in which P is the axial force in the beam, and L is its length.

ALGORITHM FOR THE NONLINEAR ANALYSIS

A computer program was developed to perform the nonlinear static analysis of any three-dimensional cable-stayed bridge subjected to its own weight and a set of user specified initial cable tensions. The Mixed Procedure II described in this paper was utilized in this implementation. The program can also perform a linear or nonlinear live load analysis that

starts from the dead load deformed state of the bridge. The algorithm for this method can he summa- rized in what follows:

Implementation (see Fig. 11)

1.

2.

Read structure data and initial cable tensions. Assume tolerance To1 < 0.01 and maximum number of iterations nitem 2 10. Initialize member stability functions; set all values equal to 1.0.

d”= Unbalanced load at the beginning of second cycle

,(I).,(1) D-

Unbalanced load

of first itwatii

Fig. 11, The nonlinear analysis technique used in the algorithm.

3.

4.

5.

6,

7.

8.

ark-dimensio~l nonlinear static analysis of cable-stayed bridges 265

Find initial cable equivalent modulus of elasticity iteration cycle, by the standard assembly from initial cable tensions procedure,

Eeg = E/[l + [(wL)‘AE/12T3]].

Compute initial member geometry, and dead load member forces {fm). WI) Check convergence: If /I W’+ ’ 1) m -c Tol,

Compute the initial matrix using lumped masses where llkllrn =max 1 <j I N Ixjl, conver-

and save it on a disk file. gence occurred, then iDfinal> = ‘D’“J and

Compute member end forces in global co- GG TO 9. If no convergence, and

ordinates (fl = [mtlr’(fn 1. i < nitem, GO TO (I). Otherwise, restart

Compute initial unbalanced joint loads by the with a large value of nitem.

standard assembly procedure 9. Compute reactions.

I II? = {P,,t. 1 - 2, L% 10. Print on files the final joint displacements, cable

tensions, Eeq, member end actions, and reactions. A flow chart of the above-described procedure is

N = no. of members illustrated in Fig. 12.

{P,,,} z externally applied joint loads = 101 in 3-D MODELING OF CABLFdTAYED BRIDGES

this case. Initialize (L)(“) = (Oj, i = 0, then start iterations.

Two three-dimensional (3-D) models were used;

i=i+ 1. the first one has a center-span length of 1100 ft, and

Divide the unbalanced load into the two side spans of 480 ft each. This model represents

required number of load increments. Each the present trend in the design of cable-stayed

increment is given by {AP’ZT) = l/ne{ I@‘)), bridges. The center and side spans of the second

n/ = no. of load increments in each cycle. model have double the lengths of the first one (i.e. the

For each load step (n), initialize {O#} = second model has a center span of 2200 ft and side

{LW-“1, n = 0. spans of 960 ft). This second model represents the

(4 For each element, form its tangent stiff- future trend in cable-stayed bridge design.

ness matrix in local coordinates The general configuration of the tower and cable

arrangement, the nodal points and deck elements of

hl = M + M. the two 3-D models are shown in Fig. 13 while Tables I and 2 list the member properties of these two

(W Compute element stiffness matrix in mathematical models. The choice of the structural

global coordinates [k] = ~~]~~r][r]. properties of the elements in these two analytical

(4 Assume the global tangent stiffness models was based on examining several proposed

matrix cable-stayed bridges in the eastern region of the United States [29-331 and in Japan [l, 34, 361.

The final choice of the models considered in this study is based on the following practical consider-

(4 Solve~A~(~~ = [~r](AlzC>~get(Ao~~~~. ations:

(4 Update displacements, (D$+ 1) > = The A-shape tower was adopted in the mathe-

I%] + (A%}. matical modeling to reduce the deformations that

(f) Update joint coordinates, and recom- may occur in the bridge deck and towers especially pute memory geometry at end of step. under unsymmetrical loading, and to increase

ad Compute member secant stiffness [km], the overall stability of the system. The A-shape, localize member end displa~ments which has a very high torsional rigidity, is the (d,) and compute member end forces most desirable one, so far, for the majority of

(fm1 = IkJ~~mI* long-span cable-stayed bridges (both those under

(h) Modify member stability functions. construction and under consideration).

(9 If last load increment in the cycle The main girders were assumed to be simply (ZY1} = {D$+,)]; GO TO (IV); other- supported and not to be fixed at the end piers or wise, n = n + 1, and GO TO (a). abutments in order to reduce the redundancy of

(IV) Compute member forces in global co- the structural system, thus reducing the sectional ordinates forces induced by the environmental loads.

3. The towers were assumed to be rigidly fixed to the piers to reduce the overall structural deformations.

4. The bridge deck was assumed to be elastically (V) Compute unbalanced forces at end of connected to the towers rather than being fixed to

266 ALY S. NAZMY and AIMED M. AEIDEL-GHAFFAR

yes

Read structure data (joint coordinates. member properties end own weight,

cable initial tension. support conditions)

Initialize member stability functions and cable equivalent modulus from initial cable tensions

1 Compute initial member geometry. initial D.L. member forces,

from which compute initial unbalanced joint loads

Form tangent stiffness matrix and solve for incremental joint displacement using either one load step or

several equal load steps with changing tangent stiffness matrix in beginning of each load step

! 1

Add incremental joint displacement to previous joint displacement end to previous joint coordinates

and recompute member geometry

Compute new cable tension and equivalent modulus at end of iteration cycle I

Compute member stress resultants and stability functions at end of cycle

I c

(Compute unbalanced joint loads at end of cycle

1 no Write a message for

nonconvergence Compute reactions

1 Print Results: (i) joint displacements.

(ii) cable tension and equivalent modulus. (iii) member end actions, (iv) reactions

(Plot deformed shape of the bridge under D.L. 1

Fig. 12. Flow chart for nonlinear static analysis.

them. In fact, the response of a cable-stayed bridge to applied loads is highly dependent on the manner in which the bridge deck is connected to the towers. If the deck is swinging freely at the towers, the induced member forces due to environmental loads, in particular the wind and earthquake loads, will be kept to minimum values, but 5. the bridge may be very flexible under service loading conditions (e.g. dead loads and live loads). On the other hand, a rigid connection between the deck and the towers will result in reduced move- ments under service loads but will attract much higher wind or seismic forces. Therefore, it is very important to provide special devices at the deck- tower connections to control the bridge’s natural period of vibration [l, 29-3 1,34-381 in order to

reduce the sizes of the towers and foundations. This was achieved by assuming that the bridge girder is connected to the tower by vertical and horizontal links (Fig. 13) whose elastic constants can be specified to reflect a wide band of practical construction options. A horizontal strut near the top of the tower was added, as shown in Fig. 13, to improve the tower behavior under static as well as dynamic loads. It was noticed that the towers without top struts, in both analytical models, experienced large bending deformations under vertical static loads due to the high compressive forces that come from the cables (Fig. 14). From a dynamic analysis conducted by the authors on these models [29,37], large distor- sion of the natural mode shapes of vibration was

Table 1. The three-dimensional cable-stayed bridge model I

A (ft2) I, (ft4) I, (ft4) 1, (ft4) E (ksf) G (ks9 Wt (kip/ft)

Girder (steel) 5.00 10.00 2500.00 15.00 4,176,OOO.OO 1,670,400.00 6.00 (6.0 for (75.0 for

central part) central part) Cross beams (steel) 1.50 1.50 600.00 6.00 4,176,OOO.OO 1,670,400.00 1.50

(3.5 at (10.0 at towers) towers)

Tower above deck level 70.00 2000.00 2000.00 1000.06 580,400.OO 232,160.OO 10.50 (R.C.) below deck level 100.00 7500.00 5000.00 5000.00 580,400.OO 232,160.OO 15.00

Tower two struts upper 50.00 150.00 900.00 150.00 580,400.OO 232,160.OO 7.50 struts deck level strut 60.00 200.00 1000.00 200.00 580,400.OO 232,160.OO 9.00 (R.C.)

Tower deck horizontal links links vertical links (steel)

3.00 0.001 0.001 0.001 4,176,OOO.OO 1,670,400.00 1.50 3.00 0.001 6.00 6.00 4,176,OOO.OO 1,670,400.00 1.50

Cable no. A (ft2) Initial tension (kips) Wt (kip/ft) E(ksf)

1,24,25,48 0.195 1950 0.1365 4,176,OOO.OO 2,11,14,23, 26,35,38,47 0.125 1250 0.0875 4,176,OOO.OO 3,10,15,22, 27,34,39,46 0.110 1100 0.0770 4,176,000.00

Cables 4,9,16,21, 28,33,40,45 0.095 950 0.0665 4,176,OOO.OO 5,8,17,20,

29,32,41,44 0.075 750 0.0525 4,176,OOO.OO 6,7,18,19,

30,3 1,42,43 0.061 610 0.0427 4,176,OOO.OO 12,13,36,37 0.200 2000 0.1400 4,176,000.00

Table 2. The three-dimensional cable-stayed bridge model II

A (ft2) 1, (ft4) I, (fP) 1, (ft? E (ksf) G (ksf) Wt (kip/ft)

Girder (steel) 7.50 12.00 12,500.OO 75.00 4,176,OOO.OO 1,670,400.00 7.25 (9.0 for (375.0 for

central part) central part)

Cross beams (steel) 1.50 1.50 600.00 6.00 4,176,OOO.OO 1,670,400.00 1.50 (3.0 at (10.0 at towers) towers)

above deck 140.00 4000.00 10,000.00 5000.00 580400.00 232,160.OO 21.00 Towers level (R.C.) below deck 200.00 15,000.00 25,OOO.OO 25,OOO.OO 580,400.OO 232,160.OO 30.00

level

Tower two upper struts struts (R.C.) deck level

strut

70.00 150.00 900.00 150.00 580,400.OO 232,160.OO 10.50

80.00 200.00 1000.00 200.00 580,400.OO 232,160.OO 12.00

Tower deck horizontal links 3.00 0.001 0.001 0.001 4,176,OOO.OO 1,670,400.00 1.50 links vertical (steel) links 3.00 0.001 6.00 6.00 4,176,OOO.OO 1,670,400.00 1.50

Cables

Cable no.

1,24,25,48 2,11,14,23, 26,35,38,47 3,10,15,22, 27,34,39,46

4,9,16,21 28,33,40,45 5,8,17,20,

29,32,41&t 6,7,18,19,

30,3 1,42,43 12,13,36,37

A (ft2) Initial tension (kips) Wt (kip/ft) E(ksf)

0.430 4300 0.3010 4,176,OOO.OO

0.270 2700 0.1890 4,176,OOO.OO

0.245 2450 0.1715 4,176,OOO.OO

0.210 2100 0.1470 4,176,OOO.OO

0.174 1740 0.1218 4,176,OOO.OO

0.133 1330 0.093 1 4,176,OOO.OO 0.455 4550 0.3185 4,176,OOO.OO

267

268 ALYS. NAZMY and AHMEDM.ABDEL-GHAFFAR

3-D CABLE -STAYED BRIDGE

3-D FINITE ELEMENT MODEL

ELEVATION m

t=

480 ft I* 1100 It 480 It

I

MODEL 1 - SPAN 1100 FT

MODEL 2 - SPAN 2200 FT

PLAN -VIEW

NODAL POINTS (DECK)

TOWER LOCAL , COORDINATES

TWO HORIZONTAL d

400 -

1

160’

MODEL 1 MODEL 2

Fig. 13. The three-dimensional finite element models I and II.

also observed when using towers without top struts. The dynamic characteristics and mode shapes were much improved by providing this top strut.

RESULTS AND CONCLUSIONS

The computer program developed as a result of this investigation was utilized to study the nonlinear behavior of the two 3-D mathematical models

described above under the effect of dead loads and a set of initial cable tensions (pretensions). Figures 15 and 16 show the nonlinear behavior of the two models under static loads, described in terms of the longitudinal displacement of the left tower top (joint lo), and the vertical displacement at the middle point on the center span (joint 24). It is evident from these figures that the cable-stayed bridge system is a hard- ening structural system in which the overall stiffness of the structure increases as the load increases. It is


3 - D CABLE-STAYED BRIDGE - MODEL 1 - SPAN 1100 FT

TOWER WITH TOP STRUT

TOWER WITHOUT TOP STRUT

STATIC DEFORMATION DUE TO D. L.

( NONLINEAR ANALYSIS )

Fig. 14. Dead load equilibrium state of model I. (a) Case of no tower-top struts. (b) Case of tower-top struts.

also obvious from these figures that the nonlinearity strong nonlinearity in the behavior of the bridge is more pronounced in the longer span model (model under loads takes place. II having a central span length of 2200 ft). This From this study, the following conclusions may be supports the fact that nonlinearity in cable-stayed drawn: bridges is of geometric nature; in addition, as the span length increases, larger deformations occur and 1. Cable-stayed bridges are highly nonlinear struc-

3-D CABLE-STAYED BRIDGE -MODEL I (NONLINEAR STATIC ANALYSIS)

I I

0.0 I - 1

-20.0 0.0 20.0 40.0 60.0 80.0 103.0 120.0 140.0 160.0 180.0 200.0 220.0

DISPLACEMENT (inches)

Fig. 15. Nonlinear static behavior of model I.


3-D CABLE-STAYED BRIDGE -MODEL II

(NONLINEAR STATIC ANALYSIS)

-40.0 0.0 40.0 80.0 120.0 160.0 ZOO.0 240.0 280.0 320.0 3600 400.0 440.0 480.0

DISPLACEMENT (inches)

Fig. 16, Nonlinear static behavior of model II.

tures under the effect of their own dead weight, where the structure stiffness increases as the load increases. The nonlinearity in this type of structure is of geometric type and needs to be handled using the finite deflection theory. As the bridge span increases, the bridge becomes more flexible and, as a result, more susceptible to nonlinear behavior. As complex geometry of the tower’s configuration is adopted in design the static behavior of cable- stayed bridges cannot be fully represented by a planar or two-dimensional analysis, Three dimensional models are essential: (i) to study the torsional behavior of the bridge under uns~metrical loading, (ii) to realistically model the A-shaped and H-shaped towers that have been recently utilized in most modem cable-stayed bridges, and (iii) to represent the actual dynamic characteristic of the bridge for wind and seismic analyses. A nonlinear static analysis is a first step towards linear (or nonlinear) dynamic analysis of the bridge which starts at the dead load deformed state. Since several important dynamic features of cable-stayed bridges do not appear except in the 3-D analysis, it is essential to start with 3-D models for nonlinear static analysis and then proceed to the piers, and most importantly the tower-deck connections.

Acknowledgements-This paper is based on the Ph.D. dissertation of the first author which was supervised by the second author and presented to the faculty of the Depart- ment of Civil Engineering of Princeton University in May, 1987. The research was supported partially by a grant (No. ECE-8501067) from the National Science Foundation with Dr. S. C. Liu as the Program Director, and partially by the Egyptian Ministry of Higher Education via a Gov- ernment Mission; this support is greatly appreciated.

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nazmy a.s. three dimensional nonlinear static analysis of cable stayed bridges 1990

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