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Shock and Vibration
Volume 2014 (2014), Article ID 153532, 11 pages
Research Article

On the Flexural-Torsional Vibration and Stability of Beams Subjected to Axial Load and End Moment

Department of Aerospace Engineering, Ryerson University, Toronto, ON, Canada M5B 2K3

Received 21 January 2014; Revised 19 June 2014; Accepted 19 June 2014; Published 13 October 2014

Academic Editor: Reza Jazar

Copyright © 2014 M. Tahmaseb Towliat Kashani 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 free vibration of beams, subjected to a constant axial load and end moment and various boundary conditions, is examined. Based on the Euler-Bernoulli bending and St. Venant torsion beam theories, the differential equations governing coupled flexural-torsional vibrations and stability of a uniform, slender, isotropic, homogeneous, and linearly elastic beam, undergoing linear harmonic vibration, are first reviewed. The existing formulations are then briefly discussed and a conventional finite element method (FEM) is developed. Exploiting the MATLAB-based code, the resulting linear Eigenvalue problem is then solved to determine the Eigensolutions (i.e., natural frequencies and modes) of illustrative examples, exhibiting geometric bending-torsion coupling. Various classical boundary conditions are considered and the FEM frequency results are validated against those obtained from a commercial software (ANSYS) and the data available in the literature. Tensile axial force is found to increase natural frequencies, indicating beam stiffening. However, when a force and an end moment are acting in combination, the moment reduces the stiffness of the beam and the stiffness of the beam is found to be more sensitive to the changes in the magnitude of the axial force compared to the moment. A buckling analysis of the beam is also carried out to determine the critical buckling end moment and axial compressive force.