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Mathematical Problems in Engineering
Volume 2015, Article ID 198390, 14 pages
Research Article

Accurate, Efficient, and Robust Q4-Like Membrane Elements Formulated in Cartesian Coordinates Using the Quasi-Conforming Element Technique

Department of Mechanics, Tianjin University, Tianjin 300072, China

Received 19 September 2014; Revised 13 January 2015; Accepted 27 January 2015

Academic Editor: Chenfeng Li

Copyright © 2015 G. Shi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


By using the quasi-conforming element technique, two four-node quadrilateral membrane elements with 2 degrees of freedom at each node (Q4-like membrane element) are formulated in rectangular Cartesian coordinates. One of the four-node quadrilateral membrane elements is based on the assumed strain field with only five independent strain parameters and accounting for the Poisson effect explicitly. There are no independent internal parameters and numerical integration involved in the evaluation of the strain parameters in these four-node quadrilateral membrane elements, and their element stiffness matrices are computed explicitly in Cartesian coordinates. Consequently, the formulation of these four-node quadrilateral membrane elements is extremely simple, and the resulting elements are very computationally efficient. These two quasi-conforming quadrilateral membrane elements pass the patch test and are free from shear locking and insensitive to the element distortion in the range of practical application. The numerical result comparison with other four-node quadrilateral membrane elements, including Q4-like plane elements with drilling degrees of freedom and the Q6-type isoparametric elements with very complicated nonconforming modes, shows that the present quasi-conforming quadrilateral membrane elements are not only reliable and robust, but also very accurate in both displacement and stress evaluations in the analysis of practical plane elasticity problems.