An approach to improve the membrane behaviour of the four-node shell element with 24 degrees of freedom DKMQ24 proposed by Katili et al. (2015) is presented. This improvement is based on a different approximation of drilling rotations, based on Allman's shape functions. Further, the element formulation is enhanced by the use of selective reduced integration of shear terms and by the proportional scaling of the penalty constant to the shell thickness. Additionaly, nodal moment corrections for the approximation of distributed normal loads are proposed. This element, called DKMQ24$_2$+, was tested on nine standard benchmark problems for the case of linear elasticity. The element is free of shear and membrane locking and shows lower numerical errors for membrane and membrane-dominated problems.