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This study investigates the free vibration of rotating stiffened toroidal shell segments in thermal environments. The shell segments are made of a functionally graded graphene platelet reinforced composite (FG-GPLRC)—advanced nanocomposite. Functionally graded (FG)-X, FG-O, and uniform distribution-type graphene platelet distribution patterns are considered. The equations of motion of rotating stiffened FG-GPLRC toroidal shell segments are derived based on variants of Reddy's third-order shear deformation shell theory (TSDT) and the smeared-stiffener technique. Then, solutions are obtained using the Rayleigh–Ritz procedure. The effects of centrifugal and Coriolis forces and the initial hoop tension resulting from rotation are considered. Various numerical examples are produced to verify the implemented scheme and demonstrate the effects of material properties, rotating speed, temperature increment, boundary conditions, geometric parameters, and stiffeners on the natural frequencies of the shell. In addition, while an incomplete version of Reddy's TSDT has been used widely in many recent studies, the present study points out significant differences between that version and the complete version of Reddy's TSDT by numerical results and discussions. [ABSTRACT FROM AUTHOR]

This paper proposes a novel nanobar–substrate medium model for static and free vibration analyses of single-walled carbon nanotube (SWCNT) systems embedded in the elastic substrate medium. The modified strain-gradient elasticity theory is utilized to account for the material small-scale effect, while the Gurtin–Murdoch surface theory is employed to represent the surface energy effect. The Winkler foundation model is assigned to consider the interactive mechanism between the nanobar and its surrounding substrate medium. Hamilton's principle is used to consistently derive the system governing equation, initial conditions, and classical as well as non-classical boundary conditions. Two numerical simulations are employed to demonstrate the essence of the material small-scale effect, the surface energy effect, and the surrounding substrate medium on static and free vibration responses of single-walled carbon nanotube (SWCNT)–substrate medium systems. The simulation results show that the material small-scale effect, the surface energy effect, and the interaction between the substrate and the structure led to a system-stiffness enhancement both in static and free vibration analyses. [ABSTRACT FROM AUTHOR]