Geometrical Description and Structural Analysis of a Modular Timber Structure

M U LT I - S C I E N C E P U B L I S H I N G C O . LT D .

5 Wates Way, Brentwood, Essex CM15 9TB, United Kingdom

Reprinted from

INTERNATIONAL JOURNAL OF

SPACE STRUCTURESVolume 26 · Number 4 · 2011

Geometrical Description and Structural Analysis of a ModularTimber Structure

by

S. S. Nabaei, Y. Weinand

1. INTRODUCTION1.1. IBOIS, the re-interpretation of timberconstructionIn recent years, the necessity of using renewable andsustainable resources in the building sector hasbecome obvious, and interest in timber as a buildingmaterial has revived [1–3]. Novel timber-derivedproducts, such as massif block panels, have emergedand the use of such products is spreading [4, 5].

Geometrical Description andStructural Analysis of a

Modular Timber StructureSeyed Sina Nabaei*, Yves Weinand

ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE (EPFL)School of Architecture, Civil and Environmental Engineering (ENAC)

Timber Construction Laboratory (IBOIS)

(Received 18/02/11, Revised version 28/06/11, Acceptation 02/10/11)

ABSTRACT: The ambitious goal of the ongoing research at IBOIS, theLaboratory of timber constructions at the Ecole Polytechnique Fédérale deLausanne (EPFL) is to develop a next generation of timber constructions madeout of innovative timber-derived products, through applying textile principleson the building scale. The presented structure is a modular composition oftimber folded panels, notably demonstrates an example of applying thegeometric techniques used to produce modular patterns and lattices to timberconstruction context. Effectively, it is shown that complex space structures canbe designed using simple connection technology between elements. Moreover,by taking advantage of advanced CAM process, complex planar timberelements are cut in large scale and assembled with high precision as for theprototype of the structure presented in this paper. Here, the basic module isconsisted of two mutually supporting timber folded panels which are slippedin, consecutively, along their cuts, to build up an arch. The folding conceptcorresponds to a new family of reciprocal frame structures, where instead oftraditional linear elements, mutually supported planar members are employed.The inter-module connection's stability is provided by contact boundarycondition over the slide joints. The fundamental mechanical properties of thestructure are examined using Finite Element Method and considering the non-linear contact boundary condition. The static behavior is studied under the self-weight load case as well as the modal dynamic response. According to analysisresults, and by aid of a CAD parametric model, structural and geometricalalternatives are proposed to improve the structural performance. A prototypebased on this geometric principal has been fabricated and assembled to explorefeasibility of the concept in the building scale.

Key Words: Timber space structure; planar reciprocal frame structure;structural system improvement; Finite Elements analysis; parametric design

International Journal of Space Structures Vol. 26 No. 4 2011 321

Since timber can be viewed as a fiber-derivedproduct, it follows that the analogy between microscale fiber structures and timber-derived woodenstructures can be explored at micro and macro-scale.The key to this approach is the underlying notion thattimber’s fibrous nature, historically perceived as aliability for a construction material, is in fact a preciousfeature that should be exploited to increase both thematerial’s functional and aesthetic value. In fact, this

*Corresponding author e-mail: [email protected]

inherent flexibility of timber-derived panels permits itto be folded into robust, lightweight structures that usematerial very sparingly. This materiality is exploredwithin a current research direction at IBOIS. The ideais to inspire from techniques of the textile industry andto re-interpret the basic units of a fabric as nodes andknots using thin flexible timber elements to proposeentangled (woven) forms as timber space structures. Aninnovative “timber knot” as well as a generalization ofthe concept into a “timber fabric” is presented in [6].The structure presented in this article comes from adifferent inspiration of textile techniques, tightlyrelated with the geometric techniques used to describepatterns and tailings.

1.2. Geometry of space structures: asurvey on “form-element” realationshipDuring the two past decades, space structures withmultiple forms and various materialities have beendesigned, where the free-form practice has been thefavorite stream. In context of timber construction,“local” inter-element relationships and the linkbetween the “local” condition and the “global” formsare of grand importance. That’s why the origin of the“form” for space structures is questioned and surveyedin the following lines, looking forward to understandthe important link between the global geometry and itslocal consequences on the discretized elements.

Effectively, four main directions dealing with thegeometry of the space structures can be recognized asfollows. • Space structures issue of a From-finding

practice: Two main structural families of thiscategory are tensile membranes and vaultedmasonry construction. Multiple techniques havebeen introduced in the literature and practiced inorder to determine a tensile or compressivegeometry: physical models of Otto for thetensile roofs, computer-based methods as Force-density method and Dynamic relaxation fortensile roofs [7–9] and Thrust network analysisfor masonry vaults as in [10, 8]. The term,“form-finding” is also used to recall a form-generation process used to find the relaxed formof grid-shells under a certain boundarycondition as in [11]. Nevertheless, what iscommon between these various techniques isthe fundamental property that the final form isthe direct result of the equilibrium and isinfluenced intensively by the materiality and theboundary condition applied.

322 International Journal of Space Structures Vol. 26 No. 4 2011

Geometrical Description and Structural Analysis of a Modular Timber Structure

• Free-form design: Here the goal is not to givereader a review of the free form design practice butrather to reveal an important “local” consequenceof such design on structural elements. Accordingto free-form approach, the design surface comes asthe input and then by means of geometric-mathematical models, it’s approximated by atypologically uniform (or at least modular) planarmesh. Although single or double curved panels arealso allowed, but planar elements constitute themajor part of the approximated mesh. The meshgeneration is clearly constrained by tolerances ofthe approximation problem from the referencesurface, as well as the economic-industrialexternal constraints [12–14]. The supportingstructure follows closely this approximated meshand thus its spatial form is indeed less influencedfrom materiality the connection technology. Inother words, “local” conditions are determined bythe “global” form.

• Topology optimization: The space structuresbelonging to this family have their form generatedby resolution of a topology or shape optimizationproblems for which a review can be found in [15].

These methods combined with evolutionaryalgorithms, often result in free-form spatialobjects, which although they might have beengenerated based on a structural criteria, fall intothe second category, where they are geometricallyapproximated in order to be built.

• Reciprocal frame structures: This family ofmodular space structures is constituted ofinterlaced linear elements, where the final form is aresult of a basic module as well as a connectiontechnology. The concept exists since centuries, forwhich a review can be found in [16] and a technicaldecomposition of the structure is proposed in [17].

The interest of such structural system lays in thefact that using identical linear beams and simpleconnection techniques, it’s possible to reproducerather complex three-dimensional space structures.

In such family of structures, the global form isnot an approximation of a design referencesurface, but the result of the basic element’stopology and the connection technology applied.Unlike the free-form design, the “global” form isdetermined by the “local” conditions.

In fact, in context of timber engineering it’srecognized that the complexity of the connectiontechnology, becomes an important feature in order todistinct different structural propositions both from

Seyed Sina Nabaei, Yves Weinand

financial and structural aspects. Usually, free-formtimber frames end up with heavy and complex steelnodes, designed to resist external loads. In suchcontext it follows that the connection technology ispreferred to be simple, whereas, for timber elements,complex geometries can be realized by means ofadvanced CAM facilities. The analogy is thusinteresting to be investigated with the reciprocalframes, where complex forms come from a basicrepeated module and simple connection technology.

The core idea which follows in coming lines is aninvestigation on a family of space structures, where thefocus, rather than on an irregular surface approximation,lays on materiality and related connection technology.According to this approach, the final form is a result ofthe geometry of connected members as well as theemployed connection technology.

The modular structure proposed in this papercomes from such exploration on form-elementrelationship. It is not only geometrically treated, butalso from mechanical point of view it isdemonstrated that its structural behavior isunderstood. Moreover a prototype is realized tocomplete the investigation.

2. PRESENTATION OF STRUCTURE2.1. The folding conceptThe folding concept presented in this paper has beeninitially examined, during an architectural workshop,“The Atelier Weinand” at IBOIS-EPFL, turning aroundthe discrete architectural geometry. A V-form basicmodule is fabricated through connecting two timberpanels and is then spatially multiplied using consecutivespatial rotations and translations to form an arch (fig 1).

The structure can be decomposed into fourprincipal typologies (fig 2) each consisted of twomirrored timber panels, joined together as a V-formmodule, through the bisector plan, by means of twohard wood dowels (fig 3). They are placed in the

middle as shear keys and two oval head screwsinserted nearer to the borders in order to avoidrelative translation and rotation. Fasteners areinserted in direction of the normal to the bisectorplan. These modules are then slipped inconsecutively along their U shape cuts, to form anarch. The inter-panel stability is provided by roto-rigidity of slide connection and axial contact ofreciprocal panels.


Figure 1. Folding concept for the modular structure designedin the Atelier Weinand workshop © Bastien Thorel.

Typ. 1 Typ. 4

4.3

m

3.4 m

7.4 m(a)

(b)Typ. 3 Typ. 2 Typ. 1

Typ. 2Typ. 3Typ. 4

Figure 2. (a) Front and top view of the prototype (b) Principaltypologies of timber panels (unfolded top view).

(a) (b)

Figure 3. Mirrored panels connection fasteners: (a) Oval headscrews (b) wooden dowels inserted in prefabricated holes.

2.2. Geometric and parametricdecomposition of global formA trapezoidal plate is introduced to be the generatormember of the form, noted as β0 and referred to as the“Base panel”. From mathematical point of view, thissingle panel is transformed by means of two classes ofoperators in order to give shape to the global form. Thefirst operations are the set of Boolean operators,representing the machining, connecting and assemblingprocess. Whereas the second type of operators, referredto as Geometrical operators, introduce rigid body

movements and consist of congruent maps, employed toplace the object in the space. Among Booleanoperations, union, intersection and remove are used.While among Geometrical operators, we mayenumerate rotation around a space vector, reflectionagainst a plan, and inversion against a spatial point ortranslation in direction of a space vector as examples ofsimple isomorphisms.

The parametric geometric transformationpermitting to determine the slide cuts between twoconsecutive modules is illustrated in fig 4. The



�1�0 �cx = �1 ∪ S1 (�1) �cv = �1 ∪ S2 (�1)

T1 = (�cv) T1 = (�cv) ∪ �cx T1 = (�cv) ∪ R2 (�cx)

R1

Q1 = R1 (Q1)

P1 = R1 (P1)Pe

Q1

S1

S2

Q1X e

Q 1

P1

P 1

P 1

P 1

W 0

Q 1

S1 (Q1)

S2 (P2)

P1

Pe

Q’2

Q1

Q 1 Q 1

P 1

P 1 P 1 P 1

Q1P1

P1

++

Qe

Q sQ 1

R1

γ π − 2γ

π − 2γ

S2 (P1)

S1 (Q1)

R2

T1

S 1 (Q

1)

ψ

ψ

T3 ¥ T1 (�cv) ∪ R2 (�cx) �’cv ∪ �’cx

P1

I1T2

T3

φ2

φ2

P 1 P 1 P 1

Figure 4. Transformation of basic trapezoidal panel (β0) into two V-form modules (M'cv ∪ M'cx).


manuscript letters (β, M, etc) stand for geometricobjects and bold capital letters as (S, R, T, etc)symbolize isomorphisms, respectively the symmetry,rotation and translations. Boolean operations arerepresented by their mathematical symbols of union,etc. By repeating the same geometric transformation, adouble cut module is obtained shown in fig 5, calledthe “Base module”, which is the base brick of themodular structure.

Base modules are then slipped in, one on top of theother, to build up a half-arch, which is trimmed by avertical plan placed at the mid line of the span and ismirrored against the same plan to produce a completearch. Next step will be to trim the arch using ahorizontal plan, and to place it on level (See Fig 6a.).This arc is then translated in direction of vectors w and2w to get the modular structure of the prototype (SeeFig 6b.). Naturally, one can continue to expand thestructure, by multiplying nw consecutive transversaltranslations.

From mathematical point of view, it’s easy to showthat each Base module can be mapped on the otherslipped Base modules of the arch by means of anisomorphism. This transformation, similar to the logicfollowed in Fig 4, is a product of spatial translationand rotations, and thus an orthogonal map.

As a result of the implementation of the geometrictransformation described above in a CAD environment,a parametric model of the modular structure is obtainedwhere the geometry is controlled by means of a set ofmeaningful scalars. This parametric description of the

structure, apart from the benefits from the design pointof view, as shown in §4.2 may lead to alternativestructural systems which show better performances interms of dynamic behavior.


Concave element(CV)

Concave element(CV)

Convex element(CX) Convex element

(CX)

U-joint

Figure 5. Base module.

Horizontaltrimming plan

w: Translation vector2ww

Trimmed part on CV element

Vertical plan of symmetrytrimming plan

(a)

(b)

Trimmed part on CX element

H

V

Figure 6. (a) Trimming of half arc by horizontal and verticalplans and mirroring (b) Translating arc to build up the

geometry of the real scaled prototype.

connection between CX and CV, a geometrical lockbetween panels. Instead the connection betweenmirrored panels in CX or CV module along their jointis noted as TYPE II. In fact this type is a moremechanical connection and depending on which typeof fastener used, the modeling considerations will bedifferent (See Fig 9).

In order to describe the TYPE I of connections, foreach slide, a local axis of coordinates is introduced:(See Fig 8) let 1 be the direction along the slide, 3 bethe normal direction to CX panel and 2 defined bymeans of right hand rule. Therefore, supposingcontact boundary condition, the degrees of freedom,1 in compression, 2, 3, 11, 22, 33 are constrainedaccording to the geometric configuration of panelswhich blocks these relative movement and rotations.The only unconstrained degree of freedom is 1 intension. This means that panels can be slipped out.

For TYPE II of connections, let 1' be the directionalong the connection between mirrored panels, 3' thevertical direction and 2' be defined by right hand rule.(See Fig 8) In such a configuration, direction 2' wouldmeasure the relative move away of mirrored panels.The constraint scheme in TYPE II depends closely toconstruction detail. In a preliminary modeling wesuppose that panels are tied together and thus degreesof freedom 1', 2', 3', 1'1', 2'2', 3'3' are constrained.

3.2. Modeling hypothesis and localboundary condition analysisAccording to the adapted modeling approach, thethickness of panels enters in the reality of modeling,permitting to model the slide connection as it isdefined from the geometric configuration. For eachslide connection, five normal contacts are definedbetween two pair of surfaces as master and slave



Figure 7. Prototype realization at EPFL © Alain Herzog.

3' 3'

22 2 31

11

33

Local system of coordinatesfor U joints: TYPE I

Local system of coordinates for the connectionbetween mirrored pannels: TYPE II

3' 1' 1'1'2'2' 2'

Figure 8. Local system of coordinates for connectionsbetween panels on a Base module.

2.3. Prototype realizationA prototype of this structure has been realized at EPFL inorder to test the structural feasibility of the concept as wellas to investigate the architectural ambiance. A selection ofphotos illustrating the project is shown in fig 7. Theproject included development of relative NC codes for 5-axis machining, as well as designing constructive detailsand fabrication of the structure for exposition. All V-formfolded modules have been manufactured from 21 mmthick three-layer cross-laminated panels and cut by meansof CNC machines at EPFL.

3. STRUCTURAL ANALYSISHere the objective is to understand the structuralsystem considering the geometric non-linearity by theaid of appropriate numerical models and to proposeeventual improvements. The diagnostics about thestructural system is consisted of the linear staticanalysis under the self-weight and the modal dynamicanalysis, in order to examine the rigidity. Based onthese observations, improvements are proposed in §4.

3.1. Local boundary condition analysis ofthe Base module connectionsIn each Base module two types of connection aredistinguished: TYPE I of connections refer to the slide


surface. Contact property is considered to befrictionless. The contact boundary condition has to be satisfied along slide joints underlying finite

deformations. A 10-node modified quadratictetrahedron element is chosen to mesh the continuummodel, referred to as C310DM ABAQUS® solidelement.

The timber is considered to be an elastichomogeneous material throughout the wholethickness. The Young Modulus, E = 8000 Mpa,Poisson’s ratio, υ = 0.3 and material density, ρ = 500 kg/m3 are concluded from documentationdisposed by the industrial provider of cross-laminatedpanels [5].

3.3. Structural analysis of a single arch Results for the global deformation field and the Vonmises stress driven from a static non-linear analysis ofa single arch under its self weight load case are shownin fig 10. It can be seen that the geometricconfiguration of the slided together modules leads to aconcentrated distribution of stress at the location ofslide cuts. This mainly happens because of bendingbehavior of the structure. Moreover, a modal dynamicanalysis for the isolated modular arch has beenrealized to have a first estimation of structural rigidityin lateral and transversal loading conditions bycomparing natural frequency values: the first globalmode is a lateral one and it has a relatively smallnatural frequency of 0.59 Hz, comparing to practicalguidelines, which advices a minimum naturalfrequency ranging between 1 to 4 Hz [18].

4. PROPOSITIONS FOR STRUCTURALSYSTEM IMPROVEMENTBased on our findings in §3, we proceed on thissection with two goals: first, to obtain a moreuniformed stress distribution in panels and to reducethe stress concentration in U-joints; Second, toincrease structural rigidity, measured by means ofnatural frequency of the first global mode.

4.1. Toward a truss system While having already a geometric superpositionconcept, one immediate remark would be to changethe current beam-like system with a more truss-likeone. Two main directions are tracked, as follows.• Addition of intermediate elements: this could

be realized by help of additional intermediateelements which are inserted properly at mid-planof the arch to connect consecutive CX-CX andCV-CV modules between each other.

• Opening the U-joint: in the initial geometricconfiguration, each module’s stability is providedby the locking effect of panels across the U-joints. In a truss system with additional


TYPE I

I.1 Axial contact between coincident faces of U-jointsI.2 Contact between top face of CV-U-joint with upper face of CXI.3 Contact between bottom face of CV-U-joint with lower face of CXI.4 Contact between bottom face of CX-U-joint with lower face of CVI.5 Contact between top face of CX-U-joint with upper face of CV

TYPE II

II.1 Joint along two mirrored panels of CXII.2 Joint along two mirrored panels of CV

I.5master

I.5slave

I.3slave

I.3master

I.4slave

I.2master

I.4master

I.2slaveCX

CV

CX

CV

CV

CX

II.1

I.3

I.2

I.5

I.4I.1

II.2

Figure 9. Base module connection typology.

intermediate panels, fixed between them andfixed to the slipped modules, it would be possibleto open the angle of U-joints. This will help toreduce the concentrated stresses (fig 11a).Applying these two main modifications on theoriginal structure, the maximum von Mises stressunder the self weight load case is reduced from14.1 Mpa for original configuration, to 1.66 Mpa,keeping the same order of magnitude ofmaximum total deformation. Furthermore, thestress has been led to intermediate elementsrather than panels and consequently, theconcentrated pattern of stress on panels has beenresolved. The natural frequency for the firstglobal mode of the structure is increased up to0.98 Hz.

4.2. Increasing panel inter-locking effect Let’s consider a CX module, picked deliberately fromthe arch. The neighborhood of this module is as in (2)(fig 12a.). According to geometric principal exposed infig 12, CX1 is connected to CV1 and CV2 across two U-joints for each. The idea is to increase the length ofthese current U-joints to make CX1 meet CV–1

(fig 12b). The whole two intersection cubes are thenremoved from CV–1 and it ends up with two extra U-joints on external part of the panels. Consequently,

CV1, connected currently with CX0 and CX1, in hisplace, intersects with CX–1. Removing the two newintersection cubes from CV–1 provides two more extraU-joints connection, placed this time, in internal partof CV module.

If we resume, the general idea is to keep the cut-pattern for CX module unchanged, although for CVmodule there would be 4 more U-form cuts: twointernal and two external (fig 12c).

To achieve this objective, the original geometricalconcept has been implemented within a parametriccomputer-aided design interface. The importantparameters determining the geometry of each typologyof modules in the original design have been identified.Then, the geometrical configuration for the montageof the Base modules is set respectfully to the heightand total span of the original structure.

(2)

Increasing the slide length, while keeping constanttotal span and height of the structure, will increase thenumber of modules. The original design is consistedof 33 slided modules. In fact, by increasing length ofjoint by 103 mm to create two more U-joints betweenpanels, 53 modules of nearly the same size is neededto respect the same height and span as the original

CV CX CV CX CV CX CV− − −− − − − − −2 1 1 0 1 1 2



µ in mm

U, Magnitude+4.123e-03+3.779e-03+3.435e-03+3.092e-03+2.748e-03+2.405e-03+2.061e-03+1.718e-03+1.374e-03+1.031e-03+6.871e-04+3.435e-04+0.000e+00

S, Mises(Avg: 75%)

+1.412e+07+1.294e+07+1.177e+07+1.059e+07+9.413e+06+8.237e+06+7.060e+06+5.884e+06+4.707e+06+3.531e+06+2.354e+06+1.178e+06+1.268e+03

S Mises in Pa

Figure 10. Global deformation field and von Mises stress in single isolated arc model..


design. Therefore, it follows that the inter-lockedversion will be 53/33 ∼ 1.7 times heavier than theoriginal one. By multiplying number of slide jointsand distributing their position across the all length ofpanel, we expect a more uniform load transferbetween modules. Indeed, the results for von Misesstress, came from an elastic nonlinear analysis,confirms this idea. The maximum von Mises stress forself-weight load case, reduces to 1.15 Mpa formaximum total deformation of 1.3 mm, which is stillacceptable. This is true even though the interlockedconfiguration is 1.7 times heavier than the originalone. The main gain is on the structural rigidity, wherethe minimum natural frequency, calculated from amodal dynamic analysis is estimated to be 5.99 Hz.Using the values of natural frequency (f) and totalmass (m) for the original configuration (marked withsubscript 0 in (3)) and the improved inter-lockedversion (marked with subscript 2), one may comparethe relative equivalent structural stiffness (k) betweenthese configurations as represented in (3), concluding

that the new slide locks make the original structure165 times stiffer.

(3)

5. CONCLUSION AND FURTHER WORKThe structure presented in this article shows anexample for the design practice when the final formis driven by the connection technology, wherecomplex modular global forms are proposed bymeans of mutually supported simple folded panels.The proposed slide connection scheme inspires anew family of reciprocal frames, where instead oflinear members (beam or bar), planar (folded orelastically deformed) members are mutuallysupported. The connection between mutuallysupported members in this concept is integratedwithin the geometry of members, unlike thetraditional reciprocal frame system where theconnector members are employed.

k

k

m

m

f

f2

0

2

0

2

0

2

165=

≅


(a)

(b)

(c)

Openned U-joint

Figure 11. Toward a truss behaviour (a) CX modulemodification and intermediate panels (b) CVmodule modification and intermediate panels(c) Isolated arch reinforced with intermediate

panels on top and bottom fibbers.

CV2 CV2

CV1

CV2

CV2CX

1

CX0

CX1

CX0

CX−1

CX−1

CV1

CV1

CV−1

CV−2

CV−1

CV−2

CV1

CX0 CX0

CX1 CX1

CV −1

CV −1

CX −1CX −1

CV −2

CV −2

(IV)

(IV)

(III)

(III): Internal supplementary U-joint typology(IV): External supplementary U-joint typology

(III)

(IV)

(II) (II)

(I)(I)

(IV)(III)

Original CV

(a)

(c)

(b)

Interlocked CV

Figure 12. Increasing panel interlocking effect: two geometricconfigurations (a) Original (b) Interlocked (c) Comparing

geometric modification brought to CV module as well as itsconnection typology.

As a further work, the same concept is employedwhere instead of folded panels curved thin deformedpanels are used. The mutual supportiveness scheme ofpanels rests unchanged but the slide connectiongeometry needs to be determined based on a form-generation relaxation analysis. Moreover on suchprototype, instead of single folded modules,continuous thin flat panels will be used whereconnections are milled and fabrication is realized byelastically deforming panels and sliding. As aconsequence, transversal extension of the structurewill be straightforward.

ACKNOWLEDGEMENTAuthors would like to thank O. Baverel from Navierlaboratory of ENPC, for his contribution to thisstudy.

REFERENCES [1] Thun, M., DETAIL Zeitschrift für Architektur +

Baudetail, 2010. 50(10): p. 982–988.[2] Weinand, Y., Innovative timber constructions. Journal of

the International Association for Shell and SpatialStructures, 2009. 50(161): p. 111–120.

[3] Herzog, T., J. Natterer, and M. Volz, Timber ConstructionManual. DETAIL ed. 2000: Birkhäuser Architecture.

[4] Dunky, M. and P. Niemz (2002) Holzwerkstoffe undLeime: Technologie und Einflussfaktoren.

[5] Buri, H. and Y. Weinand, Uebersicht Massivholzplatten, in39. Fortbildungskurs Schweizerische Arbeitsgemeinschaftfür Holzforschung. 2007, SAH SchweizerischeArbeitsgemeinschaft für Holzforschung: Weinfelden. p. 63–84.

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[7] Barnes, M.R., Form finding and analysis of tensionstructures by dynamic relaxation. International Journal ofSpace Structures, 1999. 14(2): p. 89–104.

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[10] O'Dwyer, D., Funicular analysis of masonry vaults.Computers & Structures, 1999. 73(1–5): p. 187–197.

[11] Douthe, C., O. Baverel, and J.F. Caron, Gridshell incomposite materials: Towards wide span shelters.Journal of the International Association for Shell andSpatial Structures, 2007. 48(155): p. 175–180.

[12] Eigensatz, M., et al., Paneling architectural freeformsurfaces. ACM Transactions on Graphics, 2010. 29(4).

[13] Pottmann, H., et al., Freeform surfaces from single curvedpanels. ACM Transactions on Graphics, 2008. 27(3).

[14] Glymph, J., et al., A parametric strategy for free-formglass structures using quadrilateral planar facets.Automation in Construction, 2004. 13(2): p. 187–202.

[15] Bendsøe, M.P. and O. Sigmund, Topology Optimization:Theory, Methods and Applications. Springer ed. 2003:Springer-Verlag.

[16] Popovic Larsen, O., Reciprocal Frame Architecture.2008: Architectural Press.

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[18] Bachmann, H., Vibration problems in structures:practical guidelines. 1995: Brikhäuser.



Geometrical Description and Structural Analysis of a Modular Timber Structure

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