Contact Problems Involving a Buckled Elastica

Abstract

The classical problem of the buckling of a slender rod (an elastica) is reexamined numerically and analytically taking into account contact between different parts of the rod. The lowest mode of the clamped elastica and the third mode of the pinned elastica are treated. Other modes can be treated in a similar way. It is found that for each of these modes there is a compressive buckling load $P_b > 0$ and a contact load $P_c > P_b $. The rod remains straight when $P\leqq P_b $, the rod buckles without contact for $P_b < P < P_c $, and two points of the rod are in contact for $P\geqq P_c $. The contact points move toward the ends of the rod as P increases. In each case an asymptotic formula is obtained for the location of the contact points as $P \to \infty $. In both cases a similarity solution is found for the shape of the elastica between the two contact points. In the pinned case there is a range of loads just above $P_c $ for which the distance between the ends of the rod increases as the load increases. This indicates instability and the possibility of “snap through.”