In this research, using the Gateaux differential, an efficient computational procedure is presented and a new functional with geometric and dynamic boundary conditions is obtained for orthotropic cylindrical shells. A mixed finite element (ORTHO36) is generated which is a conforming, rectangular (four-noded), isoparametric ''serendipity'' family element and has 4 x 9 degrees of freedom. The existence of first order derivatives in the functional provides the advantage of using linear shape functions. The formulation is applicable to orthotropic cylindrical shells with all kinds of boundary and loading conditions. The accuracy of the ORTHO36 element is verified by applying the method to some test problems which exist in the literature.