On the forces that cable webs under tension can support and how to design cable webs to channel stresses

Abstract

In many applications of structural engineering, the following question arises: given a set of forces f1, f2, . . . , fN applied at prescribed points x1, x2, . . . , xN, under what constraints on the forces does there exist a truss structure (or wire web) with all elements under tension that supports these forces? Here we provide answer to such a question for any configuration of the terminal points x1, x2, . . . , xN in the two- and three-dimensional cases. Specifically, the existence of a web is guaranteed by a necessary and sufficient condition on the loading which corresponds to a finite dimensional linear programming problem. In two dimensions, we show that any such web can be replaced by one in which there are at most P elementary loops, where elementary means that the loop cannot be subdivided into subloops, and where P is the number of forces f1, f2, . . . , fN applied at points strictly within the convex hull of x1, x2, . . . , xN. In three dimensions, we show that, by slightly perturbing f1, f2, . . . , fN, there exists a uniloadable web supporting this loading. Uniloadable means it supports this loading and all positive multiples of it, but not any other loading. Uniloadable webs provide a mechanism for channelling stress in desired ways.