# Investigation of the second main problem of elasticity for a layer with n cylindrical inclusions

## Authors

• Vitaly Miroshnikov National Aerospace University "KHAI", Ukraine
• Tetiana Denisova Simon Kuznets Kharkiv National University of Economics, Ukraine

## Keywords:

layer with cylindrical inclusions, Lame's equations, generalized Fourier method, infinite systems of linear algebraic equations

## Abstract

When designing structures in the form of a reinforced layer, one has to deal with the situation when the reinforcement bars are located close to each other. In this case, their influence on each other increases. In order to obtain the stress-strain state in the contact zone of the layer and the inclusion, it is necessary to have a method that would allow obtaining a result with high accuracy.

In this work, an analytical-numerical approach to solving the spatial problem of the theory of elasticity for a layer with a given number of longitudinal cylindrical inclusions and displacements given at the boundaries of the layer has been proposed.

The solution of the problem has been obtained by the generalized Fourier method with respect to the system of Lame's equation in local cylindrical coordinates associated with inclusions and Cartesian coordinates associated with layer boundaries. Infinite systems of linear algebraic equations obtained by satisfying the boundary conditions and conjugation conditions of a layer with inclusions have been solved by the reduction method. As a result, displacements and stresses have been obtained at different points of the considered medium. When the order of the system of equations is 6, the accuracy of fulfilling the boundary conditions was 10-2 for values from 0 to 1.

Numerical studies of the algebraic system of equations give grounds to assert that its solution can be found with any degree of accuracy by the reduction method, which is confirmed by the high accuracy of fulfilling the boundary conditions.

In the numerical analysis, variants of the layer with 1 and 3 inclusions have been compared. The result has shown that close placement of reinforcement bars increases stresses  on the surface of these inclusions. The values of stresses on the layer contact surfaces with inclusions have also been obtained.

The proposed solution algorithm can be used in the design of structures, the computational scheme of which is the layer with longitudinal cylindrical inclusions and displacements specified at the layer boundaries.

## Author Biographies

### Vitaly Miroshnikov, National Aerospace University "KHAI"

Ph.D. in Technical Sciences, Associate Professor, Associate Professor of the Department of Aircraft Strength

### Tetiana Denisova, Simon Kuznets Kharkiv National University of Economics

Ph.D. in Technical Sciences, Associate Professor, Associate Professor of Higher Mathematics and Economic and Mathematical Methods

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