The primary resonance of a simply supported rotating composite shafts with geometrical nonlineary is studied. The composite shaft is modeled as a thin-walled Euler-Bernoulli beam. A variational-asymptotical method (VAM) applied to anisotropic thin-walled closed-cross-sectional beams is used to describe the displacement and strain fields of the composite shafts. The geometrical nonlineary is considered in the relationships of strain and displacement of the shaft. The nonlinear extensional-bending-torsional equations of motion for the composite shaft are derived by using the Hamilton principle. In order to emphatically study nonlinear transverse bending vibration, the effects of extensional and torsional deformations are ignored. By means of the method of multiple scales the approximation solution of primary resonance of transverse bending vibration is obtained. The Galerkin method is employed to reduce the governing equations to the ordinary differential equations. By using fourth-order Runge-Kutta method the time histories, phase diagrams and power spectrums are plotted. The study shows the effect of the external damping, ply angle, eccentricity, ratios of length over radius, ratios of radius over thickness and rotating speed on nonlinear dynamic behavior of the shaft. Specifically, the numerical simulation results show that the shaft exhibits the complex dynamic behavior including periodic, quasi-periodic and chaotic motion
