Shear-deformable thin-walled composite Beams in internal and external resonance

The non-linear dynamic response of thin-walled composite beams is analyzed considering the effect of shear deformation. The model is based on a small strain and large rotation and displacements theory, which is formulated through the adoption of a higher-order displacement field and takes into account shear flexibility (bending and warping shear). The beam is assumed to be in internal resonance conditions of the kind 2:3:1, so that quadratic, cubic and combination resonances occur. In the analysis of a weakly nonlinear continuous system, the Galerkin’s method is employed to express the problem in terms of generalized coordinates. Then, the perturbation method of multiple scales is applied to the reduced system in order to obtain the equations of amplitude and modulation. The equilibrium solution is governed by the modal coupling and experience a complex behavior composed by saddle–noddle and Hopf bifurcations. The results of the analysis show that the equilibrium solutions are influenced by the shear effect, when this effect is ignored the amplitude of vibration is reduced significantly, thus altering the dynamic response of the beam. This alteration can infer in an incorrect stability prediction of the periodic solutions.