On the equilibrium of funicular polyhedral frames and convex polyhedral force diagrams

• Three-dimensional extension of graphic statics using polyhedral form and force diagrams.
• Defining the topological and geometrical relationships of 3D reciprocal diagrams.
• Design of compression and tension-only spatial structures with externally applied loads.
• Designing complex funicular spatial forms by aggregating convex force polyhedral cells.
• CAD implementation to manipulate the geometry of the force and explore its effects on the forms.

This paper presents a three-dimensional extension of graphic statics using polyhedral form and force diagrams for the design of compression-only and tension-only spatial structures with externally applied loads. It explains the concept of 3D structural reciprocity based on Rankine’s original proposition for the equilibrium of spatial frames. It provides a definition for polyhedral reciprocal form and force diagrams that allows including external forces and discusses their geometrical and topological characteristics. This paper furthermore provides a geometrical procedure for constructing a pair of reciprocal polyhedral diagrams from a given polyhedron representing either the form or force diagram of a structural system. Using this method, this paper furthermore suggests a design strategy for finding complex funicular spatial forms in pure compression (or tension), based on the construction of force diagrams through the aggregation of convex polyhedral cells. Finally, it discusses the effect of changes in the geometry of the force diagram on the geometry of the form diagram and the distribution of forces in it.