Nuclear plant reliability depends directly on its component performance. The higher heat transfer performance of nuclear plant components often requires higher flow velocities through the shell and tube heat exchangers. So, these cylindrical structures are subjected to either axial or cross flow, while the excessive flow-induced vibrations, (which are a major cause of machinery downtime; fatigue failure and high noise), limit the performance of these structures. On the other hand, these shell components often experience large amplitude vibrations that are greater than the shell thickness. Therefore, the evaluation of complex vibrational behavior of these structures is highly desirable in the nuclear industry. A semi-analytical approach has been developed in the present theory to predict the geometrical non-linearity influence on the natural frequencies of anisotropic cylindrical shells conveying axial flow. Particular important in this study is to obtain the natural frequencies of the coupled system of the fluid-structure, taking into account the geometrical non-linearity of the structure, and also estimating the critical flow velocity at which the structure loses its stability. The displacement functions, mass and stiffness matrices, linear and non-linear ones, of the structure are obtained by exact analytical integration over a hybrid element developed in this work. Linear potential flow theory is applied to describe the fluid effect that leads to the inertial, centrifugal and Coriolis forces. Numerical results are given and compared with those of experiment and other theories to demonstrate the practical application of the present method.