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Table of Contents Tensegritic Structures

9.1 Introduction The famous American Engineer, R. Buckminster Fuller is responsible 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 concept that members carrying tensile loads are much lighter than those carrying compressive 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 tensile component of high structural 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 elements, with isolated tensile elements. Fuller advocated that if this

Ca ble s (b) Vilna y Fig. 9.1 Tensegrity dome system (c) Tcnscgritic spherical shell

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-sessing 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, the cable net, is prestressed by the bars. In pneumatic shell, the envelope is 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 modules, not with straight compression elements, but with interlaced polygons, 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 Structures

3

)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 tensegrity systems27 by keeping the length of one component (cable or strut) constant and varying the length of the other component.

9.2 Tensegritic

nets

Vilnay12'2239'47 developed a different concept for tensegrity domes based on tensegrity nets. The main difference between Fullers 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 causing bars to interfere with one another. This problem is avoided in Vilnays concept; but the problem is that of longer bars, which, being in

Space Structures: Principles and Practice 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 infinite when there is no limit to 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

nets.

9.3 Tensegritic

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

No.2 Fig. 911 Spherical dome

using net no.6

Space Structures: Principles and Practice to construct a barrel vault. Similarly, Fig. 9.11 shows a spherical 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 without touching one another. In another relationship between struts and tendons, three tendons join the opposite ends of each strut and define a mirror image along it. Systems which have their struts and tendons aanged in this way are called 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 maxwells rules An interesting point about these zigzag systems is that three 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 Maxwells rule, it should have a minimum of 3J-6 members if it is to

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Space Structures: Principles and Practice

Zigzag pattern A. J. Pugh) system. Fig. Fig.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 of which can take tension only. According to Maxwells 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 Maxwells 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. Maxwell might have intended to refer to a maximum or minimum value of the length of one or more of its bars. Fullers 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 in fact became stiff.

Space Structures: Principles and Practice

This is presumably precisely the kind of maximum which Maxwell had in mind. Thus it seems clear that Fullers 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 another. Only a few examples of these studies, representing milestones in the development of tensegrity structure morphology are highlighted here. Snelsons work 27,37 is of artistic nature. Many of his sculptures are of more or less free form or of low degree of regularity. From the structural point of view they are all characterized by having a single state of prestress i.e., the tensioning of one cable prestresses the whole structure. Fullers tensegrity dome2 is a multifaceted, single-layer 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 tensegrity networks. In this configuration, bars are confined between two layers of cables or tendons. This configuration is obtained by joining together tensegrity prisms or truncated pyramids, or simplexes as Emmerich terms these objects. Flat and curved surfaces are also possible. Emmerich also investigated extensively and systematically tensegrity polyhedra, which can produce double-layer domes19. Vilnay inoduced the concept of infinite 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 carried out extensive investigations on Double-layer Tensegrity Grids (DLTGs) with non contacting bars. He also investigated the geometry of double-layer tensegrity domes by geodesic subdivision of a pyramid, employing DLTGs made of tensegritic pyramids. Grip24 extended the inventory of tensegrity polyhedra and hyper polyhedra, through the concepts of duality and space filling polyhedral networks to produce a virtually unlimited range of intricate and complex networks.

Tensegritic Structures

9.5 Characteristics

of tensegritic structures As already mentioned, tensegritic structures are similar to 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 tendons on the outside. The stmts push outwards like the pressurized air inside the balloon but are restrained by the tendons which are equivalent to the tensile skin of the balloon. Another characteristic which is common to balloon 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 under prestressing, the geometry of the structure is uniquely defined. Most of the tensegrity structures, like balloons are very sensitive to vibrations. However the vibration effects 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 conventional prestressed cable networks is that they are 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 tensegrity system is enantimorphic, each system having 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

Space Structures: Principles and Practice

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. In most of the cases the geometry of the prestressed structure will differ largely from the prepresessed structure. A geometric nonlinear analysis must be used to calculate the geometry of the net under prestressing.

9.7 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 large scale engineering. The basic concept of establishing a structure through an interaction of forces, rather than through a preconceived arrangement

Tensegritic Structures

of

components could be very important. Masts can be formed by joining several figures together as illusated in Fig. 9.14 (The thick lines represent the compression members and the thin lines represent tension members). Many lightweight lattices can be formed from tensegrity systems, especially the less complex ones.

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Space Structures: Principles and Practice

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

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Space Structures: Principles and Practice

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, and the static (or dynamic) analysis which determines member forces and nodal displacements under applied loads. An intermediate phase of investigation of stability, mechanisms and states of prestress is sometimes employed, mainly for research purposes31. Motro and his co-investigators (Ref. 18, 27, 34 and 35) applied the technique of dynamic relaxation to the shape finding problem. They also performed analysis of tensegrity prisms19 and simple double-layer networks. Vilnay, in his book 22, outlines 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 investigating mechanisms and states of prestress and for the static analysis employed a stiffness based procedure, which is computationally efficient, for the actual analysis of sizable networks. 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 acting at the nodes of these structures 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 eated 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 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 (2) the structure is determinate and under condition (3) the structure is unstable.

13

Space Structures: Principles and Practice

Hanaor2031 carried out parametric analytical investigations of double-layer tensegrity grids of varying geometries, including domes and geometrically rigid configurations (statically indeterminate, with no internal mechanisms). From these studies, he found that domes are stiffer than plane grids over a given span and geometrically rigid configurations are stiffer than geometrically flexible ones. There is a direct correlation between stiffness and load carrying capacity for a given span and topology. Hanaor20,31 investigated the various configurations 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.

Space Structures: Principles and Practice

Experimental tests were carried out by Hanaor21 and by Motro, et al.34 on small models. Hanaor performed static tests on small DLTG models of various configurations. Motro et al. performed 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.

Other aspects Structures built of infinite regular tensegritic nets having appropriate boundary conditions can be pre-stressed if atleast three cables meet at a joint. If the net has less than five cables at a joint the pre-sessed structure can carry in its 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 that the tensegrity structures display Fiillers ingenuity in designing large but easily packed and portable 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 Maxwells 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 on9.9

Space Structures: Principles and Practice

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 deployable applications, it is highly desirable that the membrane serves as a structural, as well as functional 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 supported structures while avoiding the disadvantages, several tensile structure systems have been developed. First practical example of application of the tensegrity dome was provided by a Polish Engineer, Waclaw Zalewski in 1961, three years before B. Fiiller patented his idea in 1964. Zalewski covered an auditorium in Katowice, Poland, using the first version 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 Fullers tensegrity structure and developed a radial cable dome43 called the tenstar dome.

Tensegritic Structures

16

Fig. 9.16 Schematic diagram of Tenstar Dome Ridge 15-7 ^/~Ridge cable Valley net cable cabl Ridge Fabr connector Cable domes were first adopted ic for the Gymnastic Arena (dia. 120 m) Tension Diago ring nal cable Roof section Fig. 9.17 Cable (Tensegrity) dome

Ridge cable Ponding cable as required Roof plan

Valley cable

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

Diagonal cable Hoop connector

Space Structures: Principles and Practice

Upper fabric

Strand Section Y - Y Fabric clamp (each side) FlixSE Lower fabric Section X - X Bridge rope * ---