Dynamic analysis of coupled vehicle–bridge system based on inter-system iteration method

Computers and Structures 114–115 (2013) 26–34

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Dynamic analysis of coupled vehicle–bridge system based on inter-system iteration method Nan Zhang ⇑, He Xia School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China

a r t i c l e

i n f o

Article history: Received 4 December 2011 Accepted 10 October 2012 Available online 7 November 2012 Keywords: Vehicle–bridge interaction system Railway bridges Numerical history integral Iteration method

a b s t r a c t An inter-system iteration method is proposed for dynamic analysis of coupled vehicle–bridge system. In this method, the dynamic responses of vehicle subsystem and bridge subsystem are solved separately, the iteration within time-step is avoided, the computation memory is saved, the programming difficulty is reduced, and it is easy to adopt the commercial structural analysis software for bridge subsystem. The calculation efficiency of the method is discussed by case study and an updated iteration strategy is suggested to improve the convergence characteristics for the proposed method. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction The dynamic effect of the vehicle is an important problem in railway bridge design, especially for high-speed railway and heavy-haul railway bridges. In recent years, the dynamic analysis of vehicle–bridge interaction system has been carried out for lots of cases to ensure the safety of bridge structure and running train vehicles and the riding comfort of passengers. For example, the lateral amplitude of steel plate girders with 20–40 m spans was found too large after the raise of train speed during 2000–2003 in China. To enhance the lateral stiffness of the girders, Xia et al. [1] performed numerical analysis on vehicle–bridge system to over 100 reinforcement measures and decided the final ones. Through in site experiments, the reinforcement measures were validated that they can effectively reduce the lateral amplitude as predicted. In most of the researches, the vehicle is modeled by the multibody dynamics, while the bridge is modeled by the FEM (finite element method) discretized with the direct stiffness method or the modal superposition method. In these analyses, the wheel–rail interaction assumptions are quite different, which they can be divided into three categories: (1) Moving loads. By neglecting the local vibration and the mass effect, the vehicle can be simplified into a series of moving loads. The method is widely used in analytical studies and the cases with low bridge stiffness. Only the bridge model is adopted in the method and the system can be analyzed by a time history integral method. ⇑ Corresponding author. Tel.: +86 1051683786; fax: +86 1051684393. E-mail address: [email protected] (N. Zhang). 0045-7949/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruc.2012.10.007

(2) Compatible motion relationship. The vehicle and the bridge are linked with the wheel–rail relative motion relationship. In vertical direction, the wheel-set is commonly assumed to have the same motion with the track at the wheel–rail contact point. In lateral direction, Xia et al. [1] and Xu et al. [2] used the hunting movement to define the wheel– rail relative motion, while Guo et al. [3] took the measured bogie hunting movement as the lateral system exciter. (3) Force–motion relationship. The wheel–rail interaction force is defined as the function of wheel–rail relative motion. Zhai et al. [4] adopted the Kalker’s linear theory and the Hertz contact theory to define the wheel–rail interaction force, in which the lateral/tangent wheel–rail force is the product of the creep coefficient and the wheel–rail relative velocity, the vertical/normal wheel–rail force has a non-linear relationship to wheel–rail relative compression deformation. Zhang et al. [5] simplified the Zhai’s definition to meet the linear wheel–rail relation both in lateral and vertical directions. Torstensson et al. [6] and Fayos et al. [7] modeled the rotating wheel-set and derived the wheel–rail interaction force by kinematics methods. Some researches focused on the effect of the parameters in the vehicle–bridge interaction system, including the effects of the ratio of train/bridge natural frequency, the ratio of train/bridge mass, the ratio of train/bridge length [8], the track irregularity, the bridge skewness [9], the bridge stiffness and the bridge damping [10]. The numerical method in solving the vehicle–bridge interaction equations is dependent on the wheel–rail interaction assumption. Gao and Pan [11], Li et al. [12] and Jo et al. [13] modeled the vehicle and the bridge subsystem separately, and solved them with time

N. Zhang, H. Xia / Computers and Structures 114–115 (2013) 26–34

history integral method TSI (time-step iteration), where the two subsystems meet the equivalent equations within each time-step by iteration. Xia et al. [1], Antolin et al. [14] and Yang and Yau [15] coupled the two subsystems into global equations with varying coefficients by adopting the wheel–rail interaction into the non-diagonal sub-matrices. Feriani et al. [16] and Shi et al. [17] used a complete time history iteration method in which the two subsystems was analyzed separately and linked by an interface program, but their studies only concerned the vertical interaction force for highway bridges and trucks. The lateral and the torsional interaction forces are not necessary for analysis of highway bridges but are very important for railway bridges. In this paper, an iteration method for solving the railway vehicle–bridge interaction system is proposed, considering the vertical, lateral and torsional interaction between the bridge and the railway vehicle, and adopting the track irregularity and the wheel–rail force–motion relationship (inter-system iteration, ISI). In the ISI method, firstly, the bridge subsystem is assumed rigid, while the vehicle motion and wheel–rail force histories are solved by the independent vehicle subsystem for the complete simulation time; next the bridge motion can be obtained by applying the previously obtained wheel–rail force histories to the independent bridge subsystem. Following, the updated bridge deck motion histories are combined with the track irregularities to form the new excitation to the vehicle subsystem for the next iteration process, until the given error threshold is satisfied.

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Some measured results indicated that the wheel-set yaw angle in high-speed trains is much smaller than that in the traditional trains, partly due to the special structure of yaw dampers mounted on the high-speed trains, thus the wheel-sets’ DOF in W direction (yaw angle) is not considered in the vehicle model. From assumption (A2), the vehicle subsystem can be considered as several vehicles separately. Thus the dynamic equations for an individual vehicle are:

€ V þ CV X_ V þ KV XV ¼ PV MV X

ð1Þ

where MV, CV and KV are the mass, damping and stiffness matrices of the vehicle, which are constant matrices [5]; PV is the force vector; XV is the displacement vector, containing the independent DOFs of the car-body, the bogies and the wheel-sets. There are 19 independent DOFs and 8 dependent DOFs for a 4-axle vehicle; 21 independent DOFs and 12 dependent DOFs for a 6-axle vehicle. For example, the displacement vector XV of a 4-axle vehicle is:

XV ¼ ½yC ; zC ; uC ; v C ; wC ; yT1 ; zT1 ; uT1 ; v T1 ; wT1 ; yT2 ; zT2 ; uT2 ; v T2 ; wT2 ; yW1 ; yW2 ; yW3 ; yW4 T where the subscript C stands for the car-body, T1 and T2 for the front and rear bogie, W1 and W2 for the wheel-set linked to the front bogie, W3 and W4 for the wheel-set linked to the rear bogie, respectively. 2.2. Bridge model

2. The ISI analysis method for vehicle–bridge interaction system The vehicle–bridge interaction system is composed by the vehicle subsystem and the bridge subsystem; the two subsystems are linked by the wheel–rail interaction; the given track irregularity is taken as an additional system exciter. The same coordinate systems are adopted for the both subsystems and the track irregularity: X denotes the train running direction, Z upward, and Y is defined by the right-hand rule. U, V and W denote the rotational directions about the axes X, Y and Z, respectively. The coordinate systems of both vehicle and bridge subsystem are absolute, and they have the same coordinate direction and length unit. Each rigid body in the vehicle has its independent coordinate system, with the origin in Y and Z directions at the static equilibrium position of each rigid body. According to the assumptions in Section 2.1, there is no X-DOF considered in the vehicle subsystem, so it is no need to define the origin of coordinates in X direction. 2.1. Vehicle model The following assumptions are adopted for the vehicle model and the wheel–rail interaction: (A1) The train runs over the bridge at a constant speed. (A2) The train can be modeled by several independent vehicles by neglecting the interaction among them. (A3) Each vehicle is composed of one car-body, two bogies, four or six wheel-sets and the spring-damper suspensions between the components. (A4) By the Kalker’s Linear theory, the lateral (Y) displacement of the wheel-set is the product of the creep coefficient and the wheel–rail relative velocity. (A5) By the wheel–rail corresponding assumption, the wheel-set and the rail have the same vertical (Z) and rotational (U) displacements at the wheel–rail contact point. (A6) Each car-body or bogie has five independent DOFs in directions Y, Z, U, V and W; each wheel-set has 1 independent DOF in direction Y and 2 dependent DOFs in directions Z and U.

The bridge model can be established by the FEM. The dynamic equations for the bridge subsystem can be written as:

€ B þ CB X_ B þ KB XB ¼ FB MB X

ð2Þ

where MB, CB and KB are the global mass, damping and stiffness matrices, FB and XB are the force and displacement vectors of the bridge subsystem, respectively. It is very important to note that the lumped mass method cannot be adopted for the mass matrix. Because if the diagonal elements related to the torsional (U) DOFs in MB is zero, the torsional moment of the vehicle may cause unreasonable angular acceleration for the bridge deck. In some cases, the modal superposition method may be used in modeling the bridge subsystem to reduce the number of DOFs. The equations of the bridge subsystem are expressed as:

€ B þ 2n xB X_ B þ x2 XB ¼ UT FB X B B B

ð3Þ

where nB and xB are the damping ratio and circular frequency diagonal matrices, respectively; UB is the modal matrix. For the same reason, if lumped mass method is adopted, there is no torsional mode in UB and the torsional moment and angle cannot be included in calculation. Therefore, the consistent mass matrix for the bridge subsystem is used to reflect the torsional dynamic characteristics of the bridge. 2.3. Track irregularity The track irregularity is the distance of the actual position and the theoretical position of the rail. According to the definition in rail engineering, the track irregularities are defined as:

8 yR þyL > < yI ¼ 2 zR þzL zI ¼ 2 > : u ¼ zR zL I

ð4Þ

g0

where yL and yR are the lateral irregularities for the left and the right rail; zL and zR are the vertical irregularities for the left and the right rail; g0 is the rail gauge; yI, zI and uI are the align (lateral), vertical

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and cross-lever (torsional) track irregularities adopted in the following calculation, respectively. The track irregularity causes the additional velocity and acceleration of the wheel-set, which can be expressed in a differential form:

DE DE DE @E E_ ¼ lim ¼ lim ¼ V  lim ¼V Dt!0 Dt Dt!0 DX=V Dt!0 DX @X

ð5Þ

_ _ _ _ € ¼ lim DE ¼ lim DE ¼ V  lim DE ¼ V @ E ¼ V 2 @E E Dt!0 Dt Dt!0 DX=V Dt!0 DX @X @X

ð6Þ

where E stands for the irregularity; V is the train speed. Fig. 1. The wheel–rail interaction force.

2.4. Wheel–rail interaction From assumption (A4), (A5) and (A6), the wheel–rail interaction force has three components in lateral (Y), vertical (Z) and torsional (U) directions, see Fig. 1. In Fig. 1, for an individual wheel-set, P1 and P2 are the vertical interaction force between the bogie and the wheel-set; P3 and P4 are the vertical wheel–rail interaction force; P5 is the lateral wheel–rail interaction force; G is the static axle load. Because the vehicle equations are about the gravity equilibrium position of the vehicle components, the static axle loads act on the bridge subsystem, but not contribute to the vehicle vibration equations. The vertical displacements of the upper suspension points 1, 2 and the lower suspension points 3 and 4 are:

8 z1 > > > < z2 > > z3 > : z4

¼ zJ þ id1 v J  b1 uJ

Thus the lateral wheel–rail force, P5, is in proportion to the wheel–rail relative velocity and the vehicle’s velocity terms can be moved from right-hand to the left-hand in Eq. (1):

€ V þ ðCV þ CC ÞX_ V þ KV XV ¼ FV MV X

ð10Þ

where CC is the additional damping matrix due to the wheel–rail creep force. For a 4-axle vehicle:

¼ zJ þ id1 v J þ b1 uJ

ð7Þ

¼ ðzI þ zB Þ  b1 ðuI þ uB Þ

CC ¼

¼ ðzI þ zB Þ þ b1 ðuI þ uB Þ

where zJ, uJ and vJ are the bogie displacement in Z, U and V directions; zB and uB are the bridge motions in Z and U directions at the wheel-set position and can be obtained by solving the bridge equations; 2b1 is the lateral distance between the two vertical springs/dampers, see Fig. 1; 2d1 is the longitudinal distance between the two bogies; i = 1, i = 0 and i = 1 for the front, middle (6-axle case) and rear wheel-set, respectively. Then the forces P1, P2, P3 and P4 can be expressed as:

8 P1 ¼ kZ1 ðz1  z3 Þ þ cZ1 ðz_ 1  z_ 3 Þ > > > > < P2 ¼ kZ1 ðz2  z4 Þ þ cZ1 ðz_ 2  z_ 4 Þ €

€

€

€

P3 ¼ ð12 þ bg1 ÞP1 þ ð12  bg1 ÞP2 þ G2 þ m0 ðz2I þzB Þ  IX0 ðugI þuB Þ > > 0 0 0 > > : € € € € P4 ¼ ð12  bg1 ÞP1 þ ð12 þ bg1 ÞP2 þ G2 þ m0 ðz2I þzB Þ þ IX0 ðugI þuB Þ 0

(A7) The wheel–rail normal contact force is taken as the static wheel load, G/2. (A8) TB wheel and 60 kg/m rail are considered, which are commonly adopted in Chinese railway system. The contact point is in a 1:20 cone surface for the TB–wheel and in a cylindrical surface with 300 mm radius for the 60 kg/m rail.

0

ð8Þ

0154 I44

 ð11Þ

The force vector FV in Eq. (10) is the sum of all the wheel-sets’ effect and can be obtained by the force vector PV in Eq. (1) deducting the creep force. Assuming the ith element in FV is FV,i, the nonzero FV,i elements are: 8 2k 2k X X > > > F V;5kþ2 ¼ 2kZ1 ðzIm þ zBm Þ þ 2cZ1 ðz_ Im þ z_ Bm Þ > > > > m¼2k1 m¼2k1 > > > > 2k 2k > X X > 2 2 > > 2k 2k > X X > > > ð1Þm ðzIm þ zBm Þ þ 2cZ1 d1 ð1Þm ðz_ Im þ z_ Bm Þ F V;5kþ4 ¼ 2kZ1 d1 > > > > m¼2k1 m¼2k1 > > : F V;15þm ¼ 2f 22 ðy_ Im þ y_ Bm Þ=V ð12Þ

0

where kZ1 and cZ1 are the vertical stiffness and damping coefficients between the bogie and the wheel set, respectively; m0 and IX0 are the mass and the X-inertia of the wheel-set. The lateral wheel–rail interaction force is defined by the Kalker’s linear theory:

y_ W  ðy_ IL þ y_ B Þ y_ W  ðy_ IR þ y_ B Þ þ f22 V V y_ W  y_ I  y_ B ¼ 2f 22 V

 2f 22 01515 V 0415

P5 ¼ f22

ð9Þ

where yW is lateral displacement of the wheel-set; f22 is the creep coefficient, a function of the wheel–rail normal contact force, which is the curvature of radii at the contact points in the wheel tread and the rail tread; yB is the bridge motion in Y direction at the wheel-set position and can be obtained by solving the bridge equations. By assumptions (A7) and (A8) as follows, the creep coefficient f22 is a constant:

where k = 1 and k = 2 stand for the front and the rear bogie; m = 1 and m = 2 stand for the wheel-sets linked to the front bogie; m = 3 and m = 4 stand for the wheel-sets linked to the rear bogie. The subscripts Im and Bm refer to the track irregularity and the bridge motion at the wheel-set m position. The four expressions in Eq. (12) reflect the Y, Z and U forces acting on bogies and the Y force acting on wheel-sets. The force vector FB in Eqs. (2) and (3) also can be obtained by summing the effect of all the wheel-sets of all the vehicles, the contribution of the mth wheel-set linked to the kth bogie is in Y, Z and U directions is expressed as: 8 F ¼ 2f 22 ðy_ Wm  y_ Im  y_ Bm Þ=V > > > B;km;Y > < F B;km;Z ¼ 2kZ1 ½zJk þ ð1Þm v Jk  zIm  zBm  þ 2kZ1 ½z_ Jk þ ð1Þm > > v_ Jk  z_ Im  z_ Bm   G  m0 ð€zIm þ €zBm Þ > > : 2 2 € Im þ u € Bm Þ F B;km;U ¼ 2kZ1 b1 ðuJk  uIm  uBm Þ þ 2cZ1 b1 ðu_ Jk  u_ Im  u_ Bm Þ  IX0 ðu ð13Þ

N. Zhang, H. Xia / Computers and Structures 114–115 (2013) 26–34

2.5. Interaction equations and inter-system iteration The dynamic equilibrium equations for the vehicle–bridge interaction system can be formed by the equations of the vehicle subsystem and the bridge subsystem. When the direct stiffness method is adopted for the bridge, the interaction equations are:

8 € V1 þ ðCV1 þ CC1 ÞX_ V1 þ KV1 XV1 ¼ FV1 MV1 X > > > > € þ ðCV2 þ CC2 ÞX_ V2 þ KV2 XV2 ¼ FV2 > > M X > < V2 V2 .. . > > > > € Vn þ ðCVn þ CCn ÞX_ Vn þ KVn XVn ¼ FVn > MVn X > > : € B þ CB X_ B þ KB XB ¼ FB MB X

ð14Þ

where n is the vehicles number of the train. The first n equations in Eq. (14) are for the vehicle subsystem, the last equation is for the bridge subsystem. The mass, damping, stiffness and additional damping matrices in the left-hand side of Eq. (14) are constants. The force vector FVi is the function of yIm, zIm, uIm, yBm, zBm and uBm in Eq. (12), with the subscript Im indicating the track irregularity and Bm the bridge motion, respectively. The force vector FB is the function of the above exciters and the vehicle motion in Eq. (13), with the subscript Wm indicating the wheel set motion and Jk the bogie motion, respectively. Thus Eq. (14) are coupled and can be solved by an iteration procedure. For the iteration strategies TSI and ISI, the iteration procedures are compared in Fig. 2. The wheel–rail interaction force histories are adopted for the convergence check in ISI, because they reflect the dynamic status of both the vehicle and the bridge. The operation of ISI consists of the following procedures:

29

Step 1: Solve the vehicle subsystem by assuming the bridge subsystem rigid, setting the bridge motion to zero, and using the track irregularities as the excitation, to obtain the time histories of wheel–rail forces/moments for all wheel-sets; Step 2: Solve the bridge subsystem by applying the wheel–rail interaction force histories obtained in the previous iteration loop (or Step 1) on bridge deck, to obtain the updated time histories of bridge deck movement at all joints; Step 3: Solve the vehicle subsystem by combining the updated bridge deck movements obtained in Step 2 with the track irregularities as the updated system excitation, to obtain the updated time histories of wheel–rail forces/moments for all wheel-sets; Step 4: Calculate the errors between the updated wheel–rail interaction force histories of all the wheel-sets obtained in Step 3 and those in the previous iteration loop (or Step 1) for the convergence check; If the maximum instantaneous absolute differences for all wheel-sets in the whole integral time satisfy the given threshold, the convergence check is OK, meaning the calculation is completed; otherwise, return to Step 2 to start a next iteration loop. This iteration procedure is completely different to that of TSI. In TSI, the vehicle subsystem and the bridge subsystem are solved simultaneously through the iteration process in each time-step, and the convergence check is upon the dynamic responses at the end of each time-step. While in ISI, the two subsystems are solved separately over the complete simulation time in each iteration loop, and the convergence check is performed afterwards using the continuously updated histories of wheel–rail forces/moments until the error threshold is satisfied. Based on the wheel–rail interaction assumption, the wheel–rail interaction force is the function of the wheel–rail relative motion.

Fig. 2. Iteration procedures of TSI (left) and ISI (right).

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N. Zhang, H. Xia / Computers and Structures 114–115 (2013) 26–34

Fig. 3. Illustration of ISI.

Table 1 Vehicle parameters.

a

Item

Value (m)

Item

Value

Distance of wheel-sets Distance of bogies Transverse spana of primary suspension Transverse span of secondary suspension Car-body to secondary suspension Secondary suspension to bogie Bogie to wheel-set Wheel radius Wheel-set mass Wheel-set X-inertia Bogie mass Bogie X-inertia Bogie Y-inertia Bogie Z-inertia Car-body mass

2.50 17.50 2.00 2.00 0.80 0.20 0.10 0.43 2t 2t  m2 3t 3t  m2 8t  m2 8t  m2 40t

Car-body, X-inertia Car-body, Y-inertia Car-body Z-inertia Primary suspension X-damp/side Primary suspension Y-damp/side Primary suspension Z-damp/side Secondary suspension X-damp/side Secondary suspension Y-damp/side Secondary suspension Z-damp/side Primary suspension X-spring/side Primary suspension Y-spring/side Primary suspension Z-spring/side Secondary suspension X-spring/side Secondary suspension Y-spring/side Secondary suspension Z-spring/side

100t  m2 1500t  m2 2500t  m2 0 0 20 kN s/m 60 kN s/m 60 kN s/m 30 kN-s/m 5000 kN/m 5000 kN/m 800 kN/m 200 kN/m 200 kN/m 200 kN/m

Transverse span: the transverse distance between the spring/damper in suspension system, b1 is shown in Fig. 1.

Table 2 Bridge parameters. Beam type

fH/Hz

fV/Hz

G1/kN m1

G2/kN m1

IX/m4

High-speed railway beam (A) Speed-raised railway beam (B) Common railway beam (C) Common railway low-height beam (D)

15 6 3 2.5

7 5 4 3

170 130 110 80

80 50 40 40

25 0.06 0.05 0.03

The masses, damping and stiffness of both vehicle and bridge are quite large, while the energy inputted to the vehicle–bridge interaction system is limited, which cannot excite intense vibration in high frequency, so the high frequency components in the wheel– rail force are small. Without importing the numerical dissipation, the Newmark-b method is adopted in solving the vehicle and the bridge subsystems, with c = 1/2 and b = 1/4. It can be seen in Fig. 2 that the ISI method is simpler in iteration procedure. The convergent results can be obtained in each iteration step when an unconditionally convergent iteration method is used. But ISI is not an unconditionally convergent procedure. The divergent results may be found even when an unconditional convergent iteration method is used in solving the vehicle and the bridge subsystems, which will be found in Section 3. Also in Ref. [18], in which it is concluded that a convergent result cannot be obtained for the coupled vehicle–bridge system even by using a smaller time-step

when the wheel–rail interaction is defined by the wheel–rail relative motion, such as the Assumption (A5) in this paper. One of the main advantages of adopting ISI is that the commercial structural analysis software can be used for the bridge subsystem, it is equivalent to solve Eqs. (2) or (3), making the analysis easier and more accurate. While for TSI, it is difficult to invoke external programs within the time-step, thus the matrices of bridge must be calculated explicitly. The vehicles are coupled through the bridge and must be solved simultaneously, which may lead larger memory consuming and programming difficulty in TSI. While for ISI, the vehicles run on a constant system exciter and can be analyzed separately, with the equilibrium equations of Eq. (10). The method of ISI is illustrated in Fig. 3. The mass, damping and stiffness of each DOF in the vehicle and the bridge subsystems are quite large, so the high frequency

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N. Zhang, H. Xia / Computers and Structures 114–115 (2013) 26–34

Fig. 4. Lateral, vertical and torsional track irregularity samples.

Fig. 5. Initial and final positions of the vehicle traveling through the bridge.

component is small in the vehicle–bridge interaction system. The convergent result can be obtained in several iteration steps. While for some other problems with multi coupling subsystem (other than the problem of vehicle–bridge system), it may become difficult to get the convergent result by the ISI method owing to the high frequency components in vibration. The problem can be partly solved by using a smaller time-step or larger threshold in convergence check, or adopting the numerical dissipation to reduce the high frequency vibration artificially.

is the inertia moment of beam section about X-axis, respectively. All the four types of beams are straight ones with single-bound track laid along the centerline of the bridge. The beam is divided into 32 spatial beam elements of 1 m in length, which is restrained in X, Y, Z and U directions at the fixed support and in Y, Z and U directions at the movable support. The motion equations of the bridge subsystem are expressed by the direct stiffness method, as in Eq. (14). Thus the total DOF number of the bridge subsystem is 33 ⁄ 6  7 = 191. By adopting the Poisson’s ratio 0.2, the torsional

3. Case study and discussion 3.1. General information of cases For simplicity, an individual vehicle and a bridge with single span beam are analyzed in this section. The parameters of the vehicle and the bridge are listed in Tables 1 and 2, respectively. The vehicle parameters are not from a certain type of train. Four types of beams are considered for the bridge, which are all prestressed and single bound, 32 m in span. Beam A is box-sectional, while Beam B, C and D are T-sectional. In Table 2, fH and fV are the lateral and vertical fundamental frequency, G1 is the beam weight per unit length, G2 is the secondary weight (including the rail structure and the additional devices) per unit length, and IX

Table 3 Responses of vehicle–bridge subsystem of ISI and TSI. Item

Beam A

Beam B

Beam C

Beam D

Mid span lateral disp./mm Mid span vertical disp./mm Mid span torsional disp./mrad Mid span lateral acc./m s2 Mid span vertical acc./m s2 Mid span torsional acc./m s2 Lateral w/r force/kN Vertical w/r force/kN Torsional w/r moment/kN m Car-body lateral acc./m s2 Car-body vertical acc./m s2

0.010 0.432 0.001 0.082 0.050 0.012 10.43 144.5 15.61 0.175 0.121

0.029 1.204 0.027 0.056 0.100 0.174 10.40 144.6 15.51 0.176 0.119

0.079 2.345 0.072 0.106 0.193 0.487 10.27 144.4 15.50 0.172 0.117

0.125 5.104 0.137 0.111 0.357 0.767 10.38 144.9 15.20 0.171 0.114

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Fig. 6. Lateral displacement history of bridge mid span using ISI (left) and TSI (right).

Fig. 7. Vertical displacement history of bridge mid span using ISI (left) and TSI (right).

fundamental frequencies of Beam A, B, C and D are 20.78 Hz, 2.54 Hz, 3.97 Hz and 4.40 Hz, respectively. The track irregularity data generated from the German Low Disturb Spectrum is adopted; the power spectrum density is expressed in Eq. (15).

8 Aa X2c > > > Sa ðXÞ ¼ ðX2 þX2r ÞðX2 þX2c Þ > < 2 Sv ðXÞ ¼ ðX2 þXA2vÞðXXc 2 þX2 Þ r c > > > > Av b2 X2c X2 : S ðXÞ ¼ c 2 2 2 2 ðX þX ÞðX þX ÞðX2 þX2 Þ r

c

ð15Þ

s

where Sa(X), Sv(X) and Sc(X) are align, vertical and cross-lever irregularities, respectively, with Sa(X) and Sv(X) in m2/(rad/m) and Sc(X) in 1/(rad/m). The parameters are taken as Xc = 0.8246 rad/m, Xr = 0.0206 rad/m, Xs = 0.4380 rad/m, Aa = 2.119  107 cm2 rad/m, Av = 4.032 cm2 rad/m, b = g0/2 = 0.7465 m. X is the spatial angular frequency calculated by X = 2p/Lt, where Lt is the wavelength of the track irregularity, ranging from 1 m to 80 m.

The maximum value for the lateral (Y), vertical (Z) and torsional (U) irregularities adopted in the case study are 7.57 mm, 7.15 mm and 4.79 mrad, respectively. The samples of irregularity are shown in Fig. 4. The complete histories of the train traveling through the bridge are analyzed, with the train speed of 120 km/h, the damping ratio of the bridge 0.02, and the time-step 0.005 s. The initial and final positions of the train are shown in Fig. 5. It must be pointed that the ISI method, since it adopts the direct time integration for both the two subsystems, has its inherent filtering characteristics. Thus the system with high frequency vibration may be underestimated when the time-step length or the element size is not small enough in the calculation. In the case study, the bridge in simplified into 191 DOFs, the maximum (191st) frequencies of Beam A, B, C and D are 78.1 kHz, 31.3 kHz, 20.8 kHz and 15.6 kHz, respectively. The time-step is 0.005 s or the calculated sampling frequency is 200 Hz. They are enough to meet the accuracy requirement for a railway engineering problem. Of course, it is assumed in (A5) that there’s no relative motion between the wheel-set and the bridge, which may lead to the local

Fig. 8. Torsional displacement histories of bridge mid span using ISI (left) and TSI (right).

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N. Zhang, H. Xia / Computers and Structures 114–115 (2013) 26–34 Table 4 Number of iteration steps. Beam Beam Beam Beam Beam

A B C D

ISI Steps

ISI Iteration N.

TSI Max steps

TSI Min steps

TSI Iteration N.

4 4 5 8

1252 1252 1565 2504

5 5 6 10

2 3 3 6

1203 1279 1449 2712

Fig. 9. Lateral displacement histories of bridge motion and track irregularity.

Fig. 10. Vertical displacement histories of bridge motion and track irregularity.

Fig. 11. Vertical velocity histories of bridge motion and track irregularity.

considered in maximum value statistics. For the methods of TSI and ISI, the maximum responses of the bridge mid span and vehicle car-body are listed in Table 3, in which the ISI and TSI have the same results. The classic displacement histories when the vehicle transverses Beam D are shown in Figs. 6–8. It is found from Table 3 that the dynamic responses of bridge decrease with the bridge stiffness increasing. The wheel–rail interaction force and the car-body acceleration vary quite little for different beams, because the bridge motion contributes very little to the wheel-set motion, compared to the track irregularity does, as shown in Figs. 9–13, which indicate the relative proportion of the bridge motion (solid line) and the track irregularity (dotted line) at the 1st wheel-set position when the train traverses the Beam D. From the above figures, it is found that the irregularities are much larger than the bridge motion in the lateral and torsional displacements, while the bridge motion has relative larger proportion in the vertical displacement, but it is in quite low frequency and has small effect on the wheel–rail interaction force or the vehicle response. In vertical velocity and acceleration history, the irregularities are still much larger the bridge motion. 3.3. Influence of iteration step number

Fig. 12. Vertical acceleration histories of bridge motion and track irregularity.

Fig. 13. Torsional displacement histories of bridge motion and track irregularity.

vertical vibration underestimated. Therefore, if only the macro motion status of the vehicle and the bridge are concerned, the proposed model is acceptable, but if the local motion is also concerned, the more accurate wheel–rail interaction assumption must be used. 3.2. Iteration process and result analysis The maximum instantaneous absolute difference thresholds are 10 N for the lateral and vertical wheel–rail force and 10 N m for the torsional wheel–rail moment for each wheel-set and at each timestep. Only in the time period when the wheel-sets or the car-body are coupled with the bridge, the wheel–rail force and acceleration are

The iteration numbers of ISI and TSI are shown in Table 4, where only the step numbers when the vehicle and the bridge are coupled (from Step 181 to 493) are taken into account. The column ‘‘Iteration N.’’ refers to the total number of iterations between steps 181 and 493 for both the methods. It is obvious that the ISI and TSI have similar iteration steps in the four cases. In other words, they have similar calculation efficiency. Wu [18] proved the wheel–rail displacement compatibility condition is the main reason of divergence, and the additional mass in both sides of the system equations is helpful to get the convergent result for the vehicle–bridge interaction system. It implies that the bridge mass affects the number of iteration steps to meet the convergence check. The difference of bridge stiffness causes input exciters difference between time-steps for the vehicle subsystem and may also lead to different convergent conditions. Thus, further analysis is performed for the cases with different bridge distributed mass and bridge stiffness. The number of iteration steps for different masses and stiffnesses are shown in Fig. 14, where 10– 100% of stiffness and 30–100% of distributed mass of Beam D are concerned. It is found that the number of iteration steps increases with decrease of distributed mass, from 8 with 100% mass to 23 with 30% mass. The relationship between the iteration number and the stiffness is not monotonic. When the bridge mass is over 40%, the number of iteration steps changes little with the bridge stiffness, while when the bridge mass is 30%, the number of iteration steps increases obviously with the stiffness, from 17 with 10% stiffness to 23 with 100% stiffness. 3.4. Convergence strategy For the cases with 10–20% of distributed mass for Beam D, the iteration procedure is divergent, no matter ISI or TSI is used. It is

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N. Zhang, H. Xia / Computers and Structures 114–115 (2013) 26–34

(1) Comparing to traditional methods, the iteration within timestep is avoided in ISI, so it is convenient to use the commercial structural analysis software for the bridge subsystem instead of calculating the bridge matrices directly, each vehicle can be analyzed separately, the computation memory is saved, and the programming difficulty is reduced. (2) An updated iteration strategy is proposed for ISI to improve the convergent characteristics in solving the vehicle–bridge interaction system, in which the wheel–rail force acting on the bridge subsystem is regarded as a linear combination of the wheel–rail force calculated from the vehicle subsystem in current and previous iteration steps. (3) In the ISI method, more iteration step number is needed to meet the convergence check when the bridge has smaller distributed mass.

Acknowledgements Fig. 14. Iteration step number with different bridge coefficients.

Table 5 Iteration steps versus distributed mass and combination factor. Distributed mass

k=1

k = 0.5

k = 0.2

k = 0.1

30% 25% 20% 15% 10% 5%

23 192 Divergence Divergence Divergence Divergence

14 15 18 21 33 Divergence

32 36 42 48 58 118

64 71 82 95 112 218

References

common to illustrate that the convergent characteristics is decided by the ‘‘convergent radius’’: if the evaluated system response is within the convergent region, or the error is small enough, the convergent result must be obtained by iteration, otherwise the iteration is divergent. When the difference of the evaluated response between two time-steps is too large, the convergent region may be missed. In order to reduce the step length to meet the convergent region; or to avoid skipping it, the wheel–rail force acted on the bridge subsystem in step N can be regarded as a linear combination of the wheel–rail force calculated from the vehicle subsystem in step N and step N  1: V

F BN ¼ kF N þ ð1  kÞF VN1

The research is sponsored by the Major State Basic Research Development Program of China (‘‘973’’ Program: 2013CB036203), the 111 project (Grant No. B13002), the National Science Foundation of China (Grant Nos. 51178025 and 50838006) and the Fundamental Research Funds for the Central Universities (Grant No. 2009JBZ016-4).

ð16Þ

where F BN is the wheel–rail force acted on the bridge subsystem in step N, which stands for the interaction force of any wheel-set in any direction. F VN and F VN1 are the wheel–rail forces calculated from the vehicle subsystem in step N and step N  1, respectively. 0 < k 6 1 is the combination factor, and k = 1 is adopted in the calculations. The number of iteration steps for the beams with 100% stiffness and 5–30% distributed mass of Beam D are listed in Table 5. The relationship between k and the number of iteration steps is quite complex, but it can be seen that the smaller combination factor k helps to get a convergent result. However, smaller k also causes smaller updating of the evaluated response value, which may need more iteration steps to meet the convergence check. 4. Conclusions In this paper, an inter-system iteration method (ISI) is proposed for dynamic analysis of coupled vehicle–bridge system, whose result is very close to the widely-used time-step iteration method (TSI). The characteristics of ISI are as follows:

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