Abstract
This paper describes some considerations on numerical integration schemes for stiffness matrices of isoparametric elements. In this paper, relation is investigated between dispacement models using reduced integration and mixed models based upon Hellinger-Reissner's principle and Herrmann's principle in the case of plane stress, plane strain and plate bending. The elements considered are four-node, eight-node and nine-node quadrilateral elements and three-node and six-node trianqular elements. Numerical integration schemes considered are exact integration, uniform reduced integration and selective reduced integration in which lower order integration is used only for terms of transverse shear strains and volumetric strains. Element matrices are formulated, baced upon the minimum potential energy principle for the displacement models and Hellinger-Reissner's principle and Herrmann's principle for the mixed models. From theoretical considerations, following conclusions are obtained. In general, the strain-nodal displacement relation equations of the displacement models and the mixed models have identical values at the reduced integration points. Thus, the stiffness matrices of the both models are ideutical, if they are evaluated by the same integration as reduced integration. These facts are proved theoretically and ascertained by algebraic calcurations of element matrices. The following relationships exist between the displacement models and the mixed models. (i) The displacement models using uniform reduced integration are equivalent to the mixed models based upon Hellinger-Reissner's principle, except the six-node element. (ii) The displacement models using selective reduced integration are equivalent to the mixed models based upon Herrmann's principle, but, for uniform integration the both models are not equivalent except the three-node element.
