Exact three-dimensional elasticity solutions for bending of moderately thick inhomogeneous and laminated strips under normal pressure

Abstract

An exact three-dimensional solution is presented for the deformation and stress distribution in a semi-infinite strip clamped along its two edges, and subjected to uniform normal loading of the lateral surfaces. The strip is of constant moderate thickness and composed of anisotropic elastic material which is arbitrarily inhomogeneous in the through-thickness direction. The only material symmetry assumed is that of reflectional elastic symmetry in planes parallel to the mid-plane. The important special case of an anisotropic laminated plate is given by assuming piecewise-constant properties through the thickness.The general method of solution is to reformulate the full three-dimensional elasticity equations in a way that reduces the problem to solving a system of partial differential equations in the two in-plane independent variables only, and then obtaining asymptotic solutions in terms of an aspect ratio of the thickness divided by a typical in-plane length. In general, successive terms are expressed in terms of the approximate “classical laminate” solution. In the present problem, this solution is very simple and the expansion terminates after no more than three terms. This gives a closed-form analytical solution that is valid for any aspect ratio.