In this study, the buckling of functionally graded (FG) truncated conical shells subjected to external pressures under mixed boundary conditions is investigated. The mixed boundary conditions are as follows: at one end of FGM truncated conical shell is a sleeve that prevents its longitudinal displacement and rotation, and the other end is a free support. The basic equations of FG truncated conical shells are derived using Donnell shell theory. To solve of this problem is used new approximation functions and basic equations are solved using the Galerkin method for the two cases of mixed boundary conditions. The results are compared and validated with the results available in the literature. The influences of truncated shell characteristics, material compositions and boundary conditions on the critical lateral and hydrostatic pressures are investigated. (C) 2014 Elsevier Ltd. All rights reserved.