Toward solution of problems on the stressed state of inhomogeneous thick-walled plates with the spline-collocation method

Abstract

UDC 539.3 The technology of laminated design elements in which each layer has characteristic anisotropies and in_homogeneities in plastic properties has reached a level at which they are widely used in diverse areas of machine design and structural mechanics. Such designs for composites make it possible to combine high strength and rigidity, thermal insulation, sound absorption, and other characteristics with reduced metal content. It is desirable to compute the stressed state of in_homogeneous plastic elements in order to make complete allowance for the real properties of materials by solving three-dimensional equations from the theory of elasticity, which makes it necessary to assure satisfaction of boundary conditions on all surfaces that bound an elastic object. The problem of determining specific deformations for fixed plates in three dimensions was stated and solved in [1, 6, 7, 9], et. al. However, the important engineering problem of determining the stressed state of inhomogeneous orthotropic thick-walled plates that are rectangular in plan for the case of rigidly constrained outer surfaces has not been considered. Here we solve this problem by using an approach based on the theory of elasticity, using the method of spline-collocation in one coordinate direction; we represent the solution in the form of an expansion in trigonometric series in the other coordinate directions and use numerical integration for a one-dimensional boundary-value problem for the thickness of the plate. A survey on solution of problems in the theory of plates and shells using spline functions may be found in [2]. A plate constructed of an arbitrary number of isotropic, transversely isotropic, or orthotropic layers is considered in a Cartesian rectangular coordinate system x, y, z. We seek a solution in the region 0 _< x _< a, 0 <_ y _< b, z i <- z _< zi+ 1, where i = O, M and M is the number of layers in the plate. The elastic and thermophysical characteristics of the materials in the layers are generally arbitrary functions of the coordinate z, and they may differ substantially in inhomogeneity and anisotropy in each layer of the plate. The inner and outer bounding surfaces of the plate z = z o and z = z M may be subject to nonuniform force and temperature influences. The most frequent case is the one in which the following stress vectors are specified at these surfaces: l "D oO=_q;.; ~o _q~-; _q; (~=~o); �9 ~ y=