Dynamic Analysis of a Bridge Supported With Many Vertical Supports Under Moving Load

Murat Reis, Yașar Pala, Gültekin Karadere


This study is devoted to the investigation of dynamic analysis of a bridge supported with many vertical supports under a moving load. Each vertical support is modelled as a linear spring and a linear damper. The analysis is based on Euler-Bernoulli beam theory. The present method utilises the concept of distributed moving load, spring force and damping force, and avoids the use of matching conditions. Expressing these forces in terms of the unknown function of the problem, it highly simplifies obtaining an exact solution. An important property of Dirac delta distribution function is utilised in order to reach the exact solution. Considering one and three vertical supports, the response of the supported bridge is plotted and compared to different values of parameters. In the case of an undamped bridge with no support, the results are compared with those of previous papers.


bridge; beam; damped; dynamic; support; moving load

Full Text:



Ayre, R. S.; Ford, G.; Jacobsen, L. S. 1950. Transverse vibration of a two-span beam under the action of a moving constant force, Journal of Applied Mechanics 17: 1–12.

Dahlberg, T. 2006. Moving force on an axially loaded beam–with applications to a railway overhead contact wire, Vehicle System Dynamics 44(8): 631–644.

Esmailzaden, E.; Ghorashi, M. 1992. Beams carrying uniform partially distributed moving masses. Technical report of Mechanical Engineering Dept, Sharif University of Technology, Tehran.

Esmailzaden, E.; Ghorashi, M. 1995. Vibration analysis of beams traversed by uniform partially distributed moving, Journal of Sound and Vibration 184(1): 9–17.

Fryba, L. 1972. Vibration of solid and structures under moving loads. Thomas Telford House, London, 13–33.

Garinei, A. 2006. Vibrations of simple beam-like modeled bridge under harmonic moving loads, International Journal of Engineering Science 44: 778–787.

Green, M. F.; Cebon, D. 1994. Dynamic response of highway bridges to very heavy vehicle loads. Theory and experimental validation, Journal of Sound and Vibration 170(1): 51–78.

Grigorjeva, T.; Juozapaitis, A.; Kamaitis, Z. 2006. Simplified engineering method of suspension bridges with rigid cables under action of symmetrical and asymmetrical loads, The Baltic Journal of Road and Bridge Engineering 1(1): 11–20.

Gürgöze, M. 1997. On the eigenvalues of viscously damped beams, carrying heavy masses and restrained by linear and torsional springs, Journal of Sound and Vibration 208(1): 153–158.

Gürgöze, M.; Mermertas, V. 1998. On the eigenvalues of a viscously damped cantilever carrying a tip mass, Journal of Sound and Vibration 216(2): 309–314.

Idnurm, J. 2006. Discrete analysis method for suspension bridges, The Baltic Journal of Road and Bridge Engineering 1(2): 115–119.

Inglis, C. E. 1934. A mathematical treatise on vibration in railway bridges, Cambridge University Press, Cambridge, 1–50.

Inman, D. J. 1994. Engineering vibration. Prentice Hall Inc., A Simon & Schuster Company, Englewoods Cliffs, New Jersey, 329–340.

Pala, Y. 2006. Modern Uygulamalż Diferansiyel Denklemler [Modern applied differential equations]. Nobel Publishing, Bursa, 533–536.

Stanisic, M. M.; Hardin, J. C. 1969. On response of beams to an arbitrary number of moving masses, Journal of the Franklin Institute 287: 115–123.

Timoshenko, S. 1927. Vibration of Bridges, Transactions of the American Society of Mechanical Engineers 53: 53–61.


  • There are currently no refbacks.

Copyright (c) 2008 Vilnius Gediminas Technical University (VGTU) Press Technika