Nonlinear resonant dynamics of geometrically imperfect higher-order shear deformable functionally graded carbon-nanotube reinforced composite beams

This study aims at numerically analyzing the nonlinear resonant dynamics of geometrically imperfect higher-order shear deformable functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beams with various end conditions subjected to a harmonic transverse load. Introducing a generalized displacement field including various beam theories, employing Hamilton's principle and taking into account geometrical nonlinearity and initial imperfection, three nonlinear coupled equations and associated boundary expressions are obtained for geometrically imperfect FG-CNTRC beams. These equations formulate the longitudinal, transverse and rotational motions of FG-CNTRC beams. An efficient multistep numerical solution approach based on the generalized differential quadrature (GDQ) method, a numerical Galerkin-based scheme and time periodic discretization is employed to convert the time-dependent nonlinear partial differential equations (PDEs) into a Duffing-type nonlinear set of ordinary differential equations (ODEs) which can be solved via the pseudo arc-length continuation technique. Nonlinear resonant dynamics characteristics are illustrated in the form of frequency-response and force-response curves; highlighting the influences of initial geometrical imperfection, geometrical parameters, excitation frequency and boundary conditions.