Eigenvalue buckling analysis through differential equations built on the beam cross section

This study investigates the global and local buckling of prismatic beams using a semi-analytical approximation of the equilibrium equations, linearized with respect to a pre-stressed configuration. The cross section is discretized using finite elements, while the nodal displacement is modeled analytically along the beam axis. The resulting differential equation system enables tracking the dependence of general solutions on the pre-stress level. It is shown how above a critical stress threshold, some solutions become periodic. Buckling loads can be determined by assuming a periodic solution with a specific half wavelength, allowing for the solution of a simple eigenvalue problem without discretizing the beam along its length.