Sub-harmonic and super-harmonic resonances of spiral stiffened FG cylindrical panels of variable thickness embedded within a nonlinear elastic foundation

This research devotes to study the behaviors of sub- and super-harmonic resonances of spiral stiffened functionally graded (SSFG) variable thickness cylindrical panels (VTCP) exposed to external loading. The panel considered is rested on a nonlinear elastic foundation (NEF), which is composed of two linear foundations namely Pasternak and Winkler foundations, and a nonlinear foundation with cubic stiffness. The cylindrical panels of variable thickness are reinforced with internal spiral stiffeners. Both the cylindrical panels and spiral stiffeners are considered to be continuously graded along their thickness direction. To model the spiral stiffeners, the smeared stiffeners technique is utilized, and for deriving the discretized nonlinear governing equation of the SSFG-VTCP, the improved Donnell shell theory, von-Karman equation, and Galerkin's method are applied. In continuing, to examine the sub- and super-harmonic resonances, the method of multiple scales is implemented. Also, to validate the present results, in addition to comparing the present results with that of the previous research, the response of super-harmonic resonance, which is obtained theoretically, is compared with the 4th-order P-T method.