Free Vibrations of Ring-Stiffened Toroidal Shells

Abstract

In this investigation, the free vibrations of ring-stiffene d toroidal shells are studied. The stiffening rings which are placed at meridional sections are treated as discrete members, each with different properties and unequally spaced. The shell is broken up into shell segments whose motion is studied on the basis of the Love-Reissner shell theory. Stiffness matrices for the shell segments are obtained by expanding the shell parameters in the meridional direction and integrating numerically in the circumferential direction. The stiffness matrices of the reinforcing rings are combined with those of the shell segments by imposing compatibility at the junctions. This results in a set of linear homogeneous algebraic equations which are solved by a fast converging iteration technique. A comparison of the frequencies of unstiffened toroidal shells shows good concurrence with available results. Plots are established showing the effect of the shell parameters on the frequencies of the unstiffened shells. The results indicate that the frequencies increase with increasing £ (thickness to radii ratio, h/R) and decreasing 77 (radii ratio, a/R). The five lowest frequencies and their corresponding mode shapes are determined for three different cases. When stiffening rings are added to the toroidal shells, the frequencies either remain the same or increase substantially. The changes being greater for shells with smaller rj. Moreover, the presence of additional modes is established as well as the coupling of circumferential harmonics.