Stability and response of beams in post buckling

The objective of this research work is to investigate the stability and response of buckled beams when subjected to axial loads beyond their critical buckling loads. The author established a closed-form solution for the static post-buckling response as a function of the applied axial load. The static response, which represents the equilibrium position in the post-buckling domain, for each buckling mode is dynamically disturbed in order to examine its stability. The other task was to examine the nonlinear response of a stable buckled position, which was found to be only the first, due to a primary resonance of the first mode in the presence of internal resonances. The method of multiple scales is used to find an approximate analytical solution for the response when the first mode is externally excited by a primary resonance. The modulation equations governing the amplitude and phase of the first and second modes are obtained through the standard procedure of the method of multiple scales. It was shown that the second mode is nonlinearly coupled with the first mode. Moreover, the local stability of the equilibrium solutions, which represent a steady-state response, is investigated. Frequency-response curves for the amplitude of the first and second modes are presented. Several nonlinear dynamics phenomena have been identified, such as jumping, saddle-node and Hopf bifurcations.