Space Structures - Principles and Practice- Subramanian Narayan - Volulmul 2.doc

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Table of Contents Tensegritic Structures

Introduction 9.1 The famous American Engineer, R. Buckminster Fuller is respo sponsib sible for coining the word ‘Tensegrity’ and for doing important early work on this topic. He noticed that tension and compression always coexisted in Nature. He also utilized the concep conceptt that that member memberss carryin carryingg tensil tensilee loads loads are much mu ch light lighter er than than those those carry carryin ingg comp compres ressi sive ve forces. This concept has been fully utilized in our human body which has heavy bones and slender sinews. The invisible skin of water is an example of a tensi nsile com omppone onent of high structural ral performance. Any structural system needs continuity between its components to permit forces to pass through it to the foundation. In many structures the continuity is usually achieved through a series of compression elemen elements, ts, with with isolat isolated ed tensil tensilee elemen elements. ts. Fuller Fuller advocated that if this

Ca ble s (b) Vilna y

Fig. 9.1 Tensegrity ensegri ty dome d ome system

(c) Tcnscgritic spherical shell

Tensegritic Tensegritic Structures situation is reversed, then very lightweight structural systems can be built. To these structures he gave the name ‘Tensegrity’— which was derived from the words ‘Tension-integrity’. Hence, according to Fuller the Tensegrity system can be defined as a system which is established when a set of discontinuous compression components interact with a set of continuous tensile components to create a stable volume in space8. Tensegritic structures may be composed of bars and a cable net. The bars may be arranged in such a way that no bar is connected to another. Tensegritic shells may be constructed and a stable configuration is obtained by pre-sừessing the bars against the cable net. When the shell gains its final shape no bar touches the other. An example of a tensegritic shell in the shape of a spherical dome is shown in Fig. 9.1. In many ways tensegritic shells are similar to pneumatic shells. In the case of tensegritic shells, the envelope, envelope, the cable net, is prestressed prestressed by the bars. In pneumatic pneumatic shell, the envelope envelope is pre-stressed pre-stressed by the compressed air. However, unlike pneumatic shells, no constant pumping of air is required in tensegritic shells. Hence they are attractive to the designer than pneumatic shells. Tensegritic structures may be constructed from simple modules. Elementary modules comprise three struts, four struts and six struts and can be derived respectively from a triangular prism having 3 struts and 9 cables (Fig. 9.2), a half-cuboctahedron having four struts and 12 cables (Fig. 9.3), and an octahedron having 6 struts and 24 cables (Fig. 9.4). For these three modules, all the cables have the same length (c) and all the struts have the same length(s). Hence they are called “regular” modules. During many years, only these regular systems were examined. Pugh8 gave a good description of these modules, which are completely defined by the ratio s/c. This ratio is equal to 1.47 for the 3 struts, 1.55 for the 4 struts and 1.63 for the 6 struts. Irregular shapes have also been subsequently developed. Some authors developed tensegrity system system mod module ules, s, not with with straig straight ht compres compressio sionn elemen elements, ts, but with with interl interlaced aced polygon polygons, s, whose whose elements are in compression. Since their geometry must satisfy self-stress equilibrium, design of

(Triangular prism) system 9.2 3 struts system Fig. 9.3 4 struts system Figu 9.4 6 struts (Octahedron) (Half cuboctahedron

Tensegritic Tensegritic Structures Structures

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)tensegrity system requires a form finding process. For regular systems, designed by one parameter (the ratio s/c), form finding is mono parametered. Multi-parametered form finding processes have been recently developed for irregular modules and assemblies (see Ref. 27 for the details). It has to be noted that in the assemblies of modules, more than one self-stress state exists and hence local collapse of a cable will not lead to the total collapse of the whole system. Playing on the ratio r = s/c, allows us to develop folding tensegr tensegrit ityy systems2 systems277 by keepin keepingg the length length of one compone component nt (cab (cable le or strut strut)) const constant ant and and varyi varying ng the the lengt lengthh of the the other other component.

9.2 Tensegritic

nets

Vilnay12'22’39'47 developed a different concept for tensegrity domes based on tense tensegri grity ty nets nets.. The The main main diff differe erence nce between Fuller’s dome and that of Vilnay is that in Fuller's dome the bars are relatively short. As a result, as the span increases, the curvature is reduced caus causiing bars bars to int interfe erfere re wit with one one anot another her.. This This probl problem em is avoi avoide dedd in Vilnay’s concept; but the problem is that of longer bars, which, being in

Space Structures: Principles and Practice

nets.

compression, are subject to buckling (see Fig. 9.1). The net is considered regular when in eveiy node the bar is connected to the same number of cables. The net is infi infini nite te when when ther theree is no limi limitt to the the number of nodes which can be added to a given net. Usually the nets are numbered according to the number of cables at a node. Figure 9.5 shows a net with two cables in a node and hence called Net No. 2. Similarly, Figs. 9.6, 9.7, 9.8 and 9.9 show net Nos. 3, 4, 5 and 6 respectively. Space nets can also be built using these

9.3 Tensegritic structures The general arrangement arrangement of cables and bars in the various nets can be used to construct various structures. In Fig. 9.10, 9. 10, the Net no. 2 is used

No.2 Fig. 9ề11

Spherical Spherical dome

using net no.6

Space Structures: Principles and Practice to construct a barrel barre l vault. Similarly, Similarly, Fig. 9.11 shows a spherical spher ical dome constructed out of Net No. 6. Tensegritic structures may also be classified according to their shape. Thus, if each strut is surrounded by four tendons defining the edges of a diamond shape, it is called a diamond pattern system. Similarly systems having interconnected struts which do not touch one another are called circuit pattern systems. Fig. 9.12 shows a larger circuit pattern system which has twelve decagonal circuits of struts which interweave interwea ve without touching one another. In anot anothe herr rela relati tion onshi shipp betw betwee eenn strut strutss and tend tendons ons,, three three tendons join the opposite ends of each strut and define a mirror imag imagee alon alongg it. it. Syste Systems ms which which have have thei theirr strut strutss and and tendo tendons ns aưanged aưanged in this this way are called called zigzag zigzag pattern systems, though in many cases the zigzags of tendons will almost be straight lines. Figure 9.13 shows a larger system with 90 struts and 270 tendons based on a pentagonal dodecahedron whose faces have been divided into 45 triangles. It can be seen that the struts in these zigzag system do not touch one another, continuity being achieved through their tendon networks.

9.4 Tensegritic

structures and maxwell’s rules An inte interes resti ting ng poin pointt about about thes thesee zigza zigzagg syst system emss is that that thre threee tendons and one strut meet and form a simple pinned joint at each strut end. But if a framework is designed with pinned joints, according to Maxwell’s rule, it should have a minimum of 3J-6 members if it is to

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Zigzag pattern A. J. Pugh) system. FFig. ig. 9.12 Circuit pattern system of a tensegritic 9.13 structure. (Courtesy: A. J. Pugh)

be simply stiff, where J is the number of its joints. But in this case only one strut and three tendons meet at each strut end. A zigzag pattern figure can only have 2J members, three-quarters three-quarters of which can take tension only. According to Maxwell’s rule the 90 strut figure in Fig. 9.13 should have 534 members. But it has only 360 members and it can be checked experimentally that this frame is stiff. The frame thus constitutes a paradoxical exception to Maxwell’s rule. But Maxwell does in fact anticipate such exceptions to his rule, for he states (ref. (1), p.599, collected papers, Vol. 1) ‘In those cases where stiffness can be produced with a smaller number of lines (bars), certain conditions must be fulfilled, rendering the case one of a maximum or minimum value of one or more of its lines (bars). The stiffness of the frame is of an inferior order, as a small disturbing force may produce a displacement infinite in comparison with itself.’ Maxw Maxwel elll migh mightt have have inte intend nded ed to refe referr to a maxi maximu mum m or minimum value of the length of one or more of its bars. Fuller’s Tensegrity structures have tumbuckles in some members and they cease to be stiff if the tumbuckles are relaxed. There is a limit to the length of interior bars within the net formed by the outer members, and it was at this limit that the frame fr ame in fact became stiff.


This is presumably precisely the kind of maximum which Maxwell had in mind. Thus it seems clear that Fuller’s invention corresponds to an exceptional special case anticipated by Maxwell.Morphological studies Several morphological (morphology simply means the Study of Form) studies have been conducted on tensegrity structures. Most of them have the feature that bars are not in contact with one anot another her.. Only Only a few examp example less of these these studi studies, es, repre represen senti ting ng milestones in the development of tensegrity structure morphology are highlighted here. Snel Snelso son’ n’ss work work 27,3 27,377 is of arti artist stic ic natu nature re.. Many Many of his his ‘sculptures’ are of more or less free form or of low degree of regu regula lari rity ty.. From From the the stru struct ctur ural al poin pointt of view view they they are are all all char charac acte teri rize zedd by havi having ng a sing single le stat statee of pres prestr tres esss i.e. i.e.,, the the tensioning of one cable prestresses the whole structure. Full Fuller’ er’ss tens tensegr egrit ityy dome dome22 is a mu mult ltif iface acete ted, d, sing single le-l -laye ayer r tensegrity polyhedron, in which the cables form the outside skin and the bars form an internal layer (as already shown in Fig. 9.1a). Emmerich19'31 of France was the first to conceive double-layer tense tensegri grity ty netw networ orks. ks. In this this confi configu gurat ratio ion, n, bars bars are confi confined ned between two layers of cables or tendons. This configuration is obta obtain ined ed by join joinin ingg toge togeth ther er tense tensegri grity ty prism prismss or trun truncat cated ed pyramids, or “simplexes” as Emmerich terms these objects. Flat and curved surfaces are also possible. Emmerich also investigated extensi extensively vely and systema systematic ticall allyy tenseg tensegrit rityy polyhe polyhedra, dra, which which can produce double-layer domes19. Vilnay inữoduced the concept of infinite infinite tensegrity tensegrity networks which have already been explained in sections 9.2 and 9.3. Motro35'40 produced double-layer tensegrity grids by joining together tensegrity prisms at their nodes. As a result, his grids contained bars joined together at nodes, unlike other configurations studied, but bars are still confined between two parallel layers of cables. Hanaor31 Hanaor31 carried out extensive extensive investigati investigations ons on Double-layer Double-layer Tensegri ensegrity ty Grids Grids (DLTGs) (DLTGs) with with non contact contacting ing bars. bars. He also also invest investigat igated ed the geometry geometry of double-l double-laye ayerr tensegr tensegrit ityy dom domes es by geodesic subdivision of a pyramid, employing DLTGs made of tensegritic pyramids. Grip24 Grip24 extend extended ed the invento inventory ry of tensegr tensegrit ityy polyhe polyhedra dra and hyper polyhedra, through the concepts of duality and space filling polyhedral networks to produce a virtually unlimited range of intricate and complex networks.


9.5 Characteristics Characteristics

of tensegritic structures As already already mentioned, mentioned, tensegritic tensegritic structures structures are similar similar to bubbles bubbles and hence to air-supported structures and balloons (see chapter 8). In these structures, the struts are on the inside of the system and the tend tendon onss on the the outs outsid ide. e. The The stmt stmtss push push outw outwar ards ds like like the the pressurized air inside the balloon but are ar e restrained by the tendons tendon s which are equivalent to the tensile skin of the balloon. Anot Anothe herr char charac acte teri rist stic ic whic whichh is comm common on to ball balloo oonn and and tensegrity structures is that they are already stressed. Though these structures need only to support the weight of the system, the prestress may be increased to improve the load bearing capacity of the system. If the structure can be prestressed and if it is stable unde underr prest prestres ressi sing ng,, the the geom geomet etry ry of the the stru struct ctur uree is uniq unique uely ly defined. Most of the tensegrity structures, like balloons are very sens sensit itiv ivee to vibra vibrati tion ons. s. Howe Howeve verr the the vibra vibrati tion on effec effects ts can can be improved by the addition of extra tendons. The whole system can be deformed under load only to return to its original shape when that load is removed. The main feature that distinguishes tensegrity structures from conv convent entio ional nal pres prestre tress ssed ed cabl cablee netwo networks rks is that that they they are free free standing and do not require external anchorages or continuous stiff ring beams. This feature, combined with the simplicity of the connection of the bars to the cables, makes this structural system particularly suitable for applications requiring deployability or demountability. An important point that should be remembered is that each tense tensegri grity ty syst system em is enant enantim imor orphi phic, c, each each syste system m havin havingg two two versions which are mirror images of one another, so care should be taken when this could be critical.

9.6 Stability

of tensegrltic structures To fulfil equilibrium conditions at every node, every bar has to be connected at least to three cables not in one plane or two cables where bar and cables are all in one plane. Nets with two cables at a node can be prestressed only in that plane. Only nets with three and more cables at a node can be prestressed for any desired geometry. The structure presented in Fig. 9.10 cannot be prestressed and


hence is not stable. The structures given in Figs. 9.11, 9.12 and 9.13 can be prestressed and will be stable when the length of bars and cables is appropriate. The stability of the prestressed structures in most cases is selfevident; the prestressed net No. 2 in the plane is not stable since it will collapse at any movement of any node perpendicular to this plane. The prestressed structures given in Fig. 9.11, 9.12 and 9.13 are stable because these have features common to a pneumatic structure where the cables act as the skin and the bars transfer the inner pressure. The most convenient way of prestressing is by lengthening the bars until equilibrium is reached. r eached. In most of the cases the geometry of the the pres prestr tres esse sedd stru struct ctur uree will will diff differ er larg largel elyy from from the the prepre presừessed structure. A geometric nonlinear analysis must be used to calculate the geometry of the net under prestressing.

9.7 Application Application

of tensegritic nets Since tensegrity can be defined in such general terms, there is a chance that it has very important implications outside the realm of larg largee scale scale engi enginee neeri ring ng.. The The basi basicc conc concep eptt of estab establi lishi shing ng a structure through an interaction of forces, rather than through a preconceived arrangement


of

components could be very important. Masts can be formed by joining several figures together as illusừated in Fig. 9.14 (The thick lines represent the compression members and the thin lines represent tension members). Many light lightwei weigh ghtt latt lattic ices es can can be form formed ed from from tens tensegr egrit ityy syst system ems, s, especially the less complex ones.

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Since the struts and tendons of the models represent forces in the system, they can be replaced with other components, provided those components cany those forces. For example domes can be formed using compression elements and tensile skins where the tensile skin

Fig. 9.15 9.15 Model of Tensegritic dome with tensile skin. (Courtesy. A. J. Pugh)

acts in a way similar to that of the network of tendons (see Fig. 9.15).

9.8 Analysis

and design of tensegritic structures structures Tensegrity structures are, from the structural analysis point of view, prestressed pin jointed networks similar to cable networks. Except

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in some special configurations, they are geometrically flexible-they contain internal mechanisms. As in the case of cable network, the analysis consists typically of two major phases: A shape finding phase, aimed at obtaining the equilibrium pre-stressed geometry, geometry, and the static (or dynamic) analysis which determines member forc forces es and and noda nodall disp displlacem acemen entts unde underr appl appliied loads oads.. An intermediate phase of investigation of stability, mechanisms and states of prestress is sometimes employed, mainly for research purposes31. Motro and his co-investigators co-investigators (Ref. 18, 27, 34 and 35) applied applied the technique of dynamic dynamic relaxation relaxation to the shape finding finding problem. They also performed analysis of tensegrity prisms19 and simple doub double le-l -laye ayerr netwo networks rks.. Vilna ilnayy, in his his book book 22, outlin outlines es the principles of the investigation of mechanisms and states of prestress and the shape Finding, static and dynamic analysis phases, but his procedure is not very suitable for computer implementations. Hanoar36 presented a flexibility based model for investigati investigating ng mechanisms mechanisms and states of prestress prestress and for the static anal analys ysiis empl employ oyed ed a sti stiffnes fnesss base basedd proc proced edur ure, e, whic whichh is comp comput utat atio ional nally ly effi effici cien ent, t, for the the actual actual analy analysi siss of siza sizabl blee networks. Vilnay39' Vilnay39' 47 used linear algebra to determine the nodal displacements as well as the forces induced by the prestressing. The internal forces and reactions p due to the external forces Q acti acting ng at the the node nodess of thes thesee stru struct ctur ures es can can be calculated using the following equilibrium conditions at every node: [A] {P} = {0} (9.1) where [A] is the statics matrix as defined by the geometry of the prestressed structure. In Eqn (9.1), the cables are ừeated as bars, when the prestressing action ensures that the forces in the cables will remain tensile under the given external forces. In each node there are three equilibrium equilibrium equations equations and the number of unknowns around each node is half the number of cables and bars which meets there. In solving Eqn (9.1) three situations arise: (1) when the number of unknowns is larger than the number of equations, (2) when the number of unknowns is equal to the number of equations and (3) when the number of unknowns is less than the number of equations. Under condition (1) the structure is indeterminate, under condition condition (2) the structure is determinate determinate and under condition (3) the structure is unstable.

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Hanaor20 r20’31 carrie ried out parame ametric ric ana analytical invest investigat igation ionss of double double-la -layer yer tensegr tensegrity ity grids grids of varying varying geom geomet etri ries es,, incl includ udin ingg dome domess and and geom geomet etri rica call llyy rigi rigidd config configurat uration ionss (stati (staticall callyy indete indetermi rminat nate, e, with with no interna internall mechanisms). From these studies, he found that domes are stiffer than plane grids over a given span and geometrically rigid rigid config configurat uration ionss are stiffe stifferr than than geometri geometrical cally ly flexibl flexiblee ones. There is a direct correlation between stiffness and load carrying capacity for a given span and topology. Hanaor2 Hanaor20,3 0,311 investi investigat gated ed the various various configu configurat ration ionss of double-layer tensegrity grids (DLTG) covering a circular span of diameter 27 m and ( compared them with double-layer grids. The results indicate that, so long as the principle of noncontacting bars is maintained, all DLTG configurations are heavier than the double-layer grids, due to the long bars. From the tests conducted by him21, he concluded that it is desirable to design DLTGs for bar buckling as the governing failure mechanism. High bar slenderness ratio can provide ductility and enhance force redistribution capabilities.


Experimental tests were carried out by Hanaor21 and by Motro, et al.34 on small models. Hanaor performed static tests on small DLTG DLTG models of various configurations. Motro et al. performed per formed dynamic tests on single triangular prism (simplex) models. All tests indicate good agreement with theoretical predictions. However Hanaor and Liao20 suggest that the design of tensegrity grids must be based on a nonlinear analytical model accounting for large deflections. A linearmodel was found to produce an unsafe design.

9.9 Other aspects Stru Struct ctur ures es buil builtt of infi infini nite te regu regula larr tens tenseg egri riti ticc nets nets havi having ng appropriate appropriate boundary conditions can be pre-stressed pre-stressed if atleast atleast three cables meet at a joint. If the net has less than five cables at a joint the pre-sừessed structure structure can carry in its prestressed prestressed geometry only a limited number of families of external loads. For other external loads the structures change their geometry a great deal. Structures having more than five cables at a node can carry external forces in any direction. These structures will deform only because of their elasticity under external forces. It is clear clear that that the the tense tensegri grity ty struc structu ture ress disp displa layy Fiil Fiille ler’ r’ss inge ingenui nuity ty in desi design gnin ingg larg largee but easi easily ly packe packedd and and port portabl ablee structures: an obvious saving in weight results from title fact that about three-quarters of the members are wires rattier than rods. On the other hand, if the aim is to design economical but stiff engineering structures, it is not clear that there is much point in making the outer network so sparse that the resulting frame has a number of infinitesimal nodes whose stiffness is necessarily low. Hence there is a need to study these structures experimentally, with and without additional wires which will make the tensegrity net satisfy Maxwell’s rule. One of the main technical aspects in the implementation of tensegrity structures is the roof covering. Most configurations are geometrically flexible. Hence roof covering has to be a flexible membrane. However, not many studies have been carried out on


tensegrity structures with surface membranes. It is important that the membrane forms an integral part of the design, as the problem of ensuring a crease-free, flutter-free surface, is by no means trivial. For For depl deploy oyab able le appl applic icat atio ions ns,, it is high highly ly desi desira rabl blee that that the the memb membran ranee serves serves as a stru struct ctura ural, l, as well well as funct functio ional nal role, role, reducing the size of cables or even eliminating them18,31.

9.10 Cable

(tenstar) dome In the search for solutions which retain the advantages of air support supported ed struct structure uress while while avoidin avoidingg theừ theừ disadv disadvant antages ages,, several several tensile structure systems have been developed. First practical example of application of the tensegrity dome was provided by a Polish Engineer, Waclaw Waclaw Zalewski in 1961, three years before B. Fiiller Fiiller patented his idea in 1964. Zalewski covered an audit auditori orium um in Kato Katowi wice ce,, Pola Poland, nd, using using the the first first versi version on of tensegrity dome52. David Geiger, who designed the famous cable restrained air supported roof for the U.S. Pavilion at Expo ‘70, modified the principle of Fuller’s tensegrity structure and developed a radial cable dome43 called the tenstar dome.


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Fig. 9.16 Schematic diagram of Tenstar Dome Ridge Valley 15’-7 ^/~Ridge cable net cable Ridge connector

cabl

Diago nal cable

ị Fabr Cable domes were first adopted icfor the Gymnastic Arena (dia. 120 m) Tension ring Roof

section Fig. 9.17 Cable

(Tensegrity) dome

Ridge cable Ponding cabl cablee as required

Compression ring Generalized plan of o f Tenstar Dome Post Section thro’ Ten Ten star Dome

Diagonal cable Hoop connector

Valley cable Roof plan


Upper fabric

Strand Section Y - Y Fabric (eachclamp side) ỵỉ

FlixSE™ — Lower fabric Section X - X Bridge rop rope * ---
Fig. 9.18 Cable dome details; casting, post, strand assembly44

V______ I Diagona! strand 1—^

Tensegritic Tensegritic Structures

anchorage Section Z-Z 'Ỷ

and the Fencing Aren Arenas as (dia (dia.. 90 m) of the Korean Olympic Olympic Games at Seoul, in 1986. The schematic diag diagra ram m of the the tens tensio ionn roof roof is shown own in Fig. 9.16 and the details of the roof over the Gymnastic arena are shown in


Fig. 9.17. These cable domes span span the the spac spacee using continuous tens tensio ionn cabl cables es and discontinuous compression posts. Loads are carr carrie iedd from from a cenừal tension ring ring throu hrough gh a seri series es of radia radiall ridge cables, tens tensio ionn hoop hoops, s, and intermediate diag diagona onals ls unti untill they are resolved (and (and ancho anchored red)) in a peri perime mete ter r compression ring44. The tensi ension on hoop hoopss are 14.5 m apart. Three hree tens ension hoops were used f or the Gymnastic Aren Arenaa and and two two for for the the Fenc Fencin ingg Arena. The ridge and diagonal cabl cables es separ separat atee the roof into sixteen equal

segments. For roofs with equal segments, equal hoop spacing, equal equal load loadin ingg on trib tribut utary ary areas areas,, and and corres correspon pondi ding ng verti vertical cal geometry, title corresponding members of cable domes of different diameters carry the same load as one moves from the centre of the dome outward44. By analogy, the dome behaves as two cantilever trus trusses ses not not quit quitee touch touchin ingg at the the cenửe cenửe.. This This has has resul resulte tedd in significant repetition of details and allowed the use of same castings (see Fig. 9.18) at the connection points where posts and cable meet. As a consequence, the two rings of the Fencing Arena share the same castings castings at the top and bottom of the posts and the Gymnastics Gymnastics Arena required required another set of castings for the top and bottom of the posts of the outermost hoop. The components of the roof consisted primarily of continuous, parallel 15 mm diameter prestressing strand (ultimate capacity 266 kN) and metal castings. Bridge rope with pinned sockets run between castings, cas tings, confrols the geometry and provides pr ovides equilibrium at the castings. The elements were assembled at ground and hoisted into a hanging position where they were jacked into final position using two hand held jacks at the base of the posts of successive hoops. After the cables were erected, the fabric was deployed and clamped to the top of the ridge cables. Then the vall valley ey cable cabless were were presfr presfress essed ed,, thus thus induc inducin ingg the the requi require redd prestress into the fabric. The structural behaviour of these domes was found to be highly nonlinear44. The cable, hoop, post structure, however, is statically determinate for the snow load case; if the valley cable has zero load (this may be assumed as the initial configuration for the nonlinear analysis program). Use of title parallel cables builds redundancy into the cable dome system. These low rise cable domes permit the construction of very econ econom omic ical al and and ligh lightw twei eigh ghtt dome domes. s. The The dome domess in Kore Korea, a, translucent and insulated to RIO, weigh only 14.65 kg/m2 and were constructed at a cost of approximately us $215 per sq.m44. A cable dome over an elliptical plan (78 m dia. X 92.65 m dia.) for the Illinois State University (with only one hoop cables) and a 210 m diameter Florida Sun coast dome (with four hoop cables) at St. Petersburg, Florida (Total weight of the structure = 8,258 kN, i.e., 0.24 kN/m2) have been constructed by Geiger and Associates. The details of these domes may be found in Ref. 45. It has to be noted that neither the weight nor the cost of these cable domes increase significantly with the increase in span or change from circular to super elliptical plan configurations41,44. Cable dome made its debut in Japan in Izu Shizuoka Prefecture. Weidlinger Associates designed the largest hyperbolic tensegrity dome over the Atlanta Stadium in Georgia, which has drastically changed the

existing concepts regarding construction methods used for stadia 52’ 55. The Georgia Dome roof (built in 1994) also has concentric rings. rings. But (revert (reverting ing back to

Fig. 9.19 Support system of HB cable dome

Buckminster Fuller’s

invention) they are supported by triangulated cables thus adding to the complexity. complexity. There is no doubt at all that tensegrity domes offer many advantages over more conventional skeletal types and will be used in an increasing way during the years to come. Geiger has also presented cost comparison of double-layer grids (the Takenaka Truss russ), ), air air supp suppor ortted roof roofss and and the the cabl cablee dom domes41 es41.. His His compa compari rison son incl includ udes es ener energy gy consi conside derat ratio ions, ns, roof roof memb membran ranee comparisons and insurance costs. He has illustrated the impact produced by the introduction of tensegrity domes upon lightweight low cost space structures for large span buildings. Because of the economy and ease of erection, cable domes have generated widespread interest and resulted resulted in investigations investigations and extensive testing by several other researchers50’51. Berger54 has suggested a system called composite cable dome consisting primarily of intersecting cables whereby each cable supports two struts and each strut rests on two intersecting cables as shown in Fig. 9.19. This was originally developed for the St. Pete Petersb rsbur urgg Sun dome dome (thou (though gh fina finall llyy Geig Geiger’ er’ss syste system m was was adopted for that dome). This system is simple to install and provides a considerable degree of redundancy. The advantage of this system (and any cable dome) is the predominant use of tensile members. The disadvantage is the very long and complex force flow though high strength members which results in large forces and deformations (and hence not suitable in areas with large snow

loads). To avoid these problems, Berger suggests a composite system in which the predominant uniform loads are carried to the supports in a direct way by the use of arches (see Fig. 9.20). The

Fig. 9.20 Section of arch reinforced reinforced cable dome

cables and struts in this system permit an erection process without

interior temporary supports and provide stability for the arches and the reinforcement for unbalanced loading. This composite cable dome system has been adopted for the replacement of the Unidome roof at Iowa, U.S.A. and the stadium with the new roof was opened in 1998s5.

9.11 Tension

strut dome The first tension strut dome (TSD) (Amagi dome, with a diameter of 43 m) was built in Japan in 1991 to cover the multipurpose gymn gymnasi asium um at Am Amag agii Yugash ugashim imaa Town. own. The The TSD TSD dome dome has has similarities to the cable dome, which was originated by Geiger in the U.S.A. There are, however, major differences53.

(1)

Cable domes use membrane material. They are constructed so that the rise of the domes is secured by a combination of diagonal diagonal cables and hoops as shown in Fig. 9.16 and 9.17. Whereas, the TSD dome is supported by a central lensshaped cable girder (see Fig. 9.21 b). This is suspended by a self supporting supporting tension truss which is anchored in cables below and a circular tension ring above that stabilizes the cabl cablee gứde gứderr. In othe otherr word words, s, it is a new new syst system em that that integrates two different structural systems.

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The Cable dome system introduces tension to cable members in a continuous way through various pulleys which are situated at the top of each of the posts. The TSD dome system utilizes tension in all of the upper and lower chords. These chords are

Fig. 9.21 9.21 Details of Amagi (T.S.D.) (T.S.D.) dome, Japan53

situated independently and do not make use of any slip metal fittings.


The tension to a cable dome is introduced through numerous diagonal cables in gradual stages. It is a step by step system that uses wedges to increase the tension gradually. Tension can be introduced to a TSD dome by pulling the ends of the upper chord cables in basically one movement and at one time.

FigẾ 9Ể22 Interior view of the Amagi Dome, D ome, Japan. (Courtesy. Prof. K. Ishii)

The original configuration is therefore easily monitored and maintained, and it remains more stable as a whole. The TSD dome system can be appreciated as a realistic and practical design in the field of tension dome structures. Complexities regarding slip fitting technology and construction procedures used by cable domes have been improved with the development of this TSD system. The self-supporting self-supporting tension strut Amagi dome is connected to its substructure through a complex of V-shaped steel beams. These beams are fixed to a reinforced concrete compression ring (see Fig. 9.21).

26


It is difficult to solve the problem of supporting the compression reaction force created from the laige roof tension dome at the tips of the V-shaped beams. These compressive reaction forces may be supported by a ring truss consisting of two rings: (1) an upper compression ring placed at the tips of the V-shaped beams where all the compression reaction forces concentrate, (2) a reinforced concrete compression ring placed on the substructure. However by adopting this method the continuous wave shape of the roof is distorted. Hence in the Amagi dome, the tops of folded plate wall and the tops of V-shaped beams were connected dứectìy with stays which continued along the line which extended to form the upper chords (see Fig. 9.21).Using this method, the compression reaction force can be directly transferred from the chords to the V-shaped beams-then to the stays and to the folded plate wall. With this method it was thus possible to concentrate the compression reaction force onto the reinforced concrete compression ring which has been placed on the tip of the folded plate walls. The gran grandn dnes esss of a TSD TSD dom dome lies ies in its its asto astoni nish shin ingg lightweight roof generally less than one-fifth the weight of any dome made from ordinary metal. Using the advantages of its lightness, lightness, a lift-up lift-up construction method has been adopted adopted for this dome. dom e. This This meth method od invo involv lves es the the lifti lifting ng of the the comp comple lete tely ly fabri fabricat cated ed roof roof struc structu ture re from from the the groun groundd floor floor in a singl single, e, unbroken movement. More details about this dome may be found in Ref. 53. The interior view of the completed dome is shown in Fig. 9.22.

9.12 Other

types of tensegrity domes Based on their work on flat double-layer tensegrity grids, Kono, Kunieda28 presented a numerical form finding method, a shape formation process and the static response of a new type of pretensioned double-layer dou ble-layer tensegrity grid dome. The double-layer

27


domes inve domes invest stig igat ated ed by them them are forme formedd by tetr tetrah ahedr edronon-li like ke tensegrity modules as shown in Fig. 9.23. Two adjacent modules were connected by a single single compression member (see Fig. 9.24). Each Each mo modu dule le can can be inde indepe pend nden entl tlyy pres prestr tres esse sed, d, and and the the presfressing is used for the shape formation of the cable dome. The number of infinitesim infinitesimal al mechanisms the grid dome contains contains is rela relati tive vely ly smal small. l. These These dome domess feat feature ure deplo deploym ymen entt type type construction methods and found to be greatly stiffer than initial flat grids28.

Tensegritic Tensegritic Structures Wang29 discusses discusse s a new type of cable dome, called Wang’s dome, which is a cable dome strengthened by additional cables to make the conventional kinematically indeterminate cable dome into a kinematically determinate one. It is capable of sustaining all loading conditions without relying on prestress. The additional cables work and slacken under varied loading conditions, removing all possible mechanisms mech anisms effectively. Unlike cable domes, in which the structural shape has to be adjusted during erection by proper prestress distribution, Wang’s dome does not pose any problem during erection. It can be applied to rectangular plan layout also. Wang29 also discusses about tensegrity ring beams, which can interact with a cable dome to form a truly self stressed, self equilibrium tensegrity system. The tensegrity ring beam can be constructed of tensegrity prismatic or pyramidal simplexes and by radially linking them. Besides, boundary struts must be added to reinforce reinforc e the ring after all simplexes are prestressed and connected. Wang29 also discusses new concepts such as tensegrity frames and Linear Complementary Equation (LCE) method for solving cable-Truss structures stabilized stabilized by cable tension (TSC)

Fig. 9.23 Tetrahedron like tensegrity modules used by Kono and Kunieda28

Tensegritic Tensegritic Structures Fig. 9.24 Flat double-layer grid formed by using Tetrahedron modules slackening problem of cable-strut systems. A four-bar link system as shown in Fig. 9.25a, forming a lozenge confi configur gurat atio ionn is a kinem kinemat atic ical ally ly indet indeterm ermin inat atee frame frame.. It has has inextensional inextensional displacement displacement modes and usually usually cannot be used for structural skeletons.


Fig. 9.25 Unit structure

Addition of a post and eight cable members, connecting each node of the truss to the two ends of a post (as shown in Fig. 9.25b), alters the framework kinematically determinate having one self self equi equili libr brat ated ed stre stress ss stat state. e. Introd Introduct uctio ionn of this this selfselfequilibrium equilibrium stress keeps cable members in pretension pretension and cables can act as if they were compressible. compressible. This presữessed presữessed framework was named as “Unit Unit Struct Structur uree” by Kawaguc guchi and his his associates58. Many unit structures may be assembled to form structures called “truss structure stabilized by cable tensions (T.s.c Kawaguchi and his associates have investigated the structural behaviour of this truss system s ystem by means of numerical analysis. In order to grasp the actual performance of the frame they carried out two series of tests using full scale unit structures and a partial model of vaulted T.s.c. dome58. The vaulted T.S.C., having a dimension of 13.35 m X 22.5 m in plan, constructed and tested by them is shown in Fig. 9.26. A set up of tumbuckles was arranged on each tension member for the introduction of prestress. Mero type steel ball joints were employed at the connection of truss members. PVC coated polyester membrane was employed as the roofing to cover the T.s.c. skeleton of the domeế From the tests it was found that there exists a difference in the behaviour between the nonlinear analytical model and the real model, especially after the slacke slackenin ningg of the short members membersềề They They also also conclud concluded ed that that the initial sừess of a unit structure could be irrtroduced through the operations of just one tum-buckle mounted on the longest tension member.


31

Wang29,63 has also discussed about these truss structures stabilized by cable tension, but calls them as Reciprocal as Reciprocal prism (RP) and Crystalcell pyramids (CP). According to him, RP system is formed by linking self stressed equilibrium reciprocal prisms (see Fig. 9.27) at their middle joints directly and at their upper and bottom joints by additional connecting cables respectively. A rectangular RP simplex, for example, exa mple, is made of a vertical strut, four enclosed horizontal stmts and eight edge cables connected by (frictionless) hinged joints (as already shown in Fig. 9.25). For each simplex, the elongation of the vertical stmt, either by telescoping or screw turning, will prestress the system conveniently. In the formed planar grid, most upper connecting cables

(c) Plan

Fig. 9.26 The det ails of the TSC do me test ed by Kaw agu chi Et Ass ocia tes 58

32 Space Structures Structures:: Principle Principless and Practice

(a) Square R. p. (a)

Triangular c. p.

Fig. 9.28 c.p. Simplexes (b) Pent Pentag agona onall R. R. p. p. (c) Hcxa Hcxang ngul ular ar R. p. Fig. 9.27 R.P.Simplexes R.P.Simplexes

(b) Rectangular c. p. (c) Pentagonal c. p. always slacken under downward load, but they can be used as ridge cables for the membran rane roof or under der upward load response29. Wang’s CP system is formed by connect connecting ing self-st self-stress ressed ed equilib equilibriu rium m crystal-cell pyramids (see Fig. 9.28) at their upper joints directly and at their bottom joints by additional cables. A rectangular CP simplex, for example, is composed of five inner stmts: one vertical stmt and four inclined struts and joined by eight outer cables by frict frictio ionl nles esss hing hinged ed joint joints. s. Like Like RP simplexes, there is only one state of prestress in a CP simplex, and if no cable slackens, there will be no inner mechanisms. Case studies conducted by Wang show that in comparison with a space trus trusss of simi simila larr layou layout, t, cabl cablee stru strutt grids can save about 2/5 of the self weig weight ht and and furt furthe herr redu reduct ctio ions ns are are possible using optimal design employ employing ing small small grid grid length lengths. s. Also, Also, each joint is connected with five bars

33 Space Structures Structures:: Principle Principless and Practice

at the most and hence joint design can be simplified and consequently joint weight can be lowered. The reduced grid length increases the stability of stru struts ts with with redu reduce cedd stru strutt forc forces es.. Howe Howeve verr, the the sel self weig weight ht of the system is increased only slightly29.

9.13 Suspen-dome structure

A Suspen Dome developed by Prof. Kawaguchi and his team, is a composite system consisting of a single-layer truss dome, and a tensegric system (struts and cables) as shown in Fig. 9.29. Connected to the struts suspended from the single-layer truss dome are radial cables, which radiate from the centre of the dome, and ring shaped hoop cables. Thrust of the entire dome, and sừesses of singlelayer truss members can be reduced by the tensegric system with appropriate prestress. In the case of additional loads such a


snow incre crease stre stresse ssess memb embers

loads, in the of truss truss or the s

Fig. 9.29 9.29 Suspen-dome (structural system)

defor deforma mati tions ons can can be conừ conừol olle ledd becau because se of the the tens tensil ilee force forcess produced in the cables. As describ described ed above, above, the suspensuspen-dom domee struct structural ural system system is characterized characterized by sứess and deformation deformation control by the effective use of the tensegr tensegric ic system. system. In additi addition, on, excessi excessive ve struct structural ural flexibility, as experienced in cable domes, can be prevented and the strength of the single-layer dome against budding can be incre increase ased. d. The The suspe suspen-d n-dom omee syste system, m, ther therefo efore, re, enab enable less the construction of safe and economical large- scale domes, which was impracticable with conventional methods60. Thus the suspendome has the following advantages: • reduced burden on boundary structures • increased strength against buckling and • control of structural flexibility

Space Structures: Principles and Practice The behaviour of suspen-dome has been studied extensively by Kawaguchi and his associates ass ociates at the Hosei University, Japan and its safety against buckling was verified with structural model tests60. Based on these studies, the Hikarigaoka dome was constructed in Nerima Ward, Ward, Tokyo in 1994. This dome is a 35 m span single-layer truss dome consisting primarily of H beams, as shown in Fig. 9.30. It has a height h eight of 14 m and the total weight of the roof was about 1274 kN. Only the outermost ring of the trussed dome is stiffened with a tensegrity system. Cable prestresses were applied so that the axial stress in the outer ring girder under long-term loads may be close to zero. In this structure, prestressing prestr essing steel bars are used as radial cables. The dome is supported by V-shaped V-shaped steel columns jointed continuously, with both column capitals and bases being pin-jointed in the radial direction so that movements of the roof in the radial direction are permitted. The roof was finished with steel plates and the central part of the roof was glazed to obtain sunlight. The tensegric system consists of three components: hoop

Fig. 9.30 Principal Principal components of the Hikarigaoka dome60

Space Structures: Principles and Practice cables, radial cables and struts. A method was developed and adop adopte tedd by Kawa Kawagu guch chii by whic whichh pres prestr tres esss was was appl applie iedd by extending the struts60. Static load tests and vibration tests were conducted on this full-size dome. From these tests it was concluded that (1) the desi design gn stif stiffn fness ess duly duly refl reflect ectss the the actu actual al condi conditi tion ons, s, (2) (2) no sign signif ific icant ant diff differe erence nce exist existss betw between een the natu natural ral frequ frequenc encyy characteristics of the suspen-dome and the single-layer truss dome and (3) the eigen value analysis showed good agreement with the test results in frequency and vibration mode28'60. Thus, these tests verified the effectiveness of the Suspen-dome system. Another dome, Fureai Dome, has been constructed using the Suspen- dome system in Nagano Prefecture, where there is heavy snow fall. It is proposed to measure the changes in cable tension forces due to the snow fall and temperature changes.

9.14 Flying

mast fabric roof system Another type of tensegritic structure has been adopted over the Shanghai Sports Centre, China61. When this fabric roof was planned, the American team prepared two schemes: (1) A cantilever steel trussed structure with secondary bow string arches to support a fabric roof, (2) A cabl cablee suppor supporte tedd fabri fabricc roof roof anch anchore oredd to the the perim perimet eter er concrete compression ring beam (Tenstar dome). However, a third scheme with flying mast fabric roof structure based on tensegrity concept was developed and accepted as the final solution61. The flying mast fabric roof structure layout was governed by the arrangement of the cantilever steel trussed roof structure and consists of 32 bays and 17 different types of fabric roof panels. Like large umbrellas over the heads of 80,000 spectators, 57 flyi flying ng mast mast cabl cablee and and fabri fabricc panel panel syst system emss hover hover over over the the Shanghai Stadium. Completed in October 1997, the stadium has a floor area of 150,000 m2. The roof is a saddle shaped ring that is

Space Structures: Principles and Practice a cantilevered spatial structure composed of radial and circular trusses. It is made up of 57 umbrella-like cable structures covered with an advanced fabric membrane. The panels appear to be flying because they are up in title air. The long cantilevered trusses make the effect even more striking. Birdair Inc., Buffalo, N.Y., N.Y., manufactured the panels and an d designed them with assistance from from Weidl eidlin inger ger Assoc Associa iate tess Inc., Inc., New New York. ork. The The Shang Shanghai hai Instit Institute ute of Archit Architect ectural ural Design Design and Researc Researchh designed designed the stadium. Design wind loads were established through two wind tunnel tests in China and one in Canada. A 1:35 model was made and tested under four different load cases. The The concre concrete te supe superst rstru ruct cture ure consi consist stss of an under undergro groun undd athletic centre, a parking garage, an event area, seating boxes and a hotel. The building and roof were finished in about 20 months, with as many as 3,000 workers on-site at one time. The plane projection projection of the roof is in the form of two ellipses, one inside title other. The outer ellipse measures 288.4 m in the East-W East-West est dừecti dừection on and 274.4 274.4 m North-So North-South uth;; while while the inner inner ellipse is 150 X 213 m. The highest point of the roof is 62.5 m above ground on the Western part of the roof, 41.2 m on the Eastern side; while the highest points on both the Northern and Southern sections is 31.8 m. The maximum overhang of the roof is 73.5 m. The 28,000 m2 fabric roof rests on 32 steel cantilever steel trusses that span 21 to 73.5 m, some of the longest in the world. The frapezoidal fabric panels inserted between the trusses are supported by a mast and cables (see Fig. 9.31) to provide a translucent, lightweight roof. To erect the panels, the telescopic masts were pushed up into the middle of the fabric, like an umbrella.

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71

6. 7.

8.


Marks, R.W., R.W., The Dymaxion Dymaxion World of Buckminster Buckminster Fuller, Reinhold, New Reinhold, New York, York, 1960. McHale, J., Buckminster Fuller, Fuller, Jỉệ, Prentice-Hall International, London and George Braziller Inc., New York, 1962, p.127. Pugh, A.J., An Introduction of Tensegrity, University of California Press, Berkeley! 1976, p. 122.

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