The axisymmetric response of nonlinearly elastic cylindrical shells subjected to dynamic axial loads is analysed by using an incremental formulation. The material elastic nonlinearity is modeled by the generalized Ramberg-Osgood representation. The time-dependent displacements of the shell are assumed to be governed by nonlinear equations of motion based on the von Karman-Donnell kinematic relations; moreover, both in-surface and out-of-surface inertia terms are included. The finite difference method with respect to the spatial coordinate and the Runge-Kutta method with respect to time are employed to derive a solution. Numerical results demonstrate the effect of the material nonlinearity on the deflections, stiffness matrices and dynamic buckling behavior of cylindrical shells.