This study considers the buckling of an elastic truncated conical shell having a meridional thickness expressed by an arbitrary function, subject to a uniform external pressure, which is a power function of time. At first, the fundamental relations and Donnell type dynamic buckling equation of an elastic conical shell with variable thickness have been obtained. Then, employing Galerkin's method, those equations have been reduced to a time-dependent differential equation with variable coefficients. Finally, applying the Ritz type variational method, the critical static and dynamic loads, the corresponding wave numbers, dynamic factor and critical stress impulse have been found analytically. Using the results, thus obtained, the effects of the thickness variations with a power or an exponential function, the variation of the semi-vertex angle and the variation of the power of time in the external pressure expression are studied through pertinent computations. It is observed that these factors have appreciable effects on the critical parameters of the problem in the heading. (C) 2003 Elsevier Ltd. All rights reserved.