Slip-through of a heavy elastica on point supports

In this paper, we study the deformation and stability of a heavy elastica resting symmetrically on two frictionless point supports on the same horizontal level. The static analysis finds multiple equilibria when the distance of the two point supports is smaller than a certain value. In order to determine whether these equilibria are stable, a dynamic analysis is conducted to calculate their natural frequencies. In order to take into account the sliding between the elastica and the smooth point supports during vibration, a dynamic analysis based on an Eulerian description is adopted. It is found that stable equilibrium can exist only when the half support span a is between two limits a min ( s ) and a max ( s ) . This range depends on a dimensionless ratio between the weight density and the flexural rigidity of the elastica. When an a between a min ( s ) and a max ( s ) decreases and approaches a min ( s ) , the elastica will slip away from the side. On the other hand, when an a between a min ( s ) and a max ( s ) increases and approaches a max ( s ) , the elastica will slip through between the two supports.