Graphic Statics, Graphical Kinematics, and the Airy Stress Function

1. Graphic Statics, Graphical Kinematics, and the Airy Stress Function Toby Mitchell SOM LLP, Chicago

2. Graphic Statics • Historical root of mechanics • Graphical duality of form and forces • Equilibrium  closed polygon • Vertices map to faces • Edges parallel in dual • Edge length = force magnitude • Reciprocal figure pair: either could be a structure • Modern use: exceptional cases

3. Exceptional Cases • Conventional categories of statically in/determinate, kinematically loose are inadequate • Can have determinate structure with unexpected mechanism • Can have loose structure with unexpected self-stress state • Rank-deficient equilibrium and kinematic matrices • Special geometric condition 2v – e – 3 = 1 2v – e – 3 = 0

4. Exceptional Cases Can Be Exceptionally Efficient

5. Graphic Statics: One Diagram is Exceptional Count: v* = 6, e* = 9  2v* – e* – 3 = 0 Determinate, Count: v = 5, e = 9  2v – e – 3 = -2 Indeterminate by two. P R B Y Z C B A Y R X X but must have a self-stress state to return the original form diagram as its reciprocal: 2v* - e* - 3 = m - s A Q Q Z P C Dual (Force Diagram) Structure (Form Diagram)

6. Geometry of Self-Stresses and Mechanisms ICR • 2v – e – 3 = 0 = m – s, s = 1 so m = 1: mechanism • Moment equilibrium of triangles  forces meet R R B B A A Y Y X X ICX,Y,Z Q Q Z Z ICP ICQ P P C C

7. Maxwell’s Figure 5 and V (untangled) Count: v* = 8, e* = 12  2v* – 3 = 13 = e* = 12 Underdetermined with 1 mechanism. To have reciprocal, needs a self-stress state  by FTLA, must have 2 mechanisms Count: v = 6, e = 12  2v – 3 = 9 < e = 12 Indeterminate by three. First degree of indeterminacy gives scaling of dual diagram. What about other two? D H G E L C I I H K L D B J A C J K E A G F F B Figure V. Dual (Force Diagram) Figure 5. Structure (Form Diagram)

8. ICIK Relative Centers ICBD ICEF ICIK ICBD ICGH • Additional mechanism from new AK-lines, in special position • EF – FG – GH – HE • BD – DI – IK – KB • AC – CL – LJ – JA • Already a mechanism (AK-lines consistent) ICEF ICGH D E ICEH ICDI I H ICCL L ICAC ICJL ICAJ D J A C K E ICDI ICBK I G H ICFG ICCL ICEH F L ICAC ICJL ICAJ B J A C ICBK K G ICFG F B

9. Airy Stress Function • Airy function describes all self-stress states • Discrete stress function is special case • Self-stressed truss must correspond to projection of plane-faced (polyhedral) stress function • Derivation from continuum stress function is new

10. Out-of-Plane Rigid Plate Mechanism • Plane-faced 3D meshes are self-stressable if and only if they have an origami mechanism • Can lift geometry “out-of-page” if it has an Airy function • Adds duality between ψ and out-of-plane displacement U3 • Slab yield lines, origami folding Figure: Tomohiro Tachi

11. PQ Net Reciprocal = Asymptotic Net • Asymptotic net: Force diagram • Vertex stars planar • Local out-of-plane mechanism (Airy function) • PQ net: Form diagram • Quad edges planar • Local self-stress

12. Open Problems? • Reciprocal figures for 2-parametric-dimensional meshes in 3D space • Generalization of out-of-plane motion / Airy function to non-planar surfaces • Full characterization of in-plane linkage mechanisms in exceptional geometry cases