Natural Frequencies Analysis of Moderately-Thick and Thick Toroidal Shells

Toroidal shells have traditionally found application in the pressure vessel and piping industry. Early theoretical work on these shells focused on their static properties, but with their consideration for newer applications more emphasis has been placed on their vibration properties. A major factor affecting shell vibrations is their wall thickness. The thin shell theory is firmly established for shell vibrations in the literature and is used extensively in analytical solutions and numerical analysis. With more applications of toroidal shells employed in engineering practice, such as space vehicle, liquid storage structures and water tanks, the thickness of these structures cannot be considered as thinwalled. For such applications, the corresponding wall thickness has to be moderately thick or thick and the background theory is required using the shear deformation theory or elasticity theory. The present study extends the vibration analysis of toroidal shells based on thin shell theory to the analysis for moderately-thick and thick toroidal shells. A method based on a shear deformation (Timoshenko-Mindlin) shell theory is investigated to determine the free vibration characteristics of a moderately-thick or thick toroidal shell. The analysis is developed for a shell of revolution of arbitrary meridian, and then applied to a complete circular toroidal shell. A modal approach in the circumferential direction is used in the analysis and the results are obtained using the differential quadrature method (DQM). The results are validated by comparing with previously published results and good agreement is observed in the comparison of different results. General equations are developed which enable the prediction of the vibration characteristics of moderately-thick and thick shells of revolution of arbitrary meridian. The results demonstrate the application of the shear deformation theory in determining natural frequencies for toroidal shells.