The paper begins by noting the development and recent success of membrane elements which include in-plane nodal rotations as degrees of freedom. Particular attention is given to Allman’s solution [D. J. Allman, ibid. 19, 1-8 (1984; Zbl 0548.73049)] for the three-noded triangular element and the extension of his method by Cook to four-noded quadrilaterals [D. Cook, ibid. 22, 1065-1067 (1986)]. An analysis of the number and type of internal degrees of freedom is used to point out certain deficiencies of Cook’s four-noded quadrilateral, including ambiguity in the status of one of the element’s two spurious modes, the incomplete representation of linear strain states, and the existence of locking for nearly incompressible materials. Three modifications are then proposed whose combined effect is to correct the deficiencies while preserving satisfaction of the \(C_ 0\) patch test. The modifications include least square smoothing of strains computed from displacement shape functions so as to retain only linear terms, the addition of two auxiliary strain terms whose adjoint effect is to enforce internal stress equilibrium, and spurious mode control.
Test results are described which show improved accuracy for the modified element rivaling, and in some cases exceeding, that of the eight-noded isoparametric with reduced order integration. The tests also show that in some cases accuracy tends to degenerate above certain threshold values of spurious mode control.