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SUMMARYThis paper presents a general approach to predict the influence of geometric non-linearities on the free vibration of elastic, thin, orthotropic and non-uniform open cylindrical shells. The open shells are assumed to be freely simply supported along their curved edges and to have arbitrary straight edge boundary conditions. The method is a hybrid of finite element and classical thin shell theories. The solution is divided into two parts. In part one, the displacement functions are obtained from Sanders' linear shell theory and the mass and linear stiffness matrices are obtained by the finite element procedure. In part two, the modal coefficients derived from the Sanders-Koiter non-linear theory of thin shells are obtained for these displacement functions. Expressions for the second-and third-order non-linear stiffness matrices are then determined through the finite element method. The non-linear equation of motion is solved by the fourth-order Runge-Kutta numerical method. The linear and non-linear natural frequency variations are determined as a function of shell amplitudes for different cases. The results obtained reveal that the frequencies calculated by this method are in good agreement with those obtained by other authors.