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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
-
-~i
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.
/'('Yii
___)X
Figure3
Theresultingsystemappearstobeincompatibletoa
degreethatincreasesnotproportionallyto
thediscretizationfrequency.Nevertheless,it is de-tected,thatall the
redundanciesproducingthe
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STRUCTURES: lASS
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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.
'ST ODlmalvalue
38 - - - - - - - - -1--- - 1---- - - - - - -
34 - - - - - - - - foo- - - foo-- - - - - - -33 - - - - - - - -
- - - - - - -32 - - - - - - - - foo-- - - - - - - - -31- - - - - -
- - foo- - - - - - - - - -
29-- - - - - - - - - -- - - foo- - - - - - - - - - -28 - - - - -
- - - -- - - - - - - - - - - -11 - - - - - - - - - - - - - - - -
-29-- - - - - - - - - - - - - - - - -
-
--- . --"-'- --. - -_.
~ J()LJRNALOF
IHEJ'f\lIE_RNATiONA.iA,ssociA'riNFOR'si-rE[LAr\fD~sPAl'IALsrR6TuRES'~iAss'_'0_'_
nC__ .'", .,,'. _' ,'., . _0:_' ' _o _,,, -- ':., . 0'-- -.- "...,
'.-' "__ o.. " ' --- ~ ' " _ .
Figure9
~i!5
W
~.ci.i&.Oen
~~.f,Oi&.OZO
~.ci.
~~
i.
INTERIORHEGHTOFDOME(AI m.)
Figure10
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).
8 o )X7
Figure11
137
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i - - - - - - - - -1--- - - - - - - - -- - - - - - - - - -- - -
- - - - - -- - - - - - - - -1-.- - - - - - - -- - - - - - .1-.- - -
- - - -- ffii - - - - - - - - -- - - - - - - - - if- - - - - - - -
- - - - - - - - en- - - - - - - - - - - - - - - - w
- - - - - - - - - - - - - - - - j!:i= i&.LI. - - - - - - - -
- - - - - - - - _ oo - - - - - - - - - - - - - - - =- - - - - - - -
- - - - - -- - - - - - - - - - - - - - -
Lg;ci::o - - - - 'S.:=- - ;;,;"'---1---- - - - -- -=1
1-__ i:a: - - - 1-- - - - - _ ww j!:- - - - - - -1---- - - - - -
- - i&.o- - - - - - - -1---- - - - - - -=
- - - - - -1---- - - - - - -- - - - - - - -1---- - - - - - --1
-- - - - - - - -1---- - - - - - -- - - - - - - -- - - - - - - - --
- - - - - - -- - - - - - - - -
- - - - - - - -1---- - - - - - - --1 - .. .. S!;:---- 8!RN
-
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_,,,~-,-","_".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.
-1-....--
Jii..
j:"-
'-'T~'tT'1~'"OPTIMAL. POSmONSOFTHEFOCUSFOREACHPHASE
Figure12
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.
_~~ bar~ modbar
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.
a a a a a a a a2 is.Ift -- -- -- -- -- -- -- - - -- -- -- ----
---- -- -- 2 to: _J.[;".. -- -..-- -- --i i19-- -- .i.--- -- -..--
t- -- -- _J.-- ol._--+-- i i i ii .i._ _.i. t- _J. t- nt'-- - -- i
-- -- i -- -- -- i ---
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JOURNAL OF THE INTERNATIONAL ASSOCIATION FORSHELL AND SPATIAL
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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
o1 2 3 4 5 6 7
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
-
:1
i
i
!
i
i
[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
.. ,. '.;" ..t',io".,..' " '\,'.. .'. ._" .,' I,") . . ;
-
J-'.
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.