.,~

Gcometry of Expa ndable Space Structures

by

F. Escrig and J.P. Valcarccl

Reprinted from

INTERNATIONAL JOURNAL OFSPA,CE STRUCTURES

SPECIAL ISSUE

VOLUM E 8 No. 1/2 1993

MULTI-SC IENC E PUB LlSH ING ca. LTD.107 High Street, Brent wood , Esscx CM 14 4RX. Unitcd Kingdom

Geometry of ExpandableStructures

F. Escrig1 and J.P. Valcarcel 2

Space

lSchoo1 of Architecture of Sevilla, Spain; 2Shool of Architecture of La Coruna, Spain

(Received 10 February 1990; revised version received 31 January 1992)

SUMMARY: Expandab1e Structures are a specia1kind of mechanism that canbe used in severa1 different geometries . The geometry of structures based onscissors is introduced to explain concepts necessary to design a wide range offorros like masts, archs, p1ane spatial structures, cylindrica1 and sphericalbar structures.

1. INTRODUCTION

Expandable Space Bar Structures are a specialkind of articulated struetures that ean aehieveseveral spatial configurations from completelyfolded, where all struetural elements are coneen­trated in a bundle, to widely expanded, covering agreat area. These assemblies are mechanisms thatmay work as structures by means of propermeehanic devices to fix them at desired positions.Basically there are three kinds of ExpandableSpace Bar Struetures as follows:

a) uM.BRELLA MECHANISMS, consisting ofa mast around which a radial bundle of bars isdeployed by sliding a cylindrieal or hinged jointover it (Figure 1) (Ref. 10).

b) HINGED-COLLAPSIBLE STRUTMECH­ANISMS, consisting ofa set ofbars folded in sueha way that when sueh meehanisms deploy, hingesthat connect two bars lock, and then the two barsbehave as a single eontinuous piece (Figure 2).

e) X-STRUCTURES. They are the object ofthispaper and will be described extensively below.

2. THE SCISSOR-HINGED MECHANISM

X-Structures are sets of several scissors as shownin Figures 3, 4, 5 and 6. Eaeh of these pattems areable to achieve several movable configurations

Intemational Joumal o/ Space Structures Jlól. 8 Nos. 1&21993

Figure 1

and therefore to occupy more or less surfacevolume (Figures 7 and 8).

If we conneet these pattems between them, insueh a way as to guarantee the eompatibility ofthemovement ofeaeh pieee, we obtain a eomplex sys­tem able to grow in one, two or three spatial direc­tions, building a eomplex assembly with the sameproperties of the elemental eells: expanding andfolding abilities.

As we can see, the struetural eomponents are .bars articulated at the ends and eonneeted bymeans of a joint at an inner point. This joint willconneet from two to four or even more bars.

Pattems shown aboye are ealled regular andthey may be warped and give rise to irregular units

71

Geometry 01Expandable Spa ce Structures

Figure 8Figure 7

to achievc other complex spatial configurationsthat will be explained below.To achieve this, sornecompatibility conditions between length of eachstrut of the whole, have to be satisfied.

Characteristics of regular patterns are bars ofthe same size and crossing point in the middle ofthc bars. Irregular cells lack one or both of theseproperties and give a long series of differentpossibilities. Ofcourse, foldability is only possibleif sorne restrictions are achieved. In a general casea structure like the one shown in Figure 9 can befolded only if bar lengths satisfy:

Figure 4

Figure 2

Figure 3

3. LINEAR GRIDS

Longitudinal structures may be obtained by con­necting patterns in one direction. If patterns are

Similar conditions must be satisfied whenseveral units are connected.

Depending of the kind of irregu1arity we obtaincells for different geometries and uses. We will trya tentative classification.

(1)

2-8 + 6-8 = 2-9 + 6-9

3-7 + 5-7 = 3 9 + 5 9

1-8 + 4-8 = 1-7 + 4-7

Figure 6Figure S

72 International Joumal 01 Spac e Structures Vol. 8 Nos. 1&2 1993

F. Escrigand J.P. Valcarcel

Figure ~3Figure 12

Figure 14

nection to give very stiff structures, as shown inFigure 14.

With warped cells we can obtain tapering masts(Figure 15) or curved pieces (Figure 16).

3

Figure 9

regular, growing is straight, as shown in Figures10, 11, 12 and 13.In the case oflinear grids we canuse struts with more than one intermediate con-

Figure 10 Figure 11 Figure 15

Intemational Journal of Space Structures Vol. 8 Nos. 1&2 1993 73

Geometry 01Expandable Space Structures

Figure 20

Figure 18

Figure 19

Figure 17

5. PRISMATIC GRIDS

lf we use leaned patterns as shown in Figures 19and 20 we can obtain faceted surfaces to complete

Figure 16

4. PLANE GRIDS

They are made from regular cclls and can be usedas roofs or umbrellas (Figure 17) stiffned bymean s of cables or fab rico

If we connect several plane figures we canobtain complex forms like the icosa he dron shownin Figure 18.

74 lnternational Joumal 01 Space Structures Vol. 8 Nos. 1&21993

F. Escrig and J.P. Valcarcel

covers like those shown in Figures 21 and 22 aswell as th e pyramid shown in Figure 23.

Figure 24 shows sketches of umbrellas reinfor­ced with ca bles .

Figure 21

Figure 22

Intemational Journal 01Space Structures Vol. 8 Nos. 1&21993

Figure 23

6. TWO-WAY CYLlNDRICAL GRIDS

On a cylindrical surface wc can draw a two-waygrid (Figure 25) along principal coordinates togenerate faces wherc to place scissors (Figure 26).11' the resulting patterns meet the restrictionsdefined aboye we obtain an expandable structurewith wide utility in architecture. Figures 27'and 28show a tcmporary exhibition room in Almeria(Spain) proposed by the authors.

Figure 29 shows a structure proposed by HER­NANDEZ MERCHAN (Ref. lO),which will bemade with only one length of strut. lts angularstability could be ensurcd by diagonalyzing anumber of modules with bars, cables or even thefabric cover itscll.

7. THREE-WAY CYLlNDRICAL GRIDS

If we draw on a cylindrical suríace a triangulargrid we can obtain triangles (Figure 30) where "A"trusses are curved in aplane, whereas "B" and "C'trusses are warped. As bcfore Figure 30a is a sim­plified version 01' Figure 30b representing onlyfaces tied to the scissors for the sake 01' clarity.Figure 31 shows a conliguration where "A" trussesremain plane whilc "B" and "C' are warped. Inboth cases, warping rcprcscnts an additional dif­ficulty, since scissors cannot be connected to awarped facc. Both diagonaIs lack an intermediatepoint 01' contact. Thus, threc-way foldable cylin­dric structures are a combination 01' scissors andisolated bars (Figure 32a).

The analysis that wc have made ofsuch a type ofstructures has confirrncd that they are stable, pro­vidcd that the supporl points are immobilized

75

Geometry 01Expandable Spa ce Structures

Figure 24

(Ref. 7). Another way ofsolving this problem is theuse ofcurved bars which in the folded position areforced towards a straight configuration, thus stor­ing strain energy for se1f-unfolding and foldingthe structures in thcir final positions (Figure 32b).These structures can be built with only one lengthof strut.

8. TWO-WAY SPHERICAL GRIDS

If we project a square mesh onto a spherical sur­face (Figure 33) we can obtain expandable X­Structures from patterns 11 and IV like thoseshown in Figures 35 and 36.To obtain the crossingpoint of scissors we will use the restrictions ofFigure 34: Angular segments ~ ¡ for the same nodeD¡ will be cqual. This means in the completestructure

~¡ + ~ i - I = Dp¡_1 = U i - 1

(2)

~ ¡ + ~ ¡+ I = DPi+1 = u¡

76

system of "n" equations with "m" variables, "n"bcing the number ofgrid edges connecting D i' "m"the number of nodes D¡, and u¡ thc angle resultingfrom edge sub-division,

To complete the system we must add "s"equations derived from symmetry and imposedgcometric constraints.

Finally we will have "n+ s" equations in a sys­tcm with "m" variables. If"n+s" = "m" the systemwill be determined. Otherwise it will be incom­plete or redundant.

Once obtained segments ~¡ from Figure 34 andfixed thc valucd of the angle bi'we can obtain thelength of the bars from the re1ations :

Rsin~ i\'= \' 1= (S: AI 1- COS V i+tJ¡)

(3)

Rsin~ik¡=k¡_¡= COS(b i+~¡)

R being the sphere radius for the anglc b;.

International Journal 01 Space Structures Vol. 8 Nos. 1&2 1993

F. Escrig and J P. . Valcarcel

Figure 28

~i'~I~v;~Wf,~~@},~~':¡'

J11 1! !,:IIJ~I~ I .I~ "",~ /~;é~{~~~~' :" ~. W7" r.: . / , ," 9 / ", " " ':t.''-~l..r~J . ¡'¡'ti,.'>'- ~!... ;;< ;'101. JI I.r.¡' ~F"/" , 'h e ' -C:r{{I/''i 'j ,¡

.ll11.....J ""'" ") ,..,ni".

Figure 2S

Figure 26 Figure 29

-- ...--­~.- ..- --~-- ---~--

---.-.- -

Intemation , .a Joumal of S, Figure 27

pace Str .uctures Vi 1o . 8 Nos. 1&2 1993 77

Figure 30a

Figure 30b

Geometry 01Expandable Space Structures

Figure 31

Figure 32a

With these lengths we can build the completestructure and the joint coordinates for o¡ variablewith kinematic position from Oto 90°.

Two-way spherical grids are unstable sinceangular distortions are possible. However thiseffeet can be avoided by diagonalizing a number

78

Figure 32b

of elements as indicated aboye for cylindricalgrids. A sort of different geometries can beachieved depending on strut lengths.

9. THREE-WAY SPHERICAL GRIDS

To obtain these grids with a spherical net, initial

Intemational Joumal 01 Space Structures Vol. 8 Nos. 1&21993

F. Escrig and l.P. Valcarcel

and final compatibility conditions are considered.In other words, we guarantee the geometry forboth folded and expanded positions. Howeverthese conditions, which are of course necessary,are not sufficient, the structure goes through inter-

Figure 33

Figure 34

mediate stages in which it has to be forced with anenergy input. From the point where the incom­patibility is largest, the structure returns the storedenergy by folding or unfolding and then remain­ing in one ofthe limiting positions (the only oneswith a compatible design). If the bars are notexceedingly rigid, this may be advantageous sinceself-stabilizing structures can be obtained.

The adjustment of a three-way grid on a spheremay be achieved in several ways, basical1y by pro­jecting onto a spherical surface a grid contained inthe equatorial plane, originating from a focus onthe polar axis of the sphere. Depending on the

Intemational Joumal 01 Space Structures Vol. 8 Nos. 1&2 1993

position of'the grid, structures as in Figures 37 and38 will be obtained. It is advisable to choose thefocus so that thc bar lengths are as similar as poss­ible. Figure 39 shows a six-frequency grid obtainedby projection of a triangular mesh placed in the

-"--

\ "\ '\r--

J \\ I f

\P- .....

'J V

u .- .l.. Á<, ......

Figure 35

equatorial plan from a focus placed on the bottomof the sphere.

10. GEODESIC GRIDS

To minimize distortion of two- or three-way gridsover a sphere, geodesic polyedra can be used.Deployable grids are in this case obtained by sub­stituting every edge of a polyhedron by scissorssatisfying the compatibility conditions. Moreoverthe upper and lower nodes are aligned to the cen­tre ofthe sphcre. Polyhedra bascd on any polygonand any subdivision order may be used.

In intermediate stages, this type of grids has thesame incompatibility problems previously des­cribed. Energy input is required for folding orunfolding, but it will be later returned Ior stabiliz­ing the whole.

79

Figure 41

Figure 42

evidently permit a number of variants. Moreoverother pos sibilities based on different con­figur ations will be discussed elsewhere. Figure 47shows a structure achieved with rhombic sub­division and built with only one type of strut.

82

Geometry 01 Expandable Space Structures

13. ACKOWLEDGEMENTS

This research has been achieved with financialsupport provided by DGICYT (Direccion Generalde Investigacion cientifica y Tecnologica). Figure1 has been reproduced from Ref. 10. Figures 45and 46 have been reproduced from Ref. 12 and 13.Drawings of Figures 27 and 28 belong to F.ORTEGA ANDRADE.

REFERENCES

1. CLARKE, R.e., "The kynematics ofa novel deployablespace structural system". Thirt Int. Con! on Space Struc­tures Proceedings. Surrey 1984. Elsevier. pp. 820-822.

2. ESCRIG, F., "Sistema modular para le construccion deestructuras espaciales desplegables de barras" Patenteespanola numo532117. Mayo 1984.

3. ESCRIG, F., "Expandable Space Frame structures".Third Int. Con! on Space Structures Proceedings. Surrey1984. Elsevier, pp. 845-850.

4. ESCRIG, F., "Expandable Space Structures". SpaceStructures Int. Journ. Vol. 1, numo 2. Elsevier, pp. 79­91.

5. ESCRIG, F. & P. VALCARCEL, J., "Introduccion a lageometria de las estructuras espaciales desplegables debarras". Bol. acad. ETSALa Coruna, numo 3. Feb. 1986,pp. 48-57.

6. ESCRIG, F. & P. VALCARCEL, J., "Great SizeUmbrellas solved with Expandable Bar Structures".First Int. Con! on Lightweight Structures in Architecture.Sydney 1986. pp, 676-681.

7. ESCRIG, F. & P. VALCARCEL, J., "Curvcd Expand­able Space Grids", Con! on Non-Conventional StructuresProceedings. Vol. 2. pp. 157-166. Civil-Comp Press1987.

8. ESCRIG, F. & P. VALCARCEL, J., "Estructurasespaciales desp1egab1es curvas" Informes de la Con­struccion. Madrid IET Vol. 39, Num. 393 pp . 53-71.

9. ESCRIG, F., P. VALCARCEL,J. & GIL DELAGADO,O., "Design of Expandable Spherical Grids" IASSSimp. 1989. Madrid. 16 pp .

10. HERNANDEZ, C, "Estructuras transformables. Estran1" Tecnologia y Construccion.nA. 1988. Caracas pp .103-118.

11. McNULTY, O., "Foldable Space Structurcs". First Int.Con! on Lightweight Structures in Architecture". Sydney1986. pp. 682-689.

12. P. PINERO, E., "Materia-Estructura-Forma". Hogar yArquitectura numo 40. Madrid 1962. pp. 25-30.

13. P. PINERO, E., "E structures reticulees". L'Architectured'aujourd'hui, Vol. 141, Dec 1968. pp. 76-81.

14. ZEIGLER., us. Patent 4.026.313. May 1977.

Intemational Journal 01 Space Structures Vol. 8 Nos. 1&2 1993

F. Escrig and J.P. Valcarcel

Table 1. Geodesic Icosahedron with Frequency Four. length of BarsEach chord factor is expressed in terms of the radius of the circumscribing sphere

PRISMA 4 D17= .1106 D27= .1060 D73= .1570 D74= .1478PRISMA 4 D38= .1570 D48= .1478 D85= .1570 D86= .1478PRISMA 4 D19= .1106 D29= .1060 D95= .1570 D96= .1478

PRISMA 3 1 D17= .1536 D27= .1444 D73= .1652 074= .1546PRISMA 3 1 D38= .1689 D48= .1583 D85= .1650 D86= .1549PRISMA 3 1 D19= .1570 D29= .1478 D95= .1573 D96= .1472PRISMA 3 2 D17= .1570 D27= .1478 D73= .1573 D74= .1472PRISMA 3 2 D38= .1650 D48= .1549 D85= .1689 D86= .1583PRISMA 3 2 D19= .1536 D29= .1444 D95= .1652 D96= .1546

PRISMA 2 1 D17= .1683 D27= .1577 D73= .1494 D74= .1413PRISMA 2 1 D38= .1465 D48= .1385 D85= .1677 D86= .1565PRISMA 2 1 Dl9= .1651 D29= .1545 D95= .1699 D96= .1587PRISMA 2 2 D17= .1728 D27= .1626 D73= .1738 D74= .1626PRISMA 2 2 D38= .1738 D48= .1626 D85= .1738 D86= .1626

PRISMA 2 2 Dl9= .1728 D29= .1626 D95= .1738 D96= .1626PRISMA 2 3 D17= .1651 D27= .1545 D73= .1699 D74= .1587PRISMA 2 3 D38= .1677 D48= .1565 D85= .1465 D86= .1385PRISMA 2 3 Dl9= .1683 D29= .1577 D95= .1494 D96= .1414

PRISMA 1 Dl7= .1561 D27= .1481 D73= .1103 D74= .1058PRISMA 1 D38= .1093 D48= .1047 D85= .1583 D86= .1491PRISMA 1 Dl9= .1547 D29= .1466 D95= .1583 D96= .1491PRISMA 2 Dl7= .1655 D27= .1543 D73= .1501 074= .1408PRISMA 2 D38= .1576 D48= .1484 D85= .1602 D86= .1506PRISMA 2 D19= .1738 D29= .1626 D95= .1602 D96= .1506PRISMA 3 Dl7= .1738 D27= .1626 D73= .1602 074= .1506PRISMA 3 D38= .1602 D48= .1506 D85= .1576 D86= .1484PRISMA 3 D19= .1655 D29= .1543 D95= .1501 D96= .1408PRISMA 4 D17= .1547 D27= .1467 D73= .1583 074= .1491

PRISMA 4 D38= .1583 D48= .1491 D85= .1093 ' D 86= .1051PRISMA 4 D19= .1561 D29= .1481 D95= .1103 D96= .1061

Figure 43

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Geometry 01Expandable Space Structures

Figure 44

84

Figure 45 Figure 46

Figure 47

Intemational Joumal 01 Space Structures Vol. 8 Nos. 1&21993