A comprehensive review on coupling vibrations of train–bridge systems under external excitations

Abstract

In recent years, high-speed railways in China have developed very rapidly, and the number and span of the railway bridges are keeping increasing. Meanwhile, frequent extreme disasters, such as strong winds, earthquakes and floods, pose a significant threat to the safety of the train–bridge systems. Therefore, it is of paramount importance to evaluate the safety and comfort of trains when crossing a bridge under external excitations. In these aspects, there is abundant research but lacks a literature review. Therefore, this paper provides a comprehensive state-of-the-art review of research works on train–bridge systems under external excitations, which includes crosswinds, waves, collision loads and seismic loads. The characteristics of external excitations, the models of the train–bridge systems under external excitations, and the representative research results are summarized and analyzed. Finally, some suggestions for further research of the coupling vibration of train–bridge system under external excitations are presented.

References

[1.] Daily E (2022) The total mileage of railway operation in China is more than 150000 kilometers—tamping foundation and building roads to escort the development http://www.gov.cn/xinwen/2022-01/10/content_5667326.htm Accessed 15 Jan 2022

[2.] Montenegro PA, Carvalho H, Ribeiro D, Calcada R, Tokunaga M, Tanabe M, Zhai WM. Assessment of train running safety on bridges: a literature review. Eng Struct 2021, 241 112425

[3.] Xu L, Zhai W. Stochastic analysis model for vehicle–track coupled systems subject to earthquakes and track random irregularities. J Sound Vib 2017, 407 209-225

[4.] Li Y, Xiang H, Qiang S. Reviews on coupling vibration of wind–vehicle–bridge systems. China J Highw Transp 2018, 31 7 24-37(in Chinese)

[5.] Zhai W. Vehicle–track coupled dynamics 2015 4 Beijing (in Chinese) Science Press

[6.] Zhai W. Vehicle–track coupled dynamics: theory and application 2020 Singapore Springer

[7.] Zhang N, Tian Y, Xia H. A train–bridge dynamic interaction analysis method and its experimental validation. Engineering 2016, 2 4 528-536

[8.] Zhai W, Han Z, Chen Z, Ling L, Zhu S. Train–track–bridge dynamic interaction: a state-of-the-art review. Veh Syst Dyn 2019, 57 7 984-1027

[9.] Xia H, Zhang N, Guo W. Dynamic interaction of train–bridge systems in high-speed railways: theory and applications 2018 Beijing (in Chinese) Beijing Jiaotong University Press

[10.] Zhu S, Li Y. Random characteristics of vehicle–bridge system vibration by an optimized pseudo excitation method. Int J Struct Stab Dyn 2020, 20 5 2050069

[11.] Li Y, Xu X, Zhou Y, Cai CS, Qin J. An interactive method for the analysis of the simulation of vehicle–bridge coupling vibration using ANSYS and SIMPACK. Proc Inst Mech Eng Part F J Rail Rapid Transit 2018, 232 3 663-679

[12.] Bao Y, Xiang H, Li Y. A dynamic analysis scheme for the suspended monorail vehicle–curved bridge coupling system. Adv Struct Eng 2020, 23 8 1728-1738

[13.] Zhai W, Xia H, Cai C, Gao M, Li X, Guo X, Zhang N, Wang K. High-speed train–track–bridge dynamic interactions—part I: theoretical model and numerical simulation. Int J Rail Transport 2013, 1 1–2 3-24

[14.] Zhai W, Sun X. A detailed model for investigating vertical interaction between railway vehicle and track. Veh Syst Dyn 1994, 23 Sup1 603-615

[15.] Li Y, Qiang S, Liao H, Xu YL. Dynamics of wind–rail vehicle–bridge systems. J Wind Eng Ind Aerod 2005, 93 6 483-507

[16.] Zhai WM. Two simple fast integration methods for large-scale dynamic problems in engineering. Int J Numer Meth Eng 1996, 39 24 4199-4214

[17.] Jiang Y, Wu P, Zeng J, Wu X, He Q, Wang X. Influence of bridge parameters on monorail vehicle–bridge system —a research with multi-rigid body and multi-flexible body coupling theory and park method. J Low Freq Noise Vib Active Control 2021, 40 3 1194-1214

[18.] Yu Z, Mao J, Guo F, Guo W. Non-stationary random vibration analysis of a 3D train–bridge system using the probability density evolution method. J Sound Vib 2016, 366 173-189

[19.] Zhu Y, Li X, Jin Z (2016) Three-dimensional random vibrations of a high-speed-train–bridge time-varying system with track irregularities. Proc Inst Mech Eng Part F J Rail Rapid Transit 230(8):1851–1876

[20.] He X, Shi K, Wu T. An efficient analysis framework for high-speed train–bridge coupled vibration under non-stationary winds. Struct Infrastruct Eng 2020, 16 9 1326-1346

[21.] Zhang N, Zhou Z, Wu Z. Safety evaluation of a vehicle–bridge interaction system using the pseudo-excitation method. Railw Eng Sci 2022, 30 1 41-56

[22.] Jin Z, Li X, Zhu Y, Qiang S. Random vibration analysis of nonlinear vehicle–bridge dynamic interactions. J China Railw Soc 2017, 39 9 109-116(in Chinese)

[23.] Han X, Xiang H, Li Y, Wang Y. Predictions of vertical train–bridge response using artificial neural network-based surrogate model. Adv Struct Eng 2019, 22 12 2712-2723

[24.] Xiang H, Li Y, Liao H, Li C. An adaptive surrogate model based on support vector regression and its application to the optimization of railway wind barriers. Struct Multidiscip Optim 2017, 55 2 701-713

[25.] Xiang H, Tang P, Zhang Y, Li Y. Random dynamic analysis of vertical train–bridge systems under small probability by surrogate model and subset simulation with splitting. Railw Eng Sci 2020, 28 3 305-315

[26.] Jiang L, Liu X, Xiang P, Zhou W. Train–bridge system dynamics analysis with uncertain parameters based on new point estimate method. Eng Struct 2019, 199 109454

[27.] Wan HP, Ni YQ. A new approach for interval dynamic analysis of train–bridge system based on bayesian optimization. J Eng Mech 2020, 146 5 04020029

[28.] Xiao X, Yan Y, Chen B. Stochastic dynamic analysis for vehicle–track–bridge system based on probability density evolution method. Eng Struct 2019, 188 745-761

[29.] Cai CS, Hu J, Chen S, Han Y, Zhang W, Kong X. A coupled wind–vehicle–bridge system and its applications: a review. Wind Struct 2015, 20 2 117-142

[30.] Wang M, Li XZ, Xiao J, Zou QY, Sha HQ. An experimental analysis of the aerodynamic characteristics of a high-speed train on a bridge under crosswinds. J Wind Eng Ind Aerod 2018, 177 92-100

[31.] Li X, Tan Y, Qiu X, Gong Z, Wang M. Wind tunnel measurement of aerodynamic characteristics of trains passing each other on a simply supported box girder bridge. Railw Eng Sci 2021, 29 2 152-162

[32.] He XH, Zou YF, Wang HF, Han Y, Shi K. Aerodynamic characteristics of a trailing rail vehicles on viaduct based on still wind tunnel experiments. J Wind Eng Ind Aerod 2014, 135 22-33

[33.] Guo W, Xia H, Karoumi R, Zhang T, Li X. Aerodynamic effect of wind barriers and running safety of trains on high-speed railway bridges under cross winds. Wind Struct 2015, 20 2 213-236

[34.] Li XZ, Wang M, Xiao J, Zou QY, Liu DJ. Experimental study on aerodynamic characteristics of high-speed train on a truss bridge: a moving model test. J Wind Eng Ind Aerod 2018, 179 26-38

[35.] Li Z, Yang M, Huang S, Zhou D. A new moving model test method for the measurement of aerodynamic drag coefficient of high-speed trains based on machine vision. Proc Inst Mech Eng F J Rail Rapid Transit 2018, 232 5 1425-1436

[36.] Li Y, Hu P, Xu Y, Zhang M, Liao H. Wind loads on a moving vehicle–bridge deck system by wind-tunnel model test. Wind Struct 2014, 19 2 145-167

[37.] Dorigatti F, Sterling M, Baker CJ, Quinn AD. Crosswind effects on the stability of a model passenger train—a comparison of static and moving experiments. J Wind Eng Ind Aerod 2015, 138 36-51

[38.] He X, Xue F, Zou Y, Chen S, Han Y, Du B, Xu X, Ma B. Wind tunnel tests on the aerodynamic characteristics of vehicles on highway bridges. Adv Struct Eng 2020, 23 13 2882-2897

[39.] Xiang H, Li Y, Chen S, Li C. A wind tunnel test method on aerodynamic characteristics of moving vehicles under crosswinds. J Wind Eng Ind Aerod 2017, 163 15-23

[40.] He X, Zou S (2021) Crosswind effects on a train–bridge system: Wind tunnel tests with a moving vehicle. Struct Infrastruct Eng. https://doi.org/10.1080/15732479.2021.1966056

[41.] Hu H, Xiang H, Liu K, Zhu J, Li Y (2021) Aerodynamic characteristics of the vehicles of two trains passing each other on bridge under crosswinds. J Cent South Univ, in press

[42.] Wang B, Xu YL, Zhu LD, Cao SY, Li YL. Determination of aerodynamic forces on stationary/moving vehicle–bridge deck system under crosswinds using computational fluid dynamics. Eng Appl Comp Fluid 2013, 7 3 355-368

[43.] Li F, Luo J, Wang L, Guo D, Gao L. Analysis of aerodynamic effects and load spectrum characteristics in high-speed railway tunnels. J Wind Eng Ind Aerod 2021, 216 104729

[44.] Guo J, Tang H, Li Y, Wu L, Wang Z. Optimization for vertical stabilizers on flutter stability of streamlined box girders with mountainous environment. Adv Struct Eng 2020, 23 2 205-218

[45.] Chen X, Li Y, Xu X, Tang H, Wang B. Evolution laws of distributed vortex-induced pressures and energy of a flat-closed-box girder via numerical simulation. Adv Struct Eng 2020, 23 13 2776-2788

[46.] Fang C, Hu R, Tang H, Li Y, Wang Z. Experimental and numerical study on vortex-induced vibration of a truss girder with two decks. Adv Struct Eng 2021, 24 5 841-855

[47.] Wu M, Li Y, Chen N. The impact of artificial discrete simulation of wind field on vehicle running performance. Wind Struct 2015, 20 2 169-189

[48.] Li XZ, Xiao J, Liu DJ, Wang M, Zhang DY. An analytical model for the fluctuating wind velocity spectra of a moving vehicle. J Wind Eng Ind Aerod 2017, 164 34-43

[49.] Yan N, Chen X, Li Y. Assessment of overturning risk of high-speed trains in strong crosswinds using spectral analysis approach. J Wind Eng Ind Aerod 2018, 174 103-118

[50.] Wu M, Li Y, Chen X, Hu P. Wind spectrum and correlation characteristics relative to vehicles moving through cross wind field. J Wind Eng Ind Aerod 2014, 133 92-100

[51.] Li M, Li M, Su Y. Experimental determination of the two-dimensional aerodynamic admittance of typical bridge decks. J Wind Eng Ind Aerod 2019, 193 103975

[52.] Ma C, Wang J, Li QS, Liao H. 3D aerodynamic admittances of streamlined box bridge decks. Eng Struct 2019, 179 321-331

[53.] García J, Muñoz-Paniagua J, Crespo A. Numerical study of the aerodynamics of a full scale train under turbulent wind conditions, including surface roughness effects. J Fluid Struct 2017, 74 1-18

[54.] Chen W, Zhu Z. Numerical simulation of wind turbulence by DSRFG and identification of the aerodynamic admittance of bridge decks. Eng Appl Comp Fluid 2020, 14 1 1515-1535

[55.] Li W, Patruno L, Niu H, de Miranda S, Hua X. Aerodynamic admittance of a 6:1 rectangular cylinder: a computational study on the role of turbulence intensity and integral length scale. J Wind Eng Ind Aerod 2021, 218 104738

[56.] Baker CJ. The simulation of unsteady aerodynamic cross wind forces on trains. J Wind Eng Ind Aerod 2010, 98 2 88-99

[57.] Xu L, Zhai W. Cross wind effects on vehicle–track interactions: a methodology for dynamic model construction. J Comput Nonlin Dyn 2019, 14 3 031003

[58.] Li Y, Hu P, Cai CS, Zhang M, Qiang S. Wind tunnel study of a sudden change of train wind loads due to the wind shielding effects of bridge towers and passing trains. J Eng Mech 2013, 139 9 1249-1259

[59.] Zhang J, Zhang M, Huang B, Li Y, Yu J, Jiang F. Wind tunnel test on local wind field around the bridge tower of a truss girder. Adv Civ Eng 2021, 2021 1-13

[60.] Wu J, Li X, Cai CS, Liu D. Aerodynamic characteristics of a high-speed train crossing the wake of a bridge tower from moving model experiments. Railw Eng Sci 2022, 30 2 221-241

[61.] Wang Y, Zhang Z, Zhang Q, Hu Z, Su C. Dynamic coupling analysis of the aerodynamic performance of a sedan passing by the bridge pylon in a crosswind. Appl Math Model 2021, 89 1279-1293

[62.] Yu H, Wang B, Li Y, Zhang M. Driving risk of road vehicle shielded by bridge tower under strong crosswind. Nat Hazards 2019, 96 1 497-519

[63.] Salati L, Schito P, Rocchi D, Sabbioni E. Aerodynamic study on a heavy truck passing by a bridge pylon under crosswinds using CFD. J Bridge Eng 2018, 23 9 04018065

[64.] Argentini T, Ozkan E, Rocchi D, Rosa L, Zasso A. Cross-wind effects on a vehicle crossing the wake of a bridge pylon. J Wind Eng Ind Aerod 2011, 99 6–7 734-740

[65.] Wang B, Xu YL, Zhu LD, Li YL. Crosswind effect studies on road vehicle passing by bridge tower using computational fluid dynamics. Eng Appl Comp Fluid 2014, 8 3 330-344

[66.] Li YL, Xiang HY, Wang B, Xu YL, Qiang SZ. Dynamic analysis of wind–vehicle–bridge coupling system during the meeting of two trains. Adv Struct Eng 2013, 16 10 1663-1670

[67.] Huang S, Li Z, Yang M. Aerodynamics of high-speed maglev trains passing each other in open air. J Wind Eng Ind Aerod 2019, 188 151-160

[68.] Zhang MJ, Li YL, Wang B. Effects of fundamental factors on coupled vibration of wind–rail vehicle–bridge system for long-span cable-stayed bridge. J Cent South Univ 2016, 23 5 1264-1272

[69.] Xiang H, Li Y, Chen B, Liao H. Protection effect of railway wind barrier on running safety of train under cross winds. Adv Struct Eng 2014, 17 8 1177-1187

[70.] Bao Y, Zhai W, Cai C, Zhu S, Li Y. Dynamic interaction analysis of suspended monorail vehicle and bridge subject to crosswinds. Mech Syst Signal Pr 2021, 156 107707

[71.] Nguyen K, Camara A, Rio O, Sparowitz L. Dynamic effects of turbulent crosswind on the serviceability state of vibrations of a slender arch bridge including wind–vehicle–bridge interaction. J Bridge Eng 2017, 22 11 06017005

[72.] Li Y, Long J, Xiang H, Ma H, He M, Xie J. Transverse deflection-span ratio suggested value of an urban rail transit bridge based on a wind–vehicle–bridge system. J Vib Shock 2020, 39 24 211-217(in Chinese)

[73.] Dai YQ, Dai XW, Bai Y, He XH. Aerodynamic performance of an adaptive GFRP wind barrier structure for railway bridges. Materials (Basel) 2020, 13 18 4214

[74.] Deng E, Yang W, He X, Zhu Z, Wang H, Wang Y, Wang A, Zhou L. Aerodynamic response of high-speed trains under crosswind in a bridge–tunnel section with or without a wind barrier. J Wind Eng Ind Aerod 2021, 210 104502

[75.] Xue FR, Han Y, Zou YF, He XH, Chen SR. Effects of wind-barrier parameters on dynamic responses of wind–road vehicle–bridge system. J Wind Eng Ind Aerod 2020, 206 104367

[76.] Xiang H, Li Y, Chen S, Hou G. Wind loads of moving vehicle on bridge with solid wind barrier. Eng Struct 2018, 156 188-196

[77.] He XH, Fang DX, Li H, Shi K. Parameter optimization for improved aerodynamic performance of louver-type wind barrier for train–bridge system. J Cent South Univ 2019, 26 1 229-240

[78.] Zhang N, Ge G, Xia H, Li X. Dynamic analysis of coupled wind–train–bridge system considering tower shielding and triangular wind barriers. Wind Struct 2015, 21 3 311-329

[79.] Wu M, Li Y, Zhang W. Impacts of wind shielding effects of bridge tower on railway vehicle running performance. Wind Struct 2017, 25 1 63-77

[80.] Zhu J, Zhang W, Wu MX. Coupled dynamic analysis of the vehicle–bridge–wind–wave system. J Bridge Eng 2018, 23 8 04018054

[81.] Fang C, Li Y, Wei K, Zhang J, Liang C. Vehicle–bridge coupling dynamic response of sea-crossing railway bridge under correlated wind and wave conditions. Adv Struct Eng 2019, 22 4 893-906

[82.] Kong X, Cai CS. Scour effect on bridge and vehicle responses under bridge–vehicle–wave interaction. J Bridge Eng 2016, 21 4 04015083

[83.] Zan X, Lin Z. On the applicability of Morison equation to force estimation induced by internal solitary wave on circular cylinder. Ocean Eng 2020, 198 106966

[84.] Yang W, Li Q. The expanded Morison equation considering inner and outer water hydrodynamic pressure of hollow piers. Ocean Eng 2013, 69 79-87

[85.] Zhou J, Chen L, Wang X. Hydrodynamic scaling and wave force estimation of offshore structures. Acta Mech Sinica-Prc 2020, 36 6 1228-1237

[86.] Fang C, Tang H, Li Y, Wang Z. Effects of random winds and waves on a long-span cross-sea bridge using Bayesian regularized back propagation neural network. Adv Struct Eng 2020, 23 4 733-748

[87.] Xia CY, Xia H, De Roeck G. Dynamic response of a train–bridge system under collision loads and running safety evaluation of high-speed trains. Comput Struct 2014, 140 23-38

[88.] Sun X, Bi Y, Zhou R, Zhao H, Fu X, Zhao P, Jiang Z. Experimental study on the damage of bridge pier under the impact of rockfall. Adv Civ Eng 2021, 2021 6610652

[89.] Zhang X, Wang X, Chen W, Wen Z, Li X. Numerical study of rockfall impact on bridge piers and its effect on the safe operation of high-speed trains. Struct Infrastruct E 2021, 17 1 1-19

[90.] Wan YL, Zhu L, Fang H, Liu WQ, Mao YF. Experimental testing and numerical simulations of ship impact on axially loaded reinforced concrete piers. Int J Impact Eng 2019, 125 246-262

[91.] Zhang WW, Jin XL, Wang JW. Numerical analysis of ship–bridge collision's influences on the running safety of moving rail train. Ships Offshore Struct 2014, 9 5 498-513

[92.] Sha YY, Hao H. Nonlinear finite element analysis of barge collision with a single bridge pier. Eng Struct 2012, 41 63-76

[93.] Li Y, Deng J, Wang B, Yu C. Running safety of trains under vessel–bridge collision. Shock Vib 2015, 2015 252574

[94.] Xia CY, Zhang N, Xia H, Ma Q, Wu X. A framework for carrying out train safety evaluation and vibration analysis of a trussed-arch bridge subjected to vessel collision. Struct Eng Mech 2016, 59 4 683-701

[95.] Wu M, Jin L, Du X. Dynamic responses and reliability analysis of bridge double-column under vehicle collision. Eng Struct 2020, 221 111035

[96.] Heng K, Li R, Li Z, Wu H. Dynamic responses of highway bridge subjected to heavy truck impact. Eng Struct 2021, 232 111828

[97.] Ozdagli AI, Moreu F, Xu D, Wang T. Experimental analysis on effectiveness of crash beams for impact attenuation of overheight vehicle collisions on railroad bridges. J Bridge Eng 2020, 25 1 04019133

[98.] Xu L, Lu X, Guan H, Zhang Y. Finite-element and simplified models for collision simulation between overheight trucks and bridge superstructures. J Bridge Eng 2013, 18 11 1140-1151

[99.] Xia C, Xia H, Zhang N, Guo W. Effect of truck collision on dynamic response of train–bridge systems and running safety of high-speed trains. Int J Struct Stab Dy 2013, 13 3 1250064

[100.] Abdelkarim OI, Elgawady MA. Performance of bridge piers under vehicle collision. Eng Struct 2017, 140 337-352

[101.] Do TV, Pham TM, Hao H. Dynamic responses and failure modes of bridge columns under vehicle collision. Eng Struct 2018, 156 243-259

[102.] Xia CY, Lei JQ, Zhang N, Xia H, De Roeck G. Dynamic analysis of a coupled high-speed train and bridge system subjected to collision load. J Sound Vib 2012, 331 10 2334-2347

[103.] Li P, Li Z, Han Z, Zhu S, Zhai W, Lou H (2022) Running safety evaluation of high-speed train subject to the impact of floating ice collision on bridge piers. Proc Inst Mech Eng Part F J Rail Rapid Transit 236(3):220-233

[104.] Cowan DR, Consolazio GR, Davidson MT. Response-spectrum analysis for barge impacts on bridge structures. J Bridge Eng 2015, 20 12 04015017

[105.] Fan W, Liu YZ, Liu B, Guo W. Dynamic ship-impact load on bridge structures emphasizing shock spectrum approximation. J Bridge Eng 2016, 21 10 04016057

[106.] Chen K (2017) Study on rockfall load and its influence on vehicle–bridge system. Dissertation, Southwest Jiaotong University (in Chinese)

[107.] Xia H, Han Y, Zhang N, Guo W. Dynamic analysis of train–bridge system subjected to non-uniform seismic excitations. Earthq Eng Struct D 2006, 35 12 1563-1579

[108.] Siringoringo DM, Fujino Y. Lateral stability of vehicles crossing a bridge during an earthquake. J Bridge Eng 2018, 23 4 04018012

[109.] Li Y, Zhu S, Cai CS, Yang C, Qiang S. Dynamic response of railway vehicles running on long-span cable-stayed bridge under uniform seismic excitations. Int J Struct Stab Dy 2016, 16 5 1550005

[110.] Yang TY, Tung DP, Li Y, Lin JY, Li K, Guo W. Theory and implementation of switch-based hybrid simulation technology for earthquake engineering applications. Earthq Eng Struct D 2017, 46 14 2603-2617

[111.] Gong W, Zhu Z, Liu Y, Liu R, Tang Y, Jiang L. Running safety assessment of a train traversing a three-tower cable-stayed bridge under spatially varying ground motion. Railw Eng Sci 2020, 28 2 184-198

[112.] Lei H, Li X. Input methods of non-uniform seismic excitation in coupling system of vehicle–track–bridge. Railw Sci Eng 2015, 12 4 769-777(in Chinese)

[113.] Li H, Yu Z, Mao J, Spencer BF. Effect of seismic isolation on random seismic response of high-speed railway bridge based on probability density evolution method. Structures 2021, 29 1032-1046

[114.] Zhu Z, Tang Y, Ba Z, Wang K, Gong W. Seismic analysis of high-speed railway irregular bridge–track system considering V-shaped canyon effect. Railw Eng Sci 2022, 30 1 57-70

[115.] Zeng ZP, Zhao YG, Xu WT, Yu ZW, Chen LK, Lou P. Random vibration analysis of train–bridge under track irregularities and traveling seismic waves using train–slab track–bridge interaction model. J Sound Vib 2015, 342 22-43

[116.] Xiong M, Huang Y, Zhao Q. Effect of travelling waves on stochastic seismic response and dynamic reliability of a long-span bridge on soft soil. B Earthq Eng 2018, 16 9 3721-3738

[117.] Tian ZY, Lou ML. Traveling wave resonance and simplified analysis method for long-span symmetrical cable-stayed bridges under seismic traveling wave excitation. Shock Vib 2014, 2014 602825

[118.] Borjigin S, Kim C, Chang K, Sugiura K. Nonlinear dynamic response analysis of vehicle–bridge interactive system under strong earthquakes. Eng Struct 2018, 176 500-521

[119.] Zhou W, Yu J, Jiang L, Lai Z, Zuo Y, Peng K. Component damage and failure sequence of track–bridge system for high-speed railway under seismic action. J Earthq Eng 2022

[120.] Liu Z, Jiang H, Zhang L, Guo E (2018) Natural vibration characteristics and seismic response analysis of train–bridge coupling system in high-speed railway In: Conte J, Astroza R, Benzoni, G, Feltrin G, Loh K, Moaveni B (eds) Experimental vibration analysis for civil structures EVACES 2017 Lecture Notes in Civil Engineering, vol 5. Springer, Cham, pp 831–838

[121.] Zeng Q, Dimitrakopoulos EG. Vehicle–bridge interaction analysis modeling derailment during earthquakes. Nonlinear Dynam 2018, 93 4 2315-2337

Funding

National Natural Science Foundation of China(51908472)