Simplified Analytical Approach to Vertical Response of High-Speed Train-Bridge Systems on Track Irregularities
DOI:
https://doi.org/10.22146/jcef.27096Keywords:
High-speed railway, Train-bridge system, Rail irregularity, Lumpedparameter model, Vertical displacementAbstract
This study proposes a simplified analytical approach for evaluating the vertical dynamic response of a high-speed train–track-bridge system subjected to rail irregularity excitation. In the theoretical model, the car body and bogie frame of the train, together with the rail, sleeper, and ballast of the track, are represented using a lumped-parameter method, with their degrees of freedom interconnected through spring–damper elements. The bridge is separately idealized as a simply supported Euler-Bernoulli beam. This modelling strategy enables the dynamic interaction between the moving train and the track system to be considered while maintaining a relatively simple computational framework. The dynamic interaction between the train and the track under sinusoidal rail irregularities is simulated in MATLAB/Simulink to obtain time-varying axle loads. These dynamic axle loads are subsequently applied to the bridge to compute the vertical midspan displacements using the energy-based moment–area method. In the numerical analysis, three rail irregularity profiles are considered, characterized by representative amplitude-wavelength pairs of (1.5 mm, 3.0 m), (3.0 mm, 12.5 m), and (6.0 mm, 50.0 m). The train parameters correspond to the Thalys central carriage, while the bridge adopted in the simulations is a 24 m simply supported box girder. Train speeds of 200, 250, and 300 km/h are investigated to examine the effects of operating speed and rail irregularity characteristics on the dynamic response of the coupled system. The results indicate that the dynamic axle loads and the corresponding bridge displacements are approximately 1.65 to 1.93 times greater than those induced by equivalent static loads. These findings demonstrate the strong influence of train speed and rail irregularity intensity on bridge vibrations and highlight the importance of considering dynamic interaction effects in the evaluation of high-speed railway structures. Overall, the proposed approach provides a practical and efficient tool for evaluating the dynamic performance and safety of railway bridges subjected to high-speed train operations, while avoiding the computational complexity associated with more sophisticated numerical models.
References
AASHTO, L. R. F. D., (2025). Bridge design specifications. U.S.
Bian, X., Jiang, H., Chang, C., Hu, J., & Chen, Y. (2015). Track and ground vibrations generated by high-speed train running on ballastless railway with excitation of vertical track irregularities. Soil Dynamics and Earthquake Engineering. 76, pp. 29-43. URL: https://doi.org/10.1016/j.soildyn.2015.02.009
Chopra, A. K., 2023. Dynamics of structures-6th edditon. New Jersy: Prentice Hall.
Delgado, R. M. (1997). Modelling of railway bridge-vehicle interaction on high speed tracks. Computers & Structures. 63(3), pp. 511-523. URL: https://doi.org/10.1016/S0045-7949(96)00360-4
Esveld, C. (2001). Modern railway track. Zaltbommel: MRT-productions.
Gong, X., Liang, Q., Wang, H., Lu, T., Liu, J., Chen, Z., & Zhang, R. (2025). An Inverted Hermite Kolmogorov–Arnold Network Transformer for multi‐point settlement prediction in high‐speed railway bridge piers. Computer‐Aided Civil and Infrastructure Engineering. URL: https://doi.org/10.1111/mice.70109
He, X., Wu, T., Zou, Y., Chen, Y. F., Guo, H., & Yu, Z. (2017). Recent developments of high-speed railway bridges in China. Structure and Infrastructure Engineering. 13(12), pp. 1584-1595. URL: https://doi.org/10.1080/15732479.2017.1304429
Hoa, P. D., Hung, T. V., Thang, P. B., & Hang, N. T. N. (2021). Research on the dynamic response of high-speed railway bridges. Journal of Science and Technology in Civil Engineering. 15(7V), pp. 1-12. (In Vietnamese). URL: https://doi.org/10.31814/stce.huce(nuce)2021-15(7V)-01
Knothe, K., & Stichel, S. (2017). Rail vehicle dynamics. New York: Springer. https://doi.org/10.1007/978-3-319-45376-7
Kouroussis, G., Connolly, D. P., Vogiatzis, K., & Verlinden, O. (2015). Modelling the environmental effects of railway vibrations from different types of rolling stock: a numerical study. Shock and Vibration. 1, 142807. URL: https://doi.org/10.1155/2015/142807
Lin, S., Shi, J., Li, Y., & Lin, Y. (2025). Sound and Vibration Control of Steel-Concrete Composite Beam Bridges in High-Speed Railways Using Constrained Damping Layers. KSCE Journal of Civil Engineering. 100452. https://doi.org/10.1016/j.kscej.2025.100452
Neves, M. (2010). Dynamic behaviour of high speed railway bridges with ballasted tracks. Portugal: Technical University of Lisbon.
Ngoc, T. P. (2024). Influence of train speed on ground vibrations due to high-speed train. Transport and Communications Science Journal. 75(5), pp. 1775-1788. (In Vietnamese). URL: https://doi.org/10.47869/tcsj.75.5.6
Nie, L., Jiang, L., Zhou, W., Wang, X., Jiang, Z., & Fu, H. (2024). Study on applicability of rocking piers to high-speed railway bridge-track system under strong ground motions. Structure and Infrastructure Engineering. pp. 1-12. URL: https://doi.org/10.1080/15732479.2024.2438737
Pan, J., Zhang, D., He, J., Zhang, L., Zhou, Z., Wang, S., ... & Han, Y. (2025). Experimental investigation into the dynamic response characteristics of high-speed railway bridge-track structure subjected to blast-induced vibrations. Structures. 81, 110197. https://doi.org/10.1016/j.istruc.2025.110197
Peng, K., Zhou, W., Jiang, L., Liu, S., & Liu, H. (2025). A simplified calculation method for seismic-induced residual track irregularities in high-speed railway track-bridge systems using short-time Fourier Transform. Mechanics Based Design of Structures and Machines. 1-24. URL: https://doi.org/10.1080/15397734.2025.2579734
Phong, N. H., & Chien, C. Q. (2021). A study on real measured settlement analysis and settlement forecast of high-speed railway foundation. Transport and Communications Science Journal. 72(3), pp. 277-290. URL: https://doi.org/10.47869/tcsj.72.3.4
Rigueiro, C., Rebelo, C., & da Silva, L. S. (2010). Influence of ballast models in the dynamic response of railway viaducts. Journal of Sound and Vibration. 329(15), pp. 3030-3040. URL: https://doi.org/10.1016/j.jsv.2010.02.002
Tan, Z., Gou, H., Li, W., Peng, Y., Wang, J., Pu, Q., & Bao, Y. (2025). Digital twin-based cyber-physical system for intelligent monitoring and predictive operation of long-span high-speed railway arch bridges. Structures. 82, 110609. URL: https://doi.org/10.1016/j.istruc.2025.110609
Wang, L., Li, Q., He, Y., Han, Y., Hu, P., & Cai, C. S. (2025). An efficient solution for train-track-bridge interaction analysis considering wheel-rail dynamic contact. International Journal of Structural Stability and Dynamics. URL: https://doi.org/10.1142/S0219455427501008
Wang, P. G., Chu, Y., Li, H. J., Zhao, M., & Du, X. L. (2026). Seismic response analysis of high-speed railway bridge systems spanning a V-shaped valley terrain on rigid bedrock. Soil Dynamics and Earthquake Engineering. 200, 109843. URL: https://doi.org/10.1016/j.soildyn.2025.109843
Xia, H., De Roeck, G., Zhang, N., & Maeck, J. (2003). Experimental analysis of a high-speed railway bridge under Thalys trains. Journal of Sound and Vibration. 268(1), pp. 103-113. URL: https://doi.org/10.1016/S0022-460X(03)00202-5
Xia, H., & Zhang, N. (2005). Dynamic analysis of railway bridge under high-speed trains. Computers & Structures. 83(23-24), pp. 1891-1901. URL: https://doi.org/10.1016/j.compstruc.2005.02.014
Xie, M., Wang, K., Guo, S., Yang, Z., Bai, W., & Li, X. (2025). Study on the spatial mechanical behavior of the cable tower anchorage zone in long-span curved T-shaped rigid-frame extradosed bridge for high-speed railway. Scientific Reports. 15(1), 37430. URL: https://doi.org/10.1038/s41598-025-21362-2
Xing, M., Zhu, J., Zhao, C., & Sun, Y. (2025). Study on effective frequency range of wheel-rail excitation for elevated high-speed railway. Scientific Reports. 15(1), 38685. URL: https://doi.org/10.1038/s41598-025-22394-4
Yang, Y. B., Yau, J. D., Yao, Z., & Wu, Y. S. (2004). Vehicle-bridge interaction dynamics: with applications to high-speed railways. World Scientific. URL: https://doi.org/10.1142/5541
Yau, J. D., Yang, Y. B., & Kuo, S. R. (1999). Impact response of high speed rail bridges and riding comfort of rail cars. Engineering structures. 21(9), pp. 836-844. URL: https://doi.org/10.1016/S0141-0296(98)00037-6
Zhai, W., Wang, K., & Cai, C. (2009). Fundamentals of vehicle–track coupled dynamics. Vehicle System Dynamics. 47(11), pp. 1349-1376. URL: https://doi.org/10.1080/00423110802621561
Zhai, W., Xia, H., Cai, C., Gao, M., Li, X., Guo, X., Zhang N. & Wang, K. (2013). High-speed train–track–bridge dynamic interactions-Part I: theoretical model and numerical simulation. International Journal of Rail Transportation. 1(1-2), pp. 3-24. URL: https://doi.org/10.1080/23248378.2013.791498
Zhou, Y., Hao, G., Chen, J., Wei, J., & Zheng, J. (2024). Long-term deformation monitoring of a steel-truss arch bridge using PSI technique refined by temperature field analysis. Engineering Structures. 311, 118164. URL: https://doi.org/10.1016/j.engstruct.2024.118164
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2026 The Author(s)
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Copyright is granted to authors for the purpose of providing protection for articles written to describe experiments and their results. JCEF will protect and defend the work and reputation of the author and are also willing to address any allegations of violation, plagiarism, fraud, etc. against articles written and published by JCEF. JCEF is published under the terms of the Creative Commons Attribution-ShareAlike 4.0 International License (CC BY-SA 4.0). The author holds the copyright and assigns the journal rights to the first publication (online and print) of the work simultaneously.
