2014 ijss lachauer interactive equilibrium modelling 1398776148

MULTI-SCIENCE PUBLISHING CO. LTD.5 Wates Way, Brentwood, Essex CM15 9TB, United Kingdom

Reprinted from

INTERNATIONAL JOURNAL OF

SPACE STRUCTURESVolume 29 . Number 1 . 2014

Interactive EquilibriumModelling

byLorenz Lachauer and Philippe Block

Interactive EquilibriumModelling

Lorenz Lachauer1, * and Philippe Block2

1Research Assistant, Institute of Technology in Architecture, ETH Zurich, Wolfgang-Pauli-Strasse 15,8093 Zurich, Switzerland. [email protected]

2Assistant Professor, Institute of Technology in Architecture, ETH Zurich, Wolfgang-Pauli-Strasse 15,8093 Zurich, Switzerland. [email protected]

(Submitted on 14/06/2013, Reception of revised paper 29/11/2013, Accepted on 13/01/2014)ABSTRACT: This paper presents a novel method for computer-aidedequilibrium modelling of structures in early design stages. Based on the forcedensity method, an iterative procedure is developed that enables the generationof spatial kinematic pin-jointed structures that are in equilibrium close to agiven input geometry, while satisfying additional constraints on both geometryand forces. This method forms the core of an interactive form-finding processthat consists of alternating steps of modelling and computational optimization.In each modelling step, the user is able to modify geometry, topology, externalforces and constraints of the structure. In each optimization step, equilibriumis re-established while respecting the user-defined constraints. A prototype hasbeen implemented within an existing CAD software package, and threeexamples illustrate the use of the presented method, ranging from a playfulexploration of surprising shapes to the rationalization of structural geometry.The method allows to intuitively explore the formal freedom of spatialequilibrium shapes with mixed compression and tension forces, within hard,user-defined constraints. In conclusion, it is claimed that by providinginteractive equilibrium modelling methods, the design of new, surprisingspatial forms with efficient structural behaviour is facilitated.

Key Words: computer-aided design exploration, constrained form finding,force density method, strut-and-tie models, real-time structural design tools

International Journal of Space Structures Vol. 29 No. 1 2014 25

1. INTRODUCTIONArchitectural freeform design and construction, basedon powerful CAD software, has a tradition of morethan 15 years: the icon of freeform design, theGuggenheim museum in Bilbao, Spain by FrankGehry, opened in 1997. Since then, one of the mainresearch objectives in the field of computationaldesign has been focused on bridging the gap betweenformal design freedom and hard technical constraints.Such constraints mainly arise from limitations inconstruction, fabrication and structural behaviour.Strong advances have been made in the field ofgeometry rationalization and fabrication optimization[1], while the concepts for the integration of structuralconstraints remain fragmented. Recently, various

approaches in the conceptual framework ofmorphogenesis have been developed, combiningparametric models with FEM analysis modules andgenetic algorithms [2], or combining FEM withgradient-based optimization [3]. In this paper, a novelcomputational design method, rooted in the traditionof form finding using pin-jointed truss models, ispresented.

Historically, pin-jointed truss models have playedan important role in design processes of technicallyand aesthetically innovative buildings. Great structuraldesigners used graphic statics as well as physicalmodels for the generation equilibrium shapes. Graphicstatics has been used by, for example, Gustave Eiffel,Antoni Gaudí and Robert Maillart [4]. Form finding

*Corresponding author e-mail: [email protected]

based on physical models has been applied by, forexample, Christopher Wren, Antoni Gaudí, Frei Ottoand Heinz Isler [5]. Today, those methods are still usedoccasionally in early design stages; for example, JörgConzett used graphic statics for the design of thesecond Traversiner Steg in Grisons, Switzerland [6],and Laurent Ney used hanging models in the design ofa curved suspension bridge in Kortrijk, Belgium [7].

Since the early 1970s, various computationalmethods have been developed for form finding of purecompression or pure tension structures; the forcedensity method and dynamic relaxation, together withtheir non-linear extensions, constitute the fundamentalapproaches [8]. Thrust Network Analysis combinesconcepts of graphic statics with approaches from theforce density method for the analysis and design ofunreinforced masonry structures [9]. Recently, theforce density method has been extended for the formfinding of mixed compression and tension structures[10]. Kemmler mentions the statically-geometricallycoupled method for general form-finding purposes,which is internally used in the office Schlaich,Bergermann & Partner, without describing itstechnical details. A spatial arch bridge and cable-stayed stadium roof are presented as applications ofthe method [11].

Through the availability of powerful interactivemodelling environments, various computational form-finding techniques have been implemented asmodelling methods, addressing problems of real-timeuser interaction: Kilian proposed a structural modellingsystem based on particle spring systems [12], Rippmannet al. presented a design method for compression vaultsbased on Thrust Network Analysis [13], and Tachiproposed a design process for tensegrity structuresderived from the force density method [14].

This paper presents a novel computer-aidedmodelling approach for spatial pin-jointed structureswith mixed compression and tension forces.Furthermore, the method allows to directly modify theequilibrium form, and to impose constraints on bothinner forces and geometry during the design process.The paper is structured as follows: Section 2summarizes the concepts of graphic statics, therelevance of pin-jointed models for design andconstruction, and the basics of the force densitymethod; Section 3 describes the core form-findingmethod of the proposed design process in full detail;Section 4 illustrates a prototypical implementation ofthe method within the CAD software packageRhinoceros [15]; and, finally, Section 5 presents threeapplications of the method.

2. BACKGROUNDGraphic statics is an important historic reference of anintuitive approach to structural design for pin-jointedor truss models, with direct control of forces andgeometry. Strut-and-tie models represent in anexemplary fashion the scientific link between trussmodel and construction detail. The force densitymethod and its extensions are the foundation of thenon-linear computational form-finding methodpresented here.

2.1. Graphic statics

Graphic statics has been developed in the mid-19th

century as method for structural analysis and design byCulmann [16]. Based on the concepts of vectorequilibrium and reciprocal diagrams, the methodenables to graphically calculate the inner forces of aplanar truss structures by the construction of forcediagrams. Furthermore, the method allows to findefficient structural forms using a funicular polygon,considering constraints on forces and geometry. Thefree nodes of the funicular are constrained to the lines ofaction defined by the direction and position of theexternally applied load vectors; the force diagramallows to directly control the inner forces. Recently,different approaches have been developed to extend thedesign capabilities of graphic statics to spatial systems.For example, Laffranchi [17] proposed a method for thedetermination of spatial funicular polygons for thedesign of curved bridges by introducing planes asconstraints for the free nodes, Lachauer [18] combinedgraphic techniques with dynamic relaxation for spatialbridge design. Block [9] extended the concept of thefunicular polygon to funicular surface networks withfixed projection. As in graphic statics, the free nodes areconstrained to the lines of action of the load vectors, theforces are controlled directly through force diagrams.

2.2. Strut-and-tie modelling

Funicular polygons have wide applications in the earlydesign of efficient, bending-free structures, beyond thetypology of tension structures. Truss models oftenconstitute a close link between form and constructiondetail. Based on the conceptual framework ofplasticity theory, truss models are applied as strut-and-tie models [19] or stress-fields [20] in the design andconstructive detailing of reinforced concrete. Schlaichpresented the detailed design of a cast steel componentfor the anchorage of a bridge deck in concrete basedon a truss model [19]. For masonry structures, thrustlines and thrust networks allow for the design of thestone geometry of masonry vaults [9].

Interactive Equilibrium Modelling

26 International Journal of Space Structures Vol. 29 No. 1 2014

Lorenz Lachauer and Philippe Block


2.3. Force density method

The force density method has been developed in theearly 1970s by Linkwitz and Schek [21]. In its basicform, as described in Section 2 of [22], the forcedensity method allows the form finding of equilibriumnetworks with defined topology for a given set ofloads and support points. The key concept of themethod is the replacement of inner forces and lengthsof branches by force-length ratios, called forcedensities, in order to obtain a linear system ofequations with a unique solution for the unknowncoordinates of the free nodes.

For a given network with m branches and ns nodes,the connectivity of the branches is described by abranch-node matrix Cs with dimensions m × ns. Ifbranch K connects nodes i and j, then the element (k, i)of Cs equals +1, and element (k, j) −1; all otherelements are 0. The columns of the branch-nodematrix are ordered such that one can define two sub-matrices of Cs = [C Cf], C corresponding to the freenodes, and Cf corresponding to the support nodes (foran illustration of Cs, see [22]). Each free node isinterpreted as point i with coordinates (xi, yi, zi); thecoordinates of the fixed support points are (xfi, yfi, zfi).These coordinates constitute the free coordinatevectors x, y, z of length n and the fixed coordinatevectors xf, yf, zf of length nf, hence ns = n + nf.

The force densities are defined as the force-lengthratios , forming the vector q of length m. For a setof force densities q and nodal loads px, py, pz, thecoordinates x, y, z of the free nodes are uniquely definedand obtained by solving one system of linear equations.

Linkwitz and Schek furthermore describe severalways to extend the force density method usingiterative procedures in order to incorporate additionalconstraints, such as defined branch lengths or forcevalues [21–23]. In [23], two conceptually different,iterative procedures for the optimization of givennetwork geometries are distinguished: the“Newtonverfahren” (“Newton approach”) and the“Ausgleichungsansatz” (“Variational approach”).

In the “Newton approach”, the parameter space isconstituted by the coordinates of the free nodes; the forcedensity method is used to compute the network geometryatiteration t + 1 based on its previous state at iteration t:

(Δx, Δy, Δz) = gN (xt, yt, zt) (1)(xt+1, yt+1, zt+1) = (xt + Δx, yt + Δy, zt + Δz) (2)

In the “variational approach”, the parameter spaceis constituted by the force densities and the free node

coordinates; both force densities and networkgeometry at iteration t+1 are calculated based on theirprevious states at iteration t:

(Δq, Δx, Δy, Δz) = gV (qt, xt, yt, zt) (3)(qt+1,xt+1, yt+1, zt+1)= (qt + Δq, xt+ Δx, yt+ Δy, zt+ Δz) (4)

Both methods work well for the optimization ofnetwork geometries that are close to an equilibriumstate purely in compression or tension. In the nextsection, the latter approach is adopted for designpurposes, allowing to find equilibrium networks closeto a given input geometry that consist of acombination of compression and tension branches,while imposing additional constraints on form andforces.

3. METHODThe presented method is the technical core of aninteractive form-finding approach for pin-jointedstructures within a CAD environment. The designprocess consists of alternating steps of userinteraction with the model and computationaloptimization (Fig. 1). In each modelling step, the useris able to modify network topology and geometry, andexternal forces, or to apply constraints on forcedensities and the free nodes’ coordinates. After this,the structure is generally no longer in equilibrium, or,alternatively, the imposed constraints are not satisfied.The computational optimization method iterativelyminimizes residual forces through redistribution ofinner forces and through changes of the geometry,with respect to the user-defined constraints. Ingeneral, the optimization method converges to anequilibrium state (all residual forces are below thechosen tolerance) if the user-defined constraints arefeasible. This alternating process of modelling andoptimization is repeated until the equilibrium state ofthe model satisfies external constraints, such as forinstance functionality or spatial and aestheticqualities, at which stage the form-finding processterminates.

Inspired by the concepts of graphic statics,constraints can be imposed both on force densities(hence implicitly on the forces) as on the nodes’coordinates. Force density constraints are defined aslower and upper bounds qLB and qUB. For a set ofconstrained force densities q̂, the condition qLB ≤ q̂ ≤qUB is true. Each node that is not defined as supportnode, can either be geometrically free, constrained to aline or a plane, or fixed in 3D space.

q flii

i=

For a given threshold value ε, the computationaloptimization process works as follows:1. Calculate a set of constrained force densities q̂0

from the initial network geometry x0, y0, z0

(Section 3.1);2. Calculate the nodal residuals r and the gradient

(Δq̂, Δx, Δy, Δz) with respect to the constraints(Sections 3.2 and 3.3);

3. Update the force densities q and networkgeometry x, y, z; and

4. If, ||r|| > ε, return to step II), and if ||r|| ≤ ε, thealgorithm terminates

A necessary condition is that the model forms amechanism, and the support nodes can take reaction inx-, y- and z- direction, which, after Timoshenko [24],is the case if the following inequality is true:

m < 3n. (5)3.1. Initial step

First, the coordinate differences u, v, w per branch arecalculated:

(6)

Based on the vectors u, v, w, their diagonal matrices U, V, W are constructed. This allows to define theequilibrium matrix A with dimensions 3n × 3m:

Together with the vertically stacked load vector

(8)

the vector of nodal residuals r for a given set of mforcedensities q is then formulated as

r = Aq − p. (9)

The system of equations (9) has m unknowns and3n equations, and because of (5), it is thusoverdetermined. The goal is to find a set of initial forcedensities q0 that minimizes the initial nodal residualsr0 for the given initial geometry, which can beformulated as a linear least square problem:

(10)

for which the solution can be written analytically usingnormal equations:

(11)



u Cx C xv Cy C yw Cz C z

f f

f f

f f

= +

= +

= +

AC UC VC W

.T

T

T=

=pppp

,x

y

z

q min r min Aq p|| || || || ,0 0 2 0 2= = −

q A A A p( ) .T0 1 T= −

Figure 1. Flow chart of the form-finding process consisting of alternating user-driven modelling steps and computationaloptimization steps that re-establish equilibrium.

calculate r and(∆q,∆x,∆y,∆z)

ǁrǁ ≤ εresult satisfying?

yes

no

change model calculate q0

update q,x,y,z

no

yes

initial state

final state

computational optimizationuser-driven modelling

^

^

(7)



3.2. Constraints

Inspired by an approach for geometric modellingdeveloped by Bouaziz et al. [25], constraints areimplemented using projections. In the first step of theiterative method, the constraints on the force densities,given as lower and upper boundaries qLB and qUB, areimposed on q0 by projecting them on the boundaries ofthe constrained set, if at least one constraint is violated(Fig. 2). Analytically, the constrained set of initialforce densities q̂0 is defined as:

(12)

In each iteration step, the residual vector r iscomputed from q, x, y, z, i.e. Equations (6–9), and r isdecomposed element-wise in “unconstrained”components rU and “constrained” components rC, suchthat r = rU + rC.

The idea is to resolve the “free” components rU bychanging the geometry according to Δx, Δy, Δz, and toresolve the “constrained” components rC by alteringthe force densities according to Δq. In this section, thedecomposition r = rU + rC is explained. In Section 3.3,the computation of the gradient (Δq̂, Δx, Δy, Δz) fromrU and rC is shown.

Each node that is not a support node, can either befree, constrained to a line, constrained to a plane, orfixed in space. Initially, all geometric constraints have

to be satisfied by the model, thus a node that isconstrained to a plane has to lie on this plane beforethe optimization process starts. For a given node i,depending on its degree of constraint, thecorresponding residual ri is decomposed as follows:a. If node i is a free node, a global weighting factor

0 > α > 1 is introduced to balance change ofgeometry and change of inner forces, so rUi = αri,hence rCi = (1 – α)ri (Fig. 3a).

b. If node i is constrained to a line Γ, rUi is theorthogonal projection of the residual ri onto theline. Therefore, the projection matrix PΓ isformed based on the unit vector d of the line’sdirection: PΓ = ddT. Then, rUi = PΓri, and rCi = ri –PΓri (Fig. 3b).

c. If node i is constrained to a plane Ψ, rUi isobtainedby an orthogonal projection of theresidual ri onto the plane. Therefore, theprojection matrix PΨ is formed based on anorthonormal basis d1 and d2 (two orthogonal unitvectors lying within the plane Ψ): PΨ =[d1d2][d1d2]T. Then, rUi = PΨri, and rCi = ri – PΨri(Fig. 3c).

d. If node i is defined as a fixed node, the nodeshould not move at all during the form-findingprocess, so the “free” component vanishes, rUi = 0,hence rCi = ri (Fig. 3d).

3.3. Gradient

Based on the decomposition r = rU + rC described inthe previous section, the computation of the constraintgradient (Δq̂, Δx, Δy, Δz) is straightforward. The freecomponents rU are used to compute “allowed” changesof the nodal positions within the geometric constraintsusing a stiffness k:

(13)

Veenendaal and Block [8] have summarized avariety of different definitions of stiffness matricesused in form finding. Since this method is not countingon any material properties, only “geometric” stiffnessis taken into account. In order to reduce thecomputational cost, the stiffness k is implemented as alumped geometric stiffness vector, similar as proposedby Barnes [26]. Here, the following definition for k asunitized, lumped geometric stiffness is introduced:

(14)

qq q qq q q qq q q

ˆ, if, if, if

i

iLB

i iLB

i iLB

i iUB

iUB

iUB

i

0 =

<

≤ ≤

<

x k ry k rz k r

.xU

yU

zU

∆ =

∆ =

∆ =

kC L qC L q

,T

T=

Figure 2. The diagram shows the imposing of the lower andupper bounds qLB and qUB on the force densities q for a two-dimensional case. The red dots represent states of q that arenot within the bounds, and their projection bounded states

q̂ represented as green dots.

q1

q0

q1UB

q1LB

q0LB q0

UB

with element ki having a value of +1 if the sum offorces in the adjacent branches of node i is greater than0 (“tension-dominant” node), and −1 if the sum offorces in the adjacent branches of node i is smallerthan 0 (“compression-dominant” node). Figure 4illustrates the idea behind this definition for anunloaded node. For tension-dominant nodes, theresidual points towards a close equilibrium position;for compression-dominant nodes, the residual pointsaway from a close equilibrium position, so in this casethe direction of the residual is flipped by themultiplication with −1.

The matrix L is constructed from the branch lengthsalong its diagonal, and is calculated as

(15)The remaining “constrained” components rC of the

residual vector r are resolved with Δq, for which thefollowing system of equations is formulated:

rC = AΔq. (16)Since A has dimensions 3n × m, and assumption (5)

is true, the system of equations (16) is overdetermined,so Δq is computed as linear least-squareapproximation:

Δq = min ||AΔq − rC||2, (17)whose solution can be written analytically usingnormal equations:

Δq = (ATA)−1ATrC. (18)In order to impose the force density bounds qLB and

qUB during the iterative process, the bounded forcedensity differences Δq̂ are calculated as follows:

. (19)

3.4. Implementation

To set up an interactive computer-aided designprocess, a prototype has been implemented within theCAD software Rhinoceros [15], which is a tool,specifically developed for 3D freeform design. Thebuild-in scripting language IronPython [27] with theadditional open-source library NumPy [28] has beenused for the linear algebra computations. Linear leastsquare problems, as equations (10) and (17), have been



Figure 3. Constraint decomposition of residual ri if the node i is: (a) free node, (b) constrained to a line Γ, (c) constrained to aplane Ψ, and (d) fixed.

riU = 0

riC = ri

riU = α ri

riU

Γ

Ψ ri

C = (1 - α) ri

(a) (b) (c) (d)

ri

i

ri

i

ri

i

ri

i

riC

riU

riC

= + +L U U V V W W.T T T

Figure 4. If node i is: (a) “tension-dominant” or(b) “compression-dominant”, residual ri is pointing

towards, respectively away from, a close equilibriumstate. In the latter case, the residual is flipped by the

multiplication with −1.

ri

ki = -1ki = 1

(a) (b)

kri

ri

ii kri

ˆ

,

,

,

qq q q qq q q q

q q q q

ififif

i

iLB

i i iLB

i iLB

i iUB

iUB

i iUB

i

∆ =

− <

∆ ≤ ≤

− <



solved with the function numpy.linalg.lstsq. Theorthonormal basis of planes representing nodeconstraints is obtained by the scripting commandrhinoscriptsyntax.SurfaceFrame. For solving theiterative process, an explicit Runge–Kutta methodwith adaptive step size has been used, as described byKiusalaas [29].

A pin-jointed model with applied external forces isrepresented by a set of lines and points in the 3Ddrawing space (Fig. 5), constraining objects such aslines and planar surfaces are modelled on a separatelayer. Geometric constraints on the nodes are modelledwith points with colours corresponding to theirconstraining objects (e.g. the points coloured in cyanare constrained to the cyan plan in Figure 5). Forcedensity constraints are modelled with colouredbranches; a table relates branch colours to the lowerand upper boundaries qLB and qUB.

4. DESIGN EXAMPLESFor all examples presented below, the threshold valueis defined as ε = r0/100, and the global weightingfactor α is set to 0.5. In Table 1, the legend of thegraphic symbols and conventions used in Figures 6–8is given. 4.1. A bifurcating arch bridge

The first example illustrates a playful designexploration for a bifurcating arch bridge (Fig. 6). Theprocess starts with a very rough approximation of aspatial arch bridge, inputted as a straight deck and apolygonal arch, connected to the deck with hangers.The deck’s nodes are constrained to a horizontal plane;the bridge’s dead load is represented by a set of

vertical unit forces attached to the nodes. The forcedensities of deck and arch branches (coloured inmagenta) are constrained such that the deck is force tobe in tension and the arch in compression (Fig. 6a).

The first optimization step generates a model inequilibrium with an almost planar arch and S-shapeddeck (Fig. 6b). Through the imposing of an upperforce density bound of −2.5, the structural height ofthe arch is limited. A higher bound, for example, −1,would result in smaller compression forces and thus inan arch with higher rise. A spatial configuration isenforced, by assigning a lower bound to the tensionforce in the deck; otherwise a planar arch with verticalhangers and no forces in the deck would emerge.

In the next step, shown in Figure 6c, the designercopies and rotates the first four segments of the deckand the arch, using standard CAD modelling tools tocreate the intended bifurcation of the bridge. In orderto heighten the rise of the structure, the upper bound ofthe arch’s force densities is raised to −1. The secondoptimization step generates the final equilibrium formof the bridge (Fig. 6d).

4.2. A curved suspension bridge

The example in Subsection 4.1 has been a playful, notvery controlled design exploration. For example, thenodes of the bridge deck are no longer equally spacedin the final state (Fig. 6d). To obtain geometricallycontrolled equilibrium models, additional constraintshave to be imposed. The example in this sectiondemonstrates highly constrained form finding. It isinspired by Laffranchi [16], who states that anyloaded, spatial polygon in space can be balanced bytwo funiculars.

The design process starts with a freeform curve,defining the intended pathway of the bridge (Fig. 7a).This curve is subsequently discretized as polygon,forces are added as dead load, and constructive axesare introduced (dashed lines), defining the intendedlayout of the structure in plan (Fig. 7b). Two differentconfiguration as starting point of the form-findingprocess are explored (Fig. 7c and 7e).

In order to preserve the geometry of the intendedpathway and the structural axes, the position of thedeck nodes are fixed, and the nodes of the twofuniculars are constrained to planes that are verticallyextruded from the axes in both cases. Additionally,positive force density constraints are imposed on thepart of the structure that is supposed to act as tensioncable. In the first case (Fig. 7c), the form-findingprocess generates an equilibrium model with the deckin compression, horizontally balanced by a

Figure 5. Screenshot of the CAD model used for theexample presented in Section 4.1. Supports are modelled withwhite points, geometrically fixed nodes with points in cyan;planar constraints are encoded by points and planes with thesame colour; force density constraints are imposed on the

coloured branches.

compression arch, and vertically supported by atension funicular (Fig. 7d). In the second case(Fig. 7e), the process generates an equilibrium modelalso with the deck in compression, but nowhorizontally and vertically equilibrated by tensionfuniculars (Fig. 7f). In both cases, the horizontalfunicular in compression (Fig. 7d) and the horizontalfunicular in tension (Fig. 7f) are inducing almost the

same force in the deck. Hence, the other parts of thestructure, the deck and the vertical funicular, arealmost identical in inner forces and geometry.

4.3. A pre-stressed, cable-stayed roof

The examples shown in Subsections 4.1 and 4.2 arelinear structures. This subsection presents theapplication of the method to a two-dimensional



Figure 6. A spatial, bifurcating arch bridge: (a) Initial, user-defined input geometry (b) generated equilibrium geometry of thesingle arch: (c) the designer copies and rotates the first four segments of the deck and the arch; and (d) final form-found

equilibrium geometry of the bifurcating bridge.

q ϵ ]-∞; -2.5]

q ϵ [1; ∞[

q ϵ ]-∞; -1]

q ϵ [1; ∞[

(a) (b)

(c) (d)

Table 1. Legend of the graphic symbols and conventions used in Figures 6–8

Input model with constraints and Resulting constrained model in equilibriumexternal loadsbranch with force density bounds compression force, line weight proportional to

the square root of the forcebranch without force density bounds tension force, line weight proportional to the

square root of the forceexternal force external forcesupport node support nodeunsupported node constrained to position geometry of the input modelunsupported node constrained to planeconstraining plane



structure. Inspired by an article by Göppert and Stein[30], describing the principle of pre-stressed “spokedwheel” roof structures for sports stadiums, the aim inthis example is to design a pre-stressed wheel structure

based on a given freeform surface. This is not anobvious goal as the geometry of both rings depend oneach other. For this type of structures, an outercompression ring is connected to an inner tension ring

Figure 7. An example of highly constrained form finding of a bridge: (a) the intended path geometry, defined by a freeformcurve, and (b) the discretized path with dead load and structural axes (c) constrained model, with positive force density

constraints on the cable, and second funicular on the same side of the deck, and (d) the resulting equilibrium model with thedeck in compression, horizontally supported by a compression arch, and vertically supported by a tension funicular

(e) constrained model, with positive force density constraints on the cable, and second funicular on the opposite side of the deckand (f) resulting equilibrium model with the deck in compression and the two funiculars in tension.

(e) (f)

(d)(c)

(a) (b)

q ϵ [0; ∞[

q ϵ [0; ∞[

via radial cables The dominant loading case is self-stress. To create structural depth in the vertical cross-section, usually either the compression or the tensionring is doubled. The radial cables are then alsodoubled, forming radial cable girders with additional,connecting vertical cables.

The design process starts with a 3D freeform loopsurface, representing the intended geometry of the roof(Fig. 8a). In the next step, the surface is discretized inquadrilateral polygons (Fig. 8b). The edges of the

polygons are interpreted as branches. In the currentimplementation, it is not possible to run the methodwithout application of external forces, so one branch isreplaced by two forces of same magnitude andopposing direction, representing the pre-stressingforce. The structure does not have any structuralsupports (Fig. 8c). The form-finding process generatesan equilibrium model working as pre-stressed wheel,but the spatial shape of the initial structure collapses toa plane (Fig. 8d).



Figure 8. Modelling of a spoked-wheel roof structure: (a) the intended geometry of the roof defined by a freeform surface inspace, and (b) discretized surface as polygonal model (c) structural model without any supports, externally pre-stressed, and

(d) resulting equilibrium model collapsed to a planar structure (e) geometrically constrained input model with additional columnsand supports, and (f) the resulting equilibrium model (g) refined input model with doubled compression ring and radial cable

girders, and (h) final equilibrium model.

q ϵ [0.05; ∞[

prestress force

(g) (h)

(e) (f)

(c) (d)

(a) (b)



It is the designer’s intention to stay closer to theoriginal freeform surface, so the idea is to use columnsfor balancing the out-of-plane forces of the ring. Inorder to preserve the exact input geometry of the outersurface edge, the nodes of the ring are fixed in space(Fig. 8e). The resulting equilibrium model is closer tothe initial form, only the inner edge deforms into atension hoop (Fig. 8f).

Based on this geometry, a refined model is built upby the designer. The compression ring is doubled, andthe radial cables are replaced by two layers of cables,connected by short vertical elements (coloured inmagenta). The force densities of these vertical elementsare bounded to a small positive value in order to inducea minimum tension pre-stress (Fig. 8g). Based on theequilibrium geometry yielding from the refined model(Fig. 8h), a generic spoked wheel roof constructionwith a membrane cover is rendered (Fig. 9).

5. DISCUSSION AND FUTURE WORKThe presented examples demonstrate that the methodoffers both an intuitive approach to playfully explorespatial, mixed compression and tension equilibriumforms in early design stages, but also enables formfinding of equilibrium shapes with bounds on the forcedensities and constraints on the geometry. Theequilibrium model resulting from the optimizationprocess usually forms the starting point for a following

part of the design process that deals with questions ofconstructive detailing and erection. Since the approachis limited to structures that form mechanisms, astiffening and bracing scheme has to be developedafter a satisfying equilibrium model is obtained.Subsequently, member dimensions are assigned, andanalysis regarding structural safety for live loads,stability and serviceability has to be carried out.

In general, form-finding methods are mainlyrelevant for the design of structures with highpermanent loading compared to live loads; suchpermanent loads are, for instance, high self-weight orhigh pre-stress. For structures with live loads that havea similar or higher magnitude than the permanentloads, form found geometry might serve as startingpoint for the design process; but these live loadingcases become the dominant factors for structuraldesign. By running the form-finding process withpermanent loading combined with live load cases, thedesigner could gain qualitative insight in equilibriumstates for these specific combinations. Furthermore,the designer could even use superposition strategies,combining several equilibrium states to generate form.

The method, implemented using an explicit Runge–Kutta solver with adaptive step size, performs well.Simpler third- and fourth-order Runge–Kutta methodshave been tested and work in principle, but needmuch higher number of iterations, since small stepsizes have to be used. Table 2 shows the number ofiterations and calculation times for the examplesdiscussed in the paper. The iteration number, forexample, of the example shown in Figure 8d isremarkably high. This probably is a consequence ofthe “floating” setup of the model; without anysupports, the system is determinate up to a rigid-bodytranslation. In general, the solving times are notcompetitive in the sense of a real-time simulationenvironment; this is mainly a result of the lack ofsparse matrix computation routines in the NumPyimplementation available for IronPhyton. Animplementation of the method as compiled pluginusing a fast linear systems solver should speed up thesolving time by two orders of magnitude.

Although the method allows an intuitive modellingof equilibrium forms, background knowledge instructural design is obviously required from thedesigner. A general issue with the equilibriummodelling process is the possibility to create over-constrained models easily, resulting in slowconvergence or no convergence at all. Developing ageneral approach to detect such situations is difficultand remains future work. Another source for possible

Figure 9. Rendering of a cable-stayed roof based on theequilibrium model presented in Figure 8h. The pre-stressed,spoked-wheel construction forms the primary supporting

structure for a membrane roof.

failing of the form-finding process is topologicdegeneration, i.e. branches with vanishing lengths.

Future work will also include the implementation offurther geometric constraints, such as nodesconstrained to spatial curves, non-planar surfaces orpolygonal meshes. In this case, the “free” componentof the residual will be obtained by projecting theresidual onto the tangential line or plane of theconstraining object. The gradient decompositiondescribed in Section 3.2 can easily be extended to otherconstraints such as e.g. fixed branch lengths or bounds,or even constraints, on forces directly, rather than onforce densities. Furthermore, more specific supportconditions, such as movable supports constrained togeometric objects, or supports that can take forces onlyin specific directions, will be implemented.

6. CONCLUSIONThe presented form-finding method, implementedwithin a CAD environment, allows for the intuitiveexploration of spatial, pin-jointed, kinematicstructures with mixed compression and tension forcesin equilibrium for given design loads. Through thedefinition of various constraints on form and forces,the method can also be used for the geometricrationalization of structures according to constructiverequirements. This opens up new possibilities in thedesign of efficient structural forms that visually blendbetween freeform design and known typologies oflightweight structures.

ACKNOWLEDGEMENTSWe owe thanks to Diederik Veenendaal and Dr. TomVan Mele for their suggestions and critical commentsduring the development of the method. The first authorwould furthermore like to thank Prof. Dr. RomanKemmler (Schlaich, Bergermann & Partner) forinspiring discussions and fruitful advice.

REFERENCES[1] Pottman, H., Asperl, A., Hofer, M. and Kilian, A.,

Architectural Geometry, Bentley Institute Press, Exton,2007.

[2] Tessmann, O., Collaborative Design Procedures, PhDthesis, Universität Kassel, 2008.

[3] Sasaki, M., Morphogenesis of Flux Structure, AAPublications, London, 2007, 100–109.

[4] Allen, E., Guastavino, Dieste, and the two Revolutions inMasonry Vaulting, in: Anderson, S. ed., Eladio Dieste -Innovations in Structural Art, Princeton ArchitecturalPress, New York, 2004, 66–75.

[5] Gaß, S., Experiments, Institute for LightweightStructures, Stuttgart, 1990, 1.1–1.6.

[6] Mostafavi, M., Structure as Space. AA Publications,London, 2006, 100–103.

[7] Brunetta, V. and Patteeuw, V., Ney & Partners: Freedomof Form Finding, A16 & VAi, Antwerpen, 2005, 36–38.

[8] Veenendaal, D. and Block, P., An overview andcomparison of structural form finding methods forgeneral networks, International Journal of Solids andStructures, 2012, 49(26), 3741–3753.

[9] Block, P. and Ochsendorf, J., Thrust Network Analysis: Anew methodology for three-dimensional equilibrium,Journal of the International Association for Shell andSpatial Structures, 2007, 48(3), 167–173.

[10] Miki, M. and Kawaguchi, K., Extended force densitymethod for form finding of tension structures, Journal ofthe International Association for Shell and SpatialStructures, 2010, 51(4), 291–303.

[11] Kemmler, R., Formfindung: Die Interaktion von Kraftund Geometrie, Stahlbau, 2012, 81(6), 476–480.

[12] Kilian, A. and Ochsendorf, J., Particle-Spring Systemsfor Structural Form Finding, Journal of the IASS, 2005.46(147), 77–84.

[13] Rippmann, M., Lachauer, L. and Block, P., InteractiveVault Design, International Journal of Space Structures,2012. 27(4), 219–230.

[14] Tachi, T., Interactive Freefom Design of Tensegrity, in:Hesselgren, L., Sharma, S., Wallner, J., Baldassini, N.,Bompas, P., Raynaud, J. eds., Advances in ArchitecturalGeometry, Springer, Paris, 2012, 259–268.

[15] McNeel, B., Rhinoceros - modeling tools for designers,www.rhino3d.com, 2013.

[16] Culmann, K., Die graphische Statik, Meyer & Zeller,Zürich, 1866.



Table 2. Number of branches, free nodes, iterations, and solving times for the examples presented in thepaper. The computation has been executed on a standard PC (Intel Core Duo, 2.8 GHz)

Fig. 6b Fig. 6d Fig. 7d Fig. 7f Fig. 8d Fig. 8f Fig. 8hbranches m 29 40 48 48 59 79 258free nodes n 18 24 27 27 40 40 140Iterations t 50 48 121 136 636 48 57time [s] 2.6 3.1 9.8 11.3 68.6 4.9 89.0



[17] Laffranchi, M., Zur Konzeption gekrümmter Brücken,PhD thesis, ETH Zurich, 1999, 23–28.

[18] Lachauer, L. and Kotnik, T., Curved Bridge Design, in:Gengnagel, C., Kilian, A., Palz, N. and Scheurer, F. eds.,Computational Design Modeling, Springer, Berlin, 2011,145–152.

[19] Schlaich, J. and Schäfer, K., Design and detailing ofstructural concrete using strut-and-tie models, TheStructural Engineer, 1991, 69(6), 113–125.

[20] Muttoni, A., Schwartz, J. and Thürlimann, B., Design ofConcrete Structures With Stress Fields, Birkhäuser,Basel, 1997.

[21] Linkwitz, K. and H.-J. Schek, Einige Bemerkungen zurBerechnung vorgespannter Seilnetzkonstruktionen,Ingenieur-Archiv, 1971, 40(3), 145–158.

[22] Schek, H.-J., The Force Density Method for FormFinding and Computation of General Networks,Computer Methods in Applied Mechanics andEngineering, 1974, 3(1), 115–134.

[23] Linkwitz, K., H.-J. Schek, and L. Gründig, DieGleichgewichtsberechnung von Seilnetzen unter

Zusatzbedingungen, Ingenieur-Archiv, 1974. 43(4), 183–192.

[24] Timoshenko, S. and Young, D., H., Theory of Structures.McGraw-Hill, New-York, 1945, 188–194.

[25] Bouaziz, S., Deuss, M., Schwartzburg, Y., Weise, T. andPauly, M., Shape-Up: Shaping Discrete Geometry withProjections, Computer Graphics Forum, 2012, 31(5),1657–1667.

[26] Barnes, M., R., Form Finding and Analysis of TensionStructures by Dynamic Relaxation, International Journalof Space Structures, 1999, 14(2), 89–104.

[27] Baer, S., Rhino.Python - Cross platform scripting forRhino, http://python.rhino3d.com, 2012.

[28] NumPy, http://www.numpy.org, 2013.[29] Kiusalaas, J., Numerical Methods in Engineering with

Python, Cambridge University Press, Cambridge, 2005,275–281.

[30] Göppert, K. and Stein, M., A Spoked Wheel Structure forthe World´s largest Convertible Roof – The NewCommerzbank Arena in Frankfurt, Germany, StructuralEngineering International, 2007, 4(17), 282–287.

2014 ijss lachauer interactive equilibrium modelling 1398776148

Documents