Second-order Inelastic Dynamic Analysis of Three ...

Steel Structures 8 (2008) 205-214 www.ijoss.org

Second-order Inelastic Dynamic Analysis of

Three-dimensional Cable-stayed Bridges

Huu-Tai Thai and Seung-Eock Kim1*

Department of Civil and Environmental Engineering, Sejong University, 98 Kunja Dong Kwangjin Ku, Seoul 143-747, Korea

Abstract

This paper presents a second-order inelastic dynamic analysis of three-dimensional cable-stayed bridges including bothgeometric and material nonlinearities. Geometric nonlinearity is captured by using stability functions to minimize modeling andsolution time, while material nonlinearity is considered by adopting the refined plastic hinge model. A computer programutilizing the Newmark β-method with the assumption of average acceleration is developed to predict the nonlinear inelasticdynamic behavior of the cable-stayed bridges. The accuracy and efficiency of the proposed program are verified by comparingit with SAP2000 and ABAQUS. It can be concluded that the proposed program is capable of accurately and efficientlypredicting the nonlinear inelastic dynamic response of cable-stayed bridges.

Keywords: Geometric nonlinearity, Material nonlinearity, Plastic hinge, Stability function, Cable-stayed bridge

1. Introduction

Cable-stayed bridges are widely used in bridge

engineering because of the appealing aesthetics, the

efficient utilization of structure materials, and the

relatively small size of structure members. It is well

known that the increase in the central span length of

cable-stayed bridges makes the nonlinear analysis

inevitable. Material nonlinearity comes from the nonlinear

stress-strain behavior of the materials, while the

geometric nonlinearities result from the cable sag effect,

axial force-bending moment interaction, and large

displacement. The static and dynamic behaviors of this

highly nonlinear structure have been studied extensively

in recent years. In general, these studies can be

categorized into three main types: (1) linear elastic; (2)

nonlinear elastic; and (3) nonlinear inelastic. In linear

analysis, Wilson and Liu (1991) studied the dynamic

behavior of a cable-stayed bridge by using a three-

dimensional finite element model. Their study was

compared to the measured ambient vibration of the full-

scale cable-stayed bridge. In the nonlinear elastic

analysis, Fleming and Egeseli (1980) performed the

seismic behavior of two-dimensional cable-stayed bridges,

while the three-dimensional cable-stayed bridges were

analyzed by Nazmy and Abdel-Ghaffar (1990) as well as

Abdel-Ghaffar and Nazmy (1991). In the nonlinear

inelastic analysis, Ren and Obata (1997) performed the

seismic response of a long span cable-stayed bridge.

However, their study was limited to the two-dimensional

problem. Cho and Song (2006) evaluated the global

system response of a bridge using the finite element

method. Now with the utilization of complex geometry

for the towers and the cables, it is necessary to perform

the nonlinear inelastic dynamic analysis for the three-

dimensional problem.

The purpose of this paper is to extend the application of

the stability functions and the refined plastic hinge model

for the nonlinear inelastic dynamic analysis of three-

dimensional cable-stayed bridges. A computer program

including all sources of nonlinearity is developed to

predict the nonlinear inelastic dynamic response of cable-

stayed bridges. Two earthquake records of the El-Centro

1940 and Loma Prieta 1989 are used to verify the

accuracy and efficiency of the proposed program with the

SAP2000 and ABAQUS software.

2. Formulation

2.1. Modeling of cable elements

The cables are assumed to be perfectly flexible and to

resist the tensile force only. The inclined cables of cable-

stayed bridges will sag into a catenary shape due to their

self-weight. The tension stiffness of the cable, which

varies depending on the sag, is modeled by using an

equivalent straight truss element with an equivalent

modulus of elasticity. This concept was first proposed by

Note.-Discussion open until February 1, 2009. This manuscript forthis paper was submitted for review and possible publication onAugust 1, 2008; approved on August 30, 2008

*Corresponding authorTel: +82-2-3408-3291; Fax: +82-2-3408-3332E-mail [email protected]

206 Huu-Tai Thai and Seung-Eock Kim

Ernst (1965) and has been verified by several researchers.

The equivalent cable modulus of elasticity is given as

follows

(1a)

where Eeq is the equivalent modulus of cable; E is the

Young’s modulus of cable; L is the horizontal projected

length of cable; w is the weight per unit length of cable;

A is the cross-sectional area of cable; and T is the cable

tension.

When the tension in the cable changes from Ti to Tf

during the application of a load increment, the secant

value of the equivalent modulus of elasticity over a load

increment is given as

(1b)

2.2. Modeling of cross beam, tower, and girder

members

The cross beam, tower, and girder members of the

bridges are modeled as beam-column elements. The

coupling between axial force and bending moment in

these members can be accurately captured by employing

the stability functions reported by Chen and Lui (1987).

Then a refined plastic hinge model is adopted to account

for gradual yielding. Details of the procedure are presented

in the following sections.

2.2.1. Stability functions account for second-order

effect

From Chen et al. (2001), the incremental form of

member force and deformation relationship of space

beam-column element can be expressed as

(2)

where A, Iy, Iz, L are area, moment of inertia with respect

to y and z axes, and length of beam-column element; E,

G, and J are elastic modulus, shear modulus, and torsional

constant of material; P', M'yA, M'yB, M'zA, M'zB, and T' are

incremental axial force, end moments with respect to y

and z axes, and torsion respectively. δ', θ'yA, θ'yB, θ'zA, θ'zB,

and φ' are the incremental axial displacement, the joint

rotations, and the angle of twist. S1n and S2n are the

stability functions with respect to n axis (n=y,z) given in

Chen et al. (2001).

2.2.2. Refined plastic hinge model accounts for

gradual yielding

2.2.2.1. CRC tangent modulus model associated with

residual stresses

The CRC tangent modulus concept is used to account

for gradual yielding (due to residual stresses) along the

length of axially loaded members between plastic hinges.

From Chen and Lui (1987), the CRC tangent modulus Et

is written as

Et=1.0E for (3a)

for (3b)

where Py is the axial yield force.

2.2.2.2. Parabolic function for gradual yielding due to

flexure

The tangent modulus model is suitable for the member

subjected to axial force, but inadequate for cases of both

axial force and bending moment. A gradual stiffness

degradation model for a plastic hinge is required to

represent the partial plastification effects associated with

bending. When gradually forming plastic hinges are

active at both ends of an element, the incremental force-

displacement equation can be expressed as

(4)

where

(5a)

(5b)

(5c)

Eeq

E

1wL( )2AE

12T3

---------------------+

----------------------------=

Eeq

E

1wL( )2 Ti Tf+( )AE

24Ti

2Tf

2---------------------------------------+

----------------------------------------------=

I'

M'yA

M'yB

M'zA

M'zB

T'⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

EA

L------- 0 0 0 0 0

0 S1y

EIy

L-------S

2y

EIy

L------- 0 0 0

0 S2y

EIy

L-------S

1y

EIy

L------- 0 0 0

0 0 0 S1z

EIz

L-------S

2z

EIz

L------- 0

0 0 0 S2z

EIz

L-------S

1z

EIz

L------- 0

0 0 0 0 0GJ

L-------

δ'

θ'yAθ'yBθ'zAθ'zBφ'⎩ ⎭

⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

=

P 0.5Py≤

Et 4P

Py

-----E 1P

Py

-----–⎝ ⎠⎛ ⎞

= P 0.5Py>

I'

M'yA

M'yB

M'zA

M'zB

T'⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

EtA

L-------- 0 0 0 0 0

0 kiiykijy 0 0 0

0 kiij kjjy 0 0 0

0 0 0 kiizkijz 0

0 0 0 kijzkjjz 0

0 0 0 0 0GJ

L-------

δ'


⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

=

kiiy ηA S1

S2

2

S1

----- 1 ηB–( )–

⎝ ⎠⎜ ⎟⎛ ⎞EtIy

L--------=

kijy ηAηBS2

EtIy

L--------=

kjjy ηB S1

S2

2

S1

----- 1 ηA–( )–

⎝ ⎠⎜ ⎟⎛ ⎞EtIy

L--------=

Second-order Inelastic Dynamic Analysis of Three-dimensional Cable-stayed Bridges 207

(5d)

(5e)

(5f)

The terms ηA and ηB are scalar parameter allowing for

gradual inelastic stiffness reduction of the element

associated with plastification at end A and B. This term

is equal to 1.0 when the element is elastic, and zero when

a plastic hinge is formed. The parameter η is assumed to

vary according to the parabolic function:

η=1.0 for (6a)

η=4α(1−α) for (6b)

where α is a force-state parameter that measures the

magnitude of axial force and bending moment at the

element end. The term α in this study is expressed in a

modified version of Orbison full plastification surface of

cross-section, presented by McGuire et al. (2000), as

follows

(7)

where p=P/Py, mz=Mz/Mpz (strong-axis), my=My/Mpy

(weak-axis).

If the force point moves beyond the fully yield surface,

says α>1, the member forces should be scaled down to

return the fully yield surface with the application of the

equilibrium iteration method.

2.2.2.3. Strain reversal effect

The strain reversal in the hinge is induced by the

sequential loading in the static analysis, or the change of

dynamic loading direction in the dynamic analysis. The

strain reversal can be determined by investigating the

stress state at the four corners of a section. In order to

account for the strain reversal effect, the CRC tangent

modulus Et and the stiffness reduction function η should

be modified based on the double modulus theory as

presented in Kim et al. (2000).

2.2.3. Shear deformation effect

To account for transverse shear deformation effects in

a beam-column element, the incremental force-displacement

equation can be modified as

(8)

where Ciiy, Cijy, Cjjy, Ciiz, Cijy, Cjjz are coefficients given in

Chen et al. (2001).

2.2.4. Element stiffness matrix

The end forces and displacements used in Eq. (8) are

shown in Fig. 1(a). The sign convention for the positive

directions of element end forces and displacements of a

frame member is shown in Fig. 1(b). By comparing the

two figures, we can express the equilibrium and

kinematic relationships in symbolic form as

(9a)

(9b)

where {f'n} and {d'L} are the incremental end force and

displacement vectors of a beam-column member

expressed as

{f'n}T={rn1 rn2 rn3 rn4 rn5 rn6 rn7 rn8 rn9 rn10 rn11 rn12} (10a)

{d'L}T={d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12} (10b)

and {f'n} and {d'n} are the incremental end force and

displacement vectors in Eq. (8). [T]6×12 is a transformation

matrix given in Chen et al. (2001). Using the

transformation matrix by equilibrium and kinematic

relations, the force-displacement relationship of a beam-

column member may be written as

{f'n}=[Kn]{d'L} (11)

[Kn] is the element stiffness matrix expressed as

[Kn]12×12=[T]T6×12[Ke]6×6[T]6×12 (12)

Eq. (11) is used to enforce no side-sway in the member.

If the member is permitted to sway, additional axial and

shear forces will be induced in the member. We can relate

these additional axial and shear forces due to a member

sway to the member end displacements as

{fs}=[Ks]{dL} (13)

where [Ks] is the element stiffness matrix given in Chen

et al. (2001).

By combining Eqs. (11) and (13), we obtain the general

beam-column element force-displacement relationship as

{fL}=[K]local{dL} (14)

where

{fL}={fn}{fs} (15)

[K]local=[Kn]+[Ks] (16)

2.3. Seismic response analysis

The incremental form of the equation of motion is

given by

[M]{∆u''}+[C]{∆u'}+[K]{∆u}={∆F} (17)

in which [K] is the stiffness matrix; [M] is the lump mass

matrix; and [C]=a[M]=b[K0] is the viscous damping

matrix, where a and b are mass- and stiffness-

kiiz ηA S3

S4

2

S3

----- 1 ηB–( )–

⎝ ⎠⎜ ⎟⎛ ⎞EtIz

L--------=

kijz ηAηBS4EtIz

L--------=

kjjz ηB S3

S4

2

S3

----- 1 ηA–( )–

⎝ ⎠⎜ ⎟⎛ ⎞EtIz

L--------=

α 0.5≤α 0.5>

α p2mz

2my

43.5p

2mz

23.0p

2my

24.5mz

4my

2+ + + + +=

I'

M'yA

M'yB

M'zA

M'zB

T'⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

EtA

L-------- 0 0 0 0 0

0 CiiyCijy 0 0 0

0 CiijCjjy 0 0 0

0 0 0 CiizCijz 0

0 0 0 CijzCjjz 0

0 0 0 0 0GJ

L-------

δ'


⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

=

f'n{ } T[ ]6 12×

Tf'e{ }=

d'e{ } T[ ]6 12×

d'L{ }=


proportional damping factors, respectively; {∆''u}, {∆'u},

{∆u}, and {∆F} are the incremental acceleration, velocity,

displacement, and exciting force vectors, respectively,

over a time increment of ∆t.

The Newmark β-method with the assumption of

average acceleration is adopted herein to solve, step-by-

step, the numerical solution of the Eq. (17). The detailed

algorithm of Newmark β-method, as presented in Chopra

(2001), can be summarized as the following equations:

(18a)

(18b)

in which {tu''}, {t

u'}, and {tu} are the total acceleration,

velocity, and displacement vectors at time t. Here the

integration parameters β and γ are taken as 1/4 and 1/2,

respectively, correspond to the assumption of the average

acceleration method.

By substituting Eq. (18) into Eq. (17), the final form of

the incremental equation of motion can be expressed as

(19)

Eq. (19) is solved for each time step until the considered

frame is collapsed or desired time duration ends.

3. Algorithm for Nonlinear Inelastic Dynamic Analysis

A computer program has been developed to perform

the nonlinear inelastic dynamic analysis of the three-

dimensional cable-stayed bridges subjected to its own

weight and earthquake loadings. A combination of

incremental and iterative schemes is utilized in this

algorithm. For static loading, the magnitude of the

applied load is divided into increments, whereas in

dynamic loading, the total time of the dynamic load is

divided into small intervals. Within each increment, a

solution for the equilibrium equations is solved by

iterative means, i.e., by updating the force-state parameter

and stiffness of the elements until the solution converges

and equilibrium requirements are satisfied. The flow

chart of the procedure is presented in Fig. 2.

4. Verification Studies

The computer program, 3D-PAAP, was developed

based on the aforementioned formulations to predict the

second-order inelastic seismic response of cable-stayed

bridges. It should be noted that SAP2000 provides the

cable element and can predict the second-order elastic

response, but it is incapable of investigating the second-

order inelastic response. ABAQUS, which does not

provide cable element, is capable of considering the

second-order inelastic response. Therefore, the proposed

ut ∆t+

'{ } ut

'{ } 1 γ–( )∆t u't

'{ } γ∆t ut ∆t+

''{ }+ +=

ut ∆t+{ } u

t{ } ∆t u't{ } 0.5 β–( ) ∆t( )2 u

t

''{ } β ∆t( )2 ut ∆t+

''{ }+ + +=

Figure 1. Element end forces and displacements notations.

Figure 2. Flow chart of the proposed program.

K[ ]γ

β∆t-------- C[ ]

1

β ∆t( )2--------------- M[ ]+ +

⎩ ⎭⎨ ⎬⎧ ⎫

∆u{ }=

∆F{ }1

β∆t-------- M[ ]

γβ--- C[ ]+

⎩ ⎭⎨ ⎬⎧ ⎫

ut

'{ }1

2β------ M[ ] ∆t

γ2β------ 1–⎝ ⎠⎛ ⎞

C[ ]+

⎩ ⎭⎨ ⎬⎧ ⎫

u't

'{ }+ + +


program should be verified by comparing with SAP2000

in the elastic range by using a cable element, with

ABAQUS in the inelastic range by using equivalent straight

truss element for the cable. Two earthquake records of the

El-Centro and Loma Prieta shown in Fig. 3 are used as

ground motion input data in the longitudinal direction,

which is considered to be the most destructive in cable-

stayed bridges. Their peak ground accelerations and time

steps are listed in Table 1. For each example, the initial

shapes and initial cable tensions, due to the weight of the

bridges, are first determined by a nonlinear static analysis,

and then the dynamic behavior, due to earthquake loading,

is investigated. The mass- and stiffness-proportional

damping factors are chosen based on the first two modes

of the bridge so that the equivalent viscous damping ratio

is equal to 5%.

The three-dimensional modeling of the cable-stayed

bridges taken from Song and Kim (2007) is shown in Fig.

4. The girders with the central span length of 122 m are

supported by a series of cables aligned in fan, semi-harp,

and harp type bridges. The stress-strain curve for the

cross beam, girder, and tower members is assumed to be

elastic-perfectly plastic with an initial elastic modulus of

207 GPa and a yield stress of 248 MPa. The cable

members should be valid in the elastic limit of the

material, with an elastic modulus of 158.6 GPa and a

yield stress of 1103 MPa. In the inelastic seismic analysis,

only the inelastic behavior of the girder members is

considered. The weight per unit volume of the cable and

beam-column members is 60.5 kN/m3 and 76.82 kN/m3,

respectively. The masses lumped at the bridge nodes are

calculated from the self weight of the bridges. The cable

is modeled by an equivalent straight truss element using

an equivalent elastic modulus. The cross beam, girder,

and tower members are modeled by using only one

element in the proposed program and ten elements in both

SAP2000 and ABAQUS.

4.1. Natural vibration

The vibration analysis of the cable-stayed bridges is

first performed to verify the accuracy of the proposed

program in predicting the natural periods of the bridges.

The first two natural periods along the applied earthquake

direction of fan, semi-harp, and harp bridges obtained by

ABAQUS, SAP2000, and proposed program are presented

in Tables 2 and 3. It is observed that a strong agreement

of natural periods of the bridges predicted by ABAQUS,

SAP2000, and proposed program is obtained with the

maximum difference of 0.20%.

4.2. Verification of nonlinear elastic seismic behavior

This section is focused on the verification of the

proposed program with SAP2000 in predicting the

nonlinear elastic seismic behavior of the cable-stayed

bridges. The cross beam, girder, and tower members of

the bridges are modeled using the frame element in

SAP2000 and the beam-column element in the proposed

program. The cables are modeled by using the cable

elements in both SAP2000 and proposed program.

The vertical displacement responses at the middle point

of the central span of the bridges subjected to two

different earthquake loadings of the El-Centro and Loma

Prieta are shown in Figs. 5 and 6, respectively. The

displacement responses obtained by SAP2000 and

proposed program in all cases of analysis are almost

identical. The peak vertical displacements at the middle

point of the central span of the bridges with three

different cable layouts of fan, semi-harp, and harp type

are also presented in Table 4. The difference of

displacement response of nonlinear elastic seismic

analysis in each case is very small, with a maximum

difference of 2.11% in all cases. All results obtained by

SAP2000 and the proposed program are nearly the same,

which prove the accuracy of the proposed program in

predicting the second-order effect.

Figure 3. Earthquake records.

Table 1. Peak ground acceleration and its corresponding time step of the earthquake records

Earthquake PGA (g) Time step (s)

El-Centro (1940) (Array, #9, USGS Station 117) 0.319 0.020

Loma Prieta (1989) (Capitola, 000, CDMG Station 47125) 0.529 0.005


Figure 4. Cable-stayed bridges (unit: m).

Table 2. Comparison of first two natural periods (sec) of the bridges using truss element for cable

Bridge type Mode ABAQUS 3D-PAAP (proposed) Error (%)

Fan type First 1.870 1.868 0.09

Second 1.232 1.231 0.10

Semi-Hard typeFirst 1.868 1.866 0.10

Second 1.242 1.241 0.12

Harp typeFirst 1.897 1.895 0.13

Second 1.333 1.330 0.20

Table 3. Comparison of first two natural periods (sec) of the bridges using cable element

Bridge type Mode SAP2000 3D-PAAP (proposed) Error (%)

Fan type First 1.867 1.868 0.05

Second 1.230 1.231 0.08

Semi-Hard typeFirst 1.865 1.866 0.05

Second 1.240 1.241 0.08

Harp typeFirst 1.894 1.895 0.05

Second 1.330 1.330 0.01


4.3. Verification of nonlinear inelastic seismic

behavior

The accuracy of the proposed program in predicting the

nonlinear inelastic seismic behavior of the cable-stayed

bridge is verified herein by comparing with ABAQUS.

The same structure, as presented in the previous example,

is used for verification. The cross beam, girder, and tower

members of the bridges are modeled using beam-column

elements in the proposed program and B33 beam elements

in ABAQUS. The B33 beam element of ABAQUS, as

presented in Fig. 7, has three numerical integration points

on element and sixteen numerical integration points on

cross-section. The cable members are modeled using

equivalent straight truss elements in both the proposed

program and ABAQUS since ABAQUS does not provide

a cable element.

Figures 8 and 9 show the vertical displacement

responses at the middle point of the central span obtained

by ABAQUS and proposed program for two earthquake

loadings of the El-Centro and Loma Prieta, respectively.

The peak vertical displacements at the middle point of the

central span of the bridges with three different cable

layouts of fan, semi-harp, and harp type are presented in

Table 5 with the maximum difference of 3.9%. A good

correlation of nonlinear inelastic seismic behavior in all

cases generated by ABAQUS and the proposed programs

Figure 5. Vertical displacement responses at the middlepoint of the central span of the bridges subjected to El-Centro earthquake for nonlinear elastic seismic response.

Figure 6. Vertical displacement responses at the middlepoint of the central span of the bridges subjected to LomaPrieta earthquake for nonlinear elastic seismic response.


is obtained including the slight permanent shifts in

displacement due to inelastic behavior under the Loma

Prieta earthquake. In the case of the El-Centro earthquake,

having the smallest PGA, the displacement responses of

the elastic analysis (Fig. 5) and inelastic analysis (Fig. 8)

are almost the same because the seismic behavior of the

bridges is almost in the elastic range in this case. As in

the previous example, this one also indicates that the

proposed program is able to accurately predict displacements,

which is an important index for a performance-based

seismic design. Using the same personal computer

configuration (Pentium IV 3.2GHz), the computational

Table 4. Comparison of vertical displacement response (mm) at the middle point of the central span of the bridges forelastic analysis

Earthquake type Max/min Cable layouts SAP2000 3D-PAAP (proposed) Error (%)

El-Centro

Max

Fan type 44.58 44.13 1.01

Semi-Harp type 52.33 51.65 1.31

Harp type 60.05 58.99 1.76

Min

Fan type -41.30 -41.78 1.16

Semi-Harp type -44.16 -44.95 1.78

Harp type -45.22 -46.18 2.11

Loma Prieta

Max

Fan type 66.57 66.42 0.23

Semi-Harp type 74.72 75.94 1.62

Harp type 81.44 80.91 0.66

Min

Fan type -63.18 -62.28 1.43

Semi-Harp type -77.36 -76.01 1.74

Harp type -95.99 -96.88 0.93

Figure 7. Integration point of B33 beam element.

Table 5. Comparison of vertical displacement response (mm) at the middle point of the central span of the bridges forinelastic analysis

Earthquake type Max/min Cable layouts ABAQUS 3D-PAAP (proposed) Error (%)

El-Centro

Max

Fan type 49.82 49.01 1.62

Semi-Harp type 57.43 56.80 1.10

Harp type 66.52 65.54 1.48

Min

Fan type -49.26 -49.92 1.34

Semi-Harp type -52.10 -52.74 1.24

Harp type -51.56 -51.80 0.46

Loma Prieta

Max

Fan type 58.57 58.68 0.19

Semi-Harp type 68.00 70.07 3.04

Harp type 73.92 73.39 0.72

Min

Fan type -65.46 -65.42 0.07

Semi-Harp type -73.10 -70.25 3.90

Harp type -94.07 -91.56 2.67


times of ABAQUS and proposed program are 9.7 h and

1.3 h, respectively, for the harp type bridge subjected to

the Loma Prieta earthquake. This result proves the high

computational efficiency of the proposed program.

5. Conclusions

A computer program considering both geometric and

material nonlinearities in predicting the nonlinear inelastic

seismic response of the three-dimensional cable-stayed

bridges subjected to their own weight and earthquake

loadings has been developed. The conclusions of this

study are as follows:

(1) The proposed program which provides a cable

element can accurately predict the dynamic properties of

the three-dimensional cable stayed-bridges with three

different types of cable layouts.

(2) Using a nonlinear cable element, the proposed

program compares well with SAP2000 in capturing the

nonlinear elastic seismic behavior of the bridges.

(3) The proposed program can appropriately trace the

nonlinear inelastic seismic responses in comparison with

ABAQUS by using a minimum number of elements.

(4) The longest analysis times among several analysis

Figure 8. Vertical displacement responses at the middlepoint of the central span of the bridges subjected to El-Centro earthquake for nonlinear inelastic seismic response.

Figure 9. Vertical displacement responses at the middlepoint of the central span of the bridges subjected to LomaPrieta earthquake for nonlinear inelastic seismic response.


cases are 9.7 h and 1.3 h by ABAQUS and the proposed

program, respectively. It shows that the proposed method

is more practical than finite element method, and the

proposed program can be effectively used as a powerful

tool for use in daily design.

References

Abdel-Ghaffar, A. M. and Nazmy, A. S. (1991). “3-D

nonlinear seismic behavior of cable-stayed bridges.”

Journal of Structural Engineering, ASCE, 117 (11), pp.

3456-3476.

Chen, W. F. and Lui, E. M. (1987). Structural stability:

Theory and implementation, Elsevier, New York.

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