THE METHOD OF INTEGRATING SYSTEMS OF HIGH-ORDER EQUILIBRIUM EQUATIONS OF THE MATHEMATICAL THEORY OF THICK PLATES UNDER INTERMITTENT LOADS (PART 2)

Abstract

The subject of the study is the development of an analytical method for solving boundary value problems of a variant of the mathematical theory of transversally isotropic plates of arbitrary constant thickness, which boil down to the integration of systems of inhomogeneous high-order differential equations of equilibrium. According to the developed version of the theory, all components of the stress-strain state and boundary conditions are considered functions of three coordinates. The indicated functions of the three variables are expanded into infinite mathematical series by Legendre polynomials from the transverse coordinate. The boundary conditions on the front faces of the plates are fulfilled exactly. The boundary conditions on the lateral surfaces are fulfilled according to the appropriate approximation, which is determined by a certain number of terms in partial sums of mathematical series. This makes it possible to effectively determine all components of the stress-strain state with any high accuracy. It is also important to note that the developed version of the mathematical theory takes into account vortex and potential edge effects with high accuracy. The problem here lies in the high-order systems of differential equilibrium equations with partial derivatives. This complicates their solution. Moreover, the order of systems increases with an increase in the number of members in the partial mathematical sums of the development of components of the stress-strain state into infinite mathematical series. Therefore, the solution of this problem is connected with the application of a new methodology for finding partial and general solutions. The methodology consists in the fact that the initial systems of high-order differential equations of equilibrium are reduced by various mathematical transformations to convenient homogeneous and heterogeneous systems of high-order differential equations. These equations, in turn, are reduced to homogeneous and inhomogeneous differential equations of the second order by the developed operator method. The general and partial solutions of the initial equilibrium systems are expressed through the general and partial solutions of the second-order differential equations. The skew-symmetric transverse load of the plates relative to the median plane is considered. The goal is to obtain analytical solutions of boundary value problems for plates of arbitrary constant thickness. General solutions in special functions for components of the stress-strain state (SSS) from the annular transverse load of circular and annular plates under axisymmetric deformation are obtained. Analytical solutions of the boundary value problems for the specified plates under the action of axisymmetric loads for various static and kinematic boundary conditions on the lateral surfaces were obtained. An analysis of the obtained results was carried out.The proposed methodology, the method of solving systems of high-order differential equations of equilibrium, the method of obtaining general and partial solutions can also be applied in classical and refined theories, including theories of the Tymoshenko-Reissner type.