Calculation of Constructions of Variable Thickness by Method of Steepest Descent

Abstract

The technique of calculation of structural elements of variable thickness is discussed in the article. Such constructive elements are described by differential equations with variable coefficients, for the implementation of which it is necessary to have a reliable calculation method that allows obtaining a fairly accurate solution. The most effective method for calculating such structures is the steepest descent method, developed by L.V. Kantorovich. Within the framework of this article, the idea of the method is set forth in the example of solving problems of bending a beam and a plate of variable thickness.
 A sequence is given for calculating the design of a variable MSD thickness using the example of a statically indeterminate beam, where the bending equation for a beam of constant cross section was used as the initial approximation. Then this method was generalized to a more complex two-dimensional construction - a plate of variable thickness. The problem of constructing the initial approximation for solving a partial differential equation was solved. As an example, we considered a square plate in plan, hinged on the contour. The results of the calculation were compared with the results obtained by the finite difference method.
 In solving specific problems by the method of steepest descent, it was revealed that it differs from direct methods, such as, for example, Ritz-Timoshenko, Bubnov-Galerkin, which consists in the fact that successive approximations in solving problems are not obtained in a priori chosen form, but in the form , determined by the problem itself. In the MSD, the solution is corrected qualitatively in the course of implementing the method, and when solving the problem by variational methods, we choose the approximating function and thereby set the solution configuration.
 The use of the MSD makes it possible to obtain finite formulas for determining the stress-strain state of structures of variable thickness, which will allow them to quickly implement their variant design.

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“…Граничные условия можно описать следующим образом: верхний торец свободен от нагрузок (момент и поперечная сила равны нулю), а нижний торец защемлен (перемещения срединной поверхности и угол поворота равны нулю). Геометрические соотношения Коши (связь деформаций и перемещений) можно представить, как [2]:…”