a b s t r a c tThis study focuses on the buckling of cylindrical shells with small thickness variations under external pressure. Asymptotic formulas in terms of the thickness non-uniformity parameter are derived by the combined perturbation and Bubnov-Galerkin methods. In addition to the analytic investigation based on the thin shell theory, a numerical analysis is also performed. Results from these formulas are discussed and compared with those obtained by other authors.