Accurate force evaluation with a simple, bi-linear, plate bending element

Abstract

A great deal of attention has been given to the development of simple C° continuous plate and shell elements based on the shear flexible theories for application to thick plates, sandwich or cellular plates and transversely isotropic or laminated plates. After considerable experimentation using unconventional approaches such as reduced integration, selective integration, mixed methods using discontinuous force fields, etc., it has been possible to develop simple displacement-type elements which can be reliably used. The stress recovery at nodes from such elements is often unreliable as the nodes are usually the points where strains or stresses are least accurate in the element domain. Further, nodal values can reflect severe oscillations at some difficult corner or edge conditions. In this paper, we focus attention on the optimal stress recovery from such an element. This is done after an interpretation of the displacement method as a procedure that obtains strains over the finite-element domain in a least-squares accurate fashion. If a shear flexible element is field-consistent, there are optimal locations at which bending moments and shear forces are accurate in a least-squares sense. These points are identified for the present element and used to study stresses in typical plate problems. Another difficulty faced is the rapid variation of twisting moments at free edges and corners of shear flexible plates and its influence on the shear forces at that edge. A related source of difficulty is the distinction made in Kirchhoff theory between shear forces and the effective shear reactions of that theory. The present study is seen to give accurate enough shear force and twisting moment predictions to allow one to draw the severe conclusion that the use of the Kirchhoff shear reaction at edges in classical plate theory is an ambiguous and unnecessary one and can be avoided. The findings confirm a recent suggestion that it may be more appropriate to have three (as introduced originally by Poisson) instead of the two boundary conditions (as modified by Kirchhoff) usually applied on the edge of a thin plate, especially if that edge is unsupported.