Theoretical natural frequencies and mode shapes for thin and thick curved pipes and toroidal shells

The finite element and differential quadrature methods are used to determine the natural frequencies and mode shapes of thin and thick curved pipes and toroidal shells. The methods of the study are extensively validated against previously published results. New results are given for standard thin and thick curved pipes and toroidal shells, that indicate the influence of bend angle, support conditions, and wall thickness on the free vibration characteristics. The new results serve as convenient benchmarks that should be of interest to researchers and design engineers.