On the Use of Energy Method to Determine the Stability of a Constrained Elastica

A constrained elastica under edge thrust may have multiple static equilibrium positions. It is in general difficult to determine the stability of these equilibrium positions because of the presence of unilateral constraints. In this paper we propose an energy method for this purpose. To deal with the unilateral constraints in question, we allow the contact point on the elastica to be slightly different before and after superposing virtual displacements. It is noted that certain restrictions must be imposed on the contact point change in order for the total potential to be stationary if the equilibrium position is symmetric. After linearizing the constraint equations, the matrix associated with the second variation of the total potential can be established. From the eigenvalues of this matrix, the stability of the constrained elastica can be determined. One-point-contact and one-line-contact deformations are discussed in detail. Other deformation patterns can be analyzed in a similar manner. This energy method supplements the vibration method proposed earlier by the first author, in which the contact point is allowed to change during vibration.