Vibration analysis of ring-stiffened conical–cylindrical–spherical shells based on a modified variational approach

In this paper, free vibration characteristics of conical–cylindrical–spherical shell combinations with ring stiffeners are investigated by using a modified variational method. Reissner–Naghdiʹs thin shell theory in conjunction with a multilevel partition technique, viz., stiffened shell combination, shell component and shell segment, is employed to formulate the theoretical model. The displacement fields of each shell segment are expressed as a product of orthogonal polynomials along the meridional direction and Fourier series along the circumferential direction. The ring stiffeners in shell combinations are treated as discrete elements. Convergence and comparison studies for both non-stiffened and stiffened conical–cylindrical–spherical shells with different boundary conditions (e.g., free, clamped and elastic supported boundary conditions) are carried out to verify the reliability and accuracy of the present solutions. Some selected mode shapes are illustrated to enhance the understanding of the research topic. It is found the present method exhibits stable and rapid convergence characteristics, and the present results, including the natural frequencies and the mode shapes, agree closely with those solutions obtained from the finite element analyses. The effects of the number and geometric dimensions of ring stiffeners on the natural frequencies of a submarine pressure hull are also investigated.