SUMMARYAn oblique (skew) co-ordinate system defined at the origin of the isoparametric co-ordinates is introduced as the reference co-ordinate system for assumed stresses in the formulation of a 4-node hybrid membrane element. The initially uncoupled stresses in these reference co-ordinates can be constrained to satisfy the equilibrium conditions pointwise by using the procedure of Pian and Wu', when suitable incompatible displacements are introduced. Such strategy allows the element stiffness to be formulated by using either the Hellinger-Reissner or the complementary energy functional. Numerical results show that the performances of three elements, constructed by using different sets of incompatible displacements in this paper, are practically identical and are all of slightly higher accuracy than that of the Pian-Sumihara element.'

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By using the theory of geodesics in differential geometry. inverse relations of the mapping for 8-node quadrilaterals with curved boundaries are derived and expressed in terms of the element geodesic coordinates defined at the local origin. The parameters in the expressions of the geodesic co-ordinates in terms of the isoparametric co-ordinates are suggestcd to be the distortion measures of the element. The mathematical meanings of these parameters can be discussed and explained by using the coefficient vectors in the equations delining the isoparametric co-ordinate transformation. This method of defining the element distortion measures is consistent and general, and can be applied to any other two-or three-dimensional isoparametric finite elements.