A Generalized Finite Element Approach towards the Free Vibration Analysis of Non-uniform Axially Functionally Graded Beam

Abstract

A generalized finite element approach, for the free vibration analysis of an axially functionally graded (AFG) beam, having non-uniform thickness, has been presented in the current analysis. The use of non-uniform beam element and the way of assembling the same, make the finite element model, a generalized one. The current approach can be used for beams of both uniform and non-uniform thickness, with any of the homogenous and inhomogeneous material variation. The governing equation for free vibration of beam has been derived considering Euler-Bernoulli beam theory and by using Euler-Lagrange's equation. The cross-section of the of the beam is decreasing along the length depending upon the exponential function considered for variation in thickness. The material inhomogeneity is as per the Power and Exponential law of material variation along the axial direction, taken from the literature. Mathematical modelling of geometric non-uniformity, material inhomogeneity and finite element analysis of the AFG beam, have been performed using MATLAB. The effect of geometric non-uniformity and material gradient parameters on the fundamental frequencies of vibration in different classical boundary conditions have been investigated. The efficacy of the current method has been ascertained by comparing the result of available literature.