A total Lagrangian elasticity formulation for the nonlinear free vibration of anisotropic beams

Abstract The large amplitude free vibration response of isotropic and anisotropic beams is considered using a total Lagrangian description of the motion along with Ritz-based approximations to the dynamic statement of virtual work. Results are presented for both axial and bending modes and are compared with existing results in the literature using various formulations along with a finite element solution of the Euler–Bernoulli beam with Foppl-von Karman nonlinearity. Results from the present formulation include all nonlinear strain terms and give frequency ratios that are in excellent agreement with both Euler–Bernoulli and higher order beam theories for slender beams under small amplitudes but grow noticeably higher than those of other beam models as the beam becomes thick and the amplitude of vibration increases.