Field-consistency analysis of the isoparametric eight-noded plate bending element

Abstract

The eight-node isoparametric plate bending element based on the serendipity shape functions behaves very poorly even after reduced integration of the shear strain energy. It has therefore been the subject of considerable study using various devices to improve it—mixed methods, enforcing of constraints, tensorial transformations, etc. In this paper, we shall proceed from the field-consistency paradigm to understand why the original element and even the element modified by the 2 * 2 Gaussian rule cannot achieve, consistently, the true shear strain constraints in the penalty limit of thin plate behaviour. We then derive the optimal shear strain definitions that leave the element free of all problems in the rectangular form, for most sets of practical boundary suppressions. From this, we next determine the optimum manner of co-ordinate transformation that preserves the true constraints even in the form of a general quadrilateral. This is achieved within the context of iso-P Jacobean transformations and without having to bring in tensorial or base vector definitions and transformations. This should be the simplest displacement type version of this element.