Retractable roof structuresP. E. Kassabian, BA, MEng, Z. You, BS, MS, PhD, andS. Pellegrino, LaurIng Civ, MA, PhD

j This paper presents a new concept forretractable roof structures. The newstructures consist of a foldable lattice ofbeams connected by cylindrical joints, towhich covering panels or membranes areattached. These structures fold towards theirperimeter and there is practically no limit totheir shape. Solutions to the key problemsthat have to be solved in the course of thekinematic design of this new type of structureare presented, including two different waysof connecting them to fixed foundation pointswhile maintaining their internal degree ofmobility, and how to determine the shapesof the covering panels to avoid interferenceduring retraction.

Keywords: buildings, structure and design;structural frameworks

IntroductionRetractable roofs are playing an increasinglyimportant role in the development of flexiblesports facilities that can be operated in ‘optimalconditions’, i.e. with the roof open as long asallowed by the weather, throughout the year.This requires roofs that can be opened andclosed in a few minutes, at the push of a button.A recent report by a working group of theInternational Association for Shell and SpatialStructures1 lists 23 retractable roofs, wherethe movable part typically spans in excess of50 m, of which 14 were built in the 1990s. Themost commonly used folding schemes involvelarge, rigid elements undergoing translation orrotation (Fig. 1). To reduce the amount ofground that remains covered by the roof in theretracted configuration, these elements areoften overlapped (Figs 1(a) and 1(b)) or folded(Figs 1(c) and 1(d)). Many variants on theseconcepts have been devised.1,2

2. Among the alternative approaches,Fig. 1(e) shows a solution in which the retract-able cover is a flexible membrane, whoseattachments to a cable net can be moved auto-matically. This scheme has been used a numberof times, mainly in Europe, but has proved lessrobust and durable than the earlier schemesbased on more massive elements.

3. A less conventional structural conceptwas pioneered by the Spanish engineer andinventor Pinero, who patented a system for

movable theatres that could be erected anddismantled in only a few hours and by a smallnumber of people.3 Pinero’s concept was basedon expandable trusses, forming a kind of three-dimensional lazy tongs. Fig. 2 shows a recentimplementation of this concept in a design fora dismountable roof cover for a swimmingpool.4 This structure consists of a membrane,attached to three identical expandable trussesmade from aluminium-alloy tubes joined at theends and approximately in the middle by‘scissor joints’, which are delivered to the sitecompactly packaged and with the membranealready attached to them. Fig. 3 shows twophotographs taken during the erection process.Each packaged truss is suspended from a singlepoint, using a crane, and is pulled open fromthe ground; when it is fully open, it is attachedto fixed supports. Once all the trusses havebeen deployed, side by side, they are boltedtogether.

4. A structure of this type is, of course, notautomatically retractable. Escrig and hisassociates have devised various schemes forreducing the amount of manual interventionrequired for deployment and retraction, such asreducing the number of expandable trusses toonly one, hanging from a permanent archstructure. However, the overall span is thenquite limited, and manual intervention is stillrequired to connect or disconnect the structureto or from the ground.

5. The main problem with the application ofthis particular concept to retractable roofs isthat it practically rules out the existence of anypermanent connections to a foundation. It is,however, very well suited to large structuresthat have to be packaged very small, and thisfeature has been exploited by Zeigler5 for thedesign of pop-up displays.

6. A breakthrough in the development ofconcepts for retractable roofs that work as awhole, like Pinero’s and Escrig’s structures, butwhich can be permanently attached to theirsupports was the Iris Dome concept (Fig. 4),invented by Hoberman.6–8 An important featureof this invention is that the pantograph elementsfrom which it is made are not of the type used byPinero and Escrig, but are made from non-straight elements, usually known as angulatedelements, whose properties will be explained inthe next section. These pantograph elementssupport either a membrane cover or partlyoverlapping, rigid covering elements that canmove with respect to one another like the scales

Proc. Instn Civ.Engrs Structs &Bldgs, 1999, 134,Feb., 45–56

Paper 11693

Written discussioncloses 28 May1999

45

S. Pellegrino,Reader inStructuralEngineering,Department of

Z. You, EPSRCAdvancedResearch Fellow,Department of

P. E. Kassabian,GraduateEngineer, Flint &Neill Partnership,

Engineering,University ofCambridge

Engineering,University ofCambridge

of a fish. The Iris Dome model shown in Fig. 4consists of five layers of angulated elements andeach layer forms a complete ring lying on aconical surface. Note that the bottom layer isconnected to the ground by a series of hingedelements, which allow the small, radial edgeexpansion/contraction that accompaniesretraction/deployment of the dome.

7. Further development of this approachbecame possible with the discovery9 of a largefamily of deployable bar structures which,for any plan shape, can fold towards theirperimeter. In plan, these structures are formedby a tessellation of parallelograms, which, inpractice, can be made by a series of continuousbeams with multiple kinks, which we callmulti-angulated elements, connected by cylind-rical joints. These elements are directedfrom the perimeter towards the centre of thestructure.

8. Figure 5 shows a 2 m diameter model ofsuch a retractable roof structure consisting oftwelve folded plates (plus twelve more, partiallyhidden under them) supported by a highlyredundant structure whose internal degree offreedom resembles that of the Iris Dome. Theshape and motion of this support structure arebest understood from Fig. 6, showing theretraction of a flat model based on a plan shapeof twelve regularly spaced points on a circle.This particular model is made from a series ofidentical aluminium-alloy elements. Fig. 7 showsa similar model whose joints are regularlyspaced on an ellipse, instead of a circle. Theelements that make up this structure are notall identical, because of the lower order ofsymmetry of the ellipse, but the structure canstill be folded.

9. Each of these models consists of two setsof multi-angulated elements; those running fromthe perimeter towards the centre in a clockwisesense are called clockwise elements, and thoserunning from the perimeter towards the centrein an anticlockwise sense are called anticlock-wise elements. Note that in the circular examplethere are twelve clockwise and twelve anti-clockwise identical elements. Note also that themulti-angulated elements form two concentric

‘rings’ of rhombuses, which are easiest to seein Fig. 6(b).

10. Both of these structures can be thoughtof as rather complicated variants of the simplefoldable structure shown in Fig. 8, consistingof two sets of parallel, straight rods connectedby cylindrical joints. Clearly, most structuresobtained by ‘kinking’ these straight rods will notbe foldable, and the present paper will showways of designing foldable structures formedby interconnected kinked rods.

11. The retractable roof structure shown inFig. 5 is not flat, but was obtained simply byraising each inner ring by a chosen amount withrespect to the previous ring (in this particularcase, the initial, flat layout consisted of threerings). The view remains the same in plan but adome shape is formed in elevation.

12. This paper presents solutions to thethree key problems that have to be solved in thecourse of the kinematic design of this new typeof retractable roof structure. The next sectionexplains the kinematic properties of angulatedelements, which are the basis for the design oftwo-dimensional foldable structures of generalshapes described in the following section. Theway in which these two-dimensions solutionsare used to design three-dimensional foldablestructures is also explained. The section on‘supports’ presents two different ways ofconnecting these structures to fixed foundationpoints while maintaining their internal degree

KASSABIAN ET AL.

46

(a)

(c) (d) (e)

(b)

Fig. 1. Folding schemesfor retractable roofs(courtesy of K. Ishii)

Fig. 2. Dismountablecover for a swimmingpool (courtesy ofF. Escrig)

of mobility. The section on ‘covering elements’presents the results of a kinematic study todetermine the shapes of the rigid panels that canbe attached to these foldable structures.These panels provide complete cover in thefully deployed configuration without anyinterference during retraction. A discussionconcludes the paper.

Angulated elements13. An angulated element is a pair of kinked,

coplanar rods connected by a cylindrical joint,as shown in Fig. 9. It has been shown by You andPellegrino9 that the angle a subtended by theend connectors A, B, C, D of an angulatedelement does not change when the rods AECand BED are rotated relative to one another,if either

AE ¼ DE; BE ¼ CE ð1Þ

(Type I), where in general w Þ f, i.e. thetriangles AED and BEC are isosceles triangles,or

AEDE

¼CEBE

and w ¼ f ð2Þ

(Type II), i.e. the triangles AED and BEC aresimilar. The special case where the two typesof element coincide, i.e. AE ¼ DE, BE ¼ CE andw ¼ f, was discovered by Hoberman,7 and it isthe element used in the Iris Dome.

14. More general foldable elements whichalso have the property of subtending a constantangle a are obtained by ‘cutting’ the elementshown in Fig. 9 at the scissor hinge E andinserting any number of parallelograms betweenthe triangles AED and BEC. More precisely, ageneralized angulated element (GAE) is definedas a set of interconnected angulated rods thatform a chain of any number of parallelogramswith either isosceles triangles (Type I GAE) orsimilar triangles (Type II GAE) at either end;see the examples shown in Fig. 10.

15. Foldable structures with many differentshapes can be made by forming chains of GAEs,provided that certain interface conditions aresatisfied between adjacent GAEs (see You andPellegrino9 for further details). Also, if certainglobal conditions are satisfied, a chain of GAEsthat forms a closed loop is still a foldable struc-ture. Finally, a structure obtained by connectinga tessellation of parallelograms to a foldable

RETRACTABLE ROOFSTRUCTURES

47

Fig. 3. Deployment ofroof elements over aswimming pool (courtesyof F. Escrig): twophotographs takenduring erection process

chain—where any pair of edges of the tessellationthat have a point in common and are parallel toan angulated rod of the chain are rigidly con-nected—is also a foldable structure.

Layout design16. The only layouts that will be considered

here are generated by adding tessellations ofparallelograms to (i) a ring formed by two ormore GAEs with at least one axis of symmetry,or (ii) a ring of similar rhombuses. According toYou and Pellegrino,9 all structures of this typehave an internal degree of mobility. Hence,provided that they are designed so that differentparts do not interfere, within a sufficient rangeof motion, they can provide the basis for thedesign of retractable-roof layouts.

17. Consider, for example, the layout shownin Fig. 11. Its inner ring has been generated byreflecting the Type I GAE ABCD—which hasisosceles triangles at the ends—through thesymmetry axes mm and nn. Then, a ring ofparallelograms has been connected to the out-side of this ring (it would have been equallypossible to add one to the inside). Note that theshape of any foldable structure designed bythe above method is controlled by the shape ofthe first ring, as the shape of any parallelogramthat is added is determined by the direction ofthe members of the first ring. The only optionavailable for the parallelograms is whether ornot one puts them in.

18. Once the complete layout has beendefined, all adjacent edges that run from theperimeter towards the centre, in either a clock-wise or anticlockwise sense, are made fromcontinuous, multi-angulated beams, which areconnected by cylindrical joints at all kink points.In Fig. 11 the clockwise elements are repre-sented by broken lines and the anticlockwiseelements by solid lines. The existence of aninternal degree of freedom in this structure hasalready been verified, as its layout coincideswith the intermediate configuration of the modelshown in Fig. 7.

19. Note that the process described abovedoes not have a single solution, as there aremany different ways of choosing an inner ringthat satisfies the folding conditions statedabove. Also, in general the shape of the first ringwill vary during the retraction process and, thus,a further choice that is available is the stage ofdeployment at which the shape of the firstring is defined. For example, the model shownin Fig. 7 was designed to have an elliptical firstring in the fully retracted configuration. Notethat its outer edge becomes almost a circle inthe fully deployed configuration, shown at thetop of Fig. 7. A different criterion that could beused, for example, is the minimization of thedistance travelled by a set of selected jointswhich are to be connected to foundationelements.

20. The two-dimensional solutions gener-ated by the method described above are easilyextended to three-dimensional structures, byprojecting any two-dimensional solution onto asurface with the required shape; see Fig. 12.Note that during this process each multi-angulated rod becomes curved out of its planebut, of course, all connectors between multi-angulated beams must be parallel to the direc-tion of projection, in order to maintain the samedegree of kinematic freedom as in the two-dimensional structure.

Supports21. There are two different ways of con-

necting to the ground the kinds of retractablestructure discussed above, without removingtheir internal degree of freedom.

22. Structures based on a symmetric layoutcan be connected to supports that permittranslation within the plane of symmetry. Thus,the 2 m diameter model structure shown in Fig. 5is supported along its perimeter on six supportswhich, during retraction, translate radially withoutrotating. The radial deployment of the circularstructure is shown in Fig. 13. The magnitude ofthe edge translation is small in comparison withthe translation of the inner joints, because eachring distorts less than the next inner ring. Hence,in Fig. 5, each support consists of two parallelcolumns connected by cylindrical hinges to the

KASSABIAN ET AL.

48

Fig. 4. Iris Dome(courtesy ofC. Hoberman)

ground and to the roof structure. During retrac-tion there is a small vertical motion of the wholestructure.

23. The second, less intuitive way of sup-porting any structure whose first ring consistsof similar rhombuses is to connect its elementsto fixed points, which allow rotation but nottranslation. The existence and location of suchspecial fixed points are easiest to show forregular, circular layouts, whose inner ringconsists of identical rhombuses.

24. To begin with, consider the same type ofradial motion as that which would be allowed byradially movable supports. Fig. 14 shows a plot of themotion of a single clockwise element as it deploys.The plot also shows the successive positions takenby the instantaneous centres of rotation as thismotion takes place. As can be seen, theseinstantaneous centres do not remain at a fixed pointbut instead lie on a circle whose centre is at theorigin of the plot. This suggests that, by combining

radial deployment with an appropriate rigid-bodyrotation of the whole structure about its centre, it ispossible to keep at fixed points the instantaneouscentres of motion of all multi-angulated elements thatrun in the same sense.

25. This combined motion has differenteffects on the clockwise and anticlockwise ele-ments that make up the structure, which need tobe examined separately. First, consider aclockwise element A1, A2, etc. (Fig. 15(a)):

OA1 ¼ L sinðv ¹ aÞ= sin a ð3Þ

OA2 ¼ L sin v= sin a ð4Þ

etc., and, differentiating the above expressions,the radial displacement components of A1, A2,etc. due to a small rigid-body rotation dv asso-ciated with radial deployment are

drA1 ¼ Lcosðv ¹ aÞ

sin adv ð5Þ

drA2 ¼ Lcos v

sin adv ð6Þ

RETRACTABLE ROOFSTRUCTURES

49

(a)

(b)

Fig. 5. Model ofdeployable roof (courtesyof Taiyo Kogyo Co.): (a)open, (b) closed

etc. The tangential components of displacementof these points, due to a rigid-body rotationabout O through the same angle dv, are

dtA1 ¼ Lsinðv ¹ aÞ

sin adv ð7Þ

dtA2 ¼ Lsin v

sin adv ð8Þ

etc.26. The remaining displacement vectors

have magnitude

dA1 ¼

������������������������dr 2

A1 þ dt2A1

q¼ L dv= sin a ð9Þ

dA2 ¼

������������������������dr 2

A2 þ dt2A2

q¼ L dv= sin a ð10Þ

etc., and directions at angles v ¹ a, v, etc.,respectively, with the radii through A1, A2, etc.as shown in Fig. 15(a). Thus, it can be concludedthat each multi-angulated element A1, A2, etc.rotates about the fixed point C shown in thefigure. Note that R ¼ L=ð2 sin aÞ. Next, consideran anticlockwise element (Fig. 15(b)). SinceOB1 ¼ OA1, etc., both the radial and thetangential components of displacement of thepoints B1, etc. are the same as for the

corresponding points A1, etc. Therefore, themagnitude of the displacement vectors is alsounchanged, but their directions are such thatthese displacements are now parallel. In otherwords, this element translates.

27. Thus, the final result is surprisinglysimple: the kink points on each of the clockwiseelements of the structure lie on a circle, andduring deployment/retraction they rotate aboutthe centre of this circle. The correspondingmotion for each of the anticlockwise elementsis a pure translation; see Fig. 16.

28. Further analysis shows that fixed pointsexist for any foldable structure whose first ringis a chain of similar rhombuses whose edgesform equal angles v with the sides of a polygonof any shape. These fixed points lie on a scaledversion of this polygon, rotated by v. The loca-tion of the fixed point C for the clockwiseelement A1, A2, A3 of a general plan shape isshown in Fig. 17.

Covering elements29. Practical roof structures require, in

addition to a retractable bar structure formed bymulti-angulated elements, as described above,some forms of covering superstructure. This

KASSABIAN ET AL.

50

Fig. 6. Model structurewith twelve-nodedcircular layout

covering structure can be designed in severaldifferent ways. For example, one might coverthe whole roof with a single, flexible membraneattached to some or all of the multi-angulatedelements. However, in view of the durabilityproblems that have been experienced withprevious retractable structures where mem-branes are subject to repeated stressing anddestressing, as mentioned previously, it is likelythat this type of solution would suffer fromfatigue problems, because the membranewould lose tension and form many creaseswhen the supporting structure was retracted.

30. The alternative solution that is presentedhere is to divide the cover into separate panels,each of fixed shape. These panels could beprestressed membrane elements, each attachedto a single multi-angulated element, or stiffplates of suitable shape, as shown in Fig. 5.Indeed, the plates might even replace some ofthe multi-angulated elements.

31. A method for determining the planshapes of the covering elements, so that theyleave no gaps when the structure is fully closedand do not interfere with other elements whenthe structure retracts, will be presented. Asimple, preliminary approach to this problemis to require that, regardless of the three-dimensional shape of the panels, there should

be no interference between their plan projec-tions. This approach reduces the complexityof the kinematic analysis and makes it easyto carry out interference checks but, ofcourse, poses unnecessary restrictions on thesolution. However, once the shapes of a set ofpanels that satisfy these constraints have beendetermined, it is often possible to enlarge the

RETRACTABLE ROOFSTRUCTURES

51

Fig. 7. Model structurewith twelve-nodedelliptical layout

180˚–θ

θ

Fig. 8. Simple, trellis-typefoldable structure

KASSABIAN ET AL.

52

D

A

E

E

B

C

PD

E

Q

C

B

A

O

α

φ

ψ

Fig. 9. Angulatedelement

α

α

C

JI

I

P

P

N

N

M

M

O

O

A

A

D

D

B

B

E

E

H

H

C

J

(a)

(b)

Fig. 10. Generalizedangulated elements:(a) Type I and(b) Type II

D

A

B C

n

n

m m

Fig. 11. Foldablestructure based on asymmetric ring ofelliptical shape

panels by a small amount, provided that onecarries out local interference checks whichallow for the actual, three-dimensional shape ofthe panels.

32. A general two-dimensional coveringelement that is connected to a multi-angulatedelement of known shape and with k kinks canbe defined by 2ðk þ 2Þ length parameters, Li ,plus 2ðk þ 2Þ angles, ai (see Fig. 18). Limits onthese parameters can be set by considering thefully open and fully closed configurations of theunderlying multi-angulated bar structure. Then,a kinematic simulation of the deployment/retraction of the bar structure with panelsattached to it is carried out and checked visuallyfor no interference between the panels.

33. It has been found that a general solution,valid for structures with any plan shape, is tochoose covering panels with a triangular planshape. To define the layout of these trianglesone considers the fully closed, i.e. fullydeployed, configuration of the bar structureand defines a series of adjoining triangles with acommon vertex at the centre of the structure.The sides of the triangles lie in radial directions,and their base corners coincide with the peri-meter nodes of the bar structure. Figs 19(a)and 19(c) show the resulting shapes for acircular structure and an elliptical structure,respectively. Note that the covering elementsfor the circular structure are twelve identicalisosceles triangles. Note also that, in the ellipticalstructure, some of the innermost points never reachthe centre, as other inner pionts, at the apogees ofthe ellipse, ‘collide’ first.

34. An alternative cover design obtainedfrom the kinematic analysis, which satisfies thekinematic requirements and produces a moredynamic visual effect when deployed, is shownin Fig. 19(b).

Discussion35. A new concept for retractable roof

structures has been presented. The new struc-tures consist of a foldable lattice of multi-angu-lated beams connected by cylindrical joints, towhich covering panels or membranes areattached. These structures fold along theirperimeter and there is practically no limit to therange of shapes that can be achieved.

36. A key property of these roof structuresis that they have an internal degree of mobilitythat allows them to fold without any deformationof their members. Therefore, there is no inher-ent limitation on the stiffness of their members,which can be designed to be as stiff asrequired to avoid excessive deformation underthe range of loads that will be applied to theroof during operation. For long-span construc-tion, it is likely that three-dimensional trusselements would be used, instead of multi-angulated beams. All that would be requiredin order to meet the foldability conditions

RETRACTABLE ROOFSTRUCTURES

53

Fig. 12. Twoconfigurations of atwo-dimensional foldablestructure, projected ontoa curved surface

Fig. 13. Radialretraction oftwelve-noded circularstructure

discussed earlier would be that the truss ele-ments were interconnected by means ofcylindrical joints located at the same positionsas where the joints of the multi-angulatedelements would have been.

37. A preliminary study of the gravity-induced deflections of the type of structuredescribed in this paper, including simulationswith the finite-element package ABAQUS andexperiments on a physical model similar to thatshown in Fig. 5, has been carried out.10 Thestudy showed that this type of structure behavesin a way similar to a grillage of beams connectedby momentless joints. The gravity-induceddeflections vary during the expansion of thestructure, as each beam is subject to twistingmoments of increasing magnitude as it rotatestowards the centre. Thus, the innermost points

KASSABIAN ET AL.

54

Radialdirection

Fig. 14. Position ofinstantaneous centres ofmotion during radialretraction

θ+3α

θ–αθ–α

90–α

θ+2αθ+2α

2α

α α α α α α α α

θ+α

δA1

δA2

θ θ

L L

L

R

R

L L

L

L L L L

L

LC

A5

A3=B3

A5

A4 A4

A2B2

B4

B5

A3

A2

A1A1B1

O O

(b)(a)

Fig. 15. Motion of (a)clockwise and (b)anticlockwise elements

Fig. 16. Motion ofcircular structure withfixed points: eachclockwise element rotatesabout the centres of thecircle through its joints,and each anticlockwiseelement translates

of the structure deflect downwards. Once thestructure is fully expanded, various strategiescan be adopted to make it secure under opera-tional loads. For example, some of the jointscan be latched, or the deployment actuatorscan be driven beyond the point of first ‘collision’,in order to preload the whole structure andremove the backlash from the joints. Work onthese issues is currently under way.

RETRACTABLE ROOFSTRUCTURES

55

A3

A2

A1

L1

L1

L2

L2

θ

θ

θ

θ

θ

θθ

θ

θ

γ

α

β

λ

C

O

L1(sin γ/sin α)=L2(sin λ /sin β)Fig. 17. Point C is a fixedpoint for angulatedelement A1, A2, A3

L1

L2

L2i

α2

α2i

α1

Fig. 18. Plan shape ofcovering element

(a)

(b)

(c)

Fig. 19. Coveringelements for (a),(b) circular and(c) elliptical structures

38. A general method for the conceptualdesign of this new type of retractable roofstructure has been presented. The three keysteps are as follows.

X Step 1: design of foldable support struc-ture. This is done first in two dimensions, byconsidering either a ring of two or moregeneralized angulated elements, as definedabove, with symmetry axes, or a ring ofsimilar rhombuses, and by adding to eitherof these any tessellation of parallelograms.Kinematic simulations of the motion of triallayouts are carried out, to find a layout thatmeets the requirements of a particularapplication. Once a satisfactory two-dimen-sional layout has been found, it can be pro-jected onto any curved surface withoutchanging its kinematic behaviour.

X Step 2: design of supports. Depending onwhich type of ring was chosen in step 1, thefoldable support structure can be connectedto the ground either by a set of ‘tilting’supports that allow radial translation withinthe symmetry planes or by a set of fixedsupports that allow rotation only. Thesefixed supports are located within the peri-meter of the structure, at special locationsdefined by the shape of the first ring used instep 1, whereas the tilting supports can belocated right at the edge of the structure, tosuit different applications.

X Step 3: covering elements. The plan shapesof the covering elements that are attachedto the foldable bar structure have to satisfythe kinematic requirements during deploy-ment. A simple, general solution has beenidentified, consisting of triangular panels.It has been shown that other cover shapesare also possible, although each specificcase needs to be analysed in detail.

39. A limitation of the approach pursuedin step 3 above is that no kinematic interfer-ence is allowed between the plan projections ofthe covering panels, rather than the panelsthemselves. Further work will be required toconsider the actual shape of the panels and,indeed, of the foldable support structureitself, in order to obtain designs that areoptimal in terms of both shape and structuralefficiency.

Acknowledgements40. We thank Dr S. J. Medwadowski for

helpful advice, and Professor M. Kawaguchi forhelp in obtaining some of the information pre-sented in the introduction.

41. The inventions presented in this paperare protected by two patent applications.11 Partof the research presented in this paper wascarried out by P. Kassabian during the course ofhis MEng project, and was awarded the firstprize in the ICE Student Paper Competition heldin June 1997.

43. Financial support from the Engineeringand Physical Sciences Research Council forZ. You and a Foresight Award for S. Pellegrinofrom the Royal Academy of Engineering aregratefully acknowledged.

References1. INTERNATIONAL ASSOCIATION FOR SHELL AND SPATIAL

STRUCTURES, WORKING GROUP 16. StructuralDesign of Retractable Roof Structures 1996,Draft 3.

2. KARNI E. Retractable spatial structures forswimming pool enclosures. International Journal ofSpace Structures, 1995, 10, No. 4, 231–236.

3. CALATRAVA S., ESCRIG F. and VALCARCEL J. P.Las estructuras de Emilio Perez Pinero. InArquitectura Transformable (F. Candele, E. P.Pimero, S. Colatreve, F. Escrig and J. P. Valcarcel(eds)). Escuela Tecnica Superior de Arquitecturade Sevilla, 1993, 11–32.

4. ESCRIG F., VALCARCEL J. P. and SANCHEZ J. Deploy-able cover on a swimming pool in Seville. Journalof the International Association for Shell andSpatial Structures, 1996, 37, No. 1, 39–70.

5. ZEIGLER T. R. Collapsible/Expandable StructuralFrameworks. US Pat. 4 761 929, 1988.

6. HOBERMAN C. Reversibly Expandable Doubly-CurvedTruss Structure. US Pat. 4 942 700, 1990.

7. HOBERMAN C. Radial Expansion/Retraction TrussStructures. US Pat. 5 024 031, 1991.

8. WATERS T. The unfolding world of C. Hoberman.Discover, March 1992, 70–78.

9. YOU Z. and PELLEGRINO S. Foldable bar structures.International Journal of Solids and Structures,1997, 34, No. 15, 1825–1847.

10. TEALL M. J. Deployable Roof Structure. MEngProject Report, Department of Engineering,University of Cambridge, 1996.

11. YOU Z. and PELLEGRINO S. Expandable/CollapsibleStructures. UK Pat. 9 601 450.1, 1996;International Patent Application PCT/GB97/00224, 1997.

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