An exact three-dimensional solution for normal loading of inhomogeneous and laminated anisotropic elastic plates of moderate thickness

Abstract

An exact three-dimensional solution is presented for the deformation and stress distribution in an elliptical plate under uniform normal loading of the lateral surfaces, and clamped along its edge. The plate is assumed to be of constant, moderate thickness and composed of anisotropic elastic material which is inhomogeneous in the through-thickness direction but symmetric about the mid-plane. The only material symmetry assumed is that of reflectional symmetry in planes parallel to the mid-plane. A transfer matrix method is used which, without making any further assumptions, gives the exact solution at each point in the plate in terms of the stress and displacement at the mid-plane. The two-dimensional differential equations governing these mid-plane values are found to be the same as those for an equivalent homogeneous plate whose constant elastic moduli are determined by appropriate through-thickness weighted averages of the inhomogeneous moduli. The solution of the two-dimensional problem is known for such a plate when subject to the specified surface and edge conditions, and yields a closed form analytical solution that satisfies all the governing equations and surface conditions of the full three-dimensional elasticity problem, with edge displacement conditions satisfied on the mid-plane. The important special case of an anisotropic laminated plate is given by assuming piecewise constant properties through the thickness.