14966711 Three Dimensional Nonlinear Dynamic Analysis Of Cable Stayed Bridge

사장교의 3차원 비선형 동적 해석 Three-dimensional Nonlinear Dynamic Analysis of Cable-Stayed Bridge 1)

태 후 타이 Thai, Huu-Tai

․

2)

김 승 억 ,† Kim, Seung-Eock

․

3)

김병석 Kim, Byung Suk

․

4)

조창빈 Joh, Chang Bin

요 약：본 논문에서는 3차원 사장교의 비선형 동적 해석을 개발하였다. 소성힌지이론을 사용하여 재료의 비선형성을 고려하였고, 안정함수 를 사용하여 기하학적 비선형을 고려하였다. 케이블은 등가 케이블 탄성계수를 사용하여 모델링 하였다. 구조물의 비선형 거동을 예측하기 위하여 기하학적 비선형과 재료적 비선형을 고려한 프로그램을 개발하였다. 해석 예를 통하여 제안된 프로그램의 정확성과 효율성을 입증하였다. ABAQUS 와 SAP2000의 결과와 비교함으로써 개발된 프로그램이 실무설계에 사용할 수 있고 신뢰할 수 있는 효율적인 도구임을 입증하였다. ABSTRACT：This paper presents the nonlinear dynamic analysis of three-dimensional cable-stayed bridges. The plastic hinge concept is used to model the material yielding while the stability functions are employed to capture the geometric nonlinearity. The cable is modeled

using an equivalent cable modulus of elasticity. A computer software considering both geometric and material nonlinearities is developed to predict the nonlinear behavior of the structures. Accuracy and efficiency of the proposed software are verified by comparing with the SAP2000 and ABAQUS. It is concluded that the proposed software proves to be a reliable and efficient tool for daily use in engineering design. 핵 심 용 어 : 비선형 해석, 사장교, 기하학적 비선형 KEYWORDS : nonlinear analysis, cable-stayed bridge, geometric nonlinearity

for cables to study the geometric nonlinear behavior of two-dimensional cable-stayed bridges. Nazmy and Ahmed (1990) and Kanok-Nukulchai and Hong (1993) performed the geometric nonlinear analysis of the three-dimensional cable-stayed bridges using equivalent modulus of elasticity for modeling the cable stays. Song and Kim (2006) dealt with the nonlinear inelastic problem of the three-dimensional cable-stayed bridges using the bifurcation point and limit point instability approach. However, most of these researches are limited to the nonlinear static analysis of the cable-stayed bridges. The nonlinear dynamic behavior of cable-stayed bridges should to be investigated.

1. Introduction Cable-stayed bridges are becoming very popular in bridge engineering in recent decades because of their aesthetic appearance and uniqueness. The most important feature of this kind of structures is the high nonlinearities of geometry and material. Geometric nonlinearities come from the sag effect due to the self-weight of cable stays, the interaction between axial and flexural deformations, and the large displacement due to geometric changes. Material nonlinearity occurs when the factored loads are applied. Fleming (1979) used an equivalent modulus of elasticity for cables to study nonlinear behavior of a planar system while Karoumi (1999) used the two-node catenary elements

In this paper, the nonlinear dynamic analysis of the three-dimensional cable-stayed bridges is presented. All sources of geometric and material nonlinearity have been considered in this study. A computer software which is capable of considering both geometric and material

1) 세종대학교 토목환경공학과 박사과정 ([email protected]) 2) 정회원, 세종대학교 토목환경공학과 교수 (Tel. 02-3408-3391, Fax. 02-3408-3332, [email protected])

3) 건설기술연구원 책임연구원([email protected]) 4) 정회원, 건설기술연구원 선임연구원([email protected])

262

nonlinearities is developed.

Numerical

examples are

2.2. Modeling of beam-column members

presented to verify for accuracy and efficiency of the proposed software (3D-PAAP).

The large deformations in the pylon and girder members due to the combined effect of high axial forces and large

2. Formulation

bending moments, produce a strong coupling between axial and flexural stiffness in these members. This coupling can be considered in nonlinear analysis by introducing the

2.1. Modeling of cables

concept of stability function.

The stiffness matrix

formulation of a three-dimensional beam-column element proposed by Kim et al. (2001) 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. New-Orbison’s full plastification surface of cross section, as presented by McGuire et al. (2000), is given by

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 a cable, which varies depending on the sag, is modeled by using an equivalent straight truss element with an equivalent modulus of elasticity. The concept of an equivalent cable modulus of elasticity was first proposed by Ernst (1965) and has been verified by several researchers. The equivalent cable modulus of elasticity is given by

α = p 2 + m 2z + m 4y + 3.5p 2 m 2z + 3.0p 6 m 2y + 4.5m 4z m 2y

(2)

where p = P/Py, mz = Mz/Mpz for strong-axis, and my = My/Mpy for weak-axis

Eeq =

E ( wL ) 2 AE 1+ 12T 3

(1a) To treat the strain reversal effect in the hinge due to the abrupt change in applied direction of dynamic load, the scalar parameter η, which allows for gradual inelastic

in which 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.

stiffness reduction of the element associated with plastification at member end, is modified based on the double modulus theory in Kim et al. (2006) as follows:

When the tension in cable changes from Ti to Tf during the application of a load increment, the secant value of the equivalent modulus of elasticity over the load increment is

where

η d = η dy η dz

η dy =

given as follows:

4η 0

(1+

η0

)

2

(3)

and η dz =

2η 0 1+ η 0

(4)

in which

Eeq = 1+

E 2 ( wL ) (Ti + T f ) AE 2

24Ti T

(1b)

η 0 = 1.0 for α≤0.5 and η 0 = 4 α ( 1 - α ) for α > 0.5

2 f

(5)

The axial stiffness of the cable with the equivalent modulus of elasticity is used as the axial stiffness of the truss element.

2.3. Dynamic analysis

263

The incremental equation of motion of the structures is given by

[M ][ Δu&& ] + [C ][ Δu& ] + [K ][ Δu ] = [ΔF ] where

[K ]

mass matrix;

is the stiffness matrix;

[Δu&& ]

[Δu& ]

,

,

[Δu ]

[ t Δu ] is known, [ t Δ̇u ] and [ determined by the following equations

is the lump

, and

Δ̈u ] can be

⎛ γ t γ γ ⎞ && ⎤⎦ ⎡⎣ t Δ u& ⎤⎦ = ⎡ Δ u ⎤⎦ − ⎡⎣ t u& ⎤⎦ +Δt ⎜ 1 − ⎟ ⎡⎣ t u βΔt ⎣ β 2β ⎝ ⎠ 1 1 t 1 t && ⎤⎦ = && ⎤ ⎡⎣ t Δ u ⎡ t Δ u ⎤⎦ − ⎡⎣ u& ⎤⎦ − ⎡ u 2 ⎣ βΔt 2β ⎣ ⎦ β ( Δt )

exciting force vectors, respectively, over a time increment of Δt. The viscous damping matrix of the structures [C]

[C ] = a [ M ] + b [ K 0 ]

t

[ΔF ]

are the incremental acceleration, velocity, displacement and

is defined by

c e

⎡1 ⎤ ⎡ 1 ⎤ ⎛ γ ⎞ γ +⎢ [M ] + [C ]⎥ ⎡⎣ t u& ⎤⎦ + ⎢ [M ] +Δt ⎜ − 1 ⎟ [C ]⎥ ⎡⎣ t u&& ⎤⎦ β 2β 2β ⎣ βΔt ⎦ ⎝ ⎠ ⎣ ⎦

(6)

[M ]

On

⎡ ⎤ γ 1 M ]⎥ ⎡⎣ t Δu ⎤⎦ = [ ΔF ] ⎢[ K ] + [C ] + 2 [ βΔt β ( Δt ) ⎣⎢ ⎦⎥

(9)

, in which a and b

are mass- and stiffness-proportional damping factors,

[K 0 ]

respectively, and

Finally, the displacement, velocity, and acceleration at the

is the initial stiffness matrix.

end of the time step (t+ Δt) are obtained as follows:

There are several numerical approaches to solve the equation (6). The Newmark's method with the assumption of average acceleration is used in this study to get step-by-step numerical solution of the equation (6). The detailed algorithm of the Newmark's method, as presented in Chopra (2001), can be summarized as following

⎡⎣ t+Δt u ⎤⎦ = ⎡⎣ t u ⎤⎦ + [ Δ u ] ⎡⎣ t+Δt u& ⎤⎦ = ⎡⎣ t u& ⎤⎦ + [ Δ u& ] ⎡⎣ t+Δt u && ⎤⎦ = ⎡⎣ t u && ⎤⎦ + [ Δ u && ]

equations. The velocity and displacement at time (t+ Δt)

The procedure presented in equations (8), (9), and (10) is repeated for the next time steps until the considered frame is collapsed or desired time duration ends.

can be written in terms of integration parameters of β and γ as follows: ⎡⎣ t+Δt u& ⎤⎦ = ⎡⎣ t u& ⎤⎦ + (1 − γ ) Δt ⎡⎣ t u&& ⎤⎦ +γΔt ⎡⎣ t+Δt u&& ⎤⎦

(7a)

&& ⎤⎦ ⎡⎣ t+Δt u ⎤⎦ = ⎡⎣ t u ⎤⎦ +Δt ⎡⎣ t u& ⎤⎦ + ( 0.5 − β )( Δt ) ⎡⎣ t u

(7b)

2

+ β ( Δt )

t in which ⎡⎣ u&& ⎤⎦

,

2

3. Verifications The proposed software, 3D-PAAP, considering both geometric and material nonlinearities in dynamic analysis is verified by comparing with the SAP2000 and ABAQUS. The three-dimensional cable-stayed bridges with two different cable layouts of fan-type and harp-type under earthquake loading of El-Centro 1940 (Figure 2) are used for verification of the proposed software. The geometric dimensions and material properties of the three-dimensional cable-stayed bridge, taken from Morris (1974), 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. The earthquake loading of El-Centro 1940 is applied as the ground

&& ⎤⎦ ⎡⎣ t+Δt u

⎡⎣ t u& ⎤⎦

t , and ⎡⎣ u ⎤⎦

(10)

are the total

acceleration, velocity, and displacement vectors at time t. The integration parameters of β and γ are taken herein as 1/4 and 1/2, respectively, correspond to the assumption of the average acceleration method. Using equation (7), with some efforts, the final form of incremental equation of motion can be rewritten as

(8)

264

acceleration at the support of the cable-stayed bridge in

behavior. The pylon and girder members of the bridge are

longitudinal direction.

modeled as the frame elements in SAP2000

and

beam-column elements in 3D-PAAP. The cable is modeled as the cable element in both 3D-PAAP and SAP2000 softwares.

60.96

Fan Type

SAP2000

40

Relative displacement (mm)

[email protected]

[email protected]

30

[email protected]=274.32

22.86

[email protected]=274.32

3D-PAAP

Ground motion

Harp Type

d=0.15

t=0.05

Pylon and girder: E=207GPa σ y=248MPa

1.0

2.0

t=0.05

1.0

1.0

(Pylon)

(Girder)

Cable: E=158.6GPa σ y=1103MPa

(Cable)

20 10 0 0

10

20

30

-10 -20 -30 Time (sec)

-40

Fig. 1. Cable-stayed bridges (unit: m)

Fig. 3. Displacement response at mid span (Fan type)

SAP2000

40

3D-PAAP

0.4 Relative displacement (mm)

30

0.3

Acceleration (g)

0.2 0.1 0 0

10

20

30

-0.1

20 10 0 0

20

30

-20

-0.2

-30

-0.3

-40

-0.4

10

-10

Time (sec)

Time (sec)

Fig. 4. Displacement response at mid span (Harp type) Fig. 2. El-Centro Earthquake Table 1. Comparison of maximum displacement response at the mid span (mm) using cable element

3.1. Verification 1: Nonlinear elastic dynamic analysis This example investigates the elastic dynamic behavior of the three-dimensional cable-stayed bridge using 3D-PAAP and SAP2000 softwares. It should be informed that SAP2000 is not capable of investigating inelasic

265

Cable layouts

SAP2000

3D-PAAP

Error (%)

Fan-type

33.99

33.98

0.03

Harp-type

34.54

34.07

1.36

The results for displacement response at mid span of

ABAQUS

60

3D-PAAP

two different type of cable-stayed bridges are shown in Relative displacement (mm)

Figures 3 and 4 and Table 1. It can be seen that the results obtained by the use of 3D-PAAP and SAP2000 softwares correlate very well. The maximum error in displacement at the mid span of the cable-stayed bridges is 1.36%, which is a rather small value.

40 20 0 0

10

20

30

-20 -40

3.2. Verification 2: Nonlinear inelastic dynamic analysis Time (sec)

-60

In order to verify the accuracy of the 3D-PAAP in predicting the dynamic behavior of the cable-stayed bridge in the inelastic range, the ground acceleration of 150% of El-Centro earthquake is applied to the cable-stayed bridges. The ABAQUS, with capacity of nonlinear dynamic inelastic analysis, is used to verify for the proposed software in predicting the dynamic inelastic response of the bridge. The pylon and girder members of the bridge are modeled as beam elements (B31) in ABAQUS and beam-column elements with plastic hinge model in 3D-PAAP. The cable stays are modeled by using an equivalent straight truss elements in both 3D-PAAP and ABAQUS since ABAQUS does not provide cable element.

Fig. 6. Displacement response at mid span (Harp type)

Table 2. Comparison of maximum displacement response at the mid span (mm) using truss element ABAQUS

3D-PAAP

Error (%)

Fan type

50.93

50.82

0.22

Harp type

51.65

50.42

2.38

The displacement response at mid span of two different type of cable-stayed bridges is shown in Figures 5 and 6 and Table 2. Strong agreement in result is also obtained with the maximum error in displacement at the mid span of 2.38%.

ABAQUS

60

Cable layouts

Relative displacement (mm)

3D-PAAP

40

4. Conclusions

20 0 0

10

20

A computer software considering both geometric and material nonlinearities has been developed in this paper. The accuracy of the proposed software is verified with SAP2000 in elastic range, and with ABAQUS in inelastic range through the three-dimensional cable-stayed bridges with two different cable layouts of fan-type and harp-type. It can be concluded that the proposed software is capable of predicting accurately the nonlinear dynamic behavior of the cable-stayed bridges.

30

-20 -40 -60

Time (sec)

Fig. 5. Displacement response at mid span (Fan type)

266

Acknowledgements 본 연구는 한국건설기술연구원의 하이브리드 사장교 구조 시스템 통합 기술 개발 과제의 지원에 의하여 이루어졌음을 밝히며 이에 감사드립니다.

References Chopra, A.K. (2001). Dynamics of structures: Theory and applications to earthquake engineering. Prentice Hall, New Jersey. Ernst, H.J. (1965) Der E-Modul von Seilen unter Berucksichtigung des Durchanges, DerBauingenieur, Vol. 40, No. 2, pp. 52-55. Fleming, J.F. (1979) Nonlinear static analysis of cable-stayed bridge structures, Journal of Computers and Structures, Vol. 10, pp. 621-635. Kanok-Nukulchai, W., and Hong, H. (1993) Nonlinear Modeling of Cable-Stayed Bridges, Journal of Construction Steel Research, Vol. 26, pp. 249-266. Karoumi, R. (1999) Some modeling aspects in the nonlinear finite element analysis of cable-supported bridges, Journal of Computers and Structures, Vol. 71, pp. 397-412. Kim, S. E., Park, M.H., and Choi, S.H. (2001) Direct design of three-dimensional frames using practical advanced analysis, Engineering Structures, Vol. 23, No. 11, pp. 1491-1502. Kim, S.E, Cuong, N.H, Dong, H.L (2006) Second-order inelatic dynamic analysis of 3-D steel frames, International Journal of Solid and Structures, Vol. 43, pp. 1693-1709. McGuire W., Gallagher R.H., Ziemian R.D., (2000) Matrix Structural Analysis. JohnWiley & Son, Inc. 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. Song, W.K, Kim, S.E. (2006) Analysis of the overall collapse mechanism of cable-stayed bridges with different cable layouts, Engineering Structures, In press.

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