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In this paper, in the form of expansions in Papkovich–Fadle eigenfunctions, an exact solution to the boundary value problem of the theory of elasticity is constructed for a rectangle whose horizontal sides are rigidly clamped. The expansion coefficients are determined with the help of biorthogonal functions by simple formulas.

This paper provides an exact solution to the boundary value problem of elasticity theory for a square loaded with tangential stresses along all its sides (even-symmetric deformation with respect to the central axes). The solution is represented as series in Papkovich–Fadle eigenfunctions. The coefficients of the series are determined by simple closed formulas.

In the paper, we construct an exact solution to a boundary value problem of the theory of elasticity for a rectangle in which the longitudinal sides are free, while normal and tangential stresses are given at the ends (even-symmetric deformation with respect to the central axes). The solution is represented in the form of series in Papkovich–Fadle eigenfunctions. The coefficients are determined explicitly by using functions biorthogonal to the Papkovich–Fadle eigenfunctions. We give the final formulas which have a simple appearance and can easily be used in engineering practice. The obtained solution is compared with the solution to the corresponding boundary value problem for a half-strip.