Exact solutions of boundary value problems in the theory of plate bending in a half-strip: basics of the theory

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Using the boundary value problem on the 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 as an example, we consider the main steps in constructing exact solutions to the boundary value problems of bending of thin elastic rectangular plates. The solutions are 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 via the Fourier integrals of the boundary functions specified at the half-strip end. The systems of functions biorthogonal to the Papkovich–Fadle eigenfunctions constructed here are used in this case. The exact solutions possess properties that are not inherent in any of the known solutions in the theory of plate bending. Some of them are discussed in the paper. The final formulas describing the exact solution of the boundary value problem are simple and can be easily used in engineering practice. The results that we have obtained previously when solving the boundary values problems of the plane theory of elasticity in a rectangular region form the basis for our work.