Graphic Statics, Graphical Kinematics, and the Airy Stress

FunctionToby Mitchell

SOM LLP, Chicago

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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

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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

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Exceptional Cases Can Be Exceptionally Efficient

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Static-Kinematic Duality

• 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

Local element

Ui

UjGlobaldisplacements

Vij

Local deformation (stretch)

A U = V

Qij

Resultants act on node

Pi

Must balance loads on node

B Q = P

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Fundamental Theorem of Linear AlgebraA U = V :

U = Uh + Up, UhUp= 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-stressesDual to incompatible deformations Vi : Vi

T A = 0.

A

B

CUh1

Uh2

A

B

CPi1

Pi2

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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

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Graphic Statics: One Diagram is Exceptional

Q

C

A

XR

P

ZY B

X

Z

YB

A

Q

R

P

C

Structure(Form Diagram)

Dual(Force Diagram)

Count:v = 5,e = 9 2v – e – 3 = -2

Indeterminate by two.

Count:v* = 6,e* = 9

2v* – e* – 3 = 0

Determinate, but must have a self-stress state to return the original form diagram as its reciprocal:

2v* - e* - 3 = m - s

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Geometry of Self-Stresses and Mechanisms

X

Z

YBA

Q

R

PC

ICPICQ

ICR

ICX,Y,Z

• 2v – e – 3 = 0 = m – s, s = 1 so m = 1: mechanism

• Moment equilibrium of triangles forces meet

X

Z

YBA

Q

R

PC

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Mechanisms as Design Degrees-of-Freedom

X

Z

YBA

Q

R

PC

ICPICQ

ICR

ICX,Y,Z

• Rotate 90 to get rescaling• Consistent offset = design DOFs: angles same

• Mechanism displacement vectors proportional to IC distance

X

Z

YBA

Q

R

PC

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Maxwell 1864 Figure 5 and V

Figure 5. Structure(Form Diagram)

Figure V. Dual(Force Diagram)

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

A

B

C

DE

FG

HI

JK

L

EJ

I

HL

G

K

F

C

D

A

B

Relative Centers

• Already a mechanism (AK-lines consistent)

• Additional mechanism from new AK-lines, in special position• EF – FG – GH – HE • BD – DI – IK – KB• AC – CL – LJ – JA

A

B

C

DE

FG

HI

JK

L

A

B

C

DE

FG

HI

JK

L

ICIK

ICBD

ICGH

ICEF

ICEH

ICFG

ICAC ICJLICAJ

ICDI

ICBK

ICCL

ICIK

ICBD

ICGH

ICEF

ICEH

ICFG

ICAC ICJLICAJ

ICDI

ICBK

ICCL

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

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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

Figure: Masaki Miki

Ψ(x,y)

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Discrete Stress Function from Continuum

• Integrate stress along a section cut path to obtain force

• Obtain force as jump in derivative

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r2

r1

n

τ

τ

n

σn

Restriction of Ψ(x,y) to Truss Equilibrium

Case I Case II

Ψ(x,y) on either side of bar must be planar

Force Q in bar is given by derivative jump perpendicular to bar 16

r2

r1

r2r1

or

r2

r1

Px = Q

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

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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

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Cable Net Optimization

• Clear application of self-stress

• Would prefer to have planar quadrilateral (PQ) faces

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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

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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

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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

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Thank you!

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ReferencesPellegrino 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.

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Offsets for Optimization(Parallel Redrawings)• 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)

Figures: Allan McRobie & Maria Konstantatiou

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Nodal Equilibrium is Built-In

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Compatibility of Planes

• Intersection of planes in point nontrivial for > 3 planes

• Corresponds to force equilibrium for point, moment equilibrium for hole

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