Vibration of a Beam on an Elastic Foundation Using the Variational Iteration Method

Authors: Desmond Adair, Kairat Ismailov, Martin Jaeger

Abstract:

Modelling of Timoshenko beams on elastic foundations has been widely used in the analysis of buildings, geotechnical problems, and, railway and aerospace structures. For the elastic foundation, the most widely used models are one-parameter mechanical models or two-parameter models to include continuity and cohesion of typical foundations, with the two-parameter usually considered the better of the two. Knowledge of free vibration characteristics of beams on an elastic foundation is considered necessary for optimal design solutions in many engineering applications, and in this work, the efficient and accurate variational iteration method is developed and used to calculate natural frequencies of a Timoshenko beam on a two-parameter foundation. The variational iteration method is a technique capable of dealing with some linear and non-linear problems in an easy and efficient way. The calculations are compared with those using a finite-element method and other analytical solutions, and it is shown that the results are accurate and are obtained efficiently. It is found that the effect of the presence of the two-parameter foundation is to increase the beam’s natural frequencies and this is thought to be because of the shear-layer stiffness, which has an effect on the elastic stiffness. By setting the two-parameter model’s stiffness parameter to zero, it is possible to obtain a one-parameter foundation model, and so, comparison between the two foundation models is also made.

Keywords: Timoshenko beam, variational iteration method, two-parameter elastic foundation model.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1474642

References:

[1] I. Calió, and A. Greco, “Free vibrations of Timoshenko beam columns on Pasternak foundations,” Journal of Vibration and Control, vol. 19, no. 5, pp. 686-696, 2013.
[2] M. El-Mously, “Fundamental frequencies of Timoshenko beams mounted on Pasternak foundation,” Journal of Sound & Vibration, vol. 228, no. 2, pp. 452-457, 1999.
[3] C. F. Lü, C. W. Lim, and W. A.Yao, “A new analytic symplectic elasticity approach for beams resting on Pasternak elastic foundations,” Journal of Mechanics of Materials and Structures, vol. 4, no. 10, pp. 1741-1754, 2010.
[4] C. Franciosi, and A. Masi, “Free vibrations of foundation beams on two-parameter elastic soil,” Computers and Structures, vol. 47, no. 3, pp. 419-426, 1993.
[5] P. L. Pasternak, “On a new method of analysis of an elastic foundation by means of two foundation constants,” Gosudarstvenno Izdatefslvo Literaturi po Stroitelstvu I Arkhitekture, vol. 21, 1954.
[6] T. M. Wang and J. E. Stephens, “Natural frequencies of Timoshenko beams on Pasternak foundations,” Journal of Sound and Vibration, vol. 51, pp. 149-155.
[7] T. Yokoyama, “Vibrations and transient responses of Timoshenko beams resting on elastic foundations,’ Ingenieur-Archiv, vol. 57, pp. 81-90, 1987.
[8] W. Q. Chen, C. F. Lü, and Z. G. Bian, “A mixed method for bending and free vibration of beams resting on a Pasternak elastic foundation,” Applied Mathematical Modelling, vol. 28, pp. 877-890, 2004.
[9] J. K. Lee, S. Jeong and J. Lee, “Natural frequencies for flexural and torsional vibrations of beams on Pasternak foundation,” Soils and Foundations, vol. 54, no. 6, pp. 1202-1211, 2014.
[10] A. Ghannadiasl, and M. Mofid, “An analytical solution for free vibration of elastically restrained Timoshenko beam on an arbitrary variable Winkler foundation and under axial load,” Latin American Journal of Solids and Structures, vol. 12, pp. 2417-2438, 2015.
[11] J.-H. He, “Variational iteration method – A kind of non-linear analytical technique: Some examples,” International Journal of Nonlinear Mechanics, vol. 34, no. 4, pp. 699-708, 1999.
[12] J.-H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media,” Computer Methods in Applied Mechanics and Engineering, vol. 167, no. (1-2), pp. 57-68, 1998.
[13] J.-H. He, Variational iteration method: Some recent results and new interpretations, “Computers and Mathematics with Applications, vol. 54, pp. 881-894, 2007.
[14] B. Batiha, M. S. M. Noorani, and I. Hashim, “Application of variational iteration method to heat- and wave-like equations,” Physics Letters A, vol. 369, pp. 55-61, 2007.
[15] A. M. Wazwaz, “The variational iteration method: A reliable analytic tool for solving linear and nonlinear wave equations,” Computers and Mathematics with Applications, vol. 54, pp. 926-932, 2007.
[16] Y. Lin and C. S. Gurram, “The use of He’s variational method for obtaining the free vibration of an Euler-Bernoulli beam,” Mathematical and Computer Modelling, vol. 50, pp. 1545-1552, 2009.
[17] Y. H. Chai, and C. M. Wang, “An application of differential transformation to stability analysis of heavy columns,” International Journal of Structural Stability and Dynamics, vol. 6, pp. 317-332, 2006.
[18] M. A. De Rosa, and M. J. Maurizi, “The influence of concentrated masses and Pasternak soil on the free vibrations of Euler beams - exact solution,” Journal of Sound and Vibration, vol. 212, pp. 573-581.
[19] L. S. Soares and W.K. da Silva Bezerra, “Dynamic analysis of Timoshenko beam on Pasternak foundation,” Proceeding of XXXVIII Iberian Latin-American Congress on Computational Methods in Engineering, Florianópolis, SC, Brazil, Nov, 5-8, 2017.