Free vibration analysis of rotating stiffened functionally graded graphene-platelet-reinforced composite toroidal shell segments with novel four-unknown refined theories

Abstract

In this work, novel four-unknown refined theories were used to evaluate the free vibration of rotating stiffened toroidal shell segments subjected to varying boundary conditions in thermal environments. The shell segments consist of a functionally graded graphene-platelet-reinforced composite (FG-GPLRC). The effective material properties of the composite were calculated using the modified Halpin–Tsai model and the mixture rule. The governing equations of motion for the shell were formulated within the novel four-unknown refined shell theory framework. The effects of centrifugal and Coriolis forces and the initial hoop tension resulting from rotation were all included. The Rayleigh–Ritz procedure and smeared stiffener technique were subsequently used to determine the natural frequencies of the shells with stiffeners. The advantages of the adopted shell theory result directly from the reduction of key unknowns without the need for the shear correction factor, and it can predict better results for FG-GPLRC structures. Finally, numerical examples were provided to validate the proposed solution and demonstrate the effects of four-unknown refined theories, material distribution patterns, boundary conditions, rotating speed, and temperature rise on the natural frequencies of toroidal shell segments.

Appendix

Additional results for discussion in Sects. 5.1 and 5.3 are given in Tables

Table 14 Convergence of normalized frequency parameters $\overline{\omega }$ of nonrotating simply supported FG-GPLRC toroidal shell segments

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Table 15 Convergence of normalized backward wave frequencies $\overline{\omega }_{b}$ of rotating simply supported FG-GPLRC toroidal shell segments

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Table 16 A comparison of normalized backward wave frequency parameters $\overline{\omega }_{b}$ of UD toroidal shell segments ($h/R_{2} = 1/5$)

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Table 17 A comparison of normalized forward wave frequency parameters $\overline{\omega }_{f}$ of UD toroidal shell segments ($h/R_{2} = 1/5$)

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Table 18 A comparison of normalized backward wave frequency parameters $\overline{\omega }_{b}$ of UD toroidal shell segments ($h/R_{2} = 1/10$)

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Table 19 A comparison of normalized forward wave frequency parameters $\overline{\omega }_{f}$ of UD toroidal shell segments ($h/R_{2} = 1/10$)

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Table 20 A comparison of normalized backward wave frequency parameters $\overline{\omega }_{b}$ of UD toroidal shell segments ($h/R_{2} = 1/500$)

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Table 21. A comparison of normalized forward wave frequency parameters $\overline{\omega }_{f}$ of UD toroidal shell segments ($h/R_{2} = 1/500$)

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Table 22 A comparison of normalized backward wave frequency parameters $\overline{\omega }_{b}$ of FG-X toroidal shell segments ($h/R_{2} = 1/500$)

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Table 23 A comparison of normalized forward wave frequency parameters $\overline{\omega }_{f}$ of FG-X toroidal shell segments ($h/R_{2} = 1/500$)

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Table 24 A comparison of normalized backward wave frequency parameters $\overline{\omega }_{b}$ of FG-O toroidal shell segments ($h/R_{2} = 1/500$)

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Table 25 A comparison of normalized forward wave frequency parameters $\overline{\omega }_{f}$ of FG-O toroidal shell segments ($h/R_{2} = 1/500$)

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