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 determinate (minimally rigid), statically indeterminate (rigid with overdetermined matrix), and kinematically loose (flexible) are inadequate • Can have determinate structure with unexpected mechanism • Can have flexible 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. Static-Kinematic Duality Uj A U = V Global displacements • Kinematics A U = V • Equilibrium BQ = P • Duality: A = BT • Four fundamental subspaces • Row space • Column space • Right and left nullspaces • Fundamental Theorem of Linear Algebra Ui Local deformation (stretch) Local element Vij Resultants act on node Pi B Q = P Must balance loads on node Qij

6. Fundamental Theorem of Linear Algebra A U = V : U = Uh + Up, UhUp= 0 where A Uh= 0, But A = BT UhT B = 0, Uh is dual to Pi : PiT B = 0, the mechanism-activating loads. Can repeat for B Qh= 0 self-stresses Dual to incompatible deformations Vi : ViT A = 0. Pi1 Uh1 A A C C B B Pi2 Uh2

7. Fundamental Theorem of Linear Algebra • Extended determinacy rule 2v – e – 3 = m – s includes rank-deficient cases • “Statically determinate” rank-deficient  self-stress and mechanism

8. 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)

9. 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

10. Mechanisms as Design Degrees-of-Freedom ICR • Rotate 90 to get rescaling • Consistent offset = design DOFs: angles same • Mechanism displacement vectors proportional to IC distance R R B B A A Y Y X X ICX,Y,Z Q Q Z Z ICP ICQ P P C C

11. Maxwell 1864 Figure 5 and V 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)

12. 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

13. Geometric Condition on Self-Stress • Maxwell 1864: 2D self-stressed truss is projection of 3D plane-faced (polyhedral) mesh • WHY? • If-and-only-if proof: Klein & Weighardt 1904 • Resemblance to Airy stress function noted, but lacked theoretical basis • Derive directly from continuum

14. The Airy Stress Function • Plane-stress Airy stress function • Identically satisfies equilibrium • Complete representation of continuum self-stress states •  discrete truss stress function should inherit completeness Ψ(x,y) Figure: Masaki Miki

15. Discrete Stress Function from Continuum • Integrate stress along a section cut path to obtain force • Obtain force as jump in derivative r2 τ σn n τ n r1

16. Restriction of Ψ(x,y) to Truss Equilibrium r2 r2 Case II Case I r1 Px = Q or r2 r1 r1  Force Q in bar is given by derivative jump perpendicular to bar  Ψ(x,y) on either side of bar must be planar

17. Explains Projective Condition • 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

18. Out-of-Plane Rigid Plate Mechanism • Plane-faced 3D meshes are self-stressable iff 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

19. Cable Net Optimization • Clear application of self-stress • Would prefer to have planar quadrilateral (PQ) faces

20. 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

21. Optimal PQ Cable Nets • Equal-stress net if reciprocal has equal edge lengths • Asymptotic net  planar dual • Can obtain family of optimal PQ cable nets from dual via offsets

22. Conclusions • Statics and kinematics are related by the Fundamental Theorem of Linear Algebra • The FTLA covers exceptional cases with “extra” mechanisms or self-stresses • These cases are crucial to graphic statics • The geometry of self-stressed 2D trusses is given by a plane-faced Airy stress function • This stress function is dual to an out-of-plane rigid plate infinitesimal motion • Fully stressed PQ cable nets are duals of equal-length asymptotic nets • Optimal nets can be explored via offsets

23. Thank you!

24. References Pellegrino S., Mechanics of kinematically indeterminate structures, PhD thesis, 1986; Cambridge University. Calladine, C. R., Buckminster Fuller’s “tensegrity” structures and Clerk Maxwell’s rules for the construction of stiff frames. International Journal of Solids and Structures 1978; 14: 161-172. Tachi T., Design of infinitesimally and finitely flexible origami based on reciprocal figures. Journal for Geometry and Graphics, 2012; 16; 223-234. Shai O. and Pennock G., Extension of graph theory to the duality between static systems and mechanisms. Journal of Mechanical Design, 2006; 128; 179-191. Crapo H. and Whiteley W., Spaces of stresses, projections and parallel drawings for spherical polyhedra. Beitrage zur Algebra und Geometrie, 1994; 35; 2; 259-281. Maxwell J. C., On reciprocal figures and diagrams of forces. Philosophical Magazine and Journal of Science, 1864. 26: 250-261. Baker W., McRobie A., Mitchell T. and Mazurek A., Mechanisms and states of self-stress of planar trusses using graphic statics, part I: introduction and background. Proceedings of the International Association for Shell and Spatial Structures (IASS), 2015, (this volume). Borcea C. and Streinu I., Liftings and stresses for planar periodic frameworks. Discrete and Computational Geometry, 2014; 53. Whiteley W., Convex polyhedral, Dirichlet tessellations, and spider webs. In Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination, 2013; Springer-Verlag. Fraternali F. and Carpentieri G., On the correspondence between 2D force networks and polyhedral stress functions. International Journal of Space Structures, 2014; 29; 145-159. Pottmann H., Liu Y., Wallner J., Bobenko A. and Wang W., Geometry of multi-layer freeform structures for architecture. ACM Transactions on Graphics (SIGGraph), 2007; 26. Van Mele T. and Block P., Algebraic graph statics. Computer-Aided Design, 2014; 53; 104-116. Maxwell J. C., On reciprocal diagrams in space and their relation to Airy’s functions of stress. McRobie A., Baker W., Mitchell T. and Konstantatou M., Mechanisms and states of self-stress of planar trusses using graphic statics, part III: applications and extensions. Proceedings of the International Association for Shell and Spatial Structures (IASS), 2015, (this volume). Klein F. and Wieghardt K., Über Spannungsflächen und reziproke Diagramme, mit besondere Berücksichtigung der Maxwellschen Arbeiten. Archiv der Mathematik und Physik, 1904; 8; 1-10 then 95-119.

25. Offsets for Optimization(Parallel Redrawings) Figures: Allan McRobie & Maria Konstantatiou • Offsets of reciprocal = design DOFs • Can keep structure fixed and offset dual to change forces • Keep forces fixed, change structure • Minimal-variable basis for optimization can be computed by singular value decomposition (SVD)

26. Nodal Equilibrium is Built-In

27. Compatibility of Planes • Intersection of planes in point nontrivial for > 3 planes • Corresponds to force equilibrium for point, moment equilibrium for hole