Nonlinear Behaviour of Open Thin-Walled Elastic Beams

Abstract

<p> A general, consistent, nonlinear theory for open thin-walled elastic beams is presented. The theory takes into account geometric nonlinearities caused by large rotation of the cross section of the beam. The nonlinear differential equations of deformation and response are derived by means of application of Hamilton's principle. It is found that the set of equations reduces to the results obtained by Cullimore and Gregory in the special cases of large uniform torsion of thin-walled members. A solution of a thin-walled beam, subjected to large non-uniform torsional deformation due to application of torques at the ends, is obtained. Comparison is made on the torque - rotation characteristics of a thin-walled beam subjected to large uniform torsion and large non-uniform torsion to show the effect of end constraint from warping.</p> <p> A set of nonlinear equations to study the stability of a thin-walled beam of open cross section, under axial loading (spatial stability) and lateral loading (lateral stability), is presented. Using the derived equations, the dynamic stability of thin-walled beams of symmetrical and monosymmetrical cross sections subjected to axial loads, is investigated. The regions of parametric instability, the steady state amplitudes of oscillations, once parametric instability takes place, and the non-steady state solutions, to show the growth of the parametric oscillations, are carried out.</p> <p> The effect of viscous damping on the steady state amplitude and the growth behaviour of the parametrically excited oscillations is shown. The dynamic stability of a thin-walled beam of symmetrical I section and a monosymmetrical split ring section are worked out in detail as examples.</p>