A Complete Graphic Statics for Rigid-Jointed 3D Frames. Part 2: Homology of loops

Abstract

This paper extends graphic statics by describing the forces and moments in any 3D rigid-jointed frame structure in terms of cell complexes using homology theory of algebraic topology. Graphic statics provides a highly geometric way to represent the equilibrium in bar structures. Unlike traditional matrix-based linear structural analysis which represents a structure as a set of nodes connected by bars, graphic statics imagines that the bar network defines a variety of higher-dimensional objects (polygonal faces, polyhedral cells, polytopes). These objects are related to piecewise-linear stress functions, the liftings of Maxwell, Rankine or Cremona. The requirement for such stress-functions to be plane-faced places a major limitation on the set of structures that can be analysed, as in many structures the spaces between bars do not correspond to flat polygonal regions. The CW-complexes of cellular homology provide a far-reaching generalisation of geometric notions such as polygons, polyhedra and polytopes, and their use here removes the requirement that spaces between bars must be flat. Here we demonstrate how any frame structure with bar-like members can be decomposed into a union of closed loops, each consisting of a closed circuit of bars. For general structures these loops are general closed space curves which cannot be spanned by flat polygons. Using chains of CW-complexes makes the new theory applicable to a much richer set of structural geometries. Unlike most descriptions of graphic statics, this approach is not restricted to purely axial forces. Shear forces, bending moments and torsional moments are included naturally, as described in Part 1 of this sequence of papers. Later papers will extend the approach to displacements, rotations and Virtual Work, and will give greater detail on how the loop formalism may be lifted toinvolve higher dimensional CW-complexes.

Figures (11)

• Figure 1: a) A structure in 3D labelled as a graph, with nodes $x,y,z,\ldots$ and directed edges $a,b,c, \ldots$. b) A spanning tree for the graph.
• Figure 2: A structural loop and its dual loop. The extended body and stress spaces have four dimensions, the usual three ($\mathbf{i}, \mathbf{j}$ and $\mathbf{k}$) plus an extra one $\mathbf{h}$ for the stress function. There are six basis bivector planes in the 4D stress space. Projections of the dual loop onto the planes $\mathbf{i}\mathbf{j}$, $\mathbf{j}\mathbf{k}$ and $\mathbf{k}\mathbf{i}$ give the force components. Projections onto the planes $\mathbf{i}\mathbf{h}$, $\mathbf{j}\mathbf{h}$ and $\mathbf{k}\mathbf{h}$ give the moment components.
• Figure 3: Schematic illustrating a 3D frame structure with six nodes and nine bars being decomposed into four basis loops. The force diagram thus consists of four dual loops.
• Figure 4: Sign convention: each loop has a cut at a point in a non-tree bar, and the orientation of the loop is defined by arbitrarily selecting one of the faces at the cut as the positive face.
• Figure 5: A variety of frames with the K5 graph. Some are fully 3D and some fully 2D. Top right is the 3D coning of a 2D K4.