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

Abstract

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.

4‐node unsymmetric quadrilateral membrane element with drilling DOFs insensitive to severe mesh‐distortion

“…In this section, a series of classical benchmark problems are considered. The results obtained by the present new element US-Q4θ are compared with other 4-node quadrilateral elements available in literatures 12,20,[27][28][29]32,53,[61][62][63][64][65][66] to assess the capabilities of US-Q4θ.…”

“…27 Thus, some queries and discussions concerned with their convergences were raised. 28 Shi et al 29 developed Q4-like membrane elements at the base of the quasi-conforming element technique, 30,31 whose pioneer works can be traced back to Tang et al and Chen et al 32,33 The stiffness matrices of these elements in Shi et al 29 can be explicitly computed in Cartesian coordinates. Thus, they are very computationally efficient.…”

Displacement and stress analysis of laminated composite plates using an eight-node quasi-conforming solid-shell element

“…The weak forms of the generalized equilibrium and the compatibility equation for a continuum discretized by the finite element method take the form as [40,41] …”

“…Besides, the assumed strain field of element QCSS8 need be interpolated as simply as possible to ensure the highly computational efficiency, the factors to be free from locking should be taken into account. The element assumed strain of element QCSS8 is also inspired by the idea in the development of the element assumed strain fields for the four-node C 0 plate element and the four-node quadrilateral membrane element proposed by Shi et al [35,41]. Thus, the assumed strain field of element QCSS8 takes the form as…”

“…(11) can be completely reduced to the assumed strain filed of a four-node quadrilateral membrane element [41].…”