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A rectangle with longitudinal ribs, working in tension-compression. Examples of exact solutions of boundary value problems

Menshova I.V., Vorokova M.N.

#### Abstract:

The article is devoted to examples of exact solutions of boundary value problems of the elasticity theory in a rectangle (symmetric deformation relative to the longitudinal axis). The longitudinal sides of a rectangle have strengthening ribs, which have stiffness in tension only. Relative to a vertical axis the even-symmetric and odd-symmetric deformations were studied. The solution sought in the form of a series on Fadle-Papkovich functions. The basic properties of systems of Fadle-Papkovich functions were studied in [1] earlier. For them were constructed the biorthogonality relation and was found the biorthogonal functions. Using the biorthogonal functions are not difficult to determine the required expansions coefficients. This is done in the same way as in the known solutions to Filon-Ribiere in the trigonometric series. Final expressions for the coefficients of expansions have the form of Fourier integrals from the boundary functions. A series of exact solutions converge to boundary functions as the trigonometric series for these functions (uniformly converge with them). At the same time, approximate solutions in series on the Fadle-Papkovich functions, poor converge to boundary functions usually. It was repeatedly noticed in periodical literature. Normal and tangential loads actions at the end faces of rectangle, including concentrated forces acting at the ends of the ribs. Numerical results, illustrating the effect of rib’s stiffness on the distribution of stresses and displacements in the rectangle, are given. Always it is paid a special attention to an approximate solution of the problem under consideration. An extensive overview of numerical schemes, methods and assumptions can be found in the books [2-6].

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