Control of zero‐energy modes in 9‐node plane element

For plane stress/plane strain analysis, the 9-node quadrilateral element performs better than the corresponding 8-node element, especially for non-rectangular shapes. For improved element flexibility and lower computer cost, 2 × 2 quadrature is generally preferable to 3 × 3 quadrature. Unfortunately the 9-node element contains spurious zero-energy modes when under-integrated. A method is proposed to restrain these modes without significant loss of accuracy or added cost.