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 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.1 By "admissible stress state" we mean an equilibrium state in which all bars are either in tension, or carrying no load. Thus, by "supporting", we mean supporting with all bars either in tension, or carrying no load.2 Notice that we solve this problem within the context of infinitesimal elasticity: examples of applications of the finite deformation theory to describe the geometric nonlinearity are given, for instance, by [6,18,17].

Are higher-gradient models also capable of predicting mechanical behavior in the case of wide-knit pantographic structures?

“…Finally, the problem that has been briefly presented in this paper can be investigated and many of its applications can be designed and tested. This requires accurate theoretical analysis: in the literature, several points of reference can be found [75][76][77][78][79][80][81][82][83][84][85].…”

On the existence of compression-only discrete force networks that support assigned sets of nodal forces

“…The limit analysis of such structures, subjected to a combination of fixed loads and variable forces that grow proportionally to a scalar multiplier λ, is usually performed through suitable generalizations of the well known master safe theorem of Heyman for masonry arches [7,8,9] The present work enriches this literature by presenting a linear programming (LP) procedure for the limit analysis of 'strut nets' formed by pairwise connections of the points of application of forces acting on truss models of masonry structures. The given procedure generalizes recent results dealing with cable web models [10]. The limiting values of the load scaling factor, which ensure existence of compression-only supporting truss structures, are identified with the lower bounds of the collapse multipliers of the variable forces [8,9].…”

Optimal vault problem -- form finding through 2D convex program

“…Indeed, according to Theorem 2.2 in [22] a positive matrix valued measure σ ∈ M(R 2 ; T 2 + ) whose divergence Div σ is supported on a closed set Γ must itself be supported in the convex hull of Γ, i.e. spt σ ⊂ conv(Γ).…”