Semianalytical Three-Dimensional Solutions for the Transient Response of Functionally Graded Material Rectangular Plates

Abstract

A semianalytical method for the transient response of functionally graded material (FGM) rectangular plates under various boundary conditions, by the use of state space method, differential quadrature method, and numerical inversion method of Laplace transform, is proposed in this study. The FGM rectangular plate is simply supported at two opposite edges with any classical boundary conditions on the other two opposite edges, such as simply supported, clamped, and free conditions. Comparisons show the results generated by the present method agree well with those predicted by the finite element method (FEM), no matter which variation law of material properties, functionally graded (FG) index, geometry, load, and boundary conditions are employed. Convergence studies for different numbers of sampling points along the length direction and different layer numbers along the thickness direction show that the present method has a fast convergence rate. In addition, the effects of FG index, FG direction, and thickness on the plates response are investigated. Besides the semianalytical method based on state space method (SSM) and for the purpose of validation, another analytical method is developed based on the Kirchhoff thin plate theory, the Laplace transform, and its inversion. Explicit solutions for thin FGM simply supported plates under different transient loads are derived. Comparison shows that the results generated by the two analytical methods agree with each other excellently for thin plates. Both of the two analytical methods can be used to generate benchmark results for the response of FGM rectangular thin plates subjected to transient loads. However, studies show that as the thickness increases, the results obtained by the SSM and FEM agree with each other well, whereas the Kirchhoff method generates more errors.