This article is devoted to the formulation and numerical solution of boundary-value problems in the theory of elasticity with respect to deformations. Similar to the well-known Beltrami–Michell stress equations, the Saint-Venant compatibility conditions are written in the form of differential equations for strains. A new version of plane boundary-value problems in strains is formulated. It is shown that for the correctness of plane boundary value problems, in addition to the usual conditions, one more special boundary condition is required using the equilibrium equation. To discretize additional boundary conditions and differential equations, it is convenient to use the finite difference method. By resolving grid equations and additional boundary conditions with respect to the desired quantities at the diagonal nodal points, we obtained convergent iterative relations for the internal and boundary nodes. To solve grid equations, the elimination method was also used. By comparing with the Timoshenko–Goodyear solution on the tension of a rectangular plate with a parabolic load, the validity of the formulated boundary value problems in strains and the reliability of the numerical results are shown. The accuracy of the results has been increased by an average of 15%.

Usually, planar and spatial problems of the theory of elasticity in stresses are solved using the Airy or Maxwell stress function, respectively. Solving a plane problem usually comes down to solving a biharmonic equation. In the spatial case, it can be solved by the Castigliano variational method. This work is devoted to the mathematical and numerical solution of the spatial problem of elasticity theory directly concerning stresses. In this case, the boundary value problem consists of three equilibrium equations and three Beltrami-Michell equations with the corresponding boundary conditions. At the same time, equilibrium equations are considered as missing boundary conditions. Symmetric finite-difference equations are constructed, and the problems of equilibrium of a parallelepiped under the action of domed and uniformly distributed loads are solved numerically. The reliability of the results obtained is substantiated by comparing the numerical results with the wellknown Filonenko-Borodich solution and with the results of solving the boundary value problem of the equilibrium of the parallelepiped with respect to displacements.