The critical load and the buckling modes of a transversely isotropic circular cylindrical shell under axial compression are found. It is assumed that the curvilinear shell edge is free or weakly supported. In these cases, the buckling mode can be localized near this edge and the critical load can be lower than in the case of clamped edges. The transverse shear modulus is assumed to be small, so the solution is based on the Timoshenko-Reissner (TR) model. The deformations of the edge element are described using five general coordinates. Therefore, 2(5) = 32 possible combinations of boundary conditions are considered, depending on the restraint conditions imposed on these coordinates. In 15 cases, there is a chance of buckling near the shell edge and the corresponding behavior of the functions lambda(q, g) is investigated. The role of the fifth boundary condition in the TR model, which does not exist in the Kirchhoff-Love (KL) model is studied. It is shown that, if the boundary condition H = 0 holds and g -> 0, then the results based on the TR and KL models coincide. If the restraint phi(2) = 0 is imposed and g -> 0, then the TR model produces new results as compared with the KL model.
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