deformation and stability of an elastica under a point force and constrained by a flat surface

this paper studies the deformation and stability of a pinned elastica under a point force moving quasi-statically from one end to the other. the elastica is constrained by a rigid plane wall containing the two ends. three types of equilibrium configurations can be found; they are non-contact, one-point contact, and one-line contact on the side. a vibration method is adopted to determine the stability of the calculated deformations. in order to take into account the variation of the contact region between the elastica and the plane wall during vibration, an eulerian version of the governing equations is adopted. it is found that all the point-contact deformations are unstable. on the other hand, there are two different mechanisms a line-contact deformation becomes unstable; one through a secondary buckling and the other through a limit-point bifurcation. in the secondary buckling, the length of the line-contact segment and the axial force satisfy the euler buckling criteria for a pinned-clamped column. on the other hand, when a line-contact deformation becomes unstable via a limit-point bifurcation, the axial force does not exceed the euler buckling load. the theoretical predictions are confirmed by experimental observations.