Dynamic Analysis of Axially Beam on Visco - Elastic Foundation with Elastic Supports under Moving Load

Document Type : Research Paper


School of Railway Engineering, Iran University of Science and Technology, Tehran, Iran


For dynamic analyses of railway track structures, the algorithm of solution is very important. For estimating the important problems in the railway tracks such as the effects of rail joints, rail supports, rail modeling in the nearness of bridge and other problems, the models of the axially beam model on the elastic foundation can be utilized. For studying the effects of axially beam on the elastic foundation, partial differential equations which represent the independent variables should be utilized because of the beams have infinite degrees of freedom. In this paper, solution algorithm and process of the axially beam on the elastic foundation by considering the elastic supports under moving load have been studied and equations have been analyzed as closed form. The beam model includes visco – elastic foundation and elastic supports conditions. For considering the beam element, axial force has been considered beside of shear and moment forces. The solution algorithm is that firstly the differential equations of beam on the elastic foundation with elastic supports are derived and then these equations are solved parametrically by using separation of variables and orthogonality properties of modes. This process and solution have been presented as closed form in this paper. This problem wasn’t investigated in the technical literature. This model can be utilized for the most problems in the railway tracks. The advantage of this paper is presentation of algorithm and process of parametric solution for an axially beam on the visco - elastic foundation with elastic supports.    


- Abu-Hilal, M. (2006) “Dynamic response of a double Euler–Bernoulli beam due to a moving constant load”, Journal of Sound and Vibration, vol. 297, no. 3–5, pp. 477–491.
- Akour, S. N. (2010) “Dynamics of nonlinear beam on elastic foundation”, Proceedings of the World Congress on Engineering, vol. II, London, U.K.
- Bogacz, R. and Czyczuła, W. (2008) “Response of beam on visco-elastic foundation to moving distributed load”, Journal of Theoretical and Applied Mechanics, vol. 46, no. 4, pp. 763-775.
- Chopra, A. K. (1995) “Dynamics of structures - theory and applications to earthquake engineering”, Prentice-Hall, Upper Saddle River, New Jersey.
- Clough, R. W. and  Penzien, J. (2003) “Dynamics of structures”, Computers and Structures, Berkeley, USA.
- Fryba, L. (1999) “Vibration of solids and structures under moving load”, Telford, London, UK.
- Lalanne, C. (2002) “Mechanical vibration & shock, random vibration”, III, Hermes Penton, London.
- Morfidis, K. and Avramidis, I. E. (2002) “Formulation of a generalized beam element on a two-parameter elastic foundation with semi-rigid connections and rigid offsets”, Computers and Structures, vol. 80, no. 25, рр. 1919-1934.
- Paz, M. and Leigh, W. (2004) “Structural dynamics: theory and computation”, Kluwer Academic Publishers, Norwell, MA.
- Solnes, J. (1997) “Stochastic processes and random vibrations: theory and practice”, John Wiley & Sons, Chichester, UK.
- Yang, C. Y. (1986) “Random vibration of structures”, John Wiley and Sons, New York.
- Yang, Y. B. and Chang, K. C. (2009) “Extraction of bridge frequencies from the dynamic response of a passing vehicle enhanced by the EMD technique”, Journal of Sound and Vibration, vol. 322, no. 4-5, pp.718–739.