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Graphic Statics, Graphical Kinematics, and the Airy Stress Function

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

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Graphic Statics, Graphical Kinematics, and the Airy Stress Function

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  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

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