Buckling of structures under lateral load restraint has been found to differ from the conventional bifurcation problems and cannot be handled with conventional analysis. Towards a better understanding of this type of unconventional buckling problems, this work investigates analytically and experimentally the upheaval buckling of an elastica resting on a smooth surface and restrained by a lateral constant force at mid-span. Analytical solutions for the Mode-1 and Mode-2 post-buckling responses and configurations are established based on finite deflection analysis. An energy barrier is found to exist and the snap-through buckling load depends on the restraining force and perturbation energy. The imperfection sensitivity of the system is different from the conventional buckling ones, and a characteristic imperfection is found to exist that can transform a snap-through instability into a bifurcation buckling.A set of force-restrained buckling experiments are conducted to validate the analytical solutions. The experimental setup consists of an elastic strip which is simply supported on both ends by the testing machine, a fixed support plate underneath the strip that prevents deflection of the strip in one-side, and a constant force (provided by&nbsp; a weight-and-pulley system) loaded at the mid-span of the strip. The strip is compressed under displacement-control. It is observed that the strip first shows bifurcation buckling in Mode-2 when the axial force reaches 8.18 times the Euler buckling force. When the displacement reaches a critical value, the axial force drops abruptly and the strip switches from Mode-2 buckling to Mode-1, lifting up the weight at the mid-span. To further investigate the unloading behavior of the strip, the axial displacement is then reduced. The strip is seen to remain in Mode-1 buckling with an increase of the axial force. When the axial displacement reduces to another critical value, the strip switches from Mode-1 to Mode-2 and the restraining force drives the deflection of mid-span back to zero.To investigate the effect of geometric imperfection, a group of slightly curved strips are also tested under the same experimental setup. The strips are first loaded by a weight at mid-span and then compressed axially. It is observed that for a given imperfection amplitude, the strip behaves differently depending on the weight. If the weight is too small, the strip deforms into a shape similar to Mode-1 buckling as the axial force and displacement monotonically increase. If the weight is too large, the behavior of the strip is similar to the no imperfection case, experiencing mode-switching during the loading and unloading. A critical restraining force is identified, under which the strip exhibits bifurcation Mode-1 buckling when the axial force reaches the Euler buckling force.These experiments support the analytical solutions developed for an elastica with or without geometric imperfection. This study contributes to a better understanding of load-restrained buckling problems and is expected to shed light on the elastoplastic buckling of plates and shells under lateral load restraints in general, and on the wrinkling of a thin sheet during cup-drawing in particular.

Buckling and Post-Buckling of an Elastica Under a Lateral  Restraining Force: Experimental Validation

Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2021/volume/85543

ASME 2021 International Mechanical Engineering Congress and Exposition (IMECE2021)

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