The four-noded quadrilateral with a 2 × 1 integration rule: Application to plates and other problems

Abstract

In the finite element analysis of plane problems, the Gauss integration scheme employed in conjunction with the standard four-noded quadrilateral is conventionally that which uses 4 (= 2 × 2) points. Underintegration in this case amounts to the use of a single integration point. A disadvantage of the use of the standard 2 × 2 scheme is the lack of accuracy for coarse meshes, as well as locking in the incompressible limit. The use of one-point integration, on the other hand, has the disadvantage that the resulting stiffness matrix is rank-deficient, and has to be stabilized. This work explores an intermediate possibility, viz. a two-point integration scheme, in the context of plane problems. This element has been introduced in an earlier work, and has been shown to give remarkably good results. That investigation is carried further here, with the procedure being applied to problems of axisymmetry, to plate problems, and to nonlinear problems. For plate problems in particular, it is shown that a minor modification to the integration rule for the Bathe-Dvorkin element leads to a marked improvement in results.