Three-dimensional graphic statics: Initial explorations with polyhedral form and force diagrams

Abstract

This article investigates how reciprocal form and force polyhedrons can be used to develop procedures for the design of three-dimensional trusses and funicular structures, analogous to the well-known techniques of graphic statics for two-dimensional structural systems. It demonstrates how global equilibrium of a system of forces can be established by constructing a closed force polyhedron, if the forces can be replaced by a resultant force alone, without a resultant couple. It also describes the three-dimensional equivalent of the "closing string," which is the basis in graphic statics for the construction of funicular solutions for given loads and support locations. Furthermore, it provides a procedure for constructing a constrained funicular form for a simple, determinate boundary condition. Finally, it discusses some of the difficulties involved with similar constructions and procedures for non-concurrent forces and in particular with those systems of forces that can only be replaced by a resultant force and couple.

Graphical method for the construction of pairs of 3D skew funiculars in equilibrium with coplanar loads and controlled supports' positions

“…The subdivision operation results in a redistribution of the internal force flow by branching original members of the structure with newly created members. This structural and geometric property has been illustrated in the case of 2D structures with nodes connecting at least three branches, 17 and in the case of 3D structures with nodes connecting at least four branches 11 …”

“…Some of the methods used instead to address similar 3D funicular structures 2,9,10 share common principles with graphic statics, but couple these principles to numerical simulation tools. On the contrary, in particular cases, specific fully‐graphical procedures can be defined: for example, a graphical method using polyhedron‐based 3D graphic statics has been proposed for the construction of constrained funicular forms for a simple, determinate boundary condition 11,12 …”

“…The newly created nodes and branches are located in the plane of the original branches. Therefore, transformation of the form and form diagram of a funicular with only 3‐valent nodes involving only this kind of subdivision can equally be performed on the original 2D funicular or on the skew funiculars (Figure 7d), according to the procedure detailed by Akbarzadeh, Van Mele and Block 17 . Figure 8 shows a 2D funicular structure originating from successive subdivisions of the force diagram's cells by an inner point and Figure 9 illustrates how the subdivisions can equally be performed directly on the 3D skew funicular or performed on the original 2D funicular and then projected onto the 3D skew funicular.…”

On Visual-Mechanical Synthesis in Shell Structures

“…In digital computation, the shell has been explored in poly-dimensional representation. For example, in dome typology through the shape annealing method (Mitchell 1998;Shea and Cagan 1997) and the graphic-static with polyhedral form and force schema (Akbarzadeh 2016), where a line served as a vector in computing the equilibrium of a three-dimensional shape. On the other hand, rule-based iterations were able to enrich new design variations, as demonstrated in a shell's derivative geometry such as the two-dimensional truss via evolutionary algorithm (Mueller 2014), grammarbased Graphic-Static (Lee et al 2015), and fractal-based trusses (Rian 2014).…”

The cosmic spiderweb: equivalence of cosmic, architectural and origami tessellations

“…It tailed off in popularity in the early twentieth century, but has recently experienced a resurgence of interest, especially the study of global static equilibrium for both 2D and 3D structures (e.g. [30][31][32][33][34]).…”