Nonlinear Dynamics

, Volume 93, Issue 4, pp 2315–2337 | Cite as

Vehicle–bridge interaction analysis modeling derailment during earthquakes

  • Qing ZengEmail author
  • Elias G. Dimitrakopoulos
Original Paper


A realistic simulation of train derailment is crucial when assessing the safety of a train running over a bridge during earthquake excitation. This paper presents a seismic vehicle–bridge interaction analysis that simulates directly different wheel–rail contact states including flange contact, detachment, uplifting, wheel–rail climbing-up, recontact, and ultimately, derailment. The proposed model determines the contact point and the direction of the contact forces over practical nonlinear profiles of wheels and rails. It then classifies the wheel–rail contact, as double contact, single contact or double detachment, and tackles accordingly the kinematics. The modeling of the wheel–rail contact along the normal direction hinges upon the principles of nonsmooth dynamics and accounts for continuous contacts of finite duration, impacts (instantaneous duration) and transitions from continuous contacts to detachments. The modeling of the tangential contact forces follows the nonlinear creep theory. The results verify that well-known force-based metrics such as derailment factor and offload factor yield conservative estimations of train operational safety. The analysis stresses the key role of flange contact under a large contact angle that could lead to the detachment of the other wheel of the same wheelset, and underlines the importance of a more realistic train–bridge interaction modeling during earthquakes. For the examples examined, which involve a complete three-dimensional vehicle running on simply supported bridge units, derailment occurs when a wheel rolls over the rail head (wheel–rail climbing-up). The results unveil that both the amplitude and the frequency of the earthquakes are important to the safety of trains running over bridges.


Vehicle–bridge interaction Derailment High-speed railway Nonsmooth dynamics 



This study was supported by Research Grants Council 2016, Hong Kong, the People’s Republic of China, under Contract Number: 16244116.


  1. 1.
  2. 2.
    Yan, B., Dai, G.L., Hu, N.: Recent development of design and construction of short span high-speed railway bridges in China. Eng. Struct. 100, 707–717 (2015)CrossRefGoogle Scholar
  3. 3.
  4. 4.
    Zeng, Q., Dimitrakopoulos, E.G.: Seismic response analysis of an interacting curved bridge–train system under frequent earthquakes. Earthq. Eng. Struct. Dyn. 45, 1129–1148 (2016)CrossRefGoogle Scholar
  5. 5.
    Ogura, M.: The Niigata Chuetsu Earthquake–railway response and reconstruction. Jpn. Railw. Transp. Rev. 43, 46–63 (2006)Google Scholar
  6. 6.
    Ju, S.H.: Nonlinear analysis of high-speed trains moving on bridges during earthquakes. Nonlinear Dyn. 69(1), 173–183 (2012)CrossRefGoogle Scholar
  7. 7.
  8. 8.
    Yang, Y.B., Wu, Y.S.: Dynamic stability of trains moving over bridges shaken by earthquakes. J. Sound Vib. 258(1), 65–94 (2002)CrossRefGoogle Scholar
  9. 9.
    Xia, H., Han, Y., Zhang, N., Guo, W.W.: Dynamic analysis of train–bridge system subjected to non-uniform seismic excitations. Earthq. Eng. Struct. Dyn. 35(12), 1563–1579 (2006)CrossRefGoogle Scholar
  10. 10.
    Du, X.T., Xu, Y.L., Xia, H.: Dynamic interaction of bridge–train system under non-uniform seismic ground motion. Earthq. Eng. Struct. Dyn. 41(1), 139–157 (2012)CrossRefGoogle Scholar
  11. 11.
    Cheng, Y.C., Chen, C.H., Hsu, C.T.: Derailment and dynamic analysis of tilting railway vehicles moving over irregular tracks under environment forces. Int. J. Struct. Stab. Dyn. 17(9), 1750098-1-27 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Nishimura, K., Terumichi, Y., Morimura, T., Sogabe, K.: Development of vehicle dynamics simulation for safety analyses of rail vehicles on excited tracks. J. Comput. Nonlinear Dyn. 4(1), 011001 (2009)CrossRefGoogle Scholar
  13. 13.
    Tanabe, M., Wakui, H., Sogabe, M., Matsumoto, N., Tanabe, Y.: An efficient numerical model for dynamic interaction of high speed train and railway structure including post-derailment during an earthquake. In: 8th International Conference on Structural Dynamics, EURODYN. Leuven, Belgium, pp. 1217–1223 (2011)Google Scholar
  14. 14.
    Jin, Z.B., Pei, S.L., Li, X.Z., Liu, H.Y., Qiang, S.Z.: Effect of vertical ground motion on earthquake-induced derailment of railway vehicles over simply-supported bridges. J. Sound Vib. 383, 277–294 (2016)CrossRefGoogle Scholar
  15. 15.
    Nadal, M.J.: Theorie de la stabilite des locomotives, part 2: mouvement de lacet’. 10, 232–255 (1896)Google Scholar
  16. 16.
    Wu, X.W., Chi, M.R., Gao, H.: Post-derailment dynamic behaviour of a high-speed train under earthquake excitations. Eng. Fail. Anal. 64, 97–110 (2016)CrossRefGoogle Scholar
  17. 17.
    Weinstock, H.: Wheel climb derailment criteria for evaluation of rail vehicle safety. In: Proceeding of the ASME Winter Annual Meeting, New York, pp. 1–7 (1984)Google Scholar
  18. 18.
    Shabana, A.A., Zaazaa, K.E., Sugiyama, H.: Railroad Vehicle Dynamics: A Computational Approach. CRC Press, New York (2010)zbMATHGoogle Scholar
  19. 19.
    Luo, X.: Study on methodology for running safety assessment of trains in seismic design of railway structures. Soil Dyn. Earthq. Eng. 25(2), 79–91 (2005)CrossRefGoogle Scholar
  20. 20.
  21. 21.
    Ju, S.H.: A frictional contact finite element for wheel/rail dynamic simulations. Nonlinear Dyn. 85(1), 365–374 (2016)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Comité Europeen de Normalization.: Eurocode 8: design of structures for earthquake resistance-part 2: Bridges. 2, 2003 (1998)Google Scholar
  23. 23.
    Cook, R.D.: Concepts and Applications of Finite Element Analysis. Wiley, New York (2007)Google Scholar
  24. 24.
    Chopra, A.K.: Dynamics of Structures: Theory and Applications to Earthquake Engineering. Prentice Hall, Upper Saddle River (2000)Google Scholar
  25. 25.
    Yang, Y.B., Wu, Y.S., Yao, Z.D.: Vehicle–Bridge Interaction Dynamics: With Applications to High-Speed Railways. World Scientific, Singapore (2004)CrossRefGoogle Scholar
  26. 26.
    Antolín, P., Zhang, N., Goicoleaa, J.M., Xia, H., Astiza, M.Á., Olivaa, J.: Consideration of nonlinear wheel–rail contact forces for dynamic vehicle–bridge interaction in high-speed railways. J. Sound Vib. 332, 1231–1251 (2013)CrossRefGoogle Scholar
  27. 27.
    Zeng, Q., Yang, Y.B., Dimitrakopoulos, E.G.: Dynamic response of high speed vehicles and sustaining curved bridges under conditions of resonance. Eng. Struct. 114, 61–74 (2016)CrossRefGoogle Scholar
  28. 28.
    Dimitrakopoulos, E.G., Zeng, Q.: A three-dimensional dynamic analysis scheme for the interaction between trains and curved railway bridges. Comput. Struct. 149, 43–60 (2015)CrossRefGoogle Scholar
  29. 29.
    Zeng, Q.: Analysis and Simulation of the Vehicle-Bridge-Interaction in Horizontally Curved Railway Bridges. The Hong Kong University of Science and Technology, Degree of Doctor of Philosophy in Civil Engineering (2016)Google Scholar
  30. 30.
    Carter, F.W.: On the action of a locomotive driving wheel. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. London, UK, pp. 151–157 (1926)Google Scholar
  31. 31.
    Shen, Z.Y., Hedrick, J.K., Elkins, J.A.: A comparison of alternative creep force models for rail vehicle dynamic analysis. Veh. Syst. Dyn. 12(1–3), 79–83 (1983)CrossRefGoogle Scholar
  32. 32.
    Kalker, J.J.: Three-Dimensional Elastic Bodies in Rolling Contact. Springer, Berlin (1990)CrossRefzbMATHGoogle Scholar
  33. 33.
    Shi, Z.Q., Dimitrakopoulos, E.G.: Nonsmooth dynamics prediction of measured bridge response involving deck abutment pounding. Earthq. Eng. Struct. Dyn. (2017)
  34. 34.
    Dimitrakopoulos, E.G.: Analysis of a frictional oblique impact observed in skew bridges. Nonlinear Dyn. 60(4), 575–595 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Theodosiou, C., Natsiavas, S.: Dynamics of finite element structural models with multiple unilateral constraints. Int. J. Non-Linear Mech. 44(4), 371–382 (2009)CrossRefzbMATHGoogle Scholar
  36. 36.
    Pfeiffer, F., Glocker, C.: Multibody Dynamics with Unilateral Contacts. Wiley, Singapore (1996)CrossRefzbMATHGoogle Scholar
  37. 37.
    Dimitrakopoulos, E.G.: Nonsmooth analysis of the impact between successive skew bridge-segments. Nonlinear Dyn. 74(4), 911–928 (2013)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Giouvanidis, A.I., Dimitrakopoulos, E.G.: Nonsmooth dynamic analysis of sticking impacts in rocking structures. B Earthq. Eng. (2016). Google Scholar
  39. 39.
    Glocker, C., Cataldi-Spinola, E., Leine, R.I.: Curve squealing of trains: measurement, modelling and simulation. J. Sound Vib. 324(1), 365–386 (2009)CrossRefGoogle Scholar
  40. 40.
    Antolín, P., Goicolea, J.M., Astiz, M.A.l., Alonso, A.: A methodology for analysing lateral coupled behavior of high speed railway vehicles and structures. In: IOP Conference Series: Materials Science and Engineering. Sidney, Australia, p. 012001 (2010)Google Scholar
  41. 41.
    Montenegro, P.A., Neves, S.G.M., Calada, R., Tanabe, M., Sogabe, M.: Wheel–rail contact formulation for analyzing the lateral train-structure dynamic interaction. Comput. Struct. 152, 200–214 (2015)CrossRefGoogle Scholar
  42. 42.
    MathWorks: MATLAB User’s Guide. The MathWorks Inc., Natick, MA (1994–2013)Google Scholar
  43. 43.
    Shampine, L.F., Reichelt, M.W.: The matlab ode suite 18(1), 1–22 (1997)Google Scholar
  44. 44. (2015). Accessed 08 Oct 2017
  45. 45.
    Kim, C.W., Kawatani, M.: Effect of train dynamics on seismic response of steel monorail bridges under moderate ground motion. Earthq. Eng. Struct. Dyn. 35(10), 1225–1245 (2006)CrossRefGoogle Scholar
  46. 46.
    China’s Ministry of Railways, Code for Design of High Speed Railway (2009)Google Scholar
  47. 47.
    Iwnicki, Simon: Handbook of Railway Vehicle Dynamics. CRC Press, Boca Raton (2006)CrossRefGoogle Scholar
  48. 48.
    Dimitrakopoulos, E.G., Giouvanidis, A.I.: Seismic response analysis of the planar rocking frame. J. Eng. Mech. 141(7), 04015003 (2015)CrossRefGoogle Scholar
  49. 49.
    Mavroeidis, G.P., Papageorgiou, A.S.: A mathematical representation of near-fault ground motions. Bull. Seismol. Soc. Am. 93(3), 1099–1131 (2003)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.The Hong Kong University of Science and TechnologyClear Water Bay, KowloonHong Kong

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