Graphic Statics Graphical Kinematics and the Airy Stress

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

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 2

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 2 v – e – 3 = 0 2 v – e – 3 = 1 • Rank-deficient equilibrium and kinematic matrices • Special geometric condition 3

Exceptional Cases Can Be Exceptionally Efficient 4

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

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

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

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

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 9

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

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 11

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