The reference coordinates and distortion measures for quadrilateral hybrid stress membrane element

✍ Scribed by K.-Y. Yuan; T. H. H. Pian

• Publisher: Springer
• Year: 1994
• Tongue: English
• Weight: 706 KB
• Volume: 14
• Category: Article
• ISSN: 0178-7675
• DOI: 10.1007/bf00350002

✦ Synopsis

The skew Cartesian coordinate system determined by the ]acobian of the isoparametric transformation evaluated at the origin can be shown to be a geodesic coordinate system at the origin. By using a theoryin differential geometry, inverse relations of the isoparametric coordinate transformation can be derived and expressed in terms of these geodesic coordinates. In the formulation of hybrid stress finite elements, it is suggested as a new strategy for assumed stresses that such coordinates be used as the reference coordinates. The theory described is exemplified by its applications to the 4-node hybrid stress membrane elements. A set of new distortion-measuring parameters for the quadrilateral element are also proposed based on such theory.

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InttofluctMn

In the developments of hybrid stress finite elements, different coordinate systems have been used to express the assumed stresses. In the early days, local Cartesian system, global Cartesian system, or local Cartesian system rotated to better approximate the isoparametric coordinate system have been adopted in order for stresses to satisfy the equilibrium conditions (Pian 1964;Tong and Pian 1969;Cook 1974;Spilker et al. 1981 ). After the Hellinger-Reissner principle be suggested as the formulation basis (Pian and Chen 198z), the isoparametric coordinates become currently employed. Punch and Atluri (1984, 1984a) have proposed, in their a posteriori centroidal stress concept, that the stresses be expressed in terms of the isoparametric coordinate variables but the tensor basis of stress be determined by the centroidal base vectors of the isoparametric coordinate system. Recently, Sze and Chow (1991) introduced further a skew rectilinear coordinate system, which is obtained by projecting the local Cartesian system onto the axes determined by the centroidal base vectors of the isoparametric coordinate system. In a development similar to the hybrid stress element, Robinson (1975, 1985) has also employed a skew Cartesian coordinate system in expressing the oblique, instead of the Cartesian, stresses in the construction of his stress-based membrane and plate bending elements.

Assumed stresses in the modern hybrid stess elements initially need not be chosen to satisfy the equilibrium conditions pointwise. Based on the Hellinger-Reissner principle, additional incompatible displacements may be used as the Lagrange multipliers to constrain them in a variational sense. In a recent study of the 4-node membrane element, however, Yuan et al. (1993) have found that by using a skew Cartesian coordinate system determined by the Jacobian of the mapping evaluated at the origin, the assumed stresses can be made to satisfy the equilibrium conditions pointwise. Consequently, either the Hellinger-Reissner or the complementary energy functional can be used as the basis of element formulation. The skew Cartesian coordinate system used in Yuan et al. (1993) is linearly related to the local Cartesian coordinate system through the Jacobian of the mapping evaluated at the origin, denoted by ]o. In the literature, this particular Jacobian ]0 has appeared in the studies of many incompatible and hybrid/mixed elements (for example,