An efficient quadrilateral finite element for Reissner-Mindlin plates

This paper focuses on designing an efficient quadrilateral finite element for the analysis of Reissner- Mindlin plates with the following features: (1) it has four nodes, with three degrees of freedom per node, and involves only compatible displacement functions, (2) it is locking-free, (3) it passes all the patch tests, (4) it is readily implementable into existing finite element codes, being the resultant discrete equations in the standard form of assumed displacement finite elements. With this in mind, the following steps have been taken. Motivated by the good results obtained using mixed formulations (see for example the recent paper by Zhang et al. (2004)), a mixed variational approach involving stresses and displacements as independent variables is adopted. Stress parameters are eliminated at the element level, so that requirements (1) and (4) can be met. However, in order to fulfill also requirements (2) and (3), special attention is devoted to the selection of displacement and stress approximations. As regard displacements, the so called linked interpolation for the transverse displacement field (Tessler & Hughes (1983), Auricchio & Taylor (1994)) is employed. The use of linking functions goes towards avoiding locking effects in thin-plate situations but does not suffice, in general, to remove such deficiency, since it accommodates the lack of consistency of the shear strains on the element boundary but not in its interior. Here, this drawback is circumvented by resorting to the hybrid stress approach. Assumed stresses are a priori constrained to satisfy the equilibrium equations within each element, so that the Hellinger-Reissner functional reduces to a hybrid functional which involves displacements acting on element boundaries only. This makes fully effective the linked interpolation and permits to met both requirements (2) and (3). The stress approximation, chosen based on a rational approach in the spirit of the work by Yuan et al. (1993), is coordinate invariant, has the minimum number of stress modes and has been proved to yield a rank sufficient stiffness matrix. The resulting element is stable, accurate, relatively insensitive to geometry distortions, easily implementable into existing finite element codes and computationally efficient.