Dynamic Response of a Timoshenko Beam on an Elastic Foundation Subjected to Multiple Distributed Moving Loads

This study develops and verifies a Galerkin formulation for the transient response of a simply supported Timoshenko beam on a Winkler foundation under two finite-length distributed loads. The model includes shear deformation, rotary inertia, a uniform tensile prestress, foundation stiffness, load spacing, travelling velocity, and constant or harmonic load amplitudes. Unlike a semi-infinite Heaviside step, each load is represented by the difference of two Heaviside functions, so that its physical footprint and entry and exit from the span are treated explicitly. Sine and cosine basis functions satisfying the simply supported conditions reduce the governing equations to independent two-degree-of-freedom modal systems. Laplace transformation and partial-fraction inversion give an explicit two-branch impulse-response solution; convolution with the projected moving loads then gives the final analytical displacement and rotation fields. The same retained-mode response is also evaluated by a state-transition convolution scheme, and the two forms agree to machine precision: both predict a peak midspan deflection of 2.9226547 mm at 2.400 s, with a maximum time-history difference of 3.27×10^(-13) mm. A systematic truncation study shows that 12 Galerkin modes differ from the 16-mode reference by only 0.023% in peak response and 0.012% in normalized root-mean-square error. An independent 100-element Timoshenko finite-element model gives 2.922 mm; the peak difference is 0.023%. Increasing the Winkler stiffness from 0 to 2.0 MN/m² reduces the peak response by 86.4%, while increasing tensile prestress from 0 to 120 MN produces a 42.9% reduction. Load spacing produces non-monotonic modal interference, and the velocity sweep gives a maximum dynamic amplification factor of 1.619 at 125 m/s for the selected parameters. Under undamped harmonic loading, excitation at the first flexural frequency produces the expected resonant growth during load passage. These verified results clarify the separate and interacting effects of foundation stiffness, tensile prestress, load magnitude, spacing, speed, and excitation frequency.

Timoshenko beam; Winkler foundation; distributed moving load; Galerkin method; finite-element verification; dynamic amplification; resonance.

1. Timoshenko, S. P. (1921). On the correction for shear of the differential equation for transverse vibrations of prismatic bars. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 41(245), 744–746. https://doi.org/10.1080/14786442108636264 [Google Scholar] [Crossref]

2. Cowper, G. R. (1966). The shear coefficient in Timoshenko’s beam theory. Journal of Applied Mechanics, 33(2), 335–340. https://doi.org/10.1115/1.3625046 [Google Scholar] [Crossref]

3. Frýba, L. (1972). Vibration of solids and structures under moving loads. Noordhoff International Publishing. [Google Scholar] [Crossref]

4. Kenney, J. T. (1954). Steady-state vibrations of beam on elastic foundation for moving load. Journal of Applied Mechanics, 21, 359–364. [Google Scholar] [Crossref]

5. Saito, H., & Terasawa, T. (1980). Steady-state vibrations of a beam on a Pasternak foundation for moving loads. Journal of Applied Mechanics, 47(4), 879–883. https://doi.org/10.1115/1.3153807 [Google Scholar] [Crossref]

6. Wang, R. T. (1997). Vibration of multi-span Timoshenko beams to a moving force. Journal of Sound and Vibration, 207(5), 731–742. https://doi.org/10.1006/jsvi.1997.1188 [Google Scholar] [Crossref]

7. Esmailzadeh, E., & Ghorashi, M. (1995). Vibration analysis of beams traversed by uniform partially distributed moving masses. Journal of Sound and Vibration, 184(1), 9–17. https://doi.org/10.1006/jsvi.1995.0301 [Google Scholar] [Crossref]

8. Xu, Y., Zhu, W., Fan, W., Yang, C., & Zhang, W. (2020). A new three-dimensional moving Timoshenko beam element for moving load problem analysis. Journal of Vibration and Acoustics, 142(3), 031001. https://doi.org/10.1115/1.4045788 [Google Scholar] [Crossref]

9. Assie, A., Akbaş, Ş. D., Bashiri, A. H., Abdelrahman, A. A., & Eltaher, M. A. (2021). Vibration response of perforated thick beam under moving load. The European Physical Journal Plus, 136, 283. https://doi.org/10.1140/epjp/s13360-021-01224-2 [Google Scholar] [Crossref]

10. Jafari, M., & Mohammadimehr, M. (2025). Forced vibration control of Timoshenko micro-sandwich beams with reinforced composites and piezoelectric layers on a Kerr foundation. Journal of Computational Applied Mechanics, 56(1), 15–42. [Google Scholar] [Crossref]

11. Jimoh, S. A., Ogunbamike, O. K., & Ajijola, O. O. (2018). Dynamic response of a non-uniformly prestressed thick beam under a distributed moving load travelling at varying velocity. Asian Research Journal of Mathematics, 9(4), 1–18. https://doi.org/10.9734/ARJOM/2018/41327 [Google Scholar] [Crossref]

12. Adedowole, A., & Jimoh, S. A. (2018). Dynamic analysis of a damped non-uniform beam subjected to loads moving with variable velocity. Archives of Current Research International, 13(2), 1–16. https://doi.org/10.9734/ACRI/2018/35919 [Google Scholar] [Crossref]

13. Omolofe, B., & Adedowole, A. (2017). Response characteristics of a non-uniform beam with time-dependent boundary conditions under travelling distributed masses. Journal of Applied Mathematics and Computational Mechanics, 16(2), 77–99. [Google Scholar] [Crossref]

14. Ajijola, O. O. (2025). Analysis of transverse displacement and rotation under moving load of a prestressed damped shear beam resting on a Vlasov foundation. African Journal of Mathematics and Statistics Studies, 8(1), 31–46. https://doi.org/10.52589/AJMSS-HXMZUBLY [Google Scholar] [Crossref]

15. Newmark, N. M. (1959). A method of computation for structural dynamics. Journal of the Engineering Mechanics Division, 85(3), 67–94. https://doi.org/10.1061/JMCEA3.0000098 [Google Scholar] [Crossref]

16. Bathe, K. J. (1996). Finite element procedures. Prentice Hall. [Google Scholar] [Crossref]

17. Winkler, E. (1867). Die Lehre von der Elasticitaet und Festigkeit. Dominicus. [Google Scholar] [Crossref]

18. Singh, B., & Mukhopadhyay, S. (2024). Thermoelastic vibration of a Timoshenko beam under modified couple stress theory and the Moore–Gibson–Thompson heat-conduction model. Mathematics and Mechanics of Solids, 29(3), 436–451. https://doi.org/10.1177/10812865231186127 [Google Scholar] [Crossref]