pPi

807ofes

appr a tbeintrthecomd to

2013 Elsevier Ltd. All rights reserved.

ng for

TM) foocedule onlyemplo. As se, for in

the planar triangular nite element with constant stresses andstrains. This conversion is explained in further detail in [5]. Formore formulations, examples, and details of FDM, the reader mayconsult Refs. [13,69].

e closed netep constitf the equil

of the topology network (see Fig. 2). Different colors of nodes inFig. 1a and b depend on the type of network employed. The presentstudy deals only with the closed networks of the A and B relation-ship. Therefore, steps will be referred to as rings hereafter. For fur-ther details on TM, the reader should consult Refs. [1,7].

Different basic networks may be used together in order to get acombined topology. In this case, it is necessary to dene thedesired sequence of combinations (for example AABBB). Thissequence will give the relationships between nodes of consecutive

Corresponding author. Tel.: +34 617619612.E-mail addresses: jfcarbonell@ugr.es (J.F. Carbonell-Mrquez), rjurado@camino-

s.upm.es (R. Jurado-Pia), mlgil@ugr.es (L.M. Gil-Martn), emontes@ugr.es (E.

Engineering Structures 52 (2013) 6468

Contents lists available at

Engineering

lseHernndez-Montes).can be transformed into a supercial nite elements model using shape for an open network is formed by the nodes of the contourthat dene a matrix which contains the information for the con-nectivity of the net nodes. Once the geometry and the forces inthe branches are determined by means of FDM, the cable model

the contour of the equilibrium shape [1]. First, thare those for which the nodes located in the last scontour of the equilibrium shape. The contour o0141-0296/$ - see front matter 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.engstruct.2013.02.011tworksute theibriumguess of the shape is needed in order to generate a network. Theadvantage of TMFDM is that an initial guess is not needed; onlythe connectivity of the nodes is necessary to nd the solution tothe problem in contrast to the mapping methods that are basedexclusively on geometry. The only inputs to trigger the TMFDMare the coordinates of the xed nodes, the forcelength or forcedensity ratios for the net branches and a series of topological rules

to two nodes of the following step. Case C is such that each node ata given step is connected alternatively to one or to three nodes ofthe next step [1,7]. For the three types A, B, and C, each node is con-nected to the adjacent ones on the same step. Fig. 1a and b showsthe typical relationships or basic networks A and B.

As described earlier, two possible types of networks are possi-ble: open and closed. These two types of networks are related toTopological MappingForce Density Method

1. Introduction: Topological Mappi

The use of Topological Mapping (been introduced [1] as a mapping prMethod (FDM) [2,3]. TM is applicabobtaining a rst equilibrium shapei.e. to do the form-nding processexamples presented in our literaturetension structures

r tension structures hasre for the Force Densityto FDM as a means ofying topological rules,en through the manystance, in [4], an initial

TM provides the connectivity of the network according to a setof given data: the number of nodes in the rst step, the number ofsteps, the type of relationship between consecutive steps and thealternative to close or open network.

Three types of relationships are dened between steps: A, B andC. Relationship A corresponds to a pattern in which each node at agiven step is connected to three more nodes of the following step.In the case of relationship B, each node at a given step is connectedForm-ndingSymmetry

structures. Several examples are presented in order to better illustrate the applicability of these easyprescriptions.Short communication

Symmetry preserving in Topological Map

Juan Francisco Carbonell-Mrquez a,, Rafael Jurado-Enrique Hernndez-Montes a

aDepartment of Structural Mechanics, University of Granada, Campus de Fuentenueva, 1bDepartment of Civil Engineering-Transport, Technical University of Madrid (UPM), C/Pr

a r t i c l e i n f o

Article history:Received 17 April 2012Revised 4 February 2013Accepted 11 February 2013Available online 15 March 2013

Keywords:Tension structures

a b s t r a c t

When using Topological Mrst form of equilibrium foof a regular polygon, it hasmapping procedure that isthe topology for the mesh,tern generation process berithm that has been prove

journal homepage: www.eing for tension structures

a b, Luisa Mara Gil-Martn a,

2 Granada, Spainor Aranguren, s/n, 28040 Madrid, Spain

ing (TM) together with the Force Density Method (FDM) in order to nd aension structure whose xed nodes coincide in plan view with the verticesen observed that the symmetry may be lost. The problem is related to theoduced in the process. However, if some rules are followed in constructingsymmetry can be preserved. With a symmetric mesh, the later cutting pat-es simpler, easier, and cheaper. This paper will provide one simple algo-work effectively in preserving the symmetry in the cited type of tension

SciVerse ScienceDirect

Structures

vier .com/locate /engstruct

s: (a

ineeFig. 1. Types of basic network

J.F. Carbonell-Mrquez et al. / Engrings. Therefore, if the mesh has N rings, the sequence of topologyconsists in N 1 letters indicating A or B relationships.

It is important to point out that, the xed points are assigned tothe nodes of the network through the calculation of the distancebetween consecutive xed points and the computation of theperimeter formed by them so that the distribution of nodes ofthe last ring is done proportionally to the real distance betweenthe xed points.

According to the above mentioned, the process for generatingclosed networks requires a reduced and simple data input: thelocation of the xed nodes, the force densities of the interior andexterior branches and the topology of the net given by the se-quence of combinations between rings.

2. The loss of symmetry

Let us imagine that we are dealing with a tension structure inwhich the oor plan is a regular polygon. Once the equilibriumshape of the tension structure has been found with TMFDM, thestructural shape has to be converted into a set of planar clothsfor its fabrication [7]. This is a process called cutting pattern gener-ation [10,11]. As this is not a simple process, one may take advan-tage of the symmetry of the tension structure since a non-symmetric mesh may generate different patterns leading to a moreexpensive process.

However, it is observed that depending on the adopted se-quence of relationships between rings, the mesh may or may notbe symmetric. Fig. 3 shows two meshes with symmetric supports;the only difference between them is the sequence of relationshipsbetween rings: BAA for Fig. 3a and AAB for Fig. 3b. As a con-sequence, Fig. 3a is symmetric whereas Fig. 3b is not.

Fig. 2. Types of network for the same structure: (a) type A; (b) type B. From [1].

ring Structures 52 (2013) 6468 65By observing how the mesh of Fig. 3b is built ring by ring(Fig. 4a), it can be noticed that this is always a symmetric proce-dure. According to this, the topology of the mesh will always besymmetric. However, when the last ring is reached (see Fig. 4b),their nodes must take up their prescribed distribution accordingto the space between xed nodes, as explained earlier, makingthe mesh asymmetric. Therefore, the key point to preserving thesymmetry of the topology is the spatial symmetric distributionwhich needs to be compatible with the prescribed distribution ofthe contour of the tension structure. This compatibility will bereached if the axis of symmetry of the mesh and the regular poly-gon of the contour coincide after distributing and positioning themesh nodes including the contour nodes (Fig. 3).

3. A proposed algorithm for preserving symmetry

Once the reasons for loss of symmetry in the mesh construc-tion have been explained, a new algorithm for preserving thesymmetry is presented. This algorithm will let the designerknow if the topology introduced in the FDM produces as a resulta symmetric structure without needing to visualize the entiremesh.

Fig. 1 shows how relationships A and B work. Starting from therst ring nodes, the mesh grows outwards. Thereby, the mesh maybe understood as a core linked to the exterior xed nodes by meansof a number of branches. These branches will be joined together sothat all the nodes of the contour remain linked to the core. Fig. 5ashows the case for a pentagonal-shaped structure. Here there areve rings with a sequence BBAA. The branches that link theexterior points located in the axis of symmetry are illustrated bythick lines. These branches can be plotted separately from the

) open network; (b) closed network. From [1].

axisy ax

inee(a) ContourTopolog

66 J.F. Carbonell-Mrquez et al. / Engremainder of the mesh by knowing the sequence that denes therelationship between rings, see Fig. 5b. In doing so, if one of thexed nodes of the contour belonging to the axis of symmetry isreached in a symmetric way, without altering the symmetry, theentire mesh will be symmetric.

In order to visualize some examples of the construction of thereferred branch plot, the corresponding branches are also repre-

Axis of symmetry of the polygon

(c)

Fig. 3. Two different topologies for the same structure: (a

(a) Fixed node Regular nod

Fig. 4. Building of a mesh: (a) network sketch without connections between penultimatbut last connections with an asymmetric layout.

(a) Fig. 5. (a) Topology for a pentagonal-shap(b) of symmetry is of symmetry

ring Structures 52 (2013) 6468sented for the two different meshes given in Fig. 3a and b. The plotcorresponding to mesh (a) is (c) and the one corresponding to (b) isplot (d).

It must be pointed out that the number of nodes in the rst ringcannot be any number, but rather it must be the same or a multipleof the contour number of sides. Nevertheless, this rule alone doesnot guarantee symmetry.

Axis of symmetry of the polygon and of the branch

(d)

and c) mesh and branch for BAA; (b and d) AAB.

(b) e

e and last rings; (b) complete network sketch with symmetric internal distribution

(b)

Axis of symmetry of the polygon

Core node

Branch node

Branch link

ed structure; (b) one of the branches.

A

Plot axis and first node

Plot Plot Plot Plot

Plot node corresponding to

next ring

Yes

Yes

Yes No

No

No

A B B

Asymmetric network Symmetric network

N rings Topology sequence

Is former node on axis?

Ring type Ring type

Is the branch symmetric and the last

node is on axis?

Are the plotted nodes in last

ring?

it = 1

N rings SeqVec=Topology sequence

NumRingit 1 2

NumRing1 = 1

SeqVecit A B

NumRingit+1 = 1 NumRingit+1 = 1

SeqVecit A B

NumRingit+1 = 2 NumRingit+1 = 1

it = N-1

Yes

1 2

Asymmetric network Symmetric network

NumRingN

No it = it + 1

(b) (a) Fig. 6. Proposed algorithm for analyzing the network to be used: (a) graphically and (b) analytically.

(a)

(b)

(c) B A B B A B A B B A B B

(d) Fig. 7. Symmetric tension structure: (a) plan view; (b) perspective; (c) attened pattern; (d) branch plot.

J.F. Carbonell-Mrquez et al. / Engineering Structures 52 (2013) 6468 67

From the above observations, the algorithm has already beenestablished with the procedure presented in the ow charts inFig. 6. The algorithm is presented in two different ways: (a) graph-ically, as originally conceived, and (b) analytically so that it can beeasily implemented in any computer language for its program-ming. The rst way consists of plotting a singular branch of themesh according to some given prescriptions. If the last ring isreached with a node in the axis with the plot of the branch beingsymmetric, as stated above, the network will be symmetric. Onthe other hand, the second method starts with the storage of thetopology sequence in a vector called SeqVec. Later on, an iterativeprocess controlled by the counter it, will ll a vector with the samenumber of components as the number of rings, NumRing. Each ofthese components is related to each ring and they will be equalto 1 or 2 depending on the type (A or B) of the former one. If a 1is assigned to last ring, the entire mesh results symmetric.

Although the approach is only valid in tension structures whosexed nodes coincide with the vertices of a regular polygon, thesekinds of tension structures are commonly used and, therefore,the eld of application of the method is still important.

Logically, tension structures whose xed nodes (or its projec-tions) are the vertices of a regular polygon should be symmetric.However, when applying Topological Mapping (TM) and the ForceDensity Method (FDM) for form-nding it has been observed thatin many cases the resulting tension structures become asymmetric.Preserving symmetry is therefore of great interest in the mappingof tension structures.

The reasons for this loss of symmetry have been explained and anew algorithm for preserving the symmetry has been presented.The rules for the algorithm are related to the sequence for the rela-tionship between the rings and the number of nodes of the rstring. A simple and useful rule is given in order to let the user knowif the topology to be introduced in FDM is symmetric.

Acknowledgements

This research work was carried out under the nancial supportprovided by Spanish Ministry of Education and Science as part ofthe Research Projects BIA 2007-62595. The rst author is a Spanish

68 J.F. Carbonell-Mrquez et al. / Engineering Structures 52 (2013) 64684. Example

It has been explained above that the preservation of symmetryis a basic element for making the pattern generation task easier.

Fig. 7 shows an octagonal-shaped tension structure that hasbeen materialized by means of 13 rings with 8 nodes in the rstring. The relationship between the rings is BABBABABBABB. As one may appreciate in gure (Fig. 7a), the mesh issymmetric. However there is no need to plot the entire networksince the symmetry may be evaluated with the presented algo-rithm (Fig. 7d).

Fig. 7c also shows the only pattern needed for the constructionof the whole tension structure. Using this pattern 16 times, the ten-sion structure will materialize.

5. Conclusions

Preserving the symmetry in tension structures is extremely use-ful; it makes the patterning simpler, easier, and cheaper since it al-lows building the tension structure with just a few differentpatterns.Government PhD fellow (FPU Grant AP 2010-3707). Their supportis gratefully acknowledged.

References

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[5] Lai CY, You Z, Pellegrino S. Shape and stress analysis of symmetric CRTSreectors (CUED/D-STRUCT/TR170). Cambridge: European Space AgencyContractor Report; 2001.

[6] Linkwitz K. About formnding of double-curved structures. Eng Struct1999;21(8):70918.

[7] Jurado-Pia R, Hernndez-Montes E, Gil-Martn LM. Topological mesh for shellstructures. Appl Math Model 2009;33(2):94858.

[8] Koohestani K. Form-nding of tensegrity structures via genetic algorithm. Int JSolids Struct 2012;49:73947.

[9] Snchez J, Serna M, Morer P. A multi-step forcedensity method and surface-tting approach for the preliminary shape design of tensile structures. EngStruct 2007;29(8):196676.

[10] Moncrieff E, Topping BHV. Computer methods for the generation of membranecutting patterns. Comput Struct 1990;37(4):44150.

[11] Kim J-Y, Lee J-B. A new technique for optimum cutting pattern generation ofmembrane structures. Eng Struct 2002;24(6):74556.

Symmetry preserving in Topological Mapping for tension structures1 Introduction: Topological Mapping for tension structures2 The loss of symmetry3 A proposed algorithm for preserving symmetry4 Example5 ConclusionsAcknowledgementsReferences