Buckling of a twisted and compressed rod

We consider the problem of determining the stability boundary for an elastic rod under thrust and torsion. The constitutive equations of the rod are such that both shear of the cross-section and compressibility of the rod axis are considered. The stability boundary is determined from the bifurcation points of a single nonlinear second order differential equation that is obtained by using the first integrals of the equilibrium equations. The type of bifurcation is determined for parameter values. It is shown that the bifurcating branch is the branch with minimal energy. Finally, by using the first integral, the solution for one specific dependent variable is expressed in terms of elliptic integrals. The solution pertaining to the complete set of equilibrium equations is obtained by numerical integration