Cables of a prescribed length under concentrated or distributed loads

A general analysis of inextensible perfectly flexible cables of negligible weight and a prescribed length, loaded by an arbitrary number of concentrated loads, or by a uniformly distributed load per unit horizontal length, is presented. A catenary cable under its own weight, with or without a concentrated force, is also considered. The sags under loads and tensions in all segments of the cable are determined. The results are derived in closed-form, apart from a numerical solution of a single nonlinear algebraic equation for a suitably introduced parameter in the last step of the analysis. The study complements the treatment of cables under concentrated forces available in undergraduate textbooks, which omit the consideration of cables whose length is prescribed in advance. The general analysis may be found appealing for a senior or graduate structural mechanics course, while its specialization to cables with two or three loads only is well suited for an introductory mechanics course. Expressions for a catenary cable subjected to a concentrated load at its midpoint are also derived, with only one nonlinear transcendental equation for the introduced load parameter needing to be solved numerically. Three student projects are suggested which address the existence and uniqueness of the cable configuration under a concentrated force with a specified sag and length of the cable, the lifting of two symmetrically hung loads by an upward force, and the determination of the deformed shape and tension in a catenary cable subjected to a concentrated force at an arbitrary point of the cable.