Investigation of the Stress Strain State of the Layer with a Longitudinal Cylindrical Thick-Walled Tube and the Displacements Given at the Boundaries of the Layer

Abstract

This paper proposes an analytical-numerical approach to solving the spatial prob-lem of the theory of elasticity for the layer with a circular cylindrical tube. A cylin-drical empty thick-walled tube is located inside the layer parallel to its surfaces and is rigidly fixed to it. It is necessary to investigate the stress-strain state of the elastic bodies of both the layer and tube. Stresses are given on the inner surface of the tube, and displacements, on the boundaries of the layer. The solution to the spatial prob-lem of the theory of elasticity is obtained by the generalized Fourier method with respect to the system of Lamé's equations in the cylindrical coordinates associated with the tube and the Cartesian coordinates associated with the boundaries of the layer. Infinite systems of linear algebraic equations obtained as a result of satisfying the boundary and conjugation conditions are solved by the truncation method. As a result, displacements and stresses are obtained at various points of the elastic layer and elastic tube. Due to the selected truncation parameter for the given geometrical characteristics, the satisfaction of boundary conditions has been brought to 10-3. An analysis of the stress-strain state for the elastic body at different thicknesses of the tube, as well as at different distances from the tube to the boundaries of the layer is conducted. Graphs of normal and tangential stresses at the boundary of the tube and layer, as well as normal stresses on the inner surface of the tube are presented. These stress graphs indicate that as the tube approaches the upper boundary of the layer, the stresses in the elastic bodies of both the layer and tube increase, and with decreasing tube thickness, the stresses in the elastic body of the layer decrease, growing in the elastic body of the tube. The proposed method can be used to calcu-late structures and parts, whose design schemes coincide with the formulation of the problem of this paper. The analysis of the stress state can be used to select the geo-metrical parameters of the designed structure, and the stress graph at the boundary of the tube and layer can be used to analyze the strength of the joint.

Keywords

• thick-walled tube in a layer
• lamé's equations
• generalized fourier method