Finite elements for Mindlin and Kirchhoff plates based on a mixed variational principle

Abstract

AbstractQuadrilateral finite elements with five displacements per corner node built up from four triangular elements are developed by a discretization of the Mindlin plate bending theory and based on a modified form of the mixed variational principle for stresses and displacements. The modified principle separates the contribution of shear deformation by taking the transverse shear strains as free functions. It also gives discrete element governing equations in a canonical form. Similar elements are developed based on the potential energy formulation. The elements can be readily specialized to the Kirchhoff plate theory. The discrete shear strains are transformed back to normal‐fiber rotations for enforcing general boundary conditions. The discretization procedure uses one‐dimensional polynomials instead of two‐dimensional shape functions. The discretization satisfies continuity for w, and continuity for the transverse shear strains. The Mindlin elements are tested with soft and hard boundary conditions. They are naturally free of shear‐locking. They are also capable of reproducing the boundary layer that occurs with some soft boundary conditions. Mindlin elements results tend to those of the Kirchhoff theory as the thickness decreases. Extensive numerical results are obtained for square plates some of which are compared with exact results or results obtained by other methods.