On the forces that cable webs under tension can support and how to design cable webs to channel stresses
Résumé
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 case. 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 distributing stress in desired ways.
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