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

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.

Figures (15)

• Figure 1: a) A bar subject to end forces and moments. b) A rigid body motion.
• Figure 2: a) The wedge product of general vectors $\mathbf{u}$ and $\mathbf{v}$. b) The unit bivector $\mathbf{e}_1\mathbf{e}_2$
• Figure 3: The loop representation for: a) equilibrium; b) kinematics.
• Figure 4: a) A loop of a simple moment-resisting frame structure and its dual force loop whose six projections give the components of the total stress resultant $\mathbf{R}$ acting on any cut face of the structural loop. The total moment includes the contribution from the force acting at a lever arm about the origin. By elementary equilibrium, this is the same at all positive cut faces. b) Given a positive cut face on the structural loop, an adjusted force loop can be defined whose projections give the local stress resultant where the moments are the local bending and torsional moments about the cut face. These moments vary around the structural loop. Dual to any such local force loop a generalised displacement loop $\mathbf{U}$ can be defined whose six projections give the rotation and displacement of the structure at the cut.
• Figure 5: a) A beam subject to applied end shears and end moments. b) The form diagram is an oriented loop created by connecting the beam ends with an arbitrary return path. The choice of positive face at cut J defines the loop orientation. c) Traditionally the bending moment diagram (BMD) is drawn in the plane of bending, $\mathbf{e}_1\mathbf{e}_2$ here. d) The internal bending moment $M(x)$ varies along the beam, and is plotted as a vector in the direction $\mathbf{e}_3$ normal to the plane of the frame. This is perpendicular to the traditional BMD representation, but accords with the usual vector notation for a moment.