Analytical study of the elastoplastic buckling of conical shells under external pressure

Pressure vessels are traditionally made up of cylindrical shells and hemispherical or ellipsoidal ends, but in some cases, conical sections are also present so as to ensure the transition between cylindrical sections of different radii. The buckling phenomenon is one of the main failure mode of such pressure equipments, due to the thinness of the components and the compressive stresses commonly undergone throughout standard loads like external pressure, and thus an essential dimensioning factor. If the buckling behavior of cylindrical and spherical shells has been widely investigated in the literature, the specific case of conical shells has received much less attention, all the more so in plasticity. Therefore, the present paper aims to address the problem of elastoplastic buckling of a conical shell under external pressure in an analytical way. This study is based on the plastic bifurcation theory and relies on the simplest possible hypotheses in terms of kinematics, constitutive law and boundary conditions. However, in absence of closed-form expressions, approximate solutions for the critical pressure are sought, based on the choice of appropriate shape functions in the framework of the Rayleigh-Ritz method. Unlike the case of cylindrical or spherical shells under external pressure which display uniform pre-critical stress states, the stress field appears to be heterogeneous in the length direction of a conical shell, so that three scenarios may occur. A conical shell may buckle elastically, entirely in the plastic range, or in an intermediate situation where the shell is partially elastic and plastic at the critical time. The present analytical solution is validated against reference numerical results obtained through finite element computations, considering a wide range of geometric and material parameters so as to cover all three scenarios.