Analysis of the Stress State for a Layer with Two Incut Cylindrical Supports

Abstract

The stress state of a homogeneous isotropic layer under the action of a spatial static external load is studied. Two circular cylindrical supports are cut into the body of the layer parallel to its borders. The supports and body of the layer are rigidly coupled. The spatial problem theory of elasticity is solved using the analytical-numerical generalized Fourier method. The layer is considered in the Cartesian coordinate system, the supports are considered in the local cylindrical coordinates. Stresses are set on the upper and lower surfaces of the layer. The supports are considered as cylindrical cavities in a layer with zero displacements set on their surfaces. Satisfying the boundary conditions on the upper and lower surfaces of the layer, as well as on the cylindrical surfaces of the cavities, a system of infinite integro-algebraic equations, which are further reduced to linear algebraic ones, is obtained. An infinite system is solved by the reduction method. In the numerical studies, the parameters of integration oscillatory functions are analyzed, problems at different distances between supports are solved. A unit load in the form of a rapidly decreasing function is applied to the upper boundary between the supports. For these cases, an analysis of the stress state was performed on the surfaces of the layer between the supports and on the cylindrical surfaces in contact with the supports. The numerical analysis showed that when the distance between the supports increases, the stresses σ_x on the lower and upper surfaces of the layer and the stresses τ_ρφ on the surfaces of the cavities increase. The use of the analytical-numerical method made it possible to obtain a result with an accuracy of 10^-4 for stress values from 0 to 1 at the order of the system of equations m=6. As the order of the system increases, the accuracy of fulfilling the boundary conditions will increase. The presented analytical-numerical solution can be used for high-precision determination of the stress-strain state of the presented problems type, as well a reference for problems based on numerical methods.