Summary: We examine the properties of four-noded quadrilateral elements for which integration is carried out using low-order integration rules, based on one-point integration over subelements. The purpose is to identify those rules which lead to elements having the desirable properties of high coarse-mesh accuracy and stability in the incompressible limit. A two-point rule is investigated in detail, as is its counterpart for problems of incompressible media, in which the volumetric term is integrated using a one-point rule. Numerical results indicate that the new elements perform well in general, when compared with existing enhanced strain or equivalent elements, and appear to be particularly efficient in cases in which meshes are severely distorted.