This research presents the nonlinear dynamical post-buckling analysis of an uncertain cylindrical shell. The proposed approach is adapted to the dynamical analysis of geometrically nonlinear structures submitted to a stochastic ground-based motion in the presence of both system parameter uncertainties and model uncertainties. The structure is modeled by a large finite element model using the 3D elasticity theory. The ground-based motion is represented by a Gaussian centered non-stationary second-order stochastic process. Then, a reduced-order basis is constructed using the POD (Proper Orthogonal Decomposition) analysis of a nonlinear static reference response [1] combined with selected linear eigenmodes of vibrations. The mean reduced-order nonlinear computational model is then explicitly constructed. A positive-definite operator involving the nonlinear stiffness of the structure is defined, allowing the nonparametric probabilistic approach to be used for constructing the uncertain nonlinear reduced-order computational model [2]. The dispersion parameter controlling the level of stiffness uncertainty is a scalar which has been previously experimentally identified in a nonlinear static context [3]. Finally, the instantaneous spectral density power of the dynamical response is analyzed in order to quantify the influence of both geometrical nonlinearities and random uncertainties on the stochastic dynamical response.