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Shock and Vibration
Volume 18 (2011), Issue 5, Pages 683-696

Free Vibration and Stability of Axially Functionally Graded Tapered Euler-Bernoulli Beams

Ahmad Shahba,1,2 Reza Attarnejad,1,2 and Shahin Hajilar1

1School of Civil Engineering, University College of Engineering, University of Tehran, Tehran, Iran
2Centre of Numerical Methods in Engineering, University of Tehran, Tehran, Iran

Received 6 March 2010; Revised 11 May 2010

Copyright © 2011 Hindawi Publishing Corporation. 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.


Structural analysis of axially functionally graded tapered Euler-Bernoulli beams is studied using finite element method. A beam element is proposed which takes advantage of the shape functions of homogeneous uniform beam elements. The effects of varying cross-sectional dimensions and mechanical properties of the functionally graded material are included in the evaluation of structural matrices. This method could be used for beam elements with any distributions of mass density and modulus of elasticity with arbitrarily varying cross-sectional area. Assuming polynomial distributions of modulus of elasticity and mass density, the competency of the element is examined in stability analysis, free longitudinal vibration and free transverse vibration of double tapered beams with different boundary conditions and the convergence rate of the element is then investigated.