Investigation of the asymmetric stressed-strained state of transversely isotropic cylinders under different boundary conditions at the ends

Abstract

Since solving problems of the stressed-strained state of elastic bodies in an exact three-dimensional poses considerable mathematical and computational difficulties, only for a small number of problems have the results of the solution been put in numerical form [8, 10, 15, 16, 18]. The stressed-strained state of three-dimensional bodies must be known in order to evaluate the strength capabilities of structural elements, both as a reference for models of applied theories of shells and as solutions obtained by approximate methods. In this article we consider the class of problems of the stressed-strained state of transversely isotropic hollow cylinders of finite length under asymmetric loads applied to the lateral surfaces and with different boundary conditions at the ends. The solution of that class of problems is constructed on the basis of separation of variables in the circumferential coordinate and solution of the two-dimensional boundary-value problem by the spline-collocation and discrete orthogonalization methods [4, 5, 6, 12, 13]. This approach was used earlier [1, 2, 7, 11] to solve two-dimensional problems of the theory of shells in different formulations. The influence of boundary conditions on the displacement and stress distribution in cylinders is analyzed on the basis of the results obtained. We consider transversely isotropic hollow cylinders in which the plane of isotropy coincides with the tangent to the cylindrical surface and the mechanical characteristics vary only along the normal to it. Asymmetrically distributed loads are applied to the lateral surfaces of the cylinder and boundary conditions are given at the ends. The problem is described on the basis of the three-dimensional theory of elasticity in a cylindrical coordinate system