Vol.20, No.1, 2020, pp. 15–19
UDC:

Stress state at the edge of an isotropic rectangular plate under the action of a distributed load based on a nonclassical theory is considered. When constructing a mathematical model of a plate, three-dimensional equations of the theory of elasticity are used. Displacements are represented in the form of polynomials along the coordinate normal to the middle surface two degrees higher relative to the classical theory of the Kirchhoff-Love type. As a result of minimization of the refined value of the Lagrange energy functional, a system of differential equilibrium equations in displacements and natural boundary conditions are obtained. The task of reducing two-dimensional equations to ordinary differential equations is carried out by decomposing the components of displacements and external loads into trigonometric rows in the circumferential coordinate.

The solution of the formulated boundary problem is carried out by the methods of finite differences and matrix sweep. As a result, displacements are obtained in the grid nodes, for the approximation of which splines are used. Plate deformations are found with the help of geometric relations, tangential stresses are obtained from relations of Hooke's law. One of the features of this work is that the transverse stresses are determined by direct integration of equilibrium equations of the three-dimensional theory of elasticity. Comparison of results obtained by the refined theory with data of the classical theory shows that in the zone of distortion of the stressed state, normal tangential stresses are substantially refined and transverse normal stresses that are neglected in classical theory, have the same order with maximal values of the main bending stress.

Keywords: Lagrange variation principle, mathematical model, finite difference method, stress-strain state, matrix sweep method