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Journal of Applied Mathematics
Volume 2011 (2011), Article ID 842805, 17 pages
Research Article

An Investigation on the Nonlinear Free Vibration Analysis of Beams with Simply Supported Boundary Conditions Using Four Engineering Theories

1Young Research Club, Islamic Azad University, Sari Branch, P.O. Box 48161-194, Mazandaran Province, Sari, Iran
2School of Mechanical Engineering, Sharif University of Technology, P.O. Box 14588-89694, Tehran, Iran

Received 9 May 2011; Revised 5 September 2011; Accepted 8 September 2011

Academic Editor: C. Conca

Copyright © 2011 R. A. Jafari-Talookolaei et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The objective of this study is to present a brief survey on the geometrically nonlinear free vibrations of the Bernoulli-Euler, the Rayleigh, shear, and the Timoshenko beams with simple end conditions using the Homotopy Analysis Method (HAM). Expressions for the natural frequencies, the transverse deflection, postbuckling load-deflection relation to, and critical buckling load are presented. The results of nonlinear analysis are validated with the published results, and excellent agreement is observed. The effects of some parameters, such as slender ratio, the rotary inertia, and the shear deformation, are examined as other parameters are fixed.