particle-spring systems for structural...

JOURNAL OF THE INTERNATIONAL ASSOCIATION FOR SHELL AND SPATIAL STRUCTURES: IASS

PARTICLE-SPRING SYSTEMS FOR STRUCTURALFORM FINDING

Axel Kilian1 and John Ochsendorf2

1 PhD Candidate in Computation, Department of Architecture, MIT, Cambridge MA USA 021392 Assistant Professor, Building Technology Program, MIT, Cambridge MA USA 02139

SUMMARY

Particle-spring systems are well-known in computer science for creating physical simulations. In this paper,we propose the use of particle-spring systems for finding structural forms composing only axial forces. Theequilibrium position of each particle is found using a Runge-Kutta solver, which allows the user to interactwith the simulation while it is running. Several examples illustrate the technique, beginning with two-dimensional funicular forms and extending to three-dimensional networks. The paper proposes a novelthree-dimensional design and analysis tool, which can be used by engineers and architects to find structuralforms in real time.

Key words: Form finding, hanging model experiments, tension structures, funicular form, grid shells

1. INTRODUCTION

Structural form finding is a classical problem forlong-span roof systems and lightweight structures.The methods of dynamic relaxation [1] and forcedensity [2] have been used for decades in formfinding of fabric roof systems and grid shells.Generally, these solution procedures are useful forfinding the equilibrium position of a structuralnetwork with a desired level of internal force, as inthe case of a pre-stressed membrane structure.Though such methods are highly developed for pre-stressed systems, conventional form findingmethods are not very suitable for finding the formof statically determinate structures which act inpure tension or compression under their self weight,as in the case of a network of hanging chains.Conventional finite element methods are alsounsuitable for modeling a network of hangingchains, because the large displacements involvedcause numerical instabilities and violate thenecessary assumptions of small displacements.More recently, optimization techniques using finiteelement methods have illustrated great potential forrefining a structural form to minimize bending [3].This method allows the designer to begin with an

inefficient form, such as a non-structural shape for aconcrete shell, and then search for a more efficientform by minimizing local bending stresses. In short,there are many analysis tools for refining forms, butthere are few design tools for exploring and creatingnew structural forms.

Many designers have experimented with hangingchain models and other physical methods forfinding efficient structural forms acting in puretension or pure compression. For over 40 years,Heinz Isler has promoted the use of physical modelsas the most appropriate method to discover three-dimensional forms [4][5][6]. Similarly, Frei Ottoand his colleagues in Stuttgart have developedhighly accurate physical experiments for findingstructural form [7]. In the early 20th century, AntoníGaudi employed hanging models in the form-finding processes for the chapel of the ColoniaGuell [8] and the arches of the Casa Mila [9]. Suchhanging forms are often called funicular derivedfrom the Latin word funiculus, meaning thin cord orrope, because they represent the shape taken by athin cord acting in pure tension under a given set ofloads. As Robert Hooke recognized in the 17th

century, such tension forms could be inverted to

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find the shape of structural forms acting in purecompression under the same loading. Heyman [10]has translated Hooke’s theorem from Latin: “Ashangs the flexible line, so but inverted will stand therigid arch.”

By using a chain of axial springs which do notallow bending, it is possible to model hangingcables under various loading conditions in real time.Consider the example of two masses, m and 3m,which are hanging from three weightless springs, asin Figure 1. For particular values of spring length,spring stiffness, and mass, there will be one stableequilibrium position for this system. Reducing thelength or increasing the stiffness of the springs willalter the equilibrium position so that the “sag” at thecenter is reduced. Similarly, increasing the length ofthe springs or reducing the spring stiffness willcause greater sag in the system and will produce analternative equilibrium configuration. By varyingthe length or stiffness of each spring, it is possibleto generate an infinite number of equilibriumpositions which belong to the family of funicularforms for a given loading configuration. In thecontext of graphic statics, this is equivalent tomoving the pole to a new location to create adifferent funicular polygon [11].

Figure 1. Equilibrium of simple particle springsystem

This paper presents a novel approach for theexploration of funicular forms. Particle-springsystems serve as an excellent approximation forhanging models, by using axial springs connectinglumped masses to represent the physical behavior ofweights hanging on strings. The equilibriumposition of each mass is found using an iterative

Runge-Kutta solver proposed by Baraff and Witkin[12]. Such methods are well-known in the computergraphics community, where researchers havedeveloped efficient algorithms for solving largenetworks of particles connected by elastic springs.Particle-spring systems have been used extensivelyin cloth simulations and other graphics problems,particularly for making realistic simulations for theanimation of clothing and other fabrics. After abrief description of the particle-spring method, two-dimensional and three-dimensional funicular formswill be derived using the method. Finally,advantages and disadvantages for the method arepresented.

2. PARTICLE-SPRING SYSTEMS

Particle-spring systems are based on lumpedmasses, called particles, which are connected bylinear elastic springs. Each spring is assigned aconstant axial stiffness, an initial length, and adamping coefficient. Springs generate a force whendisplaced from their rest length. External forces canbe applied to the particles, as in the case ofgravitational acceleration. To solve for theequilibrium geometry of particle-spring systems,there are many techniques with varying degrees ofefficiency and stability. The two primary classes ofsolution procedures are implicit and explicitsolvers. For solving structural models, implicitsolvers are more useful because of the high springstiffness used in the models to minimize distortionof the initial spring lengths. With high springconstants, the small changes in the length of eachelement do not converge to an answer when usingan explicit solver, making an implicit solver morepreferable [13]. The implementation used here isbased on an implicit Runge-Kutta solver, whichaims to find the equilibrium position of eachparticle. The solver is part of a particle-springsystem implemented for the Processingenvironment by Simon Greenwold [14]. Theprogramming environment uses Java, which allowsthe method to be made freely available on theinternet.

Each particle in the system has a position, avelocity, and a variable mass, as well as asummarized vector for all the forces acting on it. Aforce in the particle-spring system can be applied toa particle based on the force vector‘s direction andmagnitude. Alternatively the magnitude of the force


can be calculated using a function as in the case ofsprings. Springs are mass-less connectors betweentwo particles that exercise a force on the particlesbased on the spring’s offset from its rest length.Particles can be restrained in any dimension, so it isstraightforward to add supports by restraining thedisplacement vectors of an individual particle.

The particle-spring system is usually not inequilibrium when the simulation is started and therewill be movement throughout the system as theparticles and springs seek their equilibriumpositions. For the simulations presented here, eachparticle is assigned a mass and a gravitational fieldis applied to the entire system. Thus, the particlesfall due to gravity either until the forces in thesystem reach equilibrium or the simulation isterminated. To prevent oscillations of the particlesabout their equilibrium positions, it is necessary toapply damping to the system. Damping can beapplied as a coefficient to each spring.Alternatively, each particle can be subjected toviscosity in the surrounding environment as anotherdamping method.

Eventually the system comes to an equilibriumstate, if one exists. Even when the system no longerappears to move, the solver never stops as thesystem continues to update and refine the positionof each particle. To stop the simulation, the user candefine a limit, such as the velocity or the change inposition of each particle as a cutoff threshold for thesimulation. Papers by Baraff provide greater detailson the solution procedure, including computer codefor implementation [12][13].

Figure 2 illustrates the solution procedure for aseries of springs supporting 40 masses at equalspacing, which approximates the form of a hangingchain. The particles are initially placed in a singleline connected by springs with the outer twoparticles acting as fixed supports. As the simulationbegins (t=0.5 seconds), the line of particles beginsto fall. At t=2.0 seconds, the particles near thesupports are approaching their equilibrium positionswhile the particles near the center continue to fallvertically. The solution at t=3.0 seconds begins toapproximate a catenary, as will be discussed inmore detail later. One of the primary attractions ofthe particle-spring approach is that it allows the userto watch the system approach equilibrium and tointervene during the solution process. The user canalter the applied loading, add or subtract structural

elements, and change the support conditions inorder to discover new structural forms while thesimulation is running.

Figure 2. Solution process for a cable with fortydiscrete masses at equal spacing.

3. TWO-DIMENSIONAL SYSTEMS

For two-dimensional systems, unique funicularforms exist for given loadings and supportconditions, as in the case of a catenary formed by acable of given length hanging under its own weight.Such forms depend only on the length of eachelement and the applied loading, so that a uniquesolution exists. Combinations of hanging chainswill create more complex funicular forms as inFigure 3, which is achieved by adding “chains” ofclosely spaced particles with uniform masses.Provided that each intersection has only threesprings attached, the system is staticallydeterminate and the funicular form is unique.

Figure 3. Statically determinate funicular form in2D modeled with particle-spring simulations

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Particle spring systems can be used to effectivelymodel both statically determinate and indeterminatesystems. For indeterminate systems, it is possible todevelop compression struts which are equilibratedby tension elements and the axial spring model canaccount for these scenarios as well. Figure 4illustrates the simplest example of an indeterminatetwo-dimensional system, a point load supported bythree weightless bar elements with pinnedconnections. In this case, the force in each elementvaries with its length and axial stiffness and theexact value of force in each element can varydrastically with small changes in the geometry. Theaccompanying force polygon in Figure 4 usesgraphic statics and Bow’s notation to illustrate theinternal forces in each element of the structure, suchthat vector ab on the force polygon is equivalent tothe magnitude of the applied load AB [11]. Theperfect elastic solution gives the greatest force inelement CD and the smallest force in element AD,with tension in all three elements, as shown by theforce polygon.

Figure 4. Statically indeterminate funicular systemin 2D with the corresponding force polygon.

Many other equilibrium states are possible for thisstructural system, with each element acting as atension, compression, or zero-force member. Forexample, if the length of member CD is longer thanthe prescribed length, it must be forced intoposition, causing a state of self-stress in thestructure. In this case, a new equilibrium state willoccur with element CD in compression and withincreased tension in elements BC and AD. Thisstate of self-stress is illustrated by the alternativeforce polygon showing the corresponding locationof c’d’ in Figure 4. There are an infinite number ofvalid equilibrium states for this structure, all ofwhich can be illustrated with graphic statics, andcan be effectively modeled with particle springsystems. However, for spring elements acting incompression, the solution procedure does not

account for buckling, which must be implementedas an additional consideration.

4. THREE-DIMENSIONAL FUNICULARSYSTEMS

If one considers the term funicular to mean“tension-only” or “compression-only” for a givenloading, then it is also appropriate to identify threedimensional systems as funicular. Threedimensional funicular systems are considerablymore complex because of the multiple load pathswhich are possible. Unlike the case of a hangingcable with a single funicular form, hangingmembranes have multiple possible forms.

As an example, Figure 5 illustrates two continuoussurfaces supported on a circular base: a cone and ashallow spherical dome. Both forms can containcompression-only solutions according to themembrane theory under an applied load of gravity[15][16][17]. Therefore, for statically indeterminatenetworks of intersecting elements, there is no suchthing as a unique funicular solution. Each solutionwill depend on the exact length of each element.

Figure 5. Examples of 3D membrane systemswhich can act in pure compression.

This fact is well-known to designers who seekthree-dimensional funicular forms throughexperimentation. For example, Heinz Isler variesthe amount of fabric in his hanging membranemodels in order to alter the shape of the final design[5]. Simply by changing the length of each element,new equilibrium solutions are possible. For threedimensional networks, a spherical dome canbecome a conical shell by adjusting the length ofeach element. Both belong to the family of validsolutions for compression-only networks actingunder their own weight. Of course some solutionsperform better under varying load conditions andthe double curvature of the dome is structurally

AB

C Da

b

c

d

c'

d'


superior to the single curvature of the cone in theevent of asymmetrical live loading.

5. DESIGN EXAMPLES

To illustrate the potential for structural form findingwith particle spring systems, four examples will bepresented: a catenary, a square mesh, a 3-Dcathedral structure, and a free-form grid shell.

Example #1: A catenary of length, L, is modeled byseven masses distributed initially at equal spacingalong spring segments. As illustrated in Figure 6,the particle-spring solution varies only slightly fromthe theoretical catenary solution because of linearapproximations of the particle spring solution.Though the solution from the particle spring modelis not an exact catenary, it is a valid equilibriumsolution for a hanging chain with equal weightsdistributed at slightly uneven spacing along thechain. It is not an exact catenary because the lengthof each spring segment is slightly different due tothe varying internal forces. The difference betweenthe nodes of the particle spring system is given as apercentage of the maximum sag of the catenary. Inall cases, the particle spring solution varies by lessthan 0.01% of the sag of the cable.

Figure 6. Comparison between theoretical andcomputational solutions for the catenary

Example #2: A rectangular grid mesh with 25squares in each direction is restrained at four nodesnear each corner. Each intersection of the mesh isassigned an equal mass and the particle-springnetwork is subjected to gravitational forces, causingit to form a dome-like structure. Figure 7 illustratesthe solution procedure and the final equilibriumstate of the shell. The hanging structure has beeninverted to demonstrate the form of a compressionshell. Each support is located four squares from thenearest edge, so that the resulting structure has

upturned edges to stiffen the shell against bucklingand asymmetrical loading. Because the interiorparticles in this mesh are connected to four springs,the system is statically indeterminate and thesolution illustrated in Figure 7 contains both tensionand compression elements. In general, the particlespring system gives solutions with upturned edgeswhen the area of the surface lies outside the supportpoints, as in the case of Isler’s shells with upturnededges.

Figure 7. Simple square mesh supported near thecorners

Example #3: A skeletal cathedral structure, similarto forms designed by Catalan architect AntoníGaudi, is created from intersecting elements. Themass is lumped at a series of nodes, which arespaced evenly along each line element. Figure 8provides a view through the central nave of such astructure. The building geometry is onlyrepresented by lines of force and not as meshes or

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surfaces. For simplicity, four bays of two-dimensional transverse arches are represented,though it is straightforward to add intersectingarches between each bay to create a fully three-dimensional structure. It is also straightforward toadd additional arches and towers to this virtualmodel in the same manner that Gaudi experimentedwith physical hanging models. Using the particlespring approach, a three-dimensional structure suchas this cathedral can be created in only a fewminutes, whereas Gaudi developed his physicalmodels over many years. In addition to its merits asa design tool, this method can also be used as aninteractive analysis tool for historic masonrystructures such as Gothic cathedrals, in whichparticle-spring systems can be used to representthrust lines or thrust surfaces acting within the massof the masonry.

Example #4: A grid shell is created with irregularsupport locations, as illustrated in Figure 9. Aregular grid mesh is supported multiple times inaddition to its corner attachment points. As before,each intersection of the mesh is assigned a mass andthe structure deforms according to the appliedgravity. This is an example of a more free formdesign exploration in which the user experimentswith various support conditions. As with the squaregrid of example #2, this shell is staticallyindeterminate and the solution illustrated containselements acting in both tension and compression.

6. DISCUSSION

Finding structural form using particle-springsystems has a number of advantages. Mostimportantly, the user can change form and forces inreal time while the solution is still emerging. Theenvironment educates the user as to the effects offorces on the form of structures and provides aninteractive form-finding environment that waspreviously limited to physical models. By nature ofthe solution procedure, the particle-spring systemalways finds a possible load path for which theforces are in equilibrium. If no equilibrium solutionexists, as in the case of an unsupported structure,then the masses will continue to translate in spaceuntil the user intervenes. The primary improvementover existing finite element methods is that thisenvironment is fully dynamic, allowing the accuratecomputation of large displacements, velocities, andaccelerations for the design of structural systems.

The improvement over existing relaxation methodsis that the design environment is fully interactiveand dynamic, allowing the user to invent formsrather than analyzing existing forms.

Figure 8. Cathedral structure

Figure 9. Free-form grid shell

Particle-spring systems can help to introducestructural evaluation environments into thearchitectural design process as early as possible,allowing the designer to interact with a form and toexperiment with alternative solutions. The goal is toincrease intuitive understanding for structuralbehavior of complex forms at the early designstage. The addition of some file export capabilitiesallows for the exchange of data with otherapplications so that models created in the particle-spring environment can be exported to more


conventional design environments, such as CADand standard finite element programs. Thismethodology can allow architects and engineers toexplore forms with a structural basis in the samemanner that some designers have used physicalmodels in the past.

A number of disadvantages exist at present. Due tothe properties of axial springs they not only contractupon being stretched but also expand upon beingcompressed. This creates the possibility of tensionand compression members being present in ahanging structure at the same time. There areseveral remedies for this problem, such as removingthe compression members in order to guaranteetension-only solutions. Another solution is todynamically subdivide the compression memberinto two springs, which renders the springsincapable of taking compression forces andprovides a tension-only solution. Additionally, thechoice of an appropriate mesh pattern can allow forstatically indeterminate systems, such that eachparticle is attached to three springs or fewer forthree dimensional structures.

The solution procedures are somewhat expensivecomputationally, which limits the size of real-timesimulation at this point. Current solutions arepractical for solving up to 1,000 particles in realtime on a desktop computer. Also, as exemplified inthe case of the catenary, there is a slight loss ofprecision due to the approximation of thesimulation. Using the system to model actualproject geometry has its own challenges. Theprocess is not static but the addition of stringelements is dynamic (depending on the solver used)so that the shape and form is constantly in flux. Thefinal equilibrium shape emerges from the modeledtopology. As in the case of a physical hangingmodel, any addition disturbs the balance of theinitial shape. Finally, another disadvantage is thatfor very complex problems in combination somesolutions can become unstable. In general, thesolution procedure is quite robust.

7. CONCLUSIONS

Particle spring systems provide a powerful newmethod for structural form-finding in a designcontext. The solutions allow large displacementsand are valid beyond the realm of conventionalfinite element formulations, which assume small

displacements. This method allows for real-timediscovery of structural form rather than analysis oroptimization of an existing form. Form findingprocesses at the design stage of the process makethe designer aware of structural responses. Thisapplication represents an extension of particlespring systems beyond their initial role in characteranimation and cloth simulation within computergraphics.

There is significant future work to be done for thismethod to become a standard approach in structuralform finding. Solutions without distortions of theoriginal geometry can be accomplished by updatingthe length of each element to maintain its initiallength. Additionally, various structural elementscan be modeled with the particle-spring system,including trusses, slabs, beams, and rigid elements,which would allow users to explore a much widerrange of structural forms. Future work will alsoaddress the question of different degrees ofoptimization. In the current model, the equilibriumform is not always optimal within the chosentopology and designers may wish to experimentwith alternative solutions to achieve greater orlower efficiency. Designers may choose a lessoptimal structural form in order to accommodateother design constraints. Because design alwaysmediates between many goals and hardly ever fitsone of them perfectly, users should be able todiverge from the optimal structural solution toachieve greater design flexibility.

ACKNOWLEDGEMENTS

This paper presents the preliminary results of anongoing research project at the MassachusettsInstitute of Technology between the Department ofArchitecture and the Department of ComputerScience. The initial particle-spring system used wasimplemented by Simon Greenwold in Java as alibrary for “Processing” (www.processing.org), anenvironment developed by Ben Fry and CaseyReas. The first author also developed an earlyprototype in collaboration with Megan Galbraithand Dan Chak. In addition, the authors co-taughttwo research workshops on this topic incollaboration with Dr. Barbara Cutler, SimonGreenwold, Dr. Erik Demaine, and MartinDemaine, who each contributed substantiallythrough discussions on the topic and in making theworkshops a success. In the course of the

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workshops a more robust C++ application wasdeveloped in a group project headed by RyoShimizu. Finally, students in each workshop helpedto extend the range of structural applications forparticle spring systems, and the authors gratefullyacknowledge their contributions.

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[4] Isler, H. “New Shapes for Shells,” Bulletinof the IASS, No. 8, 1960.

[5] Chilton, J. The Engineer’s Contribution toContemporary Architecture: Heinz Isler,Thomas Telford Press, London, 2000.

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[7] Rasch, B. and Otto, F. Finding Form, AxelMenges, 1996.

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[9] Huerta, S., “El cálculo de estructuras en laobra de Gaudí,” Ingeniería Civil, No. 130,CEDEX, Madrid, 2003, pp. 121-133.

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[13] Baraff, D. “Implicit methods for differentialequations,” Proceedings of the 28th AnnualConference on Computer Graphics andInteractive Techniques (SIGGRAPH), 2001.

[14] Fry, B. and Reas, C., Processing web site,<http://www.processing.org>, accessed 12January 2005.

[15] Billington, D.P. Thin Shell ConcreteStructures, McGraw Hill, 1965.

[16] Heyman, J. Equilibrium of Shell Structures,Oxford University Press, 1977.

[17] Calladine, C.R. Theory of Shell Structures,Cambridge University Press, 1983.

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