The e!ects of large vibration amplitudes on the "rst and second coupled radial-circumferential mode shapes of isotropic circular cylindrical shells of in"nite length are examined. A theoretical model based on Hamilton's principle and spectral analysis developed previously for clamped}clamped beams and fully clamped rectangular plates is extended to shell type structures, reducing the large-amplitude free vibration problem to the solution of a set of non-linear algebraic equations. The transverse and circumferential displacements are assumed to be harmonic and expanded in the form of a "nite series of functions. The Donnel}Mushtarie shell theory, taking into account the coupling between extensional and #exural deformations is used. Then, the non-linear deformation energy is expressed by taking into account the non-linear term due to the considerable stretching of the middle surface of the shell induced by large de#ections. Tables of numerical results are given for the "rst and second non-linear modes, for a wide range of the vibration amplitude, which may be used for engineering purposes. For each value of the vibration amplitude considered, the corresponding contributions of the basic functions de"ning the non-linear transverse and circumferential displacement shapes are given, with the corresponding non-linear frequencies. Selected plots of mode shapes and bending stress distributions are presented, with an extensive discussion of the e!ects of non-linearity on the dynamic behaviour of shells.