Finite Element Studies and Dynamic Analysis of Initially Stressed Functionally Graded Plates Using a Refined Higher Order Shear Deformation Theory

Abstract

A C⁰isoparametric finite element formulation is presented to calculate the required number of lowest natural frequencies of functionally graded plates (FGPs)subjected to in-plane prestress. The material properties of the FGPs are assumed to vary continuously from one surface to another, according to a simple power law distribution in terms of the constituent volume fractions. The present theory is based on a higher-order displacement model and the three-dimensional Hooke’s laws for plate material. The theory represents a more realistic quadratic variation of the transverse shearing and normal strains through the thickness of the plate. Nine-noded Lagrangian elements have been used for the purpose of discretization using a refined Higher Order Shear deformation Theory (HOST12) that includes the effects of transverse shear deformations, transverse normal deformation and rotary inertia. The plate structure is idealized into an assemblage of nine-noded isoparametric quadrilateral elements with twelve degrees of freedom per node. Hamilton’s principle is used for the formulation. The effect of in-plane pre-stress is taken care by calculating the geometric stiffness matrix. The same shape functions are used to calculate the elastic stiffness matrix, geometric stiffness matrix and the element mass matrix. Subspace Iteration technique is applied to extract the natural frequencies. Numerical results for first seven natural frequencies are presented for rectangular plates under various boundary conditions. The results show good agreement with three-dimensional analytical formulation.