Computer simulation of buffeting actions of suspension bridges under turbulent wind

Computers and Structures 76 (2000) 787±797

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Computer simulation of bu€eting actions of suspension bridges under turbulent wind Q. Ding, P.K.K. Lee* Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong Received 28 December 1998; accepted 10 June 1999

Abstract Suspension bridges are long-span ¯exible structures susceptible to various types of wind induced vibrations such as bu€eting actions. In this paper, a three dimensional ®nite-element model formulated to deal with suspension bridges under turbulent wind is presented. In this model, all sources of geometric nonlinearity such as cable sag, force-bending moment interaction in the bridge deck and towers, and changes of bridge geometry due to large displacements, are fully considered. The wind loads, composed of steady-state wind loads, bu€eting loads and selfexcited loads, are converted into time domain by using the computer simulation technique. The Newmark-b step by step numerical integration algorithm is used to calculate the bu€eting responses of bridges. Compared with the results obtained by classical bu€eting theory, the validity of the simulation is proved. 7 2000 Elsevier Science Ltd. All rights reserved. Keywords: Suspension bridge; Bu€eting; Wind simulation; Geometrically nonlinear

1. Introduction Early this century, with the rapid development of construction material and technology, many long-span suspension bridges were built all over the world. However, these bridges exhibit special characteristics such as high ¯exibility, low structural damping and light in weight, which are very susceptible to wind actions. The Tacoma disaster in 1940 aroused engineers' attentions to the vibration characteristics of suspension bridges and the responses to wind excitations. Since then numerous researchers [1±4] have made signi®cant contributions to the stability of suspension bridges under

* Corresponding author. Tel.: +852-2540-8829; fax: +8522559-5337. E-mail address: hreclkk.hkucc.hku.hk (P.K.K. Lee).

wind actions which form the foundation of bridge aeroelasticity. Bu€eting action is a type of vibration motion induced by turbulent wind. As natural wind is not steady but turbulent in character, wind ¯uctuations in the vertical and horizontal direction are random in space and thus, the wind pressures along the bridges are random in time and space. Depending on the spectral distribution of the pressure vectors, certain modes of vibration of the bridge may selectively be excited. In fact, these wind induced bu€eting actions are related not only to wind speed, but also to the shape of the cross-section of the bridge deck and the interaction between the bridge and wind motion [5]. The turbulence induced bu€eting response will unavoidably occur at both bridge construction and completion stages. Sometimes, a large response may cause fatigue of bridge elements, discomfort to users and even endanger the safety of the bridge. With

0045-7949/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 9 9 ) 0 0 1 9 7 - 2

788

Q. Ding, P.K.K. Lee / Computers and Structures 76 (2000) 787±797

Fig. 1. Wind components at a point along the bridge axis.

high expectations on the performance of modern suspension bridges, bu€eting analysis has been one of the most important aspect of structural reliability under turbulent wind. The traditional bu€eting analysis method resorts to model superposition in a frequency domain that is limited to the linearization scope. However, it does not re¯ect the whole response procedure of bridges and hence, cannot accurately analyse fatigue of bridge elements and assess service comfortability of bridges under wind action. In this paper, a time±domain simulation method is proposed to analyse the bu€eting response of suspension bridges under turbulent wind conditions. The entire bu€eting procedure can be accurately re¯ected to provide useful information for further study on element fatigue, bridge comfortability as well as bridge safety. In this method, a three dimensional ®nite-element model, which takes into account all geometric nonlinearity, has been developed to model suspension bridges. Based on Scanlan's method [6], the wind loads including steady-state wind loads, bu€eting loads and self-excited loads are described as functions of aerodynamic parameters which can be obtained through wind tunnel tests. These wind loads are converted into time domain by using the computer simulation technique. Finally, the Newmark-b step by step numerical integration algorithm is used to calculate the bu€eting responses of bridges. Comparing with the results obtained by the classical bu€eting theory, the validity of the simulation method has been con®rmed.

vertical direction. These three wind components impose drag D, lift L and moment M on a bridge deck, as shown in Fig. 1. In bu€eting analysis, the total wind loads are made up of steady-state wind loads, bu€eting loads and self-excited loads.

2. Wind forces

2.2. Bu€eting loads

According to the classical airfoil theory, it is assumed that the wind velocity at one point along a bridge comprises three components: one for mean part U, one for ¯uctuating part u(x, t ) in the along-wind direction and one for ¯uctuating part w(x, t ) in the

Bu€eting loads are caused by the ¯uctuating part of the wind velocity of which the along-wind part u and the vertical part w are much smaller than the mean wind velocity U. On the assumption of neglecting u2 , w2 and uw, bu€eting loads per unit span can be

2.1. Steady-state wind loads Steady-state wind load is only related to the mean part of oncoming wind components. At the mean wind speed U of the oncoming ¯ow at the deck elevation, the steady-state drag, lift and moment can be expressed as follows: D ˆ 0:5rU 2 HCD

…1a†

L ˆ 0:5rU 2 BCL

…1b†

M ˆ 0:5rU 2 B 2 CM

…1c†

where r is the air density, B is the deck width and H is the deck height. CD , CL and CM are the drag, lift and moment coecients obtained by wind tunnel tests on bridge cross-section model. During the bu€eting procedure of the bridge, the steady-state wind loads are assumed to act on the bridge deck at all times. Therefore, these can be regarded as static loads which only shift the bridge's equilibrium position to a new position and are not directly related to bu€eting responses.

Q. Ding, P.K.K. Lee / Computers and Structures 76 (2000) 787±797

expressed according to the quasi-steady theory as follows:   1 u…x, t† Lb …x, t† ˆ ÿ rU 2 B 2CL …a0 † ‡ C L0 …a0 † 2 U   A w…x, t† ‡ CD …a0 † …2a† B U

Mb …x, t† ˆ

Db …x, t† ˆ

 1 u…x, t† rU 2 B 2 2CM …a† 2 U  w…x, t† 0 ‡CM …a0 † U   1 A u…x, t† rU 2 B 2 CD …a0 † 2 B U

…2b†

…2c†

0 a0 is the e€ective attack angle of the wind. CL0 and CM are slopes of CL and CM at the angle of a0 : A is the cross-wind projected area (per unit span) normal to the mean wind speed. The other symbols have the same de®nition as above. However, it is quite dicult to obtain the representative ¯uctuating parts of the wind in practice. This is the reason why the computer simulation technique is used to generate the ¯uctuating wind according to the wind spectrum on site. It is assumed that wind motion is a multi-correlated random process. At any point along the bridge deck, the wind motion is in random procession. At the same time, the wind motion at a point is also related to that at any other point. Kovacs et al. [7] have presented a method to simulate the ¯uctuating wind velocity along a bridge deck. In their method, the ¯uctuating wind velocities u(x, t ) and w(x, t ) are assumed to consist of a series of components in the frequency domain from 0.001 to 1.5 Hz. This is the scale at which wind speed has the greatest e€ect on the bu€eting responses of bridges. Each component is a complex vector, i.e., each has an absolute size and a starting phase. The simulation takes place similar to a usual random signal: the component vectors are rotated with the speed of their own angular velocity and are summed vectorially. The ¯uctuating wind velocity constitutes the real part of the resultant. The turbulent part of the momentary wind velocity in point x can be generated from Davenport's coherence formula in :

u…x, t† ˆ 2v 3 X u
u 2S…nj †Dn X ÿ cos 2pn C x ÿ x , n t ‡ f 6u X ÿ j j k kj 7 …
4t 5 j C 2 xk , nj k k

…3†

789

where C…x, n† ˆ eÿ2Knjxj=U ˆ COH 2D …x, n†; COHD …x, n† ˆ eÿKnjxj=U K is the coherence decaying coecient, S(n ) is the target spectrum, xk is the generation spot, x is the distance between two generated spots, nj is the frequency and fkj is the starting phase which is uniformly distributed between 0 and 2p: Very frequently, Simiu spectrum is chosen as the longitudinal wind target spectrum and Panofsky± McCormick spectrum is chosen as the vertical wind target spectrum. Kovacs et al. also suggested that the decaying coecient can be taken as 4.5 for longitudinal and 7.5 for transverse velocity components. The bu€eting loads along bridge decks can be calculated by substituting the generated ¯uctuation wind velocities into Eq. (2a)±(2c). 2.3. Self-excited loads The self-excited loads are caused by interaction between wind motion and the structure. It involves the interaction of aerodynamic and inertial forces with the elastic structure such that the aerodynamic forces inject additional energy into the oscillating structure and increase the magnitude of vibration sometimes to catastrophic levels. It has been a tradition that the selfexcited loads are expressed in the form of indicial functions suggested by Scanlan [8]. However, Lin [9] considered that there are some redundancies in the classical formulations. Based on the assumptions that the self-excited loads are generated by a linear mechanism, Lin suggested another simple mathematical model for self-excited forces, for investigation of the aerodynamic stability of long span suspension bridges. The self-excited loads are expressed in terms of convolution integrals between bridge deck motion and the impulse response function which is shown to be equivalent to the classical indicial function type representation. Lin's model can be summarized as: …t Mse …t† ˆ Ma …t† ‡ Mh …t† ˆ fMh …t ÿ t†h…t† dt ‡

…t 1

ÿ1

fMa …t ÿ t†a…t† dt

Lse …t† ˆ La …t† ‡ Lh …t† ˆ ‡

…t 1

…4a†

fLa …t ÿ t†a…t† dt

…t ÿ1

fLh …t ÿ t†h…t† dt …4b†

790

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Fig. 2. Suspension bridge idealization in three dimensional space.

Dse …t† ˆ Da …t† ‡ Dp …t† ˆ ‡

…t 1

…t ÿ1

fDp …t ÿ t†p…t† dt …4c†

fDa …t ÿ t†a…t† dt

where fMa …t†, fMh …t†, fLa …t†, fLh …t†, fDa …t† and fDp …t† are response functions due to unit impulse displacement a, h and p. From these equations, it is seen that the aerodynamic coupling of the modes is induced by Mh …t†, La …t† and Da …t†: Applying the Fourier transform to Eqs. (4a)±(4c) and then comparing it with Scanlan's notion in terms of aerodynamic derivatives, the relationship between transfer functions and aerodynamic derivatives are obtained as:   FMa …o† ˆ rB 4 o2 A3 ‡ iA2 …5a†   FMh …o† ˆ rB 3 o2 A4 ‡ iA1

…5b†

  FLa …o† ˆ rB 3 o2 H 3 ‡ iH 2

…5c†

  FLh …o† ˆ rB 2 o2 H 4 ‡ iH 2

…5d†

 2

 

FDa …o† ˆ rB 3 o P 3 ‡ iP 2

…5e†

  FDp …o† ˆ rB 2 o2 P 3 ‡ iP 2

…5f†

Here, Ai , H i and P i …i ˆ 1, 4† are nondimensional ¯utter derivatives obtained by wind tunnel tests on cross-section of the deck. From classical airfoil theory, the transfer functions may reasonably be approximated by rational functions, speci®cally for transfer functions of ®rst order linear ®lters. The transfer functions FMa can, therefore, be expressed as FMa …n† " 2

ˆ rB U

2

n 2pC2 X 4p2 ‡ i2pdk n ‡ C1 ‡ i Ck 2 2 n d k n ‡ 4p2 kˆ3

#

…6†

Comparing Eq. (6) with Eq. (5), the ¯utter derivatives can be obtained as: 2

n X C1 Ck ‡ 2 n2 ‡ 4p2 4p2 d k kˆ3

A3

ˆn

A2

n X n Ck dk n2 C2 ‡ ˆ 2 2p d n2 ‡ 4p2 kˆ3 k

! …7† ! …8†

The unknown parameters Ck and dk can be obtained from the least square ®t of Eqs. (7) and (8). In this paper, a total of six unknown parameters Ci …i ˆ 1, 6† are used to compose the ®rst order linear ®lters. By taking the inverse Fourier transformation of the transfer functions, the time domain expression of impulse

Q. Ding, P.K.K. Lee / Computers and Structures 76 (2000) 787±797

791

Fig. 3. Cable element at local coordinate system.

response functions can be obtained. Substituting these impulse response functions into Eqs. (4a)±(4c) yields ( B 2 2 Ma …t† ˆ rB U C1 a…t† ‡ C2 a_ …t† U ‡ C3 ‡ C4

  C5 U …t ÿ t† a_ …t† dt exp ÿ B ÿ1

…t …t

ÿ1

…9†

)   C6 U …t ÿ t† a_ …t† dt exp ÿ B

The other ®ve items including Mh …t†, La …t†, Lh …t†, Da …t† and Dp …t† can be obtained similarly. It should be noted that every set of parameters Ci …i ˆ 1, 6† for Ma …t†, Mh …t†, La …t†, Lh …t†, Da …t† and Dp …t† is di€erent.

3. Structural idealization Normally, torsional sti€ness of suspension bridges with box girder cross sections is large and hence, the decks can be idealised as three dimensional spine beam structures as shown in Fig. 2. The major structural elements of a suspension bridge are: a) Cable element: The length of cable spanning between hangers which is capable of resisting axial tensile force only. b) Hanger element: The vertical member between a cable node and the deck which is capable of resisting tensile force only. c) Beam element: The length of the deck spanning along the centerline of the bridge between a pair of nodes on the centerline. The hanger and deck el-

ements are connected by horizontal rigid arms. They have bending sti€ness in the vertical and lateral direction and are capable of resisting torsional stresses. d) Tower element: Each tower is represented by a three-dimensional framework fully ®xed against movement at its base. In this frame, every element is a beam element having both bending sti€ness and torsion sti€ness. Usually bridge elements from the towers, deck, cables and hangers can be represented by two types of ®nite elements, i.e. spatial beam elements for towers and the deck and cable elements for cables and hangers. In spite of the fact that the behavior of the material of the structural elements in a long-span bridge is linear elastic, the overall load±displacement relationship for the structure is nonlinear [10]. This overall nonlinear behavior originates from (i) the in¯uence of cable sag to its equivalent modulus of elasticity, (ii) the in¯uence of initial stresses to structural sti€ness and (iii) the in¯uence of large displacements to structural sti€ness and loads. 3.1. Nonlinear sti€ness formulation of cable elements Due to the very small bending sti€ness, cable elements can be regarded as elements capable of resisting only axial force without any bending moment. In three-dimensional analysis, cable elements are composed of two nodes that have a total of six degrees of freedom as shown in Fig. 3. For a cable element, the displacement vector fxg can be written as fX g ˆ ‰ui , vi , wi , uj , vj , wj ŠT : The nonlinear behavior of cable elements results from its sag phenomenon. The axial sti€ness of cable

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Q. Ding, P.K.K. Lee / Computers and Structures 76 (2000) 787±797

Fig. 4. Beam element at local coordinate system.

elements is a€ected by the cable sag that is greatly in¯uenced by the amount of tension in the cable. When the cable tension increases, the sag decreases and apparent axial sti€ness increases. Here, an equivalent straight chord member with an equivalent modulus of elasticity, that combines the e€ects of both material and geometric deformations, is used to account for this variation in cable axial sti€ness [11]. The equivalent cable modulus of elasticity is given by Eeq ˆ

E …rgl†2 1‡ E 12s3

…10†

in which Eeq is the equivalent modulus, E is the Young's modulus of cable material, r is the density of the cable, l is the horizontal projected length and s is the tension stress of the cable. The total sti€ness matrix of a cable element is, therefore, a combination of the geometric sti€ness matrix and the elastic sti€ness matrix. 3.2. Nonlinear sti€ness formulation of beam elements When assuming small deformations in any structural system, the axial force and ¯exural sti€nesses of members in bending are usually considered to be uncoupled. However, when deformations are no

longer small, there is an interaction between axial and ¯exural deformations in such members under the combined e€ect of the axial force and the bending moment. As a result, the e€ective sti€ness of the member decreases for a compressive axial force and increases for a tensile force. Similarly, the presence of bending moments will also a€ect the axial sti€ness of the member. In traditional linear analysis, this interaction or coupling e€ect is negligible. However, for ¯exural structures such as long-span suspension bridges, large displacements will occur. This interaction can be signi®cant and should be considered in any nonlinear analysis. In three-dimensional analysis, a spatial beam element has twelve degrees of freedom as shown in Fig. 4. For beam elements, the displacement vector fX g can be written as   fX g ˆ ui , vi , wi , yxi , yyi , yzi , uj , vj , wj , yxj , yyj , yzj T Analogous to that of cable elements, the sti€ness of a beam element is composed of two parts: elastic sti€ness and geometric sti€ness. Based on the large displacement theory, a geometric sti€ness matrix of beam elements incorporating the contribution of the axial force and bending moments has been derived by Pan et al. [12].

Q. Ding, P.K.K. Lee / Computers and Structures 76 (2000) 787±797

4. Equation of motion and method of solution

where

The equations of motion in terms of a nodal displacement vector [U ] for a bridge structural system under wind loads can be expressed in conventional matrix notation as

PMa, S ˆ rB 2 U 2 C1

    ‰M Š U ‡ ‰C Š U_ ‡ ‰K Š‰U Š ˆ ‰P Š

PMa, D ˆ rB 2 U 2

2

PMa ˆ rB U

where [M ] is a diagonal matrix containing the mass and mass moment inertia of all elements lumped at the nodes, [C ] is structural damping matrix taken as Rayleight's damping, [K ] is the structural sti€ness matrix including elastic sti€ness matrix and geometrical sti€ness matrix, [P ] is the total wind load which includes state-steady wind load ‰Pst ], bu€eting wind load ‰Pbu Š and self-excited wind load ‰Pse ] If Eq. (9) is observed more closely, it is found that the self-excited loads can be expressed in three parts: aerodynamic sti€ness part, aerodynamic damping part and motion history part, as follows:

2

0 60 6 60 6 60 6 60 6 L6 60 26 60 60 6 60 6 60 6 40 0

0 BLh, S 0 ÿBMh, S 0 0 0 0 0 0 0 0

0 0 BDp, S 0 0 0 0 0 0 0 0 0

0 ÿBLa, S BDa, S BMa, S 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

…t ÿ1

…t ÿ1

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 BLh, S 0 ÿBMh, S 0 0

2

C3

  C5 U …t ÿ t† a_ …t† dt ‡ C4 exp ÿ B )   C6 U …t ÿ t† a_ …t† dt exp ÿ B

0 0 0 0 0 0 0 0 BDp, S 0 0 0

0 0 0 0 0 0 0 ÿBLa, S BDa, S BMa, S 0 0

3 0 07 7 07 7 07 7 07 7 07 7 07 7 07 7 07 7 07 7 05 0

0 0 0 0 0 0 0 0 0 0 0 0

and the aerodynamic damping matrix, 2

0 60 6 60 6 60 6 60 6 L6 60 26 60 60 6 60 6 60 6 40 0

0 BLh, D 0 ÿBMh, D 0 0 0 0 0 0 0 0

0 0 BDp, D 0 0 0 0 0 0 0 0 0

0 ÿBLa, D BDa, D BMa, D 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 BLh, D 0 ÿBMh, D 0 0

…13b†

…13c†

The other ®ve load items in Eqs. (4a)±(4c) can also be described analogously. Combining all aerodynamic sti€ness parts and damping parts together, the aerodynamic sti€ness and damping matrix in local coordinate system can be set up. The aerodynamic sti€ness matrix of an element can be written as

…12†

0 0 0 0 0 0 0 0 0 0 0 0

…13a†

(

…11†

Ma …t† ˆ PMa, S a…t† ‡ PMa, D a_ …t† ‡ PMa

C2 B ˆ rB 3 UC2 U

793

0 0 0 0 0 0 0 0 BDp, D 0 0 0

0 0 0 0 0 0 0 ÿBLa, D BDa, D BMa, D 0 0

0 0 0 0 0 0 0 0 0 0 0 0

3 0 07 7 07 7 07 7 07 7 07 7 07 7 07 7 07 7 07 7 05 0

794

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Fig. 5. General layout of Tsing Ma Bridge (Extracted from Ref. [13]).

With respect to the motion history parts, it can be seen that the items involve convolution integrals of velocities. These series integrals can be summarized as

Ij ˆ

… tj ÿ1

 _ exp … ÿ ci U=B†…tj ÿ t † d…t† dt

…14†

  Ij ˆ exp … ÿ ci U=B†…tj ÿ tjÿ1 † Ijÿ1   ‡ exp … ÿ ci U=2B†…tj ÿ tjÿ1 † Ddjÿ1

…15†

From the above equation, it can be seen that only the quantities involving Ijÿ1 and djÿ1 at time tjÿ1 are needed to be stored for evaluating Ij : Summarily, the motion equation can be rewritten as        ‰M Š U ‡ ‰C Š U_ ‡ ‰Ke Š ‡ Kg ‰U Š

It can been seen that for calculating their values, the integral Ij must be evaluated at every time step tj which is quite time consuming. Besides, the motion history for all elements must be stored, thus occupying a large memory of the computer. To tackle these diculties, a recursive algorithm for evaluating the integral is derived as follow.

  ˆ ‰Pst Š ‡ ‰Pbu Š ‡ ‰Kse Š‰U Š ‡ ‰Cse Š U_ ‡ ‰Pseh Š

…16†

where [Ke] is the elastic sti€ness matrix, [Kg] is the geometrical sti€ness matrix, [Pst] is the steady-state wind load vector, [Pbu] is the bu€eting load vector, [Kse] is the aerodynamic sti€ness matrix due to self-excited

Table 1 The ®rst twelve vibration frequencies of Tsing Ma Bridge Mode

Frequency (Hz)

1 2 3 4 5 6 7 8 9 10 11 12

BACSB 0.067 0.108 0.138 0.151 0.176 0.212 0.229 0.230 0.234 0.237 0.239 0.275

Xu et al 0.068 0.117 0.137 0.158 0.189 0.210 0.230 0.232 0.240 0.245 0.271 0.285

Mode type

Di€erence

Lateral bending of deck Vertical bending of deck Vertical bending of deck Lateral bending of deck Vertical bending of deck Lateral move of main span cables Lateral move of main span cables Lateral move of deck and cables Lateral move of deck and cables Vertical bending of deck Torsion of deck Lateral bending of deck

1.47% 7.69% ÿ0.73% 4.43% 6.88% ÿ0.95% 0.43% 0.86% 2.50% 3.27% 11.81% 3.51%

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795

Table 2 Aerodynamic coecients C1

C2

C3

C4

C5

C6

M 0.26412 ÿ0.51993 0.16645 0.29121 13.43210 5.87260 I ÿ1.08827 ÿ2.79466 0.99064 1.06341 13.09312 229.358

Fig. 6. Generated wind history for Tsing Ma bridge.

wind loads, [Cse] is the aerodynamic damping matrix due to self-excited wind loads, [Pseh] is the self-excited wind load related to motion history. In this paper, the Newmark-b method is used to analyze the bu€eting procedure of long span suspension bridges. In the Newmark-b method the nodal acceleration between t and t ‡ Dt is assumed as linear distribution:    U z ˆ …1 ÿ g† U t ‡g U t‡Dt …17†  t is the acceleration at time t, fUg  t‡Dt is the where fUg  z is a certain accelacceleration at time t ‡ Dt and fUg eration value between t and t ‡ Dt which must meet _ ˆ fUg  z Dt the requirement of middle method fDUg Substituting Eq. (17) into motion equation (16), the nodal displacement, velocity and acceleration at time t ‡ Dt can be obtained as    U t‡Dt ˆ U t ‡ DU t …18a†    U_ t‡Dt ˆ U_ t ‡ DU_ t

…18b†

fU gt‡Dt ˆ fU gt ‡fDU gt

…18c†

1377 m. It is the longest suspension bridge carrying both road and rail trac in the world. The main support towers are concrete structures of over 200 m height. The two main cables, each of which was formed from nearly 40,000 individual steel wires of 5 mm diameter, are approximately 1 m in diameter. The typical steel box-girder deck is 41.0 m in width, carrying a dual three-lane expressway together with double rail tracks for the Airport Railway. The whole bridge is modelled by a three dimensional ®nite-element model in this study. The ®rst twelve vibration frequencies were calculated by using a selfdeveloped program BACSB (Bu€eting Analysis of Cable-support Bridges). The results compare favourably with those reported by Xu, et al. [14] as shown in Table 1. According to the site condition, the ¯uctuating wind velocities along Tsing Ma Bridge with a mean wind speed of U ˆ 60 m/s are simulated in Fig. 6. In bu€eting analysis, the coupled terms in the selfexcited expression Eq. (4) have smaller e€ect on the bu€eting response and are neglected. On the other hand, due to the lack of wind tunnel data on lateral aerodynamic derivative P i …i ˆ 1, 2, 3, 4†, only the vertical bending and the torsional motions are taken into account. Based on the wind tunnel data [13], the unknown aerodynamic coecients C1 to C6 are determined and tabulated in Table 2. In fact, the steady-state wind loads which contribute only to the determination of the equilibrium position of bridges, were found to be insigni®cant in bu€eting responses. Hence, the whole analysis process can be divided into two steps. The ®rst step is to establish a

Hence, the whole procedure during a time period can be obtained step by step. In the following study, g and b have been taken as 0.5 and 0.25, respectively.

5. Case study Ð Bu€eting simulation of Tsing Ma bridge Tsing Ma Bridge (Fig. 5) [13], an integral component of the Airport Core Projects in Hong Kong, was opened to trac in 1997. This bridge has an overall length of 2200 m with a main suspended span of

Fig. 7. Vertical response at mid-span.

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Q. Ding, P.K.K. Lee / Computers and Structures 76 (2000) 787±797

Fig. 8. Torsional response at mid-span.

Fig. 10. Torsional response at quarter-span.

static analysis to update the equilibrium positions of the bridge and the second step is to develop a time± domain analysis evaluating the bu€eting response according to the updated equilibrium position in step 1. Under steady-state wind loads with a mean wind speed of U ˆ 60 m/s, the vertical and transverse displacements as well as rotation of the bridge at mid-span are 0.04, 2.4528 and 0.019 m, respectively. The nonlinear bu€eting responses at mid-span and quarter-span under the mean wind velocity of U ˆ 60 m/s are shown in Figs. 7±10. The horizontal reaction of the main cable is shown in Fig. 11. Comparing the responses of mid-span and quarterspan, it can be seen that the peak vertical displacement at quarter-span is much larger than that at mid-span while the peak torsional response is much smaller. This phenomenon is probably due to the fact that the ®rst vibration mode has been excited which has the greatest contribution to structural response. For Tsing Ma Bridge, the ®rst vertical vibration mode shape is antisymmetrical while the ®rst rotational vibration mode shape is symmetrical. The peak vertical responses, therefore, occur at quarter-span rather than mid-span while the peak rotational responses occur at mid-span. The root mean squares of vertical responses at midspan and quarter-span are 0.085 and 0.127 m respect-

ively. These compare closely with the results derived from using the other traditional bu€eting analysis method which gives 0.094 and 0.133 m, respectively.

Fig. 9. Vertical response at quarter-span.

Fig. 11. Horizontal reaction of main cable.

6. Concluding remarks In this paper, a time-domain bu€eting method is presented to analyse the bu€eting response of long span suspension bridges based on Scanlan's bu€eting theory. Tsing Ma suspension bridge has been chosen as a case study example. In the bridge model, all geometrical nonlinearity such as cable sag, axial forcebending moment interaction in the bridge deck and towers as well as changes in the bridge geometry due to large displacement are considered. The computational results are therefore more accurate than those obtained from the traditional spectrum method which is limited in linearization scope. In addition, this method can also reveal the response process of the entire bridge which may be useful in other research studies such as the structural health monitoring system and the real-time trac control system. By comparing the results with those obtained from using the spectrum method, the proposed simulation method is veri®ed.

Q. Ding, P.K.K. Lee / Computers and Structures 76 (2000) 787±797

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