Coupled instabilities in thin-walled beams: a qualitative approach

A direct one-dimensional beam model is adopted. Kinematics is described by axis displacement, rigid rotation of the cross-section and an average measure of warping. Mechanical power is introduced as a linear functional of the kinematic descriptors and their first derivatives, hence mechanical actions naturally result as their duals. In particular, the bi-shear and bi-moment turn out to be quantities spending power on the warping and on its first derivative, respectively. Assuming as basic postulate the balance between external and internal power, local equilibrium equations for the mechanical actions are obtained. In addition to the standard inner constraint of shear indeformability, a linear relationship between twist and warping is assumed. To obtain field equations in terms of displacements, non-linear hyperelastic constitutive relations are formulated. Two coupled bifurcations for axially loaded beams are examined: in the first case no coupling occurs, in the second the beam can be sensitive to initial imperfections.