The obstacle problem in masonry structures and cable nets

TL;DR

This paper addresses the problem of designing force-supporting nets in masonry and cable structures that avoid obstacles, using linear and quadratic programming to identify feasible configurations in 2D.

Contribution

It introduces a method to determine feasible net configurations supporting prescribed forces while avoiding obstacles, including reactive forces and forces acting on segments.

Findings

Feasible regions are computed via linear programming for single obstacles.

The method extends to multiple obstacles and reactive forces.

Quadratic programming handles forces acting on line segments.

Abstract

We consider the problem of finding a net that supports prescribed forces applied at prescribed points, yet avoids certain obstacles, with all the elements of the net under compression (strut net) or under tension (cable web). In the case of masonry structures, for instance, this consists in finding a strut net that supports the forces, is contained within the physical structure, and avoids regions that may be not accessible due, for instance, to the presence of holes. We solve such a problem in the two-dimensional case, where the prescribed forces are applied at the vertices of a convex polygon, and we treat the cases of both single and multiple obstacles. By approximating the obstacles by polygonal regions, the task reduces to identifying the feasible domain in a linear programming problem. For a single obstacle we show how the region $\Gamma$ available to the obstacle can be enlarged as much as possible in the sense that there is no other strut net, having a region $\Gamma_1$ available to the obstacle with $\Gamma_1 \subset\Gamma$. The case where some of the forces are reactive, unprescribed but reacting to the other prescribed forces, is also treated. It again reduces to identifying the feasible domain in a linear programming problem. Finally, one may allow a subset of the reactive forces to each act not at a prescribed point, but rather at any point on a prescribed line segment. Then the task reduces to identifying the feasible domain in a quadratic programming problem.