НАПРЯЖЕННО-ДЕФОРМИРОВАННОЕ СОСТОЯНИЕ СФЕРИЧЕСКОЙ ОБОЛОЧКИ ПОД ДЕЙСТВИЕМ ПРОИЗВОЛЬНОЙ НАГРУЗКИ НА ОСНОВЕ НЕКЛАССИЧЕСКОЙ ТЕОРИИ

Abstract

Considered is the stress state of an isotropic spherical shell exposed to an arbitrary load based on a non-classical theory. When building a mathematical model of the shell, three-dimensional equations of the theory of elasticity are applied. 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 series in the circumferential coordinate. Displacements are represented in form of polynomials along the coordinate normal to the middle surface by two degrees higher relative to the classical theory of the Kirchhoff-Love type. Resulting from minimization of the refined value of the Lagrange energy functional, a system of differential equilibrium equations in displacements and natural boundary conditions are received. The task of reducing two-dimensional equations to ordinary differential equations is carried out by decomposing the components of the displacements and the external loads into trigonometric series as per the circumferential coordinate. The formulated boundary problem is solved by the methods of finite differences and matrix sweep. As a result, displacements are obtained in the grid nodes, for approximation of which splines are used. The shell deformations are found using geometric relationship; tangential stresses are received from the correlations of Hooke's law. One of the features of this paper lies in the fact that the transverse stresses are determined by the direct integration of the equilibrium equations of the three-dimensional theory of elasticity. An example of the calculation of a hemispherical shell rigidly restrained along the lower base contour is brought. The shell is exposed to the wind load. Comparison of the results received by the refined theory with the data of the classical theory has shown that in the zone of distortion of the stressed state, the normal tangential stresses are substantially revised and the transverse normal stresses, which are neglected in the classical theory, are of the same magnitude with the maximum values of the main bending stress. Considered is the influence of the relative thickness on the stress state of the shell. It was discovered that the shell thickness significantly increases the error of the classical theory, while determining the stresses and assessing the strength of the elements of the aircraft structures.