Hanging an Elastic Ring

A thin flexible elastic circular ring is hung at one point. The ring deforms due to its own weight. The problem depends on a non-dimensional parameter B representing the relative importance of density and length to rigidity. The heavy elastica equations are solved by perturbation for small B, by a quasi-Newton method for intermediate B, and by a homotopy method for large B. The approximate results show good agreement with numerical integration for B < 20.