Free vibrations of FGM plates with imperfections using layerwise finite element

Abstract

The initial geometrical imperfections, as deviations between the actual and intended shape, are in most cases randomly distributed in real structures and their real shape is not known in advance. Most of the studies reported up to date are based on simplified assumption that the initial geometric imperfection has a similar form to the deformed shape of the plate or sinusoidal form. Moreover, most of the mentioned studies present the analytical solutions for imperfect plates, restricred to simply supported boundary conditions. In wish to fulfil the lack of free vibration solutions based on Layer Wise (LW) plate theories, which will be able to include general form of geometrical imperfection and arbitrary boundary conditions, in this study a LW finite element solution is presented for free vibrations of geometrically imperfect FGM (Functionally Graded Material) plates. After establishing the accuracy of the present layer wise model for linear and geometrically nonlinear bending, vibration and buckling analysis of laminated composite and sandwich plates subjected to mechanical load, as well as for thermal bending and buckling of laminated composite and sandwich plates, in this paper, free vibration of geometrically imperfect FGM plates is presented. The mathematical model, based on Layer wise theory of Reddy, assumes layer wise variation of in plane displacements and constant transverse displacement through the thickness, linear strain displacement relations and isotropic/orthotropic material properties. The material properties of FGM plates are assumed to be constant in xy-plane and vary through the thickness by a power law function in terms of the volume fraction of the constituents. The effective materials properties are given by the rule of mixture. The Koiter’s model for initial geometric imperfection is adopted. In order to include various possible imperfection modes, such as sine type, local type and global type, an imperfection function in the form of the product of trigonometric function and hyperbolic function in (x,y) plane is used. The principle of virtual displacement (PVD) is used to derive Euler-Lagrange differential equation of motion. The weak form is discretized using nine node Lagrangian isoparametric finite element. The original MATLAB computer program is coded for the finite element solution. The effects of imperfection mode and amplitude on fundamental frequencies is analysed. The accuracy of the numerical model is verified by comparison with the available results from the literature. It is found that geometric imperfection greatly affect the free vibration behaviour of FGM plates and should be taken into account when designing safe and functional structures.