The refined direct stiffness method for non-conforming (C1) element method developed by author in paper [1] is used to derive the quadriiateral plane element with drilling degrees of freedom (C0) which includes Allman's conforming displacement function and non-conforning function with four internal displacements. The behavior of the proposed element in respect of coordinate invariance, spurious zero energy modes, the ability to pass the patch test and rational formulation are obtained. A number of examples is used to show the implementation efficiency, accuracy and reliability.