Three-dimensional postbuckling and vibration of vertical half-loop under self-weight

The stability and vibration characteristics of a flexible and inextensible half-loop are investigated. The loop is fixed at two base points, which are separated by a specified distance, and is only subjected to gravity loading. If the length of the loop is sufficiently small, the loop stands upright in a vertical plane. If the length is increased past a critical value, the planar equilibrium shape becomes unstable and the loop droops to one side (i.e., laterally). This out-of-plane displacement may occur smoothly (supercritical bifurcation), or the loop may suddenly jump to a severely-drooped configuration (subcritical bifurcation), depending on the constitutive law. Linearly-elastic and softening materials are considered. Prebuckled and postbuckled equilibrium states are determined numerically with the use of a shooting method. Droop caused by an applied torsional moment is also analyzed. Then small vibrations about the prebuckled (planar) states are studied. Three basic types of vibration modes occur: in-plane, out-of-plane (symmetric), and twist about a vertical axis through the center of the loop. Experiments on a fiber-optic rod and a curtain wire are carried out to qualitatively verify the numerical results for both types of constitutive laws. © 2004 Elsevier Ltd. All rights reserved.