Reduction of the Sanders-Koiter Equations for Fully Anisotropic Circular Cylindrical Shells to Two Coupled Equations for a Stress and a Curvature Function

Abstract

With the aid of the static-geometric duality of Goldenveizer (1961), Cartesian tensor notation, and nondimensionalization, it is shown that the equations of linear shell theory of Sanders (1959) and Koiter (1959), when specialized to a circular cylindrical shell with stress-strain relations exhibiting full anisotropy (21 elastic-geometric constants), can be reduced, with no essential loss of accuracy, to two coupled fourth-order partial differential equations for a stress function F and a curvature function G. Auxiliary formulas for the midsurface displacement components are also given. For isotropic shells with uncoupled stress-strain relations, the equations reduce to a form given by Danielson and Simmonds (1969). The reduction is achieved by adding certain negligibly small terms to the given stress-strain relations. For orthotropic shells of mean radius R and thickness h with uncoupled stress-strain relations, it is shown that the very short decay length of O(hR) and the very long decay length of O(RR/h) (associated with separable solutions of the form e−Rz sin nθ) depend, respectively, to within a relative error of O(h/R), only on the products of different pairs of the eight possible elastic constants.