On the use of energy method with element splitting to determine the stability of a constrained elastica

• The beam is discretized into a chain of rigid links connected by torsional springs.
• The contact region between beam and wall before and after superposing virtual displacement can be different.
• One-point and one-line-contact deformations are discussed in detail.

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 due to the presence of unilateral constraints. In this paper we propose an energy method for this purpose. The beam is decretized into a series of rigid links connected at the joints by torsional springs. 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. In order to accommodate this change of contact point we split the link near the boundary point of the contact region into two sub-links. 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 before and after superposing virtual displacement 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.