This paper deals with the elastoplastic buckling problem of thick cylindrical shells under axial compression. The main idea is to propose an analytical solution procedure which gives rise to reliable results in a straightforward and low time-consuming manner, for dimensioning purposes. Thick shells are thus considered as they are known to be less imperfection-sensitive than thin ones. First, experimental compression tests are performed thanks to a specific experimental set-up, which is designed so as to prevent again, as much as possible, from the imperfections due to boundary conditions. Then, numerical finite element computations are carried out, and it is shown that the choice of free edge boundary conditions (for which pre-critical deformations are supposed to be homogeneous) in the numerical model leads to the same critical load as obtained during the experimental tests. Next, the problem is solved analytically. The classical plastic buckling load, usually derived considering an homogeneous pre-critical state, is likely to agree with the numerical predictions in this particular case of free conditions, contrary to the cases of simply-supported or clamped edges actually displaying an heterogeneous pre-critical state. In return, the substantial thickness of the tubes considered experimentally makes it necessary to improve the well-known solution which appears to be in very bad agreement with the numerical/experimental critical values. On one hand, the issue of kinematics is discussed and, on the other hand, geometric non-linearities are taken into account in the model (pre-critical strains are no longer infinitesimal). Finally, the new analytical procedure derived in this paper shows a very good agreement with both numerical and experimental results.