J. Cent. South Univ. (2013) 20: 2830−2839 DOI: 10.1007/s11771­013­1803­5

Dynamic responses of bridge­approach embankment transition section of high­speed rail

YANG Chang­wei(杨长卫) 1 , SUN Hai­ling(孙海玲) 2 , ZHANG Jian­jing(张建经) 1 , ZHU Chuan­bin(朱传彬) 1 , YAN Li­ping(颜利平) 3

1. Civil Engineering School, Southwest Jiaotong University, Chengdu 610031, China; 2. Department of Engineering Management, Henan University of Urban Construction, Pingdingshan 467036, China;

3. Geotechnical Engineering Group, Los Angeles Department of Water and Power, Los Angeles 90099, USA

© Central South University Press and Springer­Verlag Berlin Heidelberg 2013

Abstract: Based on the vehicle−track coupling dynamics theory, a new spatial dynamic numerical model of vehicle−track−subgrade coupling system was established considering the interaction among different structural layers in the subgrade system. The dynamic responses of the coupled system were analyzed when the speed of train was 350 km/h and the transition was filled with graded broken stones mixed with 5% cement. The results indicate that the setting form of bridge­approach embankment section has little effect on the dynamic responses, thus designers can choose it on account of the practical circumstances. Because the location about 5 m from the bridge abutment has the greatest deformation, the stiffness within 0−5 m zone behind the abutment should be specially designed. The results of the study from vehicle−track dynamics show that the maximum allowable track deflection angle should be 0.09% and the coefficient of subgrade reaction (K30) is greater than 190 MPa within the 0−5 m zone behind the abutment and greater than 150 MPa in other zones.

Key words: high­speed rail; bridge­approach embankment section; numerical model; track deflection angle

1 Introduction

The control of line­shape irregularity is an important task in designing high­speed rail because it is related to the safety, reliability and ride comfort of the high­speed train. Regardless of existing railway lines or the new lines, bridge­approach embankment section has a great impact on the line­shape irregularity. It has been required that bridge­approach embankment sections should be designed in configuration of a trapezoid in the previous Chinese railway design specifications. However, some current related national specifications [1] require that a reversed trapezoid configuration should be adopted for realizing the transition from rigid abutment (culvert) to flexible subgrade. The stiffness difference between bridge and subgrade causes differential settlement in the bridge­approach embankment section. The differential settlement may result in rail bending and the tensile stress in the top surface of the reversed trapezoid section, even cracking. These will affect the smoothness of the line. When the high­speed train is passing through this section, wheel­track force increases the cumulative deformation of the track, accelerates the deterioration of the line, reduces the service life of track structure, and

may even cause instable phenomena. Hence, it is necessary to study the dynamic responses of bridge­approach embankment section under train loads and the effect of the configuration of transition sections on the dynamic responses in order to make some reasonable suggestions for solving the above­mentioned problems.

In order to ascertain the dynamic responses of bridge­approach embankment section under train loads, a high­speed rail passenger dedicated line was selected as a prototype and a three­dimensional numerical model was developed by using the finite element method (FEM) based on previous research results [2−4], where soil properties are known and the dynamic responses of bridge­approach embankment section were analyzed for a train with a speed of 350 km/h. In this work, the effects of the transition configuration on settlement, track deflection angle and the length of “reasonable reinforced zone” were investigated. Finally, the allowable track deflection angle was proposed by considering different influence factors.

2 Research scheme

Considering the track irregularity, the vertical

Foundation item: Project(41030742) supported by the National Natural Science Foundation of China; Project(2009G010­c) supported by the Technological Research and Development Programs of the Ministry of Railways, China

Received date: 2012−06−17; Accepted date: 2013−09−02 Corresponding author: ZHANG Jian­jing, Professor, PhD; Tel: +86−15928118504; E­mail: yangchangwei56@163.com

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dynamic analytical models of vehicle−track−subgrade coupled system based on the vehicle−track dynamics theory is firstly established, and then simulated by using the computer software, finally the wheel load curve is got. Second, combined with the testing data, the 3­D numerical model is established based on the design drawing of the third section in Beijing−Shanghai high­speed railway, and wheel load is applied to this model, then the dynamic responses of the transition are obtained. Third, combined with the dynamic responses of the transition, the effect of the configuration of transition section is investigated. Fourth, the relationship between dynamic indicators and track deflection angle is analyzed by the computer software ADAMS. Finally, based on the study from vehicle−track dynamics, the maximum allowable track deflection angle and the reasonable reinforced area are suggested. The research scheme is shown in Fig. 1.

3 Numerical model of bridge­approach embankment section

3.1 Vertical dynamic analytical models of vehicle− track−subgrade coupled system The vertical dynamic analytical models of

vehicle−track−subgrade coupled system developed in this study include vehicle−track and track−subgrade coupled dynamic models which are coupled by wheel/track interaction. Based on the reversed trapezoid configuration (Figs. 2(a) and (b)) transition section used in the Beijing−Shanghai High−speed railway, the

analytical model is established and simplified in Fig. 2(c). In the model, carbody is assumed as rigid body and rail is assumed as an Euler beam with continuous point supports [5−8], and track slab is simplified as a free beam with finite length based on elastic foundation. Rail, track slab, supporting layer and frictional slab are supported on continuous point springs and dampers. Cement­treated asphalt mortar between frictional slab and subgrade is simplified as continuous distributed viscous elastic units consisting of linear springs and dampers (KCA, CCA). The basebed surface layer, basebed bottom layer and subgrade are discretized into rigid blocks, and continuous point springs, dampers and stress pieces (Ki, Ci, and Smaxi) are distributed between structural layers to simulate the interactions. The parameters change gradually along bridge­approach embankment section. Other physical parameters involved are described in Refs. [9−12].

The meanings of the symbols in Fig. 2 are as follows. Mc: mass of carbody, kg; Jc: nodding inertia of carbody, kg∙m 2 ; CS2: damping coefficient of secondary suspension damper, N∙S/m; ZC(t): displacement of carbody, m; Mt: secondary suspension mass, kg; βt1(t): angular displacement of the left bogie, rad; βt2(t): angular displacement of the right bogie, rad; Zt1(t): displacement of bogie, m; CS1: damping coefficient of primary suspension damper, N∙S/m; KS1: primary suspension stiffness coefficient, N∙S/m; ZWi: displacement of wheel, m; Z0i(t): irregularity displacement of rail, m; Pi(t): wheel­rail force, kN; Mr: mass of rail, kg; EI: inertial moment of rail, N∙m 2 ; Zr(x,t): displacement of rail, m;

Fig. 1 Research scheme

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Fig. 2 Analytical models: (a) Design diagram of bridge­ approach embankment section; (b) Cross­section of I­I in Fig. 2(a); (c) Vertical dynamic analysis models of vehicle− track−subgrade coupled system

KP1: stiffness coefficient of cushion under the rail, N/m; CP1: damping coefficient of damper of cushion under the rail, N∙S/m; Kb1: stiffness coefficient of track slab, N/m; Cb1: damping coefficient of damper of track slab, N∙S/m; Kc1: stiffness coefficient of supporting layer, N/m; Cc1: damping coefficient of damper of supporting layer, N∙S/m; Kd1: stiffness coefficient of frictional slab, N/m; Cd1: damping coefficient of damper of frictional slab, N∙S/m; Mb: mass of track slab, kg; Mc: mass of supporting layer, kg;Md : mass of frictional slab, kg.

3.2 Track­subgrade dynamic analysis model The operating site DK533+440.5 located in the third

section of Beijing−Shanghai high­speed railway was taken as the research object. This operating site adopts the reversed trapezoid configuration for realizing the transition from rigid abutment to flexible subgrade. In this transition, the basebed surface layer is 0.7 m thick, the basebed bottom layer is 2.3 m thick, the subgrade is 6 m thick, and the dimensions of the track slab are 6 450 mm×2 550 mm×200 mm. The adjustment layer emulsified by cement is 300 mm thick with top and bottom widths of 2 950 and 3 250 mm, respectively. The frictional slab is 9 m wide and 0.4 m thick. The bottom boundary of the transition is 3 m long and its slope gradient is 1:2.7. Meanwhile, the slope gradient in two sides of the transition is 1:1.5. The fill material of the basebed surface layer is graded broken stone. The fill materials of both the basebed bottom layer and the subgrade are sandy stratum mixed with 28% (mass fraction) broken stones. The fill material of the reversed trapezoid transition section is graded broken stone mixed with 5% cement. The material type of the foundation is weathered granite. A specific geometric model is shown in Fig. 3. Based on the above geometric model, a numerical model of bridge­approach embankment section by using software MIDST/GTS is established, as shown in Fig. 4.

Fig. 3 Geometric model of transition section

The material parameters are got by the field tests and the lab tests, as shown in Fig. 5. The material parameters are listed in Table 1, where the deformation modulus (Ev2) of graded broken stone (GBS) in the transition section was taken from the minimum values specified in the code.

3.3 Determination of model parameters 3.3.1 Material parameters of model

The material parameters are got by the field tests

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and the lab tests, as shown in Fig. 5. The material parameters are listed in Table 1, where the deformation modulus (Ev2) of GBS in the transition section was taken from the minimum values specified in the code.

Because the abutment has higher stiffness, it is modeled by using linear elastic constitutive relationship; the GBS mixed with 5% cement, subgrade, basebed surface layer and base bed bottom layer are all modeled by the Mohr­Column (M­C) model.

Since the reflection of waves on the boundary has great influence on dynamic analysis for bridge­approach embankment section, a viscoelastic boundary is used to eliminate the effect and the parameters of the damper are listed in Table 2. 3.3.2 Train load

In this work, the related parameters of CRH­380 which is used in Beijing−Shanghai high speed railway now [13] is selected to establish vehicle model at a speed

Fig. 4 Numerical model of transition section

Fig. 5 Photos of plate loading test (K30) (a), ring sampler test (density) (b), dynamic sounding test (Evd) (c) and static deformation modulus test (Ev2) (d)

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Table 1Material parameters used in analysis

Location Material type Mode Deformation modulus, Ev2/MPa

Poisson ratio, υ

Internal friction angle, F(°)

Cohesion, C/kPa

Density, ρ/(g∙cm −3 )

Basebed surface Graded broken stone M­C 120 0.33 27 70 1.95

Basebed bottom M­C 80 0.35 20 55 1.90

Subgrade

Sandy stratum mixed with 28% broken stones M­C 36.5 0.3 10 30 2.1

Graded broken stone mixed with 5% cement

Graded broken stone mixed with

5% cement M­C 150 0.21 50 26 2.66

Friction/structural/ track slab Elastic 2.0×10 4 0.15 — — —

Abutment Concrete

Elastic 2.0×10 4 0.15 — — —

Foundation Weathered granite M­C 50 0.25 19 50 1.8

Table 2Material parameters of dampers on boundary

Location Horizontal stiffness coefficient, Kh/(kN∙m −3 )

Normal stiffness coefficient, Kv/(kN∙m −3 )

Damping coefficient of normal damper, CP/(kN∙s∙m −1 )

Damping coefficient of horizontal damper, CS/(kN∙s∙m −1 )

Foundation 1 465 1 524 529 211

Subgrade — 1 707 305 147 Friction/structural/

track plate — 9 149 619 7 978 4 984

Abutment — 1 138 313 4 664 5 573

of 350 km/h and vehicle−track−subgrade specially coupled dynamic simulation model is established by using the computer software ADAMS, where a wheel load of about 300 000 cycles is imposed to consider the role of repeated loading of high­speed train. The measuring points are set at a distance of about 72 m away from the start position to eliminate the effect of high­speed train launch and brake on the dynamic response of observation. Due to the lack of the Chinese high­speed rail track irregularity data at present [14−15], the rail track irregularity data included in the simulation software ADAMS/Rail is used to improve the simulation results. Because the ADAMS/Rail contains only two material models (i.e., flexible body and rigid body), the constitutive properties of soils cannot be simulated, non­linear springs and dampers are set uniformly between the rigid subgrade and the track along the line to consider the interactions between the track and the subgrade. The simulation model is shown in Fig. 6 and the calculation results are shown in Fig. 7.

Figure 7 shows that there are four peak values in one cycle. During 0−0.6 s, the wheel­track load is almost zero, and then the wheel load value gradually increases. At 0.72 s, the first peak value arrives. Therefore, the distance affected by the train load speed of 350 km/h is about 11.67 m. The first wheel­track load in every bogie

Fig. 6 Axis view of vehicle−track−subgrade coupling system model

Fig. 7Wheel load versus time curve

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is bigger than that in the last one. The possible reason may be the retardance of displacement of the rail.

4 Calculation results

4.1 Dynamic response of bridge­approach embankment section

4.1.1 Longitudinal distribution of dynamic response along roadway

Track deflection angle discussed in this work refers to tangent angle within the plane of the rail.

In order to study the longitudinal distribution of dynamic responses of bridge­approach embankment sections under train loading, a number of monitoring points are set along the centerline of the subgrade and located on the basebed surface. Specific conclusions from the results of these measuring points are obtained as follows.

First, both the maximum settlement and the maximum track deflection angle of bridge−approach embankment sections occur at the bridge−subgrade joint, close to the abutment, as shown in Figs. 8−10. One possible reason is that the large stiffness difference at the bridge−subgrade joint leads to a downward additional impact caused by train wheels when the train leaves the bridge and thereby increases its vertical deformation. The other possible reason is that the abutment is made of masonry material.

Second, the distribution of the dynamic stress in the basebed surface of bridge­approach embankment sections obeys the following rules: the values near the abutment are small and the values far away from the abutment are large; it has a double peak phenomenon, which is mainly related to the large stiffness of abutment and the stress redistribution action, as illustrated in Fig. 11.

Third, accelerations of basebed surface in bridge­ approach embankment sections change significantly along the longitudinal direction, the values near the abutment are large and the values far away from the abutment are small essentially, as shown in Fig. 12.

The above­mentioned three conclusions are consistent with the measured data obtained on the Qinhuangdao−Shenyang passenger dedicated line [16− 17]. 4.1.2 Lateral distribution of dynamic response along

roadway In order to study the distribution of dynamic stress

along the direction perpendicular to the route in bridge­approach embankment section under train load, the cross­section at a distance of 10.5 m behind the abutment was chosen and four monitoring points spacing 0.75 m arranged on both sides of the center line in each structural layer were selected. The results are shown in

Fig. 8 Side view of dynamic deformation (part)

Fig. 9 Distribution curve of dynamic displacement of basebed surface along longitudinal direction

Fig. 10 Distribution curve of vertical angle of rail surface along longitudinal direction

Fig. 13 and the main conclusions are presented as follows.

First, distribution of the dynamic stress along the direction perpendicular to the route of the track slabs, the basebed surface, basebed bottom, subgrade and foundation are basically consistent with each other. They all show “saddle” type distribution, the maximum values occur at the contact points between the rail and the track slab, and the minimum values occur at the centerline location. The trends are consistent with the measured results of Qinhuangdao−Shenyang passenger dedicated line [16−17].

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Fig. 11 Distribution of dynamic stress of basebed surface along longitudinal direction

Fig. 12 Distribution of acceleration of basebed surface along longitudinal direction

Fig. 13 Distribution of dynamic stress of each structure layer along lateral direction

Second, the fluctuation range of dynamic stress of the track slab is great along the direction perpendicular to the route and its distribution is very uneven. Uneven distribution of the dynamic stress will cause the uneven deformation of track structures, even cracking, which will affect the normal operation of the railway traffic. Therefore, optimizing the type of the track structure should be considered in order to improve the uneven

distribution of the dynamic stress when designing the railway approach embankment. 4.1.3 Effect of configuration of transition sections on

settlement The previous Chinese railway design specification

requires that bridge­approach embankment sections should be designed in configuration of a trapezoid; however, bridge­approach embankment sections are in the form of an inverted trapezoid specified in the interim design provisions of Beijing−Shanghai high­speed railway [18]. Therefore, the above­mentioned setting forms of bridge­approach embankment sections are considered in numerical calculation to analyze the effect of the configuration of transition sections on the settlement. Under the conditions that material characteristic parameters and boundary conditions are constant, distribution of the monitoring points is shown as follows: the first point is located at the end of the abutment; there are three points which are respectively 9, 13.5 and 19 m away from the abutment located in the basebed surface and rails. The points in rails are used to monitor the settlement. The wheel load−time history curve is shown in Fig. 7 and the material parameters are listed in Tables 1 and 2. The overall numerical model is shown in Fig. 3. The calculation results are plotted in Fig. 14.

Fig. 14 Dynamic displacement of basebed surface

The following conclusions can be obtained from Fig. 14. First, the settlement curves of the two setting forms of the transition section are generally similar. Second, the deformation slope of the trapezoidal transition is gentler than that of the inverted trapezoid. Third, the settlement of the inverted trapezoid transition is smaller than that of the trapezoid transition; however, the difference is insignificant. Hence, if the material parameters and other conditions are the same, the setting forms of the bridge­approach embankment section have little effect on the settlement. Therefore, designers can choose the setting form according to the practical circumstances.

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4.2 Relationship between dynamic indicators and track deflection angle In this work, the related parameters of HSC

high­speed passenger train were selected to establish vehicle model at a speed of 350 km/h, using single compartment and ignoring the interaction between compartments for security consideration. The other parameters of the vehicle−track−subgrade dynamic model are the same as in Section 3.3.2, the specific simulation model is shown in Fig. 6 and the scheme of track deflection angle is shown in Fig. 15, where the contour lines are not only the rail surface shape after deformation but also the distribution of the stiffness.

Fig. 15 Scheme of track deflection angle

In order to investigate the relationship between track deflection angles and various dynamic indexes of the vehicle, six indexes (wheel­track lateral force, wheel­track vertical force, carbody vertical acceleration, carbody lateral acceleration, derailment coefficients and reduction rate of wheel load) are calculated, respectively. when the track deflection angle are 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4 and 1.5 mrad. The calculation results are compared with the specifications in Chinese codes TB/T2360293 and GB 5599—85. The results are shown in Table 3, and the following conclusions are drawn.

First, four indexes (wheel­track lateral force, wheel­track vertical force, carbody vertical acceleration

and derailment coefficients) can satisfy the requirements of codes when the track deflection angle is in the range of 0.3−1.5 mrad.

Second, the carbody lateral acceleration can satisfy the requirements of codes when the track deflection angle is in the range of 0.3−1.1 mrad; but it is too large and cannot satisfy the requirements when the track deflection angle is in the range of 1.1−1.5 mrad.

Third, the reduction rate of wheel load rate can satisfy the requirements of codes when the track deflection angle is in the range of 0−0.9 mrad; but it is too large and cannot satisfy the requirements when the track deflection angle is in the range of 0.9−1.5 mrad.

Fourth, with the increase of track deflection angle, carbody vertical acceleration and carbody lateral acceleration increase non­linearly while wheel­track lateral force, wheel­track vertical force and reduction rate of wheel load increase linearly, and derailment coefficient does not change essentially and always maintains a constant value.

These results show that six indexes can satisfy the requirements of codes when the track deflection angle is in the range of 0−0.9 mrad. Therefore, it is suggested that the limitation of the track deflection angle should be located at 0.9 mrad. Combining with the results shown in Fig. 10, the reasonable reinforced zone is within 0−5 m away from the abutment.

4.3 Research on reinforcement measures and reinforcement effect The above­mentioned analytical results demonstrate

that the zone in the range of 0−5 m behind the abutment is the weak link of the bridge­approach embankment sections. During the design, if the coefficient of horizontal subgrade reaction (normally called K30) of the

Table 3 Relationship between track deflection angles and various dynamic indexes of vehicle Track deflection angle/mrad

Wheel­track lateral force/kN

Wheel­track vertical force/kN

Carbody vertical acceleration/(m∙s −2 )

Carbody lateral acceleration/(m∙s −2 )

Derailment coefficient

Reduction rate of wheel load

1.5 19.51 148 1.221 1.582 0.142 1.114 1.4 19.23 142.1 1.11 1.346 4 0.136 1.030 1.3 18.91 136 1.008 1.170 4 0.128 0.943 1.2 18.59 130.5 0.897 1.025 0.146 0.864 1.1 18.22 125.5 0.805 0.925 0.145 0.793 1.0 17.85 119 0.732 0.835 0.151 0.700 0.9 17.42 113.17 0.674 0.754 0.154 0.617 0.8 16.95 107.4 0.62 0.715 0.158 0.534 0.7 16.45 101.6 0.583 0.685 0.162 0.451 0.6 15.91 95.82 0.555 0.66 0.166 0.369 0.5 15.33 90.032 0.527 0.645 0.170 0.286 0.4 14.67 84.245 0.499 0.635 0.174 0.204 0.3 13.92 78.48 0.472 0.623 0.177 0.121

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graded broken stone mixed with 5% cement in transition sections adopts the minimum critical value of “Beijing− Shanghai High­speed rail design requirements,” it will appear that the subgrade settlement, track deflection angle and the dynamic indicators of vehicle do not satisfy the specification, which will result in a serious impact on the safety of train operation.

In order to solve the above­mentioned issues, K30

should be increased in the zone that is within 0−5 m away from the abutment and the impact on the settlement and the track irregularity should be investigated to determine the critical value of K30. Using the numerical model in Section 2.2 and the related parameters in Section 2.3, calculations for three values of K30 in the zone were performed within 0−5 m away from the abutment (170, 190 and 210 MPa) and a value of K30 in the other zones (150 MPa). From the calculations, we get the settlement of bridge­approach embankment section and the distribution of the track deflection angle along the direction of route, as shown in Fig. 16. Specific results are presented as follows.

First, Fig. 16 shows that both the track deflection angle and the settlement of the basebed surface decrease with the increase of K30. Second, the design and construction of the reasonable reinforced zone should be different from the other zones and performed specially.

Fig. 16 Distribution of vertical settlement (a) and track deflection angle (b) of basebed surface along longitudinal

direction Third, based on related codes and the overall consideration of six dynamic indicators, the designed limitations of K30 are as follows:

For the zone within 0−5 m behind the abutment, K30≥190 MPa; For the zone beyond 5 m behind the abutment, K30≥150 MPa.

5 Other research conditions

In order to research to the reasonable zone at different speeds, the same model, material parameters and research secheme listed in Section 3 are used to calculate the dynamic responses of the bridge­approach embankment section, but the speeds are 160, 250 and 350 km/h, K30=150 MPa, respectively. At the same time, when K30=150, 170, 190 and 210 MPa and the speed is 350 km/h, the reasonable reinforced zone is also calculated in this work. The concrete calculated results are shown as follows:

When the speeds are 160, 250 and 350 km/h, the reasonable zones are 2.5, 4.2 and 5.0 m, respectively. When the K30 values are 150, 270, 190 and 210 MPa, the reasonable zones are 5.0, 3.5 and 0 m, respectively.

6 Conclusions

1) The setting forms of bridge­approach embankment section have insignificant effect on the dynamic responses, thus designers can choose it based on the practical circumstances.

2) The track deflection angle should be limited to 0.9‰ in order to take six dynamic indicators into consideration.

3) Based on related codes and the overall consideration of six dynamic indicators, when the speed is 350 km/h, the designed limitations K30≥190 MPa for the zone within 0−5 m behind the abutment, and K30≥150 MPa for the zone beyond 5 m behind the abutment.

4) Based on related codes and the overall consideration of six dynamic indicators, the reasonable reinforced zones of the bridge­approach embankment are obtained.

5) According to the above­mentioned analysis results, the reasonable reinforced zone is the weak zone of bridge­approach embankment section.

References

[1] TB 10621−2009, High speed railway design specifications [S]. Beijing: China Railuay Publishing House, 2009.

[2] NGUYEN D V, KIM K D, WARNITCHAI P. Simulation procedure for vehicle­substructure dynamic interactions and wheel movements

J. Cent. South Univ. (2013) 20: 2830−2839 2839 using linearized wheel­rail interface [J]. Finite Elements in Analysis and Design, 2009, 45(5): 341−356.

[3] XU Peng. Study on vibration of train­track­subgrade coupled system and running safety of train in earthquake [D]. Chengdu: Southwest Jiaotong University, 2012. (in Chinese)

[4] ZHAI Wan­ming. The vertical model of vehicle­track system and its coupling dynamics [J]. Journal of Railway Engineering Society ,1992, 3(14): 10−21.

[5] HANDOKO Y, DHANASEKAR M. An inertial reference frame method for the simulation of the effect of longitudinal force to the dynamics of railway wheelsets [J]. Nonlinear Dynamics, 2006, 45(3): 399−425.

[6] AUCIELLO J, MELI E, FALORNI S, MALVEZZI M. Dynamic simulation of railway vehicles: Wheel­rail contact analysis [J]. 2009, 47: 867−899.

[7] PIOTROWSKI J, KIK W. A simplified model of wheel/rail contact mechanics for non­Hertzian problems and its application in rail vehicle dynamic simulations [J]. Vehicle System Dynamics, 2008, 46: 27−48.

[8] SHACKLETON P, IWNICKI S. Comparison of wheel­rail contact codes for railway vehicle simulation—An introduction to the Manchester Contact Benchmark and initial results [J]. Vehicle System Dynamics, 2008, 46: 127−129.

[9] VIL’KE V G, MAKSIMOV B A, POPOV S A. Stability of the rectilinear motion of a railway wheelset [J]. Moscow University Mechanics Bulletin, 2010, 65(2): 31−37.

[10] AN M. Two simple fast integration methods for large­scale dynamic problems in engineering [J]. International Journal for Numerical Methods in Engineering, 1996, 39(24): 4199−4214.

[11] STEENBERGEN M J M M, METRIKINE A V, ESVELD C. Assessment of design parameters of a slab track railway system from a dynamic viewpoint [J]. Journal of Sound and Vibration, 2007, 306(1/2): 361−371.

[12] SUN Yan­quan. Use of Simulation in determination of railway track vertical dynamic forces in railway vehicle acceptance produce [C]// The Third International Conference on Mechanical Engineering and Mechanics, Fugang, Japan, 2009: 1052−1061.

[13] GALVIN P, ROMERO A, DOMINGUEZ J. Vibrations induced by HST passage on ballest and non­ballast tracks [J]. Soil Dynamics and Earthquake Engineering, 2010, 30(9): 862−873.

[14] LIAN Song­liang, LIU Yang, YANG Wen­zhong. Analysis of track irregularity spectrum of Shanghai−Nanjing railway [J]. Journal of Tongji University: Natural Science, 2007, 35: 1342−1346. (in Chinese)

[15] DING Jun­jun, LI Fu. Numerical simulation of one side rail irregularity, Journal of Traffic and Transportion Engineering, 2010, 10: 29−35. (in Chinese)

[16] WANG Kai­yun, ZHAI Wan­ming, CAI Cheng­biao. Comparison on track spectra of Qinghuangdao−Shenyang passenger railway line and German railway line [J]. Journal of Sounthwest Jiaotong University, 2007, 42: 425−430. (in Chinese)

[17] QING Qi­xiang, WANG Yong­he, ZHAO Ming­hua. Research on dynamic performance of bridge (culvert)­subgrade transition section on Qin−shen High­speed railway [J]. Rock and Soil Mechanics, 2008, 29(9): 2415−2421.

[18] MENG Fan­hui, HOU Yong­feng, WU Tao. Three­dimensional numerical analysis of bridge­approach embankment section [J]. Rock­Soil Mechanics, 2007, 28: 849−854. (in Chinese)

(Edited by FANG Jing­hua)