Selective Reduced Integration of Four-Node Plane Element in Closed Form

Abstract

The exactly integrated four-node plane element is known to “lock” as elastic incompressible conditions are approached. The use of uniform reduced integration removes this problem but introduces another in the form of a zero-energy mode, which can result in “hourglass” deformations. A popular compromise is known as selective reduced integration (SRI), in which the shear contribution to stiffness is exactly integrated and the volumetric contribution is evaluated using reduced integration. This SRI approach is readily coded in a numerical algorithm but requires the evaluation of five stiffness matrix contributions per element. This paper describes a closed-form version of the SRI stiffness matrix that was generated with the help of computer algebra systems. It is shown that this “analytical” approach considerably reduces the central processing unit (CPU) time consumed during element integration.