Nonlinear Warping and Torsional Elongation in the Response of channel section beam

In last years, many paper have been devoted to nonlinear dynamics of 3D beams. In a previous paper [1] the Authors studied a nonlinear one-dimensional model of inextensional, shear undeformable, thin-walled beam with an open cross-section. Nonlinear in-plane and out-of-plane warping and torsional elongation effects were included in the model. By using a generalization of the Vlasov kinematical hypotheses, the nonlinear warping was described in terms of the flexural and torsional curvatures. The displacement field depends on three components only, two transversal translations of the shear center and the torsional rotation. By taking into account the order of magnitude of the various terms, the equations were simplified and the effect of symmetry properties has been also outlined. A discrete form of the equations was derived to study dynamic coupling phenomena in conditions of internal resonance. The results showed that warping and torsional elongation produce notable modifications in the response of the beam to harmonic excitation [2]. Unfortunately this model is very complex and the interpretation of the mechanical behavior of the system is very difficult. Aim of the present paper is to study more in detail the effects of nonlinear warping and torsional elongation that has been shown to play an important role in the nonlinear response. A preliminary study is developed to determine the different order of the kinematical quantities in a realistic beam, that will be a prototype for an experimental investigation, loaded by a static force at the free end in the direction orthogonal to the symmetry axis. A beam is considered characterized by the following nondimensional parameters : t/h=0.02, b/h=0.5, h/l=0.05, where t is the thickness of the section, b and h are the dimensions of the C cross section, being h orthogonal to the symmetry axis, and l the length of the cantilever beam. For this beam the ratio between torsional and flexural curvatures is about forty; this circumstance makes it possible to introduce a great simplification in the model developed in [1]. Through Hamilton principle, under the hypothesis of large torsional curvature and small flexural curvatures, three equations of motion are derived describing dynamics of inextensional and shear undeformable nonlinear 3D beam. An harmonic load is considered acting in the direction orthogonal to the symmetry axis and applied to the free end of the cantilever beam. A Galerkin discretization is performed by introducing the first three eigenfunctions and, by using multiple scale method and amplitude-modulation equations are obtained. Frequency-response and amplitude-load curves are evaluated to characterize the behaviour of the beam and highlight the nonlinear warping and torsional elongation contributions. A numerical investigation using a finite element model including geometrical nonlinearities, is performed to validate the mechanical model and an experimental test is also expected in the next future. 1. A. Di Egidio, A. Luongo, F. Vestroni, A nonlinear model for open cross-section thin-walled beams, Part I: Formulation, Int. J. of Non-Linear Mechanics, 2003, 38(7), 1067-1081 2. A. Di Egidio, A. Luongo, F. Vestroni, A nonlinear model for open cross-section thin-walled beams, Part II: Forced motion, Int. Journal of Non-Linear Mechanics, 2003, 38(7), 1083-1094