A Complete Graphic Statics for Rigid-Jointed 3D Frames. Part 3: Loops for Kinematics

TL;DR

This paper extends graphic statics to 3D frames by using a 4D loop formalism to describe kinematic behavior, integrating stress and displacement analysis within a topological framework.

Contribution

It introduces a 4D loop-based approach to represent kinematic variables and virtual work in 3D rigid-jointed frames, advancing the geometric and algebraic methods in graphic statics.

Findings

Unified 4D framework for stress and kinematics

Representation of displacements and rotations via projected areas in 4D

Application of algebraic homology to 3D structural analysis

Abstract

In Part 3 of this sequence of papers, the kinematic behaviour of 3D frame structures is described using the loop formalism that was developed in Part 2 to describe equilibrium. There, the notions of polygons, polyhedra and polytopes that form the geometric toolbox underlying graphic statics were replaced by the more general concept of CW-complexes from algebraic homology. The six components of the stress resultant acting on any cut face of a bar in a rigidly-jointed framework were represented by the oriented bivector areas of the six projections of a loop in a 4D-space, with three components representing the force and three components representing the moment. In this paper, projected areas of loops in 4D will represent kinematic variables, with three projected areas representing the displacement of a point on the frame, and three other projected areas representing the rotation of the structure at that point. The 4D setting for the theory consists of the usual three dimensions of physical space together with a fourth dimension for the stress function. Virtual Work then manifests as a top form (an oriented 4-volume) in this 4D setting, being the integral over the structure of the wedge product of bivectors representing the local equilibrium and kinematic variables.