1994-4 Expandable Structures With Incorporated Roofing Elements

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86. Expandable domes with incorporated roofmg elements

Space structures 4. Thomas Telford, London, 1993

J. B. VALCARCEL, F. ESCRIG and E. MARTIN, ETS Arquitectura,Spain

INTRODUCTION.1. Unfolding bar stlUctures offer an inLeresLing solution in almo l all silualionsthat require easy transport and a quick means oC proyiding roofing. Over tbe pasttwo years our team has sludied various lypoJogies of groups of bars, which alsooffer in addiLion lhe abiJity to cover a largc area when unfolded. Figs 1 and 2how an exampJe of a dome of 6 m diameler built as a model to tilO scale in the[Wo stages of enlirely foJded up and unfolded.2. The main diffcully posed wilh this lype oC lrucLure lies in covering Lheunfolded structure. Designing effective covering mechani ms thal can be foldedwith tbe complete package of bars and which cover the lructur when unfolded.is a challenge. 10 actual faer, only Lextile cover ofier effective olutions. l f one

requires a covering consisting of rigid elemenLS, Lhese musl be placed in thetructure after complete unfolding, which often resulLS in a dangerous lask oCworking at a greal height.3. Texlile covers can be designed in such a way lhal the slruclure iLSelf causiLS own folding and unfolding logether. Cerlain facl are otIered relaLing LO thisproblem ( ee ref. 5) and lhose solution developed by our team are given.However, lhe aim of this projecl is LO con del' lhe po sibility of covers formedfrom rigid elemenLS, with which grealer performance and durabiliLy can beachieved.

SYNOPSIS. In this paper the problems of roofing non-textile elements inexpandable stluCLures are sLudied. Two soluLions have been proposed. These areystems made by folded plates lhaL can be expanded wiLh the whole structure andr ofing plate elemenLS, which may be assembled on lhe ground and laLer adaptedin the fInal slage of unfolding. It i possible for Lhese elements to add to the finalsU'engLh of lhe sLructure, improving it performance. Design aspects of suchLructures are sLudied as well as, the probJems related lO lheir unfolding and theeffect of these roofing elements on the strength of the structures. At the sametime, specific examples of the possibilities of this system are proposed.

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y ALCARCEL ET AL.

Fig. 1. Fig.2.ROOFING SYSTEMS WITH RIGID PANELS4. One saluan could be unfolding me structure and altaching lO it a et of Iigidpanels joined to its nodes. As shall be seen later, fixing these rooting elementsconsiderably improves the resistance of !he unIolding structur and al o ha' !headvantage that the panels can be adjusted to achieve a nearly air tight pac . Onedi advantage, already mentioned, i lhat it is neccssary to work al a eellain heightand for prolonged periods of Lime, negaLing one of the main advantage ofunfolding struclures namely their ea e o[ erecLion.5. As faI back as 1971, lhe Spani h arehitecl Emilio Prez Picro proposed forlhe Dali Museum in Figueras, Spain, ao unfolding glass roof. This was formedby a truss of square module eonsisling of bundle which held gla s heels bymeans of ero -piee . The e sheets were lO be painled by Dali himself. The glaswork was not con trueled due to Prez Piero' dealh, bUl a model of !hetruClure was buill. Fig. 3 shows lwO phases of unfolding for thi mode!.6. Similar ystems to tbls can be used as weJI [01' unfolding lrUClureeonsisting of square cross-pieces. Fig. 4 shows the funclioning of one of thesemodulus in two sueces ive stages of unfolding. Il is worth o Ling thal in theunfolding process each of the rigid panel fit into lhe previou ones, lhuproducing a compact unit.7. This system is more convenient the more regular the ehosen module is. Forthis reason, the best advisable applications would be those trusses usingconsistenl modules such a flat 1I11 es and cylindrieal domes.8. Flat trusses (Fig. 5) are very simple structures bUl they aJIow for rather smal1spans, because lhey are not terribly effieienr from a. lruclural poinl 01' vi w. Theyare however, very uncomplicated to build and a syslem uch a thal describcdcould effectively solve the problem. In the figure, it can be en thaL is po sible lOdesign the panels in such a way as lO permit the overlapping of lhese andtherefore increase me efficiency of the covering process.

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SPACESTRUCTURES4

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Fig. 3.

Fig.4.9. Cylindr ical domes (Fig. 6) are the most useful structures. If they areconstructed from square modules, the panels can even be made to unfold in twodirections by which the problem of the top joint can be solved through anappropriate sealer.

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VALCARCEL ET AL.10. When designing suitable roofing elements U ing triangular modules a s riouproblem is encountered. A geometric incompatibility exislS which prevents thecomplete folding up of the structure (Fig. 7). Il can be easily shown thal, onclosing, the panels lrip one another up. thus preventing complete closure.Neverlheless, the authors believe tbal this uuctural solution offers such clearadvantages that the system should not be rejccted for thi reason. It is d cidedlypossible to use this structure in a similar way a other foldable slructur s. Thjscan be carried out by partially unfolding the struclure on Lhe ground, a mblingthe roofing elements in very favourable conslmction conditions, and sub equentlycontinue the unfolding process until total unfolding is reached. This strategyopens up an enormous range of possibilities, far greater than those of systemsbased on the complete folding of the truss and roofing elements.

Fig.S.

Fig.6.

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Fig.7.

Fig. 8.

SPACESTRUcrURES411. These type of elements can be used in numerous structures based ontriangular modules either of cross-pieces or bundles. Fig. 8 shows a design for ahexagonal sun-shade with the successive unfolding stages. At this point in timethe authors are working on several projects dealing with unfoldable domes withlarge spans. The authors have been able to ascertain that the way in which theyunfold is not very conventional, because the structure is kept fIat until almost thefinal stages of unfolding. Only at the end, does the structure curve up on its own,unfolding and establishing itself in its final position. This process is described in

detail in the references stated and can be observed in Fig. 11.

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VALCARCEL ET AL.12. This special manner of un(olding permits a new roofing strategy; that is, theuse of rhomboid rigid panels wilh a hinged joint. These panels could be placedover the structure in Lhe fOUIlh tage 01" unfolding, overlapping like the scales of afish. This process can be carried out almost entirely on lhe ground, so lhe aClualconstruction on site is simplified and later the unfoJdjng may be completed. Ifthen the free ends of the panels are fixed onto the corresponding node ,exceJlentstructural perfOlmance can be achieved, as will be seen latero

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Fig.9.

Fig. 10.

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Fig. 11.

SPACESTRUCTURES4

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In Fig. 12, the final state of the dome with its roofing is shown.ROOFING STRENGTH13. I f the roofing elements are conectly fixed, it is possible to count on theirstrength adding to the rigidity of the structural unit, which is of paramountimportance, because these structures tend lO have high defleclions, requiringcontrol.

Fig. 12. /

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VALCARCEL ET AL.14. As indicated in refs 3,4 the method of calculus developed is based on theuse of matr ix methods for the deployment of bars with inside joints. In thesereferences the stiffness matrix employed is given and this differs considerablyfrom the stiffness matrix for conventional applications.15. It is cIear that the s tudy of these elements must be undertaken using thefinite element method. The interaction of finite elements with a matrix programalready developed, resuIted in several problems. The solution adopted consisted

of calculating the stiffness of each one of the roofing panels and condensing thedegrees of freedom corresponding to the restraints a t the nodes. This aI lowscalculation 01' the stiffness matrix corresponding to the elements which can thenbe assembled to form the stiffness matlix of the whole stlUcture.16. The element used was a six nade tliangular type, because it aIlows greaterprecision with a less close-woven tIUSS.

""

lff:iCHT / P IN fXPANDAlJII DDMfS

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WflClff / !P iN fXPNlIJABLE SUNSllADfS

//" /\ ,/ /--- " .-

10 7Q JO... OOfJ( WfTHWr ROOfJNC PrAlrs DO"''''TH PWfS

11 1:J 11! 21~ r J ~ I Q m... SJ.M5H,Ialll/1fWrRO()fW(;PlAfE!

11m RrxY"Nj PLAlrs

Fig. 13.17. To demonstrate lhe efticiency of the strength factor of the roofing elements,two types of struclures have been considered. Firstly, a hexagonal sun-shade wasstudied, and secondly a spherical dome. In both cases, dimensions of the barswere adjusted strictly to carry the forces exerted upon them. Hollow tubes of

rectangular section were used, complying with the Spanish Standard MV-108.The design load was 60 kg/m2, corresponding to lhe simultaneous loading ol' selfweight plus snow for the area.18. The drawings show the variation in weight per m2 as the span 01' lhestructure increases, both when rigid elements in the roofing collaborate with t h skeletal structure and also when they are inoperative.

CONCLUSIONS19. The aboye explanulion indica les lhal 'cveral roofing solut ions withunfolding structures 01' rigid elemenls are pos ibJe. In sorne cases the coveringcan be incorporated in the bundle 01' bar fonning lhe Slructure, while in others a

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SPACE STRUCTURES 4slight initial unfolding will be necessary, first with the subsequent incorporationof the roofing elements and later completion of the unf01ding process. It is alsopossible in certain cases to unfold the covering near completion of the structure inorder to fit the roofing elements and subsequent1y finish unfolding.20. The resistant efficiency of these elements depends almost entirely onwhether they are fixed in their final position following unfolding. I f this is done,considerable material saving and greater efficiency in resistance and deformationof the structure will result.ACKNOWLEDGEMENTS21. The authors are grateful to scholarship receivers Jos Antonio Vzquez forhis collaboration in C.A.D. graphics and Jos M. Asorey Brandariz for his helpwith the photographs.REFERENCES.1. PREZ PIERO, E., CANDELA, F., DALI, S. "La obra de Prez Piero".Arquitectura. Madrid nO 163-164. 1972.pp 1-28.2. ESCRIG, F., P.VALCRCEL, lB . "Curved Expandable Space grids". Nonconventional Structures '87. London 1987.3. P.VALCRCEL, lB., ESCRIG, F. "Ana1ysis of curved expandable spacebar structures" Int . Symposium on 10 Years of Progress in Shell and SpatialStructures. IASS. Madrid 1989.4. ESCRIG, F., P.VALCRCEL, J.B. "To cover a Swimming Pool with anExpandab1e Structure". Inlernational Conference on Mobile and Rapid1yAssembled Structures. MARAS'91. Soulhampton 1991.5. P.VALCRCEL, J.B.; ESCRIG, F. "Expandable Slructures with Self-foldingTexti1e Cover". International Conference on Mobile and Rapidly AssembledSll1.lclures. MARAS'91. Soulhamplon 1991.6. P.VALCRCEL, J.B. ESCRIG, F.ESTVEZ, J. MARTIN, E. "Large SpanExpandable Domes" Int. Symposium; on Large Span Slruclures. IASS. Toronlo1992.

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1994-4 Expandable Structures With Incorporated Roofing Elements