First Main Problem of the Theory of Elasticity for a Layer with Two Thick-Walled Pipes and One Cylindrical Cavity

Abstract

Structures that are fixed on cylindrical inclusions are among the most common ones in machine and aircraft construction. Some of these inclusions can be modeled as thick-walled pipes with specified stress values on the inner surface. However, the literature does not provide accurate methods for calculating these structures, which indicates the relevance of posing and solving such problems. The presented paper considers the solution method for the model of the structure, which is an elastic homogeneous layer located on two pipes embedded into it and having a longitudinal cylindrical cavity that is parallel to layer boundaries. On the flat surfaces of the cavity surface layer, on the inner surfaces of the pipes, the stresses are considered known. When solving the problem, two types of coordinate systems are used: Cartesian for the layer and cylindrical for the pipes and cavity. The basic solutions in different coordinate systems are given as Lamé equations and combined using transition functions and the generalized Fourier method. An infinite system of integro-alberic equations is formed based on the boundary conditions on the upper and lower surfaces of the layer, the surface of the cavity, and the continuity conditions between the layer and the pipes. After that, the system of equations is reduced to linear algebraic equations of the second kind, to which the reduction method is applied. The problem is solved numerically with a given accuracy, which allowed obtaining the stress-strain state at any point of the elastic body. An analysis of the stress state is carried out with different values of the distance between the thick-walled pipes. On the upper and lower boundaries of the layer, and on the surface of the cylindrical surface, the stresses are considered known. The obtained results do not show a significant effect on the stress along the lower and upper surfaces of the layer. At the same time, the stresses in the layer along the surface of the pipe and layer junction decrease as the distance between the pipes increases. The obtained numerical results can be used in the prediction of geometric parameters during design.