Numerical Solution of Plane Problems of the Theory of Elasticity Directly in Stresses

Usually, the solution of a plane problem of the theory of elasticity in stresses is reduced to solving a biharmonic equation for the Airy stress function. In this paper, two (A and B) variants of plane boundary value problems of the theory of elasticity are formulated directly in terms of stresses. In the first case (A), the boundary value problem consists of two equilibrium equations and one Beltrami-Michell equation with the corresponding boundary and additional boundary conditions. In the formulation of the second plane boundary value problem (B), in contrast to the first, the equations of equilibrium differentiated with respect to x and y, respectively, are used. Symmetric finite-difference equations are constructed and the known Timoshenko-Goodier problem of stretching a rectangular plate with a parabolic load is solved for comparison. The discrete analogs of boundary value problems A and B are composed by the finite-difference method and the iterative method and the marching method are used to solve them. By comparing the numerical results of boundary value problems, A and B obtained by two methods, the validity of the formulated boundary value problems and the reliability of the obtained results are ensured.