A paradigmatic system for non-classic interactive buckling

A non-classic double bifurcation of discrete static systems, occurring along a non trivial equilibrium path, is studied. It occurs when, by varying a parameter, a pair of eigenvalues of the tangent stiffness matrix first merge and then disappear. A paradigmatic 2 degrees of freedom system with some symmetry, consisting of an inverted extensible pendulum, so far studied in literature in the linear range only, is proposed. Exact nonlinear solutions are derived, showing the mechanism which leads two distinct bifurcated paths to locally touch each other at the double bifurcation point, and then to detach from the fundamental path. The detrimental effects of the interaction on stability are discussed. An asymptotic bifurcation analysis is also carried out, able to explain how a unique buckling mode existing at the double bifurcation is able to generate, in the perturbation process, two distinct deflections on close bifurcated paths.