A boundary value problem in the theory of elasticity for a rectangle: exact solutions

We derive the formulas that describe the exact solution of the boundary value problem in the theory of elasticity for a rectangle in which two opposite (horizontal) sides are free and stresses are specified (all cases of symmetry relative to the central axes) on the other two sides (rectangle ends). The formulas for a half-strip are also given. The solutions are represented as series in Papkovich–Fadle eigenfunctions whose coefficients are determined from simple formulas. The obtained formulas remain the same for other types of homogeneous boundary conditions, for example, when the horizontal sides of the rectangle are firmly clamped, have stiffening ribs that work in tension–compression and/or bending, etc. Obviously, in this case, the Papkovich–Fadle eigenfunctions will change, as well as the corresponding biorthogonal functions and normalizing factors. To solve a specific boundary value problem, it is enough to find the Lagrange coefficients, which are determined from simple formulas, as Fourier integrals of boundary functions specified at the ends of the rectangle, and then substitute them into the necessary formulas. Examples of solving two problems (even-symmetric deformation relative to the central coordinate axes) are given: (a) The normal stresses are known at the rectangle ends, and the tangential ones are zero; and (b) the longitudinal displacements conditioned by the action of some normal stresses are known at the rectangle ends (the tangential stresses are zero). These solutions are compared with the known solutions in trigonometric Fourier series. The basis of the exact solutions obtained is the theory of expansions in Papkovich–Fadle eigenfunctions based on the Borel transform in the class of quasi-entire functions of exponential type (developed by the authors in their previous studies).