# The set of forces that ideal trusses, or wire webs, under tension can support

## TL;DR

This paper characterizes the sets of forces that ideal tension trusses or wire webs can support, providing explicit constructions and conditions in two dimensions, and discussing constraints in three dimensions.

## Contribution

It offers a complete solution for two-dimensional convex polygons, explicit construction methods for certain interior points, and extends understanding of tension support constraints in three dimensions.

## Key findings

- Force multiplets must satisfy torque conditions around vertices.
- Explicit construction procedures for tension trusses supporting given forces.
- Constraints on forces in three-dimensional tension trusses.

## Abstract

The problem of determining those multiplets of forces, or sets of force multiplets, acting at a set of points, such that there exists a truss structure, or wire web, that can support these force multiplets with all the elements of the truss or wire web being under tension, is considered. The two-dimensional problem where the points are at the vertices of a convex polygon is essentially solved: each multiplet of forces must be such that the net anticlockwise torque around any vertex of the forces summed over any number of consecutive points clockwise past the vertex must be non-negative; and one can find a truss structure that supports under tension, and only supports, those force multiplets in a convex polyhedron of force multiplets that is generated by a finite number of force multiplets each satisfying the torque condition. Progress is also made on the problem where only a subset of the points are at the vertices of a convex polygon, and the other points are inside. In particular, in the case where only one point is inside, an explicit procedure is described for constructing a suitable truss, if one exists. An alternative recipe to that provided by Guevara-Vasquez, Milton, and Onofrei (2011), based on earlier work of Camar Eddine and Seppecher (2003), is given for constructing a truss structure, with elements under either compression or tension, that supports an arbitrary collection of balanced forces at the vertices of a convex polygon. Finally some constraints are given on the forces that a three-dimension truss, or wire web, under tension must satisfy.