dynamicanalysisofcoupledvehiclebridgesystem 2013

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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 sug-

gested to improve the convergence characteristics for the proposed method.

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 hea-

vy-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 lat-

eral 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] per-

formed 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 multi-

body dynamics, while the bridge is modeled by the FEM (finite ele-

ment method) discretized with the direct stiffness method or the

modal superposition method. In these analyses, the wheel–railinteraction assumptions are quite different, which they can be di-

vided 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.

(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 rela-

tionship 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 direc-

tions. Torstensson et al. [6] and Fayos et al. [7] modeledthe rotating wheel-set and derived the wheel–rail interac-

tion 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

0045-7949/$ - see front matter 2012 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compstruc.2012.10.007

⇑ Corresponding author. Tel.: +86 1051683786; fax: +86 1051684393.

E-mail address: [email protected] (N. Zhang).

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

Contents lists available at SciVerse ScienceDirect

Computers and Structures

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p s t r u c

http://dx.doi.org/10.1016/j.compstruc.2012.10.007mailto:[email protected]://dx.doi.org/10.1016/j.compstruc.2012.10.007http://www.sciencedirect.com/science/journal/00457949http://www.elsevier.com/locate/compstruchttp://www.elsevier.com/locate/compstruchttp://www.sciencedirect.com/science/journal/00457949http://dx.doi.org/10.1016/j.compstruc.2012.10.007mailto:[email protected]://dx.doi.org/10.1016/j.compstruc.2012.10.007


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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 vary-

ing 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 interfaceprogram, 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 ver-

tical, 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 pre-

viously obtained wheel–rail force histories to the independent

bridge subsystem. Following, the updated bridge deck motion his-

tories 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.

2. The ISI analysis method for vehicle–bridge interaction system

The vehicle–bridge interaction system is composed by the vehi-

cle 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 subsys-

temsand the track irregularity: X denotesthe train runningdirection,

Z upward, and Y is defined by the right-hand rule. U , V and W denotethe rotational directions about the axes X , Y and Z , respectively.

Thecoordinate systems of both vehicle andbridge subsystemare

absolute, and they have the same coordinate direction and length

unit. Each rigid body in the vehicle has its independent coordinate

system, with theorigin in Y and Z directions at the static equilibrium

position of each rigid body. According to the assumptions in Sec-

tion 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 ) dis-

placements at the wheel–rail contact point.

(A6) Each car-body or bogie has five independent DOFs in direc-

tions Y , Z , U , V and W ; each wheel-set has 1 independentDOF in direction Y and 2 dependent DOFsin directions Z and U .

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:

MV €X V þ CV _X V þ KVX V ¼ PV 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 vec-

tor; X V is the displacement vector, containing the independent DOFs

of the car-body, the bogies and the wheel-sets. There are 19 inde-

pendent DOFs and 8 dependent DOFs for a 4-axle vehicle; 21 inde-

pendent DOFs and 12 dependent DOFs for a 6-axle vehicle. For

example, the displacement vector X V of a 4-axle vehicle is:

X V ¼ ½ yC; z C;uC; v C;wC; yT1; z T1;uT1; v T1;wT1; yT2; z T2;uT2; v T2;

wT2; yW1; yW2; yW3; yW4T

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 thefront bogie, W3 and W4 for the wheel-set linked to the rear bogie,

respectively.

2.2. Bridge model

The bridge model can be established by the FEM. The dynamic

equations for the bridge subsystem can be written as:

MB €X B þ CB _X B þ KBX B ¼ FB ð2Þ

where MB, CB and KB are the global mass, damping and stiffness

matrices, FB and X B are the force and displacement vectors of the

bridge subsystem, respectively.

It is very important to note that the lumped mass method can-

not be adopted for the mass matrix. Because if the diagonal ele-ments 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:

€X B þ 2nBxB _X B þ x

2BX B ¼ U

TBFB ð3Þ

where nB and xB are the damping ratio and circular frequency diag-onal 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 can-

not be included in calculation. Therefore, the consistent mass ma-

trix for the bridge subsystem is used to reflect the torsionaldynamic 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:

yI ¼ yR þ yL

2

z I ¼ z R þ z L

2

uI ¼ z R z L g 0

8><>: ð4Þ

where yL and yR are the lateral irregularities for the left and the right

rail; z L and z R are the vertical irregularities for the left and the rightrail; g 0 is the rail gauge; yI, z I and uI are the align (lateral), vertical

N. Zhang, H. Xia / Computers and Structures 114–115 (2013) 26–34 27


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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:

MV1 €

X V1 þ ðCV1 þ CC1Þ _

X V1 þ KV1X V1 ¼ FV1MV2

€X V2 þ ðCV2 þ CC2Þ _X V2 þ KV2X V2 ¼ FV2

.

.

.

MVn €X Vn þ ðCVn þ CCnÞ _X Vn þ KVnX Vn ¼ FVn

MB €X B þ CB _X B þ KBX B ¼ FB

8>>>>>>><>>>>>>>:

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, z Im, uIm, yBm, z Bm 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 sub-

script Wm indicating the wheel set motion and Jk the bogie mo-

tion, 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:

Step 1: Solve the vehicle subsystem by assuming the bridge sub-

system 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 itera-

tion 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 inter-

action 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 com-

pleted; 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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The masses, damping and stiffness of both vehicle and bridge arequite large, while the energy inputted to the vehicle–bridge inter-

action 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 diver-

gent results may be found even when an unconditional convergent

iteration method is used in solving the vehicle and the bridge sub-

systems, which will be found in Section 3. Alsoin Ref. [18], in which

it is concluded that a convergent result cannot be obtained for thecoupled vehicle–bridge system even by using a smaller time-step

when the wheel–rail interaction is defined by the wheel–rail rela-tive motion, such as the Assumption (A5) in this paper.

One of the main advantages of adopting ISI is that the commer-

cial structural analysis software can be used for the bridge subsys-

tem, 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 andthe bridge subsystems are quite large, so the high frequency

Fig. 3. Illustration of ISI.

Table 1

Vehicle parameters.

Item Value (m) Item Value

Distance of wheel-sets 2.50 Car-body, X-inertia 100t m2

Distance of bogies 17.50 Car-body, Y-inertia 1500t m2

Transverse spana of primary suspension 2.00 Car-body Z-inertia 2500t m2

Transverse span of secondary suspension 2.00 Primary suspension X-damp/side 0

Car-body to secondary suspension 0.80 Primary suspension Y-damp/side 0

Secondary suspension to bogie 0.20 Primary suspension Z-damp/side 20 kN s/m

Bogie to wheel-set 0.10 Secondary suspension X-damp/side 60 kN s/m

Wheel radius 0.43 Secondary suspension Y-damp/side 60 kN s/m

Wheel-set mass 2t Secondary suspension Z-damp/side 30 kN-s/m

Wheel-set X-inertia 2t m2 Primary suspension X-spring/side 5000 kN/m

Bogie mass 3t Primary suspension Y-spring/side 5000 kN/m

Bogie X-inertia 3t m2 Primary suspension Z-spring/side 800 kN/m

Bogie Y-inertia 8t m2 Secondary suspension X-spring/side 200 kN/m

Bogie Z-inertia 8t m2 Secondary suspension Y-spring/side 200 kN/m

Car-body mass 40t Secondary suspension Z-spring/side 200 kN/ma Transverse span: the transverse distance between the spring/damper in suspension system, b1 is shown in Fig. 1.

Table 2

Bridge parameters.

Beam type f H/Hz f V/Hz G1/kN m1 G2/kN m

1 I X/m4

High-speed railway beam (A) 15 7 170 80 25

Speed-raised railway beam (B) 6 5 130 50 0.06

Common railway beam (C) 3 4 110 40 0.05

Common railway low-height beam (D) 2.5 3 80 40 0.03

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


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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 diffi-

cult 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 re-

duce the high frequency vibration artificially.

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 vehi-

cle 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 pre-

stressed and single bound, 32 m in span. Beam A is box-sectional,

while Beam B, C and D are T-sectional. In Table 2, f H and f V are

the lateral and vertical fundamental frequency, G1 is the beam

weight per unit length, G2 is the secondary weight (including therail structure and the additional devices) per unit length, and I X

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

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

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

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 0.010 0.029 0.079 0.125

Mid span vertical disp./mm 0.432 1.204 2.345 5.104

Mid span torsional disp./mrad 0.001 0.027 0.072 0.137

Mid span lateral acc./m s2 0.082 0.056 0.106 0.111

Mid span vertical acc./m s2 0.050 0.100 0.193 0.357

Mid span torsional acc./m s2 0.012 0.174 0.487 0.767

Lateral w/r force/kN 10.43 10.40 10.27 10.38

Vertical w/r force/kN 144.5 144.6 144.4 144.9

Torsional w/r moment/kN m 15.61 15.51 15.50 15.20

Car-body lateral acc./m s2 0.175 0.176 0.172 0.171

Car-body vertical acc./m s2 0.121 0.119 0.117 0.114



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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 Dis-

turb Spectrum is adopted; the power spectrum density is ex-

pressed in Eq. (15).

S aðXÞ ¼ AaX

2c

ðX2þX2r ÞðX2þX2c Þ

S vðXÞ ¼ AvX

2c

ðX2þX2r ÞðX2þX2c Þ

S cðXÞ ¼ Avb

2X2cX

2

ðX2þX2r ÞðX2þX2c ÞðX

2þX2s Þ

8>>>><>>>>:

ð15Þ

where S a(X), S v(X) and S c(X) are align, vertical and cross-lever

irregularities, respectively, with S a(X) and S v(X) i n m2/(rad/m)

and S c(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 = g 0/2 = 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 ratioof 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 fil-

tering characteristics. Thus the system with high frequency vibra-

tion 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 be-

tween the wheel-set and the bridge, which may lead to the local

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).

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



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vertical vibration underestimated. Therefore, if only the macro mo-

tion status of the vehicle and the bridge are concerned, the pro-

posed 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 time-

step.

Only in the time period when the wheel-sets or the car-body arecoupled with the bridge, the wheel–rail force and acceleration are

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 inter-

action force and the car-body acceleration vary quite little for dif-

ferent 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 dis-

placements, 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 irregu-

larities are still much larger the bridge motion.

3.3. Influence of iteration step number

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 ‘‘Itera-

tion N.’’ refers to the total number of iterations between steps 181and 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 con-

dition 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 con-

vergence 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 distrib-

uted 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 de-

crease 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 itera-

tion 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, theiteration procedure is divergent, no matter ISI or TSI is used. It is

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.

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

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

Table 4

Number of iteration steps.

Beam ISI ISI TSI TSI TSI

Steps Iteration N. Max steps Min steps Iteration N.

Beam A 4 1252 5 2 1203

Beam B 4 1252 5 3 1279

Beam C 5 1565 6 3 1449

Beam D 8 2504 10 6 2712



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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 con-

vergent result must be obtained by iteration, otherwise the itera-

tion is divergent.When the difference of the evaluated response between two

time-steps is too large, the convergent region may be missed. In or-

der to reduce the step length to meet the convergent region; or to

avoid skipping it, the wheel–rail force acted on the bridge subsys-

tem 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:

F BN ¼ kF VN þ ð1 kÞF

VN1 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 V N and F V N 1 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 cal-

culations. 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 fac-

tor 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 re-

sult is very close to the widely-used time-step iteration method(TSI). The characteristics of ISI are as follows:

(1) Comparing to traditional methods, the iteration within time-

step is avoided in ISI, so it is convenient to use the commer-

cial structural analysis software for the bridge subsystem

instead of calculating the bridge matrices directly, each vehi-

cle 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–bridgeinteraction 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 subsys-

tem 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

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 Founda-

tion of China (Grant Nos. 51178025 and 50838006) and the Funda-

mental Research Funds for the Central Universities (Grant No.

2009JBZ016-4).

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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% 23 14 32 64

25% 192 15 36 71

20% Divergence 18 42 82



5% Divergence Divergence 118 218


dynamicanalysisofcoupledvehiclebridgesystem 2013

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