An Analytical Investigation of the Trapeze Effect Acting on a Thin Flexible Ribbon

Abstract

This paper systematically explores the extensional–torsional coupling due to the trapeze effect acting on a thin flexible ribbon subjected to combined tension and torsion. Kinematic relationships as well as expressions for the restoring torque associated with this effect are analytically derived. Additionally, the locus of points about which the cross sections of a twisted ribbon under tension rotate is derived. These points, called torsional centers, are found to be coincident with the centroids of the axial stress field at each station along the ribbon. More generally, it is shown that when a flexible slender member is in tension, combined transverse forces must act at the centroid of the axial stress field to produce pure bending and no twist. As a result, the elastic axis (EA) of the member shifts from the locus of shear centers to the locus of centroids of the axial stress field. A numerical model is developed to investigate the effect of the position of the EA on the prediction of steady-state deformations and natural frequencies of a rotating ribbon with tip mass. By assuming the EA to be the locus of the shear centers, the tip twist is overpredicted by a factor of 2 for small twist angles, and up to 2.5 for large twist deformations. In addition, assuming the EA to be the locus of shear centers results in an error of up to 60% in the predicted natural frequencies at large twist angles.