Abstract
In the paper, for the first time we give exact solutions to two nonhomogeneous boundary value problems of the theory of elasticity for a rectangle with free long sides. Inside the rectangle there are applied two equal concentrated forces directed oppositely along the horizontal axis (even-symmetric deformation). The method of solution is based on the use of the solution to the biharmonic problem for a smooth semi-strip and the method of the integral Fourier transform. In the first problem, the short sides of the rectangle are free; in the second, they are rigidly clamped. The solutions to both problems are constructed on the superposition principle in the form of the sum of integrals and series in trigonometric functions and Papkovich–Fadle eigenfunctions. The coefficients of these expansions are determined by simple formulas as the Fourier integrals of given boundary functions.
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