Computing static behavior of flexible rectangular von Karman plates in fast and reliable way

摘要

In this paper, several modifications of the Bubnov-Galerkin method for studying flexible rectangular plates have been constructed and validated. This is to overcome the challenges of applying the existing approaches to a specific class of problems, bordering on accuracy and machine time efficiency. Von Karman's (elliptic) equations consisting of a system of nonlinear PDEs of order eight, with respect to two functions, i.e., the deflection and the stress functions, are considered. The problem is reduced by employing several modifications of the Bubnov-Galerkin method combined with the Fourier approach (separation of variables) to a system of nonlinear ODEs. The paper deals with many of such techniques and a procedure that refines their solutions based on known discrepancies. A theorem on the convergence of the proposed iterative processes is formulated and proved. The developed iterative methods are used to solve rectangular plates problem, subjected to distributed and local loads for several kinds of boundary conditions. A comparison of the obtained solutions by various methods is carried out. The optimal calculation method in terms of accuracy and minimum machine time computation is revealed. In addition, we show that the proposed novel and efficient approaches can be used to estimate the static behavior of plates of arbitrary geometry.