A semi-analytical approach for nonlinear vibration of FG-CNTRC doubly curved shallow shells stiffened by FG-CNTRC stiffeners in thermal environment

Abstract

This paper investigates the geometrically nonlinear free and forced vibration behavior of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) doubly curved shallow shells stiffened by FG-CNTRC stiffeners under harmonic pressure loads and in uniform thermal environment. The formulation is developed based on the higher-order shear deformation theory (HSDT) combined with von Kármán-type nonlinear kinematics. An enhanced smeared stiffener approach is employed to incorporate the contribution of stiffeners into the shell's global stiffness. The governing equations are derived using the Lagrangian method, with the Rayleigh dissipation function accounting for energy loss. The harmonic balance method is adopted to determine the stress function, and the nonlinear equations of motion are solved numerically using the Runge–Kutta method. Parametric studies are conducted to assess the effects of CNT distribution, curvature, stiffener orientation, and thermal loading. The results provide key insights into the nonlinear dynamic response of stiffened FG-CNTRC shallow shells, serving as a useful reference for design applications in thermomechanical environments.