In this paper, we present a numerical analysis of the buckling of elastic thin structures by using Asymptotic Numerical Method (ANM) and Padé approximants. We consider the case of cylindrical shells submitted to external pressure and we show how the perturbation technique can be applied for conservative and non-conservative loadings. A shell formulation well adapted to thin structures and which uses the Enhanced Assumed Strain (EAS) concept and a three-dimensional constitutive law without condensation is considered in this study. Numerical examples are presented with comparison between conservative and non-conservative loading in term of critical pressure.