Initial shape of cable-stayed bridges

Compurers ci Structures Vol. 47, No. I, pp. I I I-123, Printed in Great Britain.

INITIAL

0045-7949/93 $6.00 + 0.00 Pergamon Press Ltd

1993

SHAPE OF CABLE-STAYED

BRIDGES

P. H. WANG, T. C. TSENGand C. G. YANG Department of Civil Engineering, Chung-Yuan Christian University, Chung-Li, Taiwan, R.O.C. (Received 12 December 1991) Abstract-A finite element computation procedure for determining the initial shape of cable-stayed bridges under the action of dead load of girders and pretension in inclined cables is presented. The system equation of cable-stayed bridges including the nonlinearities due to large displacement, beamxolumn, and cable sag effects is first set up and then solved using the Newton-Raphson method on increment-iterationwise. Based on a reference configuration and an assumed cable pretension force, the equilibrium configuration under dead load is found. Further, by adjusting cable forces, a ‘shape iteration’ is carried out and a new equilibrium configuration, i.e., a more reasonable initial shape, can be determined. The shape iteration is repeated until the desired tolerance is achieved. Numerical results show that a more accurate initial shape having reasonable prestress distribution and less deflection of girders can easily be determined by the proposed procedure.

1. INTRODUCTION

The concept of supporting a bridge deck by inclined tension stays can be traced back to the seventeenth century, but rapid progress in the analysis and construction of cable-stayed bridges has been made over the last 30 years. This progress is mainly due to developments in the fields of computer technology, high-strength steel cables and orthotropic steel decks. Since the first modern cable-stayed bridge was built in Sweden in 1955, their popularity has rapidly been increasing all over the world [l, 21. Aesthetic appeal, economics and ease of erection make the cable-stayed bridge the most suitable for medium-long span bridges with spans ranging from 750 ft (228 m) to about 1500ft (457 m). A cable-stayed bridge consists of three principal components, namely girders, towers and inclined cable stays. The girder is supported elastically at points along its length by inclined cable stays so that the girder can span a much longer distance without intermediate piers. The dead load and traffic load on the girders are transmitted to the towers by inclined cables. High tension forces exist in cable-stays which induce high compression forces in towers and part of girders. Since high pretension force exists in inclined cables before live loads are applied, the initial geometry and the prestress of cablestayed bridges are dependent on each other. They cannot be specified independently as conventional steel or reinforced concrete bridges. Therefore, the initial shape, i.e., the geometric configuration and the prestress distribution of cable-stayed bridges has to be determined prior to analyzing them. The purpose of this paper is to present an easy and efficient computational algorithm for finding the initial shape of cable-stayed bridges under the action of dead load of girders and pretension forces in inclined cables.

2. FINITE ELEMENT FORMULATION

2.1. Assumptions Based on the finite element concept, a cable-stayed bridge can be considered as an assembly of a finite number of cable and beam-column (for girder and tower) elements. Some assumptions are made as follows: the stress-strain relationship of all materials always remains within linear elastic range during the whole nonlinear computation, the cross-section area of elements remains unchanged during deformation, the cable element is assumed to be perfectly flexible and to possess only tension stiffness, and it is incapable of resisting compression, shear and bending forces. For the beam element, the engineering beam theory is employed and no shear strain is considered. All cables are fixed to the tower and to the girder at their joints of attachment. 2.2. Nonlinearities 2.2.1. Beam-column effect. Since high pretension force exists in inclined cables, the tower and part of the girder are under a large compression action, this means that the beamxolumn effect has to be taken into consideration for girders and towers of the cablestayed bridge. In a beamxolumn, lateral deflection and axial force are interrelated such that its bending stiffness is dependent on the element axial forces, and the presence of bending moments will affect the axial stiffness. The element bending stiffness decreases for a compression axial force and increases for a tension force. The beam-column effect can be evaluated by using the stability functions [3, 51. The beam-column element shown in Fig. 1 is utilized here. It has three element coordinates, two for end rotations, u, , u, and one for the relative axial deformation z+ = Al, where A1 is the element axial elongation or shortening. The element forces corresponding to uj are denoted by Sj

P. H. WANGer al.

112

where J_

IS,1 EIZ

-$

R, = J(S; + S;) [cot(J) + J cosec2(J)] fJR2

3,

QR3

- 2(SI + G2 + (S, S2)

qg?&-/-qR’

x

1 + J cot(J)][W cosec(J)].

For a tensile axial force, S, > 0 c _ J[J cash(J) - J sinh(J)]

Y

0

[

I-

’ - 2 - 2 cash(J) + J sinh(J) X

c, =

x,y : Local coardinotes system. X,V : Glob01 coordinates system u,

q.

R, =

: Element coordinates. : System coordinates.

Fig. 1. Beam-column

element.

42 2 4 i0

2 - 2 cash(J) + J sinh(J) 1 1 _ EAR,, ’ 4s 3I2

where

in which S, and S, are the end moments and S, is the axial force. When the material behaves linearly elastically and the beam-column effect is not considered, the element stiffness matrix KEjk has the form

KEjk=[KE]=7

J[sinh(J) - J]

0

0 0 A/I

1 ,

where E is the modulus of elasticity, A is the crosssectional area, Z equals the moment inertia of the cross-sectional area and 1 is the element length. When the beam-column effect has to be taken into consideration, the beam-column element stiffness matrix can be modified by employing the stability functions [3, 51 and has the following form

The stability function C,, C, and R, can be expressed in terms of the element axial force S, and the end moments S, and S, as follows. For a compressive axial force, S, < 0

R.,,,,= J(S: + S:)[coth(J) + J cosech2(J)] - 2(S, + S2)2 + (S, S,)[l + J coth(J)] x [2J cosech(J)]. 2.2.2. Cable sag e$ect. The inclined cable stay of cable-stayed bridges is generally quite long and it is well known that a cable supported at its end and under the action of dead load and axial tensile force will sag into a catenary shape (Fig. 2). The axial stiffness of a cable will change with changing sag. When a straight cable element for a whole inclined cable stay is used in the analysis, the sag effect must

4R2 t

c = J[sin(J) - J cos(J)] ’ 2 - 2 cos(J) - J sin(J) c, =

R, =

J[J - sin(J)] 2 - 2 cos(J) - J sin(J) 1 EAR, ’ ’ - 4(S, (3P

Fig. 2. Cable element with sag.

Initial shape of cable-stayed bridges be taken into account. On the consideration of the sag nonlinearity in the inclined cable stays, it is convenient to use an equivalent straight cable element with an equivalent modulus of elasticity [5-71, which can well describe the catenary action of the cable. When no sag effect exists, the straight cable element with one coordinate (relative axial deformation) ui = Al, has the stiffness matrix as follows: KEik = [KE] =

II 1 [rol,EA I

’

Table 1. Transformation coefficients of the plane cable element u

where E is the effective cable material modulus of elasticity, A is the cross-sectional area, and I is the element length. The cable stiffness vanishes and no element force exists for U, < 0, i.e., when shortening occurs. In the analysis of cable-stayed bridges, the sag effect of the inclined cable stays has to be taken into account, when the whole inclined cable is replaced by a single cable element. The sag effect can be included by using the equivalent modulus [S-7]. The concept of an equivalent modulus of elasticity was first introduced by Ernst [4]. If the change in tension for a cable during a load increment is not large, the axial stiffness of the cable will not significantly change and the cable equivalent modulus of elasticity can be considered constant during the load increment and is given by

1 + (wL)*AE’ 12T’

in which E, is the equivalent cable modulus of elasticity, w is the cable weight per unit length, L is the horizontal projected length of the cable and T is the tension force in the cable. If the tension in the cable varies largely during a load increment, the axial stiffness of the cable will significantly change and the equivalent modulus of elasticity over the load increment is given by E 1 + (wL>*(Tj + T/ME’ 24T!T* 1 /

where Ti and T,are the initial and final tension in the cable over the load increment. The cable equivalent modulus of elasticity combines both the effects of material and geometric deformation. The value of the equivalent modulus is dependent upon the weight and the tension in cable. Hence, the axial stiffness of the equivalent element combining cable sag and cable tension determined by the above equations is the same as the axial stiffness of the actual cable. 2.2.3. Large displacement effect. In general, cablestayed bridges have a larger span and less weight than that of conventional steel and reinforced concrete bridges. Large deflections may easily appear in cablestayed bridges. Hence the large displacement effect

R2

%

1

Ax --.

-- AY I

Ll

L2

1

B

a

AX

AY

r

r

RI

R2

an_8

Ll L2 RI

H,

-H3 4

HI H,

symmetric

H, 4

6

R2

H,+A$, H2Tf_$,

H3=-.

-H,

HZ

Ax Ay I

I = ,/m which equals deformed element length. Ax equals the projection of element length on the X-axis. Ay equals the projection of element length on the Y-axis.

has to be considered and the equilibrium equations must be set up based on the deformed position [8,9]. In the present study the motion of structural elements during large deflections is described by the exact transformation coefficients ah that relate the Table 2. Transformation coefficients of the plane beamcolumn element a Ll

L2

L3

i

1 2 3

For j=

-B -B -C

R2

R3

A A -D

0 0

B B c

-A -A D

0 1 0

L2

L3

R1

R2

R3

H, -H, 0 -H2

-H, -Hz 0 H,

1

I,2 Ll

a Ll L2 L3

RI s

B ar.8

-Hz

H,

0

H2

0 0

Rl

E, =

Rl

.i

E Eeq =

L2

Ll

for ‘I ”

for a, Q 0,

113

symmetric

R2 R3

H2

0 0 0 0 0

0

Forj=3 Ll

L2

L3

a Ll L2 L3

B

RI

R2

R3

@,./I H,

Rl R2 R3

H =Ax2-Ay= L p, 14 H =-AXAY 4 p., I’

fh H5

symmetric

0 0 0

-H, -H4 0

H,

-H4 -H, 0

4 Hs

1 Ax= 2Ax Ay H2=-__ > H,=r-71, 14 +f_!$

0 0 0

0 0 0

P. H.

114

WANG

local element coordinates and the global system coordinates. The nonlinear transformation coefficients of the first order and the second order, ajUand u,~,~ for the straight cable element and the beam-column element can be found in [8,9] and are listed in Tables 1 and 2. With the help of the nonlinear transformation coefficients aja and aj,,a, the tangent system stiffness matrix will be built up with the standard procedure by assembling the element stiffness matrices (see the next section). 2.3, General system equation In the following analysis of this paper, the nonlinearities induced by beam-column, cable sag and large displacement effects will be taken into account, but all materials behave linearly and elastically. The general system equations for a finite element model of structural systems in nonlinear static analysis can be derived from the virtual work principle and have the following form [8] Ki.W~-~Sja,=O,

tl = 1,2 ,...,

N

DOF,

(1)

EL

where l$i = a Wj/aq. are the basic vectors, LZ,= hj/aq, are the transformation coefficients, P, = Kj. bO, are the generalized external forces, T, = EEL Sjaj, are the generalized internal forces, Kj are external nodal load vectors, Wj are the displacement vectors corresponding to K, Sj = KEjkuk + S,” are the generalized element forces, S,” are the initial generalized element forces, uj are the generalized element coordinates, qUare the generalized system coordinates, KEjk is the element stiffness matrix, ZEL is the summation over all elements, and N equals the number of degrees of freedom (DOF). The superscript j denotes the nodal number. The subscripts a, /I, y, . . , denote the number of system coordinate and j, k, 1, . . . , the number of element coordinate. The index summation convention is used here for the super- and subscripts. The letters printed in boldface type, e.g. Kj, b’,, represent vectors. The dot notation between vectors means scalar product. In the analysis of cable-stayed bridges, the material behaves linearly elastic and the deflection is large, but with a small strain. In eqn (1) b’,, ai,, uj may be nonlinear functions of the system coordinates q. when large deflections occur and the nodal load vectors K may also become functions of qZ, if they are not displacement independent. Equation (1) represents a set of nonlinear algebraic equations and can be solved by the load increment methods or by the iteration methods. 2.4. Linearized system equation For the sake of incrementally deflection problem, the linearized has to be derived. By taking the Taylor’s expansion of the general (I), the linearized system equation interval is obtained as follows: AP; + .Pz = ‘K&. Aq;

solving the large system equation first order of the system equation for a small load

for P; < P, < P",+I,

(2)

et al.

where AP: = Pi+’ - P: are load increments, Aq; = qi;f’ - q; are displacement increments, ,,P; = P: - Z, Sya$ are unbalanced forces in statics, 2K:a = Z, KE~ka;a”,p + C, .$~a~~B- n J. ” J, B -"K$ . “b’, is the system stiffness matrix, a,,“= ai.‘)aq, are transformation coefficients of second oGer and Wi,s= ab’,/aqs. The superscript n denotes the number of load step and the two means iteration matrix of second order. It is well known that a pure increment computation will bring larger numerical errors. An increment-iteration computation procedure is recommended in order to get more accurate solutions.

3. SHAPE FINDING

PROCEDURE

The initial shape of a cable-stayed bridge provides the geometric configuration as well as the prestress distribution of the bridge under the action of the dead load of girders and the pretension force in inclined cable stays. The relation for the equilibrium conditions, the specified boundary conditions, and the requirements of architectural design should be satisfied here. For the following shape-finding computation, only the dead load of girders is taken into account and the dead load of cables and towers are neglected, but cable sag nonlinearity induced by cable dead load is included. The computation for shape finding starts with zero (or very small) tension forces in inclined cables. Based on a reference configuration (the architectural designed form) having no deflection in girders and zero prestress in any element, the equilibrium position of the cable-stayed bridge under dead load action is first determined iteratively by the NewtonRaphson method. Although this first determined configuration satisfies the equilibrium conditions and the boundary conditions, the requirements of architectural design are not fulfilled. Since the bridge span is large and no tension forces exist in inclined cables, quite large deflections and very large bending moments may appear in the girders. A shape iteration has to be carried out in order to reduce the deflection and to smooth the bending moments in the girder. For shape iteration, the cable axial forces determined in the previous step will be taken as initial element forces for inclined cables and a new equilibrium configuration under the action of dead load and such cable initial forces will be determined again. During shape iteration several control points (nodes intersected by the girder and the cable) will be chosen for checking if the convergence tolerence is achieved or not. In each shape iteration the ratio of the vertical displacement at control points to the main span length will be checked, i.e. vertical displacement at control point G 6.7. main span The shape iteration will be repeated until the convergence tolerance cr, say 10d4, is achieved. Numerical experiments show that the iteration computation

Initial shape of cable-stayed bridges converges monotonously. When the convergence tolerance for shape iteration is reached, the computation will be stopped and the initial shape of the cablestayed bridges is found. 4. COMPUTATION SHAPE

ALGORITHM FINDING

cable including sag effect and for the beamcolumn, (ii) calculate the transformation coefficients ujm,ujll,s, (iii) set up the tangent system stiffness matrix, *K$, (iv) solve the linearized system equations by the Newton-Raphson method to find the equilibrium configuration. After solving the linearized system equations: is found. Set i = I), (‘IAq; = Aq;, cogi;= qj, and ci+‘)q; = (“q; + ‘“Aq; . If

AP, + uP; = *Kjl,. Aqj, Aq;

II‘%“a II < ~ (convergence tolerance)

Il(i)4ns((.

then (9 i+ I = G+‘)qi;, and begin the next load step) and solve else (i=i+l, Ci,p; = 0’2Q . WAq; bith (it i’q; = (i)qi;+ (i)Ag;; and repeat convergence check.)

of elasticity

(4) Shape iteration check if the convergence tolerance

(ii

is achieved or not, (ii) if convergent, then the equilibrium configuration is the desired shape, else take the determined cable axial forces as initial cable force and repeat steps (3) and (4). 5. NUMERICAL

- I= 45.0ft4

Tower

- above girder : I = 20ft4;

5.1. Unsymmetrical cable-stayed

I=

200ft4;

A=3ft2 A --IOft2

- A= l.lft2

cable

W = 18.0 kips ft-’ W ~0.3

bridge

This unsymmetrical cable-stayed bridge is taken from [lo, 111.Its geometrical form and physical properties are shown in Fig. 3. The shape iteration control points are chosen at node 3. Until the initial shape is found, shape iteration is performed four times and in each shape iteration, five to six cycles of equilibrium iteration are needed. The geometrical configuration of the bridge and the moment, shear and axial force diagrams of the bridge girder are plotted in Fig. 4.

; A= 8.0ft2

below girder:

RESULTS

Three different types of cable-stayed bridges are taken from the literature and their initial shapes will be determined by the previous described shape finding procedure. Here, a convergence tolerance c = 1O-4 is used for both the equilibrium iteration and the shape iteration. From the numerical results, it is seen that the deflection and the maximum bending moment in girders are strongly reduced, and a more accurate initial shape having less deflection and more uniform prestress distribution of girders can be found.

E= 4 x IO6 ksf

Girder

Dead load - girder

and the

vertical displacement at control point G cs main span

6) calculate the element stiffness matrices for the

Cable

Determine the element deformations element forces.

FOR

The algorithm for shape finding computation of cable-stayed bridges is briefly summarized: (1) Input the geometric and physical data of the bridge. (2) Input the dead load of girders and the zero (or very small) initial force in cables. (3) Equilibrium iteration

Modulus

l

115

kips ft-’

Fig. 3. Unsymmetric cable-stayed bridge.

P. H. WANG et al.

116 -----

Reference configuration

-

Before shape iteration

-

After

shape iteration

(displacement

Shapes enlorged by IO x)

65,240

-71,272

G5.900

Bending moment diagram (kips ftl

Shear diagram fkips)

-7778 -9400 Axial force diagram (kips)

Fig. 4. Initial shape, moment, shear and axial force diagrams of the unsymmetric cable-stayed bridge.

variations on axial forces in the cable-stays during the shape iteration is given in Table 3. The deflection at node 3 and the maximum bending moment in girders are listed in the Table 4. The dead load stresses determined by Tang [lo] are also listed in Tables 3 and 4 for comparison. The pretension forces in cables between nodes 3-5 and 5-10 converge from 8410 and 10,163 kips to 10,010 and 12,043 kips. The vertical displa~ment at The

Table 3. Cable forces of the unsymmetric cable-stayed bridge NSI

CEI

1 6 2 5 3 5 4 5 Ref. [lo]

Cable No.

Force (kips)

Cable No.

Force (kips)

3-5 35 3-5 3-5 3-5

8410 9764 9977 10,010 9680

5-10 5-10 S-10 5-10 5-10

10,163 11,754 12,004 12,043 11,500

NSI = No. of shape iteration. CEI = cycles of equilibrium iteration.

the control point node 3 is reduced from 2.452 to 0.0137 ft. The maximum positive and negative bending moment occurs at nodes 2 and 3. They are reduced from 65,240 at the node 2 and -75,900 at node 7 to 44,361 kips ft at the node 2 and -71,272 kips ft at the node 3, respectively. The maximum shear force and axial force in girder have slightly increased after shape iteration, because the pretension force of cables increases (see Fig. 4). The dead load stress (pretension force called in this study) of cables between the nodes 3-5 and 5-10 determined by Tang [lo] are 9680 and 11,500 kips, respectively and the maximum positive and negative moments of the girder are 66,400 kips ft at the node 2 and -73,000 kips ft at the node 3, respectively. From Table 3 we see that the result presented by Tang [IO] is between the results of the first and the second shape iteration in the present study. It is evidently seen that the present shape-finding procedure offers a more accurate initial shape having less defiection and more smooth moment distribution in the girders than the

Initial shape of cable-stayed bridges

117

Table 4. Deflections and maximum moments of the unsymmetric cable-stayed bridge Vertical deflection (fQ

Maximum moment in girder (kips ft)

NSI

CEI

Node

Deflection

Node

Positive moment

Node

Negative moment

1

6

2 3 4

5 5 5

3 3 3 3

2.452 0.391 0.0649 0.0137

2 2 2 2

65,240 47,584 44,799 44,361

7 3 3 3

-75,900 - 64,826 - 70,397 -71,272

-

2

66,400

3

-73,000

Ref. [lo]

-

NSI = No. of shape iteration. CEI = cycles of equilibrium iteration.

Table 5. Initial shape of the unsymmetric cable-stayed bridge Element forces Reference configuration Node No. 1 2 3 4 5 6 7 8 9 10 11 12

X

Determined configuration X

Y

0.00

0.00

100.00 200.00 300.00 400.00 400.00

0.00 0.00 0.00 - 80.00 -40.00

2: 45o:oo

000 8’00 0:OO

500.00 550.00 600.00

::: 0.00

b-6

tower -Es cable

Element No.

-1.61 0.00 99.98 2.67 199.98 2.45 299.98 1.69 399.97 - 79.93 399.93 - 39.96 399.97 0.00 400.00 8.00 449.97 -0.13 499.97 0.00 549.97 0.11 599.97 0.00

at i50ft

Girder,

Y

Rending moments (kipsft) Axial force (kips)

3-5 5-10 l-2 2-3 3-4 4-7 7-9 9-10 10-l 1 1l-12 6-5 7-6 8-7

+*2OOft-G

4.320.000

Left 0.727596E 0.443612E + -0.712718E + 0.284778E + -0.305430E + -0.169593E + -0.121953E + 0.139023E + -0.496326E + -0.88314lE + 0.154043E -

O.lOOlOOE+ 05 0.120435E + 05 0.385399E + 01 0.988764E + 01 -0.929008E + 04 -0.929123E + 04 -0.940489E + 04 -0.9404648 + 04 0.766184E + 00 0.408150E + 00 -O.l12415E+05 -O.l12418E+05 -0.136632E + 05

6

at Isoft

ksf

-E=4,320,000ksf

Girder

- I = 131.0ft4:

Tower

- I = 24.4,

A= 3.44ft2

40.0,

50.0ft4

(from

top to bottom)

-A=2.18,2.45,2.90ft2 Cable

Dead

- exterior

: A = 0.4S2ft2

interior

: A = 0.174ft2

load - girder cable

W= 6.0 kips ft-’

: exterior interior

W= 0.22 IV= 0.085

I kips

ft-’

kipr ft-’

Fig. 5. Symmetric harp cable-stayed bridge.

Right 11 -0.443612E + 05 0.712718E + 05 -0.284778E + 05 0.390289E + 05 0.169594E + 04 0.121953E + 05 -0.139023E + 05 -0.813088E 04 0.949942E 04 0.496326E + 06 0.345533E +

____I(

05 05 05 05 04 05 05 09 03 04 03

P. H. WANGet al.

118

previous methods proposed by other authors. The final geometric configuration and the element forces of the unsymmetrical cable-stayed bridges are listed in Table 5.

Table 6. Cable forces of the symmetric harp cable-stayed bridge NSI CEI 1 2 3 4

5.2. Symmetric harp cable-stayed bridge This symmetric harp cable-stayed bridge is taken from [l 1, 121. Its geometrical and physical properties are given in Fig. 5. The shape iteration control points are chosen at nodes 4, 5, 10. The vertical displacements at these nodes are checked during shape iterations. Four cycles in shape iteration are needed until the initial shape is found and six to seven cycles of equilibrium iteration in each shape iteration are performed. The geometrical configuration of the bridge and the moment, shear, axial force diagrams of the bridge girder are plotted in Fig. 6. The cable forces, vertical deflections at the control points and maximum positive and negative bending moments in girders of the

-----

Reference

-

Before

-

After

Cable No.

Force (kips)

Cable No.

Force (kips)

Cable No.

Force (kips)

8-9 8-9 8-9 8-9

2862 2842 2724 2650

76 76 76 76

1556 1912 2063 2148

2-3 2-3 2-3 2-3

1393 1852 2066 2175

6 6 6 7

NSI = No. of shape iteration. CEI = cycles of equilibrium iteration.

symmetric harp cable-stayed bridge are listed in the Tables 6-8. After shape iterations the cable forces converge from 2862, 1566 and 1393 kips to 2650,2148 and 2175 kips in the cable element between the nodes 8-9, 6-7 and 2-3, respectively. The vertical deflection at nodes 4, 5, 10 is strongly reduced from 1.607 to -O.O816ft, 3.369 to -0.102 ft, 4.257 to -0.090 ft, respectively.

configuration

shape shape

iteration iteration

Shapes enlarged

(displacement

by 20 x 1

29.500

Bending moment

u -937

\

-53oo -6370

diagram

u Shear

diagram

p’ Axial

(kips)

i_i force

diagram

4

(kips)

Fig. 6. Initial shape, moment, shear and axial force diagrams of the harp cable-stayed bridge.

450.00 300.00

600.00 450.00 450.00 750.00

150.00 450.00 0.00

1000.00 900.00

1100.00 2000.00 1550.00

1550.00 1850.00

125o.oO 1400.00 1550.00

1550.00 1700.00

: :

7 !

:I:

12 f:

::

17 18 19

2:

10 10 IO

0.176 3.369 -0.072 -0.102

449.94

600.00 750.00 449.92 150.00 44;: 9tkl:OO 1OOtKJO 1100.00 1550.02 2000.00 1849.99 1550.09 1250.00 1400.00 1550.07 1699.99 1550.00

:z - 133:33 0.00

-2ooo:g 0.00 0.00 0.00 --200.00 0.00 0.00 - 133.33 0.00 0.00 -66.67 0.00 0.00

450.00 300.00

- 133:34 -0.10 -0.08 - 66.67 0.21 0.00

0.00 0.21 -66.67 -0.08 -0.10 - 133.34 0.29 0.00 - 199.99 -0.09 -0.01 -0.09 - 199.99 8.E

Determined configuration x Y

X:: - 66.67

Reference configuration X Y

:

Node NO.

5 5

Node

Deflection

-0.133 -0.090

0.067 4.257

Deflection

: 6 7

1 2 3 4

z l-3 16-13 19-16 21-19

l-4 4-5 5-10 10-l 1 11-12 12 17 17-18 18-21 21-20 20-15 15-14

ZO 12-13 17-16 IS-19 19-20 1615 13-14 8-7 2-1 7-2

z 3-4

8-9

Element No.

11 7 11 11

Node

Node I 1 1 10

moment Positive 29,550 9758 10,023 12,276

0.265004E + 04 0.214803E + 04 0.2174958 f 04 0.216434E + 04 0.207151E + 04 0.270385E -l- 04 0.270385E t- 04 0.20715lB + 04 0.216434E + 04 0.217495E + 04 0.2148038 + 04 0.265004E + 04 -0.242164E + 04 -0.4383688 + 04 -0.6369918 + 04 -0.6342648 + 04 -0.436429E + 04 -0.2471018 + 04 0.225982B i- 00 0.22598213 + 00 -0.24710115 + 04 - 0.4364298 + 04 -0.634264E + 04 -0.636991E + 04 -0.438368E + 04 -0.2421648 + 04 -0.217396E + 04 -0.388884E + 04 -0565268E + 04 -0.217396E + 04 -0.388884E + 04 -0.565268E + 04

Axial force (kinsl . __

-

80,645 34,480 22,482 17,724

Negative moment

0.OOOOOOE + 00 -0.126024E + 04 -0.742814E + 04 -0.1682938 i-05 -0.1226688 + 05 -0.1060lOE + 05 -0.177241E + 05 0.122758E + 05 - 0.1772428 f 05 -0.106010E + 0.5 -0.122668E +- 05 -0.166947E f OS -0.742813E f 04 -0.126023E + 04 -0.343802E + 04 -0.206629E + 04 0.1346428 + 03 0.343802E + 04 0.206629E + 04 -0.134640E + 03

-

O.l26024E+O4 0.7428 14E + 04 0.166947E + 05 0.122668E + 05 0.106OlOE + OS 0.177241 E + 05 -0.122758E + 05 0.177242E + 05 0.106010E + 05 0.122668E + 05 0.168293E f 05 0.742813E + 04 0.126023E + 04 -0.132422E - 08 -0S11318E-08 0.343802E + 04 0.206629E + 04 -0.620184E - 08 -0.3438028 + 04 - 0.206629E + 04

-

-

Element forces _. Bending moments (kips ft) Left Right

NSI = No. of shape iteration. CEI = cycles of equilibrium iteration.

CEI

NSI

Maximum moment in girder (kips B) -___-

Table 8. Maximum bending moments of the symmetric harp cable-stayed bridge

Table 9. Initial shape of the symmetric harp cable-stayed bridge

5

Node

0.102 1.607

4

6

:.

Deflection

3 6 4 -0.047 4 7 4 -0.0816 NSI = No. of shape iteration. CEI = cycles of equilibrium iteration.

Node

CEI

NSI

Vertical deflection (ft)

Table 7. Deflections of the symmetric harp cable-stayed bridge

?

G &

Y

se

P. H. WANGet al.

E-_-------6

at ISOft ------&2oOft+

Girder) tower -Es

6 at ISOft -4

4,320J300kSf

Girder

-E=4,320,000ksf - I= 131.0ft4: A= 3.44ft’

Tower

- I= 24.4,

40.0,

- A= 2.18,

2.45 ,2.S0tt2

cable

Coble

- exterior

(from top t0 bottom]

: A =0.452 ft2

interior Dead load - girder cable

50.0ft4

: A = 0. 174ft2 W= 6.0 kips ft-’

: exterior W= 0.221 kips ft-! interior

W= 0.085kips

ff’

Fig. 7. Symmetric radiating cable-stayed bridge.

-----

Reference configuration

-

Before shape iteration

-

After shape iteration

(displacement

Shapes enlorgad by 20 x1 36,942

-49.360

Sending moment diagram (kips ftf

651

-659 Shear diogrom (kips)

-4420

Axial force diagram (kips) Fig. 8, Initial shape, moment, shear and axial force diagrams of the radiating cable-stayed bridge.

121

Initial shape of cable-stayed bridges Table 10. Cable forces of the symmetric radiating cablestayed bridge NSI CEI 1 2 3

Cable No.

Force (kips)

Cable No.

Force (kips)

Cable No.

Force (kips)

8-9 8-9 &9

2571 2597 2549

7-9 7-9 7-9

1396 1586 1656

2-9 2-9 2-9

1004 1170 1198

15 4 5

NSI = No. of shape iteration. CEI = cycles of equilibrium iteration. Table 11. Deflections of the symmetric radiating cable-stayed bridge Vertical deflection (ft) NSI

CEI

Node

1 2 3

15 4 5

4 4 4

Deflection

1.203 0.0515 3.85 x 1O-4

Node

Deflection

Node

5 5 5

2.745 0.115 -0.0168

10 10 10

Deflection 3.83 0.172 -0.0183

NSI = No. of shape iteration. CEI = cycles of equilibrium iteration. Table 12. Maximum bending moments of the symmetric radiating cable-stayed bridge Maximum moment in girder (kips ft) NSI

CEI

Node

Positive moment

1 2 3

15 4 5

11 I1 11

36,942 13,253 11,880

Negative

Node

moment

1 10 10

-49,400 - 16,747 - 18,120

NSI = No. of shape iteration. CEI = cycles of equilibrium iteration.

The maximum positive and negative bending moments in girder are reduced from 29,550 at the node 11 and - 80,645 at the node 1 to 12,276 kips ft at the node 11 and - 17,724 kips ft at the node IO. The moment and the shear force in girders are strongly reduced and become more smooth in this example. The axial force in girders increases slightly (see Fig. 6). The final geometric configuration and the element forces of the symmetric harp cable-stayed bridge are listed in Table 9. 5.3. Symmetric radiating cable-stayed

bridge

This symmetric radiating cable-stayed bridge is taken from [I 11. Its geometry is shown in Fig. 7. Its geometric and physical properties have the same values as that of the previous example. The shape iteration control points are chosen at the nodes 4, 5, 10. For this example three cycles of shape iteration are needed. In each shape iteration, 4-15 cycles of equilibrium iteration are needed. Similarly, the geometric conligurations and the moment, shear and axial force diagrams are plotted in Fig. 8; their cable forces, vertical deflections at the shape iteration control points and maximum bending moments in the girder are given in Tables 10-12. The cable forces converge from 2571, 1396 and 1004 kips to 2549, 1656 and 1189 kips in the cable elements between the nodes 8-9 7-9 and 2-9. The vertical deflections at nodes 4,5, 10 are reduced from

1.203, 2.745, 3.83 to 0.0004, -0.0168, -O.O183ft, respectively, and the maximum positive and negative moment from 36,942, -49,400 kps ft to 11,880, -18,120 kips ft. The moment distribution in the girder becomes more uniform. The shear force is also reduced in this example and the axial force increases slightly (see Fig. 8). The final geometric contiguration and the element forces of the symmetric radiating cable-stayed bridge are listed in Table 13. 6. CONCLUSION

A finite element computation procedure for finding the initial shape of cable-stayed bridges under the action of the dead load of girders and the pretension in inclined cable stays is presented. The nonlinear&s induced by beam-column, cable sag and large displacement effects have been taken into account. The nonlinear system equation is solved incrementiterationwise. Two iteration cycles have been introduced: one is the usual equilibrium iteration and the other is the shape iteration for finding the initial shape, which can mostly achieve the architectural designed form and satisfies all the equilibrium and boundary conditions. Numerical results show that the proposed shape fmding procedure of cable-stayed bridges works very efficiently and the computation converges monotonously. A more accurate and reasonable initial shape having less deflection of the girder and more

Reference x

450.00 300.00 450.00 600.00 750.00 450.00 150.00 0.00 450.00 900.00 1000.00 1100.00 1550.00 2000.00 1850.00 1550.00 1250.00 1400.00 1550.00 1700.00 1550.00

Node No.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

0.00 - 66.67 0.00 0.00 - 133.33 0.00 0.00 - 200.00 0.00 0.00 0.00 - 200.00 0.00 0.00 - 133.33 0.00 0.00 -66.67 0.00 0.00

0.00

configuration Y 450.0 1 300.01 449.99 600.01 750.01 449.95 150.02 0.01 449.88 900.01 1ooO.01 1100.01 1550.13 2000.00 1849.99 1550.06 1250.00 1400.00 1550.02 1699.99 1550.00

0.00 0.19 -66.67 -0.00 -0.17 - 133.33 8.13 0.00 - 199.99 -0.18 0.54 -0.18 - 199.99 0.00 0.81 - 133.33 -0.17 0.00 - 66.67 0.19 0.00

a-9 7-9 2-9 94 9-5 9-10 12-13 17-13 18-13 13-20 13-15 13-14 8-7 7-2 2-l 14 4-5 5-10 IO-11 11-12 12-17 17-18 18-21 21-20 20-15 15-14 69 3-6 l-3 1613 19-16 21-19

Element No.

shape of the symmetric

configuration Y

13. Initial

Determined X

Table

(kips)

Axial force

cable-stayed

0.254935E + 0.165617E + 0.118929E + 0.112332E + 0.152lllEfO4 0.271472E + 0.271472E + O.l52lllE+O4 0.112332EfO4 O.l18929E+O4 O.l65617E++ 0.254935E + -0.232965E + -0.370743E + -0.442095E + -0.44205lE + - 0.374649E + -0.24808OE + 0.215465E + 0.215467E + - 0.24808OE + -0.374649E + -0.44205lE + -0.442095E + -0.370744E + -0.232965E + -0.57506lE + -0.57506lE + -0.57506lE + -0.57506lE + -0.57506lE + -0.57506lE +

radiating

04 04 04 04 04 04 04 00 00 04 04 04 04 04 04 04 04 04 04 04 04

04 04

04 04 04 04

bridge

0.80392OE + 0.136554E + 0.113986E + 0.107682E + 0.102512E + 0.1812lOE + -0.11879OE + 0.1812lOE + 0.102513E + 0.107682E + O.l15575E+05 0.136554E + 0.803916E + 0.823638E 0.559794E -0.6595OOE + -0.120192E + -0.194694E 0.6592038 + 0.120173E +

-

-

-

Right

Left

Bending moments (kips ft)

forces

0.363798E - 11 -0.803920E + 04 -0.136554E + 05 -0.115575E + 05 -0.107682E + 05 -0.102512E +05 -0.1812lOE + 05 0.118790E + 05 -0.1812lOE + 05 -0.102513E +05 -0.107682E + 05 -O.l13986E+05 -0.136554E + 05 -0.803916E + 04 0.6595OOE + 02 0.120192E + 03 0.158839E + 03 -0.659203E + 02 -0.120173E + 03 -0.158868E + 03

Element

05 04 08 06 02 03 05 02 03

04 05 05 05 05 05 05 05 05 05

7

a

2

n”

P

?

.T

Initial shape of cable-stayed bridges uniform distribution of bending moment in the girder can he determined very easily and quickly. research was sponsored by a grant under No. NSC 79-0410-E033-04 from the National Science Council, Taiwan, Republic of China.

5. J. F. Fleming, Nonlinear static analysis of cable-stayed

7. 8.

REFERENCES

Concept and

9.

W. J. Podolny and F. Fleming, Historical development of cable-stayed bridges. J. Struct. Div., ASCE 98, 2079-2095

(1972).

A. Ghali and A. M. Neville, Structural Analysis: A Unified Classical and Matrix Approach. Chapman & Hall, New York (1978). H. J. Ernst, Der E-Modul von Seilen unter Beruecksichtigung des Durchhanges. Der Bauingenieur 40, No. 2 (1965).

bridges. Comput. Struct. 10, 621-635 (1979). and N. V. Raman, Nonlinear analvsis of cable-stated bridges. IABSE Proc., P-37/80, pp. 265-216 (1980). A. S. Nazmy and A. M. Abdel-Ghaffar, Threedimensional nonlinear static analysis of cable-stayed bridges. Comput. Struct. 34, 257-271 (1990). K.-H. Schrader, MeSy Einfuehrung in das Konzept und Benutzeranleitung fuer das Programm MESY-MINI. Technisch-Wiss&chaftliche Mitt&ng Nr. 78- 11, Institut Fuer Konstruktiven Inginieurbau, Ruhr-Universitaet Bochum (1978). M. T. Wu, Y. L. Chen and. P. H. Wang, Shape finding of cable systems using nonlinear displacement analysis approach. J. Chinese Inst. Engng 11, 583-594 (1988). M. C. Tang, Design of cable-stayed girder bridges. J. Struct. Div., ASCE 98, 1789-1802 (1972). N. F. Morris, Dynamic analysis of cable-stayed bridges. J. Struct. Div., ASCE 100, 971-981 (1974). B. E. Lazar, M. S. Troitsky and M. M. Douglass, Load balancing analysis of cable stayed bridges.

6. A. Rajaraman, K. Loganathan

Acknowledgement-This

N. J. Gimsing, Cable Supported Bridges, Design. John Wiley, Chichester (1983).

123

10. 11. 12.

J. Struct. Div., ASCE 98, 1725-1740

(1972).