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Chapter 1

An Introduction to Tensegrity

1.1 Basic Tensegrity Principles

Anthony Pugh1 gives the following definition of tensegrity:

A tensegrity system is established when a set of discontinuous compressive components interacts with a set of continuous tensile components to define a stable volume in space.

Tensegrity structures are distinguished by the way forces are distributed within them. The members of a tensegrity structure are either always in tension or always in compression. In the structures discussed in this book, the tensile members are usually cables or rods, while the compression members are sections of tubing. The tensile members form a continuous network. Thus tensile forces are transmitted throughout the structure. The compression members are discontinuous, so they only do their work very locally. Since the compression members do not have to transmit loads over long distances, they are not subject to the great buckling loads they would be otherwise, and thus they can be made more slender without sacrificing structural integrity.

While the structures discussed in this book aren't commonly seen, tensegrity structures are readily perceptible in the surrounding natural and man-made environment. In the realm of human creation, pneumatic structures are tensegrities. For instance, in a balloon, the skin is the tensile component, while the atoms of air inside the balloon supply the compressive components. The skin of the balloon consists of atoms which are continuously linked to each other, while the atoms of air are highly discontinuous. If the balloon is pushed on with a finger, it doesn't crack; the continuous, flexible netting formed by the balloon's skin distributes this force throughout the structure. And when the external load is removed, the balloon returns to its original shape. This resilience is another distinguishing characteristic of tensegrity structures.

Another human artifact which exhibits tensegrity qualities is prestressed concrete. A prestressed concrete beam has internal steel tendons which, even without the presence of an external load, are strongly in tension, while the concrete is correspondingly in compression. These tendons are located in areas so that, when the beam is subjected to a load, they absorb tensile forces, and the concrete, which is not effective in tension, remains in compression and resists heavy compressive forces elsewhere in the beam. This quality of prestressed concrete, that forces are present in its components even when no external load is present, is also very characteristic of tensegrity structures.

In the natural realm, the structural framework of non-woody2 plants relies completely on tensegrity principles. A young plant is completely composed of cells of water which behave much like the balloon described above. The skin of the cell is a flexible inter-linkage of atoms held in tension by the force of the water in the contained cell.3 As the plant is stretched and bent by the wind, rain and other natural forces, the forces are distributed throughout the plant without a disturbance to its structural integrity. It can spring back to its usual shape even when, in the course of the natural upheavals it undergoes, it finds itself distorted far from that shape. The essential structural use the plant makes of water is especially seen when the plant dries out and therefore wilts.

1.2 Applications of Tensegrity

The qualities of tensegrity structures which make the technology attractive for human use are their resilience and their ability to use materials in a very economical way. These structures very effectively capitalize on the ever increasing tensile performance modern engineering has been able to extract from construction materials. In tensegrity structures, the ethereal (yet strong) tensile members predominate, while the more material-intensive compression members are minimized. Thus, the construction of buildings, bridges and other structures using tensegrity principles could make them highly resilient and very economical at the same time.

In a domical configuration, this technology could allow the fabrication of very large-scale structures. When constructed over cities, these structures could serve as frameworks for environmental control, energy transformation and food production. They could be useful in situations where large-scale electrical or electromagnetic shielding is necessary, or in extra-terrestrial situations where micrometeorite protection is necessary. And, they could provide for the exclusion or containment of flying animals over large areas, or contain debris from explosions.

These domes could encompass very large areas with only minimal support at their perimeters. Suspending structures above the earth on such minimal foundations would allow the suspended structures to escape terrestrial confines in areas where this is useful. Examples of such areas are congested or dangerous areas, urban areas and delicate or rugged terrains.

In a spherical configuration, tensegrity designs could be useful in an outer-space context as superstructures for space stations.

Their extreme resilience make tensegrity structures able to withstand large structural shocks like earthquakes. Thus, they could be desirable in areas where earthquakes are a problem.

1.3 Early Tensegrity Research

Key contributions to the early development of tensegrity structures appear to have come from several people. Some historians claim Latvian artist Karl Ioganson exhibited a tensegrity prism in Moscow in 1920-21 though this claim is controversial.4 Ioganson's work was destroyed in the mid-1920's by the Soviet regime, but photographs of the exhibition survived. French architect David Georges Emmerich cited a different structure by Ioganson as a precedent to his own work.5 In the article "Snelson on the Tensegrity Invention" in Lalvani96, tensegrity pioneer Kenneth Snelson also cites the Russian constructivists, of which Ioganson was a member, as an inspiration for his work.

The word "tensegrity" (a contraction of "tensile-integrity") was coined by American entrepreneur Buckminster Fuller.6 Fuller considered the framing of his 1927 dymaxion house and an experimental construction of 1944 to be early examples of the technology.7

In 1948, after attending lectures by Fuller at Black Mountain College in North Carolina, Kenneth Snelson made a catalytic contribution to the understanding of tensegrity structures when he assembled his X-Piece sculpture.8 This key construction was followed by further contributions by Fuller and others of his circle.9 Independently, in France, in 1958, Emmerich was exploring tensegrity prisms and combinations of prisms into more complex tensegrity structures, all of which he labeled as "structures tendues et autotendantes" (prestressed tensile structures).10

Emmerich, Fuller and Snelson came out with patent claims on various aspects of the technology in the 1960's,11 and all continued developing the technology. Fuller's primary interest was adapting the technology to the development of spherical and domical structures with architectural applications in mind.12 He also used tensegrity structures to make some philosophical points.13 As an architect, Emmerich was also interested in architectural applications and designed at least one dome as well.14 Snelson is primarily interested in the artistic exploration of structure using the medium of tensegrity.15 All three developed tower or mast structures which continue to be a source of fascination for tensegrity enthusiasts but only recently have found practical application in the development of deployable structures.16

While many tensegrity models were built and achieved quite a fame for themselves, for instance through a notable exhibition of Fuller's work at the Museum of Modern Art in New York,17 and a retrospective on Snelson's work at the Hirschorn Museum in Washington, D.C.,18 the bulk of production structures which Fuller and his collaborators produced were geodesic domes rather than tensegrity structures.19

It seems probable that part of the reason that tensegrity structures didn't get farther, even in circles where there was a strong interest in practical applications of tensegrity, was the apparent dearth of powerful and accurate tools for carrying out their design. Fuller's basic tensegrity patent has quotations of member lengths, but no indication of how one would compute the lengths.20 Probably the lengths were computed after the fact by measuring the tendon lengths of a finished structure.

An early exception to this dearth of information on tensegrity calculating was Hugh Kenner's work Geodesic Math21, which went into an exact technique for the very simple tensegrity prism and outlined an approximate technique for dealing with some simple spherical structures. His technique for designing prisms is presented in Section 2.2 as an introduction to tensegrity calculations since these simple structures provide an avenue for developing an intuitive feel for what tensegrity is all about.

First-hand accounts of the early history of tensegrity can be found in Coplans67, Fuller61 and Lalvani96.

1.4 Recent Tensegrity Research

Civil engineers have taken an interest in tensegrity design. An issue of the International Journal of Space Structures22 was devoted to tensegrity structures. In that collection, R. Motro noted in his survey article "Tensegrity Systems: State of the Art":

...there has not been much application of the tensegrity principle in the construction field. ...examples...have generally remained at the prototype state for lack of adequate technological design studies.23

The primary obstacles to the practical application of tensegrity technology which these researchers have identified are:

1.Strut congestion - as some designs become larger and the arc length of a strut decreases, the struts start running into each other.24
2.Poor load response - "relatively high deflections and low material efficiency, as compared with conventional, geometrically rigid structures."25
3.Fabrication complexity - spherical and domical structures are complex which can lead to difficulties in fabrication.26
4.Inadequate design tools - as Motro's statement above suggests, lack of design and analysis techniques for these structures has been a hindrance.

Double-layer designs, introduced by Snelson and Emmerich and further investigated by Motro and Hanaor,27 begin to deal with the first obstacle. Poor load response (the second obstacle) is still a problem in their configurations, and they don't have much advice on fabrication techniques (the third obstacle). They have developed tools to deal with the fourth obstacle. These tools are based on earlier work by J. H. Argyris and D. W. Scharpf analyzing prestressed networks.28

In what follows, reference is made to this work; however, the techniques presented here are somewhat different and take advantage of some special characteristics of tensegrity structures. Hanaor's work is the source of the "double-layer" terminology used to describe some of the structures presented here. Appendix A discusses relationships between Motro's and Hanaor's work and some of the ideas presented here.

Mention should also be made of David Geiger's "Cabledome" technology and Matthys Levy's related spatially triangulated tensegrity dome technology which have provided the bulk of practical applications of tensegrity in construction. Campbell94 provides a description of these two approaches which are dependent on a peripheral anchorage for their structural integrity.

Much interesting theoretical work has come from the University of California at San Diego. A particularly interesting result, described in Masic05, shows that any affine transformation of a tensegrity structure retains tensegrity properties. This is a result pertaining mostly to tensegrity designs with hubs modeled simply as single points. When hubs are modeled more completely, an affine transformation would not be appropriate in many situations due to the distortion it would produce in the hubs.

1.5 Other Space Frame Technologies

Other space frame technologies can be roughly sorted into three categories. The first category contains those space frame technologies where member lengths are very homogenous. They are typically realized as planar trusses perhaps connected at an angle with other planar trusses. Biosphere 229 is an example. Their faceted shape means they contain less space per unit of material than a spherical shape would. Makowski65 contains a variety of examples.

The second category contains those which are typified by the geodesic domes30 and Kiewitt domes.31 Geodesic domes share many qualities of tensegrity domes. The primary difference is the requirement of these technologies that all components be able to sustain both tensile and compressive forces.

The third category is typified by the circus tent. Here a tensile network (the tent fabric) is supported at various locations by large poles. Anchors and supporting cables usually also play a role. These structures can almost be considered a sort of tensegrity since elements of the structure are either in tension all the time or compression all the time. Their compressive elements are much fewer and much more massive than in the usual sort of tensegrity. Many times these poles disrupt the internal space of the structure substantially. The tensile network has a catenary shape to it between the compressive supports. This means it encloses less space than it would if it were supported as in the usual spherical tensegrity with many struts embedded in the network. For a variety of examples of structures in this category, see Otto73.

1.6 Book Scope and Outline

The discussion here centers on tensegrity structures of a particular type. They are composed of discrete linear members: the tensile members can be thought of as cables which pull two points together, while the compression members can be thought of as sections of rigid tubing which maintain the separation of two points. The tensile members are continuously connected to each other and to the ends of the compression members while the compression members are only connected to tensile members and not to other compression members.32 The primary motivation for this work is to outline mathematical methods which can be applied to the design and analysis of this sort of tensegrity.

In addition to single-layer tensegrity structures, double-layer structures are presented. In particular, highly-triangulated methods of tensegrity trussing are discussed which can be applied to domical, spherical and more general tensegrity designing. These double-layer tensegrities are designed to be effective in larger structures where trussing is needed.

While a lot of the discussion centers around highly symmetric, spherical structures, the derivation and analysis of truncated structures like domes are also treated. The development of techniques for these less symmetric applications makes tensegrity a much more likely tool for addressing practical structural problems.

Finally, sections on analyzing member forces and clearances in tensegrity structures are included. This analysis is a large element of concern in any engineering endeavor and also of interest to anyone who seeks an understanding of the behavior of tensegrity structures.

1 Pugh76, p. 3. See also the last footnote in this chapter which cites Kanchanasaratool02's elegantly succinct definition and the footnote in Section 6.2.3 which cites Wang98's rigorous and descriptive definition.

2 The qualification "non-woody" is used to exclude trees. The woody elements of a tree are made to undergo both tension and compression, much as is required of the structural elements of a geodesic dome.

3 Donald Ingber has pointed out (personal communication, October 8, 2004) that this very simple view of the living cell does not reflect very well the results of modern research in applying tensegrity principles to the analysis of cell structure. No doubt this example would benefit from the attention of a biologist and the details would change as a result. For a look at how tensegrity principles have been applied to the analysis of living cells, see Ingber98.

4 See note 1 for Section 2.2.

5 Emmerich88, pp. 30-31. The structure Emmerich references is labeled "Gleichgewichtkonstruktion". He states:

Cette curieuse structure, assemblée de trois barres et de sept tirants, était manipulable à l'aide d'un huitième tirant detendu, l'ensemble étant déformable. Cette configuration labile est très proche de la protoforme autotendante à trois barres et neuf tirants de notre invention.
This apparently means he doesn't recognize Ioganson's invention of the tensegrity prism. Gough98's thorough examination of the exhibition photographs unfortunately doesn't mention Emmerich's work.

6 See the description for Figure 1 in U.S. Patent No. 3,063,521, "Tensile-Integrity Structures", November 13, 1962. Kenneth Snelson sometimes uses the description "floating compression" in preference to the term tensegrity.

7 Fuller73, Figs. 261, 262 and 263, pp. 164-165.

8 See Lalvani96, pp. 45-47. Fuller immediately publicized Snelson's invention, but via a variation on Snelson's X-Piece which used tetrahedral radii rather than an X as the compression component (Fuller73, Figs. 264-267). Snelson didn't publish until a decade later when he filed his patent (U.S. Patent No. 3,169,611). Emmerich characterizes Fuller's contribution to Snelson's invention as that of a "catalyst" (Lalvani96, p. 49).

It seems both Fuller and Snelson catalyzed this tensegrity revolution by bringing together their relevant ideas and experience and fabricating artifacts that stimulated further innovations. The next important step, which Fuller took, was to start using the simple linear compression components which are used to fabricate the structures studied in this book. In the summer of 1949, the same year that Fuller found out about Snelson's work, Fuller fabricated the tensegrity icosahedron (Section 2.3) which is an outgrowth of the "jitterbug"/cuboctahedron framework whose dynamics he had been exploring. Kenneth Snelson gives the following description of the sequence of events (see "Re: Tensegrity on and on", bit.listserv.geodesic news group, December 6, 2005):

1)  I made the X-Module column December 1948, Pendleton, Oregon; obviously it was extendable through added modules; a space-filling system.
2)An astonished Fuller when I showed it to him, July, 1949 did NOT say to me it was "tensegrity" as he later claimed since there was no such coinage until five years later.
3)He insisted it ought not to be X's but rather the central angles form (as in his knock-off "Tensegrity Mast") so I built at his request the curtain-rod column shown with him holding it in the well-known photo at BMC. (Again, it was I, not he, who actually built that structure.)
4)During that summer Bucky created the first six-strut which was a first step toward the tensegrity domes. It was an important discovery but remarkably he saw it only as a step toward tensegrity domes.

The photograph Snelson refers to in #3 above is reproduced in Fuller73, Fig. 264, and should not be confused with a more formal studio photograph from the same year reproduced as Fig. 265. See "Re: Tensegrity on and on", bit.listserv.geodesic news group, March 6, 2006, which has extracts of an e-mail of the same date from Kenneth Snelson to the author. The captions for these two photographs were apparently inadvertently switched in Fuller73.

9 Fuller73, Figs. 264-280, pp. 165-169.

10 Lalvani96, p. 29.

11 R. Buckminster Fuller, U.S. Patent No. 3,063,521, "Tensile-Integrity Structures", November 13, 1962. David Georges Emmerich, French Patent No. 1,377,290, "Construction de Reseaux Autotendants", September 28, 1964, and French Patent No. 1,377,291, "Structures Linéaires Autotendants", September 28, 1964. Kenneth Snelson, U.S. Patent No. 3,169,611, "Continuous Tension, Discontinuous Compression Structure", February 16, 1965.

12 Fuller73, Figs. 268-280, pp. 165-169. See also Lalvani96 and Wong99, pp. 167-178, for further discussion of the Fuller-Snelson collaboration and controversy.

13 See Fuller75, Fig. 740.21, p. 407, for an example.

14 Emmerich88, pp. 158-159.

15 See the "Sculpture" section of Snelson's website,

16 For example, Skelton97.

17 Geodesic D.E.W. Line Radome, Octe-truss and Tensegrity Mast - one-man, year-long, outdoor garden exhibit, November 1959. Also, at least one tensegrity was exhibited inside. See Fuller73, p. 169, illus. 280. This exhibit was also significant in that Kenneth Snelson also participated with a vitrine of his own, and the exhibit marked his return to making tensegrity structures after pursuing other occupations for ten years. See Lalvani96, p. 47.

18 See Snelson81. The exhibition was in Washington, D.C., June 4 to August 9, 1981.

19 See Section 1.5 for a comparison of geodesic dome and tensegrity technology.

20 See Figure 7 in U.S. Patent No. 3,063,521, "Tensile-Integrity Structures", November 13, 1962.

21 Kenner76.

22 Vol. 7 (1992), No. 2.

23 Motro92, p. 81.

24 Hanaor87, p. 35.

25 Hanaor87, p. 42.

26 Hanaor87, p. 44.

27 See photos in Lalvani96, p. 48, and the references in Appendix A.

28 Argyris72.

29 Kelly92, p. 90.

30 Fuller73, pp. 182-230.

31 Makowski65.

32 These structures would be described as Class 1 tensegrity structures using the definition cited in Kanchanasaratool02. That definition is:

A tensegrity system is a stable connection of axially-loaded members. A Class k tensegrity structure is one in which at most k compressive members are connected to any node.
"Connection" doesn't seem like quite the right word here and could be a typographical error. Substituting "continuously-connected collection" yields a better description. "node" is synonymous with "hub".