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GENERATIONOF FOLDABLE DOMESFORMEDBY BUNDLEMODULESWITHQUADRANGULARBASE

EmilioMartinGutierrez,Juan B.PerezValcarcelDoetorsofArchiteeture.ProfessorsofDepartmentofTechnologyofConstruetionofA CorunaUniversity

SUMMARY

Thepurposeof thisarticleis todeseribeapossiblewayof configw-ingdomesformedbyquadrangularbasebundlemodules.As for thefoldable systemsthe correspondingjoints shouldpermit certaincontrol/edmovementswhichin theirturnwouldcausetheprocessesoffoldingandunfoldingof theframeworkThebasemoduleis basical/ycomposedoffour barsconnectedwitheachotherneartheircentralpointbymeansofanauxiliaryelementandpins.Apartfromtheabovementioned,it canincorporateauxiliaryelementsfor stiffeningandfasteningapossibletextilecovering.

Keywords: Foldable Structures,Domes,Trusses.

1. INTRODUCTION. BACKGROUND

Thequadrangularbundlemodulepresentsa seriesof propertiesthatmakeit especiallycomplex.Actuallytheonlyfoldablestructures,madeupaftertheaforementionedmodule,oweto theSpanisharchitectEmilioP6rezPifiero.in 1964hecreatedamobileexhibitionpayilionhavingfour typesofprecinctswithamaximumvolumetricsizeof 1.40x1.00xl.80m.whenfoldedand12.60x9.50xl.00m.when completelyunfolded.Amongthe condi-tioningfactorsof theprojectwere:Themobilityoftheexhibition;thenecessityof itsadaptationtothephysicalcharacteristicsof differentplaces;easeofassembling,disassemblingandtransporting;andlittle time for projectingand realization(fivemonthsin total).The majorityof theelementsconstitutingthemoduleareof aluminiumsheetsof0.50xO.90m.,sothattherepercussionoftheweightof thestructureandtheroofisabout12.50kglm2.itshouldbetakenintoaccountthanoncethebasesystemisunfoldedit isnecessarytointroduceaddi-tionalbarstoprovidethestabilityofthecomplex.

The secondproposal(notrealized)comesup in1971aftera collaborationwithSalvadorDaUforthepreparationoftheTheatreMuseumofFigueras.It treatsa nonstructuralmechanismwhichwouldunfoldaccordingto a verticalplanuntilcoveringthemouthintheformof 18m.highand10m.widemid-pointarch.Thesystemshouldincorporate36rigidpanelscoveringtotallyoneof itsfaceswithjointsuptothetoptoserveasabasefora layoutofDaH.

The two projectsrespondto completelyplaneconfigurations,aconditionthatprovidesinadvancethegeometricalcompatibilityof thesystembothatthefinal anddifferentintermediatestagesof theprocessof unfolding.In thecaseof thedomesthisproblemtumsouttobeverycomplexasit isshowninthefollowingparagraphs.

2. GENERATIONOF THE SYSTEMIN UN-FOLDEDPOSmON

Figure1

Having analyzed different altematives,theprocedurepresentedby F6lixEscrigfromSevilleUniversitywas chosento be adaptedfor the

133

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resolutionof quadrangularbundlemoduledomes.In this caseapiane grid of a squaremodule,situatedatthelevelof thesite,is projectedoverasphericalareatakingas a focusthe pointthatminimizesthe possibledimensionalerrors ofadjustmentcorrespondingtodifferentjointsof thesystem(Fig.1). .

In resolutionwith the differentparametersthatinitiallydefinetheprojectionprocess,theonesthataredirectlyconnectedwith thecreativeprocesshave been chosen,I.e. those that would beinterestingfromthemerelyformalpointofview:

Sideof thesiteor of thebasereticlethatissupposedtobecoveredbythefoldingstructure.

Discretizationfrequencyor thenumberof thefractionsofthepreviousparameter.

Domethicknessin unfoldedposition.Thismagnitudedefinesthefinal distancebetweenthetwoframeworknodesthataremaintainedoverthevertical,drawnthroughthecenterofthesphere.

Theinteriororupperpoleheightof thesphereovertheplanethatcontainstheinitialreticle.

Thedeterminationof coordinatesisrealizedbytheintersectionof thespherewith thestraightlinesbetweenthecenterandthenodesofthereticle.Theresultantpositionsarestoredin a matrixformandtransferredto a PC-aideddesignenvironmentbymeansof interchangefiles.

Whentheprojectionis over,thedeterminationofthe angularsegments,comprehendedfrom thecenterof the sphere,is approachedfrom anequationsystemthatacquiresboththeconditioning

134

factorsof unfoldabilityandthederivativesof thesphericalconfiguration(Fig.2):

. Thesphericalsurfacecontainsall thecrossingofthetrussbeams(C).Thementionedpositionsshouldbededucedby subdivisionsof angularsegmentscomprehendedbetweentheprojectedpointsclosetoeachotherandthecenterof thespherecomplyingwith the conditionsoffoldability.

. Thenodesof theprojectedreticleareof type(D),includingthesuperiorpole.Asaresultthesuperior(A) and inferior(B) nodesof thestructuralsystemwill belocalizedontheradiiwhich,whenpassingthroughthecenter,crossthepositions(D).

. In orderto obtaina feasiblefolding,theequalityof consecutiveangularsegmentsshouldbe fulfilled:Xbi=Xbj;(ij) beingthemagnitudesthatareproducedonbothsidesofthesameposition(D).

. The semiopeningof thecross(di) mustbealwayssuperiorto thecorrespondingangularsegment(Xbi);inthecontrarycasethestructureisbent,hencethefoldingisnolongerfeasible.

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Figure3

Theresultingsystemappearstobeincompatibletoa degreethatincreasesnotproportionallyto thediscretizationfrequency.Nevertheless,it is de-tected,thatall the redundanciesproducingthe

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mentioned incompatibility are related to thediagonalsparallel to the bisector plane. As aconsequence,it is possible to avoid suchexpressionswithoutendangeringthefoldabilityofthe grid (Fig.3). This matteronlyrelatesto themajoror minoradequacyof theresultingstructureto thesphericalformwhich,judgingby the finalresults, tums out to be visually completelyirrelevant.

Reconstructingthe equationsystemwith suchcriteria,anindeterminatesetis obtainedwhereanadditionalexpressionis precisedin anycase.Theoptimalsupplementaryconditionrequirestheorthogonalityof X-shapedcrossedconformingthefirsttruss,neededfordevelopingaquadrantwhichis alsosymmetricalin regardtothebisectorplane.Taking this into account,a compatibleanddeterminatesetis obtainedthatcanberesolvedapplyingthe directmethodof Gauss,properlyreinterpretedbymeansofapivottechnique,asthequantityof nuHtermsthatpresentsthematrixcoefficientsishigh.

Figure4

j

The correspondinglongitudesof the bars aredeterminedfromthepreviousanglesapplyingthesinetheoremtothetrianglesformedbythecenterof the sphere,the interiorarticulationandtheextrernejoints(Fig.4).in ourworkthecorrespon-dingsemiopeningof thecrosshasbeendeducedinitsturnonthebasisof thedomethicknessdefinedatitshighestpointandwhentotallyunfolded.

Beforeproceedingto thedomegenerationfromthepreviouslongitudes,adetaileddeseriptionisneededfor theelementalmoduleattendingitsform,consti-

tutingelements,numerationpattemand coveringpossibilities.Eachframeworkmoduleis conformedby a trussof four barsconfiguringa quadrangularbase,prism. The mentioned scheme can beoptionallycompletedby cablesonthesuperiorandinferior faces and with a centralpieceproperlystiffenedto thetrussextremesthatcouldserveasa

sustenanceof thecoveringtextile(Fig. 5).

Figure5

Startingfrom the previousconfigurationthedeterminationof the final coordinateis laid outwhich,in its turn,is structuredin threedifferentroutines:Configurationof the first module,formationof subsequenttrussesandprocessingofthelastmoduleof eachlayer.in thefirstcasethelocalizationof theprincipalcrossnodesis realizedby meansof intersectionof circumferencesandsimplerelationsofproportionality.Onthecontrary,thetransversalcrossrequiresadditionalconditionscompatiblewiththeinitialhypothesisandwiththeconfigurationof the dome:Conservationof thecrossaperture,belongingto thesameplanethatcontainsthe centerof the sphere,and perpen-dicularitywithrespectoftheprincipalcross.

As a continuationa doublepathis established,displacingus alongthe differentlayersof theframework,fromthe,secondto thelastone,andoverthemodulesthatformthem,fromthefirsttothelastbutone.Eachoneis builtbeginningfromthe neighbouringmodulesituatedon its left,resolvingthepositionof thecentraljoint by the

135

r VOL. 43 (2002) n.140." r~.

intersectionof threespheres(Fig.6).Thelasttrussof eachlayerrequiresa specialtreatmentas itsprincipalcrossmustbenecessarilysituatedin thebisectorplaneto adequatelysatisfythe requi-rementsofsymmetry.

5

Figure7

In thelasttworoutinesmorethannecessarycondi-tionsarededuced,whichobligesto quantifythepossibleerrorsof geometricaladjustments.Forminirnizingthementioneddivergences,theprocessis resumedin twonestedloopsthatcoverrespec-tivelythepossibleareasfor theinteriorheightofthedomeandforpositioningthedenominatedfocusof projeetion.Combiningtheseparameterswithdifferentdomethicknessesand variousdiscre-tizationfrequencies,wecometotheconcIusionthatthemethodof projectionis viableandleadsto acompatiblesystemwhentotallyunfolded(Fig.7).

136

Thatis,choosingadequatelythevaluesofthemag-nitudesthatconstitutetheproblem,it is possibletodelirnitthemaximummaladjustmentsinsuchawaythattheywill notsurpasstheproperdimensionaltolerancesinherenttotheassemblingofthenodes.

INTERIOR HEIGHT OF DOME (Aim.)

Figure8

As for the interiorheigbtthevalidityrangeofcertainamplitudeisdeduced.Onlynoticeablyplaneconfigurationsandcoveringscloseto a semisphe-ricalshapearediscarded(Fig. 8).Figure9 showstheextremesituationsthatdefinethementionedarea.

On the otherhandthe optimalpositionof theprojectionfocusis placedaroundtheinferiorpoleofthesphere,sligbtlybelowit,whenthereisnoanymathematicaldependenceof anyof the involvedparameters(Fig. 10).

Thedomethicknessdoesnotsignificantlyaffectthefinal levels of the error, nor influences thedeterminationof the optimalfocus exceptfor thegridscIosetotheplaneconfiguration.

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A majorfrequencyof discretizationusuallyincre-mentsthevaluesof maladjustments,as muchastheyareaccumulatedaccordingto thenumberoflayers.Nevertheless,in usualsituationsthepro-blemsofcompatibilitysurpassingacceptablerangesarenotdetected.

Whenthepreviousoperationsarefinished,startstheprocedureofcornpletingthequarterofthedorneby rneansof consideratiohsof symrnetryandconfiguringthecompletefrarneworkby rneansofrotationroutines.In anycase,thelinksdefinedin.thedesignprocessareautornaticallyassignedfromthe developedinformaticsapplicationand thedifferenthypothesisofpossibleloadsarestructuredforanalyzingthebehaviourof thefrarneworkinthesituationofservice.

3. GENERATION OF INTERMEDIATE POSI-TIONS IN THE PROCESS OF UNFOL-DING

Thedefinitionof thestructuralsystemmustalsoconsiderthe differentphasesof the apertureprocess,withthepurposeof showingitscompati-bilityandanalyzingtheparticularitiesin itsbeha-viouralongthepath.

Forthisreasonthepreviousgenerationroutinesareusedbutvaryingconsequentlythepositionof twoinitialpointsof thesystem(1,2).Thefinaldistancebetween.thesepointscoincideswiththethicknessofthedome,whilein thecompactpackagepositionitreachesthesumof exteriorandinteriorlongitudesofthefirstangularsegment(Ll, L3).

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. ..', ~._._.........- VOL.43 (2002fn. 140. A' _,,,~-,-","_".n_"__",,,,._,._,,Thefirstconelusionof theinvestigationis thatthestraightlines34and56,whichconnectthesuperiorandinferiorextremesofthetransversalcrossinthefirstmodule,donotcrossin theoriginof coor-dinatesA, butdo it in thepositionA' situatedbelowthepreviousone(Fig. II). The distancebetweenthetwopointsincreaseswiththeprocessof foldingin sucha waythatthejumpsbetweenconsecutivefocusesA' areproducedwithsimilarmagnitudes(Fig.12).Thepercentageerrorineachphasedescribesadiagramelearlydescendingtotheoptimalposition,nearwhich it is practicallyahorizontalplateauofvariablewidth.

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'-'T~'tT'1~'"OPTIMAL. POSmONSOFTHEFOCUSFOREACHPHASE

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Themaximumerrorsareproducedin themodulesof thelastlayer.However,aireadyin thesecondlayertheerrorlevelsarenotnegligibleandlatertheyaccumuIateandgrowwiththeassemblingofthesubsequentmodules.Thissituationsdiscardsapossiblealternativewhichwouldconsistin main-tainingcertainnodesdisconnectedfromeachotherduringtheunfoldingofthegrid.

POSmON OF THE FOCUS FOR FORMINGTHE TRANSVERSAL

CROSS OF THE RRST MODUlE (m~

1110,810,434

Figure13

Ontheotherhandthelevelof theresultingerrordescribesa curvedpathwith an intermediatemaximumwhichcannotbeassumedwiththeusual

138

tolerancesoftheassembling(Fig.13).Theseresultspointouta possiblehypothesisaccordingtowhichthesystemisnottotallycompatiblethroughouttheseriesprecisingcertainenergeticinflowduringthecentralphases.

Thefactthattheerrorsproducedinthesecondlayeraresignificant,continuesto pointout one firstmodule,therealcoordinatesof whicharenotyetevaluated.If thehypothesisformulatedbeforeiscorrect,it canbealsodeducedthatthementionedmodulewill be,in someway,forcedduringthementionedphases;a questionwhichhasnotyetbeen contemplatedin the known processofgeneration.

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edge i ulesi

Figure14

Foranalyzingthispossibility,aquarterof thedomeis assembledwherealltheconnectionsbetweenthemodulesof contiguousbandsareuntiedexceptfortheonesthatcorrespondtothenodesplacedonthebisectorplane.It meansthatthe sequencesofmodulesarecreatedintheL-form(rowandcolumnof equalorder)whichareconnectedonlywiththepreviousandfollowingseriesby meansof thenodeson the45 degreesymmetryplane.All theeliminatedlinksaresubstitutedby newficticiousbarswiththepurposeofachievingtheconvergenceof itsextremenodestothesamepositionasofthedeformedone(Fig.14).it meansthatthesystemissubjected,throughaprocessof cakulus,tocharac-teristiceffortsnecessaryforobtainingtheadequateconnectionswithminimalenergeticinflow.

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Parallellyandconsideringthecomplexunfoldingofthewholedome,it is notpossiblefor theedgesXand Y to be separatedfrom the correspondingplanes XZ and YZ, a phenomenonwhich wasobservedbefore.To resolve it, the insertionofficticiousbars,whichlink eachpointof theedgeX -(Y) with its orthogonalprojectionover the planeXZ (YZ), is similarly used. The mentionedprojectionshavetobecompelledin theirtransversaldisplacement,to assurethattheedgepointswill befinallysituatedin theadequateplane.

At the endof theca1culusprocessthe correctconfigurationof theframeworkis obtainedandtheevaIuationof both the eriergyof defonnation,associatedwitheacheffort,andtheenergyrelativetothetotalityofthegrid,ismadepossible.

1000

900

800

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PHASE OF UNFOLDING

Figure15

8

Taking intoconsiderationtheresults,it is deducedthat the structure is not strictly compatiblethroughoutits aperture.However,the lattercandevelopif an externalinflow of energy,havingvariablemagnitudeaccordingto thephasethatthesystemgoesthrough,is produced.The mentionedparameterpresentsa variationcurvewith anintennediatemaximumvalue, elearly acceptablewithadjustmenttousualrealizationprocedures,anda path, descendingin accordancewith themovementtowards any of the two extreme

situations(Fig. 15).On theotherhand,as it isshownonthediagramtherequirementsnoticeablydescendwiththeincreaseofthedomethickness.

Figure16

Theenergyintroducedintothesystemis usedforreachingafeasiblepositionateachmoment,actingoverits.barsbasicallythroughthemechanismsofflexure.The associateddefonnationsimplicateminimalstressresponsein thepiecesalwaysbeingbelowthelimitofelasticityofmateriaL.As aresult,thedescribedproposalprovesthattheprocessesoffoldingandunfoldingarefeasiblewitha reasonableconcurrenceof energyandshowsthatduringitsdevelopmentthesystempresentsa correctbeha-viourinastatealwaysfarfromplasticenvironment(Fig.16).Thisdoesnotmeananylimitationinthepractice.Even if the structurewas totallycompatible,anenergeticinflowwouldbenecessarytoovercomethefriction.

9 4. GEOMETRICAL COMPATmILITY INTHE CENTRAL NODE OF THE TRUSS

Thedescribedmodu1esaredefinedinthefirstplacebythefactthattheirfourprincipalbarsarejoinedwith eachotherby meansof their intennediatenode.This connectionis resolvedthroughtheinterpositionofacylindricalpiece.Thismechanismmakesthe mentionedbarsmovelaterallywithrespectto their theoreticalintersection.Theeccentricitycan be of a local character,whencurvedorbrokenpathbarsareused,orconsistinahomogeneoustranslationofthewholestraighttube.Thesecondpossibility,shownonthediagram,leadstotheextremenodes,havingthesamescaleasof

139

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[9] Eserig, F.; Valeareel,J. P.; Sanehez,J."Deployablestructuressquaredin plandesignandconstruction".InternationalSymposiumonSpatial Structures:Heritage,PresentandFuture.Vol. 1,pp.483-493.Milan,1995.

[10]Valeareel,J. P. "Cupulas.desplegablesdegrandeslucesconmodulosdeaspas".i En-cuentroInternacionalde estructurasligerasparagrandesluces.FundacionEmilio PerezPifiero.pp.109-136.1992.

[11]Valeareel,J. P.; Eserig,F. "La obraarquitec-tonicadeEmilioPerezPifiero".BoletinAcade-micodela EscuelaT. S. deArquitecturadeA Comfia.N 17,pp.35-45.1993.

Figure19

[12]Valeareel,J. P.; Eserig, F.; Martin, E."Expandabledomeswithincorporatedrootingelements".Four InternationalConferenceonSpaceStructures.Vol. 1,pp.804-812.Surrey,1993.

[13]Valeareel,J. P.; Eserig,F.; Martin,E."Expandablespacestructureswithself-foldingtextilecover".InternationalConferenceonRapidlyAssembledStructures.Vol. 8,pp.283-295.Southampton,1991.

[14]Valeareel,J. P.; Eserig, F.; Estevez,J.;Martin,E. "Largespanexpandabledomes".InternationalCongresson InnovativeLargeSpanStructures.Vol. 2,pp.617-627.Toronto,1992.

141

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the interiorjoint, thatin its tum causeseccen-tricitiesinthementionedconnections.

Figure17

Thediamet~roftheinteriorjointpieceturnsouttobe fundamentalfor thepossibleevolutionof themoduleinsofarasaminoraperturecorrespondstoaminordimension:Withtherotationof thebarsacontactmaybeproducedbetweenthembeforethepositionoftotalunfoldingisreached(Fig.J 7).

The parametersthatdefinetheproblemcanberelatedthroughthefollowingexpression:

tan 13=sina . Dc =cos13 .(~ +sina

)Db sma

. De Diameterofthejointpiece.Db Diameterofthebars.a Angle formedby the bar with its

horizontalprojectionovertheinferiorfaceofthemodule(Fig.18).

..

Figure18

140

lt isnecessarytotakeintoaccountthatagratevalueDeofdiameter,pushingasideaestheticalquestions,implicatesmajoreccentricitiesin thebehaviourofthe forces of the element.For reducingthementioneddimensionto acceptablelimits thedescribedangleis tobeincreased,whichmeanstoalterthethicknessofthedomeorthediscretizationfrequency.

The applicationprogramdesignedin accordancewiththisworkpermittedtoconfiguredomesasit isdescribedintheexampleof Fig.19,thatwouldco-vertheprecinctof35x35m.withthepossibilityofincorporatinginteriorandexteriorcoveringtextIles.

s. REFERENCES

[I] Eserig,F. "Estructurasespacialesde barrasdesplegables".Informesde la Construcci6n.Vol.36,nO365,pp.35-46.Madrid,1984.

[2]Eserig,F. "Expandablespaceframestructu-res".ThirdInternationalConferenceon SpaceStructures.pp.845-850.Surrey,1984.

[3]Eserig, F. "Expandablespacestructures".InternationalJournalofSpaceStructures.Vol.I,nO2,pp.79-91.1985.

[4]Eserig,F.; Valeareel,J. P. "Curvedexpanda-ble spacegrids".InternationalConferenceonthe Design and Constructionon Non-conventionalStructures.Vol. 2, pp. 157-166.Edinburgh,1987.

[5]Eserig,F.; Valeareel,J. P. "Geometryof ex-pandable space structures".InternationalJournalof SpaceStructures.Vol 8. nO1-2,pp.71-84.1993.

[6]Eserig,F.; Valeareel,J. P. "To covera swim-ming pool with an expandablestructure".InternationalConferenceonRapidlyAssembledStructures.Vol. 8, pp.273-282.Southampton,1991.

[7]Eserig,F.; Valeareel,J. P.; G,o. "Designofexpandablesphericalgrids". Ten yearsofprogressin shell and spatial structures.CEDEX-IASS.Vol.4.Madrid,1989.

[8]Eserig, F.; Valeareel,J. P.; Sanehez,J."Deployablecoveron a swimmingpool inSeville".Journalof the InternatIonalAsso-ciationforShellandSpatialStructures.Vol.37,nO120,pp.39-70.1996.