The construction of an integral representation of the solution of the equilibrium equations of Timoshenko-type theory for shells of complex geometry

Abstract

A method of constructing an integral representation of the solution of the equilibrium equations of timoshenko-type theory for thin or shallow isotropic shells of complex geometry is proposed. The method involves the following steps: writing the equilibrium equations for a fundamental solution of the three-dimensional theory of elasticity—the Kelvin vector in a curvilinear system for coordinates, normally to the middle surface of the shell; selecting a differential operator corresponding to the given theory of shells from the exact equilibrium equations for the Kelvin vector and constructing an integral representation of the vector of displacements of elements of the shell using Green's formula for the differentil operator of the given theory of shells. It is shown that problems of determining the parameters of the stress-strain state of a shell in differential and integral formulations are equivalent, with an error which is small in the context of approximations of the theory. One method of constructing integral equations for the displacement vector of the elements of a shell of constant thickness is proposed.