zhaiwanming-2

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Train/Track/Bridge Dynamic Interactions:

Simulation and Applications

W.M. ZHAI* C.B. CAI

Train & Track Research Institute Train & Track Research Institute Southwest Jiaotong University Southwest Jiaotong University

Chengdu 610031 Chengdu 610031 P. R. of China P. R. of China

[email protected] [email protected]

SUMMARY

This paper describes a numerical simulation technique that is used to investigate the dynamic train/track/bridge interaction. Two dynamic models are established to simulate the dynamic responses of a train running on the bridges with the ballasted track and the non-ballasted slab track, respectively. Effect of the track structure and the wheel/rail interaction on the system dynamics is considered in the models. The influence of track random irregularities on train/track/bridge dynamic interactions is investigated. The proposed simulation technique is applied to practical construction engineering in the Chinese first special railway line for passenger transport. The structural design of three extraordinary large bridges with non-ballasted tracks in this line is evaluated through a detailed simulation in the design stage and results show that these bridges are able to satisfy the demand of dynamic performance for the high-speed transport.

1 INTRODUCTION

When a train passes through a bridge, vibrations of the train and the bridge structure will be simultaneously induced, which might affect the comfort of passengers in the train, even threaten to the running safety of the train. Dynamic train/bridge interactions intensify dramatically with the


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increases of train speeds. Therefore, it is very important to investigate the train-bridge coupled vibrations in higher operating speeds. For example, it was observed in Chinese railway lines that some simply supported bridges –especially with the span length of 32 m– vibrated more strongly than before, because the operating speeds of trains were raised from 80~100 km/h to 140~160 km/h in recent years. Some measures for strengthening the bridges are then needed. However, it is not easy to determine how to strengthen them and what is the most economic way if the dynamic simulation is not carried out. Another example comes from a China’s new railway line from Qinhuangdao to Shenyang, which is now in construction and will become the first special railway line for passenger transport in China. Because the design speeds of the infrastructure are above 200 km/h, dynamic performance of vehicles, tracks and bridges on this line is of considerable interest. Simulation of train/bridge dynamic interactions is necessary so as to enable us to assess the feasibility of the structural design in advance.

Many studies have been carried out on the analysis of train/bridge interactions for various purposes [1-4]. The majority of studies have been based on the assumption that the displacement of a wheel is equal to that of the bridge beam under the wheel. They didn’t consider the wheel/rail interface, in which a wheel might loss its contact with the rail due to track irregularities with short wavelength in higher speed situation. Besides, these studies didn’t consider the dynamic effect of track structures on the train/bridge interactions. Different tracks, e.g. the ballasted track and the non-ballasted track, may have different influences on the dynamic performance of the whole train-track-bridge system.

In this paper the track structure will be incorporated to the dynamic model of train/bridge interactions and the wheel/bridge beam interface will be replaced by the wheel/rail interface. Therefore the train/bridge dynamic interaction is also called as the train/track/bridge dynamic interaction.

2 TRAIN/TRACK/BRIDGE INTERACTION MODELS

Two types of models are established to simulate the dynamic train/track/bridge interactions. One is for the bridges with general ballasted tracks, as shown in Fig.1. Another one is for the bridges with the special non-ballasted slab track, see Fig. 2. The reason to consider the slab track is that Chinese Railways (CR) has adopted this type of track structure on two extraordinary large bridges in Qinhuangdao-Shenyang line and dynamic analysis for these bridges will be performed in the paper as the application example.


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Fig. 1. Train/track/bridge interaction model I (with ballasted track).

Fig. 2. Train/track/bridge interaction model II (with slab track).

The train is composed of given number of vehicles and each vehicle is modeled as a 10-degree-of-freedom multibody system with the primary and the secondary suspensions. The train is assumed to run in the horizontal direction at a constant speed (V).

In the track model, the rail is described as a Bernoulli-Euler beam discretely supported at each fastening point. The slabs are simply modeled as continuous elastically supported beams with free ends in the vertical plane. In order to consider the dynamic effect of train entering into or leaving from the bridge [5], general track model [6,7] is taken into account at the two ends of the bridge.

The bridge beams can be described as simply supported Bernoulli-Euler beams with damping. Generally five spans are sufficient to simulate the dynamic problem of a train traversing a bridge. However, only three spans are plotted as an example in the Fig. 1 and Fig. 2.

The interface between the train and the bridge is the wheel/rail interface, which can be described by the non-linear Hertzian contact theory commonly used in vehicle/track interaction problems (see e.g. [7]):

(1)

where P is the wheel/rail force, G is the Hertzian wheel/rail contact coefficient, Zw and Zr are the displacements of the wheel and the rail under the wheel.

( )2/31

⎥⎦⎤

⎢⎣⎡ −= rw ZZG

P

V

V


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3 NUMERICAL IMPLEMENTATION

Equations of motion of the vehicle-track system could be found in [7] and will not be given here. For the slab track, equation of motion of the slab is given by

(2)

where Zs(xs,t) is the vertical displacement of the slab, EsIs is the bending stiffness of the slab, ρs is the slab mass per unit length, cs is the slab damping per unit length, Qsb is the force (in per unit length) between the slab and the bridge beam, Frsi(t) is the rail pad force on the i-th rail support point, xi is the longitudinal coordinates of the i-th support point, n0 is the number of rail support points on one slab and δ(x) is the Dirac delta function.

Equations of motion of the bridge beams in model I and in model II are similar but with little difference: For model I:

(3)

For model II:

(4)

where Zb(xb,t) is the vertical displacement of the bridge beam, EbIb is the bending stiffness of the bridge beam, ρb is the beam mass per unit length, cb is the damping of the bridge beam, nk and ns are the number of rail support points and number of the slabs on one span of the bridge beam.

Equations (2)～(4) are the fourth order partial differential equations and can be simplified to a series of second order ordinary differential equations by use of the Ritz’s method. And then all the equations for the individual vehicle, track and bridge components are combined to formulate a coherent set of ordinary differential equations, which can be numerically solved with the direct time integration methods.

Due to the very high degrees of freedom of the train-track-bridge dynamic system, the application of conventional time integration methods, such as the Runge-Kutta method, or the Newmark-β method, requires a large amount of computer storage and a long solution time. A new fast explicit time integration method developed by Zhai [8] is employed to solve such a large system. This method has been widely applied to the vehicle-track dynamic analysis and proved to be with high computational efficiency and good accuracy. Use of this integration method, the program for simulating the

( ) ( ) ( ) ( ) )(δ )(,,,, 0

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k

−=∂

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∂ ∑=

ρ


5

train/track/bridge dynamic interactions can be implemented on common personal computers with enough fast solution speed.

4 TRACK RANDOM IRREGULARITY AND ITS INFLUENCE ON DYNAMIC TRAIN/TRACK/BRIDGE INTERACTIONS

The dynamic interactions between bridge structures and passing trains not only depend on the property of bridge, such as the span length and the stiffness of the beam, but depend on the track irregularity as well. Different track irregularities have different influence on the train/track/bridge interaction.

The track geometric irregularity Z0 can be incorporated in the train-track -bridge interactive system from the wheel/rail interface:

(5)

The track irregularity Z0 could be measured in spatial field. For new lines at the design stage, however, the track geometry can not be obtained from measurements. Instead, the equivalent track irregularity spectrum is usually known according to the design standard of the designed line. The problem is that the spatial spectrum could not be directly input to the train-track-bridge system by using equation (5). So it is necessary to transform the spatial spectrum in frequency domain into general geometry variations with the running distance (or in time domain). That is done in the paper and an example is given below.

Consider the German high-speed track spectrum: Vertical profile:

m2/rad/m (6)

Alignment: m2/rad/m (7)

Cross-level: m2/rad/m (8)

))()((/)( 222222

222

scr

cvc

bASΩ+ΩΩ+ΩΩ+Ω

Ω⋅Ω⋅=Ω

))(()( 2222

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cvv

ASΩ+ΩΩ+Ω

Ω=Ω

))(()( 2222

2

cr

caa

ASΩ+ΩΩ+Ω

Ω=Ω

( )2/3

01

⎥⎦⎤

⎢⎣⎡ −−= ZZZG

P rw


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Fig.3. A sample of high-speed railway track vertical irregularity.

where Ωc=0.8246 rad/m, Ωr=0.0206 rad/m, Ωs=0.438 rad/m, Aa=2.119×10-7 mrad, Av=4.032×10-7 mrad. By means of some necessary mathematical transformation to equations (6)～ (8), track irregularities varied with distance are obtained. Fig.3 shows a sample of the vertical irregularity.

Fig. 4 to Fig.6 illustrates the difference of train-track-bridge system responses when a train passes through a bridge with and without the track irregularity shown in Fig. 3. The bridge considered is a simple beam bridge with three spans and the span length is 32 m. The train speed is 250 km/h in this computation. It can be seen from Fig. 4 and Fig. 5 that the difference is very obvious either for the car body acceleration (hence the ride comfort) or for the wheel/rail dynamic force. The track irregularity almost has no effect on the displacement of the bridge beam (Fig. 6), but has some influence on the bridge acceleration. If no track irregularity is considered in the computation, the acceleration of the bridge beam at the middle point of a span is 0.507 m/s2, whereas this value will become to be 0.722 m/s2 when the track irregularity is considered.

Fig. 7 and Fig. 8 compare the difference of the dynamic responses induced by irregularities of the German high-speed track and by irregularities measured at the Chinese fast-speed test track section on Zhenzhou-Wuchang line. Although the car body acceleration of vehicle on the Chinese test track is somewhat less than that on the German high-speed track, the wheel/rail dynamic forces on the Chinese test track are much larger than those on the later.

(a) No track irregularity considered (b) Track irregularity considered

Fig. 4. The influence of track irregularity on car body acceleration.

Z 0 (m

m)

0 100 200 300 400 500-6-3036

Distance (m)

100 150 200-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

a c (m

/s2 )

S (m)100 150 200

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

a c (m

/s2 )

S (m)


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Fig. 5. The influence of track irregularity on wheel/rail force.


Fig. 6. The influence of track irregularity on displacement of bridge beam at middle of span.

(a) German high-speed track irregularity used (b) Chinese fast-speed track irregularity used

Fig. 7. Comparison of car body acceleration of train on different tracks.

0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

Z b (m

m)

T (s)0 1 2 3 4 5

-5

-4

-3

-2

-1

0

1

2

Z b (m

m)

T (s)

100 150 20025

50

75

100

125

Bridge section with 3 spans

P (k

N)

S (m)100 150 200

65

70

75

80

Bridge section with 3 spans

P (k

N)

S (m)

100 150 200 250-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Bridge section

a c (m

/s2 )

S (m)100 150 200 250

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Bridge section

a c (m

/s2 )

S (m)


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(a) German high-speed track irregularity used (b) Chinese fast-speed track irregularity used

Fig. 8. Comparison of wheel/rail force of train on different tracks.

The above results have indicated the great influence of the track geometry on dynamic train/track/bridge interactions in high-speed situation. Compared with the influence of track irregularity, the influence of bridge vibration on the ride quality and on the dynamic wheel load is much smaller as long as the stiffness of bridge beam is in the normal design range. Otherwise if the bridge stiffness is not large enough the ride quality of a passing train will be obviously worsen. As an example, if the bending stiffness of beam decreases by 50%, the car body acceleration of train will be increased form 1.18 m/s2 (normal case) to 1.69 m/s2.

5 APPLICATION IN PRACTICAL ENGINEERING

5 .1 Background of eng ineer ing

In 1999 China’s first special railway line for passenger transport only was put into construction, and will be built up in 2003. That is Qinghuangdao-Shenyang line, in which there is a 66.8 km long high-speed test section from Shanhaiguan to Suizhong. The design speed of the whole line is 200 km/h, and in the test section the design speed of the infrastructure is up to 300 km/h. Because this is CR’s first railway line with so high design speeds, the system dynamic behavior, especially the train-bridge vibration property, becomes one of the most important problems concerned. There are several bridges located in the high-speed section; among them there are two extraordinary large bridges, i.e. Shahe Bridge and Gouhe Bridge. Non-ballasted track will be laid on these two bridges. Another extraordinary bridge that will be laid with non-ballasted track is Shuanghe bridge (beyond the test section). Fig. 9 shows an example of the configuration of the non-ballasted slab track on a box-shape beam bridge, in which there is a concrete asphalt layer (CA layer) with the thickness 0f 50 mm under the slab.

100 150 2000

50

100

150

P (k

N)

S (m)100 150 200

0

50

100

150

P (k

N)

S (m)


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Fig. 9. System configuration of slab track on bridge.

CR requires the authors to carry out the dynamic simulation of a train passing through these extraordinary large bridges with high speeds and to give out the evaluation results for the design of the bridges.

5 .2 Parameters o f t ra in , t rack and br idges

Only some basic dynamic parameters can be presented in the paper. Table 1 lists the parameters used in the vehicle model of the high-speed train.

For the slab track, the rail mass is 60 kg/m, the rail pad stiffness is 60 MN/m, the mass of slab per unit length is 1115.62 kg/m, the fastening spacing is 0.6 m.

For the bridges, the total length of one bridge is approximately 500 m, the span length is 24 m, the height of beam is 2.2 m, the moment of inertia of beam cross section is 4.87 m4, and the weight of beam per unit length is 204.7 kN/m.

Table 1. High-speed train parameters. Parameter Power car Trailer Unit unsprung mass (per wheelset) 1900 1400 kg primary suspended mass (per bogie) 4297 3500 kg secondary suspended mass 63807 47400 kg car body mass inertia 1349328 1288200 kg·m2 bogie frame mass inertia 4033 3470 kg·m2 primary suspension stiffness (per wheelset) 2.3396 0.55 MN/m primary suspension damping (per wheelset) 21 12 kN·s/m secondary suspension stiffness (per bogie) 0.8858 0.4 MN/m secondary suspension damping (per bogie) 63.75 80 kN·s/m bogie distance 11.46 18.0 m wheelbase 3.0 2.4 m wheel diameter 1.04 0.915 m static wheel load 98.1 73.6 kN

CA layerSlab

Bridge beam

Concrete base


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5 .3 Eva l uat ion s tandard for running sa fe ty and r ide comfor t o f t ra in on br idge

Table 2 gives the standards used in the simulation for evaluation of the running safety and ride quality of a train on a bridge. Most of these come from CR’s standards. In Table 2, the reduction rate of wheel load is defined as (P-P0)/P0, where P0 is the static wheel load. CR uses this index as one of the evaluation indices for the running safety.

Table 2. Evaluation standards used in the simulation. Evaluation indices Allowable limit car body vertical acceleration 1.25 m/s2 (comfortability standard) ride comfort in Sperling index 3.0 wheel/rail force 300 kN (the design load of slab track) reduction rate of wheel load 0.65 ratio of displacement to span of beam 1/1800 acceleration of bridge beam 3.5 m/s2

5 .4 S imula t ion re su l t s and the ir eva luat ion

Dynamic responses of vehicles, tracks and bridges are simulated when a train passes through the bridges with speeds of 200~300 km/h. The train is composed of one power car and six trailers. Some examples are given in Fig. 10 to Fig. 13, in which case the train speed is 200 km/h and the bridge beam is the box-shape one with the slab track on it. The system excitation was input by use of the

Fig. 10. Time history of the car body acceleration. (a)Power car, (b) Trailer

25 50 75 100 125 150 175 200-1.2-0.60.00.61.2

a c(m

/s2 )

S（m）

(a)

60 80 100 120 140 160 180 200 220 240-1.2-0.60.00.61.2

a c (m

/s2 )

S（m）

(b)


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Fig. 11. Time history of the wheel/rail force of the power car.

Fig. 12. Time history of the acceleration of the slab at the middle of span.

Fig. 13. Time history of the displacement of the bridge beam at the middle of span.

Table 3. Simulated results of system dynamic indices. Train speed Evaluation indices

200 km/h 250 km/h 300 km/h car body acceleration of power car (m/s2) 0.488 0.521 0.646 car body acceleration of trailer (m/s2) 1.096 1.166 1.243 ride comfort index of power car 1.834 2.091 2.120 ride comfort index of trailer 2.322 2.663 2.642 wheel/rail force (kN) 121.93 136.30 147.64 reduction ratio of wheel load 0.236 0.389 0.505 ratio of displacement to span length of bridge beam 1/3403 1/3249 1/3200 acceleration of bridge beam (m/s2) 0.45 1.324 1.615

25 50 75 100 125 150 175 2006080

100120140

P (k

N)

S（m）

50 100 150 200 250 300-6-3036

a s (m

/s2 )

S（m）

50 100 150 200 250 300 350-3-2-101

Z b (m

m)

S（m）


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German high-speed track irregularity shown in Fig. 3. Complete results of the evaluation indices are given in Table 3.

The above results show that the car body acceleration and the ride comfort index increase as the train speed increases and do not exceed their allowable limits at the speed range concerned. Furthermore the ride comfort index is less than 2.75, which means the ride comfort belongs to the grade ‘GOOD’ according to the CR’s standard. The wheel/rail force fluctuates between 50～150 kN in the range of speeds from 200 km/h to 300 km/h, which is far less than the maximum value of the design load. The reduction ratio of wheel load, (P-P0)/P0, increases from 0.236 at speed of 200 km/h to 0.505 at 300 km/h, but still satisfies its safety limit (0.65).

It can also be seen from the simulated results that the evaluation indices for the bridge satisfy the allowable limits very well, even in the speed of 300 km/h. For example, in this speed the ratio of displacement to span length of the bridge beam is only 1/3200, far less than 1/1800, and the vertical acceleration of the beam is 1.615 m/s2, less than 3.5 m/s2, the allowable limit.

Based on the simulation, it is concluded that when the high-speed train runs on the designed non-ballasted bridges with speeds of 200～300 km/h all the dynamic indices satisfy the evaluation standards for running safety and ride comfort of the train on the bridge. It has been demonstrated that the structural design of Shahe, Gouhe and Shuanghe bridges on the Qinghuangdao-Shenyang fast-speed line is reasonable and feasible.

6 CONCLUDING REMARKS

The paper has presented numerical models for the simulation of dynamic train/track/bridge interactions. The feature of this study is that the systematical train/track/bridge dynamic model, in which the effect of the track structure and the wheel/rail interaction on the system dynamics is included, replaces the conventional train/bridge interaction model.

It has been shown that the track random irregularity has great effect on the train/track/bridge dynamic interaction, especially in the high-speed operation. The influence of the bridge vibration on the ride quality of a train on a bridge is not big by comparison with the influence of the track geometry. Thus, the really existing track random irregularity should be considered in the simulation of the dynamic train/track/bridge interactions. On the other hand, if the bending stiffness of the bridge beam is not sufficient large, the bridge may induce big vibration when a train passes over the bridge, and the ride quality in the train will decrease obviously.

The dynamic simulation technique with the proposed models has been applied to the practical construction engineering of China’s first passenger special transport line. The structural design of three extraordinary large bridges with non-ballasted tracks in this line has been evaluated from the


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dynamic aspect in the design stage. It has been demonstrated that these bridges are able to satisfy the demand of dynamic performance for the high-speed transport. At present time, these bridges have already been built and the track structures are now being laid out on the bridges. The first high-speed running test will be carried out in May of 2002. All the dynamic indices will be measured and then be compared with the simulated results to validate the simulation method.

By the way, unified software using the presented method is now being designed under the support of the Ministry of Railways, China, so as to provide a powerful dynamic analysis tool for the design of bridges on new high-speed lines or fast-speed lines. ACKNOWLEDGEMENTS

The research work presented in the paper was finically supported by National Natural Science Foundation of China (NSFC), and also by Foundation for University Key Teacher by the Ministry of Education, China. The application example is a part of the research projects (grant No. 99G06 and 99G16) of the Ministry of Railways, China and the research results have been adopted in the structural design of Shahe bridge, Gouhe bridge and Shuanghe bridge in Qinhuangdao-Shenyang line. REFERENCES 1. Chu, K.H., Garg, V.K. and Dhar, C.L. Railway-bridge impact: Simplified train and bridge model. Journal of the

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Dynamics, 1980, 9(4): 207-236. 3. Dahlberg, T. Vehicle-bridge interaction. Vehicle System Dynamics, 1984, 13(1): 187-206. 4. Diana, G. and Cheli, F. Dynamic interaction of railway systems with large bridges. Vehicle System Dynamics, 1989,

18(1): 71-106. 5. Zhai, W.M. and True, H. Vehicle-track dynamics on a ramp and on the bridge: simulation and measurements.

Vehicle System Dynamics, 1999, 33 (Supplement), 604-615. 6. Zhai, W.M. and Sun, X. A detailed model for investigating vertical interactions between railway vehicle and track.

Vehicle System Dynamics, 1994, 23 (Supplement), 603-615. 7. Zhai, W. and Cai, Z. Dynamic interaction between a lumped mass vehicle and a discretely supported continuous

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