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A Technology for Designing Tensegrity Domes and Spheres Robert Burkhardt Tensegrity Solutions Box 426164 Cambridge, MA 02142-0021 USA [email protected] This material in this prospectus, along with a revision history, is available on the internet at: http://bobwb.tripod.com/prospect/prospect.htm. Last revision: October 10, 2005

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Page 1: A Technology for Designing Tensegrity Domes and Spherestsoln.tripod.com/prospect/prospect.pdf · A Technology for Designing Tensegrity Domes and Spheres ... tensegrity, space frame,

A Technology forDesigning Tensegrity Domes and Spheres

Robert BurkhardtTensegrity Solutions

Box 426164Cambridge, MA 02142-0021

[email protected]

This material in this prospectus, along with a revision history,is available on the internet at:

http://bobwb.tripod.com/prospect/prospect.htm.

Last revision: October 10, 2005

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Abstract

A technology based on tensegrity techniques is proposed for the economical construction oftough, rigid, large-scale domes. Sample structures, possible applications, relatedtechnologies and plans for future research are also presented.

The technology is of the double-layer type where an outer and an inner layer of tendons areinter-connected by struts and additional tendons. Struts and tendons are organized toemphasize triangulation and lateral transmission of exogenous loads within the spaceframe. It is hypothesized, and preliminary testing has verified, that this arrangement willresult in a structure with improved stiffness. If more thorough testing verifies thishypothesis is correct, this technology will overcome a shortcoming which plagues tensegritydesigns, while retaining the ethereality and resilience which recommends these designs.

The development of a structural technology to economically encompass large areas wouldbe useful for warehouses, permanent or temporary protection for archaeological and othervulnerable sites, large-scale electrical or electromagnetic shielding and exclusion orcontainment of flying animals or other objects. Over cities, structures based on such atechnology could serve as frameworks in which environmental control, energytransformation and food production facilities could be embedded.

Summary Advantages

• Improved Rigidity• Ethereal• Resilient• Equal-Length Struts• Simple Joints

Key Words: tensegrity, space frame, structure, truss, dome, architecture,suspension, tensile, software

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CONTENTS v

Contents

1 Introduction 8

2 Outline of the Technology 9

2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Summary of Research Results . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Tensegrity Truss Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Form-Finding Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 Analysis of Exogenous Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.6 Design Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Possible Applications 20

4 Related Technologies 21

5 Comparison with Related Technologies 22

6 Future Directions 24

6.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6.2 Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6.3 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

7 Tensegrity Solutions: Background Information 25

8 References 26

9 Panel Comments on Proposal to NSF 28

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CONTENTS vi

9.1 Panel Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

9.2 Comments of First Respondent . . . . . . . . . . . . . . . . . . . . . . . . . 28

9.3 Comments of Second Respondent . . . . . . . . . . . . . . . . . . . . . . . . 29

9.4 Comments of Third Respondent . . . . . . . . . . . . . . . . . . . . . . . . . 30

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LIST OF FIGURES vii

List of Figures

1 4ν Double-Layer Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 2ν Double-Layer Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Tensegrity Tripod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 6ν Octahedral Tensegrity Network . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Spherical Assembly of Tensegrity Tripods . . . . . . . . . . . . . . . . . . . . 13

6 Planar Assembly of Tensegrity Tripods . . . . . . . . . . . . . . . . . . . . . 14

7 10ν Octahedral Tensegrity Network . . . . . . . . . . . . . . . . . . . . . . . 17

8 10ν Double-Layer Dome (Side View) . . . . . . . . . . . . . . . . . . . . . . 18

9 8ν Double-Layer Dome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

10 4ν Double-Layer Dome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

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1 INTRODUCTION 8

1 Introduction

The technology discussed in this prospectus is aimed at facilitating the economicalconstruction of tough, rigid, large-scale domes. The technology is based on the tensegrityapproach to space frame design which strictly segregates compressive and tensile forcesamong the members of the frame. This minimizes the number of members which need tosustain compressive forces. The minimization of these more massive and visually intrusivecomponents results in most frame members being lighter-weight more-transparent tensilecomponents.

Since the introduction of the tensegrity concept in the 1940’s, many models have beenbuilt.[See Fuller73 pp. 164-169 for several early examples.] However, as R. Motro notes inhis survey “Tensegrity Systems: State of the Art”:

...there has not been much application of the tensegrity principle in theconstruction field. ...examples...have generally remained at prototype state forlack of adequate technological design studies.[Motro92, p. 81]

This is not to say that there have been no practical structures built using the tensegrityprincipal. Effective use of tensegrity principles has been made by Geiger and Levy todesign roofs which cover large areas.[Campbell94 reviews these approaches.] However theseapproaches result in essentially composite structures where a substantial portion of thestructure is fabricated using non-tensegrity technologies. Thus these structures do not takecomplete advantage of what the tensegrity technique has to offer.

Structures based on the technology proposed here maximally employ tensegrity principles.Tensegrity methods derive maximal structural performance from the materials employed.This is due to their maximization of the tensile components of the structure which allowsthem to take full advantage of the progress has been made in deriving larger tensilestrengths from materials.[See Davis70 pp. 2-3 for data on metals, and Jang94 for the hugetensile strengths obtained from fibers of various materials.] The resulting light weightallows tensegrity structures to encompass very large areas with minimal support at theirperimeters, obviating the “heavy anchorage devices”[Motro87, p. 43] needed for supportwith some cable-based technologies, or extensive support structures needed by thecomposite structures discussed above.

This prospectus focuses on the construction of tensegrity domes since these are most usefulin a terrestrial context. However, since the technology proposed conceives domes astruncated spheres, the technology can easily be applied to the design of spheres as well.

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2 OUTLINE OF THE TECHNOLOGY 9

2 Outline of the Technology

2.1 Motivation

One characteristic of materials research has been the increasing tensile performance whichresearchers have been able to extract from materials through various fabricationtechnologies. In the construction sector, this performance can be harnessed by structuraltechnologies which rely predominantly on tensile components. Tensegrity designs representsuch a technology.

The primary obstacles to the practical application of tensegrity technology have been:

1. Strut congestion - as some designs become larger and the arc length of a strutdecreases, the struts start running into each other.[Hanaor87, p. 35]

2. Poor load response - “relatively high deflections and low material efficiency, ascompared with conventional, geometrically rigid structures.”[Hanaor87, p. 42]

3. Fabrication complexity - spherical and domical structures are complex which canlead to difficulties in fabrication.[Motro87, p. 44]

4. Inadequate design tools - lack of design and analysis techniques for these structureshas been a hindrance.

2.2 Summary of Research Results

Much progress has been made by Tensegrity Solutions and others in overcoming theseobstacles. The research of Tensegrity Solutions in the area of tensegrity design representsthe focusing of results from several disciplines (computer science, mechanical engineering,mathematical programming, economics) on problems in this area. This research hasresulted in flexible techniques for designing and analyzing tensegrity structures as well asseveral design innovations which apply to these structures. Many of the problems relatedto the practical realization of tensegrity structures have already been solved by TensegritySolutions. Specifically, the following solutions have been obtained:

1. Mathematical programming techniques and other algorithms have been implementedin computer software to compute unloaded member lengths, clearances betweenmembers, relative prestress forces, effects of exogenous loads and perform otheranalyses. In this context, a general method for formulating the mathematical

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2 OUTLINE OF THE TECHNOLOGY 10

programming problem was developed which provides maximal design flexibility. Thisflexibility has allowed, among other things, the design of approximately sphericalstructures where all the struts have equal lengths.

2. A method for organizing tensegrity trusses into spheres has been developed. Themethod is based on the geodesic subdivisioning[Kenner76, Chapter 5] of anoctahedron. The triangular faces of the octahedron are subdivided into smallertriangles using a grid generated by subdividing the edges of the octahedron. Thenumber of subdivisions for an edge is called the frequency of a particularsubdivisioning. Frequency is also referred to by the Greek letter ν (nu). Hence, atruss based on an octahedron whose edges are subdivided into 10 segments would bereferred to as a 10ν octahedral truss. When the average length of the membersremains constant, greater frequency means more members and hence a largerstructure, linear dimensions increasing proportionally to the frequency.

3. A method for truncating spheres to obtain domes has been developed. In turn,methods have been developed to truncate domes to obtain half shells and largeropenings in the dome.

4. A method for deriving a planar, circular foundation with evenly spaced footings forthe dome has been developed. (See Figure 8 for an example application of themethod.)

5. Truss topologies are utilitzed which maximize tendon triangulation and minimize thenumber of struts per unit area. (The truss topology utilized is one of manydeveloped in a planar context by Kenneth Snelson in the 1950’s.)

6. Assembly procedures have been developed which will be suitable forcommercial-scale structures.

7. Small-scale spherical models have been assembled to test computed values forunloaded member lengths and examine behaviors of the structures. A photograph ofa model of a sphere assembled using the proposed double-layer tensegrity technologyis shown in Figure 1.

8. Two spherical models have been assembled to test potential commercial realizationsof these structures. They used metal components instead of the simpler models’dowels and fishing line. A photograph of one of these models is shown in Figure 2.

9. Two larger-scale dome models composed of wooden stakes and nylon twine havebeen constructed to test the proposed methodology for dome design and investigatefabrication techniques. A photograph of the 4ν model which uses 3-foot struts isshown in Figure 10. A photograph of the 8ν model which uses 4-foot struts is shownin Figure 9.

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2 OUTLINE OF THE TECHNOLOGY 11

Figure 1: 4ν Double-Layer Sphere

10. Techniques have been developed for representing these structures schematically in asystematic and intelligible way.

11. A preliminary analysis has been done of the response of these structures toconcentrated loads.

12. Software has been developed to automatically generate description files forvery-large-frequency versions of these structures.

2.3 Tensegrity Truss Technology

The starting point of Tensegrity Solutions’ system for designing a tensegrity truss is thetensegrity tripod[Burkhardt04] and tensegrity networks derived from geodesicpolyhedra.[Pugh76, pp. 28-30] A tensegrity tripod encompasses six tendons. Threetendons, referred to here as “apex” tendons, bind the three struts together at one end toform a triangle which represents the apex of the tripod. Each of the other three tendons,referred to here as “interlayer” tendons, connects the apex end of one strut to the thenon-apex end of an adjacent strut in the tripod. This forms an additional three triangles,each triangle containing an apex tendon, an interlayer tendon and a strut. Figure 3 showsa tripod, Figure 4 shows a tensegrity network based on a 6ν subdivisioning of anoctahedron and Figure 5 shows a spherical assembly of tripods embedded in the tensegritynetwork. To enhance diagrammatical clarity, the back halves of Figures 4 and 5 have beensliced away. A dome can be induced by truncating the sphere.

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2 OUTLINE OF THE TECHNOLOGY 12

Figure 2: 2ν Double-Layer Sphere

Figure 3: Tensegrity Tripod

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2 OUTLINE OF THE TECHNOLOGY 13

Figure 4: 6ν Octahedral Tensegrity Network

Figure 5: Spherical Assembly of Tensegrity Tripods

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2 OUTLINE OF THE TECHNOLOGY 14

Figure 6: Planar Assembly of Tensegrity Tripods

In addition, each inner tendon triangle where the struts of three tripods converge can beviewed as the apex of an inward-pointing tripod. Adding the corresponding interlayertendons to complete these tripods provides more interlayer triangulation, and thus morereinforcement of the structure. Finally, the outer layer of tendons can be completed byconnecting the apexes of the original outward-pointing tripods using tendon triangles. Thiscompletes the truss. It is hypothesized that this arrangement will result in a structure ofsuperior stiffness. In this configuration, each strut is secured by 12 tendons. It is worthnoting that this is precisely the minimum number of tendons Fuller has experimentallyfound to be necessary to rigidly fix one system in its relationship to a surroundingsystem.[Fuller75, pp. 105-107]

Figure 6 shows what the complete truss looks like in a planar context. The truss looks thesame regardless of which side of the plane it is being viewed from. This apparently planarsection of truss can also be conceived as a small section of the surface of a very largedouble-layer sphere. In this context, the small section looks the same regardless of whetherit is being viewed from inside or outside the sphere.

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2 OUTLINE OF THE TECHNOLOGY 15

Tensegrity Solutions’ technique can be applied to any tensegrity network which can bedividing into alternating polylaterals.[A “polylateral” is a subset of a tensegrity. Thesubset is a ring composed of hubs of the tensegrity and includes tendons whichcontinuously connect the hubs pair-wise. A “hub” is any location where tendons and strutsare connected to each other. Each included tendon connects two hubs, and each hub isconnected to two tendons. The minimum polylateral is a triangle of three hubs connectedby three tendons.] However, in the interests of maximizing stiffness and minimizing strutdensity, alternating triangles seem the best way to go. In at least one situation, Hanaor hasexperimentally verified the preferability of triangular networks on the basis ofstiffness.[Hanaor87, p. 40]

In the spherical context, the limitation to networks of alternating triangles essentiallyeliminates all but the octahedron as possible bases for a network. Figure 7 diagrams a 10νversion of such a network as it appears when projected onto a sphere. This has itsdisadvantages since the octahedron has a much more faceted look to it than theicosahedron. An icosahedron would be the preferred basis if it were feasible. This meansadjustments must be made in the shape finding procedure so a spherical looking result isobtained. An octahedral basis also yields a lot of member activity at the vertexes whichmust be resolved. Tensegrity Solutions has developed design methodologies whicheffectively solve both these problems.

2.4 Form-Finding Technology

In addition to developing an effective double-layer topology for tensegrity domes,Tensegrity Solutions has developed very flexible techniques for form finding. Thesetechniques are based on the mathematical programming view of tensegrity as propoundedin Burkhardt04. Form finding is viewed as the solution of a non-linear mathematicalprogramming problem whose most general statement is:

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2 OUTLINE OF THE TECHNOLOGY 16

minimize o ≡ w1l21 + · · ·+ wM l2M

Coordinatevalues

subject to Member constraints:

±l2

M+1 ≥ ±l2M+1

· · ·±l

2

N ≥ ±l2N

Symmetry constraints:

0 = s1(· · ·)· · ·

0 = sK(· · ·)

Determinacy and other constraints:

0 = d1(· · ·)· · ·

0 = dQ(· · ·)

where:

M = number of members in the objective functionN = number of members in the modelK = number of symmetry constraintsQ = number of determinacy and other constraints

The phrase “Coordinate values” appearing under “minimize” indicates that the coordinatevalues are the control variables of the minimization problem. These are the values whichare changed (in accordance with the constraints) to find a minimum value for o.

lm stands for the length of member m, and wm is the constant weight on the second powerof lm in the objective function. wm is negative if member m is a strut, positive otherwise.These weights provide maximum control over the characteristics of the resulting solution.Just about any feasible tensegrity configuration can be reached by choosing the membersto include in the objective function and their weights appropriately.

In the member constraints, ln is a positive constant value. “-” precedes l2n and l2

n if membern is a strut and “+” precedes them if member n is a tendon. In the other constraints,sk(· · ·) and dq(· · ·) are (most likely linear) functions of the coordinate values.

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2 OUTLINE OF THE TECHNOLOGY 17

Figure 7: 10ν Octahedral Tensegrity Network

Tensegrity Solutions has developed exact and penalty methods for solving this problem.The technique for the exact method is based on ideas in Luenberger’s book[Luenberger73]and some matrix techniques. The penalty method is implemented as described inLuenberger[Luenberger73, pp. 278-280]. The penalty method is especially advantageous ininitial iterations since it doesn’t require the constraints to be exactly met. The exactmethod yields more precise results and can be used to fine tune a result. In both cases, thedesign software allows the solution to be derived using either the PARTANmethod[Luenberger73, pp. 184-186] or the Fletcher-Reeves method[Luenberger73, p. 182] inconjunction with a custom line-search technique.

A corollary to this research showed that the relative prestress forces in a structure’smembers (also known as “self-stress”) can be computed from the Lagrange multipliers ofthe resulting solution instead of going through the more standard procedure of solving thematrix of force vectors.

2.5 Analysis of Exogenous Loads

To analyze the effects of exogenous loads on a structure’s members, a different conceptualframework is used. For this purpose, a structure is viewed as a flexibly-jointed set of elasticand fixed-length members, the tendons being the elastic members, and the struts being thefixed-length members. The solution of the mathematical programming problem outlinedabove and the accompanying prestress forces provide a valid initial unloaded solution forthe equations developed to compute the effects of exogenous loads. The vectors

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2 OUTLINE OF THE TECHNOLOGY 18

Figure 8: 10ν Double-Layer Dome (Side View)

representing the exogenous load are then introduced, and the equation system is solvedusing Newton techniques to arrive at values for the resultant member forces.

2.6 Design Examples

A representation of a dome which utilitizes Tensegrity Solutions’ technology appears inFigure 8.

A photograph of a lower-frequency dome which has actually been built is shown inFigure 9. It is composed of 102 one-inch by one-inch by four-foot (25.4 mm by 25.4 mm by1.22 m) hardwood stakes and 570 nylon-twine tendons of various lengths.

A photograph of a very-low-frequency dome is shown in Figure 10. It is composed of 27one-inch by one-inch by three foot (25.4 mm by 25.4 m by 0.91 m) hardwood stakes and174 nylon-twine tendons of various lengths. At this low frequency, the interior space isfairly minimal in comparison with the volume of the dome as a whole. This model wasconstructed mainly to test software predictions and develop assembly procedures.

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2 OUTLINE OF THE TECHNOLOGY 19

Figure 9: 8ν Double-Layer Dome

Figure 10: 4ν Double-Layer Dome

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3 POSSIBLE APPLICATIONS 20

3 Possible Applications

This research was inspired by Buckminster Fuller’s claim that structures based ontensegrity technology could be used to build domes which could provide environmentalcontrol for entire cities. Brain-storming sessions have yielded additional possibilities for thestructural technology outlined in this prospectus. The resulting list represents an initialattempt to identify applications for which the technology would be suitable. As thetechnology develops and is tested against them, some or all of the applications may bewinnowed out and other suitable applications not in the list may become apparent. Thecurrent list is as follows:

• Superstructures for embedded substructures allowing the substructures to escapeterrestrial confines where this is useful (e.g. in congested or dangerous areas, urbanareas, flood plains or irregular, delicate or rugged terrains).

• Economic large-scale protection of storage, archaeological, agricultural, construction orother sites.

• Refugee or hiking shelters.

• Frames over cities for environmental control, energy transformation and foodproduction.

• Large-scale electrical or electromagnetic shielding.

• Exclusion or containment of flying animals or other objects.

• Spherical superstructures for space stations.

• Earthquake-resistant applications. These structures are extremely resilient and testingwould very likely show they could withstand large structural shocks like earthquakes.Thus, they would likely be desirable in areas where earthquakes are a problem. Apneumatic structure (Fuller has characterized tensegrities as pneumatic structures, seeFuller75, pp. 369-431.) performed very well during recent earthquakes in Japan. (SeeSaka97, pp. 143-145.)

• Low-environmental-impact shells for musical performances.

• Indoor/outdoor pavilions for trade shows etc.

• Supports to hold sunscreen protection for vulnerable amphibians.

• Watersheds to keep rain water from percolating through contaminated soils intogroundwater, perhaps temporarily during in-situ remediation.

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4 RELATED TECHNOLOGIES 21

• Frames for hanging plants or other objects to dry.

• Pergola, trellis, or topiary framework.

• Micro-meteorite protection, sun-shielding for Martian colonies.

4 Related Technologies

At a gross level, related non-tensegrity technologies (RNTTs) for economically enclosinglarge areas can be roughly sorted into two categories:

1. The first category contains space frame technologies where the members are veryhomogeneous. They are typically realized as planar trusses perhaps connected at anangle with other planar trusses. Biosphere 2[Kelly92, p. 90] is an example. Theirfaceted shape means they contain less space per unit of material than a sphericalstructure built with the same amount of material. Makowski’s book[Makowski65]contains a variety of examples.

2. The second category contains those technologies which are typified by geodesicdomes[Fuller73, pp. 182-230] and Kiewitt domes.[Makowski65] Geodesic domes sharemany qualities of tensegrity domes. The primary difference of these from tensegritytechnologies is the requirement of these technologies that all components be able tosustain both tensile and compressive forces.

Related tensegrity technologies (RTTs) can be divided into five categories. They areordered with the technologies which have yielded more practical structures toward thebeginning.

1. Technologies typified by the circus tent represent a sort of tensegrity approach. Herea tensile network (the tent fabric) is supported at various locations by large poles.Anchors and supporting cables usually play a role. The compressive elements are fewand massive. This technique has resulted in a variety of practical structures. Forexamples, see Otto73 and Otto82.

2. Pneumatic technologies, where a large fabric structure is supported by internalpressurization, are also tensegrity technologies. In this case the internallycompressed air molecules serve as the compression members, and the structure’s skinserves as the tension members. This technique has also resulted in a variety ofpractical structures. For examples, see Otto82.

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5 COMPARISON WITH RELATED TECHNOLOGIES 22

3. The technologies embodied by the “Cabledome” of Geiger and the “spatiallytriangulated” approach of Levy have been utilized by architects to develop practicalroof coverings for functional buildings, Campbell94 reviews these technologies indetail. These technologies are basically a technique for designing large-scalerim-supported circular roofs. Cables at the perimeter suspend a smaller circle ofvertical posts which in turn provides a perimeter for supporting a yet smaller circleof vertical posts and so on until the center of the roof, the smallest circle of theseries, is reached.

4. Oren Vilnay[Vilnay90] has developed a theoretical technology for tensegrity designwhose symmetries are similar to those of #3. In his single-layer technology, thestruts extend between perimeters instead of being vertically suspended within aperimeter as in the previous technology.

5. Emmerich, Hanaor and Motro[Emmerich88, Hanaor87, Motro87, Hanaor92,Motro92] have developed a method for fabricating trusses from tensegrity prisms.This technology is similar to the one proposed here, but differs in details which maybe structurally significant. For the most part, the resulting network for the outerand inner layers of these structures is identical with the dome proposed here.However, the way the struts and tendons are connected between the layers isdifferent. Research has concentrated on the examination of planar trusses and hasn’tas yet dealt concretely with the design of spheres or domes.

Hanaor has also done significant work developing an algorithm for computingmember lengths in tensegrity designs. That procedure is based on the methodologyin Argyris72. Pellegrino86 has an alternative methodology for computing memberprestress forces.

5 Comparison with Related Technologies

The the tensegrity approach has the following advantages over the non-tensegrityapproaches:

1. The separation of members into purely tensioned components and purely compressedones means the purely tensioned members can be as light weight as current materialtechnology allows.

2. Only tendons are connected to struts. This simplifies joints considerably.[Hanaor87,p. 41] [There are tensegrity designs where struts are flexibly interconnected to otherstruts. See Pugh76, p. 34. Their joints are correspondingly complex.]

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5 COMPARISON WITH RELATED TECHNOLOGIES 23

3. The predominance of light-weight tendons in the designs allows the structure to beless visually intrusive, lighter and probably more economical.

The relationship of the tensegrity approach proposed here over related tensegritytechnologies can be summarized as follows:

1. It is hypothesized that the well triangulated design of the technology described inthis prospectus will yield a stiff structure with superior performance characteristics.As noted in Section 2.1, lack of stiffness and lack of resistance to concentrated loadsis one of the notable problems with tensegrity structures in general. Acomprehensive comparison of the proposed technology with the other technologies,particularly RTT #5, needs to be carried further. Preliminary tests have shown theproposed technology resists concentrated loads much more effectively as frequencyincreases and will thus be suitable for commercial-scale applications where RTT #5would not. The resistance to concentrated loads certainly differentiates thistechnology from RTT #1 and #2.

2. Since the topology of the structures proposed here is based on truncated spheres,entire structures with roofs and side walls can be fabricated. This overcomes a majorshortcoming RTT #3 which basically can only provide a structure for a roof andrequires an existing structure based on another technology to support the roof. Thisshortcoming makes RTT #3 unsuitable for many of the applications discussed inSection 3. It is also an advantage over RTT #5 as currently proposed althoughTensegrity Solutions’ dome fabrication technology can easily be extended to giveRTT #5 the same capability.

3. Preliminary investigations have shown that having 12 tendons securing each strut,and having the structure supported by a large number of independent struts meansthe failure of one of them has very little impact on the structure. None of the otherRTTs exhibit this degree of redundancy. All of them have members whose failurewould be expected to have a major impact on the shape of the structure they areembedded in.

4. The technology proposed here allows for the dome to have a planar, circularfoundation with evenly spaced footings. This could simplify construction in manysituations and give an aesthetically pleasing appearance. This is an advantage overRTT #5.

5. The fact that the structure requires foundation footings only where the base strutsmeet the ground means a minimal foundation is required. This is an advantage overRTT #3 which is basically only a technology for a flat roof and requires asubstantial additional non-tensegrity perimeter structure to support it.

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6 FUTURE DIRECTIONS 24

6. The assembly process for the structure is such that the higher-tension tendons arefairly slack when installed and are brought to full tension when high-leveragetendons of lower tension are installed. This eases the construction process. Thetripods are fabricated individually first and form a natural and compact “atom” forthis construction process. This is an advantage over RTT #3 and #4.

7. As stated in Section 2.2, Tensegrity Solutions’ tensegrity design software allowsstructures to be designed where all the struts have identical lengths and can therebyall have identical design. This substantially eases fabrication for these substantialcomponents. The tendons have a variety of lengths. However this is not a problemsince there is not a great loss of mass production advantage in having to cut cable(or whatever linear material is being used to fabricate the tendons) to variouslengths. This is an advantage over RTT #4. It is anticipated that futureinvestigations will show that this identical strut length feature creates nodisadvantage as far as exogenous load handling, more particularly that the structureshares the usual characteristic of an effective tensegrity design that loads aretransmitted away from concentrations.

6 Future Directions

6.1 Applications

Tensegrity Solutions’ most fundamental need is for the development or location of aconcrete design problem where the proposed technology can be applied to enoughadvantage that it would be adopted.

6.2 Computation

Another focus of future research will be to utilize the software tools developed byTensegrity Solutions to analyze the load-bearing capabilities of domes designed usingTensegrity Solutions’ and alternative technologies. Of particular interest is how thesecapabilities change with the frequency of a dome. For the technology to be useful, it mustexhibit desirable characteristics at very large scales.

To carry out this research, Tensegrity Solutions’ computing facilities need to be upgraded,and the software needs to be enhanced as well. A machine able to accommodateC++ software, supply abundant random-access memory and advanced floating-pointcapability is needed. Current plans are to acquire hardware based on the Alpha chip withLinux and/or Windows NT-based software.

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7 TENSEGRITY SOLUTIONS: BACKGROUND INFORMATION 25

6.3 Design

Another critical need is for outside consultants to assist in the details of the design oftensegrity solutions and comment on the analysis techniques used by Tensegrity Solutions.Tensegrity Solutions’ primary expertise is theoretical. Outside consultants are needed toidentify appropriate ways of realizing the foundation, joinery and surfacing of a solution. Inaddition, they can suggest ways in which Tensegrity Solutions’ procedures can be betterbrought into line with standard architectural and structural engineering practices.

7 Tensegrity Solutions: Background Information

Tensegrity Solutions is a spin off of Software Services which provides computer-relatedinstruction and application software design and coding services and has provided computersupport for the research activities outlined in this prospectus. These activities in turn haveprovided Software Services with a laboratory for testing various approaches to softwaredesign and as a test bed for various optimization products.

Robert Burkhardt is the principal of Tensegrity Solutions. He received a B.A. in economicsfrom the University of California at Irvine in 1975 and attended graduate school ineconomics for five years at the Massachusetts Institute of Technology. His fields ofconcentration at MIT were monetary economics and econometrics. The background inoptimization and mathematical programming received at MIT is responsible for much ofhis approach to tensegrity design.

He has taught computer science courses at Boston University and the Lowell InstituteSchool (Computer Graphics, C Programming, C++ Programming, X WindowProgramming, BASIC). He has fulfilled contracts for software design and implementationfor Computervision Corporation (interactive graphics metafile display) and Carvajal, S.A.(software to create, update, maintain and process data for a Spanish-language dictionary).He worked as a software engineer for Gamesville.com for three years.

He has been doing research in the area of tensegrity design since 1981 and on double-layerstructures since 1985. Many of the results of this research are summarized in Section 2.2.He has gathered together the results of much of this research into a book.[Burkhardt04]

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8 REFERENCES 26

8 References

Argyris72 Argyris, J. H. and D. W. Scharpf, “Large Deflection Analysis ofPrestressed Networks”, Journal of the Structural Division, AmericanSociety of Civil Engineers, Vol. 98 (ST3), pp. 633-654 (1972).

Burkhardt04 Burkhardt, R. W., A Practical Guide to Tensegrity Design, Cambridge,Massachusetts: Tensegrity Solutions, 2004.

Campbell94 Campbell, David M. et al., “Effects of Spatial Triangulation on theBehavior of ‘Tensegrity’ Domes,” in John F. Abel et al., ed., Spatial,Lattice and Tension Structures: Proceedings of the IASS-ASCEInternational Symposium 1994, American Society of Civil Engineers,1994.

Davis70 Davis, LeRoy and Samuel Bradstreet, Metal and Ceramic MatrixComposites, Boston: Cahners Publ. Co., Inc., c1970.

Emmerich88 Emmerich, David Georges, Structures Tendues et Autotendantes, Paris,France: Ecole d’Architecture de Paris la Villette, 1988.

Fuller79 Fuller, R. B., Synergetics2: Further Explorations in the Geometry ofThinking, New York: MacMillan Publishing Co., Inc., 1979.

Fuller75 Fuller, R. B., Synergetics: Explorations in the Geometry of Thinking,New York: MacMillan Publishing Co., Inc., 1975.

Fuller73 Fuller, R. B. and R. Marks, The Dymaxion World of BuckminsterFuller, Garden City, New York: Anchor Books, 1973.

Hanaor92 Hanaor, A., “Aspects of Design of Double-Layer Tensegrity Domes,”International Journal of Space Structures, Vol. 7, pp. 101-113 (1992).

Hanaor87 Hanaor, A., “Preliminary Investigation of Double-Layer Tensegrities,”in H.V. Topping, ed., Proceedings of International Conference on theDesign and Construction of Non-conventional Structures (Vol. 2),Edinburgh, Scotland: Civil-Comp Press, 1987.

Jang94 Jang, Bor Z., Advanced polymer composites, Materials Park, Ohio:ASM International, 1994.

Kelly92 Kelly, K., “Biosphere 2 at One,” Whole Earth Review, Winter 1992,pp. 90-105.

Kenner76 Kenner, H., Geodesic Math and How to Use It, Berkeley, California:University of California Press, 1976.

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8 REFERENCES 27

Luenberger73 Luenberger, D. G., Introduction to Linear and Nonlinear Programming,Reading, Massachusetts: Addison-Wesley Publishing Co., 1973.

Makowski65 Makowski, Z.S., Steel Space Structures, London: Michael Joseph Ltd,1965.

Motro94 Motro, Rene et al., “Form Finding Numerical Methods for TensegritySystems”, in John F. Abel et al., ed., Spatial, Lattice and TensionStructures: Proceedings of the IASS-ASCE International Symposium1994, American Society of Civil Engineers, 1994.

Motro92 Motro, R., “Tensegrity Systems: The State of the Art,” InternationalJournal of Space Structures, Vol. 7, pp. 75-84 (1992).

Motro87 Motro, R., “Tensegrity Systems for Double-Layer Space Structures,” inH.V. Topping, ed., Proceedings of International Conference on theDesign and Construction of Non-conventional Structures (Vol. 2),Edinburgh, Scotland: Civil-Comp Press, 1987.

Otto73 Otto, Frei, ed., Tensile structures; design, structure, and calculation ofbuildings of cables, nets, and membranes, Cambridge, Massachusetts:MIT Press, 1973.

Otto82 Otto, Frei, Naturliche Konstruktionen, Stuttgart: DeutscheVerlags-Anstalt, 1982.

Pellegrino86 Pellegrino, S. and C. R. Calladine, “Matrix Analysis of Statically andKinematically Indeterminate Frameworks”, International Journal ofSolids and Structures, Vol. 22, No. 4, pp. 409-428 (1986).

Pugh76 Pugh, A., An Introduction to Tensegrity, Berkeley, California:University of California Press, 1976.

Saka97 Saka, T. and Y. Taniguchi, “Damage to Spatial Structures by the 1995Hyoguken-Nanbu Earthquake in Japan,” International Journal of SpaceStructures, Vol. 12, Nos. 3&4, pp. 125-147 (1997).

Vilnay90 Vilnay, Oren, Cable Nets and Tensegric Shells: Analysis and DesignApplications, New York: Ellis Horwood, 1990.

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9 PANEL COMMENTS ON PROPOSAL TO NSF 28

9 Panel Comments on Proposal to NSF

The following comments were received when an initial version of this prospectus was sentin response to a solicitation by the National Science Foundation’s Small BusinessInnovation Research Program.

9.1 Panel Summary

Proposal Number: 9660694Principal Investigator: BurkhardtCompany Name: Safeware Service (sic)

Panel # and Name: 23-5, Civil and Mechanical Systems - StructuresPanel Date: September 23-24, 1996

SUMMARY

Further research on the static’s and dynamics of “tensegrity” systems is needed becausesuch systems are being widely used for longer spans.

The P. I. does not appear to have the expertise to perform an engineering evaluation of“tensegrities”. However he has thought about their geometry and may have developedinnovative forms. He is encouraged to team-up with a structural engineering firm (thereare many notable ones in the Boston/Cambridge area) and resubmit an SBIR proposal.

9.2 Comments of First Respondent

1. The scientific/engineering/educational significance of the proposed research.

The proposal relates to the design space frameworks constructed from axially loadedmembers. The so called tensegrity structures are one type. The research would improvea design scheme assembled by the PI.

Rating=Fair

2. The soundness of the research plan to establish the probable technical and commercialfeasibility.

Based on the work of others, software has been written to design/analyze tensegrityspace frames. The plan is to upgrade this software and validate it. Small dowel andfishing line models would be tested.

Rating=Fair

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9 PANEL COMMENTS ON PROPOSAL TO NSF 29

3. The uniqueness/ingenuity of the proposed concept or application as technologicalinnovation.

The uniqueness here has to do with some aspects of the software developed by the PI.

Rating=Fair

4. The potential of the proposed concept for significant commercial applications.

Marketing the software could be a commercial application. Further, the likelihood ofdesigners considering the use of tensegrity space frames would be enhanced if the PIpublished his work.

Rating=Good

5. The educational and professional experience of the Principal Investigator, other keystaff, and consultants in relation to the proposed research; the time commitment of thePrincipal Investigator (NSF requires a minimum of one month); and the availability ofinstrumentation and facilities.

The PI has programming background but no engineering training. One would tend toquestion his ability to make informed structural design judgements. Software Servicesis a one person firm so design assistance is unavailable. Small dowel and fishing linemodels would be tested.

Rating=Fair

Total Score 11 (Max 25)OVERALL RATING: Fair

9.3 Comments of Second Respondent

1. The scientific/engineering/educational significance of the proposed research.

I concur that “tensegrity” systems have the potential of making longer spans feasible.

Rating=Very Good

2. The soundness of the research plan to establish the probable technical and commercialfeasibility.

Tasks 4, 4A and 5 are not well defined.

Rating=Fair

3. The uniqueness/ingenuity of the proposed concept or application as technologicalinnovation.

The proposed “double layer dome system” appears innovative, different from theGeiger/Levy systems.

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9 PANEL COMMENTS ON PROPOSAL TO NSF 30

Rating=Very Good

4. The potential of the proposed concept for significant commercial applications.

A thorough study of the PI’s system in collaboration with a structural engineering firmwould clarify its potential.

Rating=Good

5. The educational and professional experience of the Principal Investigator, other keystaff, and consultants in relation to the proposed research; the time commitment of thePrincipal Investigator (NSF requires a minimum of one month); and the availability ofinstrumentation and facilities.

The P.I. should team up with a structural engineering firm. It’s unclear whether theP.I. understands prestressing and the different types of trusses that are denoted as“tension”.

Rating=Good

Total Score 16 (Max 25)OVERALL RATING: Good

9.4 Comments of Third Respondent

1. The scientific/engineering/educational significance of the proposed research.

The objective of the proposal is to theoretically and experimentally evaluate a newspace frame technology.

Rating=Fair

2. The soundness of the research plan to establish the probable technical and commercialfeasibility.

The research approach is poor and practically non-existent.

Rating=Poor

3. The uniqueness/ingenuity of the proposed concept or application as technologicalinnovation.

The research proposal looks at computer modelling, with very poor technical andstructural approach.

Rating=Poor

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9 PANEL COMMENTS ON PROPOSAL TO NSF 31

4. The potential of the proposed concept for significant commercial applications.

The concept has no potential for commercial applications.

Rating=Fair

5. The educational and professional experience of the Principal Investigator, other keystaff, and consultants in relation to the proposed research; the time commitment of thePrincipal Investigator (NSF requires a minimum of one month); and the availability ofinstrumentation and facilities.

The PI has expertise in economics and computer programming. No background instructures.

Rating=Fair

Total Score 8 (Max 25)OVERALL RATING: Poor

Additional analysis of respondent information provided by Tensegrity Solutions:AVERAGE OF THREE RESPONDENT TOTAL SCORES: 11.3 (Max 25)OVERALL RATING CORRESPONDING TO THIS AVERAGE: Fair