On expansions in Papkovich–Fadle functions in the problem of plate bending

Abstract:

On the example of the boundary value problem of bending of a thin elastic semi-infinite plate in which the long sides are free, while a self-balanced bending moment and a generalized shearing force are specified at its end, the main stages of constructing exact solutions to the boundary value problems of bending of thin elastic rectangular plates are considered. The solution is constructed in the form of series in Papkovich–Fadle eigenfunctions. The unknown expansion coefficients are determined in the same way as in classical periodic solutions in trigonometric series, and have the same structure, i.e. are expressed in terms of the Fourier integrals of the boundary functions given at the end of the half-strip. Herewith the systems of functions that are constructed in the work and that are biorthogonal to the Papkovich–Fadle eigenfunctions are used. The exact solutions have properties that are not inherent in any of the known solutions of plate bending theory. Some of them are discussed in the article. The final formulas describing the exact solution to the boundary value problem are simple and can easily be used in engineering practice. The work is based on the results obtained by the authors earlier in solving boundary value problems of the plane theory of elasticity in a rectangular domain.