A generalized finite element approach to the free vibration analysis of non-uniform axially functionally graded beam

Abstract

The present study aims to present a generalized finite element approach to the free vibration analysis of an Axially Functionally Graded (AFG) beam characterized by non-uniform thickness. Application of the non-uniform beam element and assembling make the finite element model a generalized one. The current approach can be applied to beams with both uniform and non-uniform thicknesses and any of the homogenous and inhomogeneous material variations. The governing equation for free vibration of beam was derived from Euler-Bernoulli beam theory as well as Euler-Lagrange equation. As observed throughout the study, the cross-section of the beam decreased along the length depending on the exponential function related to variations in thickness. Material inhomogeneity is determined according to the power and exponential law of material variation along the axial direction, taken from the literature. Mathematical modeling of the geometric nonuniformity, material inhomogeneity, and finite element analysis of the AFG beam was achieved using MATLAB software. The effects of geometric non-uniformity and material gradient parameters on the fundamental frequencies of vibration in different classical boundary conditions were also evaluated. A comparison of the results of available literature can guarantee the efficacy of the proposed method.