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

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

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 = 0 2v – e – 3 = 1

Exceptional Cases Can Be Exceptionally Efficient

Graphic Statics: One Diagram is Exceptional Count: v* = 6, e* = 9  2v* – e* – 3 = 0 Determinate, P Count: v = 5, e = 9  2v – e – 3 = -2 Indeterminate by two. B R Y Z C B R A Y 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 Structure (Form Diagram) Dual (Force Diagram)

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

Maxwell’s Figure 5 and V (untangled) 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? 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 D H G E L C I I H K L D B J A C J K E A F G F B Figure 5. Structure (Form Diagram) Figure V. Dual (Force Diagram)

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

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

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-of-plane displacement U3 Slab yield lines, origami folding Plane-faced 3D meshes are self- stressable if and only if they have an origami mechanism

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

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