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In this work the non-linear free vibrations of a simply supported cylindrical shell made of a functionally graded material are analyzed using Donnell and Sanders nonlinear shell theories, considering an internal fluid and thermal effects. The fluid is assumed to be incompressible, non-viscous and irrotational. Thus, it is possible to describe its effects by a velocity potential. First, to discretize the shell, the transversal displacement field, which is determined by a perturbation procedure and takes into account the coupling between non-linear vibration modes and the linear vibration mode, is derived. Then it is used, together with Donnell theory, to obtain analytically the compatible axial and circumferential displacement fields. These displacement fields satisfy the boundary and continuity conditions of the shell. Next, the obtained displacement field is assumed as an optimal modal solution for both Donnell and Sanders …