In this study transverse shear forces are included, as well as rotational deformations, in the functional and in the new mixed finite element matrix. In deriving the new functional we use the Gateaux differential. Displacements, rotations, in-plane axial forces, in-plane shear force, transverse shear forces, bending moments, and torsional moments are the basic unknowns of the formulation. Loading conditions on the shell also can be distributed moments as well as distributed forces. A simple and efficient C0 quadrilateral isoparametric element is developed which has 4 x 13 degrees of freedom. The existence of the first-order derivatives in the functional provides the advantage of using linear shape functions. The accuracy of the S13 element is verified by applying the method to some test problems from the literature. Both the functional and the S13 shell element are original in the literature. The suggested cylindrical shell element is much more compatible with the stiffeners formulated using mixed finite elements from the literature.