Elastic buckling of a column under axial compression and constant lateral load restraints has been found to differ from the conventional bifurcation problems and thus requires better understanding. The buckling of an elastica resting on a smooth surface, loaded by a pair of axial forces and restrained by a constant force in the middle, acting perpendicular to the surface, is investigated. To this end, the mode-1, mode-2 and bifurcation post-buckling responses of the elastica are established, respectively.

The post-buckling mode is found to depend on the levels of the restraining and (more importantly) the axial force. For relatively small axial force, only mode-1 buckling is possible and the corresponding post-buckling behavior is characterized by a snap-through from the initially straight state to an unstable equilibrium configuration and then to a stable one. As the axial force exceeds a certain level (about four times the Euler buckling force, i.e., w/o restraining), mode-2 buckling becomes possible. The corresponding post-buckling configuration is unstable for relatively small additional axial force, but locally stable for relatively large additional axial force. It is further found that the unstable mode-2 post-buckling equilibrium configuration can switch to the globally stable mode-1 configuration automatically as a snap-through. Bifurcation becomes possible only when the axial force reaches an even higher level (about 8.18 times the Euler buckling force). The bifurcation mode corresponds to the buckling mode of a column of half-length with one end simply supported and the other clamped and thus the critical axial force is independent of the restraining force. Below this critical level of bifurcation force, energy barriers are found to exist for any post-buckling modes and so an external perturbation is needed to trigger buckling. The minimum possible buckling load corresponds to the mode-1 buckling and increases with the restraining force. The actual buckling load depends on when the system is perturbed and the amount of perturbation energy.

Imperfection sensitivity of the system is also investigated and the system is found to be very sensitive to the initial geometrical imperfections. A specific imperfection amplitude (termed as *characteristic imperfection amplitude*) is found to exist that can transform this unconventional buckling problem into a conventional bifurcation problem. The *characteristic imperfection amplitude* is found to depend on the properties of the structure (i.e., length and bending stiffness of the column) and the restraining force.

This study is expected to bring some insights to the elastic and plastic buckling of structures under lateral restraining loads, which are commonly encountered but involve not well-understood phenomena.


Buckling of an Elastica Under a Lateral Restraining Force

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Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2020/volume/84478

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