In many applications of structural engineering, the following question arises: given a set of forces
f
1
,
f
2
, …,
f
N
applied at prescribed points
x
1
,
x
2
, …,
x
N
, 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
x
1
,
x
2
, …,
x
N
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
f
1
,
f
2
, …,
f
N
applied at points strictly within the convex hull of
x
1
,
x
2
, …,
x
N
. In three dimensions, we show that, by slightly perturbing
f
1
,
f
2
, …,
f
N
, 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.
Improved Convex Hull Algorithm Applied to Body Size Measurements
