Reduction of excessive energy in the four-noded membrane quadrilateral element. Part I: Linear theory-compressible materials

The objective of the paper is to set forth, in a consistent manner, reasons for the appearance of excessive energy in the four-noded membrane quadrilateral element and to propose a formulation leading to simple and reliable elements that are less sensitive to distortions of the geometrical shape. By presenting the differential geometry, emphasis is placed upon those geometrical attributes which are inherently related to the quadrilateral. A modified version of the functional of Hu-Washizu is employed for the discretization. Appropriate approximations for the displacements/rotations are chosen and the physical meaning of the various parameters is identified. A systematic procedure is followed for the approximation of stress and strain. The convergence of the formulation is investigated by examining the inf–sup condition and applying the patch test. Also, results of numerical examples and comparisons with other elemental formulations are presented.