Since a knowledge of the free vibration characteristics of the structure is essential to obtain its dynamic response, free vibration analyses were performed on tanks and liquid-filled shells by many investigators. Tedesco et al [268] summarized the results of a comprehensive, computer based, numerical investigation of lateral free vibrations of cylindrical liquid storage tanks. Gupta et al ([74], [75], [76]) used variational principles to obtain a functional describing coupled oscillations between a linear elastic body and a liquid of small wave heights. They introduced a complementary Rayleigh's quotient to obtain the natural frequencies of circular cylindrical tanks partially filled with liquid and oscillating in an axisymmetric manner.
Tani et al [265] analyzed the linear free vibration of partially liquid-filled shells of revolution using the finite element method. They used thick isoparammetric shell elements for the axisymmetric shell and isoparammetric quadrilateral elements and line elements for the liquid interior and exterior surfaces, respectively. They reported that the static liquid pressure has a significant effect on the natural frequency of LNG tanks. Singhal and Williams [258] discussed the free vibrations of thick circular cylindrical shells and rings using the energy method. They carried out experimental investigations on several models in order to assess the validity of the analysis and reported that comparison with experimental values showed very close agreement.