14966613 Practical Nonlinear Dynamic Analysis Of Cable Stayed Bridges

구

조

Practical Nonlinear Dynamic Analysis of Cable-Stayed Bridge ∗

Thai Huu-Tai

•

Kim Seung-Eock

∗∗

1. Introduction Cable-stayed bridges are widely used in bridge engineering in recent decades because of their aesthetic appearance and uniqueness. The theory and formulation of the nonlinear dynamic analysis of cable-stayed bridge have been presented in detail in Tai et al. (2007). The purpose of this paper is to focus on the case studies of the behavior of a cable-stayed bridge. The results of static behavior, natural period as well as dynamic response of the cable-stayed bridge generated by SAP2000, ABAQUS, and proposed software are compared. The good results obtained in all cases of analysis prove that the proposed software can efficiently be used in predicting the nonlinear behavior of cable-stayed bridges subjected to static and dynamic loading.

2. Formulation

2.1. Modeling of cables The cables are assumed to be perfectly flexible and to resist the tensile force only. The tension stiffness of a cable is modeled by using an equivalent modulus of elasticity proposed by Ernst (1965). The secant value of the equivalent modulus of elasticity, when the tension in cable changes during the application of a load increment, given by Nazmy and Ahmed (1990) is used in this paper.

2.2. Modeling of beam-column members The large deformations in the tower and girder members due to the combined effect of axial forces and bending moments produce a strong coupling between axial and flexural stiffness in these members. This coupling can be considered in nonlinear analysis by using the stability functions. The stiffness matrix formulation of a three-dimensional beam-column element proposed by Kim et al. (2006) is applied in this paper. The gradual yielding due to flexure can be traced by using the parabolic function. The yielding level at the end of member is determined by using the New-Orbison yield surface. To treat the strain reversal effect in the hinge due to the abrupt change in applied direction of dynamic load, the modified double modulus theory proposed by Kim et al. (2006) is used in this study.

∗ ∗∗

PhD Student • Dept. of Civil & Environmental Engineering, Sejong University • E-mail: [email protected] Member, Professor • Dept. of Civil & Environmental Engineering, Sejong University • E-mail: [email protected] - Presenter

13

구

조

2.3. Dynamic analysis The Newmark β-method with the assumption of average acceleration is used in this study to solve for step-by-step numerical solution of the equation of motion. The detailed algorithm of Newmark β-method was presented in Chopra (2001). The integration parameters β =1/4 and γ =1/2 are chosen which correspond to the assumption of the average acceleration method. 3. Case studies The proposed software is verified for accuracy in predicting static behavior of a cable-stayed bridge as well as its vibration and nonlinear response under earthquake loading of El-Centro 1940. The geometric dimensions and material properties of the three-dimensional cable-stayed bridge are presented in Figure 1. Mass- and stiffness-proportional damping factors are chosen such that the equivalent viscous damping ratio is equal to 5%. The gravity load due to the weight of the structure is applied first to the structure as lumped masses at the nodes.

Fig. 1. Cable-stayed bridges (unit: m) 3.1. Static behavior The load-deflection curves at the mid span of the bridge generated by ABAQUS and 3D-PAAP are compared in Figure 2. It can be seen that the ABAQUS generally overestimates the capacity of the bridge whereas the proposed software can accurately predict the capacity of the bridge. 3.2. Dynamic response The first two natural periods of the cable-stayed bridge are compared in Table 1. It can be observed that a strong agreement of period of the study bridge generated by SAP2000 and 3D-PAAP is obtained. Table 1. Comparison of first two natural periods Mode First Second

SAP2000 9.81 4.07

14

3D-PAAP 9.83 4.09

Error (%) 0.21 0.37

구

조

P δ

600 500

P (kN)

400 ABAQUS-2el. per girder member

300

ABAQUS-5el. per girder member ABAQUS-10el. per girder member

200

ABAQUS-20el. per girder member ABAQUS-40el. per girder member

100

3D-PAAP-1el. per girder member

0 0

-200

-400

-600

-800

-1000

-1200

δ (mm)

Fig. 2. Static behavior of cable-stayed bridge Table 2. Comparison of maximum displacement (Nonlinear elastic analysis) Displacements (mm) Top of tower (Horizontal) Mid span (Vertical)

SAP2000 41.79 41.59

3D-PAAP 42.65 39.82

Error (%) 2.06 4.26

The displacement responses at the top of tower and the mid span of the cable-stayed bridge are shown in Figure 3 and Table 2. It can be seen that the results obtained by using SAP2000 and 3D-PAAP software correlate well. SAP2000

40

3D-PAAP

20 10 0 -10 0

5

10

15

20

25

30

-20 -30 -40

3D-PAAP

30 Displacement (mm)

Displacement (mm)

30

SAP2000

40

20 10 0 -10 0

5

10

15

20

25

-20 -30 -40

Time (sec)

a. Horizontal displacement at the top of tower

Time (sec)

b. Vertical displacement at the mid span

Fig. 3. Displacement elastic response Table 3. Comparison of maximum displacement (Nonlinear inelastic analysis) Displacements (mm) Top of tower (Horizontal) Mid span (Vertical)

ABAQUS 41.08 46.91

15

3D-PAAP 41.91 44.71

Error (%) 2.01 4.71

30

구

조

The nonlinear inelastic seismic responses obtained by ABAQUS and 3D-PAAP are presented in Figure 4 and Table 3. As can be observed from those, the difference of displacement response of second-order inelastic analysis is not remarkable with the maximum error at the mid span of 4.71%. 40

50

ABAQUS

3D-PAAP

40

3D-PAAP

30

20 10 0 -10 0

5

10

15

20

25

30

-20

Displacement (mm)

Displacement (mm)

30

ABAQUS

-30 -40

20 10 0 -10 0

5

10

15

20

25

30

-20 -30 -40

Time (sec)

-50

a. Horizontal displacement at the top of tower

Time (sec)

b. Vertical displacement at the mid span

Fig. 4. Displacement inelastic response 4. Conclusions A computer software considering both geometric and material nonlinearities has been developed in this paper. The efficiency and accuracy of the proposed software are verified by comparing with the SAP2000 and ABAQUS through static behavior, natural period as well as dynamic response of the cable-stayed bridge. The good results obtained in all cases of analysis prove that the proposed software is capable of predicting accurately the nonlinear dynamic behavior of the cable-stayed bridges. Acknowledgements This paper is a part of the result from the "Standardization of Construction Specifications and Design Criteria based on Performance ('06~'11)", the "Construction & Transportation R&D Policy and Infrastructure Project". References 1. Chopra, A.K. (2001). Dynamics of structures: Theory and applications to earthquake engineering. Prentice Hall, New Jersey. 2. Ernst, H.J. (1965) Der E-Modul von Seilen unter Berucksichtigung des Durchanges, DerBauingenieur, Vol. 40, No. 2, pp. 52-55. 3. Kim, S.E, Cuong, N.H, Lee, D.H. (2006) Second-order inelatic dynamic analysis of 3-D steel frames, International Journal of Solid and Structures, Vol. 43, pp. 1693-1709. 4. Nazmy, A.S., and Ahmed, M. (1990) Three dimensional nonlinear static analysis of cable-stayed bridges, Journal of Computers and Structures, Vol. 34, No. 2, pp. 257-271. 5. Tai, T.H, Kim, S.E, Kim, B.S, and Joh, C.B (2007) Three-dimensional nonlinear dynamic analysis of cable-stayed bridge. Proceedings of Korean Society of Steel Construction Conference, pp. 63.

16