The construction of advanced numerical methodologies for the prediction of the dynamical behavior of complex uncertain structures represents an important current challenge. In the present work, structures undergoing large displacements and high strains are investigated. Of particular interest is the analysis of the post-buckling dynamics of a cylindrical shell submitted to an horizontal seismic excitation. The nominal (i.e. without uncertainties) computational model of the cylindrical shell is large, i.e. comprising about 4 200 000 degrees of freedom, obtained with the finite element method using three-dimensional solid elements.Anonlinear reduced-order modeling is first carried out. Then, model uncertainties (on geometry, material properties, etc.) are introduced using probabilistic methods and the corresponding stochastic reduced-order nonlinear computational model is obtained. The identification of its parameters is next carried out using nonlinear static post-buckling data. Finally, a numerical nonlinear dynamic analysis of the uncertain shell is performed in a seismic context, for which the base of the cylindrical shell is submitted to a prescribed rigid shear displacement, modeled through a centered non-stationary Gaussian second-order stochastic process. The stochastic displacement field is then calculated and the effects of uncertainties and of nonlinearities are analyzed in details.