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. The paper is dedicated to the classical problem of elasticity theory – to the solution of the biharmonic equation in a semi-strip (rectangle) and to some conclusions that follow from the analysis of exact solutions of the biharmonic problem. The formulas describing the distribution of stresses and displacements in the semi-strip (rectangle) with free longitudinal sides and stresses set on the end face, as well as the mathematical apparatus used in this case can be found in the works of authors such as [1, 2]. The solutions are represented as the expansions in the Fadle–Papkovich functions. The coefficients of the expansions are determined as the simple Fourier integrals from the known boundary functions. The exact solutions in the rectangle have some unusual properties that do not exist in any of the known exact solutions of elasticity theory. For example, they are not unique and, therefore, they may describe the residual stresses. Thus, solving the biharmonic problem for the rectangle, thus we constructed the theory of residual stresses – one of the fundamental problems of elasticity theory. The obtained solutions allow us to understand what it means to set boundary conditions on the rectilinear boundary of a domain; because she is already not rectilinear after application of the load. What is the angular point of an infinite wedge and how it differs from the angular point of the rectangle? Has got the angular point in the rectangle the singularity always, if the type of boundary conditions is changed at the angular point? And so on. Based on the developed mathematical apparatus, it is possible to obtain the exact solutions of various mixed boundary value problems in the rectangle. For example, when there is a crack in the rectangle, when the part of the boundary of the rectangle is rigidly fixed, and the part is free, etc. The developed methods can be transferred to other coordinate systems: to polar coordinate system (cambered beam), to oblique (triangle, trapezium). They can be generalized to three-dimensional problems (rectangular parallelepiped).

THE FEATURES OF ANALYTICAL SOLUTIONS OF BOUNDARY VALUE PROBLEMS OF THE ELASTICITY THEORY FOR FINITE DOMAINS WITH ANGULAR POINTS OF A BOUNDARY