Analysis for an anisotropic hollow cylindroid subjected to arbitrary loadings

Abstract

In this paper, the solution for the hollow cylindroid is analytically derived under two assumptions that (1) the cylindroid is consist of various anisotropic elastic material and (2) only both sides of the hollow cylindroid are subjected to arbitrary load, which will be expanded to complex form of Fourier series with period 2. To simplify an analysis, mapping function, which is mapped elliptical boundaries of the cylindroid to unit circle, and complex stress functions proposed by S. G. Lekhnitskii are introduced. These stress functions have undetermined coefficients, however, these coefficients can be determined by taking into account the boundary conditions for resultant stresses at both sides of the hollow cylindroid. In particular, undetermined coefficients in stress functions will be determined by comparing with complex Fourier coefficients for those resultant stresses. In the case of isotropic cylindroid, analytical solution for similar problem was already derived, however, there is one limit of application. This limit is related to the cross-sectional shape of hollow cylindroid. That is, if both sides of the hollow cylindroid did not have a same focus of an ellipse, analytical solution for isotropic case was not able to derive. On the other hand, the solution derived from this study does not have such a limit. So some numerical examples are shown by some figures and tables not only anisotropic case but also isotropic case.