Vehicle–bridge interaction analysis by the state-space method and symplectic orthogonality

  • J. F. Wang
  • J. C. Pan
  • J. T. Zhang
  • G. R. Ye
  • R. Q. XuEmail author


The dynamic properties of bridges can be extracted from the dynamic responses of the vehicles passing on these bridges. This paper proposes a method for the vehicle–bridge interaction analysis of continuous beam bridges with different spans and variable cross sections using numerical methods that are high in computational efficiency. Herein, the vehicle is simplified as a spring–damper–mass system and coupled to the bridge by its interactional force in the governing equations based on the Timoshenko beam theory. According to the symplectic orthogonality of the state vectors, the orthogonality of the mode shapes of the Timoshenko beams is proved, and the dynamic responses of the continuous beam bridges with different spans and variable cross sections can be solved by the mode superposition method. More complicated factors, such as harmonic load on vehicles, noise in measurement, and roughness of pavements, can also be conveniently taken into account. Finally, the proposed method is demonstrated using some numerical examples and applied to a real bridge. The results indicate that the method is convenient, efficient, and precise for engineering applications.


Vehicle–bridge interaction State-space method Symplectic orthogonality Structural health monitoring 



This work was supported by the National Natural Science Foundation of China (Nos. 51478422 and 51878603).


  1. 1.
    Li, J., Su, M., Fan, L.: Natural frequency of railway girder bridges under vehicle loads. J. Bridge Eng. 8(4), 199–203 (2003). CrossRefGoogle Scholar
  2. 2.
    Kong, X., Cai, C.S., Deng, L., Zhang, W.: Using dynamic responses of moving vehicles to extract bridge modal properties of a field bridge. J. Bridge Eng. 22(6), 04017018 (2017). CrossRefGoogle Scholar
  3. 3.
    Lou, P.: Finite element analysis for train–track–bridge interaction system. Arch. Appl. Mech. 77, 707–728 (2007). CrossRefzbMATHGoogle Scholar
  4. 4.
    Zhu, Z., Yu, Z., Jiang, L., Gao, M.: Analysis of bridge-ground vibrations induced by moving loads of high-speed train. J. Vib. Eng. 25(5), 548–555 (2012). (In Chinese)Google Scholar
  5. 5.
    Chen, L., Jiang, L., Tao, L., Yu, Z.: Seismic response analysis of high-speed vehicle-bridge considering soil–structure interaction. Rock Soil Mech. 33(10), 3162–3170 (2012). (In Chinese)Google Scholar
  6. 6.
    Willis, R.: Report of the Commissioners Appointed to Inquire into the Application of Iron to Railway Structures (Appendix B). William Cloves and Sons, London (1849)Google Scholar
  7. 7.
    Stokes, G.G.: Discussion of a differential equation relating to the breaking of railway bridges. Trans. Camb. Philos. Soc. 8(5), 707–735 (1849)Google Scholar
  8. 8.
    Frýba, L.: Vibration of Solids and Structures Under Moving Loads, 3rd edn. Thomas Telford Ltd, London (1999)CrossRefGoogle Scholar
  9. 9.
    Michaltsos, G.T.: Dynamic behaviour of a single-span beam subjected to loads moving with variable speeds. J. Sound Vib. 258(2), 359–372 (2002). CrossRefGoogle Scholar
  10. 10.
    Zhu, X.Q., Law, S.S.: Precise time-step integration for the dynamic response of a continuous beam under moving loads. J. Sound Vib. 240(5), 962–970 (2001). CrossRefGoogle Scholar
  11. 11.
    Shiau, T.N., Huang, K.H., Wang, F.C., Hsu, W.C.: Dynamic response of a rotating multi-span shaft with general boundary conditions subjected to a moving load. J. Sound Vib. 323(3), 1045–1060 (2009). CrossRefGoogle Scholar
  12. 12.
    Mofid, M., Akin, J.E.: Discrete element response of beams with traveling mass. Adv. Eng. Softw. 25(2–3), 321–331 (1996). CrossRefGoogle Scholar
  13. 13.
    Mofid, M., Shadnam, M.: On the response of beams with internal hinges, under moving mass. Adv. Eng. Softw. 31(5), 323–328 (2000). CrossRefGoogle Scholar
  14. 14.
    Azam, S.E., Mofid, M., Khoraskani, R.A.: Dynamic response of Timoshenko beam under moving mass. Sci. Iran. 20(1), 50–56 (2012). CrossRefGoogle Scholar
  15. 15.
    Wang, F.Y., Gao, Y.Q.: On frequency sensitivity and mode orthogonality of flexible robotic manipulators. IEEE/CAA J. Autom. Sin. 3(4), 394–397 (2016). MathSciNetCrossRefGoogle Scholar
  16. 16.
    Yang, Y.B., Lin, C.W., Yau, J.D.: Extracting bridge frequencies from the dynamic response of a passing vehicle. J. Sound Vib. 272(3), 471–493 (2004). CrossRefGoogle Scholar
  17. 17.
    Yang, Y.B., Lin, C.W.: Vehicle-bridge interaction dynamics and potential applications. J. Sound Vib. 284(1), 205–226 (2005). CrossRefGoogle Scholar
  18. 18.
    Lin, C.W., Yang, Y.B.: Use of a passing vehicle to scan the fundamental bridge frequencies: an experimental verification. Eng. Struct. 27(13), 1865–1878 (2005). CrossRefGoogle Scholar
  19. 19.
    An, N., Xia, H., Zhan, J.W.: Experimental verification of bridge system identification based on dynamic response of passing vehicle. In: Xia, H., Takemiya, H. (eds.) Environmental Vibrations: Prediction, Monitoring, Mitigation and Evaluation, Vols I And II, pp. 1101–1105. Science Press, Beijing (2009)Google Scholar
  20. 20.
    Siringoringo, D.M., Fujin, Y.: Estimating bridge fundamental frequency from vibration response of instrumented passing vehicle: analytical and experimental study. Adv. Struct. Eng. 15(3), 417–434 (2012). CrossRefGoogle Scholar
  21. 21.
    Bu, J.Q., Law, S.S., Zhu, X.Q.: Innovative bridge condition assessment from dynamic response of a passing vehicle. J. Eng. Mech. 132(12), 1372–1379 (2006). CrossRefGoogle Scholar
  22. 22.
    Zhan, J.W., Xia, H., An, N.: Damage diagnosis method for railway bridges based on train dynamic responses. China Railw. Sci. 33(3), 35–39 (2012). (in Chinese)CrossRefGoogle Scholar
  23. 23.
    Yang, Y.B., Chang, K.C.: Extracting the bridge frequencies indirectly from a passing vehicle: parametric study. Eng. Struct. 31(10), 2448–2459 (2009). CrossRefGoogle Scholar
  24. 24.
    Chen, S.Y., Xia, H.: An identification method for fundamental frequency of bridge from dynamic responses due to passing vehicle. Eng. Mech. 26(8), 88–94 (2009). (in Chinese)CrossRefGoogle Scholar
  25. 25.
    Gomez, H.C., Fanning, P.J., Feng, M.Q., Lee, S.: Testing and long-term monitoring of a curved concrete box girder bridge. Eng. Struct. 33(10), 2861–2869 (2011). CrossRefGoogle Scholar
  26. 26.
    González, A., Obrien, E.J., McGetrick, P.J.: Identification of damping in a bridge using a moving instrumented vehicle. J. Sound Vib. 331(18), 4115–4131 (2012). CrossRefGoogle Scholar
  27. 27.
    Zhang, Y., Lie, S.T., Xiang, Z.H.: Damage detection method based on operating deflection shape curvature extracted from dynamic response of a passing vehicle. Mech. Syst. Signal Process. 35(1–2), 238–2354 (2013). CrossRefGoogle Scholar
  28. 28.
    Nagem, R.J., Williams Jr., J.H.: Dynamic analysis of large space structures using transfer matrices and joint coupling matrices. Mech. Struct. Mach. 17(3), 349–371 (1989). CrossRefGoogle Scholar
  29. 29.
    Li, J.Q., Leng, X.L., Fang, T.: Evolutionary random response problem of a coupled vehicle-bridge system. Arch. Appl. Mech. 72, 536–544 (2002). CrossRefzbMATHGoogle Scholar
  30. 30.
    Yao, W.A., Zhong, W.X., Lim, C.W.: Symplectic Elasticity. World Scientific, Singapore (2009)CrossRefGoogle Scholar
  31. 31.
    Lardies, J.: Modal parameter identification based on ARMAV and state-space approaches. Arch. Appl. Mech. 80, 335–352 (2010). CrossRefzbMATHGoogle Scholar
  32. 32.
    Shen, X.D.: Dynamic and static problems for composite beams with partial interaction. Doctorial Dissertation, Zhejiang University (2012) (in Chinese) Google Scholar
  33. 33.
    Singh, M.P., Abdelnaser, A.S.: Random vibrations of externally damped visooelastic Timoshenko beams with general boundary conditions. J. Appl. Mech. Trans. ASME. 60(1), 149–156 (1993). CrossRefzbMATHGoogle Scholar
  34. 34.
    Clough, R.W., Penzien, J.: Dynamics of Structures. McGraw-Hill, New York (1993)zbMATHGoogle Scholar
  35. 35.
    Honda, H., Kajikawa, Y., Kobori, T.: Spectra of road surface roughness on bridges. J. Struct. Div. 108(9), 1956–1966 (1982). CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • J. F. Wang
    • 1
  • J. C. Pan
    • 1
  • J. T. Zhang
    • 1
  • G. R. Ye
    • 1
  • R. Q. Xu
    • 1
    Email author
  1. 1.Department of Civil Engineering, Zijingang CampusZhejiang UniversityHangzhouChina

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