Assumed strain finite element for natural frequencies of bending plates

Abstract

Purpose This paper aims to describe the formulation of a new finite element by assuming the strain field rather than the displacement field and by using the Reissner–Mindlin plate theory for the free vibration analysis of bending plates. This quadrilateral element consists of four-nodes and twelve degrees of freedom. The suggested element is based on assumed functions of the strain field that satisfy the compatibility equation. Design/methodology/approach After the proposition of the new element, several numerical tests for plates with regular and distorted meshes are presented to assess the performance of the new element. In addition, a parametric study is carried out to analyze the effects of biaxial loads on the natural frequencies of square plates with various boundary conditions. Detailed discussions are proposed after each benchmark problem. Findings The formulated element has verified the shear locking test and passes the patch test. The obtained results from the developed element show an excellent accuracy and fast convergence, and the natural frequencies are in excellent agreement when compared with analytical and other available numerical solutions. Originality/value The present element is simple in its formulation and has been proven to be applicable to thin or thick plate situations with sufficient accuracy. This element with full integration is free from shear locking, however, the numerical results provided by the standard four-node plate element R4 element show locking phenomena in thin plates. In addition to these features, the imposition of the compatibility conditions and the rigid body modes allow obtaining a finite element with higher-order terms for displacements field, which can increase the performance of the finite elements.