The problem of determining the deformation of a longitudinally corrugated, long cylindrical shell under external pressure is considered. The topics that are covered can be summarized as follows: the formulation of a boundary value problem for the incremental approach as a normal system of differential equations under appropriate boundary conditions, the determination of postbuckling behavior characteristics for cylindrical shells using the discrete orthogonalization method, and an analysis of deformation for both closed and open cylindrical shells. In particular, we consider the stability and postbuckling behavior of both isotropic and composite shells. The solution is based on the relationships for the cubic version of nonlinear Timoshenko-type shell theory. A comparison is made with the well-established quadratic version, as well as analytical solutions where applicable. The necessity for using more precise equations to examine the postbuckling behavior of shells is shown. Using this higher-order approach, it is possible to determine the postbuckling behavior with much greater accuracy.