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 = 0
2v – e – 3 = 1

4. Exceptional Cases Can Be Exceptionally Efficient

5. Static-Kinematic DualityUj
Kinematics A U = V
Equilibrium B Q = P
Duality: A = BT
Four fundamental subspaces
Row space
Column space
Right and left nullspaces
Fundamental Theorem of Linear Algebra
Global
displacements
A U = V Ui
Local
deformation (stretch)
Local
element
Vij
Resultants act on node Pi
B Q = P
Qij
Must balance
loads on node

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. A B C
Uh1
Uh2 A B C
Pi1
Pi2

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, 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)

9. Geometry of Self-Stresses and MechanismsX 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

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

11. Maxwell 1864 Figure 5 and V Count: v = 6, 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)

12. 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)

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 r1 n τ σn

16. Restriction of Ψ(x,y) to Truss Equilibriumor
Case I
Case II r2 r1
Px = Q
 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
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 iff they have an origami mechanism

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: Tachi T., Design of infinitesimally and finitely flexible origami based on reciprocal figures. Journal for Geometry and Graphics, 2012; 16; Shai O. and Pennock G., Extension of graph theory to the duality between static systems and mechanisms. Journal of Mechanical Design, 2006; 128; Crapo H. and Whiteley W., Spaces of stresses, projections and parallel drawings for spherical polyhedra. Beitrage zur Algebra und Geometrie, 1994; 35; 2; Maxwell J. C., On reciprocal figures and diagrams of forces. Philosophical Magazine and Journal of Science, : 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; 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; 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

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