Numerical use of {[formula omitted]}-approximation method thermoelastic anisotropic thin shell theory equations represented in a special form

Abstract

A linear uncoupled quasi-static theory of thermoelastic anisotropic thin shells of constant thickness is formulated. In the derivation of equations and boundary conditions, a variant of the {m,n}-approximation method, which is based on the variational principle of the thermoelasticity theory, is used. According to this method, the unknown functions are represented in the form of a series of Legendre polynomials in the transverse coordinate that agree with the force boundary conditions on the facial surfaces. From the system of equations of the constructed theory for generalised displacements, strains, and stresses, a system of equations for generalised displacements is obtained; this system is solved for the second derivatives of generalised displacements with respect to one of the Gaussian parameters of the middle surface. For a system of partial differential equations represented in such a form, the known techniques for the reduction of systems of two-dimensional equations to normal systems of ordinary differential equations, which can be solved by standard numerical methods, can be applied. The system of equations for generalised displacements is reduced to a normal system of ordinary differential equations in the case of a plane deformation of a cylindrical panel. Through the use of these equations and the S. K. Godunov method of orthogonal successive substitutions, the bending problems of a planar beam subjected to a mechanical load or a thermal load and a circular cylindrical panel exposed to a thermal load are numerically solved. It is demonstrated that the proposed equations are particularly suitable for the analysis of boundary effects in planar beams and circular cylindrical panels.