Displacement-based finite elements with nodal integration for Reissner-Mindlin plates

An assumed-strain finite element technique is presented for shear-deformable (Reissner-Mindlin) plates. The weighted residual method (reminiscent of the strain-displacement functional) is used to enforce weakly the balance equation with the natural boundary condition and, separately, the kinematic equation (the strain-displacement relationship). The a priori satisfaction of the kinematic weighted residual serves as a condition from which strain-displacement operators are derived via nodal integration, for linear triangles, and quadrilaterals, and also for quadratic triangles. The degrees of freedom are only the primitive variables: transverse displacements and rotations at the nodes. A straightforward constraint count can partially explain the insensitivity of the resulting finite element models to locking in the thin-plate limit. We also construct an energy-based argument for the ability of the present formulation to converge to the correct deflections in the limit of the thickness approaching zero. Examples are used to illustrate the performance with particular attention to the sensitivity to element shape and shear locking.