Characterization of truss structures under tension
Given a set of forces applied at prescribed points under what constraints on the forces does there exist a truss structure (or wire web) with all elements under tension that supports these forces? We prove that the existence of a web is guaranteed by a necessary and sufficient condition on the loading forces which corresponds to a finite dimensional linear programming problem. When such a problem has solution, then at least one web exists which is formed by the wires connecting the terminal points pairwise.
Figure 1. a) The web generated by the linear programming problem: it connects the terminal points pairwise. b) An equivalent web in which all the closed loops have been replaced by open webs, apart for one minimal loop.
Figure 2. If the points where the forces are applied form a convex polygon a), then the displacements corresponding to the extreme rays of the cone of admissible displacements provided by the linear programming problem are clam-shell movements. These are obtained by breaking the polygon into two non-overlapping subpolygons connected at one vertex, as in b). The clam-shell movement then consists of fixing one subpolygon, say the lower triangle, and infinitesimally rotating the other subpolygon anticlockwise, so the clam opens slightly.
Related publications
G. Bouchitté, O. Mattei, G.W. Milton, P. Seppecher. On the forces that cable webs under tension can support and how to design cable
webs to channel stresses (in preparation).