Treatment of hourglass patterns in low order finite element codes

Numerical codes which use a one‐point quadrature integration rule to calculate stiffness matrices for the 2‐D quadrilateral element and the 3‐D hexahedral element, produce matrices which are singular with respect to a number of displacement patterns, other than the rigid body patterns. In this paper an economical method is derived to remove this singularity and which also produces accurate flexural response. For rectilinear element geometry the method is equivalent to the incompatible model element of Wilson et al.⁷ For non‐rectilinear element geometry a slight modification of the scheme is required in order to assure that it passes the patch test. The method of this paper can also be used in finite difference codes which experience similar difficulties.