Mapping relationship between dynamic responses of high-speed trains and additional bridge deformations

Abstract

Deformations of high-speed railways accumulate over time and affect the geometry of the track, thus affecting the running safety of trains. This article proposes a new method to map the relationship between dynamic responses of high-speed trains and additional bridge deformations. A train–track–bridge coupled model is established to determine relationship between the dynamic responses (e.g. accelerations and wheel–rail forces) of the high-speed trains and the track deformations caused by bridge pier settlement, girder end rotation, and girder camber. The dynamic responses are correlated with the track deformation. The mapping relationship between bridge deformations and running safety of trains is determined. To satisfy the requirements of safety and riding comfort, the suggested upper thresholds of pier settlement, girder end rotation, and girder camber are 22.6 mm, 0.92‰ rad, and 17.2 mm, respectively. This study provides a method that is convenient for engineers in evaluation and maintenance of high-speed railway bridges.

Keywords

Additional bridge deformation high-speed railway mapping relationship safety threshold train–track–bridge coupled model

1. Introduction

By 2025, 38 million meters of high-speed railway will cover the main residential areas in China (He et al., 2017). A large percentage of high-speed railway is bridge, subjected to deformations, such as pier settlement, girder end rotation, and girder camber due to concrete creep, in addition to the design deformations under dead and live loads (Ju, 2013; Lee and Billington, 2010; Strauss et al., 2017; Wang, 2014). The bridge deformations accumulate over time (Bian et al., 2010; Chen et al., 2014), thus affecting the track regularity and dynamic response of the train (Ju et al., 2014; Kimani and Kaewunruen, 2017). To ensure the safe operation of high-speed trains, it is necessary to control the vibrations of the trains under long-term bridge deformations.

In the literature, train–track–bridge models have been established to study the effects of bridge deformations on the dynamic responses of high-speed trains (Greco and Lonetti, 2018). Many research works focused on the influence of bridge pier settlement on vehicle running safety. Yau (2009) conducted an incremental–iterative procedure to investigate the effects of bridge pier settlement on the dynamic responses in a track–bridge model. The results showed that the foundation settlement generally had little influence on the dynamic responses of the bridge but drastically affected running safety of the train. Wang et al. (2016) established a three-dimensional finite element model to evaluate the safety of trains under bridge pier settlement and girder camber, considering dynamic effects and nonlinearity. In terms of the safety and comfort of the train, girder deformation and train speed are the main influencing factors. Zhang et al. (2017) investigated how the high-speed railway bridge pier settlement and train speed affect the accelerations of the train, wheel unloading rate, and fastener force of the track. The results showed that the vertical acceleration increased with pier settlement and train speed, and the settlement exceeded the maximum allowable limit in the standards. The influences of additional bridge deformation caused by concrete shrinkage and creep (Yang et al., 2014), temperature (Li et al., 2012; Tian et al., 2017; Wang, 2014), pier settlement (Zhong and Yang, 2017), and frost heave deformation (Guo, 2016) were investigated. Many existing studies consider the coupling effect among train, track, and bridge (Gou et al., 2019b; He et al., 2018; Li et al., 2018), but neglected the effects of track deformations on the dynamic responses of the trains. To accurately consider the effect of track deformation on running safety of high-speed railway, a mapping relationship between the pier settlement and the rail deformation is derived theoretically, and the variations of vehicle dynamics indices under pier settlement are analyzed (Chen et al., 2015, 2018a). Each of the existing models was developed based on a specific type of bridge deformation. To consider multiple types of bridge deformations, Gou et al. (2018c, 2019b, 2019c) presented an analytical model to link various bridge vertical deformations and the track geometry, and conducted field tests to investigate the dynamic behaviors of the bridge and the safety and comfort of the train (Gou et al., 2018a, 2018b, 2018d, 2019a).

Based on the above studies on the dynamic responses of the train–track–bridge system, the thresholds of bridge deformations were studied to ensure safe operation and riding comfort of high-speed trains. According to the limits of vehicle dynamic responses, parameter studies (Cao et al., 2015; Chen et al., 2015, 2018b; Gou et al., 2019b) were conducted to analyze the vehicle dynamic responses in the train–track–bridge system with additional bridge deformations. By changing the speed of the train and the amplitude of additional bridge deformation in the finite element models, the thresholds of bridge deformations are obtained through multiple trial calculations. Xia et al. (2018) assumed that the dynamic responses of the train under track irregularity that is caused by bridge pier settlement followed normal distribution, calculated dynamic responses of the train under different levels of settlement, and used the probability method to analyze the safety and comfort thresholds of pier settlement with 95% confidence. Chen et al. (2018a) determined the allowable limit of multipier settlement using the relationship between the multipier settlement and the vehicle dynamic indices.

However, in the literature, there is limited research on the influence of the type of ballastless track and bridge deformation on the running safety of trains. In addition, most of the existing research works attempted to determine the thresholds of bridge deformations through parametric study. For example, the speed of trains or the amplitude of a type of bridge deformation was varied in a range, which involves intense computation cost and is time consuming. The previous research (Chen, 2017) indicated that the waveforms of vertical acceleration of the train and wheel–rail force caused by pier settlement were almost the same. Thus, it is rational to speculate that there is a mapping relationship between the additional bridge deformation and the dynamic responses of the train. Then, given the mapping relationship, the influences of different bridge deformations on the dynamic responses of high-speed trains can be described quantitatively.

The contributions of this study are introduced. Compared with the previous studies on the dynamic responses of trains under a specific bridge deformation condition, this article presents a new method to map the quantitative relationships between dynamic responses of high-speed trains and multiple types of bridge deformations. Next, the evaluation criteria of bridge additional deformations of high-speed trains for running safety are investigated to ensure the running safety and riding comfort of high-speed trains. The presented method is expected to eliminate the need to perform the train–track–bridge coupled calculation, which involves expensive computation cost and thus, greatly facilitate real-life engineering practices.

2. Train–track–bridge coupled model

This study establishes a train–track–bridge coupled model using ANSYS and SIMPACK. SIMPACK is a multibody dynamics simulator that can integrate multiple bodies whose motions are coupled with each other. Two types of train models are considered, which are the CRH380B and CRH2 trains. The China railway track system II slab ballastless track is considered. The bridge is a 32-m simply supported girder bridge. The selected trains, track, and bridge are representative in high-speed railways in China. The track and bridge models are established in ANSYS and then imported in SIMPACK for integration with the train model. Figure 1 shows the flow chart for the train–track–bridge model. The bridge, track, and rail are modeled in ANSYS for finite element analysis. The analysis results are imported in SIMPACK for multibody coupled dynamic analysis. More detailed information of the models is elaborated in the following sections.

2.1. Train model

Each train is composed of the car body, bogie, wheelset, primary and secondary suspensions, axle box, yaw damper, lateral damper, antirolling torsion bar, rotating arm, lateral stop, and traction rod. In SIMPACK, the car body, bogie, and wheelset are treated as rigid bodies, which are connected to the geodetic coordinate system by the general rail track joint. The axle box is simulated by a rigid body and connected to the wheelset. Primary and secondary suspensions, axle box, yaw damper, lateral damper, antirolling torsion bar, rotating arm, lateral stop, and traction rod are simulated by force element. Figure 1 shows the train models. Table 1 shows the train parameters. Figure 2 shows the adopted curves of the yaw damper force and the lateral stop force.

Table 1.

Parameters of train models.

Parameter	CRH380B	CRH2	Unit
Mass of wheelset	1517	2000	kg
Rolling inertia of wheelset	693	720	kg m²
Nodding inertia of wheelset	118	80	kg m²
Yawing inertia of wheelset	693	980	kg m²
Mass of bogie	2953	3200	kg
Rolling inertia of bogie	1846	2592	kg m²
Nodding inertia of bogie	1293	1753	kg m²
Yawing inertia of bogie	2792	3200	kg m²
Mass of car body	38,884	31,600	kg
Rolling inertia of car body	92,500	123,444	kg m²
Nodding inertia of car body	1,756,000	1,866,900	kg m²
Yawing inertia of car body	1,728,000	1,609,725	kg m²
Longitudinal stiffness of primary spring	919,800	980,000	N/m
Lateral stiffness of primary spring	919,800	980,000	N/m
Vertical stiffness of primary spring	886,500	1,176,000	N/m
Longitudinal stiffness of secondary air spring	131,000	158,760	N/m
Lateral stiffness of secondary air spring	131,000	158,760	N/m
Vertical stiffness of secondary air spring	195,000	189,140	N/m
Vertical damping of secondary air spring	30,000	40,000	N-s/m
Joint stiffness of yaw damper	35,000,000	8,820,000	N/m
Stiffness of antirolling torsion bar	4,150,000	2,000,000	N-m/rad
Wheel rolling diameter	920	860	mm
Wheelbase	2500	2500	mm
Distance between bogie pivots	17,375	17,500	mm

2.2. Track–bridge model

The bridge, track, and rail are modeled in ANSYS using solid elements, as shown in Figure 3. The bridge has three 32-m simply supported spans supported on the four piers with the bottom fixed. The bridge girders have the box section consistent with the real bridge. The piers and girders are made of concrete: elastic modulus = 34.5 GPa, density = 2400 kg/m³, and Poisson’s ratio = 0.2. The rail is made of steel: elastic modulus = 210 GPa, density = 7800 kg/m³, and Poisson’s ratio = 0.3. Mesh size convergence analysis is conducted to determine the appropriate mesh sizes that balance the computation accuracy and efficiency.

With the ANSYS model, finite element analysis is performed to determine the mass, stiffness, and geometric information of the bridge, track, and rail, which is included in the “.sub” and “.cdb” files. The files are then imported into SIMPACK in the forms of flexible body and flexible track (flextrack), respectively. The flexible body/track files (.fbi) of the bridge and track are generated by the flexible body input file generator of finite element multi-body system program interface in SIMPACK. The flexible body needs to choose the mode order or frequency range to determine the calculation mode of the bridge. The flexible track is directly coupled with the trains on the track, as a part of the track foundation.

Makers are generated at the boundary conditions set and the connection with the outside, such as pier bottom nodes and fastener nodes of beam-track connection. The bottom of the pier is fixed by constraint between the pier bottom and the geodetic coordinate system. The fastener spring is simulated by adding a spring-damper parallel cmp to achieve the bridge–track interaction.

The finite element model of the multibody dynamics system is realized by SIMPACK finite element interface module (FlexModal). SIMPACK uses the modal synthesis method to simulate the elastic deformations of flexible bodies.

2.3. Preliminary model verification

The train model and bridge–track model are verified before they are coupled in SIMPACK. A preload of 46 kN is applied to the train to obtain the maximum residual acceleration (Cui, 2009; Miao et al., 2006). The maximum residual acceleration is used to verify the equilibrium, which is indicated by a maximum residual acceleration less than 0.01 m/s². The maximum residual acceleration is 1.17 × 10⁻¹⁰ m/s² for the train and 1.62 × 10⁻⁶ m/s² for the bridge–track model, indicating that the train and bridge–track models are in equilibrium.

The bridge–track model is introduced into the train model in SIMPACK, and the foundation under rail is replaced by the flexible track to link the bridge and the rail. The contact between the rail and the train along the normal direction is defined by the Hertz nonlinear elastic contact, and the contact along the tangential direction is defined by the Kalker nonlinear contact. In SIMPACK, the quasi-elastic contact is used to simulate the wheel–rail contact. Based on the wheel–rail contact force, software automatically adopts rigid contact to analyze the nonjumping on the rail (the normal wheel–rail force is positive); as the train has a jump on the rail (the normal wheel–rail force is negative), the wheel–rail contact is simulated by the elastic contact. Then, the wheel–rail relationship is regenerated to realize the interaction between the train and the rail. In this way, the train–track–bridge coupled vibration model is established.

The responses of the multibody system are solved by time-domain integration (time integration) in SIMPACK. Based on the input interpolation node of the wheel–rail contact force, the finite element nodes are performed displacement interpolation to determine the wheel–rail contact point at that moment. Wheel–rail contact force element 78 is added to the wheel–rail contact point. The contact force of the flexible track is assigned to the corresponding finite element node by interpolation to solve the dynamic responses of the wheelset and the flexible track at the next moment. As the contact point satisfies the allowable error of the compatibility conditions of force and displacement, the wheel–rail force is converged in this time step, and the responses are used as the initial condition for the next time step. The differential/algebraic system solver algorithm is used to solve the motion equation of the system, and the time step is automatically determined by the algorithm based on the residuals. The simulation time is set to 1.45 s. The sampling frequency is set to 200 Hz. The maximum step size is 0.1 s.

2.4. Track irregularity

The random track irregularity is generated using the German low-interference track power spectrum. The wavelength of the random track irregularity is in the range of 3 m–25 m. The track irregularity due to additional bridge deformations includes the pier settlement, girder end rotation, and girder camber. The track irregularity due to bridge deformations is determined using a bridge–track deformation mapping model developed in the authors’ previous study (Long, 2019). With the mapping model, given a bridge deformation, the track deformation is determined. The investigated pier settlement and girder camber are in the range of 5 mm–30 mm with a 5-mm interval. The girder end rotation angle is in the range of 0.2‰ rad to 1‰ rad with a 0.2‰ interval, as shown in Figure 4.

3. Bridge deformation thresholds

The threshold of bridge deformation is determined using the wheel unloading rate according to the limits of running safety and riding comfort (TB10621-2014, 2015). The wheel unloading rate associated with the bridge deformations is determined by subtracting the wheel unloading rate without considering the bridge deformation from that considering the bridge deformation.

3.1. Pier settlement

Figure 5 shows the influences of train type and speed on the wheel unloading rate under pier settlement. The wheel unloading rate increases with the pier settlement and train speed. At 350 km/h, the wheel unloading rate of CRH2 exceeds the limit (TB10621-2014, 2015), as the pier settlement reaches 21.7 mm. The wheel unloading rate of CRH380B does not exceed the limit in the investigated ranges of pier settlement and train speed. These results indicate that the CRH2 train is more sensitive to the pier settlement than the CRH380B train.

3.2. Girder end rotation

Figure 6 shows that the wheel unloading rate increases with the girder end rotation and the train speed. The wheel unloading rate of CRH2 exceeds the limit, as the girder end rotation reaches 0.79‰ rad. The wheel unloading rate of CRH380B does not exceed the limit within the investigated ranges.

3.3. Girder camber

Figure 7 shows that the wheel unloading rate increases with the magnitude of girder camber and the speed of trains. The wheel unloading rate of CRH2 exceeds the limit when the girder camber reaches 16.7 mm, whereas the wheel unloading rate of CRH380B does not exceed the limit within the investigated ranges.

4. Deformation threshold formulae

The finite element analysis results presented in Section 3 show that the dynamic responses of the CRH2 train are more sensitive to bridge deformations than the CRH380B train. Thus, the CRH2 train is selected to determine the thresholds of bridge deformations.

4.1. Pier settlement

Figure 8 shows that the vertical acceleration and wheel–rail force of the CRH2 train increase with the pier settlement and reach the maximum values at the settled pier. The two peaks of the acceleration are caused by the front and rear bogies passing the settled pier.

An interesting finding is that the dynamic indices of the train follow a consistent trend with the second derivative of track deformation, as shown in Figure 9. The peaks of the dynamic indices and the second derivative curves correspond to the same location, which is the settled bridge pier in this case.

The change of the vertical wheel–rail force shows strong correlation with the second derivative of track deformation as shown in Figure 10. A formula is obtained through curve fitting

Δ Q = 31353 Y_{r}^{″}

(1)

where ΔQ is the change of the vertical wheel–rail force and

Y_{r}^{″}

is the second derivative of track deformation.

At the same time, the second derivative of track deformation is associated with the pier settlement, as shown in Figure 11. A formula is determined through linear curve fitting. The coefficient of determination (R²) is 0.9989, indicating a good correlation. The formula is

Y_{r \min}^{″} = - (1.4209 d + 0.9832) \times 10^{- 5}

(2)

where

Y_{r \min}^{″}

is the minimum second derivative of track deformation and d is the settlement.

Combining equations (1) and (2), the relationship between the reduction of vertical wheel–rail force and pier settlement is

Δ Q = - 0.4455 d - 0.3083

(3)

The wheel unloading rate is expressed as

U = [P - (Δ Q + P_{\min})] / P

(4)

where U is the wheel unloading rate, P is static wheel weight, and P_min is the minimum vertical wheel–rail force without bridge deformations.

The static wheel weight and the minimum vertical wheel–rail force of the CRH2 train are 58.0 kN and 33.6 kN, respectively.

Combining equations (3) and (4), the relationship between the wheel unloading rate and pier settlement is

U = 0.0077 d + 0.4263

(5)

The limit of wheel unloading rate of the CRH2 train is 0.6

U = 0.0077 d + 0.4263 \leq 0.6

(6)

The threshold of pier settlement is

d \leq 22.6 mm

(7)

4.2. Girder end rotation

Similar to the trends in Figure 9, the trend of the wheel–rail force is consistent with the second derivative of track deformation under girder end rotation, as shown in Figure 12.

Similar to Figures 10 and 11, the change of the vertical wheel–rail force shows strong correlation with the second derivative of track deformation (Figure 13(a)), which is correlated with the girder end rotation angle (Figure 13(b)).

With the data in Figure 13, a formula is obtained through curve fitting

Δ Q = - 7.7173 R - 3.2259

(8)

where R is the girder end rotation angle. To satisfy the limit of wheel unloading rate, the girder end rotation angle should satisfy

R \leq 0.920 ‰ rad

(9)

4.3. Girder camber

Similar to the trends in Figures 9 and 12, the trend of the wheel–rail force is consistent with the second derivative of track deformation under girder camber, as shown in Figure 14.

Similarly, the change of the vertical wheel–rail force shows strong correlation with the second derivative of track deformation, which is correlated with the girder camber. To save space, the curves are not provided. Similar to equations (3) and (8), a formula is obtained through curve fitting for the girder camber

Δ Q = - 0.5947 C - 0.1750

(10)

where C is the girder camber deformation value. To satisfy the limit of wheel unloading rate, the camber should satisfy

C \leq 17.2 mm

(11)

4.4. Comparison

The bridge deformation thresholds determined using the curve fitting formulae are in reasonable agreement with the results obtained from the parametric study, as shown in Figure 15. The discrepancy is due to the fitting errors because the coefficient of determination (R²) in the curve fitting is less than 1.0. The discrepancy is acceptable, indicating that the thresholds of bridge deformations can be determined by the presented formulae. Also, given a measured bridge deformation, the train’s dynamic indices can be determined using the presented formulae.

5. Conclusions

This study establishes a train–track–bridge coupled model and applies the model to determine the threshold of bridge deformations. The following conclusions are drawn:

According to the limit of the wheel unloading rate, the thresholds of pier settlement, girder end rotation, and girder camber are 22.6 mm, 0.92‰ rad, and 17.2 mm, respectively, at 350 km/h.

The second derivatives of track deformation with respect to distance are linearly proportional to the change of the vertical wheel–rail force. The investigated bridge deformations are linearly proportional to the second derivative of track deformation with respect to distance.

The proposed formulae provide reasonable predictions of the thresholds of the additional bridge deformations including the pier settlement, girder end rotation, and girder camber due to concrete creep.

The wheel unloading rate of the trains increases with the additional bridge deformations and train speed. The wheel unloading rate of CRH2 train is more sensitive to the bridge deformations than that of the CRH380B train.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The research was funded by the National Natural Science Foundation of China (Grant No. 51878563), the Sichuan Science and Technology Program (Grant No. 2018JY0294 and 2018JY0549), and the Ministry of Science and Technology of China (Grant No. KY201801005).

ORCID iD

Hongye Gou

References

Bian

Jiang

Chen

(2010) Accumulative deformation in railway track induced by high-speed traffic loading of the trains. Earthquake Engineering and Engineering Vibration 9(3): 319–326.

Cao

Xia

, et al. (2015) A numerical method to predict the riding comfort induced by foundation construction close to a high-speed-line bridge. Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit 229(5): 553–564.

Chen

Zhao

, et al. (2014) Cumulative settlement of track subgrade in high-speed railway under varying water levels. International Journal of Rail Transportation 2(4): 205–220.

Chen

Zhai

Cai

, et al. (2015) Safety threshold of high-speed railway pier settlement based on train–track-bridge dynamic interaction. Science China Technological Sciences 58(2): 202–210.

Chen

Zhai

Tian

(2018a) Study on the safe value of multi-pier settlement for simply supported girder bridges in high-speed railways. Structure and Infrastructure Engineering 14(3): 400–410.

Chen

Zhai

Yin

(2018b) Analysis of structural stresses of tracks and vehicle dynamic responses in train–track-bridge system with pier settlement. Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit 232(2): 421–434.

Chen

(2017) Influence of pier settlement on dynamic performance of running trains in high-speed railways. PhD Thesis, Southwest Jiaotong University, China.

Cui

(2009) Refined simulation study between vehicle and bridge based on multi-body system dynamics and finite element method. PhD Thesis, Southwest Jiaotong University, China.

Greco

Lonetti

(2018) Numerical formulation based on moving mesh method for vehicle–bridge interaction. Advances in Engineering Software 121: 75–83.

10.

Gou

Zhou

, et al. (2018a) Experimental and numerical investigations of the dynamic responses of an asymmetrical arch railway bridge. Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit 232(9): 2309–2323.

11.

Gou

Long

Bao

, et al. (2019a) Dynamic behavior of hybrid framed arch railway bridge under moving trains. Structure and Infrastructure Engineering 15(8): 1015–1024.

12.

Gou

Yang

, et al. (2019b) Effect of long-term bridge deformations on safe operation of high-speed railway and vibration of vehicle–bridge coupled system. International Journal of Structural Stability and Dynamics 19: 1950111.

13.

Gou

Long

Bao

, et al. (2018b) Stress distributions in girder-arch-pier connections of long-span continuous rigid frame arch railway bridges. Journal of Bridge Engineering 23(7): 04018039.

14.

Gou

Ran

Yang

, et al. (2019c) Mapping vertical bridge deformations to track geometry for high-speed railway. Steel and Composite Structures 32(4): 467–478.

15.

Gou

Yang

Leng

, et al. (2018c) Effect of bridge lateral deformation on track geometry of high-speed railway. Steel and Composite Structures 29(2): 219–229.

16.

Gou

Zhou

Chen

, et al. (2018d) In-situ test and dynamic response of a double-deck tied-arch bridge. Steel and Composite Structures 27(2): 161–175.

17.

Guo

(2016) Study on the effect of subgrade frost heaving on the deformation properties of track structure and its vehicle dynamic behavior in high-speed railway. MS Dissertation, Southwest Jiaotong University, China.

18.

Gai

(2018) Simulation of train–bridge interaction under wind loads: a rigid-flexible coupling approach. International Journal of Rail Transportation 6(3): 163–182.

19.

Zou

, et al. (2017) Recent developments of high-speed railway bridges in China. Structure and Infrastructure Engineering 13(12): 1584–1595.

20.

(2013) 3D analysis of high-speed trains moving on bridges with foundation settlements. Archive of Applied Mechanics 83(2): 281–291.

21.

Leong

(2014) Safety of Maglev trains moving on bridges subject to foundation settlements and earthquakes. Journal of Bridge Engineering 19(1): 91–100.

22.

Kimani

Kaewunruen

(2017) Free vibrations of precast modular steel-concrete composite railway track slabs. Steel and Composite Structures 24(1): 113–128.

23.

Lee

Billington

(2010) Modeling residual displacements of concrete bridge columns under earthquake loads using fiber elements. Journal of Bridge Engineering 15(3): 240–249.

24.

Zhu

(2012) Dynamic response of bridges to moving trains: a study on effects of concrete creep and temperature deformation. Applied Mechanics and Materials 193–194: 1179–1182.

25.

Zhou

, et al. (2018) An interactive method for the analysis of the simulation of vehicle–bridge coupling vibration using ANSYS and SIMPACK. Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit 232(3): 663–679.

26.

Long

(2019) Mechanism of mapping relationship between additional bridge deformation and track irregularity of high-speed railway. MS Dissertation, Southwest Jiaotong University, China.

27.

Miao

Xiao

Jin

(2006) Research on modeling and analysis of a complex multibody system by using Simpack. Mechanical Science and Technology 25(7): 813–816 [in Chinese].

28.

Strauss

Wan-Wendner

Vidovic

, et al. (2017) Gamma prediction models for long-term creep deformations of prestressed concrete bridges. Journal of Civil Engineering and Management 23(6): 681–698.

29.

TB10621-2014 (2015) Code for design of high speed railway [in Chinese].

30.

Tian

Zhang

Xia

(2017) Temperature effect on service performance of high-speed railway concrete bridges. Advances in Structural Engineering 20(6): 865–883.

31.

Wang

, et al. (2016) Safety and stability of light-rail train running on multispan bridges with deformation. Journal of Bridge Engineering 21(9): 06016004.

32.

Wang

(2014) Influences of concrete creep and temperature deformation on vehilce travelling across bridge. Applied Mechanics and Materials 556–562: 655–658.

33.

Xia

Zhang

Guo

(2018) Dynamic analysis of train–bridge system under differential settlement and scouring effect of foundations. In: Xia

Zhang

, and Guo

(eds) Dynamic Interaction of Train–Bridge Systems in High-Speed Railways. Berlin: Heidelberg, pp. 471–535.

34.

Yang

Chen

Zhang

, et al. (2014) Dynamic analysis of train-rail-bridge interaction considering concrete creep of a multi-span simply supported bridge. Advances in Structural Engineering 17: 709–720.

35.

Yau

(2009) Response of a train moving on multi-span railway bridges undergoing ground settlement. Engineering Structures 31(9): 2115–2122.

36.

Zhang

Shan

Yang

(2017) Effect of bridge-pier differential settlement on the dynamic response of a high-speed railway train–track-bridge system. Mathematical Problems in Engineering 2017: 1–13.

37.

Zhong

Yang

(2017) Dynamic effect of foundation settlement on bridge–vehicle interaction. Engineering Structures 135: 149–160.