Tensegrity Structures and their Application to ArchitectureValentín Gómez Jáuregui Tensegrity Structures and their Application to Architecture “Tandis que les physiciens en sont déjà aux espaces de plusieurs millions de dimensions, l’architecture en est à une figure topologiquement planaire et de plus, éminemment instable –le cube.” D. G. Emmerich “All structures, properly understood, from the solar system to the atom, are tensegrity structures. Universe is omnitensional integrity.” R.B. Fuller “I want to build a universe” K. Snelson . Belfast. Date of submission: September 2004. Queen’s University.School of Architecture Queen’s University Belfast Tensegrity Structures and their Application to Architecture Valentín Gómez Jáuregui Submitted to the School of Architecture. in partial fulfilment of the requirements for the MSc in Architecture. . I would like to express my gratitude to Manolo García. El Lichis.” I really do not know what would be his opinion if English was not his mother tongue. I could not forget John Williams. i . for their words and continuous support. Danielle Dickinson and Elspeth Wales from Buro Happold and Santiago Guerra from Arenas y Asociados. which is my case. who has been with me in every moment for so long. from Sidell Gibson Partnership Architects. therefore. In writing this dissertation. James Joyce wrote in a letter: “Writing in English is the most ingenious torture ever devised for sins committed in previous lives. Avelino Samartín and Jörg Schlaich.Tensegrity Structures and their Application to Architecture I. When I was in trouble or when I needed something that I could not achieve by my own. When writing the chapters on the applications and I was looking for detailed information related to real constructions. I have crossed through diverse difficulties. Acknowledgements In the month of September of 1918.to some professionals as great and important as Javier Manterola. Acknowledgments I. of architecture. and this is the moment to say ‘thank you’ to all of them. I was helped by Nick Jay. I have been helped and encouraged by several people. who are contributing with remarkable works to the evolution and progress of engineering and. Indeed. for me it has been an honour to be helped by them during the development of this dissertation. Joaquín Sabina y Miguel Ángel Hernando. more or less 86 years ago. Everyone in this list is indebted –as I am. and the idiomatic problem was just one more. I also gratefully thank the following for their help in answering the questionnaires. showing me some sources that I would never have been able to achieve. Javier Torres Ruiz in particular. in particular. Williams I am.Tensegrity Structures and their Application to Architecture I. who opened the gates of tensegrity to me when I knocked on the door of his office. Raymond Gilfillan. for checking the manuscript patiently. It is necessary to mention Prof. The simplest way to thank them is to produce the impressive list below: Enrique Aldecoa Sánchez del Río Roslyn Armstrong Sergio Cabo Bolado Andrew Cowser Ghislain A. and to Dr. K. Université de Liège and Queen’s University Belfast. the person responsible for the design and construction of the Tower of ii . who were ready to assist me at any time. Acknowledgments I wish to thank the dedication of the professors who taught me during my primary schooling and later during my specialised education as an engineer in the Universidad de Cantabria. their kind intentions are acknowledged. from Schlaich Bergermann und Partner. grateful to three more people who have had a major influence with their fruitful contributions: Mike Schlaich. which was never closed. I cannot forget the role of Roslyn Armstong and Ailbhe Hickey. Fonder Carolina García-Zaragoza Villa Jose Antonio Gómez Barquín Santiago Guerra Soto Celso Iglesias Pérez Daniel Jáuregui Gómez Juan José Laso de la Riva Mª Elena Linares Macho Javier Manterola Armisén Almudena Monge Peñuelas Arturo Ruiz de Villa Valdés Avelino Samartín Quiroga Jörg Schlaich Mike Schlaich Javier Torres Ruiz Su Taylor Chris J. and for guiding me during all these months. for their many useful suggestions. who has been collaborating with the computation of some of the new proposals exposed in this work and has generously shared everything he knows about tensegrity. who worked with the latter and so kindly explained to me everything about the calculations and design of this tower. and. “La murria aprieta con juerza”. Last but not least. For everything. Arturo Ruiz de Villa. I finish with another cite from Araby: “My body was like a harp and her words and gestures were like fingers running upon the wires. as is said in my birthplace. fue por tu amor. He has personally answered some of my questions and doubts about the origins of floating compression. Special thanks to Liona for helping me from her computer and sending me all the items that I required. I started this acknowledgment quoting James Joyce.” What are tensegrity structures. And. many thanks to my mother. has helped me with this dissertation. she has been my left hand and. at the point where my own resources failed me. especially. I had the support of my family and friends. the discoverer of the tensegrity structures. thinking of her. si algún día después de amar amé. some fingers of the right one. During all my life. Lucía”. who facilitated me with precious information about it. She has been for almost three years a very important part of my life. and Bob Burkhardt.Tensegrity Structures and their Application to Architecture I. Consequently. even giving me access to some of his articles still in press. Acknowledgments Rostock. but beautiful harps in space? iii . I must sing with Serrat: “Si alguna vez amé. replying to all of my endless e-mails and checking some of the chapters concerning his experience. probably. I must mention Julie. and for the last months. I received invaluable help from Kenneth Snelson. . especially in the so-called biotensegrity. an in-depth research has been carried out. iv . and to find out what the derivations of these phenomena are. as well as their advantages and weakness when applying them to Architecture. With this premise. and to define the fundamental forces at play. Abstract II. the steps that followed the progress of the studies and the evolution until the present day. in spite of its particular flexibility and relatively high deflections. it is essential to understand the structural principles of floating compression or tensegrity. Abstract Tensegrity is a relatively new principle (50 years old) based on the use of isolated components in compression inside a net of continuous tension. this work makes an effort to gather more information from various fields.Tensegrity Structures and their Application to Architecture II. The main aim of this work is to prove that it is possible to find some applications for such an atypical kind of structure. Some references about precedent works that have been important for the development of tensegrity structures have also been mentioned. in such a way that the compressed members (usually bars or struts) do not touch each other and the prestressed tensioned members (usually cables or tendons) delineate the system spatially. other than Architecture. therefore. trying to make the controversial origins clearer. Once this point is established. as well as the polemic about the fatherhood of the discovery. the continuous tension-discontinuous compression has also been shown to be a basic principle of nature. Moreover. the characteristics of these structures are described. In order to achieve the intended purpose. rather than detailed drawings proposed for a real project. as an illustration of the possibilities and potentials of tensegrity structures. some proposals designed by the author are shown. This is because the dissertation also has the aim of serving as a guide to future investigators who could find useful references along with the sources cited.Tensegrity Structures and their Application to Architecture II. an intense research on patented works tried to find out more feasible possibilities already invented. Abstract Many experts have been working for the past decades on the subject. Finally. Precedent and current works founded on tensegrity are presented in this thesis. When looking at the bibliography. v . distinguishing between false and true tensegrities. it might be noted that this research has been based on a large number of previous publications. the definition is crucial to accept or refuse the legitimacy of using the term. Besides. .......4....3......... 36 4......................... Why a dissertation about tensegrity structures? ................................i II.............................. 19 3...................................................2...1..... 2 1........... Background and History 6 2............... What is Tensegrity?...........4..............................................................3 Tensegrity in Inorganic Chemistry 3............................... Acknowledgements ................................4.....4....................1...............3........ 8 2........... List of illustrations ...................3..3.............. Table of Contents III................... General Characteristics....................................................... Definitions and Basic Principles 36 4............. Abstract.....1............4.......................................... 21 3........................................2............ Tensegrity as a universal principle ...2...............................................2 Tensegrity in Biology 3...................................... Main Concepts 4................................ Table of contents .. 40 4.............4..................... Introduction 1 1.............. Equilibrium Analysis vi ............... What are the objectives of this work? ......... 15 3..................................4................................. 36 4.............4........... The Skylon 3................................................ The evolution...................Tensegrity Structures and their Application to Architecture III..... Definitions .......................................3................. Basic Principles ...................... 3 2..... Table of Contents I......2................................1..........1.. Divergences ................................ Introduction .............................. 10 2............... The Creation of the Simplest Configurations 4.................ix 1............. 6 2............... Cable-Domes 3...........1. Introduction .............4..................................... The origins.... The controversy ... Materials and tension. 29 3.iv III...............................................................3...4 Tensegrity in Anatomy 4.............3...................2......... Some analogies 4.....3...............4................ Some precedents................... Precedents and Key Studies 19 3...2..... 1 1..................................................vi IV.......................... Tensegrity in Macrocosm and Microcosm 3..... Suspended roofs and tensile structures 3......... 43 4........................... 19 3....4................................3......................................4..1............ Personal proposals .. Furniture (tables.2.. 69 6.2.......1.......2....... Domes 73 6.... chairs.........3.1.................1........2.......... Some other applications for tensegrity towers 6..7..... 59 5..........5.... lamps.... Roof structures 87 6..2.... Submarines (skin fabric) 6........ Tents-like structures 91 6.....2.4. “Zigzag” configuration or “Type Z” 5... Arches 90 6......3.. Cylindrical systems 5............ Tensegrity pyramidal roof from Truncated 6..2... Sculptures 6.. Disadvantages 5.3...3... 71 6...........3.....7..2..4....2... Deployable structures ......1... Star systems 5....... 66 5............. Different applications besides Architecture 94 6.... 73 6. Towers 80 6.....1.............3.7.. Outer space structures 92 6..............3... Rhombic configuration 5............2.........5.......... Nomenclature ... 51 4.................Tensegrity Structures and their Application to Architecture III................ Typologies and classification.....1..............2.. Properties 4.2.......3........3..2.....2........ Lightning rod from the Helix Tower 6..... Advantages and applications for domes 6........5..2.................3.... etc) 6.2. 58 5.....................................2.... Footbridge by assembly of modules 6..........1...3.......3.. “Circuit” configuration 5....... Calculation of the load response 6...2............7..1..... Different proposals for domes 6.. Assemblies of simple structures 58 5.....3.......... Table of Contents 4.. Irregular systems 5.....1...2.6.... Other suggestions to develop vii ....1....2. Tower of Rostock 6.4..2... Features ..1.....2...2...2......3.........3... Introduction ..2.2...................2.2.. Conglomerations 5............2. Spherical systems 5......2........2..2.......... Tensegrity dome from the Truncated Icosahedron 6.................2.. Advantages 4..... Roofing for Stadiums by assembly of modules 6.2..3............5..........2..1..................1.4..7........1. Grids 5...1... Vertical Masts (horizontal beams) 5.................... Different proposals for towers 6........ Toys 6.............2..3.......4.5...5.... Assemblies..2..3.. Applications 71 6..3. Classification ...................3....3.....1.. Actual examples . 97 6...2.......1................ ................1.. Further research ............ C............................................................. 164 Extended Bibliography................................ 108 Original tensegrity patents..................Tensegrity Structures and their Application to Architecture III............... 106 8.................................. Discussion and conclusions 106 8.......................3..... Questionnaires and Interviews 102 7............ 104 8.........2....... Table of Contents 7.......................................... H............................. B........... 181 VI...... Questionnaires to professionals 7.................................... G........ 159 Plans and renders.............................. 192 viii ... 155 Questionnaire........................................... E...................................................................... D...... F......................................1............................... Questionnaires to the general public 7..................................... 117 Personal correspondence .................. I.....1................ Appendices A..............................................................1......... Bibliography ..................................... Questionnaires to specialists 7..... Questionnaires ............................................1................................... 112 Other tensegrity patents................................................................................. Interviews ................. 156 Tensegrity Models ... 102 7..... 107 V.... 138 Deflection of the expanded octahedron..................... Motro’s correspondence ............... Discussion and conclusions........1.......................................2...........................................2............................................ . in Kuanshien (China) Illustration taken from IL (1985) Fig. Illustration drawn by the author.Tensegrity Structures and their Application to Architecture IV.9 : “T-Octahedron Dome” Positions and effects of exogenous loads. Fundamental concepts.1 : “30’ Cantilever” by Snelson (1967) Illustration donated by the artist to the author.2. Illustration taken from Gengnagel (2002) Fig. Illustration taken from CISC (2003) Fig. Fig.3 : “Z3-1 mat prismatique 4B racemique” by Emmerich.4 : “Dragon” by K.2.2 : “Elementary Equilibrium” or “Simplex” Illustration drawn by the author.1. World record of cable-stayed bridges in 1983. Rambouillet (France) Illustration taken from Emmerich. Snelson.2. Illustration taken from Gengnagel (2002) Fig.2 : “Mini-Skylon in chess game” Sculpture made by the author (2000) Chapter 2 Fig.2.7 : “U.3. List of illustrations Chapter 1 Fig. Pavilion for Expo ‘67” by Fuller in 1967. Manterola.3 : a) “Suspension bridge”. Illustration drawn by the author.3.5 : “Skylon” Illustration taken from King and Lockhart (2004) ix . Fig.5 : “Monument à la forme futile” by Emmerich.2.1.S.S. Fig. b) “Cable-stayed bridge”.2.10 : Fuller’s figures. Fig.2. Illustration taken from frenchculture.2.6 : “Geodesic Tensegrity Dome” by Fuller in 1953.2 : Deformation under compression and under tension. Illustration taken from Snelson (2004) Fig. Pavilion for Expo ‘67” Illustration taken from CISC (2003) Fig. Illustration taken from Burkhardt (1994-2004) Fig.2. Fig. 1966.1 : “Structure-Sculpture” by Ioganson. Illustration taken from Búrdalo (2004) Fig.3. Illustrations taken from Fuller (1961) Chapter 3 Fig.org Fig.8 : “U. Illustration drawn by the author.1 : An-Lan Bridge. List of illustrations IV.3.2. Fundamental concepts.4 : “Barrios de Luna Bridge” by J.3. ) Illustration taken from Setzer (1992) Fig.8 : “Analogy of the wire wheel” Illustration drawn by the author Fig. The boundary has no compressed component. Fig. belongs to the boundary.10 : Kenner’s derivation of the “Simplex” Illustration drawn by the author. Fig.4.8 : “Raleigh Arena” by Nowicki.3 : “Octahedron” False tensegrity.4. Illustration taken from Kawaguchi et al. Illustration taken from Ingber (1998) Fig.4.9 : “Generation of the Simplex” by Snelson. Illustration taken from Snelson (1965) Fig.6 : “Georgia Dome” Detail of the compressed ring.4.7 : Equilibrium of a post supported by cables.11 : Author’s derivation of the “Simplex” Illustration drawn by the author. Model built by the author.5 : “Georgia Dome” spanning 233.4. Fig.4. List of illustrations Fig.12 : Set of forces acting on a strut.6 : “Skylon”.4.4. Illustration taken from Fuller (1975) Fig.5x186m (by Weidlinger Asscts.15 : Diagram showing the role of tensegrity in heart functions. assembly of three struts.3.4. it flattens itself and its nucleus when it attaches to a rigid surface (left) and retracts into a spherical shape on a flexible substrate (right).2 : Snelson with a double planar structure (1961) Illustration donated by the artist to the author.9 : “Music Pavilion” by Frei Otto (1955) Illustration taken from Atelier Warmbronn (2003) Fig.3.7 : “Analogy of the balloon” Illustration drawn by the author Fig.13 : “Tensegrity model of a cell” Like a living cell.3.3.3. Fig.12 : “Cable-dome diagram” Symbols for structural members.14 : “Springs model” Illustration from Ingber (2003) Fig.10 : “German Pavilion for Expo'67” by F. Illustration taken from Gossen et al.3. (1997) Fig.16 : “Tensegrity Thoracic Vertebrae” Illustration taken from Levin (2002) Chapter 4 Fig. Representation of the different elements.11 : Roof diagram for a Cable-Dome.3. The compressed square. Illustration taken from Buchholdt (1985) Fig. Illustration drawn by the author. Fig.3. Otto (1967) Illustration taken from Stanton (1997) Fig. Illustration drawn by the autor Fig. Model built by the author. Illustration drawn by the author x .4 : “Expanded octahedron” Pure tensegrity. (1997) Fig. Illustration taken from Setzer (1992) Fig.3. Illustration taken from Lab (1998) Fig.3.3.4.4.1 : Some Fuller’s tensegrities. Fig.Tensegrity Structures and their Application to Architecture IV.4. Illustration drawn by the author after Pugh’s drawings.6 : “8v Double-Layer Tensegrity Dome” Model built and published by Burkhardt (1999-2004) Fig. by Ali El Smaili.3 : “Cuboctahedron” Illustration drawn by the author Fig.6. by Sneslon.5.7 : “Needle Tower II” 30m height.1 : “Apex of Truncated Icosahedron” Model made by the author Fig.4 : “Tensile skin dome” Illustration from Shelter Systems (1996-2001) Fig.7 : “Bi-dimensional double-layer grid” by Laboratoire de Mécanique et Génie Civil. rotation translation and shear.4. Illustration drawn by the author Fig.13 : Set of forces acting on a cable.1 : Classification of Spherical Systems. Fig.6 : “V expander” Illustration taken from Motro and Bernard (2002) Fig. List of illustrations Fig. (Holland) Illustration taken from Snelson (2004) Chapter 5 Fig. Illustration taken from Snelson (2004) Fig.5.5.Tensegrity Structures and their Application to Architecture IV.6. Illustration taken from Snelson (1965) Fig.5 : “Snowdon Aviary” Illustration taken from Ford (2004) Fig.4.4. Arnhem.4.2 : Generation of T-Prims. Schlaich (2003) and Ruiz de Villa (personal correspondence) xi .16 : “Easy-K Installation” by Snelson in 1970. Maintaining the prestress: Rotationrotation. Fig.6.5.5. Illustration drawn by the author Fig.6.3 : Detail of Fuller’s patent.9 : “Tensegrity Tower of Rostock” Illustrations taken from M.15 : “Expanded octahedron” or “Icosahedron” Illustration drawn by the author.5. Fig. Illustration drawn by the author after Pugh’s drawings. Illustration taken from Smaili (2003) Chapter 6 Fig. Montpellier.6. Cancelling the prestress: Local suppression and global suppression.8 : Continuous tension-discontinuous compression Columns.6. Fig.6.4 : “Expanded octahedron” and “Truncated Tetrahedron” Illustration drawn by the author.6. Fig.5 : Deployed Single Layer System (n8S4C12) and Double Layers System (n16S8C32) Illustration drawn by the author.8 : Folding Mechanisms.14 : Right-handed and Left-handed Simplex (“Dextrorse” and “Sinistrorse”) Illustration drawn by the author.5.6. Illustration taken from Fuller (1954) Fig. Fig.5.2 : “Dome projected by Snelson” Illustration taken from Snelson (1965) Fig. 6. List of illustrations Fig.17 : Tent-like Marquee.18 : “Lunar applications” Fuller exhibition.6.6. Octet truss. N.13 : Ice Rink Roof in Munich.15 : “Shade Structure” Illustration taken from Daniel Ng (2001-2004) Fig.6.14 : Tensegrity Arch supporting membrane. Illustration taken from Adriaenssens and Barnes (2001) Fig.6.Tensegrity Structures and their Application to Architecture IV. Illustration taken from Burkhardt (1999-2004) and Snelson (2004) Fig.6.6. Three of his basic structures: Tensegrity mast.10 : “La Plata Stadium” Illustration from Weidlinger Associates (2002) Fig. Illustration taken from Janberg (1998-2004) Fig.6.12 : Tensegrity Arches. Illustration drawn by the author Fig.6. Geodesic dome.20 : “Sixty Strut Tensegrity Sphere” by Fuller. Illustration drawn by the author Fig. Picture taken by the author (2004) Fig. Modern Museum.6. Illustration taken from BRUW System (2002) xii .6. Illustration taken from Fuller (1961) Fig.11 : “Double-layer grid” by the Laboratoire de Mécanique et Génie Civil in Montpellier.Y. shopping centre in Edinburgh.16 : “Shade Structure” Basic structure. by Schlaich.19 : Tensegrity Furniture Illustrations taken from Koennig (2004) and Werta (2003) Fig. Tensegrity Structures and their Application to Architecture Chapter 1 Introduction . . systems and natural phenomena have been qualified as tensegrity when. the discoverer of the floating compression.1 shows a tantalizing sculpture by Kenneth Snelson. a lot of structures. What is Tensegrity? The definition of this term is essential to the consideration of some structures as real or false tensegrities. 1. why the author wants to deal with this subject and how the research is organised.1. actually. Several definitions have been established by different experts. 1. in such a way that the compressed members (usually bars or struts) do not touch each other and the prestressed tensioned members (usually cables or tendons) delineate the system spatially. 1 . Introduction Before discussing in any detail the contents of this dissertation. This point is further explained in chapters 2. the fig. it would be desirable to explain what the topic is about. The bars floating in the air. As an illustration. In any case. making an attempt to explain it as simply as possible. without any contact with a ‘solid’ support are truly very impressive. they were not.Tensegrity Structures and their Application to Architecture Chapter 1. really like to contemplate such a ‘magic’ phenomenon that they do not understand. the best way to understand how a tensegrity system works is to have a look at a model or. to build one. People. as he called it. During the last two decades. 3 and 4. The author. what the purpose is. Introduction Chapter 1. even better. in general. suggests that tensegrity is a structural principle based on the use of isolated components in compression inside a net of continuous tension. de Ing.T. The atypical and fascinating way it worked motivated the author to discover something more about this structure and about tensile structures in general.S.Tensegrity Structures and their Application to Architecture Chapter 1. the first exercise in the course of Advanced Calculation of Structures (E. 1. The Skylon was a sort of sculpture. Introduction 30’ Cantilever Snelson (1967) Mini-Skylon in chess game Sculpture made by the author (2000) Illustration donated by the artist to the author. responded in a smart manner: “Food for thought” 1.2). 2 . In October of 2000. 1. in 1951. London. de Caminos de Santander) was a reflection about the equilibrium of the Skylon (cf. 1 Personal correspondence: excerpt from a letter to the author. a symbol erected for the Festival of Britain's South Bank Exhibition. Professor Javier Torres Ruiz. At this point. it might be interesting to establish how and when the author’s own interest in tensegrity structures started. 8 Jul 2004. 3. Why a dissertation about tensegrity structures? The engineer from Stuttgart Jörg Schlaich. when asked about tensegrity structures. The co-ordinator of the course.5). fig. In fact.2. he started building some models of a mini-Skylon made with the two knights of a chess game (fig. What are the objectives of this work? When reading J. possibilities to develop. Whatever the case may be. This reminded him a passage of Seamus Deane’s “Reading in the dark”: “I’d switch off the light. Stanley Black’s dissertation (1972). and the School of Architecture has permitted him to choose the topic as the central point of this dissertation. the preliminary objectives were adequately defined and very close to the overall aim of this investigation: to find out if it is possible to use tensegrity structures in Architecture and..” The feelings were more or less the same. books to read. the author found some incomplete facts about 3 . the book open.3. a personal exploration of these systems has allowed the writer to better understand their behaviour..Tensegrity Structures and their Application to Architecture Chapter 1.4). 1. if that answer is affirmative. the uncertainty about the final product was present every night in his mind.. Hence. the novel opening to endless possibilities in the dark. due to the large amount of resources already read. Since that moment. chapter 6 and Appendix H) Despite the fact that it is an ambitious purpose. methods to choose. the various ways the plot might unravel. to try and understand the best way to do it and suggest proposals (cf. the author felt very empathetic with one of his expressions about his own work: “one is ‘groping in the dark’ with little idea of the final result” (p. re-imaging all I had read. some other objectives have been sought. During the research. but also explained to him something else about other similar structures as interesting as the Skylon: tensegrity structures. Introduction not only showed him the sources where to find more information. specialists and publications. the sculptor Kenneth Snelson (cf. chapter 3 & 6). As a result the collateral intention of this investigation is: To study the origins of tensegrity. chapter 4). enquiring personally to its discoverer. not only related to Architecture but also to other dissimilar fields. To investigate the use of structures similar to tensegrity in previous studies. Appendix B) and shed light on some polemic aspects about the authorship. chapter 7) 4 . To revise the history and progress of this kind of structure. tracing a line of the time and pointing out the most relevant authors. chapter 5). To define the structural characteristics and fundamental concepts of the socalled continuous tension-discontinuous compression. required clarification. works or patents (Appendix C) and compare them to some of the suggested proposals in order to attest the feasibility of their potential (cf. in his opinion. To estimate how widespread the knowledge about tensegrity structures actually is among architects and engineers by means of interviews and questionnaires (cf. chapter 2). chapter 4) and to set up a general classification for these systems (cf. chapter 2 & 3). which could serve as a guide for further investigators (cf. the present and the future of tensegrity that. describing its properties. To establish a clear and generally accepted definition of tensegrity (cf. Introduction the past.Tensegrity Structures and their Application to Architecture Chapter 1. original patents included (cf. highlighting the advantages and indicating as well its weak spots (cf. there are several appendices containing relevant information. the absence of appropriate infrastructures. Appendix G) and. 5 . Unfortunately. Introduction To achieve a wider professional awareness and encourage consideration of tensegrity structures in Architecture and Engineering.Tensegrity Structures and their Application to Architecture Chapter 1. It is worthwhile highlighting that at the very beginning some experimental studies and load testing of models were programmed. In addition. the author worked with models in depth (cf. an attempt was made to compute the final geometry in more detail. but which could be peripheral and could disturb the main theme of the study. once the design was established. budget and time suggested abandoning the idea. as a feasible application for modern works. Instead. Some excerpts of the author’s personal correspondence (cf. Appendix D) and some other unpublished works are also included. . Tensegrity Structures and their Application to Architecture Chapter 2 Background and History . . 1 Illustration taken from Gengnagel (2002) As a precaution. était manipulable à l'aide d'un huitième tirant detendu. The origins. Background and History Chapter 2. so knowledge of its mechanism and physical principles is not very widespread among architects and engineers. Three men have been considered the inventors of tensegrity: Richard Buckminster Fuller. 6 . Snelson-16 Feb 1965.1. As it will be explained in chapter 7. controversy and polemic will always be present when arguing about its discovery.Tensegrity Structures and their Application to Architecture Chapter 2. 2. fig. assemblée de trois barres et de sept tirants.1). one of the most curious and peculiar aspects of tensegrity is its origin. David Georges Emmerich and Kenneth D. it is not a commonly known type of structure. Although all of the three have claimed to be the first inventor. R. "Cette curieuse structure. (See Appendix B). Emmerich-28 Sep 1964. Background and History Tensegrity is a developing and relatively new system (barely more than 50 years old) which creates amazing. However. see Appendix D. was created by a certain Karl Ioganson 2 in 1920 (cf. In order to obtain a further explanation of this sculpture. As Emmerich (1988) explains: Structure-Sculpture Ioganson. these names have been mentioned in chronological order of their granted patents: Fuller-13 Nov 1962. Motro (1987. Snelson1. lightweight and adaptable figures. called "Gleichgewichtkonstruktion". 2. giving the impression of a cluster of struts floating in the air. where Snelson gives his personal opinion. 2 It must be contrasted that in Motro (1988) the same author called him Johansen. 2003) mentions that Emmerich (1988) reported that the first proto-tensegrity system. The most controversial point has been the personal dispute. lasting more than thirty years. seven cords and an eighth cable without tension serving to change the configuration of the system. D. #2 and #3 of Appendix A). in addition to being a charismatic and a nonconforming architect. 2. with three struts and nine cables (cf. 7 . during the summer of 1948. Motro (see Appendix A). he achieved a new kind of sculpture. As the artist explains. creating different sculptures (see photos #1. and after that summer. does not allow Ioganson's “sculpturestructure” to be considered the first of this kind of structures. the "Elementary Equilibrium". 1895-1983) and K. the absence of prestress. Fuller (Massachusetts. Snelson (Oregon. B. As the latter explains in a letter to R. Background and History l'ensemble étant déformable. USA). All the same.Tensegrity Structures and their Application to Architecture Chapter 2. he started to study some three-dimensional models. cosmologist. 1927). which is one of the characteristics of tensegrity systems. Fuller was a new professor in the Black Mountain College (North Carolina.2). Cette configuration labile est très proche de la protoforme autotendante à trois barres et neuf tirants de notre invention. mathematician. influenced by what he had learnt from Fuller and other professors. engineer. He adds that this configuration was very similar to the proto-system invented by him. Snelson was an art student who attended his lectures on geometric models. fig. which can be considered the first tensegrity structure ever designed. asking for his Elementary Equilibrium Simplex Illustration drawn by the author. but maintaining its equilibrium. poet and inventor (registering 25 patents during his life)." This means it was a structure consisting of three bars. between R. When he showed it to Fuller. org 2. Background and History opinion. but independently. 3 ” At the same time.2. Z3-1 mat prismatique 4B racemique Illustration taken from frenchculture. probably inspired by Ioganson's structure. which were exactly the same kind of structures that were being studied by Fuller and Snelson (Vesna. “Creating this strange name was his strategy for appropriating the idea as 3 In contrast to other authors. before meeting Kenneth Snelson. the professor realized that it was the answer to a question that he had been looking for. Actually. In Fuller’s (1961) words: “For twenty-one years. he defined and patented his "reseaux autotendants" (see Appendix B). naming and development of both the multi-dimensional vectorial geometry and the three dimensional Tensegrity. he always wrote “Tensegrity” starting with a capital T. I had been ransacking the Tensegrity concepts.Tensegrity Structures and their Application to Architecture Chapter 2. As a result. so by calling these structures with the denomination he chose. for so many years. I had been unable to integrate them. after some time he started to consider it as “my Tensegrity”. started to study different kinds of structures as tensile prisms and more complex tensegrity systems. five and six axes symmetrical Tensegrity. he coined this term in 1955 as a contraction of “Tensional-Integrity”. 2000). tensile and selfstressed structures (see fig 2. David Georges Emmerich (Debrecen-Hungary. thus to discover multi-dimensional four. (…) Despite my discovery. which he called "structures tendues et autotendants".3). 8 . he let people think that it was his invention. and serving as an illustration of how important it was considered. 1925-1996). The controversy Even though at the beginning Fuller mentioned Snelson as the author of the discovery. when in 1980 Fuller wrote a 28-page letter to Snelson. and not the inventor. 1967. According to Vesna (2000). Schneider. Paradoxically. In fact.” However. but some years later he changed his mind. This situation pushed Snelson to insist on acknowledgement during an exposition of Fuller’s work in 1959. “Synergetics” and failed to do so again in his correspondence with Burkhardt (see both references in Bibliography). Therefore his contribution to tensegrity was credit and recognized publicly. Obviously. (See Appendix D) 9 . he had patented those universal laws in 1962. asking his student to remain anonymous for some time. and must be officially mentioned in my formal recital of my "Tensegrity" discovering thoughts.Tensegrity Structures and their Application to Architecture Chapter 2. Fuller (1961) referred to Snelson again: “(…) an extraordinary intuitive assist at an important moment in my exploration of the thus discovered discontinuous-compression. A couple of years later. 2004). at the Museum of Modern Art (MOMA) in New York. 4 Expression suggested by Snelson instead of battle of egos. continuous-tension structures was given me by a colleague. he never mentioned Snelson in one of his most important and renowned books about tensegrity. The accuracy in reporting 4 by both men continued furthermore. he always thought that if he had not catalyzed Snelson’s discovery. his art student was certainly confused. Background and History his own”. quotes Snelson in various publications (Coplans. in those letters they tried to clarify the authorship of the discovery. at the end of 1949 Fuller wrote to Snelson saying that his name would be noted in history (see Appendix A). 1977. Tensegrity would have never been invented as a new structure. Kenneth Snelson. because Fuller affirmed that inventors can’t invent the eternal principles-cosmic laws of the universe. Snelson. in answer to a Snelson’s one-page letter. The evolution. Snelson likewise acknowledges his own debt to Fuller's visionary work” (1968). the synergy (a word so often used by Fuller) created by both the student and professor. As quoted by Stephen Kurtz’s: “If Fuller acknowledges his debt to Snelson for the invention of the tensegrity principle.kennethsnelson. He avoided very deep physical and mathematical approaches. 2. Although acknowledgement is very important for the two of them. After the brief moment of acknowledgment in the MOMA. 5 See Keneth Snelson’s web page (www. he focused his work on the sculptural and aesthetic aspect. perhaps it would be better to pay more attention to the possibilities of these structures than to the past controversy. Snelson was once again keen to continue working with tensegrity as an essential part of his sculptures. resulted in the origin of tensegrity. due to his artistic background and his opinion in relation to the difficult application of tensegrity systems. which he has been creating until the present day.net) 10 . This process provided him the facility to develop very different configurations. In the author’s opinion. gathered and summarised in his web page 5. especially for Snelson (the only one still alive).Tensegrity Structures and their Application to Architecture Chapter 2. Background and History Who invented tensegrity? It is evident that the answer is not evident.3. Even though he commenced studying the fundamental concepts of tensegrity. Dragon Illustration taken from Snelson (2004) asymmetrical and non conventional. the “thirty-islanded Tensegrity sphere” (icosahedron). he is the only person capable of engineering his constructions. respectively (Fuller 1961).3 & 2. the construction of tensegrity systems requires a fine and delicate technique that he has been improving over the years. Fuller’s denominations. studying the different possible typologies of tensegrity. Fuller and Emmerich took a different approach. they looked for possible applications to architecture and engineering.4). They did it using models and empiric experiments as their main tools. Background and History applying his intuitive knowledge and achieving impressive sculptures that are spread all over the world (cf. fig. Consequently. actually. 11 . He also discovered the “six-islanded-strut icosahedron Tensegrity” (expanded octahedron)6 . creating such Tensegrity systems as the “vector equilibrium” (cubo-octahedron). 1966. On the other hand. the inventor from Massachusetts studied some simple compositions. the “six-islanded Tensegrity tetrahedron” (truncated tetrahedron) and the “three-islanded octa-Tensegrity”. The actual process whereby Snelson erects his works is a science and an art in itself. Moreover.Tensegrity Structures and their Application to Architecture Chapter 2. 6 In quotation marks. mainly spherical and one-dimensional systems: masts (cf. a hierarchy of premier Tensegrity structures was created and the comprehensive laws of universal tensegrity structuring were completed. figs.2. 2. and in contrast to Snelson.5). Just after viewing Snelson’s sculpture. this work was developed by other people. as it is stated by Fox (1981). five and six each. Monument à la forme futile Illustration taken from Emmerich. Subsequently. and produced a family of four Tensegrity masts characterised by vertical side-faces of three. four. kept on looking for new designs. figs. and. applications and methods of construction. started to explore this new structural system. he was forced to build the Montreal bubble at Expo ’67 (cf. the final application of Tensegrity was not as successful as he thought it would be.7 & 2. 12 . He made several attempts to design geodesic tensegrity domes (cf. fig.6) (although they lacked of stability due to the absence of triangulation).235). 2.989). 1975a).341. 2. Bucky (as Buckminster Fuller was also known). However.8) as a geodesic dome but without using Tensegrity principles due to time and budget reasons. Background and History Geodesic Tensegrity Dome Illustration taken from Gengnagel (2002) Thus. Henceforth. in addition. while Fuller patented his “Geodesic Domes” in 1954 (US 2. as he intended. 1967. Emmerich patented the “Stereometric Domes” in 1967 (US 3. some people who were influenced by Fuller’s work. and patented 7 some of his works connected to this subject (Fuller. he was never able to produce the Tensegrity dome which could cover a whole city. looking for any application to architecture and 7 By coincidence.Tensegrity Structures and their Application to Architecture Chapter 2.682. started to publish his studies on the subject in 1973: Topologie des structures discrètes. the Olympic Stadium of Munich by Fritz Leonhardt. Goosen et al. to have a more complete perspective of the different aspects of tensegrity in terms of publications and studies. Stanley Black (1972) wrote an unpublished study which tried to recall the main concepts known at that time and to figure out some possible systems and configurations. Incidence sur leur comportement mécanique. René Motro. Pavilion for Expo ‘67 Illustration taken from CISC (2003) engineering 8 . 1988.S. For instance.Tensegrity Structures and their Application to Architecture Chapter 2. tensile structures became more popular in the 1970s. Although it was a good attempt. 1992). probably one of the most important specialists in tensegrity at present.. Setzer. Extended Bibliography by subjects. the basis of tensegrity were not very clear at that moment. and his final design was not a reflection of a true tensegrity system. It will be explained in the next chapter that after some first attempts of tent-shaped structures by Frei Otto during the 60s. Pavilion for Expo ‘67 Illustration taken from CISC (2003) U. e. but something more similar to Levy and Geiger’s works (Geiger. J. 13 . 1997. Background and History U.g. It was an internal note for the Laboratory of Civil Engineering of the University of Montpelier (France) about the mechanical behaviour of this kind of 8 See Appendix I.S. Frei Otto and Jörg Schlaich in 1972. Autotendant icosaédrique. ix. some authors made an effort to develop the field opened by their predecessors. both from the University of California (Berkeley). it also lacks the treatment of behaviour of tensegrity under load. Some years later.…). xi). Anthony Pugh and Hugh Kenner (see bibliography). little reliable information had been published on the subject. T-Octahedron Dome Illustration taken from Burkhardt (1994-2004) 14 . in 1976. 3. while Anthony Pugh refers to Snelson in several paragraphs of his book (pgs. and explores their potentials. Nevertheless. which shows how to calculate “to any degree of accuracy” the pertinent details of geodesic and tensegrity regular structure’s geometry (lengths and angles of the framing system). Robert Burkhardt started an in-depth investigation and maintained a correspondence with Fuller (1982) in order to obtain more details about the geometry and mathematics of tensegrity. both of the authors realized that. which is interesting for the variety of models that it outlines and his strict classification and typology. On the other hand. Kenner developed the useful “Geodesic Math and How to Use It”. It is important to note that there is conflicting information in both books: Kenner affirms that Snelson’s work was “unknown to Tony” (pg. Background and History structure. Pugh wrote the “Introduction to Tensegrity”. apart from some of Fuller’s writings (see Bibliography).Tensegrity Structures and their Application to Architecture Chapter 2. From this time forth. Even though the latter work is more explicit in geometric and mathematic subjects. continued this work with different lines of attack. this laboratory and engineer became a reference in terms of tensegrity research. During the 1980s. On the one hand. Divergences Nevertheless. Watt. Williams.) have also studied the physics. However. Kono. Recently. 20 years later. etc. R. A. Passera. apart from the authors mentioned above. is a very complete. Pellegrino. Other important investigators have been Ariel Hanaor (1987. Using the mathematical tools of group theory and representation theory and the capabilities of computers. Pedretti. including some that have never been seen before. Williamson. useful and continuously revised Practical Guide to Tensegrity Design (Burkhardt. topological and algebraical points of view) and mechanics of tensegrity structures. Other authors (S.E. W.G. 1992).Tensegrity Structures and their Application to Architecture Chapter 2. Buckminster Fuller. who defined the main bidimensional assemblies of elementary self-equilibrated cells and Nestorovic (1987) with his proposal of a metallic integrally tensioned cupola. Skelton. they have drawn up a complete catalogue of tensegrities with detailed prescribed types of stability and symmetry. Since it is not always possible to read all the publications that are appearing in relation to a specific field. 1994-2004). Connelly and Back (1998a. the resourceful and charismatic inventor.O.M. Background and History The final result. something more universal and abstract. only the most relevant will be pointed out in the next paragraphs. M. 2. Y. The most recent works will be referred to again in chapter 6.4. D. Tibert. looked for something else. more 15 . 1998b) have aimed to find a proper threedimensional generalization for tensegrities. and Motro and his group in Montpellier. A. several works have been adding to the body of knowledge. mathematics (from geometrical. there have not been many works seeking to apply this new knowledge to any field in particular. he wrote: “All structures. Fuller extrapolated this phenomenon to the total Universe. it is explained. 2. Universe is omnitensional integrity” (Fuller. Although he never refused to apply tensegrity to technical fields. Contrary. elements under tension don’t need a great deal of matter. Fuller (1975b) thought that there is no limit ratio in tension. properly understood. I gave this phenomena the name “tensegrity. something that would be able to achieve a major universal law.” (ibid. both of them tried to make the ideas of such a hectic inventor clearer. how compression obliges the components of a structure to become thicker in order to avoid buckling. In his book Synergetics. it serves as an illustration that. making a rather complicated metaphor. and support enormous amounts of tension with very narrow sections (cf. fig. 713. The structural integrity of Universe is tensional as Kepler discovered. so we could have very great lengths and no section at all. He was not very readable. 1982) Therefore. Tensegrity discoveries introduce new and very different kinds of structural principles which seem to be those governing all structuring of Universe.Tensegrity Structures and their Application to Architecture Chapter 2. from the solar system to the atom.04) “This structural scheme of islanded spheres of compression.10). Background and History generic. are tensegrity structures. especially with the discovery of new materials which are resilient and strong. Edmonson (1987) wrote her Fuller explanation. 700. the solar systems and the atoms. in his opinion tensegrity was the base of the Universe: both. both macrocosmic and microcosmic. In these publications. macrocosm and microcosm.” (Fuller. were structured following the tensegrity principles. following the ideas exposed in Synergetics. until the point of considering the sphere as the best shape to support compression loads. convinced about the advantages and basic principles of tensegrity. after Fuller was deceased. while Applewhite (1986) prepared the Synergetics dictionary: the mind of Buckminster Fuller. which are only mass-attractively cohered. 1975b.08) “I simply found that the Universe is compressionally discontinuous and only tensionally continuous. also characterizes the atomic nucleus's structural integrities. this is the game that the 16 . this is the manner in which the solar system coheres. but the male goes off to be the hunter and the fighter. which is applied to the human race. He thrusts. also tried to obtain an atomic configuration. to attract. The female and her young and so forth are the great continuity of that family. I just. He is the island. 1981) 9 Perhaps it is a coincidence. Illustrations taken from Fuller (1961) On another scale. generally speaking. And a man is compressive. personally find then that the woman is tensive. it is curious how he tried to explain everything making use of tensegrity principles. and always coexist. Background and History Universe is playing: Gravity. Fuller’s figures. a “portrait” of the atom. Just the sex act. What we call being female is to pull—to walk away.Tensegrity Structures and their Application to Architecture Chapter 2. This is really very fundamental in social behaviour. In this way. Now.9 Finally. where compression and tension are always separated. 1989) 17 . the Earth and the Moon are invisibly cohered and. I find the male tending to do this—to punch. He becomes islanded.” (Fuller. she pulls. She does the other way. She pulls in. but Snelson. And it’s just very fundamental. he was convinced that the atomic attraction (especially the invisible interaction between atoms. like Fuller. He is a hunter. is a good illustration: “I also then point out to you the difference between the male and the female. Just fundamentally. The following example. She is central. but his approach was from an artistic and geometrical perspective (Snelson. The male then becomes discontinuous. nuclei and electrons) is another type of tensegrity. so it is necessary to have a clear and concise definition that avoids confusion. other examples of tensegrity in Nature are shown: cell structures and their behaviours (Ingber 1993. relationships and interactions between diverse elements in the environment can be considered as tensegrity. 10 Kenneth Snelson: excerpt from an e-mail to the author.Tensegrity Structures and their Application to Architecture Chapter 2. internal structure of the radiolaria (marine protozoa). where sun pushes and plants pull. support system of the spine and some other components of the skeleton (Levin 1982). In conclusion. 1998. it is feasible to involve these kinds of principles to a wide range of phenomena. 3 Aug 2004. Another good example of the extension of the term tensegrity to other fields was the participation of René Motro in a seminar at the Collège International de Philosophie of Paris. This will be the aim of chapter 4. anatomic organisms. Tensegrity structures are endoskeletal prestressed structures -. In the next chapter. other authors (Wilken. and named as. it is possible to affirm that depending on the definition of the word “tensegrity”. The course was dedicated to. 2003). and had the contribution of biologists. systems. “Tensegrity”.and that restriction leaves out endless numbers of items. sculptures. where female is continuously attracting and males are discontinuously pushing. historians and Hellenists. in preypredator. Snelson is very clear: “Yes. so what's the point in using that word?” 10 Following Bucky’s line of thinking. 2001) compared this “push and pull” strategy to living organisms in Nature (including vegetables and plants) to describe the three possible classes of life looking for tensegrities: in photosynthesis-radiation. if everything is tensegrity then tensegrity is nothing of any particular sort. and finally in student-teacher. Fuller declared that everything in the universe was tensegrity. Background and History In contrast to this opinion. Structures. where the first is pulling in new knowledge while the latter one is pushing out information to someone else. (See Appendix D) 18 . As I've also said elsewhere. . Tensegrity Structures and their Application to Architecture Chapter 3 Precedents and Key Studies . . 000 psi. but its 10. This chapter will attempt to explain how it is possible to achieve such a modern and contemporary structure from its more original beginning. the first mass production of steel. Efficient “push-and-pull” structures would have been inconceivable before the 18th Century due to the incapability to obtain effective behaviour of material under tension. (1 psi = 0. That steel was able to reach 50. 3. until that moment. Precedents and Key Students Chapter 3.068 Atm) 19 . resulted in many new possibilities and.000 psi in compression of stone masonry. Edmonson (1985) states that. 1 psi = pounds per square inch. Materials and tension Due to the fact that the main support of these structures is the continuum tension. the investigation of materials suitable for traction efforts has been crucial. Introduction. said Fuller (ibid).069 bar = 6.000 psi 1 in traction was not comparable with the 50. its evolution and development are strongly connected to other events and circumstances. only the tensile strength of wood had been exploited (mainly in ships’ construction). according to Edmonson (ibid). in both compression and traction. the building of the Brooklyn Bridge opened an innovative era of tensional design. “Tension is a very new thing”. However. in 1851.89 KPa = 0. changed this situation greatly.1.2. Despite the fact that the origins of tensegrity were exposed in the previous chapter. Precedents and Key Studies 3.Tensegrity Structures and their Application to Architecture Chapter 3. ).2. As illustrated in figure 3. their weight. It is made of bamboo rope cables. they were probably the first system that took advantage of tensile properties of materials.Tensegrity Structures and their Application to Architecture Chapter 3.D. but also to decrease their cross-section and. Although they were made from rope and wood. It should not be forgotten that the first suspension bridges. six An-Lan Bridge out of hardwood and the centre one out of granite (cf. Illustration taken from IL (1985) In any case. weight and performances of materials. Precedents and Key Students From the author’s point of view. These new materials not only served to increase the resistance of the components. fig. consequently.1). which hang from seven piers. this statement is not completely accurate. based on a tensile structural concept. the behaviour of elements under a load is different depending on the type of load. it is evident that the development of steels and other alloys led to unpredicted outcomes in terms of resistance. it has the tendency to augment its cross-section (due to 20 . in Kuanshien (China). 300 A. were invented many centuries ago. which is the oldest suspension bridge in use (app. An example is the An-Lan Bridge. and their load-bearing capacity was incapable of supporting heavy loads. which enabled engineers and architects to create new designs and new structural concepts. However. 3. when a lineal element is compressed allong its main axis. On the contrary.a). G. Some precedents. After this first attempt. According to Tibert (1999). more importantly. 2 Philip Drew (1976) refers to him as “Shuchov”. the new materials discovered during the 19th and 20th centuries. whose compressed elements must be more resistant to buckling. Before and after the discovery of tensegrity in 1948. which means it loses its straight shape (fig. some other structures were proposed during the 1930s. especially their tensile strength. Shookhov 2 in 1896. 3.b). Illustration drawn by the author. and whose tensioned members have to better resist the traction forces. 3. 3. Precedents and Key Students Poisson’s ratio effect) and to buckle. the first cable roofs were designed by V. some works were conceived to adopt the most recent resources and to take advantage of their most privileged properties.Tensegrity Structures and their Application to Architecture Chapter 3. This Russian engineer built four pavilions with hanging roofs at an exhibition in Nizjny-Novgorod (Russia). it tends to become thinner and.3. the innovation in materials is essential for the future of pre-stressed structures. For this reason. 21 . As has just been commented on. but they were not very important examples.2. Deformation under compression and under tension.2. it “reaffirms” its straight axe (fig. when the same element is tensioned in the same direction. permitted the revolution of thinking in terms of architectural and engineering design. Apart from the suspension bridges. A very good example is the Barrios de Luna Bridge (fig.4) in Asturias (Spain). 22 . 3. which shows this principle perfectly in both of its two towers and main span of 440 m.3.3.a. some other types of bridges elevated the importance of tension to the same level that compression had had during the preceding centuries. Cable-stayed bridge Illustration drawn by the author. 3. 3. fig. by Javier Manterola. which make use of the stressed cables to support the deck and also put it under compression. Precedents and Key Students Suspension bridge Illustration drawn by the author. which were observed above and in fig.b).Tensegrity Structures and their Application to Architecture Chapter 3. Thus the deck is prestresed and put in equilibrium (cf. This is the case with cable-stayed bridges. fig. Illustration taken from King and Lockhart (2004) 23 . In 1951. The Skylon. Illustration taken from Búrdalo (2004) 3. World record of cable-stayed bridges in 1983.3. Precedents and Key Students TE N S I O N COMPRESSION Barrios de Luna Bridge J. Philip Powell and Hidalgo Moya (helped and inspired by their former Felix Samuely) designed the Skylon (cf.5). 3. which was selected as the best proposal and built near the “Skylon” Dome of Discovery. just three years after the official discovery of tensegrity. a staple of international exhibitions grounds. Manterola. the Festival of Britain's South Bank Exhibition took place in London.1. a competition was organised to erect a “Vertical Feature”. In that occasion.Tensegrity Structures and their Application to Architecture Chapter 3. You felt there weren't enough wires to hold it up. a beacon of technological and social potentialities and. finally. Once the whole structure was assembled. he pumped up these jacks and raised the pylons. 1995) 24 . and seemed to float 40 feet above the ground.. This reduced the number of wires needed to anchor the Skylon and halved the amount of oscillation in the structure. the father of the idea: “By an amazing stroke of genius [Felix Samuely] arranged a system of hydraulic jacks underneath the three smaller pylons.6. According to Moya. The 300 foot high spire was a cigar-shaped aluminium-clad body suspended almost invisibly by only three cables.Tensegrity Structures and their Application to Architecture Chapter 3. 3. was composed of a cradle of prestressed steel wires and three splayed pylons. as it is shown in fig. Precedents and Key Students Some authors (Cruickshank. which made it tremendously exciting." (Cruickshank. 1996) state that this needlelike structure was a monument without any functional purpose. 1995. This lack of support made the structure look tremendously hazardous. F w1 w2 E B A Skylon Illustration drawn by the author. a reference for future engineers and architects. Burstow. This put tension or stresses into all the cables and by doing that the whole thing became a stressed structure. The structure. but it became a symbol for the festival. As an illustration. If it does it in D. If one of the wires (w1) is attached to the ground in B.6. the post is in an instable equilibrium (any movement of F will lead it to fall down). the exploitation of cables in traction was not only improved. which explains the condition for stability of a post (pin-joint to the ground in A) supported by stressed cables.4. and the rest of the points are in association with the nomenclature of fig. In paragraph 4. the system is in a stable equilibrium.Tensegrity Structures and their Application to Architecture Chapter 3. it collapses. at the same level. it tends to return to the upright position. it has been demonstrated that the conditions for the equilibrium of a strut in a three-dimensional space are susceptible to the point of application of the ends of the wires that fix it. a diagram inspired by Francis (1985) is presented in fig. the equilibrium of the strut will depend on the position where the other string (w2) is held: If it is fixed in a point C below the level of A. but also that of other elements such as membranes.7. when there is any disturbance of this situation.4 the equilibrium analysis will be further explained. As a consequence. Suspended roofs and tensile structures During the 1950s. if it is held in a point E above the level of the ground. the cables are w1 and w2.3. 3.7. 3. Equilibrium of a post supported by cables. In the diagram of Skylon in fig. Illustration drawn by the autor 3. materials and tissues. Precedents and Key Students The cause of the feeling of not having enough cables to hold the zeppelin-like shape element is due to the stable equilibrium obtained by means of its particular configuration. 3. in other words.. In contrast.2. 25 . and was completely fascinated by the innovative idea. Kassel (Germany) in 1955 (fig. His name was Frei Otto and that was the first comprehensive documentation on suspended roofs (Drew. 1999). to further increase the research into tensile architecture (see Appendix I. some important works were developed exploiting the tensile properties of materials. 3.10) and the celebrated Olympic Stadium in Munich in 1972. 3. the first large cable net structure with fabric cladding. These projects are important for the development of tensegrity structures since this kind of membrane can be adopted as the tensile component of tensegrities. As a result. For instance. That same year. he started a systematic investigation that was presented as his doctoral thesis in 1952. whose structure was calculated by Jörg Schlaich.Tensegrity Structures and their Application to Architecture Chapter 3. the German pavilion at the World’s fair in Montreal 1967 (fig.9). cotton-polyester mix. glass fibre. acrylic panels. architecture had a brief look at the drawings and plans during a exchange Illustration taken from Buchholdt (1985) trip to the USA. PVC. there was an early four-point tent as a Music Pavilion of the Bundesgartenschau. 1973). 26 . Among these projects. 1976. 3. and in 1964 was included in The Institute of Light Surface Structures at the University of Stuttgart. Precedents and Key Students In 1950. but also polyurethane. The Development Centre for Lightweight Construction was founded by him five years later in Berlin. at Raleigh (North Carolina) was designed by Matthew Nowicki following his intuitive concepts of suspended roofs (fig. especially steel. Tibert.8). Hence. a German student of Raleigh Arena Nowicki. Otto 1967-69. the State Fair Arena. etc. polyester. Pugh (1976) built a dome made out of wooden struts and plastic skin. and Appendix C). invented by David Geiger in 1986 3 (see Bibliography: Geiger 1988. In fact.2).3. it should be recalled that Fuller (1964) patented a similar kind of structure.3. The reason behind this argument will be shown in the subsequent chapter (paragraphs 4. 3 Even though Geiger did not refer directly to Buckminster Fuller. 27 . the denomination of “tensegrity” has been extended to include any sort of pin-connected structure in which some of the frame members are wires in tension or bars only in compression. Motro (2003) is quick to identify them as false tensegrities since they have a compressed member in the boundary. Otto (1967) Illustration taken from Stanton (1997) 3. which he later called “Aspension”. Since then. Precedents and Key Students the latter being the component in tension that supported the compression members of the structure. This is the case of the “Cable-Domes” or “Wire Wheel Domes“. This can be seen in Appendix C. and converges to an internal ring in order to join all of them. several domes have been built following this technique. As W.Tensegrity Structures and their Application to Architecture Chapter 3.3 & 4. Williams (2003) points out. “Music Pavilion” by Frei Otto (1955) Illustration taken from Atelier Warmbronn (2003) “German Pavilion for Expo'67” by F. Despite the fact that some architects and engineers consider these roof structures as tensegrities.4. where a group of radial tensegrity beams is attached to an external ring in compression. Cable-Domes. O. the biggest dome in the world to date. They are. followed by the Redbird Arena in Illinois. when asked about the subject. 3. (1997) 4 Kenneth Snelson: excerpt from an e-mail to the author. Petersburg (1988). exoskeletal structure? I think not. 28 . the first oval cable-dome (1988).Tensegrity Structures and their Application to Architecture Chapter 3. regardless what people wish to call them. which is a one of Roof diagram for a Cable-Dome Illustration taken from Gossen et al. essentially.11) The first cable-domes were designed by Geiger: for the Olympics in Seoul (1986). the sculptor responds in a clear manner: “The (…) domes you cite can not be considered tensegrity. Did the world need a different name for that kind of solid rim. bicycle wheels. See Appendix D.” 4 Admitting that they are different to tensegrities. it is evident that at least they are inspired by their principles: compressed struts that do not touch each other and are linked only by means of cables (cf. Precedents and Key Students Snelson does not regard them as real floating compression systems. same with a spider web. fig. and the Tayouan Arena in Taiwan (1993). Indeed. the Florida Suncoast Dome in St. 3 Aug 2004. in the meantime some scientists. 1997). cited in Gossen et al. 4. inextensional modes of deformation (Pelegrino.. 1989).Tensegrity Structures and their Application to Architecture Chapter 3. as an answer to a general question about the nature of structure. from macrocosm to microcosm. about the structure of nature (Burrows. 29 .5 & 4. Tensegrity as a universal principle. Or even more. Nevertheless. starting with Fuller and Snelson. The origins of tensegrity are linked to sculpture. It might be interesting to note that. mainly civil and mechanical engineers are trying to research its properties and applications. and at present. (1997) 3. they were related to architecture and mathematics.6 in next chapter).4. these structures are not very determinate in classical linear terms and have several independent mechanisms. or in other words. because of the sparseness of the cable-dome network. subsequently. conceive tensegrity as a basic principle in the Universe. Precedents and Key Students this type. is the Georgia Dome in Atlanta (1992) by Levy and Weidlinger Associates (see figs. Cable-dome diagram Illustration taken from Kawaguchi et al. also described in paragraph 2. and with the condition that the continuum of tensions is always surrounding the “islands” or components in compression. fig. for instance. professor of pathology at Harvard Medical School. The main one was contributed by Donald E. which is very convenient for gravitational and atomic examples (Motro. it is necessary to establish some important concepts. In order to do the transposition of tensegrity to subjects other than material ones. Tensegrity in Macrocosm and Microcosm. 3. 2003) Kurtz (1968) mentioned that Snelson notices all ways of connection through tensegrity: in Astronomy (a planet to the sun). Harris about the elasticity of cells.Tensegrity Structures and their Application to Architecture Chapter 3. Ingber.2 Tensegrity in Biology. As was explained in chapter 2. In order to illustrate this fact.4. Precedents and Key Students 3.10 of chapter 2). In addition to the last proposal. it has been stated by the author that in “Tensegrity”. several suggestions have been put forward by different specialists from different fields. Fuller’s writings are continuously referring to tensegrity as the essential pattern of the universe (cf. in a stable equilibrium. in the early 80s.1.4.4. it occurred to him that a view of the cell 30 . Compression and traction can be. associated with repulsion and attraction respectively. After some comments by Albert K. under a precise distribution of elements or components. only if it does it corresponding to a particular field of forces. “atom” was mentioned 12 times and terms related to the “nature” 13 times. in atomic physics (an electron to the nucleus) and in mechanics (a cable to a rod). he cited the word “universe” or anything else related to the universe in 19 occasions. Tensegrity can generally be considered as a structural principle. 2. a journal article written in 1961. A.13) Tensegrity model of a cell .” (Ingber. Madri and J. 2003a). Like a living cell. mechanochemical transduction and morphogenetic regulation can be found elsewhere. it flattens itself and its nucleus when it attaches to a rigid surface (left) and retracts into a spherical shape on a flexible substrate (right). 1998). explains Ingber (ibid). he added: “A discussion of how tensegrity may be used for information processing. Jamieson a theory about the subject in 1981 (cf. the biologist suggested that cells and nuclei 31 . Illustration taken from Ingber (1998) “The tensegrity model”. For example. 1993) Despite the fact that it was only a preliminary hypothesis. based on several experimental works. “suggests that the structure of the cell's cytoskeleton can be changed by altering the balance of physical forces transmitted across the cell surface”.D. 3. Precedents and Key Students as a tensegrity structure could easily explain such behaviour (Ingber.Tensegrity Structures and their Application to Architecture Chapter 3. and subsequently published with J. In other publication. some new discoveries have proved that the proposition is valid and mathematical formulations of the model predict many aspects of cell behaviour (Ingber. The only discordance with the established tensegrity principles is that. it has been generated a tensegrity model composed of six rigid bars connected to a continuous network of 24 viscoelastic pre-stretched cables (Voigt bodies) in order to analyse the role of the cytoskeleton spatial rearrangement on the viscoelastic response of living adherent cells (Cañadas et al. Precedents and Key Students do not behave like viscous water balloons. According to Vesna (2000). which has been demonstrated by Andrew Maniotis recently. but are physically connected by tensile filaments.14. Inberg (2003a) accepts flexible springs instead of rigid elements. thus. viruses. tissues. some other experts have been working on this hypothesis. Oddou and Isabey (1999) proposed a quantitative analysis based on a theoretical model of a 29 element tensegrity structure5. Following this line of research. in contrast with other authors. studying its nonlinear mechanical behaviour under static conditions and large deformations. This configuration and use of materials confer different elasticities and. and other living Springs model Illustration from Ingber (2003) creatures.Tensegrity Structures and their Application to Architecture Chapter 3. some studies strongly suggested that tensegrity have Diagram showing the role of tensegrity in heart functions. 2002) 32 . proteins. as it is showed in fig 3. not only cells but also an incredibly large variety of natural systems are constructed following the tensegrity model: carbon atoms. The same year. Ingber discovered that.. Illustration taken from Lab (1998) 5 More recently. behaviours under tension or compression. Wendling. water molecules. but this must be the aim of further research. This leads to volume changes that appear to be experimentally corroborated by recent measurements.) holds sway. 1999). it seemed that while organic chemistry (cells. “Suppose for fun. even inorganic substances can be based on floating compression.. (1998) 9 See Bibliography: Zanotti and Guerra (2002) 10 CRN: continuous random network. 3. etc.” (Tsu et al. 1997) 8 See Bibliography: Eckes et al. pollen grains. 2003) have proposed a new tensegrity model for an amorphous silicone (a-Si:H) consisting of tensile and compressive agents that act to globally redistribute the effects of locally created defects.Tensegrity Structures and their Application to Architecture Chapter 3. according to some new findings.3 Tensegrity in Inorganic Chemistry. 6 7 See Bibliography: Ingber (1998) The buckminsterfullerenes or “bucky balls” are spherical groups of 60 carbon atoms (Carbon-60). this can be used to build better new heterogeneous structures and substances. named like that after it was suggested that its structure is similar to that of a geodesic sphere. water molecules. it is very interesting that.. 1998).. we assign CRN10 the compressive role.4. 3. To date.15. pp. So in a simplistic topological sense. and the CLOs11 the tensile role. the CRN is like a stiff rod. 33 . Some authors (Tsu et al. widely rely on tensegrity. carbon atoms6 or buckminsterfullerenes7. See fig. 11 CLOs: ‘‘chain-like objects’’. and the CLOs like flexible (but strong) cables. it has been discovered for example that the function of tensegrity in the transmission of endocrines in the heart is essential because it facilitates integration of force and strain changes from area to area (Lab. 2003. Precedents and Key Students implications for all types of cell transplants requiring cell isolation (Thomas et al. 2000. vitamins8. However. Yamada et al. The composite structure is in a ‘‘prestressed’’ state where cables pull against rods in a multilateral relationship. Other authors (Volokh et al. as a result. the inorganic things seemingly do not have the benefits of this principle.138) As a result. viruses.. proteins9.. 2000) have been using the same theory applied to living cells with similar results and. invented by Buckminster Fuller (Lu. Precedents and Key Students 3. 34 .4 Tensegrity in Anatomy. 1982) Tensegrity Thoracic Vertebrae Illustration taken from Levin (2002) The latter sentence refers to one of the main properties of tensegrity systems. to anatomy. He focused his reflection in the system of the human spine. A Model for Biomechanical Support of the Body”. Meyers. In spite of having been used only as an example to illustrate the models.” (…) “External forces applied to the system are dissipated throughout it so that the "weak link" is protected. Discontinuous Compression. The forces generated at heelstrike as a 200 pound linebacker runs down the field. It is very common to find the term “tensegrity” applied to biomechanics and. but a set of compression components suspended within a continuous tension network. The clavicle has been traditionally recognized as acting more as a compression strut. when he wrote “Continuous Tension. could not be absorbed solely by the os calcis but have to be distributed—shock absorberlike—throughout the body. tendons and bones in animals and humans. The scapula does not press on the thorax. the scapulothoracic and the sacroiliac joints. which will be exposed in next chapter. the capacity to distribute the forces. cited by Gordon. some sources (Heller. The entire support system of the upper extremity is a tension system being supported by the musculature interweaving the spine. The first reference to tensegrity in this subject was proposed by Stephen M. 2004) make use of the term to explain the relationship between muscles. as it would in a tensegrity model (…) We therefore can see in readily discernible anatomical studies that the tensegrity system is utilized in two of the major support joints of the body.” (Levin.4. They claim that the skeleton is not just a frame of support to which the muscles. ligaments and tendons attach. which deserves to be quoted in length: “We can examine the scapulothoracic articulation. especially. 2002. Wikipedia. 2004. Levin in the early 1980s. for example. thorax and upper extremity into a tension support system. and indeed the remainder of the body.Tensegrity Structures and their Application to Architecture Chapter 3. has in itself the physical shape and properties of a continuous tension-discontinuous compression structure. in order to make clear difference between each subject and to find out what are its main advantages and disadvantages. According to Wilken (2001). amoeboid protozoa that produce intricate mineral skeletons. He also observed that radiolaria. employed this principle as well. from the point of view of some specialists. the sensory neurons are always sensing information (continuously pulling) while the motor neurons are only occasionally involved in some motor action (discontinuously pushing). and has been applied to so many fields of Science that it is perhaps loosing its main meaning. 1961). It has become a basic principle of Nature. it has been recently proposed that the central nervous system also functions as a tensegrity. something that was mentioned by Fuller 30 years before (Fuller. stances and movements) developed by Native American shamans. tensegrity will be defined. Finally. Precedents and Key Students Nevertheless. In summary. Levin declared that the methane molecule. something else rather than just a spatial structure made of struts and strings. described and characterized.Tensegrity Structures and their Application to Architecture Chapter 3. Tensegrity has even been used to denominate the modernized version of some movements called “magical passes” (a series of meditative stretches. because it connotes the two driving forces of the magical passes (Castañeda. 1996). 35 . In next chapter. it can be concluded that floating compression is. one of the simplest organic substances. . Tensegrity Structures and their Application to Architecture Chapter 4 Definitions and Basic Principles . Tensegrity Structures and their Application to Architecture Chapter 4. Definitions and Basic Principles Chapter 4. Definitions and Basic Principles 4.1. Introduction. For so many years, some authors have been trying to find a “definitive definition” of tensegrity, which is unambiguous and accepted by the whole scientific community. It is essential to specify precisely what a tensegrity structure is because, depending on the different definitions, we will be able to consider some kinds of structures as real or false tensegrities. As was mentioned in previous chapters, there are a lot of cases where the term “tensegrity” is being used incorrectly to denominate any type of structure based on compressed and tensioned components. Obviously, this is a mistake, as tensegrity is a very distinct principle. As an illustrative and peculiar example, two very curious patents will be mentioned: the “Female condom employing tensegrity principle” (Glenn and Tam, 2002) and the “Sports catch glove with stiffner” (Goldsmith, 1998) 1 . Of course, none of them is really a tensegrity application at all. In chapter 6, more examples of false tensegrity will be shown which relate to such applications. 4.2. Definitions. In order to show the evolution of the analysis of these systems, different definitions will be explored in a chronological order. The first descriptions, which were explained in the chapter 2, were given by the authors of the patents, trying to describe what they had discovered. Obviously, 1 Both patents are referred to in Appendix C. 36 Tensegrity Structures and their Application to Architecture Chapter 4. Definitions and Basic Principles in those days it was very difficult to generalise and find a complete definition that could summarise such a complex entity as tensegrity. In the article called Tensegrity, Buckminster Fuller (1961) explained very profusely the principles and main concepts that govern the tensional-integrity systems, but he did not give any precise definition. In his patent, he describes this kind of structures as “a plurality of discontinuous compression columns arranged in groups of three nonconjunctive columns connected by tension elements forming tension triangles” (Fuller, 1962, p.1). However, he gives a very short explanation, which has been passed to the annals of the history of tensegrity: “The Some Fuller’s tensegrities Illustration taken from Fuller (1975b) compression elements become small islands in a sea of tension” (ibid). Some years later, he wrote in Synergetics an extended explanation: “Tensegrity describes a structural-relationship principle in which structural shape is guaranteed by the finitely closed, comprehensively continuous, tensional behaviors of the system and not by the discontinuous and exclusively local compressional member behaviors” (1975b, 700.011) The other “father” of tensegrity, David G. Emmerich, declared in his patent that his invention could be further described in a non-limitative manner with reference to several examples, shown by accompanying drawings. In this way, he avoided the difficult task of giving a strict description. 37 Tensegrity Structures and their Application to Architecture Chapter 4. Definitions and Basic Principles Perhaps Kenneth Snelson is clearer in his definition. In his patent, he explained: “The present invention relates to structural framework and more particularly, to a novel and improved structure of elongate members which are separately placed either in tension or in compression to form a lattice, the compression members being separated from each other and the tension members being interconnected to form a continuous tension network.” (1965, p.1) Even although he prefers to call them “Floating compression structures”, he describes them as follows (thus collaborating with the previous description): “Tensegrity describes a closed Snelson with a double planar structure structural system composed of a set Illustration donated by the artist to the author. of three or more elongate compression struts within a network of tension tendons, the combined parts mutually supportive in such a way that the struts do not touch one another, but press outwardly against nodal points in the tension network to form a firm, triangulated, prestressed, tension and compression unit.” (Snelson, 2004) Additionally, as mentioned in previous chapters, he made a very clear distinction: “Tensegrity structures are endoskeletal prestressed structures -- and that restriction leaves out endless numbers of items”.2 Some years later, Anthony Pugh gave the following characterisation of tensegrity, which has been accepted almost universally by the rest of the specialists, due to its well adapted constitution for an extended definition, possibly the first one of its kind: “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” (1976, p.3). 2 Kenneth Snelson: excerpt from an e-mail to the author, 3 Aug 2004. See Appendix D. 38 identifying other important characteristics: tensegrity structures are self-supporting and rigidified by self-stressing (something that had already been advanced by Emmerich and Kenneth). the state of 39 .g. to which a state of prestress is imposed that imparts tension to all cables”. adding later. 93) There are further and more complex definitions depending on the perspective of the authors. The wider definition given by Wang and Li (1998. Definitions and Basic Principles It was not until the 90s that Schodeck (1993) realized that a definition based on redundancies and degrees of movement may be a better description than the ambiguous notions formulated at that moment. E.” (pp. Therefore.Tensegrity Structures and their Application to Architecture Chapter 4. 2003) is the following: “Tensegrity systems are free-standing pin-jointed cable networks in which a connected system of cables are stressed against a disconnected system of struts and extensively. Ariel Hanaor described tensegrity structures as “internally prestressed. free-standing pin-jointed networks.. 2002) gave a narrower interpretation: “A tensegrity structure is any structure realised from cables and struts. While Miura and Pellegrino (cited in Tibert. any free-standing pin-jointed cable networks composed of building units that satisfiy aforesaid definition. Bin-Bing Wang (1998) went beyond the previous definition. in which the cables or tendons are tensioned against a system of bars or struts”. being a Class k tensegrity structure if at most “k” compressive members are connected to any node. Kanchanasaratool and Williamson (2002) state that a tensegrity system is a stable connection of axially-loaded members. a traditional tensegrity structure is a class 1 structure because only one compression member makes a node. he labelled tensegrities as rigid structures made of discontinuous rods in compression and continuous cords in tension in which each component has one degree of member redundancy. “as well as imparting tension to all cables. ” (p. thus providing ?rst-order sti?ness to its in?nitesimal mechanisms. 40 . relational structure (between the different components). René Motro (2003) tried to distinguish two different concepts. in compression and in tension).18) The other description.3. As a result.Tensegrity Structures and their Application to Architecture Chapter 4. as Motro suggests: System: In relation to the theory of systems. and the system has self-equilibrium stability. as all three describe the same structure: “Patent based definition: Tensegrity systems are spatial reticulate systems in a state of selfstress. it is possible to distinguish true and false tensegrity due to their respective characteristics.” Finally. total structure (associating relational structure with characteristics of components) and form (projected on to a three-dimensioned referenced system). Each node receives one and only one compressed element. The first one is established on the basis of patents (see preceding definitions and Appendix B). General Characteristics If this last definition is accepted as being sufficiently comprehensive and concise to define the term. Definitions and Basic Principles prestress serves the purpose of stabilising the structure. but has additional factors: the compressed elements are included inside the continuous set in tension. René Motro suggests the following: “Extended definition: Tensegrity system is a system in a stable self-equilibrated state comprising a discontinuous set of compressed components inside a continuum of tensioned components. It would also be possible to state the following. has some common points with Pugh’s’ definition. it has components (two kinds.19) 4. All their elements have a straight middle fibre and are of equivalent size. the extended one.” (p. Tensioned elements have no rigidity in compression and constitute a continuous set. He makes the distinction between the “patent based” and the “extended” definition. Compressed elements constitute a discontinuous set. it is independent of external forces (even gravity) or anchorages due to its self-stress initial state. it can be a strut. 4. See the models of next figs. the action lines lying on the boundary surface are tension lines. and a compressed element is inside when the points between its ends do not belong to the boundary (or envelope). and self-equilibrated because it doesn’t need any other external condition. because the key is that the whole component has to be compressed or tensioned depending on its class. a membrane. and the tensegrities. such as the torus. with different “insides” and “outsides”. Thus. and the tensioned components are creating an “ocean” of continuous tension.3 & 4. an air volume. Components: in contrast to the term “element”. In order to avoid controversial systems. in a tensegrity system. Inside: This is a crucial point since it will permit the differentiation of two sorts of structures: the conventional. a cable. It is stable even in orbit. where this role is played by the tension. an assembly of elementary components. Motro defines a system as one of tensegrity when all its compressed components are inside the system. Compressed or tensioned components: instead of compression and tensile components. where compression is the basis of the load support. Definitions and Basic Principles Stable self-equilibrated state: Stable because the system can reestablish its equilibrium after a disturbance. Continuous tension and discontinuous compression: because the compressed components must be disconnected.Tensegrity Structures and their Application to Architecture Chapter 4. etc.4 as examples: 41 . Thus. this is the point that allows us. for example. to consider the biggest dome in the world. in their opinion it is in the range of pre-stressed systems as a “cable dome” and not as a “tensegrity dome”.5 & 4. as it was explained in the previous chapter. in the boundary of the system. Georgia Dome Illustration taken from Setzer (1992) Georgia Dome Illustration taken from Setzer (1992) 42 . since it has a compression ring surrounding the net of cables and struts. Definitions and Basic Principles Octahedron .6). as a pure or as a false tensegrity.Tensegrity Structures and their Application to Architecture Chapter 4. Nevertheless. Model built by the author. and. the “Georgia Dome” in Atlanta (see figs. 4. Some purists don’t consider that it belongs to this type of structure. consequently. Expanded octahedron Model built by the author. This last characteristic could possibly seem superfluous for people who are not very familiar with these structures. Tensegrity structures are based on a completely different approach. In a concrete. they do not have to be supported as they are self-equilibrated and pre-stressed.4. the technique of construction and the philosophy of building have been very simple: everything was held in place by weight. How can they be in that position only attached by the wires? 4. Moreover. but the actual shape of the dome is responsible for maintaining its stability. wood or steel dome. Until the last century.4. “In fact. the tension created by the attraction of the Earth is replaced by the multidirectional tension of their members. they are made as a “system of equilibrated omnidirectional stresses” (Kenner. Definitions and Basic Principles 4. One of its unique aspects is the surprising and not always understood equilibrium of islanded struts floating in the air. 1987). Basic Principles Since the mid 20th Century. 43 . the only thing to do is to reinforce the weak points. so they are not depending on gravity factors for their own equilibrium.1. it has been accepted that tensegrity is a new and very particular structural principle. 1976). so the continuities of stress were basically compressive. After this consideration.Tensegrity Structures and their Application to Architecture Chapter 4. tension must be primary” (Edmonson. Main Concepts. Instead of the “weight and support” strategy. For instance. the weight is much lower because we distinguish between “skin” and “bones”. Furthermore. each component of a stone dome is pulled by tension “downward” through the structure. Fuller (1975b) affirms that the example of Nature shows that tension must be included in every design since the beginning of its conception. but the compressive continuity is still in charge of sustaining most of the load. 1976.7. elastic. Snelson and Von Baeyer. seeking for analogies in order to understand their principles in a clearer way. 1982. people have been looking for mechanical and structural explanations. 2004) admit that inflatable constructions are tensegrities because they are self-equilibrated systems composed by an exterior tensile component which embraces the atoms of gas behaving as discontinuous components in compression (cf. fig. tensegrities and pneumatics. In this research.4.). Both. are compressible. Analogy of the balloon Illustration drawn by the author. The most common of them has been the comparison between tensegrity and pneumatic structures. lightweight and local-load-distributing structures. for compression and tension respectively. 44 . Edmonson. Nevertheless. since the birth of the floating compression. 1975b. Motro. Some analogies. the main members will be struts and cables. 1976.2. 1987. In fact. so pneumatics are only an extension of the proper definition. expandable. when we deal with proper tensegrities. Burkhardt. For 50 years. several authors (Fuller. selfbalanced. we consider the compressed components other than air or gas. Kenner. Definitions and Basic Principles 4.Tensegrity Structures and their Application to Architecture Chapter 4. Pugh. 1961. 4. 2003. 1989. the axle load of wheel.3. applied on the hub. while the whole rim stays in compression. 45 . is hung up from the spokes at the top. The effect is that the rim tries to belly out. as a result. the roof of the Leningrad Sports Palace) and have been improved in recent times as shown in paragraphs 4.8. during the 1960s and 70s there was much development in pre-stressed steel construction and. so the horizontal spokes keep it from deforming (cf. In contrast to general opinion.3 and 3. fig. as he thought it inaugurated a new era of thinking in terms of comprehensive tensions and discontinuous compression. Nonetheless. gravity is also secondary in terms of stability in the wire wheel. 4. Definitions and Basic Principles As a consequence.a).Tensegrity Structures and their Application to Architecture Chapter 4. Fuller turned to this example very often. “cycle wheel” roofs appeared on a huge scale (e. the main load-transfer system of the wire wheel is not the forces of compression supported by the vertical spokes of the bottom. As for pneumatics and tensegrity.g. in fact.3. which works in traction (cf. 4.8. fig. According to Francis (1980). the second analogy applied to explain the fundamental concepts of tensegrity is the comparison with the wire wheel (also mentioned in chapter 3)..b). Analogy of the wire wheel Illustration drawn by the author. joining the four extremes defined by them. 1967. his patent (see Appendix B) employed X-shaped modules to generate several masts of continuous tension-discontinuous compression and to explain the generation of the simplest tensegrity structure: the “Simplex”. 1981). This antique toy is simply based on two crossed sticks with a tensioned string around it. This is basically a two-dimensional structure. probably. 4. Generation of the Simplex Snelson. It is not a coincidence that Snelson achieved his first tensegrity sculpture (see photo #3 of Appendix A) from kite-like modules out of plywood. Illustration taken from Snelson (1965) 46 . The most primitive case of stressed structures is not the wire wheel but. The Creation of the Simplest Configurations. Fox. Moreover. and because it does not exhibit very intuitive principles. they can’t be judged as tensegrities.4. Due to the complexity of such an interesting type of structure. “Elementary Equilibrium” or “ThreeStruts T-Prism” (cf. maybe it is better to explain the generation of the easiest tensegrity designs. which can’t be considered tensegrity because the two rods in compression are touching each other in the middle of the kite.3.9).Tensegrity Structures and their Application to Architecture Chapter 4. 4. the kite (Coplans. fig. Definitions and Basic Principles and consistent with the previous quote by Snelson. 712. With the rods in this position. 4. although designed by the author in a more developed graphic.10. Definitions and Basic Principles Hugh Kenner (1976) obtained the “Simplex” from a different approach. He explained it by means of evolution of a system consisting of a single clothesline attached to two trees and supported in the mid way by two poles.10. 47 . Finally. When the two poles are very oblique.Tensegrity Structures and their Application to Architecture Chapter 4. the support of the trees can be substituted by fixing the ends of the rope to the ground (fig. 4.d).10 illustrates his main idea. there is the risk of sliding. his explanation is inspired in the “Fig. In such a situation. so they have to be attached to the ground (fig.10. although he does not make reference to Fuller’s diagram that appeared in Synergetics.e. a tent without centre-pole has been originated. Figure 4. if we join the two ends of the rope by a third pole and we add 4 more strings as in fig.c). we obtain the stable and self-sufficient “Symplex”. 4. perfectly stable. Kenner’s derivation of the “Simplex” Illustration drawn by the author.01 Tensegrity Behavior”. Definitions and Basic Principles In addition to this latter explanation and to Snelson’s description. 4. 4.b). graphically shown in fig. 4. we push the other two corners as in fig. Beginning with a kite-like module (fig. and we know its basic design. we add the third pole between these two points and tie its ends to the corners of the kite lying on the Author’s derivation of the “Simplex” Illustration drawn by the author. Equilibrium Analysis. 4. Once it has been described how a very simple example of tensegrity can be set up. it is less difficult to have an idea of the 48 .a). ground.d). As pointed out above.Tensegrity Structures and their Application to Architecture Chapter 4.11. 4. we can remove the string between them (fig.11.11.c. Finally. it is necessary to separate the two struts. when we fix two of its corners to the ground. in order to consider this configuration as tensegrity.11. So.4. the author dares to add another method to create the “Elementary Equilibrium”. which are in contact at their middle point.11. 4.4. and fix this situation by attaching two tendons to the ground (fig. To understand the self-equilibrating behaviour of continuous tensiondiscontinuous compression systems. Each strut is acted upon by the tension of the cords. 2 of Emmerich´s patent in Appendix B). 1989). because otherwise the rod would be affected by a bending moment and would not be in equilibrium.12 serves as an illustration of the static forces involved in this kind of analysis. 49 .1 and shown in fig. and it is usually necessary to use computer programs to accomplish this task. added to the relatively small weight of each component. Illustration drawn by the author should have at least three cables attached to the node (the conditions of stability of posts supported by only two cables was explained in paragraph 3. in each end of the strut we Set of forces acting on a strut. there is a three-dimensional equilibrium of tensions and compressions at each node.Tensegrity Structures and their Application to Architecture Chapter 4. The resultant of each triad of forces at each node.3. Definitions and Basic Principles major principles that govern it. This is also remarked by Snelson: “I know I need a minimum of three wires on any end of any stick” (Snelson and Von Baeyer. 1993). i. it is necessary to develop a static analysis of the tension and compression forces acting on each node (Schodeck. As it is a three-dimensional system. Each vertex must be in equilibrium in order to provide the whole structure with stability.e. has to be in line with the axis of the strut. Sometimes a mechanical study can be very complex because of the geometry and number of elements of the structure. The figure 4. A particularity of tensegrity structures is that the forces acting on them are visible in a sense.13). Where there is a strut. As a consequence. In other words. after thirty years. Clerk Maxwell showed that b bars assembled into a frame. Snelson affirms about his sculptures: “I am showing you. tension and compression. would be simply stiff if b = 3j. However. According to Calladine (1978). which are attached to the ends of two struts 3 and influenced by. He also predicted that their stiffness will "be of a low order" and permit at least one state of "self-stress" in the frame.Tensegrity Structures and their Application to Architecture Chapter 4. in a tensegrity structure the two types of forces in essence. 1977). at least. are completely separated and you can see them in their pure state. some tensegrity structures have fewer struts than are needed to satisfy Maxwell's rule. but it is not very usual. rediscovered by Maxwell. what structural space really looks like” (Schneider. Definitions and Basic Principles The same reasoning could be applied to the wires (fig. For instance. having j joints. and where there is 3 Sometimes they are attached to a node responsible for joining a set of cables. but are actually rigid structures. Illustration drawn by the author It should be noted that the equilibrium conditions of the continuous tension-discontinuous compression systems were already anticipated by Möbius and. 4. which is usually a pretensioning force. the other two cables in each node.6. 50 . there is pure compression. for the very first time. particular tension. each string is in equilibrium if it is put under a Set of forces acting on a cable. and are not "mechanisms" as it could be expected. 51 . they will be described. chapter 3.1.Tensegrity Structures and their Application to Architecture Chapter 4. The precise and detailed configuration of the “floated compression” structures. or if preferred. Features. (Kenner. 1976). is nullified (Perlberg. Properties: They are very lightweight in comparison to other structures with similar resistance. although new tendons can be added to consolidate the structure. make it possible to accept the assumption that they have very special characteristics. they have a high resistance in comparison to other structures with similar weight. Definitions and Basic Principles a cable. Wang (2003) states that this characteristic is not inherent. In the following pages. 4. but members “at the high tide” of a compressional aspect. especially to Motro (2003). The systems are stable in any position. Tension and compression always coexist. basis of the conventional architecture. They have no redundant parts. it is more suitable to refer the reader to the bibliography. so they don’t need to be anchored or leaned on any surface. They don’t depend on gravity due to their self-stability.5. Because of the complexity of the subject. It is not the aim of the present work to explain in depth the extended laws that govern the finite and infinitesimal mechanisms of tensegrity. as for example tensegrity grids are heavier than conventional structural grids. with their main advantages and disadvantages. In contrast. there is pure tension 4. 1977) 4 Fuller and Edmonson argue that there is neither pure tension nor pure compression members. The force of gravity.5. 4. Tensegrity Structures and their Application to Architecture Chapter 4. Pugh. The degree of tension of the pre-stressed components is proportional to the amount of space that they occupy (Muller. 1976). Using the analogy of the balloon. The compression is located in specific and short lines of action. This means that they exist as right and left-handed mirror pairs. 4. If the self-stressing is higher in a tensegrity system. grids or conglomerates made of the same or different figures. 1976. The majority of tensegrity systems are enantiomorphic. Snelson 2004). 1971). For an illustration. the tension forces in the skin are greater and it is harder to deform it (Pugh. As the components in compression are discontinuous. Elemental tensegrity modules can be joined in order to create masts. if a balloon is more inflated. “dextrorse” and “sinistrorse” respectively (Kenner. 1976. so they are not subject to high buckling loads. see fig. its load-bearing capacity is higher too. 52 . Definitions and Basic Principles Right-handed and Left-handed Simplex Illustration drawn by the author. they only work locally.14. 75. the tendons stretch “Expanded octahedron” or “Icosahedron” Illustration drawn by the author. They have the ability of respond as a whole. so the whole system has the capacity to multiply the elasticity of the tendon by 600 times. in the expanded octahedron (fig. 0. cited in Flavin. (Vesna 2000) Due to the previous characteristic. 1996) They have the property of synergy where the behaviour of the whole systems is not predicted by the behaviour of any of their components taken separately. (Fuller. (Kenner.15). 82) The resilience (flexibility) or stiffness of the structure depends on the materials employed. and by their method of assembly. but experiences no torque?” (Fuller. so local stresses are transmitted uniformly and absorbed throughout the structure. Definitions and Basic Principles Due to this discontinuity in compression. Levin. they are very sensitive to vibrations under dynamic loads. For example. Elasticity inherent multiplication to them: is When separating two struts by a certain distance. the stretching of the tendons is much less than this amount. if the struts are separated by 1%. that uses compression and tension. Tensegrity is the answer to the question: "What's the minimal structure that can support a weight and oppose horizontal forces.00166% (600 times less!). 4. 1976). they don’t suffer torque at all. They can be very flexible or very rigid and quite strong. For an 53 .Tensegrity Structures and their Application to Architecture Chapter 4. Snelson.5. so the members of a tensegrity structure are precisely positioned to best withstand stress. The fact that these structures vibrate readily means that they are transferring loads very rapidly. In Vesna’s and Fuller´s words (2000). so the loads cannot become local. (Kenner. and buckling is very rare due to the short length of their components in compression. so there are no points of local weakness. experience a rotation around this axe (Kenner. 1976. 1976. Perhaps. under axial load. The direction of this rotation depends on the handedness of the system (enantiomorphic characteristic explained above). The response to the loads is non linear. 1997). Some tensegrities. Smaili. 2003. Wang 2003). the deflection of the expanded octahedron modelled by cables and beams finite elements (Mijuca.Tensegrity Structures and their Application to Architecture Chapter 4. They don’t suffer any kind of torque or torsion. tensegrity demonstrates ephemeralisation. or the capability of doing more with less. it is possible to use materials in a very economical way. 4. can be seen in Appendix E. ‘ethereal’ is more adequate than ‘ephemeral’. 2004). This is very useful in 54 . but their stiffness increases rapidly as the load is higher. 1998). 1976) Due to the ability to respond as a whole. Tensional forces naturally transmit themselves over the shortest distance between two points. They are more flexible under light loads. like a suspension bridge (Kenner.2. Definitions and Basic Principles example. offering a maximum amount of strength for a given amount of building material (Ingber. Advantages: The multidirectional tension network encloses fortuitous stresses where they take place. (Holland) Illustration taken from Snelson (2004) 55 . For large tensegrity constructions. Thus. Easy-K Installation Snelson in 1970. This conception implies the option of the endless extension of the assembled piece (Muller. 1971).16. 2003). the process would be relatively easy to carry out. which are stable by themselves. An example is the illustration of fig. since the structure is self-scaffolding (Whelan. Arnhem. 1981). Further explanations will be provided in the next chapter. The spatial definition of individual tensegrity modules. they would be desirable in areas where earthquakes are a problem. permits an exceptional capacity to create systems by joining them together.Tensegrity Structures and their Application to Architecture Chapter 4. Definitions and Basic Principles terms of absorption of shocks and seismic vibrations (Smaili. 4. 1987. after experimental research. very economical. Pugh (1976) estimated. Kenner (1976) proposed shell analysis as the best way. Spherical and domical structures are complex. although this is a bit distant from structural reality. In foldable systems. Definitions and Basic Principles Burkhardt (1994-2004) sustains that the construction of towers. as compared with conventional. domes. pp. the arc length of a strut decreases). etc.Tensegrity Structures and their Application to Architecture Chapter 4. geometrically rigid structures” (Hanaor. at the same time. There was a lack of design and analysis techniques for these structures. The same author stated. 2002). (Burkhardt. 45) The fabrication complexity is also a barrier for developing the floating compression structures. For 56 . 4. Skelton and Sultan have explored the use of tensegrity structures as sensors and actuators (Tibert. As some designs become larger (thus. bridges.3. which can lead to problems in production. the struts start running into each other. the design and calculation of these figures was relatively simple. employing tensegrity principles will make them highly resilient and. 2004) The inadequate design tools have been a limitation until now. The kinematic indeterminacy of tensegrities is sometimes an advantage. Disadvantages: According to Hanaor (1997) tensegrity arrangements need to solve the problem of bar congestion. incorrectly. that as the connections between struts and tendons are pinned joints. Consequently. In spite of this evidence. “relatively high deflections and low material efficiency.5. only a small quantity of energy is needed to change their configuration because the shape changes with the equilibrium of the structure. works well enough to design and calculate tensegrities. 57 . 1993). the pre-stress forces should be high enough. Burkhardt has been working on a computer program that. 5 See Appendix D: Personal correspondence with Robert W. seemingly.Tensegrity Structures and their Application to Architecture Chapter 4. has been developed by René Motro and his group at the Laboratoire de Génie Civil in Montpellier. Definitions and Basic Principles the past ten years. which could be difficult in larger-size constructions (Schodeck. “Tensegrité 2000”. In order to support critical loads.5 And recently new software. Burkhardt. classification and assemblies .Tensegrity Structures and their Application to Architecture Chapter 5 Typologies. . Other known systems are not excluded from these circumstances. “twist element”.. As a result. “(3. PM) tensegrity structures consisting of N compression members (i. . .9. it is essential to describe.e. Typologies. at same time. Williamson and Whitehouse (2000) employ just numbers. “3 struts T-prim”. “3 struts single layer”.. At present. The former considers a general class of (N. based on the definition of the geometry by means of basic and systematic rules. classification and assemblies When developing a new field of knowledge. colons and comas in brackets. P1. For instance.1. the author has come across a couple. which is clear and systematic. the simplest tensegrity system has already been denominated as “simplex”.Tensegrity Structures and their Application to Architecture Chapter 5. 5. struts) and S tensile members 58 . 9 tendons”. would give enough information about them. but at present there are some discrepancies among the authors and specialists.2. Typologies.1)” and so on. while Motro (2003) uses numbers and letters in intuitive way. classification and assemblies Chapter 5. some authors have tried to create a definitive nomenclature. “3 struts. Nomenclature It has been stated throughout previous chapters that the definitions for the highlighted examples are not categorised in a standard way. P2. S. “elementary equilibrium”. which are very logical and similar. This would permit the categorisation of floating compression systems and. Tensegrity systems are not an exception. denominate and categorize in order to develop a complete and extensible classification of the subject in question. Typologies. cables).e. the last example. a variant of these systems.2) would be (3. classification and assemblies (i.2. C = Cables or tensile components.3). The latter organizes them following an alphanumeric code. the nomenclature is not very useful when speaking about the most common figures. explaining each term with the initials followed by the number of items. The structure has M stages with PM struts per stage.g. In any case. The author finds several advantages of the latter in comparison to the former. thus sometimes it is necessary to define the number of nodes explicitly.Tensegrity Structures and their Application to Architecture Chapter 5. As an illustration. the “elementary equilibrium” (fig. S = Struts or compressed components. uses nodes where only cables are jointed. In conclusion. SS = Spherical system (homeomorphic to a sphere) if this is the case For instance. proposed by Kono and other experts.2. in some cases it is essential to know whether the system is regular and inscribed in a sphere (e.2. being listed: n = Nodes. 5. Moreover.b & 2. in order to truncate it to transform it into a dome) or is irregular and non spherical.b & 2. 5. would be expressed as “n6-S3-C9-R-SS”. R = Regular system or I = Irregular system depending on the case. the author will adopt Motro’s nomenclature to describe and denominate the cited tensegrities. Finally.9. the “simplex” (fig.2). Classification It is probable that the first classification of tensegrities was carried out by 59 . Sometimes it is difficult to make a distinction between the different stages that compose a continuous tension-discontinuous compression system. 5. as is explained in fig. he described the three basic patterns that can be used to configure spherical or cylindrical tensegrity structure: Diamond pattern. but it is still very helpful. It is true that he did it almost exclusively related to polyhedra. 5. but not in an in-depth manner. Some of the following points are based on chapter 4 of Motro’s book “Tensegrity. 5. Spherical systems These systems are homeomorphic to a sphere.1. Finally. The former is self-stable since there is a pre-existing tensile stress or isometric tension. In that section some grids. Armstrong. but so far the author has only come across a general division into two broad structural classes: prestressed and geodesic tensegrities. etc. Typologies. masts and domes were described. This classification was based on the relative position of the struts of the figures.2. e. Then. Anthony Pugh (1976) was the first person to show a thorough catalogue of tensegrity systems. classification and assemblies Fuller and his collaborators. the latter finds equilibrium as a result of the triangulation of its structural members. To achieve a clearer classification it could be useful to have an account of some other configurations and geometries. 2003a. he related the way of joining systems together and the construction of larger figures.1. on the complexity of the compressed components (single elements or groups of struts). 2004).Tensegrity Structures and their Application to Architecture Chapter 5. he described the simplest figures superficially (both 2D and 3D).g. Structural Systems for the Future” (2003). orientated along geodesic lines or minimal spherical paths (Ingber. on the number of layers or stages. depending on the relative position of their tendons (passing through their centres or not). First. Circuit pattern and Zigzag pattern. all cables can be mapped on a sphere without intersections between them and all the struts are inside 60 . 1. 5.2. They are some of the most common floating compression figures. Rhombic configuration.2.Tensegrity Structures and their Application to Architecture Chapter 5.1.1.a). 5. Typologies. If a relative rotation is intoduced between the upper and lower polygons. a tensegrity prism is obtained (fig. The name of these types of figures responds to the way that they are constructed.a). folded following the diagonal (fig. fitting the following classification: 5. 61 .2.b). Classification of Spherical Systems Illustration drawn by the author after Pugh’s drawings. 5. Tensegrity prisms (T-Prisms) are included in this section. This corresponds to the Diamond Pattern established by Pugh. classification and assemblies this cable net creating a spherical cell. Each strut of a rhombus system represents the longest diagonal of a rhombus formed by four other cables. T-Prisms or Prismatic tensegrities are generated from a straight prism where the cables are horizontal or vertical and the struts are diagonal between the vertices of the two different levels (fig. a triangular prim a=30º (cf. 5.3) The most known exemplars of the rhombic configuration are the “simplex” and the “expanded octahedron” (also so-called “icosahedric tensegrity”). in a square configuration a=45º. for. Generation of T-Prims Illustration drawn by the author after Pugh’s drawings. paragraph 5. the less stable and more flexible is the T-Prism.2. depends on the number of struts (n=number of edges of the polygon) and is given by the formula demonstrated by Roger S. Typologies. the higher the number of struts. Tobie in 1967: a = 90º . or “twist angle”. classification and assemblies Kenner (1976) states that the rotation angle (a).180º/n For instance. 62 . in a hexagon a=60º. and so on.2). fig. Each prismatic tensegrity system comprises of a single layer of struts. in a pentagon a=54.Tensegrity Structures and their Application to Architecture Chapter 5. In any case and according to Pugh (1976). but other figures can be built by adding more stages and thus creating a kind of cylindrical rhombic system (cf. and then sketching the struts and tendons onto the grid (Armstrong.a. 5. snub cuboctahedron. snub icosidodecahedron. closing the rhombus generated by the struts and cables of the diamond pattern tensegrities (figs.2. circuit systems are also able to generate geodesic tensegrity spheres or domes if the breakdown frequency is a multiple of two. cube. As can be seen in the fig. 5. due to its three pairs of struts.4. the biggest tensegrity polyhedron described in that catalogue. 2004). e. and the latter (n12-S6-C24-R-SS) is a typical two layer tensegrity system.b & 5. etc. Anthony Pugh gave a complete list of figures built following this method. 5. classification and assemblies The first one (n6-S3-C9-R-SS) has been shown in preceding graphics (figs. figs. defining a grid of triangles for each face.2. icosahedron.15 and Appendix E). parallel two by two (cf. 4. for instance.1. with a very strong symmetrical component.1. 5.Tensegrity Structures and their Application to Architecture Chapter 5. Typologies. The procedure consists of dividing the polyhedra in question following the rules of geodesic spheres. was designed using 672 struts and 1344 tendons. Cuboctahedron Illustration drawn by the author 63 . 5. Moreover. “Circuit” configuration. In this second class.3. 4.c).2).4. Several regular and semi- regular polyhedra can be built related to this class.2 & 2. the compressed components are conformed by circuits of struts.1. the cuboctahedron is composed of four circuits of three struts (every circuit interweaving with each other) and the cables defining the edges of the polyhedron.g. the eight-frequency truncated tetrahedron. As Motro (2003) remarked.d). a circuit system is more rigid than a rhombic one with the same number of struts. it is not always possible to attain a balanced geometry and.4. 5. “Zigzag” configuration or “Type Z”. 5. This is understandable since the former evolves from the latter.4.2. if some of the cables are changed in such a way that they form a ‘Z’ of three non aligned tendons (fig. but it becomes more compact and there is contact between its compressed elements. the “zigzag” configuration is obtained.a) is changed and the cables are fixed following the zigzag pattern. Typologies. 5. the result is a “truncated tetrahedron” (fig. When using a rhombic system as a basis. therefore.Tensegrity Structures and their Application to Architecture Chapter 5. if the configuration of an “expanded octahedron” (fig. It is important to remark that the substitution of the cables must be coherent in order to preserve the stability of the system. classification and assemblies According to Pugh’s experience.1.1. sometimes the figures do not have a perfect definition of the polyhedron in question. Due to the orientation of the struts that converge in each 64 . For instance.b) Expanded octahedron Truncated Tetrahedron Illustration drawn by the author.3. 5. 5. except that the breakdown frequency has to be a multiple of 3. taking as a basis one of the rhombic system. additional cables can be inserted into the original system to obtain the perfect geometry. it can be appreciated that a certain distortion of the regular polygons can arise. a cylindrical mast is obtained. Typologies.b. Another possibility could be proposed by inserting a small spherical node instead of the central strut. Star systems Even though they are also spherical cells. as is represented in fig. Deployed Single Layer System (n8S4C12) Double Layers System (n16S8C32) Illustration drawn by the author. a star system is created. Cylindrical systems There is also a variation of the rhombic configuration. they are considered as a derivation of the preceding class. obtained by adding other layers of struts to the initial layer. and subsequently closed all around itself again. In any case. 65 .Tensegrity Structures and their Application to Architecture Chapter 5.2.5. if a vertical strut is inserted in the centre following the main axis of symmetry and linked to the rest of the cables by means of tendons.3. classification and assemblies face.2. the process is similar to that of circuit systems. If the aim is to create a geodesic figure.a shows the deployed bunch of bars and tendons of a four-strut rhombic cell. 5. the resulting tower will be more or less tall. 5.2. 5. Fig. 5. If a second line is added. For example. Depending on the number of layers. In the following sections some possibilities will be contemplated.1. a high percentage of Kenneth Snelson’s sculptures could be regarded as irregular structures since they are not governed by any rule defined in this or other studies. For instance. Obviously.3. 5.3.Tensegrity Structures and their Application to Architecture Chapter 5. Vertical Masts (horizontal beams) One-dimensional systems can be generated by adding the different modules following an axis that rigidly dictates the geometry. a planar structure is created with advanced characteristics in relation to one-dimensional beams. this performance is strongly dependent on the way the different modules are joined.3.2. Assemblies More complex systems may be achieved by joining the elementary cells described above. Several straight towers have been built over the years. while contrarily not many floating compression beams have been regarded in such a way. although the list is not exhaustive. classification and assemblies 5. Irregular systems In this section numerous figures are included that do not fit into previous classifications. although the cable-domes illustrated in previous chapters have improved this behaviour.2. 5. 66 . Grids By assembling tensegrity cells in two dimensions.4. Obviously. 5. as will be explained in the next chapter. Typologies. the reason is the lack of resistance of these structures to bending moment. producing unique configurations too (hexagonal T-Prism excepting) After these considerations. it was stated that these grids had an overall good agreement in structural response and the major advantages of simple joints. 67 . concluding for example that triangular grids are more rigid than square grids. Type Ib . several grids have been constructed attempting to avoid the lack of stiffness of the simple agglomeration of T-Prisms. classification and assemblies This is the reason why Ariel Hanaor’s works are a reference and guide for specialists in tensegrity structures. more general conclusions were obtained. However. led by René Motro. or that the efficiency of material utilisation was similar in the three grids.Type I applied to even-sided polygons. Since 1998. Hanaor (1987) basically defined three types of connections: Type I – Modules share only nodes Type Ia . Further investigation was required. producing symmetric configurations.Type I applied to odd-sided polygons (right handed and left-handed modules). several geometric studies and load tests were carried out by him and other collaborators. he started in the 1980s by studying the geometric configuration of double-layer tensegrity grids (DLTGs) by means of the Tprisms defined above (which generated planar surfaces) and T-pyramids (for curved surfaces). it has been accomplished since then and the main research has been carried out at the Laboratoire de Génie Civil in Montpellier. Type II – Modules also share portions of the base polygons. They also recognised that those structures suffered large deflections and had low material efficiency compared with conventional rigid structures.Tensegrity Structures and their Application to Architecture Chapter 5. At that stage. producing unique configurations. Typologies. topology. was built at the end of 2000. classification and assemblies Vinicius Raducanu. Bi-dimensional double-layer grid Laboratoire de Mécanique et Génie Civil. applied to different geometries: bi-. This steel structure was constructed according the Eurocode3 building standard for a 160daN/m2 external downward load. etc. Typologies. Montpellier Illustration drawn by the author 68 . certainly the simplest one (an extension of the module shown in fig. the “V expander” (fig. It therefore proved the feasibility of this kind of grid that had a surprising rigidity. V expander Illustration taken from Motro and Bernard (2002) A prototype of a bi-directional layout.7). Raducanu and Motro (2001) patented the system.and quadri-directional tensegrity grids. who worked on the Tensarch Project at Montpellier. covering 82m2 and weighting 900 kg. The group used the same mechanical principle. based his research on points of view such as analogy. 5. tri.5. presented in Appendix C.6). geometry. As a result.Tensegrity Structures and their Application to Architecture Chapter 5. as it would take many chapters to explain the advantages. so they have a three-dimensional shape. the field of application of these systems has been extended noticeably. As a conclusion. solid. because the compressed components are not the single struts but the frames and chains of bars composed in between the two layers. coverings. At the moment they have not been applied in any definite field. In fact.4. 69 .Tensegrity Structures and their Application to Architecture Chapter 5.3. classification and assemblies In every grid the extended definition given in chapter 4 is respected. possibilities and potentials of deployable tensegrities. therefore. much of the future of floating compression relies on this significant characteristic. Motro and his collaborators stated that tensegrity grids are feasible. etc. Conglomerations Finally. they proposed their application to walls (allowing the insertion of integrated architectural systems). 5. They are tensegrity solids without any predominant direction.3. discrete pneumatic structures and rigid enough depending on their function. 5. In fact. Deployable structures Without any doubt. Typologies. roofs. adaptive. The author considers that it is more suitable to refer to an extended bibliography (Appendix I) which mentions the main specialists that are dealing with the subject. It is not the aim of this work to deal with this subject in depth. it is necessary to mention these systems although they have been barely studied. folding tensegrity structures have been one of the main research topics for the past ten years. e. Skelton. Hanaor. 5. 1997.8).Y.Tensegrity Structures and their Application to Architecture Chapter 5. rotation translation and shear. Knight et al. A. Indeed. paragraph 6. Cancelling the prestress: Local suppression and global suppression Maintaining the prestress: Rotation-rotation.E. Folding Mechanisms. Bouderbala). Stern. V.6). Ali El Smaili. Furuya and H. 2000.E. classification and assemblies Pellegrino and Tibert’s works (1991. Smaili (2003).E.. the research of René Motro and his laboratory is extremely important (especially A. Raducanu and M. Illustration taken from Smaili (2003) 70 .2.g. Once again. R. Typologies. making an almost exhaustive revision of all the options that these systems offer and of all their potentials (fig. 2003). some of the results have been patented and are referred to in chapter 6. Other experts have done recent research in foldable floating compression structures. Pak. Appendix C and in the bibliography (Skelton. 2003) have been very useful from the point of view of deployable masts and other structures disposed for the conquest of Space (cf. H. . Tensegrity Structures and their Application to Architecture Chapter 6 Applications and proposals . . of the struts in compression. simple elements or more complex structures. this point is controversial since even in the group of the “real” tensegrities there are “pure” and “non pure” floating compression systems. Schodek (1993). Applications and proposals. software and infrastructures. maybe the most important figure in the topic. or not. this chapter will deal with the task of showing the applications of this material with exemplars. First. the most important examples of works already built will be presented.Tensegrity Structures and their Application to Architecture Chapter 6. Introduction. and then he extrapolates this argument to other structures (see last paragraph of Appendix A).1. both the “real” and the “false” tensegrity structures. this does not mean that it has any special structural worth. He refers to Mario Salvadori’s opinion about tensegrity Vs conventional beams. the author will present some of his own proposals to apply the continuous tension-discontinuous compression principle to architecture. due to the limitations of time. In any case. As Snelson wrote to the author: 71 . The sculptor Kenneth Snelson. Applications and proposals Chapter 6. It might be interesting to note that some professionals. for instance Daniel L. according to the definitions of chapter 4. affirm that even though a tensegrity sculpture is a fascinating spatial exploration. budget. As soon as these examples are shown. is really convinced about the unfeasibility of applying these structures to any architectural or engineering construction. these designs are not as developed and defined as professional works. depending on the contact. It should be recalled that. Once the basic fundaments and basic systems have been described. 6. As an illustration. in 1899. it would not change the fact that numerous people are working in the subject.” 2 Certainly. articles and papers are being circulated in different journals and conferences. a “dim future for the automobile”. 3 Aug 2004. the Literary Digest predicts.Tensegrity Structures and their Application to Architecture Chapter 6. it is true that some of the Fuller’s announcements and propositions seemed like humbugs. and this is a great truth. “Tensegrity is now applicable to architecture as an established structural system. like the possibility “to bridge the Grand Canyon with tensegrity” (ibid) or to cover a whole city with a geodesic dome. None have shown there is the slightest structural advantage in its use for such purposes. and according to the California Energy Commission (CEC. claiming it will never “come into as common use as the bicycle”. (See Appendix D) 72 . the most significant examples will be recalled. Jules Verne said “Anything one man can imagine. especially in his case. 2003). (See Appendix D) Kenneth Snelson: excerpt from an e-mail to the author. in the preface of Motro’s last book (2003). The author does not consider himself authorised to make severe judgements. Even if Snelson’s opinions were true. Applications and proposals “It is my belief based on long experience and making endless numbers of tensegrity structures of all shapes and sizes that the principle in itself is impractical for building buildings. Kawaguchi. As you know many architects and engineers have worked toward that end and still do. President of the IASS (International Association for Shell and Spatial Structures).” 1 “[They] are also very flexible and I know of no instance where they've been put to use for any practical purpose. In the following paragraphs. while it can be applied to other fields as well” concluded M. However. 1 2 Kenneth Snelson: excerpt from an e-mail to the author. and more than a few publications. it is not the author’s intention to despise or disdain any suggestion. Fifty years of it now. other men can make real”. 20 Jul 2004. this concept can be applied to relatively small spans because if this is increased the curvature is also smaller and the components come into contact. where the author shows possible truncations of a tensegrity truncated icosahedron). 6. It is a really significant fact that he did not mention any tensegrity dome. Figure 2.2. (chapter 2) has shown one of the first geodesic tensegrity domes.1.6. 6.2. Design and Construction of Braced Domes” edited by Z. Applications and proposals 6. Several authors have proposed different kinds of domes following the continuous tension-discontinuous compression fundaments.1. Domes. Snelson has a definite opinion about this configuration. Actual examples. one of the most important books about domes is “Analysis. attributable to the facility to obtain a dome from a sphere or spherical polyhedron (see Appendix G. This point serves to illustrate the degree of recognition of these structural constructions.2.Tensegrity Structures and their Application to Architecture Chapter 6. as in one of Apex of Truncated Icosahedron Model made by the author 73 . although he mentioned Fuller’s patents as well as Pugh and Kenner’s studies. by Buckminster Fuller in 1953. Different proposals for domes Most of the works and studies in tensegrity have been done in relation to spherical or polyhedral configurations. Perhaps.1. According to Hanaor (1987).S Makowski (1984). but can be resolved by adding more wires between them and connecting other apexes of the dome.. Some of the domes obtained from truncated polyhedra. but especially tensegrities. (See Appendix D) 74 . any structure. This situation is not very convenient. and in other e-mail he cites it again saying that.7.2. (See Appendix D) Kenneth Snelson: excerpt from an e-mail to the author. 6. It is this detail that Snelson pointed out and that the author took into account for the design of some domes in section 6. “His tensegrity domes (. with the absence of triangulation. which refers to a construction puzzle that will be referred to in section 6.2. 23 Aug 2004. he states that this truncated sphere is “as soft as a marshmallow”. Applications and proposals his letters to the author3. each apex is formed by five struts creating a pentagon of tendons.1.3. each of the arches is formed by these towers Dome projected by Snelson Illustration taken from Snelson (1965) 3 4 Kenneth Snelson: excerpt from an e-mail to the author. A different kind of dome using floating compression was the one shown by Kenneth Snelson in his patent of 1965. 3 Aug 2004. it could barely hold itself up. but on the x-shaped towers that he first discovered.Tensegrity Structures and their Application to Architecture Chapter 6. have the connections of the struts forming a polygon different of the triangle.) are as shaky and floppy as a Tensegritoy” 4. In fact. In the example of fig. loses an important factor for stability. It was not based on polyhedra.. due to the absence of triangulation. Applications and proposals bending adequately. it was not very successful. fig. 6. patented by Geiger ten years later. Apart from his extensive list of models and configurations. mutually strutted by straining pieces.2. The subject will not be developed here as it was the main point of section 3. 6. except that it was not fixed to a rigid and prestressed 75 . since it is a grid with one or several layers. Obviously. J. but it was the pioneer in the “wire wheel domes or cable domes. fig.4). as it was abandoned and never used for practical purposes. Oren Vilnay established a new concept. Detail of Fuller’s patent Illustration taken from Fuller (1954) In 1977. Motro 2003). Anthony Pugh (1976) also proposed some interesting models. large spans (Hanaor 1987.3) where the tendons had been substituted by a plastic skin that took the role of the continuous tension component (cf. one of them a relevant application. The structure was also very similar to the cable domes.Tensegrity Structures and their Application to Architecture Chapter 6. This was a geodesic tensegrity dome inspired by Fuller’s patent (1954) (cf. using regular planes nets (single layer) which can be employed to produce curved surfaces with small curvature and hence. which can also induce the buckling of the bars. Some years later. Stanley Black (1972) proposed a new configuration based on tensegrity trusses fixed to a peripheral rim.3. However. a double network of prestressed cables. 6. The innovation cannot be considered a dome. Miodrag Nesterovic (1987) published the project for a “Metallic Integrally Tensioned (Tensegrity) Cupola”. as is explained in fig.3. the author considers that these systems are not as spherical as desirable. due to the extreme length of their struts. Finally. Apart from his other work cited in the Bibliography.2.2. As it was not self-stable. which are illustrated in the Appendix C. as well as additional wires in order to obtain the crucial triangulation that contributes to the stability of the structures. Applications and proposals concrete ring.Tensegrity Structures and their Application to Architecture Chapter 6. At the same time. it is considered a “false tensegrity” like the rest of the domes. He considers double-layer domes. resulting in more sophisticated bi-dimensional assemblies of cells or double layer floating compression cupolas. In any case. some other authors (Huegy. 6. but to two pin connections at different levels. where an outer and an inner layer of cables are inter-connected by bars. the mechanical behaviour of these curved grids was not satisfactory enough. When generating a single or double curvature from them. Burkhardt’s research on this subject. which is constantly being revised. who defined a dome made of tensegrity modules and built on a node-on-cable principle (Hanaor. They can be perfectly judged as tensegrities.1. 1996) have patented other types of tensegrity domes. It is important to mention Robert W. due to the internal prestress of 76 . Those first studies were further developed in the Laboratoire de Génie Civil de l’Université de Montpellier. were propounded by René Motro (1987). Wesley. he started to develop “A Technology for Designing Tensegrity Domes and Spheres” (1999-2004). referred to in the previous chapter. the member force analysis had been carried out by Burkhardt taking into account the endogenous factors. the first double layer tensegrity grids. it is possible to obtain a space structure similar to a dome. although their components in compression are in contact with each other. This was not the case for a proposal by Ariel Hanaor. 1990. Calculation of the load response In the latter study. 1987). Advantages and applications for domes Burkhardt (1999-2004) summarises the main advantages of tensegrity domes. due to external loads. The analysis of the response to exogenous forces is achieved by adding an independent force vector to the forces present at a hub. Motro.2.3. he computes the endogenous state by applying the Principle of Minimum Potential Energy. This is an important contemplation. in other words. 6. (cf. etc.Tensegrity Structures and their Application to Architecture Chapter 6. since the interferences can change the behaviour under loads. fig. etc. Once the total geometry is defined. improved rigidity. It can be obtained by minimizing a weighted sum of the second powers of the lengths of the members. 2. In this study.9 of chapter 2) By means of a mathematical programme procedure. as well as generate bending moments in the struts. Smaili. and the exogenous factors. 2003) have also thought about this point for the considerations in folding tensegrity systems. where positive and negative weights are used for tendon and strut lengths respectively. he first calculates the geometrical design of the structure. Other authors (Le Saux at al. high lightness. the distances separating the different elements of the structure. gravity. perhaps inspired by some of other Fuller’s ideas: 77 . foundations. for instance: use of equal-length struts and simple joints.1. The following are some of the possible applications that he points out. Applications and proposals the structure itself. the clearances in tensegrities are also considered. extreme resilience. 2003. 1999. so the new configuration is derived by solving a system of equations rather than by solving the problem as before. fig. have been made by Shelter Systems (1996-2001) and Daniel Ng (2001-2004). delicate or rugged terrains). in congested or dangerous areas.4). 6.5). Applications and proposals Superstructures for embedded substructures in order to escape terrestrial confines where this is convenient (e. 78 . following the tensile skin domes projected by Pugh (see fig 6. similar to the Snowdon Aviary in London. agricultural. Some similar proposals. Cedric Price and Frank Newby (cf. The author proposes a comparable structure in the following section. Economic large-scale protection for storage. Exclusion or containment of flying animals or other objects. by Tony Armstrong-Jones (Lord Snowdon). although some of their constructions are not pure tensegrity. electrical or electromagnetic shielding or other delicate sites.g. construction. Tensile skin dome Illustration from Shelter Systems (1996-2001) Snowdon Aviary Illustration taken from Ford (2004) Frames over cities for environmental control. archaeological. energy transformation and food production.Tensegrity Structures and their Application to Architecture Chapter 6. flood plains or irregular. urban areas. Refugee or hiking shelters. or topiary framework. Supports to hold sunscreen protection for vulnerable amphibians. trellis. The author suggests another possibility: Portable and foldable structures: Due to the particular characteristics of tensegrity. etc. Some other interventions in a smaller scale. trade shows. domes using this principle could be very useful in: 79 . indoor/outdoor pavilions for expositions. Watersheds to keep rain water from percolating through contaminated soils into groundwater. sun-shielding for Martian colonies or spherical superstructures for space stations. while Wang (2003) estimates very opportunely the necessity of dynamic analysis to explore the mechanical properties further. Kenner (1976) pointed out the fact that both. some other possibilities related to lunar stations were suggested by Literati (2001). Micro-meteorite protection. As was mentioned by Pugh (1976) and Fuller (1975b). perhaps temporarily during in-situ remediation. Nevertheless. as frames for hanging plants or other objects to dry.Tensegrity Structures and their Application to Architecture Chapter 6. pergola. Applications and proposals Earthquake-resistant buildings. shelters. frame and skin should have analogous flexibility. fairs. Following Frei Otto’s conceptions. which will be dealt with in detail below. etc. low-environmental-impact shells for musical performances. Thus. entrances to events. bridges. these structures are extremely resilient and testing would very likely show they could withstand large structural shocks like earthquakes. they would likely be desirable in areas where earthquakes are a problem. In addition. A more unadventurous catalogue of applications can be found in Ariel Hanaor's article “Tensegrity: Theory and Application” written in 1997. Medwadowski.Tensegrity Structures and their Application to Architecture Chapter 6. further research must be done in order to have a better understanding of their load resistance and to improve the techniques of folding. plication and transport. Stefan J. In Motro’s last book (2003).2. a number of which are in existence. Towers. 80 . continuous tension-discontinuous compression principles seem to be appropriate for application to domed structures.2. the past president of the IASS. However. until very recently the one notable field of application was the tensegrity dome.” (Preface I) Once the potential of the domes have been related. Applications and proposals Devastated areas (disaster relief) Nomadic people Field hospitals In conclusion. stated: “Apart form the tower. the tower will be the subject of the following paragraphs. 8v Double-Layer Tensegrity Dome Model built and published by Burkhardt (1999-2004) 6. Different proposals for towers Without any doubt. fig. The different arrangements are shown in fig. In his patent.7) with some function.C. under certain circumstances (where large scale towers are constructed) serve as a central shaft for the passage of an elevator suspended by cables from the horizontal beam 57. 81 .1. 6. Needle Tower II (1969) and Penta Tower (2001-03). He had already designed a tensegrity mast the year after the tensegrity principles were discovered. Tetra Tower (1963-2001). fig. Snelson wasn’t as sure as he is at present about the unfeasibility of applying his discoveries to any field in particular. His first proposals were described in 1960 in his patent Needle Tower II Illustration taken from Snelson (2004) of Continuous tension-discontinuous compression structures (Snelson. 4. 6. he has built several masts over the past 40 years: 4-way tower (1963). They are configured as assemblies of the T-prisms that were explained in chapter 5. Applications and proposals 6. the main contributor to the development of tensegrity towers has been the artist Kenneth Snelson. In fact. 4.Tensegrity Structures and their Application to Architecture Chapter 6. 1965).” (pp. E. This mast was shown by Fuller in his book “The Dymaxion World of Buckminster Fuller”. lines 58-63) Following these drawings. Column (1969-81). fig. 6.9 of chapter 4). three of them based on X-shaped modules (cf. In those papers four different columns were designed. as it is quoted: “The passageway formed by the crossed compression members which follows the axis N—N might. he suggested a possible function for the X-shaped tower (cf. Needle Tower (1968).2.11 of chapter 4) and other one based on the “simplex” or three legged structure (cf.7.2. 3) in the 1950s and some obelisks built by Burkhardt (2000-2004). fig.Tensegrity Structures and their Application to Architecture Chapter 6. In any case. the most 82 . the author has not found any more examples of “pure” tensegrity towers. 2. Applications and proposals Continuous tension-discontinuous compression Columns Illustration taken from Snelson (1965) Sneslon It should be pointed out that Snelson proposed another possibility: he intended to build a sculpture as tall as the Eiffel Tower some day. 4. Apart from some other masts erected by Fuller and Emmerich (cf. and one of the reasons is that these sculptures are self-scaffolding (cf. fig. 1981).16). but the obstacle was money (Whelan. Technically it would be moderately easy to construct. the tower consists of six “simplex” or twist modules (8. apart from self-weight and wind. Jörg Schlaich. food for thought” 5. According to M. so the bars of one level enter in contact with the bars of adjacent levels. stated that tensegrity has “no real practical use. his consulting firm Schlaich Bergermann und Partner. Since this construction did not have to support any external load. made of three steel tubes (Ø=273 mm. excerpt from letter received the 4th July 2004.9). led by his son Mike Schlaich. fig. t=12 to 40 mm) and six high-strength steel cables. three of them horizontal (Ø=30 and 50 mm) and three other thicker diagonal wires (Ø=50 and 75 mm) (cf.2. has raised the highest tensegrity tower of the world (62. personal correspondence). defined and analysed by Mike Schlaich and his group in Stuttgart. 5 Personal correspondence. but it was really designed. as was admitted by him in an e-mail to the author (see Appendix D. Tower of Rostock. Applications and proposals relevant example of tensegrity towers. Schlaich (2003). Each of these twist modules are disposed in a similar way to that of the “Needle Tower” (alternating right-handed and left-handed “simplex”). but the difference is that they are rotated 30 degrees. it was decided to use the floating compression principle. despite it being a “false” tensegrity is the Tower of Rostock.3 m height each).2. Marg und Partner. landmark and visual reference of the Rostock fair and the International Garden Exposition (IGA 2003).3 m) The Tower of Rostock (see fig 6. one of the greatest engineers at present.Tensegrity Structures and their Application to Architecture Chapter 6. only fancy sculptures. 83 . 6. However. The original idea corresponded with the architects von Gerkan. 6.2.9) was conceived as a symbol. Schlaich (2003) and Ruiz de Villa (personal correspondence) 84 . Applications and proposals Tensegrity Tower of Rostock Illustrations taken from M.Tensegrity Structures and their Application to Architecture Chapter 6. Tensegrity Structures and their Application to Architecture Chapter 6. as at the joints. Motro would estimate that the Tower of Rostock comprises of a continuous net of wires and three compressed components. while the total security against break-up is the same. because the tower is so light that its own weight can be neglected. Nevertheless. a computer program that served to accomplish the geometric non-linear analysis. thus it was necessary to use third order the theory for large deformations. Finally. and there is more deformation. as was discussed in chapter 5. anchorage and upper needle (a stainless steel needle placed on top of the tower). as proved in Schlaich’s article. from them. the system would be identified as a “true” tensegrity. the dynamic analysis started studying the vibrations of the structure and. employing “Sofistik”. this factor would be enough to consider the structure as “false” tensegrity. each of them made of a chain of six struts. the wind determinates the degree of pre-tensioning. the bars have to support a bigger tension as they are rigidly connected. Thus. Since the three components in compression do not touch each other. If this value is smaller. 25 August 2004 (See Appendix D) 85 . there is less deflection and less movement in the tower. e-mail to the author. the pretensioning was decided to be 1100 kN for the diagonal cables (30% of the tensile strength of the cables). This aspect was important because of the highly pretensioned state of the structure. In contrast. 6 Personal correspondence. The structure was calculated by Arturo Ruiz de Villa 6. This type of structure is only under the action of the wind and its selfprestress. It would be a tensegrity ‘class 2’. because at most two compressive members are connected to any node. an aerodynamic study was developed in order to discern the influence of the wind. Some other details were analysed using a 3D finite element model. Applications and proposals From a very strict point of view. if high pre-tensioning arises. In fact. 000). h=2m) that provides the weight necessary to prevent the tower from ‘blowing in the air’.3. The tower is fixed to a pile cap in concrete (Ø=8m. 5. Applications and proposals With 1100 kN on these cables. light and graceful structures can still be produced. Computer software is available.Tensegrity Structures and their Application to Architecture Chapter 6. At the same time. 2. the maximum displacement on the top of the tower is 850 mm (1200 mm on the top of the needle). It is not a problem for contractors to fabricate and erect floating compression structures with the required precision. for its construction was around 500. It is possible to construct large-scale tensegrity structures of this kind. (Schlaich. the author dares to add to the list some other fields where 86 .2. 2004) 6. which in Schlaich’s opinion is quite an expensive amount due to the absence of application of the tower. due to its lightness. it is anchored to the ground by means of 6 drilled piles of Ø=500 mm. and final price. In this field many practical. Some other applications for tensegrity towers After considering the conclusions derived from the construction of the tower in Rostock. which describes the structural design and analyses these structures. The expensive cost of additional design and production labours can be compensated by savings in material and weight. The initial budget. 4.000 € (£ 330. The potential of tensegrity for roof structures is considerable. 3. The responsible engineer of the Tower of Rostock called attention to some conclusions: 1.2. Tensegrity Structures and their Application to Architecture Chapter 6.2. etc. The first one was the “tensegrity structures to atrium roofs” for the Reuters HQ in London. Skelton and Sultan have been exploring the use of tensegrity structures as sensors and actuators due their kinematic indeterminacy (Tibert. tensegrity towers can be employed to support antennas. they could serve perfectly for this application. but it was never built. The lightness of these structures could minimize the visual impact of these energetic installations. as was mentioned in chapter 4. For instance. and they tolerate certain small movements. a study should be carried out in relation to any vertical structure that can damage the visual landscape of an area. Wind parks: Even if it seems unfeasible. Aesthetic elements: In general. Roof structures. there should be some study to analyse the effects of turbines installed on the top of a single or a group of tensegrity towers. mobile telephone transmitters. or at least the author has not found any significant examples. Communications: In situations where the margin of displacements is not very strict. 6. receptors. 2002). First of all. The second one was the new Stadium in 87 . by the architects Sidell Gibson Partnership and Buro Happold Engineers. radiotransmitters. it is necessary to say that there are not any roof structures based on the principles of continuous tension-discontinuous compression. Applications and proposals tensegrity towers or columns could be useful: Lightning conductors: As it is not required to have these elements in a completely static situation.3. The author has come across only two references related to tensegrity roof structures. Applications and proposals La Plata (Argentina). some other experts are working at present in this field. Double-layer grid the Laboratoire de Mécanique et Génie Civil in Montpellier Illustration drawn by the author 88 . The result was an interesting debate about the load bearing capacity of these grids. it is more similar to a cable-dome structure than to a conventional roof structure. The first studies for the design of tensegrity grids were carried out by Snelson (cf. For instance. but he did not find applications other than his own sculptures. The design adapts the La Plata Stadium Illustration from Weidlinger Associates (2002) patented Tenstar tensegrity roof concept to the twin peak contour and the plan configuration. It is worthwhile to remark that the structural engineers are Weidlinger Associates. 4. Ariel Hanaor (1987) started researching the double layer planar configurations in the 1980s. based on a prizewinning concept developed by architect Roberto Ferreira. As one can anticipate. and consequently. proposing different systems for assembling tensegrity prisms (detailed in chapter 5).2).Tensegrity Structures and their Application to Architecture Chapter 6. fig. who also worked on the analysis of some Snelson’s sculptures and of the “Georgia Dome” in Atlanta. As already explained in chapter 5.11 & 5.8) and foldable tensegrity systems. Tensegrity Arches Illustration taken from Burkhardt (1999-2004) and Snelson (2004) 89 . figs. For the past few years. Finally. Tibert (2002) reports that Skelton. this kind of grid has its most feasible possibilities in the field of walls. roofs and covering structures. Pinaud and Chan analysed planar tensegrity structures and concluded that they can be perfectly efficient in bending. 6. Helton. their main projects have been focused in the development of double-layer tensegrity grids (cf.Tensegrity Structures and their Application to Architecture Chapter 6. Applications and proposals Maybe the most outstanding research is by René Motro and the Laboratoire de Génie Civil in Montpellier. Adhikari. it is correlated with Frei Otto’s projects in the Kuwait Sports Centre or the Schlaich’s Ice Rink Roof in Munich (fig.13). presently. until now no applications of arches have been brought into being. They have been investigating the use of tensegrity arches and spline beams to support large-span membranes. In some way. 2003).12. there is a research project carried out in the Tor Vergatà University (Rome). Applications and proposals 6. to build a tensegrity arch in order to estimate the effective actions of the wind.2. Nevertheless. is quite remarkable. in this case the properties of the floating compression 90 . Ice Rink Roof in Munich. it is true to say that some of these elements using continuous tension-discontinuous compression principles have been constructed and erected. Illustration taken from Janberg (1998-2004) Tensegrity Arch supporting membrane. Illustration taken from Adriaenssens and Barnes (2001) Finally. It is projected to span a distance a 50 m by assembling expanded octahedrons (Motro. X-modules by Robert Burkhardt (b) and “simplex” again by Snelson (c).Tensegrity Structures and their Application to Architecture Chapter 6. the research achieved by Adriaenssens and Barnes (2001). It might be interesting to note that.4. Arches As for roofs. in collaboration with Italian and French institutions. However.6. 3. Some examples are given in fig. showing arches respectively based on “simplex” by Maxim Schrogin (a). 16 in order to show a clearer perspective without the tensile membrane that covers the inner space.2. but there is also an absence of self-stability and pre-stress in the structure. Tent-like structures This section is dedicated to tent-like structures and shadow roofs. the 91 . Shade Structure Illustration taken from Daniel Ng (2001-2004) Shade Structure Illustration drawn by the author Figure 6. Applications and proposals arch are very suitable to accommodate the asymmetric loads and avoid the stress concentrations in the tensile structure. due to the torsional freedom of the arch that equilibrate the stresses (fig.5. The same system has also been represented in fig. In this case. It can be noted that the system has no self-equilibrium stability since it needs to be anchored to the ground in three points and the struts are not stable if they are not resting on the ground. which show typical examples of false tensegrity.14) 6. 6. there are compressed components in the boundary and the strut-strut contact or similar contemplations.Tensegrity Structures and their Application to Architecture Chapter 6. 6. Moreover. created and presented by Daniel Ng (2001-2004).15 shows an example of a so-called Tensegrity Shade Structures. 6. exhibitions. so useful and interesting in expositions. In 1961.17) is very different to the previous one in terms of stability. Outer Space structures Since the beginning of the “tensegrity era”.Tensegrity Structures and their Application to Architecture Chapter 6. Picture taken by the author (2004) Nevertheless. one of the most recurring applications found for the floating-compression has been its use in moon-colonies.6.2. Applications and proposals absence of prestress is definitive to deny the denomination of tensegrity to this shadow roof.. Besides.6. etc. Another case produced in Edinburgh (fig. this marquee is perfectly tensed and conformed without the action of the cables. as will be proposed in following paragraphs. which play a role in giving more rigidity to the system. Tent-like Marquee. each module is attached to the ground by means of a fixed basement. if the other tent is not stable. Buckminster Fuller revealed his new inventions: potential prototypes of 92 . the author would like to emphasize that floating compression could be applied to tent-like structures. not necessary in true tensegrities. Lunar applications Fuller exhibition. many proposals along the same lines have been given by different people.Tensegrity Structures and their Application to Architecture Chapter 6. Modern Museum. N. since one of the particular characteristics of tensegrity structures is that they don’t depend on gravity. foldable.18). Basically. (cf. 1961). 93 . 6. Geodesic dome. so they are stable in any position. prestressed. It is not very surprising to arrive at these conclusions. fig. Illustration taken from Fuller (1961) Since then. but not without avoiding the task of evaluating the consequences of their propositions in-depth. Octet truss. omni-triangulated. etc. “spherical nets in which local islands of compression act only as local sprit-stiffeners” (Fuller. Applications and proposals satellite and moon-structures conceived as tensional integrity.Y hree of his basic structures: Tensegrity mast. extremely light. (Appendix C). 6. In this case. However. this section is dedicated to show some other functions that have been found with continuous tension-discontinuous compression.Tensegrity Structures and their Application to Architecture Chapter 6. 7 See also “Deployable Antenna” patented by Knight et al. Applications and proposals Maybe the exception is the research of Tibert and Pellegrino. 94 . in any case. mainly foldable reflector antennas7 and masts (2002) and the latter has dealt with large retractable appendages in spacecraft (1995). wind. dehydration. It could be claimed that in this case we are not dealing with conventional architecture. it is still architecture. This project sought the improvement of new structural concepts that experience completely different external loads (1/6 of Earth’s gravity.). 2001). meteorite impacts. The former has been studying deployable tensegrity structures for Space applications.7. the technology needed to obtain a source of energy. light. etc). etc.2. different risks (like pressure containment. it is lunar architecture and some day it will be necessary to face circumstances of this kind. (2002) and “Deployable Reflector Antenna” by Ian Stern (2003). a very defined project has been carried out from another approach (Literati. but a resource to achieve another objective: the establishment of a selfsustainable society in the moon.g. Floating compression options are included in a long list of suggestions. moonquakes. etc) and different environmental conditions (atmosphere. etc. radiation. Different applications besides Architecture As a curiosity and illustration of interesting initiatives. the utilisation of lunar regolith to produce concrete. Recently. There is no doubt about it. e. tensile integrity structures were not the starting point. There are other exemplars like the 95 .2.7. Applications and proposals 6.Tensegrity Structures and their Application to Architecture Chapter 6. 2003b.1. ornaments are some examples of attractive applications (fig. Miller (1977) and Barber (2003a. toys and leisure. Puzzles. tables.7.2. lamps.2. 6.19). 2003c & 2004) in Appendix C. Furniture Chairs. For more models. In Appendix C it is possible to see the characteristics of these patents by Kittner and Quimby (1988) and Mogilner (1972). see the patents of Wiesner (1973). Tensegrity Furniture Illustrations taken from Koennig (2004) and Werta (2003) 6. Applications and proposals “Skwish Classic”. he created the 60-strut stainless steel and wire sphere. 96 .2. but it might be interesting to remark that the “Tensegritoy” has been developed from the first patent. University of Sixty Strut Tensegrity Sphere Fuller Illustration taken from BRUW System (2002) Wisconsin-Madison (fig.4. 6.Tensegrity Structures and their Application to Architecture Chapter 6. Sculptures In addition to Kenneth Snelson’s sculptures.20). UCSD Flow Control Lab (2004) is currently studying how to optimize the compliance properties of a compliant tensegrity fabric in order to reduce the skinfriction drag induced by a turbulent flow. installed hanging from the roof of the Engineering Centers Building atrium. This is the instrument that the author has been using for the configuration of the simplest models as it is very useful for spherical tensegrity polyhedra. As a result of such a fascinating facet.2.3. Such a surface could move in response to the pressure and skin-friction fluctuations of an overlying wall-bounded turbulent flow.7. 6. and could be applied to the external surface of the submarines.7. Submarines. 6. even Buckminster Fuller dedicated some of his efforts to this field of art. Tensegrity dome from the Truncated Icosahedron The original idea was to take advantage of one of the configurations shown in Appendix G (tensegrity models. have been incorporated in order to give more rigidity and minimize the typical deflections of floating compression structures. taking advantage of the vertices of the polyhedra. Personal proposals In the following sections some potential applications will be explained. Three floors are considered. art pavilion. some additional cables. while the respective drawings. Therefore. this system is not triangulated.3.1.16). Sheet 1). 97 . where a Tensegrity Truncated Icosahedron is truncated again to conform to a dome (3/4 of a sphere). In the design.3. In fact. these floors have also the purpose of making the structure firmer by connecting the apex located in the same horizontal plane. etc. It should be noted that the designs are ideas that might be developed further. G.Tensegrity Structures and their Application to Architecture Chapter 6. Applications and proposals 6. In the drawings (Sheet 1 of Appendix H). rather than detailed drawings proposed for a real project. Therefore. figs. offices. nothing is defined in more detail because the main purpose is to show feasible applications rather than developments in depth. a small dome is proposed to contain an atypical architectural space. dedicated to versatile uses such as an exposition centre. 6.15 & G. shown in the plans (cf. so it is not as stable as desirable. plans and images will be presented in Appendix H. The dimensions are also relative. but the diameter could be approximately 20 meters.14. they do not achieve the requirements of professional projects since they are not necessary at this stage. As is mentioned by Snelson. G. Lightning rod from the Helix Tower Taking the “simplex” as a basis.3 & H.10-G. but adding smaller modules each time. a new configuration for a tower is proposed in Sheets 2 & 3. a helicoidal tower is obtained.5). fig. If this step is repeated. H. covered with insulating material. 6. The 98 . Roofing for Stadiums by assembly of modules This is maybe the most ambitious proposal. it could be dangerously exposed and would disturb the elegant figure of the tower. It was created by adding one of these “simplex” modules over another module and rotating it 30º until the end of one strut enters in contact with the end of other one at the lower level.3.5). G.13).2). there is a needle whose function is to receive the lightning in case of a discharge in the surroundings (cf.3.2. On the top of the mast. The conduction of the lightning is through a cable. fig. G.Tensegrity Structures and their Application to Architecture Chapter 6. A study for the Lightning rod is shown in Appendix G (figs. 6. inserted into the struts until it arrives at ground level. Following this process. the skin of the dome has been set up as transparent. When working with the models. which is called the Helix Tower. Applications and proposals In the ‘render’ images (cf. 1 & H.3. the result is a pyramidal Helix Tower that will be employed for the lightning rod (fig.4). He accidentally discovered a tensegrity figure made with six struts and 20 cables (cf. H. Appendix H. the author was trying to define the shape of a system that would be able to support a certain asymmetric hypothesis of external loads. but the reason for that is just to make the inner distribution of the space clearer. If the conductor was installed vertically along the axis of the tower. additional cables and cladding would be necessary to give more rigidity and protection.7). figs.8 & G. Due to their adaptability. 6. The fact that they are foldable could facilitate their preconstruction and assembly. etc.6 & G. G. Applications and proposals model was peculiar because of its capacity to be folded and unfolded (cf. elliptical. a pyramidal configuration is achieved. 5 and 6 it is possible to see the different steps of folding. the author achieved a conglomeration of tensegrities by assembling several tensegrity tetrahedra based on the faces defined by their truncated vertices (cf. Even though every module is self-stable and could rise on its own over the stands. the possibility of adding a cable attached to a marquee or balcony has been considered.9). G.4. in order to transport them to the site. Once they are located in their place. based on these modules as part of the roofing of stadiums in general. In sheets 4. Pyramidal roof from assembly of Truncated Tetrahedra After some experiments with models in space (cf. they would be optimum for covering the stands of stadiums with different shapes: circular. figs. G. and also the lengths and the relative member prestress force magnitudes. In the mentioned examples. square. G.3. Some examples are provided (Sheets 7 & 8. H7 & H8). a new system was conceived. where they could be unfolded and prestressed. Sheet 9). in order to balance the weight of the cantilevered roof (see graphics). which could easily cover triangular areas. The latter are relative values to be scaled up or down (everything multiplied by a single positive constant) depending on how much the structure is to be prestressed. figs.7.Tensegrity Structures and their Application to Architecture Chapter 6.1) in this way. only the option of the marquee has been contemplated. 99 . fig. When joining the Truncated Tetrahedra (cf. H6. Taking this consideration into account. 9 & H.).11). some images (cf. Footbridge from assembly of “Simplex” modules This last example is conceived as a small footbridge in an urban context. Sheet 10). etc. for instance. H. etc. Applications and proposals In order to show a comprehensible example of this application.14 shows an additional cable-stayed structure to support this footbridge for longer spans. linking two buildings or spanning narrow rail lanes or roads. Even though the new structure is self-stable. but obviously they might be investigated in more detail: moon stations. H. structures for seismic areas. it could be supported in just four points (cf.13) show the final configuration of the structure and other installations (deck. 6. The main structure of the footbridge could be easily installed by means of a crane. marquees to cover parking places. Transparent elements have been chose in order to achieve understandable perspectives. railing. additional cables would be necessary to give more rigidity and to permit larger spans. Other suggestions to develop In this chapter. because of its lightness. The author also suggests studying the feasibility of marquees for entrances. wind parks.Tensegrity Structures and their Application to Architecture Chapter 6. although it could also be considered for other possibilities.10) have been included.3. Other figures of Appendix H (H. It is the result of assembling several “simplex” along their main axis (cf. communication towers.. 100 . Figure H. some possibilities have been suggested. The transparency of the cladding is intended to give a clearer and more aesthetic perspective of the assembly. lights.12 & H. Moreover.3. 6.5. fig.3.. Applications and proposals It might be interesting to project a tensegrity structure conceived as an exclusion or containment of flying animals or other species. The main advantage would be that. For instance. The author would like to point out the possibility of generating a similar structure using large bars as isolated components in compression with the tensile surface working as the prestressed component. Some other applications could depend on the evolution of the investigations on foldable tensegrity structures.2). etc. by Tony Armstrong-Jones (Lord Snowdon). floods and so on. As a result. they could be used for disaster relief in areas devastated by earthquakes. thus. The latter might be conceived as a transparent membrane skin or. fig. However. hurricanes. G. 101 . it would not need to be anchored at all and. due to the lightweight and self-stability of the structure. field hospitals. could be transported easily without the inconvenience of changing the animals from their habitat. perhaps more suitably in terms of conservation. like any other proposal mentioned in this work. Cedric Price and Frank Newby.Tensegrity Structures and their Application to Architecture Chapter 6. a further research must be carried out to develop these potential applications. as a cable net with dense grid. bridges. it could be considered the shape of a Truncated Octahedron (cf. by installing deployable systems in the form of temporal dwellings. The idea came after seeing the Snowdon Aviary in London. . Tensegrity Structures and their Application to Architecture Chapter 7 Questionnaires and Interviews . . Tensegrity Structures and their Application to Architecture Chapter 7. Questionnaires and Interviews Chapter 7. Questionnaires and Interviews During the preparation and planning of the present work, it was decided to carry out both qualitative and quantitative research techniques in order to understand some important points related to the topic in question. The characteristics and results of the studies will be listed in the following paragraphs, although the preliminary interviews with some professors will not be included as they served only as a help to focus the dissertation. 7.1. Questionnaires Three kinds of questionnaires have been organised and prepared, however, only two of which have been utilised. 7.1.1. Questionnaires to professionals: The first aim of this study was to discover if the knowledge of tensegrity structures, and their basic principles, are widespread among architects and engineers. The second aim was to gather more information about tensegrity from those professionals that had any knowledge of it. Therefore, a general and basic questionnaire, which is attached in Appendix F, was prepared and sent by email to professionals of both subjects, architecture and engineering, to various places in Europe. As the author anticipated a low rate of response, it was decided to send them to the departments of structures in the three universities where he carried out his studies: Universidad de Cantabria (Spain), Université de Liège (Belgium) and Queen’s University Belfast (Northern 102 Tensegrity Structures and their Application to Architecture Chapter 7. Questionnaires and Interviews Ireland). In addition, due to other circumstances, it was also sent to the Universidad Politécnica de Madrid, Universidad Politécnica de Cataluña, Universidad Politécnica de Valencia (Spain) and University of Bath (UK). In effect, the rate of response was very low; the questionnaire was sent to 139 e-mail addresses and only 21 answered (15% aprox.). Even though it was remarked in the cover letter of the questionnaire that no knowledge of tensegrity was needed in order to answer it, the author considers that this low number of replies could be due to the unawareness of the subject. The results are clear enough: only 10 of 21 had heard about tensegrity before. Some of them did not have a clear concept but a vague idea, or even recognized that they did not know that much about it. Taking into account that they are specialists in structural subjects, it is easy to deduce that it is not a commonly known type of structure and not very well known among architects and engineers. It should be emphasized that some of these questionnaires, (included in the 10 positive answers) were addressed to experts that have been dealing with tensegrity structures. In this case, the result was to obtain more information about tensegrity rather than studying the number of professionals aware of floating compression principles. Some of these experts were Chris J K Williams (University of Bath), Celso Iglesias and Avelino Samartín (Universidad Politécnica de Madrid) 7.1.2. Questionnaires to specialists: A similar survey was carried out, but this time with some changes since they were addressed to experts that have been working with tensegrity structures. In this instance, the questions were very similar but, obviously, the answers expected from them were to be more concise. Some of these questionnaires were sent to 103 Tensegrity Structures and their Application to Architecture Chapter 7. Questionnaires and Interviews Schlaich Bergermann und Partner, because of their participation in the Tower of Rostock; to Arenas y Asociados, for the same reason; and finally to Buro Happold and Sidell Gibson Partnership, for a common project in Blackwall Yard (London) which had a tensegrity roof. The information obtained from their replies is included in the text. The author is proud to report that this questionnaire has been filled and returned by two of the most important engineers of the world at present: Javier Manterola Armisén and Jörg Schlaich. Their collaboration has been very useful and doctrinal, and at the same time a privilege and an honour. 7.1.3. Questionnaires to the general public It is obvious that this type of structure is tantalizing, since it is not very instinctive in the way it works and how the struts can be “floating” in the middle of a group of cables. Therefore, the author had the idea of confirming the impression of excitement that people have when seeing a tensegrity structure. An informal survey was carried out in order to discover the most predominant opinions, but it was abandoned not much later because the unique opinion was generalized. Every single person that saw any of the models thought that it was “really amazing”, “gorgeous”, “stunning”, using these or similar expressions. Therefore, it was not worthwhile to gather all these opinions when the point of view was basically the same. 7.2. Interviews Once the major work of research was completed and after gathering a large amount of information from diverse sources, the author looked for a new phase in the process of completing the data already obtained. 104 Tensegrity Structures and their Application to Architecture Chapter 7. Questionnaires and Interviews The interviews were established by means of several and continuous emails with the experts in question, due to their geographic inaccessibility. The most important results are reflected along the main discourse and collected in the Appendix D of Personal Correspondence, where only the answers of the specialists are shown because the questions can be easily inferred from them. There is principally contact with four different people: Kenneth Snelson, the sculptor who discovered the tensegrity fundaments in 1948 and who kindly shared his knowledge with the author; Mike Schlaich, Civil Engineer responsible of the design of the Rostock Tower; Arturo Ruiz de Villa, Civil Engineer responsible of the calculation of the same tower; and finally, Robert W. Burkhardt, the author of the publication “A practical guide to tensegrity design”, who has been collaborating in the calculations of some of the models proposed by the author. It might be interesting to note that another interview was sent to René Motro to Montpellier (France), but unfortunately he failed to reply, which seems to be something usual for this outstanding researcher. In any case, all of these personal correspondences have been really fruitful and profitable, and it is not an exaggeration to recognize that the author did not expect such an important source of knowledge. 105 Tensegrity Structures and their Application to Architecture Chapter 8 Discussion and conclusions Tensegrity Structures and their Application to Architecture Chapter 8. Discussion and conclusions Chapter 8. Discussion and conclusions This concludes the main body of research and design work developed during the last months. Personally it is considered that the objectives programmed at the beginning have been achieved. 8.1. Discussion and conclusions Throughout the research work, the author has come across a large number of references about tensegrity, and other structures, in Nature. It seems like the floating compression is present in every single atom of our Universe, which recalls some of the quotations of the first pages. Moreover, natural principles are not only a constitutive of biotensegrity, or vice versa, but also of some other examples in the history of Architecture. Antoni Gaudí, Santiago Calatrava and Frei Otto are only some of these cases. Take, for instance, the studies developed by the latter: he used the structural fundaments of soap films, spider webs, vertebral spines, oil drops, etc. to achieve an improvement in his designs. He invoked biological functionalism to support the concept that lightweight is a real measure of structural effectiveness (Drew, 1976). The author realized that, to date, some scientific methods followed the sequence: researching ? developing systems/theories ? finding them in Nature. Tensegrity is not an exception. The experience of the architects mentioned above shows that maybe it is more logical to follow this other sequence: researching in Nature ? finding systems/theories ? developing them in other fields. 106 Tensegrity Structures and their Application to Architecture Chapter 8. Discussion and conclusions In his manifesto of futurist architecture, written in 1914, Antonio Sant’Elia (cited in Drew, 1976) predicted a new architecture with new qualities: revolutionary, elastic, light, expandable, active, mobile and dynamic. Thus, he identified the most important features of tensegrity structures. Needless to say few things can be achieved without more investigation, but tensegrity could be one of the structural systems of the future. From the author’s point of view, an important step was reached by finding several examples of tensegrity prototypes that could be applied to Architecture and Engineering. His own proposals could serve as an illustration to the feasibility of tensegrity as a lightweight structure to cover large spans, bridge shorter distances or support light infrastructures. Of course, a much more detailed structural investigation would be necessary, but at least the presupposed idea of tensegrity as an inapplicable system has been disproved. 8.2. Further research In chapter 6, some possibilities have been briefly pointed out, but obviously they could be investigated in more detail: moon stations, communication towers, wind parks, marquees for entrances, marquees to cover parking places, structures for seismic areas, etc. Some other applications could depend on the evolution of the investigations on foldable tensegrity structures. As a result, they could be used for disaster relief in areas devastated by earthquakes, hurricanes, floods and so on, by installing deployable systems in the form of temporal dwellings, bridges, field hospitals, etc. However, as for other proposals mentioned in this work, further research must be carried out to develop these potential applications. 107 Tensegrity Structures and their Application to Architecture Appendices . . html R. great circle. If it's his name. play aimed at making mobile sculptures. He talked about it publicly as "my tensegrity" as he also spoke of "my octet truss". we are now in the year 1990 and not 1948 so all of this fades into the dim footnotes of history.Tensegrity Structures and their Application to Architecture Appendix A. The attraction at Black Mountain was the Bauhaus master himself. International Journal of Space Structures. Motro International Journal of Space Structures Space Structures Research Centre Department of Civil Engineering University of Surrey. Of course. which it was not. only an early. in so many words. the painter Josef Albers who had taught at the German school and immigrated to the U. in 1933 to join the faculty of that tiny liberal arts college (fifty students that summer) in the Blue Mountains of North Carolina. 108 . to others it is trivia." Whenever an inventor defends his authorship the issue invariably turns out to be important only to the author himself.S. A second-year art student at the University of Oregon in 1948. What Bucky did. claimed to have been its inventor. in Motro (2003) and in http://www. Buckminster Fuller. Guildford Surrey GU2 5XH Dear Mr. fifteen miles from Asheville. Albers picked me to help because I had shown special ability in his three-dimensional design class.. but I'd like. There was no such thing as a tensegrity or discontinuous compression structure in his collection. version of his geodesic dome. etc. Motro: I regret it has taken me so long to respond to your letter about the special issue of Space Structure dedicated to tensegrity. the "tensegrity" discovery resulted in a way from play. especially because Mr.and besides. He spoke and wrote about it in such a way as to confuse the issue even though he never. published in November 1990. Motro. isn't it his idea? As many new ideas do. As you probably know. in this case. Motro’s correspondence from Snelson From Kenneth Snelson to R. dymaxion. I am not an engineer but an artist so I don't really feel qualified to write for an engineering journal. however. "But that was long ago in another land -. I have long been troubled that most people who have heard of "tensegrity" have been led to believe that the structure was a Bucky Fuller invention. Nonetheless I know something about this particular form of structure from making so many sculptures over the years which use the principle which I prefer to call floating compression. somehow. during his long and impressive career. to set the record straight. There is a line somewhere in a theater piece which goes. turned up two weeks into the session. was strong on publicity and. for his own purposes. But since he rarely accredited anyone else for anything. unknown to most of us in those early days. Motro’s Correspondence from Snelson Appendix A. I took a summer off to attend a session in North Carolina at Black Mountain College because I had been excited by what I had read about the Bauhaus. Josef Albers asked me to assist the new faculty member in assembling his assortment of geometric models for his evening lecture to the college. a substitute for a professor of architecture who cancelled a week before the summer began. none of this is all that surprising. successfully led the public to believe tensegrity was his discovery. a powerful strategy for appropriating an idea.net/snelson/rmoto. Fuller.grunch. Maybe you're acquainted with the tale of Buckminster Fuller and me. was to coin the word tensegrity as he did octet truss and geodesic dome. the wench is dead. though one could imagine that they might just as well lead to something other than to the first tensegrity structure. modular. were to become electrified "Fullerites". But. Oregon. oneto-another. cardboard. It was the effort to make the pieces move which resulted in their spinal-column. perhaps to variations on Calder mobiles. metal from tin cans. that before Buckminster Fuller came along. for example. literally. While Albers' teachings were imparted as useable ideas in public-domain. I made numbers of small studies. He had that cult-master's kind of charisma. Photo #2 One step leading to the next. and. If I pushed on them lightly or blew on them. I blush for it now. but it was true. I was quite amazed at what I had done. building things. I believed. small mobile sculptures mostly. Albers counselled me that I demonstrated talent for sculpture. I had already become the first in a trail of students from colleges and universities who. more importantly. an Indian rope trick. The three small works which are of interest here were concerned both with balance of successive modular elements hinged one-to-another and stacked vertically as seen in photo #1. clay. Thus. using thread. while forfeiting mobility. after all. which I did."his" geometry. of Alexander Graham Bell's early space frames. Students joked that. at least as invisible as marionette strings. In my Fullerian trance the descent into the real world was greatly confusing. because he claimed so. Were they structures or sculptures? They incorporated the attitudes of both Fuller and Albers. later. but amplifications of the familiar balancing toys seen often in novelty shops. I returned home to Pendleton. As the summer quickly went by with most of the small school monitoring Fuller's classes I began to think I should try something three-dimensional rather than painting. I saw that I could make the structure even more mysterious by tying off the movement altogether. wire. I managed to gain something even more exotic. suspended one-to-the-next by means of thread-slings as shown in photograph #2. Bucky's lessons were laden somehow with the sense that the ideas were proprietary -. At the end of the summer session. nor anything at all about crystallography.Tensegrity Structures and their Application to Architecture Appendix A. In photo #2 one can see module-tomodule sling tension members replacing the wire hinges connecting the modules shown in photo #1. I had learned much about geometry from Fuller as well as art and design from the Bauhaus. no human had ever noticed. as I said. for example. causing the connections to be more or less invisible. making my parents miserable by moping and spending hours in the basement. solid elements fixed in space. etc. My small discoveries in these two pieces were logical enough. kite-like modules out of plywood. Motro’s Correspondence from Snelson During his lecture that evening Professor Fuller mesmerized us all with his ranging futurist ideas. We were young and looking for great issues and he claimed to encompass them all. They were. I thought of these threads as adding a note of mystery. they swayed gently in a snake-like fashion. held together only by tension members. I spent the autumn at home. that to inscribe the diagonals of the square faces of a cube was to define two interlocking tetrahedra within. property. over the years. hadn't Bucky invented the triangle? None of us knew. making "X". Photo #1 In the autumn of 1948. of course. Photo #3 109 . replacing the clay weights with additional tension lines to stabilize the modules one to another. he could now span the Grand Canyon. a true space-filling system. so he told me at the end of the summer. they should be shaped like the central angles of a tetrahedron. The name of Ken Snelson [his underline] will come to be known as a true pioneer of the realized good life and good will. triumphantly holding the new structure I had built. that is like spokes radiating from the gravitational center. In Time magazine he declared that.whatever that may have meant. Rather than the X-module for compression members. That you were excited by the latter. but demonstrates how the important events happen where the atmosphere is most favorable. the ready reproduction of which. #3). into spontaneous articulation of the solution.G.Tensegrity Structures and their Application to Architecture Appendix A. with "his" tensegrity. Motro’s Correspondence from Snelson Still confused about my purposes and direction in school. Those were not yet the years when students easily contradicted their elders. also demonstrates the importance of good faith of colleagues of this frontier. holding it in his hands. He was quite struck with it. studying it for a very long moment. Photo #3 Next day I went into town and purchased metal telescoping curtain rods in order to build the "correct" structure for Bucky. properly incorporated in fundamental structures. and rejected it in favor of the kite-like X modules because they permitted growth along all three axes. who had been seeking this structure in Energetic Geometry. even though Bucky penned the following in a letter to me dated December 22. partly because I felt relieved that he wasn't angry that I had employed geometry (Buckminster Fuller's geometry) in making art. The event was one of those 'It happened' events. Of course the irony was that I had already used that tetrahedral form in my moving sculpture #2.(com-pressure) and continuous tension structural advantage. turning it over. If you had demonstrated this structure to an art audience it would not have rung the bell that it rang in me. He then asked if I might allow him to keep it. may advance the spontaneous good will and understanding of mankind by many centuries. When I showed him the sculpture. Next day he said he had given a lot of thought to my "Xcolumn" structure and had determined that the configuration was wrong. The absorption process began early. That original small sculpture disappeared from his apartment. grand claims in magazine articles and public presentations. it was clear from his reaction that he hadn't understood it from the photos I had sent. The classes depressed me even further. I hated it and did very poorly. I corresponded with Bucky and I told about my dilemma and also sent photos of the sequence of small sculptures. to the vertices of a tetrahedron. 1949: "In all my public lectures I tell of your original demonstration of discontinuous pressure .. I enrolled for engineering that winter ('48'49) at Oregon State College. but I gave it to him. let alone their professors. The rest of the story is one of numerous photographs and statements in print. It hadn't been my intention to part with it. He must have understood from the letter how confused and depressed I was at school for he suggested I return for another Black Mountain Summer Session. He also described it as a structure which grows stronger the taller you build it -." 110 . .in which right makes light in a prototype structure. rather than only along a single linear axis. I felt a little wistful but not at all suspicious of his motive as he had his picture taken. E. When we got together again in June I brought with me the plywood X-Piece (Fig. at least in its purest form is not likely to prove highly efficient or utilitarian. that ought to be enough. This forced Bucky's hand. Especially I picked up where I had left off with the neglected X-module which was left unnoticed for an entire decade. As a result. this is but a footnote to a storm in a teapot. One of the ironies of this not-too-unusual tale in the history of teacher-student relationships. Over the years I've seen numbers of fanciful plans proposed by architects which have yet to convince me there is any advantage to using tensegrity over other methods of design. any number of students labored to constructed their own "masts". The ghost of Bucky Fuller continues to muddy the water in regard to "tensegrity". about my part in tensegrity. of course. Kenneth Snelson 140 Sullivan Street New York. transitory as it was. In those years. I have continued to make sculptures which now stand in public sites in many places.Tensegrity Structures and their Application to Architecture Appendix A. since I know where the ideas came from. A year later. "The moment you tell me that the compression members reside interior of the tension system. Sorry there are none in England or France. "Ken. but he regarded himself as a man of the sea. so why not find a way to put it to some use. In that it was dated three days before Christmas. in 1959 I learned that Fuller was to have a show at the Museum of Modern Art in New York and included in it was to be a thirty-feet high tensegrity "mast"." He was speaking metaphorically about this type of structure in general. he managed successfully to put under wraps my original form. I was not mentioned. was quite fortifying and enabled me to once again pick up my absorbing interest in this kind of structure with the feeling that now I was free and on my own. At last. surprisingly. old man. this type of structure. No. the highly adaptable X form. Whether or not you are able to use this narrative about the beginnings of tensegrity. January 1951 he published a picture of the structure in Architectural Forum magazine and. As I said earlier. curator at the Museum. Calling it a mast seemed especially obtuse. for no other purpose than to unveil the exquisite beauty of structure itself. but all were built using the tetrahedral form. is that by Bucky's transposing my original "X" module into the central-angles-of-the-tetrahedron shape to rationalize calling it his own. mood. of course. none of the many students in schools where he lectured ever got to see it. as I describe them. of physical forces at work. That moment of recognition at the Museum of Modern Art in November 1959. you can afford to remain anonymous for a while. he had denounced it as the "wrong" form. Consciously or unconsciously we respond to the many aspects of order in nature. force-diagrams in three-dimensional space. as he said. I see the richness of the floating compression principle to lie in the way I've used it from the beginning. generous. He could not have lived with himself with the blatant theft of my original system." Finally. New York 10012 111 . Sincerely. As the engineer Mario Salvadori put it to me many years ago. For me. my credit for tensegrity found its way into the public record. he replied. With some persistence and with the lucky aid of Bucky's assistant I was able to get word to Arthur Drexler. I wish you the very best with your special issue on the subject. Motro’s Correspondence from Snelson Bucky's warm and uplifting letter arrived about six months after I first showed him my small sculpture. I suppose he was in a festive. I can tell you I can build a better beam than you can. I no longer felt anonymous. As I see it. in his public lectures and never in print. Usually the philosophy is akin to turning an antique coffee-grinder into the base for a lamp: it's there. and besides. I tell myself often that. these studies in forces are a rich source for an art which celebrates the aesthetic of structure. When I posed the question some years later why he accredited me. Patent No. 3.611.B. Construction de réseaux autotendants. discontinuous compression structures. 1. D. An illustration of the following patents will be presented in this Appendix: FULLER. D. Patent No. 112 .Tensegrity Structures and their Application to Architecture Appendix B. U. U. September 28.377. November 13.G. EMMERICH.291. (1965) Continuous tension. 1964 EMMERICH. September 28.S. 3. (1964). Original Tensegrity Patents.S. 1.521.169. R. 1964 SNELSON. (1962) Tensile-Integrity Structures. K. Original Tensegrity Patents Appendix B. French Patent No. 1965. 1962. (1964). French Patent No. February 16.290.063. Structures linéaires autotendants.G.377. Original Tensegrity Patents Extract of Fuller’s patent 113 .Tensegrity Structures and their Application to Architecture Appendix B. Tensegrity Structures and their Application to Architecture Appendix B. Original Tensegrity Patents Extract of Emmerich’s patent 114 . Tensegrity Structures and their Application to Architecture Appendix B. Original Tensegrity Patents Extract of Emmerich’s patent 115 . Original Tensegrity Patents Extract of Snelson’s patent 116 .Tensegrity Structures and their Application to Architecture Appendix B. S. B. U.741. U. 4.962. L. (1979). Patent No. S. Tensegrity module structure and method of interconnecting the modules..483. C.994. 4. D473.Tensegrity Structures and their Application to Architecture Appendix C. April 22. U. Patent No. U. (2003). April 4. Compression-tension strut-cord units for tensile-integrity structures.S. (2003b) Table composed on non-contacting compression members and tension members.S.520. 1990 KITRICK.S.S. Tensegrity Deployable antenna using screw motion-based control of tensegrity support architecture. III.155.S. 2003. (1988) Roof structure.S.441.736. (1998) Sports catch glove with stiffner. U.553. 1988. 1980 KITTNER. Patent No. Other Tensegrity Patents Appendix C.S.717. DUFFY.M. 2003. 1998 HUEGY. January 13. G. Patent No.676. 4. CRANE.M.715. and TAM.148. March 22. 2003 MILLER. June 17. U. March 18. 2003/0009974 A1. February 20. 6. (1990) Spiral helix tensegrity dome. 2003. 4.901. GLENN.S.S. Patent No. C. C. G. 2002 LIAPI. Patent No.731. 1979. 1988 KNIGHT. BARBER.T.F. August 27. Patent No. U. J. February 17. Patent No. Other Tensegrity Patents An illustration of the following patents will be presented in this Appendix: AUSTIN.W. U. U. Patent No. E. BARBER. April 4.. (1980). U. GOLDSMITH. U. D487. Piece of furniture. C. November 11. 2002. GEIGER. Tensegrity Unit.S.S. Patent No. 2004. (2002) Female condom employing tensegrity principle.S. U. Patent No.S. 5. D481. Female condom employing tensegrity principle. (2003c) Chair composed on non-contacting compression members and tension members. February 24. Structure and Method for construction. A.T. U. U.880. and ROONEY. D.801.D.D. K. J. 2002 BARBER. 4. (2002). Patent No. 2002/0038658 A1. April 12. BARBER. Patent No.T. (2004) Four-strut support composed of non-contacting compression members and tension members. G. (2003a) Lamp composed of non-contacting compression members and tension members. D471. US2002038658. 117 . L.R (1988). (2002). Patent No. G. April 10.J.207. and TAM.T.A. R. and QUIMBY.H. G. U.614.E. U. R.S. R. 4. déposée le 9 avril 2001 aux noms de C. RADUCANU.928. 2003. Patent No.901. (1986). April 2. (1972). and MOTRO. Patent No. August 26.S. Patent No. Deployable tendon-controlled structure. – Ste Tissage et Enduction Serge Ferrari.502.Tensegrity Structures and their Application to Architecture Appendix C. October 3.590. U. 3. 1975. U. (2001) Système a autoéquilibre stable pour élément de construction.R. Tensegrity Structure Puzzle. 118 .132. S.695. W. 1996. April 1. Patent No.542. 6. (1996) Tension braced dome structure.551. September 30.J. (1997). (1975) Stressed structure for supporting weight. Patent No. 1986. Other Tensegrity Patents MOGILNER. NELSON.L. U. WIESNER.S.642. (2003) Deployable reflector antenna with tensegrity support architecture and associated methods. Patent No.A.A.N. Telescoping strut members and tendons for constructing tensile integrity structures. 5.S.S.S.502. I. SKELTON. 3. G. W. July 1. demande de brevet français nº 01 04 822.S.617. TERRY. V. 5. 1997 STERN. 1972. U. Other Tensegrity Patents Extract of Fuller’s patent 119 .Tensegrity Structures and their Application to Architecture Appendix C. Other Tensegrity Patents 120 .Tensegrity Structures and their Application to Architecture Appendix C. Tensegrity Structures and their Application to Architecture Appendix C. Other Tensegrity Patents 121 . Other Tensegrity Patents 122 .Tensegrity Structures and their Application to Architecture Appendix C. Other Tensegrity Patents 123 .Tensegrity Structures and their Application to Architecture Appendix C. Other Tensegrity Patents 124 .Tensegrity Structures and their Application to Architecture Appendix C. Other Tensegrity Patents 125 .Tensegrity Structures and their Application to Architecture Appendix C. Other Tensegrity Patents 126 .Tensegrity Structures and their Application to Architecture Appendix C. Tensegrity Structures and their Application to Architecture Appendix C. Other Tensegrity Patents 127 . Tensegrity Structures and their Application to Architecture Appendix C. Other Tensegrity Patents 128 . Other Tensegrity Patents 129 .Tensegrity Structures and their Application to Architecture Appendix C. Tensegrity Structures and their Application to Architecture Appendix C. Other Tensegrity Patents 130 Tensegrity Structures and their Application to Architecture Appendix C. Other Tensegrity Patents 131 Tensegrity Structures and their Application to Architecture Appendix C. Other Tensegrity Patents 132 Tensegrity Structures and their Application to Architecture Appendix C. Other Tensegrity Patents 133 Other Tensegrity Patents 134 .Tensegrity Structures and their Application to Architecture Appendix C. Other Tensegrity Patents 135 .Tensegrity Structures and their Application to Architecture Appendix C. Tensegrity Structures and their Application to Architecture Appendix C. Other Tensegrity Patents 136 . Tensegrity Structures and their Application to Architecture Appendix C. Other Tensegrity Patents 137 . 2. Deflection of the expanded octahedron Appendix E. 1997) Fig. 1997) 155 . the deflection of the expanded octahedron modelled by cables and beams finite elements (Mijuca. can be seen in next figures: Fig. no deformation” Illustration taken from (Mijuca. “Tensegrity. “Tensegrity. 1997). “Tensegrity. 1997) Fig. mid-deformation” Illustration taken from (Mijuca. 1.Tensegrity Structures and their Application to Architecture Appendix E. no deformation” Illustration taken from (Mijuca. 3. Deflection of the expanded octahedron In order to show the Elasticity Multiplication of tensegrity systems. in Rostock (Germany). Correspondence with Kenneth Snelson. 138 . D.Tensegrity Structures and their Application to Architecture Appendix D. Robert W. However. Oregon. Personal correspondence D. Fernand Leger. Correspondence with Robert W. Personal correspondence Appendix D. Correspondence with Arturo Ruiz de Villa Valdés.-Ing. Arturo Ruiz de Villa is a Civil Engineer. he is working in “Arenas Y Asociados” (Santander). Eugene. Mike Schlaich is a Civil Engineer at the University of Stuttgart (Germany). Paris.. de Ingenieros de Caminos Canales y Puertos de Santander. degree in the E.1.4. class "Bauen mit Seilen" (building with cables). Black Mountain College.3. He is also a partner and managing director of the company “Schlaich Bergermann und Partner”. he has been lecturing at the University of Stuttgart.2. At present. Burkhardt.C. He was the person responsible for the calculation of the Rostock Tower while he was working in Schlaich’s consulting “Schlaich Bergermann und Partner”. D. For further references. He was the director of the design of the Rostock Tower. Switzerland (Dipl. which Fuller popularised as tensegrity. and at the Suisse Federal Institute of Technology (ETH) in Zürich. Kenneth Snelson is a very recognized sculptor. his sculptures have done more to spread the concept of tensegrity than anybody else.T.S. and in his web page he shows very interesting points about tensegrity applications (see Bibliography). D. Correspondence with Mike Schlaich. Art studies in University of Oregon. Universidad de Cantabria (Spain). N. see chapter 2. ETH). He discovered floating compression in 1948. Black Mountain. Burkhardt is the author of the publication “A practical guide to tensegrity design”. Since the year 2000. As you know many architects and engineers have worked toward that end and still do. 18 Jul 2004 13:09:37 . Kenneth Snelson Asunto: Re: From Jauregui. Yes. The web is wonderful but it's also a place where anonymity makes the contact seem a waste of time.es> kenneth snelson <k_snelson@mindspring. you should look there for your research to the extent that my work is part your thesis. just look at the range of my work compared with that of either of those two guys.Snelson .gomez@alumnos. it is my belief based on long experience and making endless numbers of tensegrity structures of all shapes and sizes that the principle in itself is impractical for building buildings.0400 Dear Mr. as a student.0400 Dear Mr. Emmerich labeled me a "defeatist" because I said that tensegrity is not a sound building strategy. Fifty years of it now. Personal correspondence Asunto: Re: To K.com> Sun.Questionnaire Para: De: Fecha: "VALENTIN GOMEZ JAUREGUI"<valentin.unican. Best wishes for success in your dissertation. It would seem from your request that in your imagination I am sitting by a window somewhere trying to think of what I might do to kill time.unican. I'm happy to say this is not the case. Tensegrity dissertation Para: De: Fecha: "VALENTIN GOMEZ JAUREGUI"<valentin. If my response to your inquiry was harsh it's because I receive many emails from people with rather pointless ideas which I spend time trying answering. Fuller gained much of his fame as a salesman selling tensegrity snakeoil. I appreciate that you are interested in my work.es> kenneth snelson <
[email protected] Structures and their Application to Architecture Appendix D. Jáuregui. I will be send you pictures for your paper. 20 Jul 2004 08:18:03 . I've grown weary of it especially since I never hear a word from them again. In brief though. None have shown there is the slightest structural advantage in its use for such purposes. However.gomez@alumnos. Jauregui. There is much to read and much to learn and. I am overwhelmed with projects that take all of my time. I do not have empty hours to fill out forms or questionnaires. claiming he could "bridge the Grand Canyon with tensegrity". They produced nothing useful nor enduring 139 .com> Tue. That is one of the benefits of publishing so many of my ideas and articles at my website. "1961SpringSt.Tensegrity Structures and their Application to Architecture Appendix D.es> Kenneth Snelson <k_snelson@mindspring. same with a spider web.com> Tue.and that restriction leaves out endless numbers of items. It's perhaps analogous to people's trying to achieve perpetual motion in the nineteenth century. "Cantilever30'1967. 3 Aug 2004 15:16:00 . exsoskeletal structure? I think not. Fuller declared that everything in the universe was tensegrity. Tell me which pictures you need and I'll try to find time to locate files large enough for print. Most important: it's not a triangulated structure 2) the other domes you cite can not be considered tensegrity.unican. Some of his disciples often show my work with a sly implication that it is Fuller's. As I've also said elsewhere. I've made this point in my writings which you probably have come across in your research.jpg" 3 IMAGE/JPEG 86218 bytes. See my 90' tower at the Kroeller Mueller in Holland or "Easy Landing" in Baltimore Md. Attached is a photo of the artist as a young man back then with one of my experiments. 1)Bucky's "tensegrity dome" or sphere is by its nature as soft as a marshmallow. Tensegrity structures are endoskeletal prestressed structures -. Did the world need a different name for that kind of solid rim. I was fascinated with these when I first made them in 1961.0400 1 TEXT/PLAIN 6852 bytes Adjunto mostrado debajo 2 IMAGE/JPEG 78626 bytes. Personal correspondence with tensegrity. These planes are also very flexible and I know of no instance where they've been put to use for any practical purpose. Yes.KenPlanar. I am also struck by your having picked up somewhere on Bucky's endless claims of having invented everything in the universe. Two of my planar pieces are in sculpture collections.gomez@alumnos. 140 . some other points Para: De: Fecha: "VALENTIN GOMEZ JAUREGUI"<valentin. Kenneth Snelson Asunto: Re: From Jauregui. essentially.jpg" Dear Mr. bicycle wheels. Maybe that's what you referred to in your question about atom models. regardless what people wish to call them. Enough said except that there are many theses from architectural students around the world that have been infected with the tensegrity fever. It really comes down to this: until you actually build a few of these structures you won't understand the issues involved. Jáuregui. no way to avoid that as long as one stays with discontinuity. if everything is tensegrity then tensegrity is nothing of any particular sort. Where did you get the idea he had produced an atom model? If he did it's news to me. so what's the point in using that word? As for my friend Rene Motro's double-layer planes. They are. a matter for accuracy in reporting? Also.Tensegrity Structures and their Application to Architecture Appendix D. It six years His statements about tensegrity's magical property. "MariaGough. As you note. the one in the far background of the famous Constructivist Exhibition photo. Koleichu for the Guggenheim Museum exhibition a few years ago and was said to be the "first tensegrity". "a battle of egos" about Fuller and me. Once again. 23 Aug 2004 15:03:24 . from other address Para: De: Fecha: "VALENTIN GOMEZ JAUREGUI"<valentin. I don't know why." How perfectly would have dated his using the word "tensegrity at least before he coined it. They need heavy and robust end-fixtures in order to absorb the powerful tension forces that pull outwardly with great cumulative force. The other Ioganson piece. "" Adjunto mostrado debajo 2 APPLICATION/MSWORD 35840 bytes. Jaurequi. (still in sculpture goofy. In any case. Short compression struts mean long tension lines which mean extreme elasticity. "".doc" Dear Mr. no. It's the only cantilever I've done whose name is Cantilever. The struts can't be all that lightweight because they must support enormous compression loads. was "replicated" from the photo by a Mr. that presumably make it a supernaturally efficient structure is.0400 1 TEXT/PLAIN 5787 bytes. I was not influenced by Mr. Best wishes.com> Mon." At least his charlatanism was charming.unican.es> Kenneth Snelson <k_snelson@mindspring. If I've repeated here what I said in my last message I wouldn't be surprised. The short-compression-members assertion is somehow analogous to Bucky's glib answer when someone during a lecture challenged him about an echo-chamber effect inside one of his domes: "No problem at all: just place a sponge at the focal center to absorb the sound. You use the expression. Personal correspondence Here attached also is a photo of the "30' Cantilever" which I guess is the piece you are referring to. but here goes.gomez@alumnos. It was also the subject of an article in "October" magazine by Maria Gough. for Bucky to have kept repeating the silly tale that print on the web) "I told Ken. nonsense. now at Stanford. rather. "". again. the sculpture by Ioganson that Rene Motro focuses on is not a prestressed structure. Is it not. very short compression members. My thoughts about it are included in the attached letter to Ms Gough about Ioganson whom she discussed in the article which will be included in her upcoming book. Ioganson. 141 . Kenneth Snelson Asunto: Re: From Jauregui. when I first saw his wood that what he had discovered was tensegrity. " I strongly challenge your assertion that their method was somehow "science" as opposed to my blind approach. A piece like "Mozart I" would today sell for from four to five hundred thousand dollars. You say about Fuller's domes: "However. I trust that you'll be able to include the facts of this message and still find a way to get where you're going in your thesis. he was never able to produce a Tensegrity dome which could cover the whole city. Show me any tensegrity structure whose tension network is not fully triangulated and I'll show you a flaccid structure as is the case for most of Connelly's and Black's inventory. fiftypercent of the selling price. studying the different possible topologies. If I had been a brilliant mathematician instead of a skillful and inventive model builder. Again: "On the other hand. What makes there work science? Other than the comparative output the main difference is that they were pursuing the goal of utility and neither succeeded in that. Best wishes. Personal correspondence You say. I think this covers most of what I see as problems in Chapter 2. There are simply too many variables including the ultimate "tuning" of the piece which can only be done in the field. Why is it that you and others characterize artists as naifs whose work is frivolous whereas men who call what they do "science" are trusted to have profound understanding of heavyweight matters? Fuller built his tensegrity dome based on measurements of his small models. his tensegrity domes are not triangulated and therefore are as shaky and floppy as a Tensegritoy. the final application of Tensegrity was not as successful as he thought it would be. Despite all his celebrating of triangulation. Kenner developed the useful "Geodesic Math and How to Use It" which shows how to calculate "to any degree of accuracy" the pertinent details of geodesic and tensegrity structure's geometry. man. Emmerich or Kenner have to tell you about how these structures work. Fuller and Emmerich took the scientific approach.. Fuller made grandiose claims with no testing whatsoever. No one has developed a computer program or algebraic formula that can design tensegrity with any degree of accuracy. even his cigar-strut "Geodesic Tensegrity Dome" you show sitting in that workspace could barely hold itself up. Science goes from theory to proof by testing.using mathematics. Kenneth Snelson 142 . as he intended. after costs are subtracted. Galleries take. My bet is that Emmerich worked also with models and then built his larger pieces based on them as reference. the cost of fabricating an outdoor piece is roughly twenty-five percent of the gallery's selling price. depending on the size of the work.. empirically. my widely varied collection of works simply would not exist. measuring what actually turned out. Omitting the truth about tensegrity won't improve the quality of your scholarship. Pertinent details? What does that mean? and for what variety of tensegrity structures? You ask about the fabrication costs for my sculptures: Roughly. Fuller. Look at the facts: how many genuine tensegrity works did Fuller produce in his lifetime? Also compare the "structures" analytic study at my website against anything either Pugh." My God.Tensegrity Structures and their Application to Architecture Appendix D. "On the other hand. or three-strut. tensegrity module. etc. 2003 What a surprise to learn that you were in the audience at my talk at Michigan. came across what one now calls a three-way. your illustration #10 “spatial constructions.Tensegrity Structures and their Application to Architecture Appendix D. I’m bad with names so when I got Ms Schwartz's email about pictures I regret I didn't make the connection that you are the author of that excellent paper. I do wish you had introduced yourself because I have your 1998 October 84 piece. Personal correspondence Mozart 1 Kenneth Snelson (1982) Stainless steel (7x9x9m) Stanford University. He then. according to your paper apparently struggled with those three octahedral variations. the entire three-way structure is the module if used as such. The individual sticks 143 . Bucky Fuller’s claims about these structures are off the wall. June 17. which tell us something about his focus on crosses. Don’t know the end result of his plan. Karl Ioganson. CA Illustration taken from Snelson (2004) Dear Maria Gough. I take it that your request for the particular pictures you wish to use in your upcoming book are in regard to the Karl Ioganson theme of your original paper? I appreciate your involvement with the Constructivists and their art and history and I want not to detract from your fine scholarship but I do wish you had talked with me beforehand since you try to deal with the subject of tensegrity and I don’t think you had the best of sources. I found the "In the Laboratory. By the way." article fascinating and informative when a fellow from Latvia named Juris Sils faxed it to me in connection with an exhibition of Karl Ioganson’s work he was trying to make happen. by some unrecorded steps. But in themselves. you compare the Stenberg brothers’ and Ioganson’s aspirations with their actual achievements and you award Mr. in the course of the twentieth century. Similarly. But the forces within the system need to be so huge that the structure becomes inefficient for supporting any external loads. is number IX indeed what Koleichuk says it is? Once you see his model it looks like the piece there in the background. since even I wouldn’t have been able to verify it as such. “floating compression” goes back to. a geodesic dome is synonymous with tensegrity.” The unfortunate fact is that tensegrity is not and never was functional except for the function in my sculptures of permitting viewers to admire the nature of pure structure. the very word has become garbled. that expanded object would be a modular structure. 1962. that he himself placed no emphasis on it. Would he not have asked. sticks positioned and strings properly attached. I thought. or x-modules. As I no doubt said at Michigan. No one on Earth would have been able to discern the nature of IX without prior acquaintance with the tensegrity primary. This is less than convincing since he would have had sticks and strings. “It is as if they are floating in a net of. except that he had studied my work. My standard descriptive name. When I saw the 1992 catalog of the Guggenheim show with Mr. too. say like my “Needle Tower”. At the end of your paper. tensegrity works the way it does because it is an equilibrium of contesting forces within a closed system. Koleichuk’s reconstructions of Ioganson’s small sculptures. The hint that he had studied me was your quote from Koleichuk which is an appropriated paraphrase. at least. enormous functional significance. but I think it’s important for you as well as for me and for the sake of your splendid scholarship. will that work?” It doesn’t hold water that Karl Ioganson was thwarted by inadequate materials. primitive works composed of two-way. To him. For example the engineer Mathys Levy calls his great dome in Atlanta “tensegrity” whereas it actually is a beautifully designed giant bicycle wheel. If indeed it is. a real estate agent or a greedy entrepreneur had figured out a way to make tensegrity into a reasonable building system. it’s curious since even I wouldn’t have been able to make out from that famous jumbled 1921 exhibition photo that Ioganson’s piece (marked number IX) was indeed a three-strut tensegrity structure. is it not uncanny that Ioganson nor anyone else left a comment about this surprising object. beyond all other attributes. that he apparently quite abandoned his amazing discovery with no follow-up? Did none of the other artists or visitors think it represented a remarkable phenomenon? Wouldn’t one expect him to take a next step.” I then considered. Attached are two pictures from early studies. that Koleichuk would have no way of guessing at the object. Ioganson the prize: “(Karl Ioganson) invents a new principle -. that I was using at the beginning. Perhaps some political pressure ended his quest or perhaps his inventiveness or inquisitiveness simply had its limits. Your paper argues that he didn’t have sufficiently high-tech materials in order to move forward. “Well. As far as we can tell. or several of them. absent of discussion. or Bucky Fuller’s or David Emmerich’s. Over the past fifty years. any next step that would let us know he had a grasp of what was going on with the structure? Apparently not. Yes. “What if I use four sticks instead of three.. the startling discovery just sat there among his other works and those of his colleagues. Bucky Fuller exploited his puffed up tensegrity claims shamelessly even though he knew better. A module is a whole object or closure that. Wires” Coming across one’s own words mouthed by a stranger is eerie indeed.Tensegrity Structures and their Application to Architecture Appendix D. I very much look 144 . the materials he already was using. The materials are wooden dowel sticks painted silver and string. when attached or interconnected with similar objects can create a more complex form.a prototensegrity principle -. If Ioganson had made another three-strut tensegrity. So. and tension-spoke bicycle wheel with its major load-bearing rim is not tensegrity no more than is a spider web.. and connected them together in one of several ways. only parts. neither the individual struts nor the individual tendons are modules. the Harvard microbiologist Donald Ingber invokes tensegrity as a buzzword to bolster a contested theory of cell structure. By now. the country would be dotted with novelty shopping centers or MacDonalds supported by tensegrity golden arches since. or even an unreasonable one. I regret that this letter grew much longer than I possibly imagined when I started out. novelty is great for commerce. if a clever architect. Personal correspondence are not modules but simply compression struts.that would come to have. clear and highly intelligent. Kenneth Snelson 145 . Ms Gough. Warmest wishes for the book.Tensegrity Structures and their Application to Architecture Appendix D. Personal correspondence forward to the publication of your book but I hope you have the chance to work out these problems before it goes to press. I realize my discussion here sounds harsh but it isn’t meant to be hostile. I’m sure your book will be much more complete than your thesis which. only corrective. Out of fairness. Please let me know your purpose in choosing my 1967 stainless steel X-Piece for illustration. to me came across as forceful. “Easy Landing” or other major piece for the benefit of those who know nothing of my work and might take it that I stopped producing way back then. I would much prefer you include a photo of something really representative such as “Needle Tower”. es> 146
[email protected]. Saludos.ein Tensegrityrekord. Nosotros hemos proyectado la torre de Rostock.de Asunto: Antwort: Articulo Tensegrity Para: De: Fecha: "VALENTIN GOMEZ JAUREGUI" <valentin. 19 Jul 2004 14:23:05 +0200 Valentin: La revista Alemana se llama "Stahlbau" la editora es "Ernst und Sohn" y salió en Octubre 2003: [x] M. Heft 10. Schlaich Bergermann und Partner Hohenzollernstr. Asunto: Antwort: Dissertation Tensegrity Para: "VALENTIN GOMEZ JAUREGUI" <valentin. 7 Jul 2004 19:02:24 +0200 Estimado Valentín.de Mon.Schlaich: Der Messeturm in Rostock .schlaich@sbp. sc. Para que sepas: hacía finales del año saldrá una versión inglés de este artículo en la revista: JOURNAL OF THE INTERNATIONAL ASSOCIATION FOR SHELL AND SPATIAL STRUCTURES: IASS. Mañana te mandamos un artículo (en aleman) sobre la estructura.de Wed. Dr.schlaich@sbp. Stahlbau 72 (2003).1. Mike Schlaich -------------------------------------------------------------------Mike Schlaich. techn. Ernst & Sohn (in German).sbp. Mike ------------------------------------Mike Schlaich.unican. techn. D-70178 Stuttgart fon: +49-711-6487114 fax: +49-711-6487166 e-mail: m. Personal correspondence Asunto: tensegrity Para: De: Fecha: valentin. Saludos de Stuttgart. que con sus 62m de altura probablemente es la torre tensegrity más alta hasta ahora.gomez@alumnos. Dr.Tensegrity Structures and their Application to Architecture Appendix
[email protected]. sc.es m.es> m. gracias por tu
[email protected] http://www. No obstante. "". Personal correspondence De: Fecha: m. techn.schlaich@sbp. Dr.de Mon. El coste neto (y el presupuesto) de la torre era de 500. Saludos. 30 Aug 2004 08:52:32 +0200 debajo 1. Te adjunto un nuevo artículo en ingles que saldrá pronto en la revista del IASS.pdf" Estimado Valentin. "". definida y analisada completamente por Schlaich Bergermann und Partner y también nosotros propusimos en su día utilizar tensegrity. Mike Schlaich ------------------------------------Mike Schlaich.000€.Tensegrity Structures and their Application to Architecture Appendix D.1 TEXT/PLAIN 2822 bytes. "messeturm_IASS. Para torres creo que tensegrity es demasiado flexible ( y por lo tanto caro) para servir mucho. sc. La torre ha sido disenada. 147 . los arquitectos eran de gran ayuda ya que nos aconsejaron y también establecieron todos los contactos con el cliente. "" Adjunto mostrado 2 APPLICATION/OCTET-STREAM 476776 bytes. Las dos últimas están tomadas de la portada de una revista y de un libro de Schlaich. superando una de 30 m que existe en EEUU. pero cuando estuve en Alemania con Schlaich participé en el proyecto de una torre tensegrity. 2 IMAGE/PJPEG 519144 bytes.es> Arturo Ruiz de Villa Valdés <arturo. La primera es mía y puedes utilizarla como quieras.com Mon. Mejor contéstame a esta dirección.es arturo. "rostock-2.ruizdevilla@ono. También se realizaron modelos de elementos finitos para los detalles. Está rematada por una antena de acero inoxidable.jpg" Asunto: Re: Torre Tensegrity Rostock Para: De: Fecha: VALENTIN GOMEZ JAUREGUI <valentin. "rostock-3. Tiene una altura de 61. Qué programas usaste para el calculo de la torre? Para el análisis global de la estructura se calculó con el programa Sofistik realizando un cálculo no lineal (geométrico) en grandes deformaciones (teoría de tercer orden). "". 19 Jul 2004 11:24:45 +0200 Hola Valentín: He recibido tu correo relativo a las estructuras tensegrity. Un saludo.com> Wed. "". 25 Aug 2004 18:07:47 +0200 Hola Valentín: Espero que no sea demasiado tarde.Tensegrity Structures and their Application to Architecture Appendix D. Creo que ya lo sabes por
[email protected]. ya que está formada por 6 módulos tensegrity de 8. Si tienes alguna duda. "rostock-1. "". La torre en cuestión no es tensegrity pura. También me ha comentado que habías recibido contestación de Mike Schlaich. Arturo Ruiz de Villa Valdés PD: en el siguiente mail te envío un par de fotos de la torre. no dudes en preguntarme. tales como nudos de barras.jpg" 3 IMAGE/PJPEG 1106612 bytes. pero ahí van las respuestas: 1.3 m superpuestos unos encima de otros en los que las barras comprimidas del inferior tocan las del superior. placas de anclajes y para la aguja o antena de coronación. Personal correspondence Asunto: Torre Tensegrity 1/2 Para: De: Fecha:
[email protected]" 4 IMAGE/PJPEG 1116400 bytes.8 m que la convierten en la más alta del mundo (por lo menos que tengamos constancia).gomez@alumnos. 148 . Sabes cuáles son los movimientos y desplazamientos de la mentada torre? El desplazamiento máximo de la punta de la aguja es de 1200 mm bajo las cargas máximas de viento en servicio (sin mayorar). con lo que si se disminuye el tesado. como habrás visto en el artículo. Cuanto mayor es el tesado inicial de los cables. El pretensado se fijó de manera que bajo la carga de viento máxima en servicio (sin mayorar) ningún cable se destesara (1100 kN para los cables diagonales = aproximadamente el 30% de la carga de rotura del cable). Éste es un asunto sensible. las tensiones en las barras aumentan. 6. es tan decisivo el factor viento? El viento y el pretensado son las dos acciones principales sobre la estructura. La idea del encepado es colocar un peso en la base que evite literalmente que "el viento se lleve la torre por los aires". Del mismo se obtuvo un coeficiente dinámico que permitió considerar la respuesta dinámica de la estructura frente al viento y así poder mayorar su acción (Cd = 1. Personal correspondence 2. una estructura más flexible por efecto de los cables provoca un aumento de las flexiones de compatibilidad de las barras (que están rígidamente unidas entre ellas). o vosotros la modificasteis en función de la estabilidad del diseño? 149 . Estos nudos se mueven 850 mm. mayor es la rigidez de la estructura y menores sus deformaciones. Este estudio reveló que dado el carácter zig-zagueante y relativamente irregular de la estructura no eran de esperar fenómenos de resonancia y acoplamiento de los vórtices turbulentos. Este encepado se apoya en 6 pilotes de 500 mm. pues tiene gran influencia en el coste y en la deformabilidad de la estructura. Sin embargo. Qué pretensado. La torre es tan ligera que su peso propio es despreciable frente a las otras cargas. 5. Marg und Partner?). 3. Por otra parte. Fue un calculo estático. La estructura está anclada a la cimentación mediante barras pretensadas. o también dinámico? Se analizaron los modos propios de vibración de la estructura y a partir de ellos se hizo un estudio aerodinámico de la influencia del viento. aproximado. 4. Resumiendo fue un cálculo cuasiestático. es de acero inoxidable y está sujeta por seis cables anclados en los tres nudos superiores del último módulo de la torre. que permanece constante (mira la gráfica 8 del artículo de Mike). Qué tipo de cimentación se usó para estabilizarla? La torre está anclada en un encepado circular de hormigón que tiene un diámetro de 8 m y un canto de 1. El viento es la acción externa principal y totalmente determinante en el diseño. no se aprecia influencia del pretensado en la seguridad global de la estructura frente a rotura de los cables. tienen los cables para evitar el efecto del viento? Siendo una estructura eminentemente "hueca". pues condiciona el pretensado. Esta antena. El diseño de la torre vino definido por la oficina de arquitectos (von Gerkan.3).Tensegrity Structures and their Application to Architecture Appendix D. dado lo ligera que es.5 m (aunque por cuestiones arquitectónicas se recreció hasta unos 2 m). También es cierto que se puede recurrir a soluciones mixtas (por ejemplo la torre de Rostock no es tensegrity pura para reducir los movimientos. Recuerdo que debido al fuerte viento que hace allí se modificó el diámetro de la torre para que tuviera más inercia. pues los elementos en compresión se tocan). creo que costó 500. Arturo Ruiz de Villa Valdés 150 . Crees que torres de este tipo podrían usarse como estaciones de repetición. También se tuvieron que tantear diferentes diámetros de cables y barras. La torre es. el "Warnow Halle". son elementos muy vistosos que merecen la pena la inversión. receptores o similares. Sabes cuál fue el presupuesto y el precio real de la torre? Crees que el factor económico es poco conveniente para este tipo de construcciones? Espero no meter la pata con esto. pero a nada que sean algo estrictas veo inviable su empleo. pero mejor pregunta a Mike. sin duda. 8. También es cierto que en lugares emblemáticos como exposiciones. si se busca dulcificar el impacto estético de una antena puede que sean una buena solución. Yo creo que Mike propuso varias soluciones tensegridad y finalmente los arquitectos seleccionaron la altura y número de módulos. Personal correspondence Los arquitectos querían construir una torre que sirviera de referencia y símbolo del recinto ferial junto al edificio principal de exposiciones. No puedo asegurarte si la idea de realizar una tensegrity fue de Schlaich o de los arquitectos. recintos feriales etc. Por otra parte. 7. Un saludo. cara pues los cables lo son y además porque exige un proceso constructivo muy preciso. De lo que sí estoy seguro es de que en el proyecto constructivo sólo intervinieron en la iluminación y pavimentación del encepado y zócalos de apoyo de la torre.Tensegrity Structures and their Application to Architecture Appendix D. También podrían usarse como pararrayos en zonas urbanas. así como en la orientación de la torre. o las oscilaciones que sufren las harían desaconsejables? Desconozco las limitaciones de movimientos y oscilaciones de este tipo de estructuras. antenas.000 Eur. Snelson admits to knowing about the Gough article and doesn't seem to contest it though when he says "far background" and puts "replicated" in quotes I sense a certain amount of skepticism. I know where to get copies of the U.gomez@alumnos. 25 Aug 2004 22:00:47 -0400 Hi Valentín.com> Sun. If you read my historical essay you'll note that I also mention Emmerich who is referring to a completely different structure by Ioganson which must be what Snelson (and Rene Motro) are referring to in regard to not being pre-stressed. It is a standard tensegrity 3-prism exactly like the one I discuss in Section 2. 29 Aug 2004 12:38:21 .BUFFALO. Since Snelson doesn't directly contest it. I don't have a copy handy. Personal correspondence Asunto: Re: Tensegrity Dissertation Para: De: Fecha: VALENTIN GOMEZ JAUREGUI <valentin.com> Wed. As far as tensegrity is concerned. 151 . and the one Emmerich refers to is not.html Valentín. Sorry I can't send a copy of the article. and the Guggenheim seemed to think it valid if they displayed the replications as such. but Figure 13 is as I say. patents on the web.2. The claim seemed reasonable to me. Bob Asunto: Re: These sur Tensegrite Para: De: Fecha: " List for discussion of Buckm inst er <GEODESIC@listserv. and I'd imagine you can find pictures of the constructivist exhibition elsewhere though maybe not at the resolution that would allow you to judge Gough and Koleichu's claim. Emmerich and Fuller. The French ones I wasn't able to find. but I'm not that curious so don't bother sending them. I just use a dialup connection and I'll be patient if I know it's something worthwhile.unican.0400 Fuller's works" Ref: http://www. The Gough article is the only place where I have seen a claim that Ioganson originated a tensegrity prism.com/users/bobwb/tenseg/book/cover. you might as well treat it as valid. Ioganson just did that one structure and really didn't develop the form like Snelson.Tensegrity Structures and their Application to Architecture Appendix D.es> Robert W Burkhardt <
[email protected]> Bob Burkhardt <bobwb@lycos. Certainly the prism Gough exhibits must be prestressed if only a small amount.S. If you email it and it's over 500K let me know a day ahead to expect it and what size it is. It's just hair splitting. I don't know how controversial this replication is. but Figure 13 is just a picture of a reconstruction of a structure Gough (and/or collaborators) claims to have reconstructed from a picture of an exhibition.channel1. I'd be glad to look over your dissertation if you care to email it or whatever. see the bridge design I did recently (http://www. It is also important to remember that tensegrity structures are prestressed and that the effect of an exogenous load on one can vary quite a bit depending on the magnitude of the prestress. I don't think anyone has objected to resilience as a property of structure. Personal correspondence Yes I'd have to disagree with Snelson based on my experience. I'd agree with him as far as algebraic formulas are concerned as those have been found only for some simple structures. I know he made this claim in the past. For example.4 comment #2).3. Of course none of them has produced the array of interesting structures that he and Emmerich have.6).com/users/bobwb/synergetics/photos/x2l11chain6. and I use meta-constraints (Section 7. Some people see this as a defect as the materials are in a sense fighting against themselves instead of just against gravity. Many times I don't appreciate the full implications of a structure until I have it assembled and I've learned a lot by putting them together. I tend to state a lot of my conclusions (if I have any worthwhile ones) in the body of my exposition without reserving a special section for them. I think my methodology is very general (see Section 7. My use of nylon makes the load response of my structures somewhat problematic.2. but I think application of less elastic material would help a lot here.2. I think his tuning capability is somewhat limited.com/users/bobwb/synergetics/photos/x3dblprism1. but I'm not a civil engineer so maybe I don't understand the technical meaning of resilience. but I'm surprised if he's still saying it since there are so many engineers that say otherwise. or the X-Module arch (http://www. This works out for me since I think it's best to have them very close to the procedure they apply to.channel1. It is something like Hanaor's comment: "relatively high deflections as compared with conventional.2 in releases of July 29. 2004 or later) to good effect. I think his procedure amounts to minimizing a sum of second powers of lengths and perhaps that leads to the sturdiest structures (see the end of Section 7. For the most part I think iterative techniques are necessary. From what he's said. geometrically rigid structures" (Section 1.channel1. 152 . accurate. I'm not going to bother explicitly rebutting Snelson's claim since I think the book does that implicitly. 2004 or later). Getting effective load response is perhaps more difficult. efficient and allows tuning on the computer though I'd agree field work is very valuable.6 in releases of July 16. but I see it as a virtue (prestressed concrete is pretty popular right?) since it adds so much resilience.html). I sent him a copy of the first edition of my book but perhaps it didn't make much of an impression. I think these considerations may arise in many quarters due to the early experience with single-layer structures which were pretty wobbly although even there the deresonated tensegrity domes do pretty well I think. but perhaps by-products of the characteristics that lead to resilience.Tensegrity Structures and their Application to Architecture Appendix D. I've found there's a sort of art to setting up the mathematical programming problem and initializing it in various settings. but I think with the double-layer designs progress has been made there.html). If you see anything you think I've omitted I'd be glad to hear about it. On the other hand. and I don't think he can make the radical sort of experiments that I can numerically. I think I've done pretty well. Bob Asunto: Re: Some other points Para: De: Fecha: " List for discussion of Buckm inst er <GEODESIC@listserv. simpler approaches may do very well.ofram. Personal correspondence You found the French patents on line? I guess after all I'm curious enough that if you have electronic copies send them along thanks or best of all tell me where to find them on the net. I may not be familiar with it. there may be lots of other packages out there that do the job. but the interface is closely tailored to support the tensegrity work since that's all I use it for.com> Sun. 153 . My main emphasis is on design. For a lot of the tensegrities where the struts touch I think the geometry is fairly well determined and though my extremal approach works there.Tensegrity Structures and their Application to Architecture Appendix D.EDU> Bob Burkhardt <
[email protected] Fuller's works" Hi Valentín. I don't think the design difficulties are so great as with those where the struts don't touch and I am very interested in exploring these latter sorts unconstrained by problems with design software which I think my software allows one to do. 29 Aug 2004 15:13:16 . I may look into developing it for release after the revisions for the 2nd edition of the book are done. But a lot of my advantage is the way I've tailored my software to apply extremal techniques specifically to tensegrity.I can't find it). I haven't tried to distribute mine as I'm not ready to yet though I think I've set out the theory and practice behind it pretty well in my book. It would be difficult for other people to use I imagine. That I think is where I would find the most difficulty getting satisfaction out of other packages. but there too perhaps a commercial package could do better. For analysis. I can't say I'm familiar with other software packages. My customdeveloped extremal software is somewhat generic. and I imagine my analysis theory and practice (Chapter 7 and 7. I will be curious to see what you get from your software.maybe he still has them there somewhere -. and all of them admit structures where the struts touch. This Tower of Rostock reminds me of Tristan Sterk's towers (which were at www. If you have an internet link to this guy's work. I cite three fairly diverse definitions in my book. and I haven't heard of anyone who rules out a structure as a tensegrity in that regard except for your allusions.3 in particular) could benefit from the attention of a civil or mechanical engineer. If it's not Tristan's work. I'd be curious to see it.BUFFALO. I consider tensegrities where the struts touch to be true tensegrities.com though the sublink seems to have disappeared . I looked around and couldn't find anything. I think the custom extremal analysis software I use works well. Perhaps the main consideration for me is that the commercial packages are beyond my budget even when I've spotted one that might work. unican. I've been working on my French lately.es> Bob Burkhardt <bobwb@lycos. Bob VALENTIN GOMEZ JAUREGUI <valentin. Bob Asunto: Re: Renders Para: De: Fecha: VALENTIN GOMEZ JAUREGUI <valentin. Your software hasn't been helpful? The modules are folded by changing lengths of specific tendons I take it? Once you are more secure in the design. but if it's inadequate I'll ask for the British patents. Asunto: Re: Modules Figures Para: De: Fecha: Val. Bob stadium have an stadium design. It looks like you've got a sort of deterministic tensegrity where my sorts of procedures aren't necessary to find tendon lengths. I'd think any canned software could tell you stresses etc.Tensegrity Structures and their Application to Architecture Appendix D. Very simple but it should work. Thanks for sending it along.es> Bob Burkhardt <bobwb@lycos. Thanks for sending these along.unican..com> Tue.0400 154 . I hope you will post it on the net. I'll try out one of the modules eventually and see what I come up with. but they both look good to me. Looks like an interesting concept. Looks good so far.gomez@alumnos. 07 Sep 2004 16:46:24 . but mine can handle it as well although it's sometimes tricky to set up the problem so it knows what I'm talking about without running into singularities. Personal correspondence I'll see how I do with the French patents. I think I understand the better than the bridge. 08 Sep 2004 09:04:09 .com> Wed. The lower-left view in the first picture confused me for a minute since visually two struts look combined into one. I did a similar thing when analyzing a Geiger dome. You interesting approach to tensegrity.0400 Hi Val. Thanks for sending me the French ones.gomez@alumnos. Belfast (Northern Ireland) 156 . I am finishing my dissertation which is titled “Tensegrity Structures and their Application to Architecture”. Questionnaire Belfast. in my opinion the views of current engineers and architects are very important and could give me a wider perspective of what I am researching now and what they may have already studied. Throughout my thesis (you can find the table of contents on the following page). If you have any queries or problems with the questionnaire. Yours sincerely.Tensegrity Structures and their Application to Architecture Appendix F.p. buildings and public works. and some specialists are attempting to apply them to functional shapes. which will also include some history of Tensegrity. Hoping to hear from you very soon. this shall be facilitated. However. My name is Valentín Gómez Jáuregui..a. as it would be interesting to see how much is known about these structures in our profession. On a smaller scale.s. If you wish to read the final work. Questionnaire Appendix F. a MSc Architecture student at Queen’s University Belfast. Valentín Gómez Jáuregui. as I’m trying to gather all my information in the next month. I am trying to do something similar. the basic principles. It would help me immensely if you could fill out the brief interview that you can find enclosed. these types of structures are currently being studied and experimented upon. As you already know. some precedents and current examples. I would like to thank you in advance for your collaboration and I would be very grateful if you could reply a. At the moment and until mid September. Finally. Even if you do not know a lot about the subject. 18 June 2004 Dear Professor. This is the reason why I am addressing this letter to you. I would like to record your opinion. just let me know and I will send you a text version of the thesis. I will be carrying out theoretical and experimental research. please don’t hesitate to contact me. If you wish to remain anonymous. I would like to ask some simple questions about your personal details. journals. Please. Further education: 5. Do you usually travel with the motivation of visiting any architectural or public work. Age: 3. think. Questionnaire SECTION A Firstly. write the answer that proceeds. Degree: 4. Location: 4.3.3.. dam. other case. any information given by you will be referenced as "some architects/engineers/. Do you wish to remain anonymous? (In that case. etc. If you do not wish to answer. Location: 5. Profession 5.1. Name: 2.. 1. do not feel forced to and skip the question. could you write any recent example(s)? 11.2.1.) related to architecture or engineering? 8. If yes. If yes. etc? 10. Current profession: 5. I would like to ask you a few questions about general issues concerning your attitudes. Formation 4. If yes.2. bridge..") SECTION B Now. please. write Y (YES) or N (NO). Precedent professions and locations: 6. which? 157 . Do you usually read any other publication(s) related to other different subjects? 12.. such a building. 7.Tensegrity Structures and their Application to Architecture Appendix F. which? 9. in order to ask to them or. Sex: 4. Do you usually read any publication (books. where? 19. Do you know any professional or specialist dedicated to work/study about Tensegrity Structures? 17. please. Have you ever heard about any real or practical application of this sort of structures in any architectural or engineering work? (Y or N – If NO. as it would be interesting to see how much is known about these structures in our profession. And what personal opinion(s) have you about them? 16. where? 21. not in photographs or videos)? (Y or N – If NO. Would you like to receive a text version of the dissertation when it is finished? Thank you very much for your collaboration. go to question 21) 20.Tensegrity Structures and their Application to Architecture Appendix F. do not be troubled if you do not know a lot about the subject. I just would like to record your opinion. Questionnaire SECTION C Finally. 13. please. Have you ever seen any of them (in reality. go to question 22) 14. suggestion or question? 23. go to question 19) 18. Have you ever heard about Tensegrity structures before? (Y or N – If NO. If yes. Have you any other proposal. Please. please. Would you be able to suggest any possible application for this kind of structures (even if at first glance could seem unfeasible)? 22. 158 . If yes. I would like to ask you about the main subject of this letter. the Tensegrity Structures. What do you know about them? 15. Tensegrity Truncated Tetrahedron Tensegrity Truncated Octahedron Assembly of two Truncated Tetrahedron by strut-strut contact.Tensegrity Structures and their Application to Architecture Appendix G. Tensegrity Models All the models shown in this Appendix have been made by the author in 2004 by means of “Tensegritoy” elements. Assembly of Truncated Tetrahedron by strut-cable contact. 159 . Tensegrity Models Appendix G. Each strut is 30 cm length. Unfolded Foldable Module for Stadium.Tensegrity Structures and their Application to Architecture Appendix G. Tensegrity Models Foldable Module for Stadium. Folded 160 . Tensegrity Models Study of conglomerations for the Pyramidal Roof Option 1 Assembly of three Truntated Tetrahedra Option 2 Assembly of three Truntated Tetrahedra Option 3 (Chosen) Assembly of three Truntated Tetrahedra 161 .Tensegrity Structures and their Application to Architecture Appendix G. Tensegrity Structures and their Application to Architecture Appendix G. Tensegrity Models Study of masts for the Lightning Rod Study of masts Upper perspective Study of masts Upper perspective Study of masts Horizontal disposition Study of masts Frontal view 162 . Tensegrity Structures and their Application to Architecture Appendix G. Tensegrity Models Generation of domes from a Tensegrity Truncated Icosahedron Tensegrity Truncated Icosahedron Model made by the author (2004) Dome from Truncated Icosahedron Model made by the author (2004) Dome from Truncated Icosahedron Model made by the author (2004) 163 . Plans and renders Appendix H. Lightning rod from the Helix Tower 3. Tensegrity dome from the Truncated Icosahedron 2. Roofing for Stadiums by assembly of modules 4. Plans and renders Index: 1. Tensegrity pyramidal roof from Truncated 5.Tensegrity Structures and their Application to Architecture Appendix H. Footbridge by assembly of modules 164 . Tensegrity Structures and their Application to Architecture Appendix H. Plans and renders Tensegrity Truncated Icosahedron Tensegrity Truncated Icosahedron 165 . Perspective from below Lightning Rod. Perspective from above 166 . Plans and renders Lightning Rod.Tensegrity Structures and their Application to Architecture Appendix H. Perspective from below Lightning Rod. Perspective from inside Roofing for elliptical Stadium. . Perspective from outside Roofing for elliptical Stadium. Perspective from inside 167 .Tensegrity Structures and their Application to Architecture Appendix H. Plans and renders Roofing for elliptical Stadium. Plans and renders Pyramidal roof from assembly of Truncated Tetrahedra Pyramidal roof from assembly of Truncated Tetrahedra 168 .Tensegrity Structures and their Application to Architecture Appendix H. Tensegrity Structures and their Application to Architecture Appendix H. Plans and renders Footbridge from assembly of “Simplex” modules Footbridge from assembly of “Simplex” modules 169 . Tensegrity Structures and their Application to Architecture Appendix H. Plans and renders Footbridge from assembly of “Simplex” modules Footbridge from assembly of “Simplex” modules 170 . channel1.channel1. California: University of California Press.com/users/bobwb/synergetics/photos/index. Munich: Fakultät für Architektur.20. R. Topping.htm#sec:app Accessed December 2003-August 2004. C.2). GENGNAGEL. EMMERICH.chiroweb. 181 . (2002) “Tensegrity Models”. ed. No.W. R. EMMERICH. pp. Technische Universität München. Edinburgh. BURKHARDT. International Conference on Space Structures. Dynamic Chiropractic.. BURKHARDT. H. Vol. (1994) A Practical Guide to Tensegrity Design.channel1. [on-line]. (1999-2004) A Technology for Designing Tensegrity Domes and Spheres. (1966) "Reseaux".html Accessed December 2003-August 2004. University of Surrey. [on-line].html Accessed December 2003-August 2004. BURKHARDT. G.html KENNER. Cambridge (USA): Software Services. 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February 1981.net/snelson/rmoto.wisc. Vol.angelfire.July 2004. International Journal of Space Structures. Mass. International Journal of Solids and Structures.14. (USA): Birkhäuser Boston.htm SNELSON. Cambridge.kennethsnelson. Art News.S.April 1989.html Accessed August 2004 CALLADINE. E.edu/news/headlines/2002/Aug12. pp. Also available in http://www. 148. R. Kenneth Snelson Exhibition: The Nature of Structure. New York: Garland. Ontario (Canada). Portfolio and Art News Annual. January .ca/expo67. K.). and in Motro (2003). EDMONDSON.4. January . pp.).net/icons/art. Straddling the Abyss Between Art and Science”.cisc-icca.grunch.html Accessed July 2004. http://www. Also available in http://www. New York (USA).112127. (2004) Kenneth Snelson.R. (1987) A Fuller Explanation: The Synergetic Geometry of R. Buckminster Fuller. [on-line].kennethsnelson. (1981) “Kenneth Snelson.rwgrayprojects. http://www.com/mt/marksomers/42. New York (USA). Also available in http://www.C. 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London http://www. 7-10 April 2003. Unpublished BArch dissertation.se/~gunnart/TibertDocThesis.lirmm. (2001-2004) Camp Elsewhere's Tensegrity Shade Structures. (2003) Pliage/Dépliage de systèmes de Tensegrité. G. USA: Geiger Engineers.pdf TIBERT.htm Accesed June 2004. Structural Dynamics. Foldable tensegrity systems LE SAUX C. Norfolk (USA).cfm?RecordID=1 Accessed April-August 2004. DANIEL NG. [on-line]. S. The Crown Coliseum. Septembre 1999. (1972) Cords and countercords.mech.pdf Applications (False “tensegrities” included) BLACK. (1999) “Strut-Strut contact on numerical modelling of tensegrity systems folding”.uk/London_Zoo_Aviary..kth. SMAILI. Proposal for a tensile building system. (2003) “Deployable Tensegrity Masts”. BOUDERBALA M. 2001.au/Vizlab/viz_showcase/williamson_darrell/ Accessed May 2004. http://www. and Camp Elsewhere.org.. USA: Clownland Enterprises.kth. Extended Bibliography http://anusf. http://www. Queen’s University Belfast. CHEN. Université Montpellier II.anu. 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