Coupled instabilities in a two-bar frame: a qualitative approach

A direct one-dimensional beam model is kinematical ly described by the axis displacement, the rotation of the cross-sections and an average measure of their warping. The mechanical power is introduced as a linear functional of the kinematic descriptors and their first derivatives: the mechanical actions natural ly result as duals of the former ones. By means of the balance between external and internal power, the local balance equations for the mechanical actions are obtained. Inner constraints of shear indeformability and of a linear relationship between twist and warping are assumed, and non-linear hyperelastic constitutive relations are formulated. Thus, field equations in terms of displacements are obtained, and the various possibilities of buckling in a two-bar frame are examined. The critical value of the load multiplier is found for both the in-plane (single) and the out-of-plane (coupled) bifurcations.