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 = 0
2v – e – 3 = 1

4. Exceptional Cases Can Be Exceptionally Efficient

5. 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)

6. 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

7. 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)

8. 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)

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
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

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