A variant of the finite element method in elasticity theory (Q1899527)

The problem of analyzing the stress-strain states (sss) of structures is frequently solved using finite elements in displacements. Combined elements do not always possess sufficient accuracy, however, as applied to shell structures. Equilibrium elements based on the principle of minimum of additional energy and approximations of the stress fields exhibit good accuracy. The methods of combined and equilibrium elements have not, however, found broad development and application due to difficulties in their implementation. A limited number of studies are focused, as a rule, on plates under an external load. In this paper the author attempts to refine the equilibrium element method and to develop it for other effects. The efficiency of the method is checked for square rigidly fixed plate with Poisson's ratio \(\gamma =0.3\) under a uniformly distributed load \(P\) and force \(F\) concentrated at the centre. The solution is sought on the basis of Kirchhoff's theory. The computational results illustrate the high accuracy of equilibrium elements not only with respect to forces, but also to displacements. The direct stiffness method in equilibrium elements makes it possible to simplify significantly implementation of the equilibrium element method and to include equilibrium elements in libraries of geometrically combined elements. Good possibilities and prospects are therefore opened for equilibrium elements and two-sided analysis of the sss of structures.