1c stevenson phd thesis by Carolina (Stevenson) Rodriguez

THE SWIVEL DIAPHRAGM: a geometrical examination of an alternative retractable ring structure in architecture

Carolina M. Rodriguez Bernal

Thesis submitted to the University of Nottingham for the degree of Doctor of Philosophy

August 2 0 0 6

___________________________________________________________ ______________________________________________________________________________THE SWIVEL DIAPHRAGM: a geometrical examination of an alternative retractable ring structure

ABSTRACT

Since ancient times, deployable structures have been applied in architecture for a wide range of purposes. Over the last four decades, substantial scientific research regarding the structural and kinematical characteristics of these structures has been carried out. With advances in technology and the invention of stronger materials, new applications, for example retractable roofs and transformable enclosures have become possible. Several systems come under the classification of deployable structures. The present research focuses on the study of one group of structures named deployable lattice structures with rigid links [DLSRL]. These structures are formed mainly by rigid elements (i.e. bars or plates) linked by hinges that allow them to transmit movement from one element to the next. Research by others indicates that the demand for buildings with deployable structures such as DLSRL appears to be on the 1

increase, particularly in applications like retractable roofs or transformable enclosures . However, to date, built architectural projects have been quite limited. Possible reasons for this are, that there may be a high cost involved in the development of these type of structures, specialised literature on the subject is not widely available within the design sphere, their architectural potential has not been fully appreciated by designers and/or the existing systems may be very complex and not completely appropriate for architectural purposes. This thesis seeks to explore the last two possible reasons mentioned above. One of the main aims of this work is to develop alternative simpler systems, based on those already existing, that could potentially be easier to use within architecture. This work includes a historical background of the existing DLSRL systems, followed by a more detailed exploration of retractable ring structures. Retractable ring structures were chosen for further study since they showed more possibilities of variations that could potentially serve to create new systems. Computer generated and physical models were used to experiment with different morphological and geometrical configurations and alternative support conditions for retractable ring structures. As a result of this exploration, an new type of retractable ring structure, named: the â&#x20AC;&#x2DC;Swivel Diaphragmâ&#x20AC;&#x2122;, was proposed. Compared with existing DLSRL system, the Swivel Diaphragm showed improved characteristics, such as the option of fixed in position supports and interconnection between components. The swivel diaphragm was developed further, i

ABSTRACT

thus yielding two sub-systems: the ‘peripheral swivel diaphragm’ and the ‘central swivel diaphragm’. It was found that by interconnecting a series of swivel diaphragms within grids, a large range of designs with different deployment qualities were possible. In so doing, five methods of interconnecting swivel diaphragms in two and three dimensions, based on geometrical tessellations, were proposed. The resulted grid structures showed to have great potential for application within transformable devices for building surfaces. To illustrate this potential, five hypothetical design solutions using the swivel diaphragm within kinetic canopies, transformable enclosures and a retractable roof were suggested. It is considered by the author that the present thesis contributes to the existing knowledge of deployable structures by offering an alternative type of retractable ring structure that shows applicability for architectural purposes. This work presents a detailed explanation of how to manipulate the geometrical and morphological characteristics of the swivel diaphragm in order to adapt it to desired design conditions. Showing the flexibility and simplicity of the proposed systems aims to motivate or inspire designers and clients to consider their use in architecture. This work seeks to be of particular interest for audiences related with architecture, industrial design, civil engineering, mechanical engineering and geometry. Key words: transformable, kinetic, retractable, foldable, unfoldable, dismountable and convertible structures.

ABSTRACT

LIST OF PUBLISHED PAPERS

§ RODRIGUEZ, C. & CHILTON, J. (2005). Transformable and Transportable Architecture with Scissors Structures, Transformable Environments III, Eds. Kronenburg R. And Klassen F., Taylor & Francis, ISBN 0415343771. § RODRIGUEZ, C.; WILSON, R. & CHILTON, J. (May. 2005). Kinetic canopy using the swivel diaphragm mechanism, Proceedings of the II Latin-American Symposium on Tension Structures, Caracas, Venezuela. Extended abstract p. 22, paper on CD-ROM. § RODRIGUEZ, C.; CHILTON, J. & WILSON R. (Sep. 2004). Flat Grids Designs Employing The Swivel Diaphragm. Proceedings of the International Symposium on Shell and Spatial Structures from Models to Realization, Montpellier, France. Extended abstract p.316-317 and paper on CD-ROM. § RODRIGUEZ, C. & CHILTON, J. (Apr. 2004). Transformable and Transportable Architecture with Scissors Structures, Proceedings of the 3rd International Conference on Portable Architecture and design, Toronto, Canada, p. 87-93. § RODRIGUEZ, C. & CHILTON, J. (Dec. 2003). Swivel Diaphragm a New Alternative for Retractable Ring Structures, Journal of the International Association for Shell and Spatial Structures, Vol. 44 n.3, p. 181-188.

§ RODRIGUEZ, C. & CHILTON, J. (Oct. 2003). Swivel Diaphragm a New Alternative for Retractable Ring Structures, Proceedings of the International Symposium on New Perspectives for Shell and Spatial Structures, Taipei, Taiwan. Extended abstract p. 254255 and paper on CD-ROM.

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LIST OF PUBLISHED PAPERS

ACKNOWLEDGEMENTS

I would like to express my gratitude to my supervisors: Dr. Robin Wilson and Professor John Chilton for their support and guidance throughout this research. I would like to also extend my thanks to the IASS for acknowledging my work with the Hangai Prize Thanks to Marsh and Grochowski Architects for allowing me to use ideas of depoyable structures within their projects. Most of all my gratitude to my parents, Carmen I. Bernal and Lino Rodriguez, my brother Lino S. Rodriguez, my partner, David Stevenson for their help and patience. I am also grateful for the support of Manuel Bernal, Vicente Bernal and Lucia Bernal. Also thanks to the rest of my family and my friends .

ACKNOWLEDGEMENTS

CONTENTS

ABSTRACT.............................................................................................................. i LIST OF PUBLISHED PAPERS...................................................................................... iii ACKNOWLEDGMENTS............................................................................................ iv 1.

INTRODUCTION.......................................................................................... 1.1. Deployable structures in architecture..................................... 1.2. Scope and research questions.............................................. 1.2.1. Scope...................................................................... 1.2.2. Research questions................................................. 1.3. Objectives of the research..................................................... 1.4. Research outline....................................................................

1 1 6 6 7 8 8

HISTORICAL BACKGROUND........................................................................ 2.1. Introduction............................................................................ 2.2. Pantographic structures or structures that use scissormechanisms.......................................................................... 2.2.1. Applications of scissor-mechanisms in ancient times....................................................................... 2.2.2. Modern applications of scissor-mechanisms in architecture............................................................. 2.2.3. Other research on scissor-mechanisms .................. 2.3. Deployable Reciprocal Frames............................................. 2.3.1. Reciprocal frames in ancient times......................... 2.3.2. Modern applications of reciprocal frames.............. 2.3.3. Recent research on reciprocal frames.................... 2.4. Discussion..............................................................................

10 10

MORPHOLOGICAL CHARACTERISTICS OF DEPLOYABLE LATTICE STRUCTURES WITH RIGID LINKS.................................................................... 3.1. Introduction ........................................................................... 3.2. Pantographic structures or structures that use scissormechanisms ......................................................................... 3.2.1. General principals................................................... 3.2.2. Straight-scissor bars with central pivot...................... 3.2.2.1. Morphology............................................... 3.2.2.2. Examples of fixed position supports.......... 3.2.2.3. Examples of grid designs.......................... 3.2.3. Straight-scissor bars with an offset pivot .................. 3.2.3.1. Morphology............................................... 3.2.3.2. Examples of fixed position supports.......... 3.2.3.3. Examples of grid designs..........................

10 11 12 22 25 25 27 27 29

31 31 33 33 35 35 35 37 38 38 39 40

CONTENTS

CONTENTS .

3.2.4. Angulated-scissor bars............................................. 3.2.4.1. Morphology............................................... 3.2.4.2. Examples of fixed position supports.......... 3.2.4.3. Examples of grid designs.......................... 3.2.5. Multi-angulated-scissor bars.................................... 3.2.5.1. Morphology.............................................. 3.2.5.2. Examples of fixed position supports.......... 3.2.5.3. Examples of grid designs.......................... 3.2.6. Interconnecting-scissor bars (ring grids)................... 3.2.6.1. Morphology............................................... 3.2.6.2. Examples of grid designs ......................... 3.2.7. Deployable plates based on angulated, multiangulated or interconnecting-scissor bars ............ 3.2.7.1. Morphology............................................... 3.2.7.2. Examples of grid designs.......................... 3.3. Hybrid deployable lattice structures with rigid link................... 3.3.1. Combinations of straight-scissor bars with central pivot and straight-scissor bars with an offset pivot................................................................................... 3.3.1.1. Morphology............................................... 3.3.2. Combinations of straight-scissor bars with central and/or offset pivot with angulated, multi -angulated and/or interconnecting-scissor bars................ 3.3.2.1. Morphology............................................... 3.3.3. Irregular retractable ring structures with angulated, multi-angulated and/or interconnecting-scissor bars........................................................................ 3.3.3.1. Morphology............................................... 3.3.3.2. Examples of fixed position supports.......... 3.3.3.3. Examples of grid designs.......................... 3.3.4. The â&#x20AC;&#x2DC;base structureâ&#x20AC;&#x2122; ............................................... 3.3.4.1. Morphology............................................... 3.3.4.2. Examples of grid designs.......................... 3.3.5. Combinations with all types of scissor-mechanisms and deployable plates........................................... 3.3.5.1. Morphology ............................................. 3.4. Deployable reciprocal frames............................................... 3.4.1. Deployable reciprocal grids.................................... 3.4.1.1. Morphology............................................... 3.4.1.2. Examples of grid designs ......................... 3.4.2. Deployable reciprocal rings..................................... 3.4.2.1. Morphology............................................... 3.4.2.2. fixed position supports............................... 3.4.2.3. Examples of grid designs.......................... 3.4.3. Deployable reciprocal plates.................................. 3.4.3.1. Morphology............................................... 3.4.3.2. Options for supports.................................. 3.5. Discussion............................................................................... vi

42 42 44 46 47 47 48 52 53 53 53 56 56 58 59

59 59

60 60

61 61 65 66 67 67 68 68 68 69 69 69 70 70 70 73 73 74 74 75 76

CONTENTS

CONTENTS 4.

ARCHITECTURAL POTENTIAL OF DEPLOYABLE LATTICE STRUCTURES WITH RIGID LINKS.............................................................................................. 77 4.1. Introduction............................................................................ 77 4.2. Transformable versus transportable architecture..................... 77 4.3. Potential uses and architectural significance......................... 78 4.3.1. Coexistence with the environment.......................... 83 4.3.2. Participation with the observer................................. 83 4.4. Structural Morphology............................................................. 84 4.4.1. Type of components............................................... 84 4.4.2. External supports...................................................... 85 4.4.3. Material, scale, modularity and frequency.............. 86 4.5. Discussion............................................................................... 86 EXPLORATION OF ALTERNATIVE RETRACTABLE RING STRUCTURES.................. 5.1. Introduction............................................................................ 5.2. Advantages and disadvantages of fixed position supports within existing retractable rings...................................................... 5.2.1. Retractable rings with angulated, multiangulated and/or interconnecting-scissor bars...... 5.2.2. Deployable reciprocal rings..................................... 5.2.3. Retractable rings with deployable reciprocal plates....................................................................... 5.2.4. Retractable rings with the ‘base structure’............... 5.3. Experimental models of retractable ring structures................ 5.3.1. Experimental model one......................................... 5.3.2. Experimental model two.......................................... 5.3.3. Experimental model three....................................... 5.3.4. Experimental model four......................................... 5.4. Discussion...............................................................................

88 88

THE PERIPHERAL SWIVEL DIAPHRAGM ........................................................ 6.1. Introduction............................................................................ 6.2. General characteristics.......................................................... 6.3. Polygonal shapes................................................................... 6.4. Deployment angle u.............................................................. 6.4.1. Displacement of the joints during the deployment. 6.4.2 The change point..................................................... 6.5. Relationship between joints.................................................... 6.5.1. The kink angle a....................................................... 6.5.2. Dimensions between joints type A, F and B............. 6.6. Shape and size of the rigid elements..................................... 6.7. Discussion...............................................................................

100 100 100 103 105 107 111 113 113 115 119 123

THE CENTRAL SWIVEL DIAPHRAGM................................................................. 7.1. Introduction............................................................................ 7.2. General characteristics.......................................................... 7.3. Polygonal shapes...................................................................

124 124 124 127

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89 89 89 90 92 92 95 95 97 97 99

CONTENTS

CONTENTS 7.4. Deployment angle u.............................................................. 7.4.1. Displacement of the joints during the deployment.. 7.5. Relationship between joints.................................................... 7.5.1. The kink angle a....................................................... 7.5.2. Dimensions between joints type A, S, F and B.......... 7.6. Shape and size of the rigid elements..................................... 7.7. Discussion...............................................................................

128 131 135 135 136 138 140

GRID DESIGNS WITH THE SWIVEL DIAPHRAGM............................................. 8.1. Introduction........................................................................... 8.2. Methods of interconnection in two-dimensional grids............ 8.2.1. The fractal method.................................................. 8.2.2. The puzzle method.................................................. 8.2.3. The equivalent link method..................................... 8.2.4. The polypus method............................................... 8.3. Three-dimensional grids......................................................... 8.3.1. The parallel-rings method........................................ 8.4. Discussion..............................................................................

141 141 141 144 147 155 157 160 160 163

PROPOSED APPLICATIONS.......................................................................... 9.1. Introduction............................................................................ 9.2. Physical modelling and three-dimentional exploration.......... 9.3. Experimental projects............................................................. 9.3.1. Kinetic canopies...................................................... 9.3.1.1. Experimental project proposing a kinetic canopy with rigid covers............................ 9.3.1.1.1. Architectural aspects.................. 9.3.1.1.2. Morphological aspects.............. 9.3.1.2. Experimental project proposing a kinetic canopy with flexible covers........................ 9.3.1.2.1. Architectural aspects.................. 9.3.1.2.2. Morphological aspects.............. 9.3.2. Transformable enclosures........................................ 9.3.2.1. Experimental project proposing a transformable faรงade............................. 9.3.2.1.1. Architectural aspects................. 9.3.2.1.2. Morphological aspects.............. 9.3.2.2. Experimental project proposing a transformable enclosure for a botanic garden..................................................... 9.3.2.2.1. Architectural aspects................. 9.3.2.2.2. Morphological aspects.............. 9.3.3. Retractable roof...................................................... 9.3.3.1. Architectural aspects..................... 9.3.3.2. Morphological aspects................. 9.4. Discussion/ Evaluation............................................................

164 164 164 171 172

viii

172 172 173 176 176 177 181 181 181 181

185 185 185 188 188 188 191

CONTENTS

CONTENTS 10.

GENERAL CONCLUSIONS AND RECOMMENDATIONS.................................. 10.1 Conclusions and outcomes ................................................. 10.2. Lessons learned................................................................... 10.3 Recommendations for future work.......................................

194 194 198 199

APPENDIX A.......................................................................................................... APPENDIX B........................................................................................................... APPENDIX C.......................................................................................................... APPENDIX D........................................................................................................... LIST OF FIGURES..................................................................................................... LIST OF TABLES....................................................................................................... REFERENCES.........................................................................................................

200 203 204 205 207 217 218

Attached to this dissertation is a CD including videos of the structures discussed in each chapter. The videos are organized within a macromedia flash presentation, which requires flash player to be seen. (Flash player is available for free download at www. macromedia. com)

CONTENTS

INTRODUCTION

1.1. DEPLOYABLE STRUCTURES IN ARCHITECTURE The term Deployable Structures encompasses a wide range of systems. In this thesis this term describes structures that have the ability to change in size and shape through a process of deployment. Other terms such as transformable, kinetic, retractable, foldable, un-foldable, dismountable and convertible might also be found in other research to describe these kinds of system. For centuries, deployable structures have been applied in architecture for a wide range of purposes. For example, prehistoric nomadic cultures built a variety of deployable and transportable dwellings that employed timber frames linked with joints and covered with textiles. According to Gantes, “Small, light and compact structures such as nomad shelters from Iran, tepees from America and yurts from Mongolia or Beber tents from African deserts, their design imposed by the limited means of transportation and limited construction materials and equipment at the time, are some of the first historical examples of deployable structures” (Gantes, 2001). Furthermore, the Romans used foldable membranes to provide temporary shade in their coliseums (Escrig,1996). In medieval times, many inventors such as Leonardo Da Vinci (1452-1519), Palladio (1508-1580) and Verantius (1504-1573) developed numerous machines, movable bridges, umbrellas and other artefacts with deployable structures (Escrig,1996). Despite these antecedents, it was not until the twentieth century that deployable systems were to have a more important role in architecture. Since the early 1920s, deployable devices have been applied within portable buildings for emergency housing, commerce, industry, education, healthcare and military purposes. World-renowned inventors such as Buckminster Fuller (1895-1983) envisaged their application in futuristic projects. For instance, the 4D tower (1928), a lightweight, prefabricated, multi-storey apartments tower to be delivered anywhere in the world by airship. And the 4-D Dymaxion House (1929), a temporary, and transportable hexagonal plan house made from lightweight steel, ‘duraluminium’ and plastic and suspended from a central mast (fig. 1.1a). Its name was a result of the combination of words: Dy(namic)max(imum)ion and what Fuller denominated the fourth-dimension, which refers to transport and telecommunications (Baldwin, 1997).

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b) a) Fig. 1.1. a)Buckminster Fuller with his model for the Dymaxion House, 1929; b)Installation of a magnesium-framed geodesic dome designed by Buckminster Fuller,© Buckminster Fuller Institute, pictures taken from: http://www.designmuseum.org

Alongside these practical projects, substantial theoretical research was carried out on the integration of mobility within architecture and the environment. As expressed by Cook: ” The future environment will be where you may find it. If there is an upsurge in designing for mobility of society and mobility of facilities, so that the previous limitations of location and institution are overridden, we shall reach a point where the whole territory is part of a responsive environment ” (Cook, 1970). The pinnacle of this academic framework came with the European radical AvantGarde movement of the 1960s and 1970s. The idealistic work of groups such as Archigram, The Utopie Group, Super studio, Haus-Rucker Co. and Archizoom opened the intellectual discussion behind the dynamism and freedom of this new type of architecture. This modern movement introduced revolutionary concepts for a new way of living, which in turn reinforced the integration of mobility and speed within architecture (fig. 1.2). ”Critically, Archigram’s approach to architecture was fun, as illustrated by two of the group’s most memorable projects: Ron Herron’s 1964 cartoon drawings of a Walking City, in which a city of giant, reptilian structures literally glided across the globe on enormous legs until its inhabitants found a place where they wanted to settle; and the crane-mounted living pods that could be plugged in wherever their inhabitants wished in Peter Cook’s 1964 Plug-in City.” (© Design Museum, http://www.designmuseum.org) Transformable and transportable buildings turned to be very compatible with the dynamic and constantly evolving contemporary life style of that period. Dwellings were envisaged as 'environment-controlling' machines where the architectural space could be modified and customized on a daily basis. In this context, deployable structures captured the interest of researchers and designers. This idea also helped to encourage academic research on other capabilities of deployable and transportable structures 2

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such as speed of construction, reusability, ease of transportation and the compact size during storage.

Fig. 1.2. The Walking City, 1964,ÂŠ Ron Herron, Archigram, picture taken from: http://www.designmuseum.org and http://www.arch.tu-dresden.de, respectively

Recognized contemporary architects, in the second half of the twentieth century, concentrated on evolving these ideas and experimenting with new materials and structural mechanisms in order to develop their own distinctive proposals for transformable and transportable buildings. Amongst others are Frei Otto, pioneer in the construction of temporary and foldable lightweight fabric structures(Otto, 1974), and Santiago Calatrava, renowned for his kinetic buildings where movement plays a very symbolic and figurative role (Tzonis, 1999), (Caltrava, 2001) (fig. 1.3). Other architectural and engineering practices such as Festo have experimented with manufacturing and constructing projects with deployable structures.

Fig. 1.3. Planetarium of the Valencia Science Centre, Spain, 1991, picture taken from Tzonis,1999 3

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With advances in technology and the invention of stronger materials (such as fibre reinforced materials), new applications have become possible, for instance, the construction of large retractable covers for sport and entertainment venues where a variety of outdoor and indoor activities are performed. As argued by Jensen, nowadays retractable roofs are very desirable and a lot of the new stadiums incorporate them, since they could potentially maximise the monetary profits of the venue (Jensen, 2004). Parallel to the above developments, over recent years a new potential application of deployable structures has started to emerge. With the current awareness of climate change and global warming, there has been a growing interest amongst architects and clients for 'environmentally friendly buildings' that can efficiently adapt to diverse and changing weather conditions. For example transformable faรงades (faรงades that incorporate movable or adaptable devices) are becoming popular since they help to regulate internal climate conditions in unison with the external environmental changes, thus reducing the energy consumption for heating and cooling. As argued by John and Clements-Croome, currently, the market for deployable structures appears to be on the increase. (John & Clements-Croome, 2004). Until now scientific research, mostly accumulated over the last four decades, studied and classified deployable structures according to their morphology, complexity, functionality and/or kinematic characteristics. There are various existing classifications of deployable structures by different authors. For example, Hernandez and Zalewski (Hernandez & Zalewski, 1993), Escrig (Escrig, 1996) and Gantes (Gantes, 2001). The present research makes use of a classification developed by Hanaor and Levy at The Israel Institute of Technology (Hanaor & Levy, 2001) (fig. 1.4). This classification divides deployable structures, according to their kinematical features, into: structures with rigid links and structures with deformable links. According to their morphological features, structures with rigid links are divided into: lattice or skeletal structures and continuous or stressed- skin structures, whist structures with deformable links are divided into: strutcable systems and tensioned membrane. This classification differentiates deployable structures according to the type of components employed to achieve the motion and the load-bearing role played by the elements of the structure. Rigid link structures employ bars or plates linked by hinges to transmit movement throughout their elements. They can be assembled to form single-layer grids (SLG), double-layer grids (DLG) or spine structures. In contrast, deformable structures use flexible components such as cables or fabric to transmit movement throughout its elements. In lattice or skeletal structures the frame performs the major load-bearing function whilst in stressed- skin structures this role is played by the covering surface. 4

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Morphology Lattice DLG

Continuos

SLG

Spine

Pantographic (scissors)

Plates Folded Plates

Peripheral Scissors Linear deployment Angulated scissors (Retractable roofs) Masts and Arches

Rigid links

Radial Scissors

Radial deployment Others

Kinematics

Bars

Curved surface

Ruled Surface

Reciprocal grids (Dismontable)

Articulated joints

Strut-cable systems

Tensioned membrane

Deformable

Fabric

Hybrid

Pneumatic

Low pressure

Tensegrity

Others Ribbed

High pressure

Deployable lattice structures with rigid links

Fig. 1.4. Classification of deployable structures according to Hanaor and Levy (Hanaor & Levy, 2001)

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1.2. SCOPE AND RESEARCH QUESTIONS 1.2.1. SCOPE This research focuses primarily on the study of deployable lattice structures with rigid links [DLSRL] (highlighted in figure 1.4). This subject has drawn the interest of the author because after a comprehensive study of the literature available, it was noticed that a large amount of information on the structural capabilities of these types of system was available. However, compared with other types of deployable structures the architectural and aesthetic characteristics of DLSRL (such as the quality of the spaces enclosed by these structures or their participation with the observer) have been overlooked. It was found that since the 1960s considerable research has been undertaken on deployable lattice structures with rigid links, particularly on pantographic systems. The majority of this research reflects a strong interest in the structural, mathematical and geometrical aspects of these structures and methods to improve their construction and deployment process. More than 70% of the information on this subject reviewed for this research was produced at engineering or technological institutions such as The Deployable Structures Laboratory (DSL) at The University of Cambridge, UK; The Deployable Structures Group at The Department of Civil Engineering, Oxford University, UK, The Massachusets Institute of Technology , USA; The Faculty of Civil Engineering at The National Technical University of Athens and The Department of Engineering at The University of Queensland, Australia. It was also noticed that despite the large number of studies on this field, built architectural projects have been quite limited. As it is illustrated in chapter two of this thesis, the most representative examples have been by Emilio Perez Pi単ero (1936-1972) (Fundation Calasparra, http://www.regmurcia.com), Felix Escrig (http://personal. telefonica.terra.es/web/escrig-sanchez/proyectos.htm) and Charles Hoberman (http://www.hoberman.com), and

the most common applications have been in

temporary shelters and retractable roofs. This scarcity of built examples suggests that there are one or various factors, still not clearly specified, restricting or retarding the development of practical applications. However, the demand for buildings with deployable structures seems to be on the increase for applications such as retractable roofs, as stated by Jensen at the DSL, The University of Cambridge, UK:

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â&#x20AC;&#x153;Though many deployable structures have been proposed for retractable roofs, so far they have found limited use. This has often been caused by their many moving parts, complex hinges, discontinuous load paths or failure to provide adequate cover. In some of the structures that have been proposed, the sliding of different parts of the structure against each other causes unwanted friction, in others the repeated folding and unfolding of membranes has caused material failure. Despite these problems there is an increasing interest in these structures as they promise to provide designers with visually unique and efficient structuresâ&#x20AC;? (Jensen, 2004). 1.2.2. RESEARCH QUESTIONS The author considers that deployable lattice structures with rigid links have significant qualities to offer to the design of contemporary architecture. They could be used not only in transportable architecture, but also in transformable architecture. In addition to their conventional use in roofs, they could also be applied in other parts of the building; for example: faĂ§ades or internal partitions. The author suggests that possible reasons for the lack of built architectural examples with deployable lattice structures with rigid links are:

The existing systems are relatively new and specialised literature on the subject is not widely available within the design sphere.

The existing systems and mechanisms are perceived as being very complex to use within architecture.

The capabilities and architectural potential of the existing systems are not fully appreciated by designers.

There is a relatively high cost involved in their development, which may have some influence, since knowledge in this subject is likely to be achieved only by means of 'trial and error'. This may represent a bigger challenge for architects, engineers researchers and clients when one considers the cost implications of projects that may appear at first to be 'experimental'.

The author believes that the existing types of DLSRL show significant potential for further development into new types of structures that could prove appropriate for architectural purposes. In order to demonstrate this, the present work tackles two main research questions:

Is it possible to develop one or various alternatives, and preferably simpler DLSRL systems, based on those already existing, that could be easier to use within architecture?

Do such systems have potential to be applied within architecture? 7

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1.3. OBJECTIVES OF THE RESEARCH The present work aims to contribute to the existing knowledge of DLSRL, from an architectural point of view, by undertaking the research questions presented in section 1.2.2. In so doing, this work focusses on the following principal objectives: 1. To find within the existing DLSRL possible gaps in the research or areas that could serve as the basis to develop alternative systems with combinations or variations of the existing systems. 2. To identify factors that, if taken into account when designing, might help to fully exploit the capabilities of DLSRL. 3. To propose alternative DLSRL systems that demonstrate competitive attributes compared with existing ones, such as a simpler geometrical and morphological configuration, simpler components or less complex fixed position supports. 4. To develop methods and tools to manipulate the characteristics of the proposed systems in this thesis, that could be used by the architects or designers during the process of design. 5. To suggest applications in architecture of the systems developed during the research .

1.4. RESEARCH OUTLINE This thesis is divided in three main parts. In general terms part 1 identifies gaps of research within existing DLSRL, part 2 explores alternative DLSRL that could fill in those gaps and part 3 suggests applications of those alternative DLSRL within architecture. PART 1: EXISTING DLSRL Chapter 2 (HISTORICAL BACKGROUND) studies the historical evolution of DLSRL and reflect on their possible future development and applications. Three characteristics of DLSRL are identified as key aspects that could be manipulated by the designer in the process of improving the existing systems or create new ones. These aspects are: Morphology of the system components, option for fixed position supports integrated with the structure and examples of grid configurations. Chapter 3 (MORPHOLOGICAL CHARACTERISTICS OF DLSRL) studies DLSRL with emphasis on the three key aspects identified in chapter 2. Chapter 4 (ARCHITECTURAL POTENTIAL OF DLSRL) suggests architectural factors, which if taken into account when designing, might help to take full advantage of the capabilities of DLSRL. It highlights past uses of DLSRL in architecture, identifies systems that have seen little or no application in architecture and suggests more suitable uses for each type of system in different areas of architecture. 8

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PART 2 : EXPLORATION WITH ALTERNATIVE DLSRL Chapter 5 (EXPLORATION WITH ALTERNATIVE DLSRL) compares existing retractable ring structures in terms of their options for fixed position supports. It presents four experimental models developed through variations and combinations of the existing systems. It is considered in this chapter that two of the experimental projects (named: the peripheral swivel diaphragm and the central swivel diaphragm) show promising capabilities compared with existing systems. Hence, it is decided to study them further. Chapter 6 (THE PERIPHERAL SWIVEL DIAPHRAGM)

studies geometrical and

morphological characteristics of the peripheral swivel diaphragm. Advantages of the peripheral swivel diaphragm are identified. For example, the ring can be supported from many different points forming almost any regular or irregular shape and using joints with only one degree of freedom. The components of the system can be modified to reach different deployment stages and be designed to provide full coverage of the ring or to suit desirable aesthetic characteristics. Chapter 7(THE CENTRAL SWIVEL DIAPHRAGM) studies geometrical and morphological characteristics of the central swivel diaphragm. It was found that the central swivel diaphragm offered very similar advantages to the peripheral swivel diaphragm. It is noticed, however, that the central fixed position support presented in this system could limit, to a certain extent, its design possibilities. Chapter 8 (GRID DESIGNS WITH THE SWIVEL DIAPHRAGM) presents five methods, developed in this research, to interconnect swivel diaphragms and form grids. Four of the methods are for grids with two-dimensional movement and one method for grids with three-dimensional movement. PART 3: PROPOSED APPLICATIONS Chapter 9 (PROPOSED APPLICATIONS) presents five experimental projects developed for this research that show possible applications of the swivel diaphragm within kinetic canopies, transformable enclosures and a retractable roof. It evaluates these applications based on the architectural and morphological aspects of DLSRL suggested in chapter 4, which are: coexistence with the environment, participation with the observer, external supports, material, scale, modularity and frequency. Chapter 10(CONCLUSIONS AND RECOMMENDATIONS) contains a diagram with a modified classification of DLSRL showing the contribution of this research to the existing body of knowledge. It presents general conclusions, objectives achieved and recommendations for future research.

CHAPTER 1

HISTORICAL BACKGROUND

2.1. INTRODUCTION The purpose of this chapter is to provide a general historical overview of existing deployable lattice structures with rigid links [DLSRL]. This will help to comprehend the evolution of these systems and predict possible directions that future development may take. It will also serve to identify which types of structures that could be adapted or improved further to suit architectural purposes. In doing so, the first part of this chapter looks at two main types of DLSRL: pantographic structures (or structures that use scissormechanisms) and deployable reciprocal frames. More detailed information regarding the geometrical characteristics of each system will be given in chapter three. The second part of this chapter analyses possible gaps in the research and key elements that could be exploited in order to develop alternative systems .

2.2. PANTOGRAPHIC STRUCTURES OR STRUCTURES THAT USE SCISSOR-MECHANISMS Pantographic structures use the principle of the ‘pantograph’ to transmit movement (figure 2.1). According to Gantes, “ In a pantograph each rod of the framework has three nodes, one at each end, connected to end nodes of other members through hinges, and one at an intermediate point, connected to the intermediate node of another member by a pivotal connection. These pivots allows free rotation between the two bars about the axis perpendicular to the plane of the pantograph, but restricts all other degrees of freedom”(Gantes, 2001, p. 20). The mechanism used to connect the components of a pantograph will be described in this thesis as a scissor-mechanism or scissor-unit. It is also found in research by other as scissor-hinge or scissor-like-element.

End hinge / pivoting joint 1 Rod/bar 1 Intermediate hinge /pivoting joint Rod/bar 2 End hinge/ pivoting joint 2 a)

Fi g. 2 .1. a) Configuration of a scissor-mechanism, b) Deployment of a pantograph 10

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The term “scissor” was adopted fairly recently to describe this mechanism, due to its similarities with the common cutting device. Multiple interconnected pantographs form an assembly that transports forces from one element to the next. Throughout this process the entire system expands or contracts and thus changes in size and shape. 2.2.1. APPLICATIONS OF SCISSOR-MECHANISMS IN ANCIENT TIMES For centuries pantographs and scissor- mechanisms have been closely associated with terms such as deployable, foldable, mobile, portable and transportable. It is difficult to date the invention of the scissor-mechanism precisely; however, history suggests some of its extensive and varied uses. There is evidence that the ancient Egyptian, Middle Eastern and Greek cultures employed it in elementary folding chairs (fig. 2.2) and umbrellas. As expressed by Sanchez and Escrig, ”It [the scissor-chair] is built with three rods connected at an intermediate point. The scissor-chair is represented in the oldest engravings...the umbrella can be also an Egyptian invention because of its similarities with the palm tree. But it is in the Middle East where it was very well described” (Sanchez & Escrig,1997, p.24).

Fig. 2.2. Applications of the scissor-chair in ancient times a).Greek stool, picture taken from:.http://www.acornantiquesexmoor.co.uk b) Egyptian stool, picture taken from:http://www.dolcn.com

Since ancient times Mongolian communities have constructed portable Yurts using grids of straight wooden bars linked by scissor-mechanisms. This is a practice that continues to this day (fig. 2.3). The flexibility of this system allows them to build flat lightweight grids, easily packed into small volumes, which can later be quickly unfolded and bent to form the yurts. As expressed by Escrig: “It [the Yurt] is a tent stiffened by means of diagonal struts that move freely in rhombic grid and that can be folded in a bundle or extended as a fence”(Sanchez & Escrig,1997, p.24). It is also known that during the renaissance, inventors such as Leonardo Da Vinci (1452-1519) applied scissor-mechanisms to their machines, mobile bridges and umbrellas (Escrig, 1996) (fig. 2.4).

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Fig. 2.3. Construction process of a modern Mongolian Yurt, pictures form: http://www.ares.cz

Fig. 2.4. Leonardo Da Vinci’s machines employing deployable mechanisms, Pictures taken form: Candela et al.,1993

Despite the above examples, it was not until the 19th century and the emergence of the industrial revolution that the widespread use of this mechanism became commercially popular through its domestic and industrial use in, for instance, lift doors, lamps, chairs and clothes racks. These articles proved to be very successful, to such an extent, that modern versions of most of them are still in daily use (Mollerup, 2001). 2.2.2. MODERN APPLICATIONS OF SCISSOR-MECHANISMS IN ARCHITECTURE In architecture structures with scissor-mechanisms only started to receive further academic attention in the second half of the 20th century, as part of a new wave of revolutionary ideas that shaped contemporar y design. Architectural movements such as Archigram, reintroduced the idea of Portable Architecture as an integral part of their proposed style of life for the future ( Cook, 1999). One of the first architects interested in this topic was Buckminster Fuller (1895-1983). He used the concept of “deployability” in structures as part of his synergy theor y. With a geometric model named “Jitterbug”, he explained the essence of deployability within geometry. With this model Fuller illustrated how the cuboctahedron can be transformed into smaller polyhedrons such as the icosahedron, octahedron and tetrahedron, by a process of deployment. Applying the concept of deployability, he designed the first known unfolding geodesic dome for the USA

Army in 1958

(Edmondson, 1987).

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Fig. 2.5. Section of a Fuller’s deployable dome , pictures taken form: Baldwin, 1997

One of Fuller's students, the architect Emilio Perez-Piñero (1936-1972), took up work on this topic in the early 1960s. He became a pioneer in the construction of automatic deployable domes and different types of grids formed from pantographs. Furthermore, Perez-Piñero proposed a variation of the original scissor-mechanism where the intermediate pivoting joint was offset from the centre of the bar (fig. 2.6b).

Central pivoting joint

Offset pivoting joint

Fig. 2.6. a)Original scissor-mechanism, b) Variation with an offset pivoting joint

Using this variation he designed cylindrical and spherical shelters with two layers of pantographs. During his life he carried out a series of projects with deployable structures, patented many of his inventions and won several prizes for his work. One of his first projects was a movable theatre presented at the I.U.A. (International Union of Architects) in London in 1961. (fig. 2.7). It consisted of a deployable aluminium dome 11m high, 32m in diameter and 3,000 kg of weight (Candela et al., 1993). This project was never built, however, it won an architectural prize for ingenuity.

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Fig. 2.7. Model of the Perez-Piñero’s deployable cover for a transportable theater, pictures taken from: Candela et al., 1993

His first built project was a movable exhibition pavilion initially installed in1964 in Madrid and later in San Sebastian and in Barcelona, Spain. It consisted of a 8.000 m2 cover divided into 12 x 9m modules each weighing 500kg. Each module was supported on wheels and individually deployed. After this project Perez-Piñero built various scaled prototypes of deployable domes and three-dimentional frames (fig. 2.8), including one commissioned by Salvador Dali to exhibit some of his work (Candela et al., 1993). The architectural application of various of these prototypes is not completely clear in the literature available. Little is known regarding the conclusions drawn from these prototypes in terms of their structural efficiency during and after deployment. According to Gantes, “Piñero’s structures can be distributed over space and closed up to a fairly compact bundle. The members remain unstressed in the bundled and deployed state, except for their own dead weight. Furthermore, the geometry is such that ‘no stresses’ occur during deployment” (Gantes, 2001, p.20). Despite this, the stability of the structure can only be reached at the deployed state by using locking devices such as cables or intermediate members between the nodes. According to Gantes, “Such external mechanisms are an acceptable solution for a small, simple unit, or for a combination of simple units that are deployed one by one and assembled afterwards. However, for large structures consisting of several single units assembled in their

Fig. 2.8. Prototypes of Perez-Piñero’s deployable structures exhibited at the Emilio PerezPiñero Foundation, Calasparra, Spain, pictures taken form: http://www.regmurcia.com 14

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collapsed form and deployed at once, external stabilizing is problematic.” “This problem constitutes the major disadvantage of a broad class of deployable structures. In spite of this, however, Piñero established deployable structures as an interesting and viable form of space structures” (Gantes, 2001, p.21). Recognized contemporary engineers and architects, such as Theodore Zeigler and Santiago Calatrava have been inspired by Perez-Piñero's inventions. As expressed by Gantes: “In 1974 Theodore Zeigler, having recognized the weaknesses of Piñero’s structures, filed a patent of his own design (Zeigler,1976) ....The important change introduced by Zeigler is that his structure is supported in an erected form and does not require additional members or cables or any type of external locking mechanism.” (Gantes, 2001, p. 21). In subsequent patents Zeigler improved his dome design by variations in its geometrical shape and type of connections (Zeigler,1977,1981,1984) . Santiago Calatrava, world renowned for his dynamic designs inspired by animal movement, made his own contribution to the field of scissor-structures with his doctoral thesis entitled: On the Foldability of Space Frames, which he presented at the Swiss Federal Institute of Technology (ETH) in Zurich,1981(Caltrava, 2001). In this work, he explored possible ways to fold three-dimensional space frames, built out of rigid rods and movable joints, into two dimensions and then into one (fig.2.9).

Fig. 2.9. Model of Calatrava’s deployable dome, pictures taken from Tzonis, 1999 15

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Another particularly active architect in this field is Felix Escrig (Escrig, et al.,1987,1993,1996,1997,2000,2002). With the collaboration of Jose Sanchez and Juan Perez Valcarcel, he has studied and built, for more than fifteen years, various structures employing scissor-mechanisms. His work has focussed on the analysis and design of cylindrical and spherical grids built with a series of scissor-units. One of the biggest projects of this kind was designed and built by Escrig and Sanchez in 1994 to cover the 52 x 25m San Pablo Olympic swimming pool in Sevilla, Spain (fig. 2.10). This structure was formed by two linked domes that were deployed separately. Each dome comprised a series of aluminium scissor-bars and a textile cover. When fully folded, each dome occupied an area of 2.5 x 2.5 m by 5m high. The domes were suspended from a crane and deployed manually. During erection, the deployment was completed within four days (Felix Escrig web page: http://personal. telefonica.terra.es/web/escrig-sanchez/proyectos.htm).

Fig. 2.10. Deployable cover for the San Pablo Olympic swimming pool in Sevilla, Spain, 1994, pictures courtesy of Felix Escrig

Escrig has also developed covers using an innovative system that involves curved scissor-beams, which he patented. One of the most important constructions of this kind was designed and built by Escrig and Sanchez in 1998 to cover the Jaen auditorium in Jaen, Spain (fig.2.11. and 2.12.). This structure was formed from curved-scissor-beams spanning 42 metres. It was designed to slide upon rails, aided by electric motors within 20 minutes.

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Fig. 2.11. Deployable cover for the Jaen auditorium, Jaen, Spain, 1998 Pictures courtesy of Felix Escrig

Fig. 2.12. Model of the deployable cover for the Jaen auditorium, Jaen, Spain, 1998 Pictures courtesy of Felix Escrig

One of the most important recent developments of the scissor-mechanism was made by Charles Hoberman. In 1990 he founded The Hoberman Association; a company that specialized in the design of diverse movable artefacts. Most of these objects employ a mechanism invented by Hoberman. This new mechanism, commonly known as angulated-scissors is an alteration of the original idea of the scissormechanism. Instead of using rigid straight bars, Hoberman’s structures use bars bent at a particular angle w (fig. 2.13). This simple variation allows the realisation of a vast spectrum of polygonal and polyhedral shapes that are not possible with straightscissors. Many of Hoberman's structures, such as the Expanding Geodesic Sphere and Expanding Hypar, have been very popular and sold successfully worldwide as sculptures and children’s toys. Some of his sculptures measure up to five-storeys high (15m approx.) and weigh up to 2,500kg. Hoberman uses aluminium bars and stainless steel connectors to build these structures and computer-controlled motors to induce and control the motion (Hoberman’s web page: http://www.hoberman.com). 17

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Straight bar

Bar bent at a particular angle w

Fig. 2.13.a) Original scissor-mechanism, b) Angulated-scissors: variation where the bars are bent at a particular angle

Hoberman has also designed more complex architectural structures such as the Iris Dome (fig. 2.14) and the Hoberman Arch (fig. 2.16). The Iris Dome is a design for a retractable roof that opens and closes like the iris of an eye. This structure remains on the drawing board; however, a similar dome 6m in diameter was built by Hoberman for the German pavilion of the Expo 2000 in Hanover, Germany (fig. 2.15).

1 Closed

4 Open

Fig. 2.14. Hoberman’s Iris dome, © Charles Hoberman, pictures from: Hoberman, 1996

Fig. 2.15. Deployable dome for the German pavilion of the Expo 2000 in Hanover, Germany, pictures from: a) courtesy of © Charles Hoberman, b) from: http://www.hoberman.com

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The Hoberman Arch (fig. 2.16), a semi-circular aluminium latticework backed by translucent panels, was installed for the stage of the Olympic Medals Plaza for the Salt Lake 2002 Olympic Winter Games. This is the largest unfolding structure of its kind built to date, measuring 10.7m high and 21.3m in diameter. Due to the complexity and scale of the design it represented a big challenge from the engineering point of view. Hoberman developed the initial structural concept and the company Buro Happold was commissioned to develop it further. As stated in Buro Happold’s report: ”the arch is comprised of two main parts: a matrix of movable panels and a static arch that supports those panels. The concept of the screen was to use 96 panels, each with a skeletal frame constructed from aluminium box sections, which were clad in translucent fibre-reinforced panels. Four different shaped panels were used, which were radially arranged and layered over each other to form an almost solid screen whilst closed. The outermost panels attach to 13 radial slides on the static arch. The lower sections of the panels are supported by ‘trolleys’ which ride on tracks housed in the stage’s turntable located beneath the Arch. Thus all movable elements on the perimeter of the arch are directly attached to static supports. The total weight of the moving elements of the screen is about 6,800kg, the static portion an additional 6,800kg.“ (unpublished report courtesy of Buro Happold, p.3). Retraction was achieved by a combination of cables, pulleys, pin sliders and special hinges that worked coordinately during the opening and closing of the arch. Since the screen was positioned vertically there was a concern regarding wind loads, hence it was recommended that the screen would need to be retracted if the wind speed exceeded 40mph.

Fig. 2.16. Hoberman Arch, Salt Lake, USA, 2002, picture courtesy of © Charles Hoberman 19

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After Hoberman’s invention of angulated-scissors, further research on the subject, with particular emphasis on applications within the aerospace industry and sports venues, was carried out at the Deployable Structures Laboratory (DSL), at the University of Cambridge, UK, by P. E. Kassabian, Zhong You and Sergio Pellegrino (fig. 2.17 and 2.18). In their research they formulated a series of continuous beams with multiple kinks, connected by cylindrical joints named multi-angulated-scissors (Kassabian, You & Pellegrino, 1997) (fig. 2.19). This research proposed the use of structures of this kind in the design of retractable roofs. Multi-angulated-scissors have proven more convenient for structures with a high frequency of hinges. According to Kassabian, You and Pellegrino, “These structures [retractable roofs] fold along their perimeter and there is practically no limit to the range of shapes that can be achieved. A key property of these roof structures is that they have an internal degree of mobility that allows them to fold without any deformation of their members.”

Fig. 2.17. Prototype of a deployable ring, pictures taken from: http://www-civ.eng.cam.ac.uk

Fig. 2.18. Prototype of a retractable ring structure, pictures taken at The Deployable Structures Laboratory, Cambridge, UK. Courtesy of Sergio Pellegrino

Angulatedscissor bar

Multi-angulatedscissor bar

b) Fig. 2.19.a) Angulated-scissors, b) Multi-angulated-scissors

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Rodriguez and Villate at the National University of Colombia looked at alternative options to interlink rings of angulated-scissors (fig. 2.20). In this research they proposed to attach rings next to each other within flat grids creating new shapes for the interconnecting-scissors (fig. 2.21). These grids where proposed to be used within buildings as part of day lighting control systems (Rodriguez , 2000).

Fig. 2.20. Computer model of a retractable grid with square rings and interconnectingscissors, picture from: Rodriguez, 2000

Multi-angulatedscissor bar

Angulatedscissor bar

w w a) Fig. 2.21. a) Angulated-scissors, b) Interconnecting-scissors 21

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Figure 2.22 summarizes the evolution of the scissor-mechanism

a) Straight-scissor bars with central pivot, used by Buckminster Fuller

b) Straight-scissor bars with an offset pivot, proposed by Perez-Pi単ero

d) Multi-angulated-scissor bars, proposed by Kassabian, You and Pellegrino

c) Angulated-scissor bars, proposed by Hoberman

e) Interconnecting-scissor bars, proposed by Rodriguez

Fig. 2.22. Evolution of the scissor-mechanism

2.2.3. OTHER RESEARCH ON SCISSOR-MECHANISMS Other recent research on this subject has focussed primarily on the study of the stability of structures that use scissor-mechanisms, their geometrical characteristics, their process of construction and deployment, the design of the cladding and the methods to support them during operation. Stability within structures with scissor-mechanisms has been one of the key issues of research. Large-size projects containing a considerable number of bars and hinges, subjected to long-term use, require a detailed study to ensure proper and safe behaviour of the mechanisms before, during and after deployment. Stability in smallscale structures can be achieved by additional locking devices, such as cables or bracing bars, fixed to the intermediate members of the structure in its final stage of deployment. However, large-scale structures experience what Theodore Ziegler named as the 'self-locking phenomenon' (Gantes, 2001). This is a problem that occurs within large spherical domes where the hinges interlock each other at a specific point restricting the deployment. Zeigler patented a design that incorporated a special device that helps to overcome this problem (Zeigler, 1976).

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Furthermore, Perez-Valcarcel and Escrig (Perez-Valcarcel & Escrig, 1999) developed innovative methods to solve problems of angular instability presented in large-scale structures with deployable meshes of scissor-mechanisms. Langbecker and Albermani (Langbecker, 1999) developed a kinematic formulation to determine the structural response of certain foldable structures (such a single and double curvature structures with scissor-mechsnisms) in the fully deployed configuration under static loading. Moreover, Gantes at the National Technical University of Athens, Greece, detected incompatibilities between the members of certain pantograph chains that produce locking in the structures and consequent stress. Subsequently, he formulated a theoretical method for obtaining geometric constraints for deployable units, maintaining the desired features of deployability and 'stress-free' conditions (Gantes, 2001). There are other studies and PhD thesis that concentrate on studying problems of stability of structures with scissor-mechanisms, including: Krishnapillai and Zalewski (mentioned in Gantes, 2001); Logcher and Rosenfeld (Logcher & Rosenfeld, 1988) at the Massachuset Institute of Technology, USA and Raskin and Roorda (Raskin & Roorda, 1996) at the University of Waterloo, Ontario, Canada. Searching for more efficient geometry, morphology and connections for structures with scissor-mechanisms, Shan (Shan, 1993) applied formex algebra (a mathematical system that provides a medium for configuration processing) to manipulate and change conditions such as joint numbers, support arrangements and element connectivity within pantograph structures. Chen and You (Chen & You, 2001) propose structures using 3-dimensional over-constrained linkages, known as 'Bennett' linkages, instead of 2-dimensional foldable scissor-hinges. Ferrugia, Bezzina and Cole (Ferrugia, Bezzina & Cole,2002) proposed scissor-mechanisms that used joints with two degrees of freedom, which they named â&#x20AC;&#x2DC;calix dupletâ&#x20AC;&#x2122;. Using this joint they proposed curved spacial foldable structures similar to those previously designed by Escrig and Sanchez (fig. 2.11and 2.12). Other research has concentrated on examining alternative ways to improve the construction and deployment process of structures with scissor-mechanisms. For instance, Kawaguchi, Hangai and Nabana (Kawaguchi, Hangai & Nabana, 1993) proposed an analytical procedure to obtain the quickest way to erect a scissor structure and the optimum processes to fold it into a desired shape. Tsutomo and Tokai (Tsutomu & Tokai, 1997) included supplementary cables in scissor arches to make deployment easier. In addition, a project developed at the Vrije University of Brussels, Belgium (Block & Van Mele, 2003) studied the advantages offered by structures with scissormechanisms combined with structural membranes.

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Escrig and Perez-Valcarcel (Escrig & Perez-Valcarcel, 1987) have highlighted difficulties of covering structures employing scissor-mechanisms with surfaces attached to the principal frame. Textile covers that are easy to fold with the complete package have been used in the past. They appear to be compatible with deployable structures due to their flexibility and lightweight nature. However, textile materials experience problems during the deployment process due to repeated application and release of tension, hence they generally have a short-term life. Alternative solutions have been considered by Perez-Valcarcel, Escrig and Martin. They have studied effective covering solutions for curved expandable space grids implementing rigid elements that unfold together with the main structure (Perez-Varcárcel, Escrig, & Martín, 1994). In the case of angulated scissors, Hoberman experimented, in his Iris dome with layers of covering panels that glide over one another in synchronicity with the scissor structure to form a continuous skin-envelope when the dome is fully closed (Hoberman, 1990). Jensen at the Deployable Structures Laboratory, at the University of Cambridge, UK, explored on his research various shapes of covering elements for structures with multi-angulatedscissors, which will be explained further in chapter three (Jensen, 2004) (fig. 2.23).

Fig. 2.23. Prototype of a retractable plate structure by Jensen, pictures taken at The Deployable Structures Laboratory, Cambridge, UK. Courtesy of Sergio Pellegrino.

Some research has been carried out on ways to attach structures with scissormechanisms to fixed position supports. In this thesis, fixed position supports refer to the joints/elements that operate/pivot within in a permanent position and are used to support the DLSRL. Kassabian, You and Pellegrino the Deployable Structures Laboratory (DSL), at the University of Cambridge, UK, developed two methods of supporting retractable rings with multi-angulated-scissor, which are explained more in detain in section 3.2.5. Subsequently, You proposed a general method to create closed loops with fixed position supports named: the ‘base structure’, which is explained in section 3.3.4 (You, 2000).

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2.3. DEPLOYABLE RECIPROCAL FRAMES 2.3.1. RECIPROCAL FRAMES IN ANCIENT TIMES Reciprocal frames are three-dimensional grillage structures composed of mutually supporting beams. According to Chilton: "…In reciprocal frames…. each beam in the grillage is placed tangentially around a central closed curve so that it rests upon the preceding beam and this procedure is continued until the ring is complete". (Chilton, 1994) (fig. 2.24).

Fi g. 2.24.Three-dimensional view of a reciprocal frame, Picture from: Choo, Couliette & Chilton, 1994

The name ‘reciprocal frame’ was introduced by Brown (Brown, 1989) and derives from the way the beams are mutually or reciprocally supported. They can also be found in other research named as Nexorades (Baverel, 2000) or mutually supported elements (MSE) (Rizzuto & Saidani,2004). The idea of reciprocal frames have precedents in history. Native tribes in North America constructed their dwellings named teepees using wooden poles that met around a single point at the top (fig.2.25). According to Popovic, “the Indian teepee has a great similarity with the reciprocal frames. In both cases, the structure consists of mutually supporting beams and a roof opening” (Popovic, 1996, p. 2-2). Other ancient cultures such as native Eskimos and Navajo Indians built shelters with structures akin to reciprocal frames. However, according to Popovic, “It seems, that the country where the real reciprocal structure was born is Japan. In traditional Japanese architecture structures very similar to the reciprocal frame have been used for centuries” (Popovic, 1996, p. 2-4). There is evidence that the Japanese have used this principle of reciprocity in temples, bridges and shrines since the twelfth century (Popovic, 1996).

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a) b) Fig. 2.25. a) Example of a native North American teepee, pi ct ur e t ak en fr om : http://www.moderntradingpost.com , b)Example of a modern teepee, picture taken from: http://www.pbs.org

Two and three-dimensional structures very similar to reciprocal frames we re studied and applied in the design of roofs for churches and cathedrals by the medieval architect Villard de Honnecourt (Fl.c.1220s-1230s) (Popovic,1996). According to Popovic, “ an example of this is the chapter house at Lincoln [UK], designed by Alexander and built between 1220-1235” (Popovic, 1996, p. 2-7) (fig. 2.26a). During the Renaissance inventors such as Leonardo Da Vinci (1452-1519),

Sebastiano Serlio

(1475-1554) and John Wallis (1616-1703) explored various designs of grillages using reciprocal frame like configurations. Examples of these are the sketches drawn by Da Vinci in Vol.1 of the Codice Madrid and the Codex Atlantico, of beam grillages and temporary bridges designed with planar reciprocal frame-like assemblies (Popovic, 1996) (fig. 2.26b) .

Fi g. 2.26. a) Plan view of the roof of the chapter house at Lincoln cathedral, b) Flat beam grillage by Leonardo Da Vinci, pictures from: Popovic, 1996 26

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2.3.2. MODERN APPLICATIONS OF RECIPROCAL FRAMES Modern examples of roofs with structures similar to reciprocal frames can be found. For example, in Casa Negre, San Juan Despi, Barcelona, Spain (1915) and Casa Bofarul, Pallararesos, Taragona, Spain, (1913-1918) by the Spanish architect Jose Maria Jujol (Popovic, 1996). In recent years various examples of three-dimensional reciprocal frames like structures have been built, including the Spinning House (1985), the house Yu (1990) and the Seiwa Bunraku Puppet Theatre (1993) by the Japanese architect Kazuhiro Ishi (Popovic, 1996), the Pavilion (2002) and the Community Hall in Colney, UK by Graham Brown (http://www.south-norfolk.gov.uk). This last project was awarded the 2004 South Norfolk Design Award in the New Building Category (fig. 2.27 a).

Fi g. 2.27. a) The Community Hall by Brown. picture from: http://www.south-norfolk.gov.uk b) The pavilion by Brown. picture from: http://www.mts.net

“Since 1988, a number of small polygonal buildings have been built in the UK using reciprocal frame structures (Chilton, Wester and Yu, 1994). The spans in these buildings were in the range of 4.2 to13 metres and in all cases timber beams were used...with the exception of two buildings, which have a circular plan, all the plan forms were seven to twelve side regular polygons” (Popovic, 1996, p. 2-18). 2.3.3. RECENT RESEARCH ON RECIPROCAL FRAMES In recent years there has been a growing interest in the study of reciprocal frames. Research on this subject has assessed geometrical and morphological characteristics of such systems and their potential applicability in architecture. Due to the geometrical configuration of the reciprocal frames, it is possible to incorporate movement into the structure to make it deployable. According to Ban Seng Choo, Couliette and Chilton, “as the reciprocal frame roof has the same configuration as a camera shutter, it is easy to visualize its opening and closing.....during the retraction process, like the leaves of an

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iris diaphragm, each beam rotates individually about its external support, opening and closing the structure”(Choo, Couliette & Chilton, 1994) (fig 2.28). These types of structure are described in this thesis as: deployable reciprocal rings.

Fi g. 2.28.Plan views of deployable reciprocal ring structure, Picture from: Choo, Couliette & Chilton, 1994

In later research by Popovic and Chilton possible problems of deployable reciprocal rings where highlighted. “when the beams approach the closed position, the pitch of the beams increases very rapidly....this could be a problem when deep beams are used, as in the case when spanning large distances. The reciprocal frame becomes a retractable roof structure when each beam rotates around its outer support and also slides along its neighbouring supporting beam. Therefore, the supports need to be designed as hinges, with certain boundary conditions” (Popovic,1996). A parallel study by Wilkinson and Choo analysed the feasibility of very similar retractable roofs with reciprocal frames within sports stadiums (Wilkinson, 1997). Research by Ariza and Villate (Ariza, 2000), at The National University of Colombia, took a different approach, examining the foldability of reciprocal grids using only modules composed of four elements. They found that such modules are foldable when built with all the joints rotating either clockwise or anti-clockwise (fig. 2.29). In this thesis these structures are described as: deployable reciprocal grids.

b) modules of four elements

a) Fi g. 2.29. a) Deployable reciprocal grids, b)modules of four elements (pictures courtesy of Ariza) 28

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Despite the above research on deployable reciprocal frames, there is, as yet, no known building that employs such structures, however, the idea continues to be explored within academia. Geometrical and morphological capabilities of the reciprocal frames mentioned above are explained more in detail in chapter three.

2.4. DISCUSSION After the historical overview presented in this chapter it can be argued that the application of deployable lattice structures with rigid links in architecture is relatively recent. The most representative academic research on this area has been carried out since the late 1950s for pantographic structures and the 1990s for deployable reciprocal frames. Apart from numerous scale prototypes, there are limited built examples of these types of structures in architecture. In the case of retractable reciprocal frames, there is no known project built to date. The majority of built projects with pantographic structures have been by Perez- Pi単ero, Hoberman and Escrig and collaborators. Past research on deployable lattice structures with rigid links has mainly focussed on the study of the stability of these structures during and after operation, their geometrical characteristics, their process of construction and deployment, the design of elements for cover and the methods to support them from fixed position supports. It can be seen in this chapter that the original concept of the scissor-mechanism has gradually evolved into new, more elaborate configurations that involve offset pivots, angulated bars, multi-angulated bars or interconnecting-scissor bars. Similarly, developments within reciprocal frames have lead to the creation of deployable reciprocal rings and deployable reciprocal grids. These advances have been carried out by means of geometrical and morphological variations of the original systems. This suggests that in future research, new systems may result from further variations of those existing. The author considers that there are three key aspects of DLSRL that could be manipulated in order improve the existing systems or create alternative ones. These aspects are: Morphology of the components of the structure: The key elements in DLSRL are the joints. The bars can potentially adopt diverse forms as long as the relationship between the joints is preserved. This constitutes a powerful tool for design, since it allows one to create alternative structures with very different aesthetic qualities and type of movement by only changing the shape of the bars.

CHAPTER 2

Fixed points of supports integrated within the structure: One of the main difficulties with the majority of DLSRL is the way of attaching them to fixed position supports. In these systems all the elements (mainly bars and joints) are in constant movement. Hence, it is difficult to find more that one point from which to support the structure during its deployment process. Existing systems could be improved by integrating fixed position supports within the design of the structure. Configuration of grids: Due to the nature of the interconnectivity within DLSRL with rigid links, when they are working within a network, the movement produced in one point is spread smoothly throughout the rest of the structure. Hence, grids formed from these structures offer a wide range of possibilities in design. New grid designs can result from alterations of the shape and/or direction of movement within existing configurations, or by changing the type of connection used to spread the motion. The next chapter studies in more detail the three aspects mentioned above within each system classified as a DLSRL.

CHAPTER 2

MORPHOLOGICAL CHARACTERISTICS OF DEPLOYABLE LATTICE STRUCTURES WITH RIGID LINKS

3.1. INTRODUCTION The purpose of this chapter is to illustrate the main morphological and geometrical characteristics of the systems classified as deployable lattice structures with rigid links [DLSRL]. The classification developed by Hanaor and Levy at The Israel Institute of Technology (IIT) (Hanaor & Levy, 2001), previously shown in fig. 1.4, is one of the most complete classifications of deployable structures found by the author. However, it ommits some systems within DLSRL, such as multi-angulated scissors or rings with deployable reciprocal plates. Hence, taking the classification by IIT as base, a more elaborate classification of DLSRL was developed for this research (fig. 3.1). This new classification divides DLSRL according to their morphology into three main types: pantographic structures (or structures that use scissor-mechanisms), hybrid deployable lattice structures with rigid links and deployable reciprocal structures. It also divides these structures according to their type of movement or kinematics into: structures with radial deployment (or retractable rings) and structures with linear movement. This chapter aims to show the differences and similarities between the various types of DLSRL. This will serve as a platform to experiment with alternative systems resulting from a combination of existing systems. Three particular aspects of each system are studied within this chapter: morphology of the system, examples of grid designs and examples of fixed position supports. Each section provides diagrams, developed for the present thesis, illustrating the main morphological characteristics of the system, their type of deployment and options for fixed position supports. Information regarding the geometry and operation of each system has been taken from different sources as referenced in each section.

CHAPTER 3

Morphology Lattice structures 3.3. Hybrid DLSRL

3.4. Deployable reciprocal frames

3.3.1.Combinations of straightscissor bars with central pivot and an offset pivot

3.4.1. Deployable reciprocal grids

Linear deployment

3.2. Pantographic structures (Scissor-mechanisms)

3.2.2. Straight-scissor bars with central pivot

3.2.4. Angulated-scissor 3.3.2. Combinations of straightscissor bars with central and/or bars offset pivot with angulated, multi-angulated and/or interconnecting-scissor bars

Bars

Radial deployment (retractable rings)

Rigid links

Kinematics

3.2.3. Straight-scissor bars with an offset pivot

3.2.5. Multi-angulated scissor-bars (Concentric rings) 3.3.3. Irregular retractable rings structures with angulated, multi-angulated and/or interconnecting-scissor bars 3.4.2. Deployable reciprocal rings 3.2.6. Interconnecting scissor-bars (ring grids)

Plates

3.3.4. The â&#x20AC;&#x2DC;Base structureâ&#x20AC;&#x2122;

3.2.7. Deployable plates based on angulated or multi-angulated or interconnecting-scissor bars

3.3.5.Combinations with all types of scissor-mechanisms and deployable plates

3.4.3. Deployable reciprocal plates

Retractable rings Fig. 3.1. New classification of deployable lattice structures with rigid links based on the classification of deployable structures by Hanaor and Levy (shown in fig. 1.4). 32

CHAPTER 3

3.2. PANTOGRAPHIC STRUCTURES OR STRUCTURES THAT USE SCISSOR-MECHANISMS 3.2.1. GENERAL PRINCIPALS In a basic scissor-mechanism or scissor-unit, the rigid components (i.e. bars, rods or plates) pivot about an axis of rotation perpendicular to the plane on which they operate (fig. 3.2a). They are linked by an intermediate pivoting joint or hinge that allows only one degree of freedom. If the central joint is fixed in position to a support the rigid components can only rotate about this support. However, when various scissormechanisms are linked together forming a pantograph (fig.3.2 b) and any of the joints is fixed in position to a support F, the overall system experiences a controlled deformation during the deployment process resulting in movement of all other points relative to that fixed support. With some scissor-mechanisms, such as straight-scissor bars with central pivot, it is only possible to fix one of the joints to a support. To control the movement of the structure other joints are forced to follow a fixed path, described in this chapter as constraint path. For example in the pantograph of figure 3.2b) joint F is fixed in position to a support and the rest are constrained to move in a vertical direction. Constraint path Axis of rotation Intermediate pivoting joint fixed in position Rigid component F

Fixed position support.

F b)

Fig. 3.2. a) Operation of a basic scissor-mechanim fixed in position by the intermediate joint b) Deployment process of a pantograph fixed in position to a support

Structures that use pantographs are formed from two layers of elements, which operate in parallel planes. They are represented with the colours black and red in this chapter. The geometry and deployment characteristics of pantographs depend on the relationship between the joints A, P and B in each scissors-mechanism that comprises such a structure (fig. 3.3). The angle w formed between joints APB differentiates each type of scissor-mechanism (see table 3.1). If the angle w =180ยบ and AP is the same length as PB, the unit is a straight-scissor with central pivot. If the angle w =180ยบ but AP is of different length to PB, the unit is a straight-scissor with an offset pivot. If the angle w <180ยบ, the unit is an angulated-scissor. When the rigid element comprises more that 33

CHAPTER 3

three joints it could be a multi-angulated-scissor or a interconnecting-scissor. Structures with combinations of various types of scissor-mechanisms have been classified in this chapter within the category of hybrid deployable lattice structures with rigid links. Complex grids can be designed by interconnecting series of identical or dissimilar pantographs. Most of the two-dimenional grids illustrated in this chapter are based on polygonal patterns and the three-dimensional grids on polyhedral geometry.

End pivoting joint 1 A

A1 Rigid component 1

w P

intermediate pivoting joint

PB Rigid component 2 End pivoting joint 2 B1

B Fig. 3.3. Basic scissor-mechanism

Type of scissor-mechanism

Angle w

AP versus PB

Q=number of joints in each rigid element

Straight-scissors with central pivot

w=1808

AP = PB

Q=3

Straight-scissors with an offset pivot

w=1808

AP = PB

Q=3

Angulated-scissors

w<1808

AP *,] PB

Q=3

Multi-angulated-scissors

w<1808

AP *,] PB

Q> 3

Interconnecting-scissors

w *1808

AP *,] PB

Q>3

Table. 3.1. Differences between scissor-mechanisms

The following nomenclature is used in this chapter to describe the various components of the scissor-mechanisms: F = fixed position support w = angle between the intermediate joint and the end joints = end joint = intermediate joint = constraint path

CHAPTER 3

3.2.2. STRAIGHT-SCISSOR BARS WITH CENTRAL PIVOT A

A1 AP

P d

c B1

d1 Fig. 3.4. Straight-scissor bars with central pivot

3.2.2.1. Morphology Straight-scissor bars with central pivot (fig. 3.4) are the most elementary types of scissormechanisms. According to Santiago Calatrava, pantographs formed from identical units of this kind are fully foldable in two directions if they fulfil the following basic conditions (Candela et al., 1993, p.39): AP = AD = PB1= DB1

(3.1)

and

w =1808

(3.2)

Hence: d1= 2 AP sin (a/2)

(3.3)

d2 = 2 AP cos (a/2)

(3.4)

b = 18082 a

(3.5)

b= d

(3.6)

a= c

(3.7)

Area of APB1D = d1*d2 2

(3.8)

Detailed information regarding other geometrical characteristics of this type of system can be found in the work of Escrig, Calatrava, Zeigler and Gantes, (Escrig,1996); (Calatrava, 2001) (Zeigler,1976,1977,1981,1984); (Gantes, 2001) . 3.2.2.2. Examples of fixed position supports This type of scissor-mechanism can be used to form planar or three-dimensional pantographs. In planar pantographs the joints follow different paths, during the process of deployment, depending on the fixed point chosen to support the frame and the movement path used as the constraint. Figure 3.5 shows two identical planar grids that use straight-sicssor bars with central pivot. Grid (a) has a left corner fixed position support and the lower most joints are constrained to move along a horizontal path. Grid (b) has a central fixed position support and the central horizontal joints are constrained to move along a horizontal path. 35

CHAPTER 3

Constraint p a t h

Fig. 3.5. Deployment process of two identical planar grids using straight-scissor bars with central pivot with different types of fixed in position supports and movement paths

Figure 3.6 illustrates the different loci for the joints in these two identical grids. Notice that the position of the joints in the initial and final stages of deployment is the same for both grids. However, in grid (b) (with a central support) the joints trace straight or circular paths, whilst in grid (a) (supported in the corner) the joints follow irregular curved paths. This is of relevance when designing the type of support for a particular grid design, since the support conditions may vary depending on the joint chosen to be fixed in position. Fixed position support C onstraint path

Fixed position support

Fig. 3.6. Diagram of the joint displacements in two identical planar grids using straight-scissor bars with central pivot with different types of fixed in position supports and movement paths. Note: Colours indicate different joints 36

CHAPTER 3

3.2.2.3. Examples of grid designs Grids with straight-scissor bars that operate on one plane work efficiently for small-scale structures with only a few pantographs. However, large-scale grids with a greater number of pantographs present an extremely low structural efficiency, since the depth of the structure relies entirely on the depth of the individual bars. In addition, according to Gantes, “In the case of two-way grids [or grids that operate on one plane] there are problems of instability due to non-trianguation” (Gantes, 2001, p. 23). Threedimensional space grids with this type of scissor-mechanism demonstrate a better performance compared with two-dimensional assemblies. The simplest way to assemble various units within a three-dimensional configuration is by forming a triangular prismatic base unit referred to in past reseach as a trissor (threescissor)(Gantes, 2001) (fig. 3.7.b). According to Gantes, “the trissor is defined as a mobile assembly of six struts joined together in three pairs of scissor-like-elements” (Gantes, 2001, p. 22). Space grid structures can be formed by interconnecting series of trissors (fig. 3.7.a). However, the larger the structure, the greater the depth d required for its structural stability . There are different ways to constrain or lock the deployment of the grid when it reaches the required depth. For example, additional devices such as cables, fabrics or retractable bars can be fixed to the end of the scissor-units.

Additional cable to restrict deployment

b) Trissor

Fig. 3.7. a) Deployment process of a space grid with trissors b) Configuration of a Trissor 37

CHAPTER 3

3.2.3. STRAIGHT-SCISSOR BARS WITH AN OFFSET PIVOT 3.2.3.1. Morphology In straight-scissor bars with an offset pivot (fig. 3.8) the segment AP is of different length to the segment PB. This variation allows pantographs comprising identical units of this kind to achieve curvature (fig 3.9). A

a P

Offset pivoting joint

b/2 b

b/2

d D

l B

Fig. 3.8. Straight-scissor bars with an offset pivot

According to Escrig, cylindrical configurations formed from straight-scissor bars with an offset pivot, based on regular polygons with centre in S (such as the example of fig. 3.9) are foldable if they fulfil the following basic conditions (Candela, 1993, p. 110):

AP+ PB1 = AD +DB1 AP = AD =

SP sin f cos ( b/2 +f/2)

(3.9)

PB1 = DB1 =

(3.11)

Base polygon (decagon) 1

(3.10)

w =1808 SP sin f Cos ( b/2 +f/2)

A1 P

(3.12)

A B1

c = 1808

f/2 f 2

4 Fig. 3.9. Deployable arch with straight-scissor-bars with an offset pivot

CHAPTER 3

In this type of cylindrical configuration the maximum stage of deployment is reached when c = 180ยบ (fig. 3.9). At this point the system adopts a polygonal shape in the outer perimeter directly proportional to the value of the angles a and l. From inspection, these angles can be found from the following equations : a = 2arcsin (PB/ AP)

(3.13)

l = 180 8 - arccos (PB/ AP)

(3.14)

3.2.3.2. Examples of fixed position supports Figure 3.10 shows two identical cylindrical structures formed from straight-scissor bars with an offset pivot. Structure (a) is fixed in position at one of the last end joints F of the structure and deploys following a linear path described between the last two end joints (joints A and F). Structure (b) is fixed in position at the intermediate joint F of a central scissor-mechanism and deploys following two circular paths described by the end joints of the same scissor-mechanism (A and B). As may be seen in figure 3.11 in both grids the joints describe curved lines during the deployment process. However, the patterns of movement are completely different. It is of particular importance to take this into account when designing the support conditions, since the stability of the structure, the spread of the motion throughout its components and the aesthetics qualities of the deployment may be affected depending on the position of the fixed joint. A F 1 B A F 2

B A

B 3

F B

A 1

A 2

F A 3 a)

F A F

4 Constraint p a t h

Fig. 3.10. Deployment process of two identical cylindrical grids using straight-scissor bars with an offset pivot and with different types of fixed in position supports and movement paths 39

CHAPTER 3

Fixed position support

Constraint p a t h a)

Fig. 3.11. Diagram of the joint displacements in two identical planar grids using straight-scissor bars with an offset pivot using different types of fixed in position supports and movement paths

3.2.3.3. Examples of grid designs Consecutive arches formed by straight-scissor bars with an offset pivot can be linked with additional scissor-units used as bracing to create cylindrical structures (fig. 3.12 and 3.13).

Base cylindrical space frame Arch of straight- scissor bars with an offset pivot

2 a)

3 b)

Fig. 3.12. a) Three different alternatives of bracing, b) Half cylindrical configuration using straight-scissor bars with an offset pivot

Surfaces with double curvature are also possible by connecting arches of straightscissor bars with an offset pivot in diverse directions (fig. 3.14). Detailed descriptions of the three-dimensional geometry of these structures can be found in the work of Escrig, Perez-Valcarcel and Martin, Zeigler and Gantes (Escrig, Perez-Valcarcel & Martin, 1993); (Zeigler,1976,1977,1981,1984); (Gantes, 1993, 2001)

CHAPTER 3

Scissor-units locked in open position and used as bracing

Fig. 3.13. Deployment process of a half cylindrical structure using arches of straight-scissor bars with an offset pivot

Rails to direct the movement of the structure during deployment Inter-connected arches of straight- scissor bars with an offset pivot 1

Fig. 3.14. Half spherical configuration with arches of straight-scissor bars with an offset pivot interconnected in different planes, pictures from: Candela, 1993

CHAPTER 3

3.2.4. ANGULATED-SCISSOR BARS 3.2.4.1. Morphology Angulated-scissor bars were first proposed by Hoberman to form closed polygonal configurations with radial deployment (Hoberman, 1996). Each scissor-unit comprises two rigid angulated elements linked at an intermediate joint P and linked to other scissor-units by end joints (i.e. inner joint A and outer joint B) (fig. 3.15.The angulated bars are distributed in two layers (black and red), when the elements in one layer move clockwise, the elements in the other layer move anti-clockwise. Inner joint

Angulated-scissor bar 1

Intermediate pivoting joint

Angulated-scissor bar 2

P b

Outer joint

Fig. 3.15 Angulated-scissor bars

Series of these units are connected to form a ring configuration that expands from a closed to an open stage (fig. 3.16). The terms closed and open are used in this document to describe the two absolute states of deployment of the ring. In the equations in this chapter no consideration is given to the sizes of the bars or hinges. However, for clarity in the diagrams, the rings are shown open and closed at the point where the hinges touch each other (fig. 3.17). If AP =PB, when the ring is closed, the inner joints A coincide at the centre of the polygon and the radius Rclosed from the centre to the outer joints determine the size of the ring at this stage. When the ring is open, the inner and outer joints, A and B respectively, coincide at the perimeter of the polygon and its size is determined by the radius Ropen from the centre to the outer joints. The angle w formed between joints A, P and B is commonly known as the kink angle (Jensen, 2004). This angle determines the shape of the polygon obtained at the absolute states of deployment. Numerous regular and irregular polygonal configurations can be obtained using angulated-scissor bars (fig. 3.16). From inspection, it is possible to formulate the value of the kink angle w necessary to build a desired regular polygon with identical angulated-scissor bars. This value is inversely proportional to the number of edges n of the final polygon: w = 180ยบ- 360ยบ / n

60ยบ [ w < 180ยบ

(3.15) 42

(3.16) CHAPTER 3

w 1 Closed Decagon, w=1448

4 Open

w 1 Closed Nonagon, w=1408

w 1 Closed Octagon, w=1358

w 1 Closed Heptagon, w=128.578

4 Open

w 1 Closed

4 Open

Hexagon, w=1208 w

1 Closed Pentagon, w=1088

w 1 Closed Square, w=908

4 Open

w 1 Closed Triangle, w=608

4 Open

Fig. 3.16. Examples of regular polygonal configurations with angulated-scissor bars 43

CHAPTER 3

The most elementary assembly that could be formed with identical angulated-scissor bars is a triangular ring, where the value of w is 60Âş. As this value increases, the number of edges of the polygon also increases. In small polygonal configurations the size of the ring will not experience a considerable change between the closed and the open state of deployment; therefore the values for Ropen and Rclosed are almost identical. Jensen has proposed the following equations to find the values of Ropen and Rclosed in any polygonal configuration with angulated or multi-angulated-scissor bars, where k is equivalent to the number of segments in one angulated bar (Jensen, 2004, p.32) : AP= PB =Ropen sin (f/2)

(3.17)

Rclosed = Ropen sin (k f/2)

(3.18)

In the above equations the bars are considered to be lines of zero thicknesses and the hinges are considered as having no finite size. Similar equations with considerations of bars and hinges sizes can be also found in Jensenâ&#x20AC;&#x2122;s work (Jensen,2004).

f Rclosed

B A

Ropen

S S

1 Closed

4 Open

Fig. 3.17. Closed and open positions of a hexagonal ring with angulated-scissor bars

3.2.4.2. Examples of fixed position supports Figure 3.18 shows two identical hexagonal rings formed from angulated-scissor bars. Ring (a) is fixed in position at the intermediate joint of one of the scissor-units and the inner and outer joints of that unit (A and B) are constrained to follow a circular path with centre on the fixed joint F. Ring (b) is fixed in position at the outer joint of one of the scissor-units and the inner joint of the same unit (i.e. A,B) is constrained to follow a movement path describe between these two joints (i.e. F,A). Notice that the joints in the two rings follow completely different movement patterns (fig 3.19) .

CHAPTER 3

1 Closed a)

4 Open

Constraint p a t h

F A

1 Closed

4 Open

b) Fig. 3.18. Deployment process of two identical hexagonal rings using angulated-scissor bars that have different types of fixed position supports and movement paths Constraint path

Fixed position support

F F

Fig. 3.19. Diagrams of the joint displacements in two identical hexagonal rings using angulatedscissor bars that have different types of fixed position supports and movement paths 45

CHAPTER 3

3.2.4.3. Examples of grid designs Hoberman discovered that by intersecting various deployable rings in different planes, it is possible to form complex double curvature shapes, such as hyperboloids or polyhedral configurations, which he named: expanding spheres (Hoberman, 1990,1996 and www.hoberman.com), (fig. 3.20). The most common geometrical shapes used by Hoberman come from Platonic solids or Archimedean and/or Catalan polyhedrons. In these three-dimensional assemblies a special hinge design(i.e. a ball joint) that connect the rings between each other is needed, since this joint should allow every ring to operate on a different plane.

Base polyhedral frame (cuboctahedron)

Hexagonal ring of angulatedscissor bars

a) Special joint that links various rings

Fig. 3.20. a) Polyhedral configuration formed from rings with angulated-scissor bar, b) Deployment process of an expanding sphere, based on a cuboctahedron

CHAPTER 3

3.2.5. MULTI-ANGULATED-SCISSOR BARS 3.2.5.1. Morphology Kassabian, You and Pellegrino discovered that in deployable structures formed from two or more concentric rings of angulated-scissor bars(fig. 3.22) it is possible to link two or more of the angulated-scissor bars and form a multi-angulated-scissor bar (Kassabian, You & Pellegrino, 1997) (fig. 3.21). Inner joint

Multi-angulated-scissor bar

w d

Intermediate joint P w B

b D

P1B

P2 P1

Outer joint

Fig. 3.21. Multi-angulated-scissor bars

Structures made from identical series of these bars and that have equal kink angles w between the joints are described in this chapter as regular. In ring configurations the multi-angulated bars are distributed in two layers of elements (black and red). During the deployment, when the elements in one layer move clockwise, the elements in the other layer move anti-clockwise from and towards the centre S (fig. 3.22). Notice that the degree of motion of each type of joint is different. For example, the inner most joints move further from the initial to the final stages of deployment compared with the outer.

1 Closed

3 Open

Fig. 3.22. Decagonal ring configuration with multi-angulated-scissor bars

CHAPTER 3

The final state of retraction in a regular ring with multi-angulated-scissor bars occurs when the inner joints meet at the centre S. However, the final state of expansion depends on the number of the rings within the structure. When the outer ring is fully deployed (meaning that the inner most and outer most joints meet) the deployment of the whole structure concludes. The values for Ropen and Rclosed and ropen in any regular polygonal configuration formed from multi-angulated-scissor bars with identical segments k (where AP=PP1=P1B, etc.) can be found with equations 3.17, 3.18 and 3.19, proposed by Jensen (Jensen, 2004, p.32) (fig. 3.23) : ropen = cos[(k-1) f/2]

(3.19)

k = number of segments in the multi-angulated bar f

Rclosed

Ropen

ropen f

Inner most joint Outer ring 1 Open

Outer most joint

3 Closed

Fig. 3.23. Octagonal ring configuration with multi-angulated-scissor bars, based on Jensenâ&#x20AC;&#x2122;s diagram (Jensen, 2004, p.32)

3.2.5.2. Examples of fixed position supports When rings formed from angulated or multi-angulated-scissor bars expand and retract from and towards the central point S with no fixed position supports, the joints can be constrained to follow radial lines centred on S (fig. 3.22, 3.24 and 3.25).

0 closed

5 open

Fig. 3.24. Deployment process of a hexagonal ring configuration with angulated-scissor bars

CHAPTER 3

6 4 3

Constraint path for the inner and outer joints

4 3

1 2 1

S Constraint path for the intermediate joints

Fig 3.25. Diagram of the joint displacements in an hexagonal ring with angulated-scissor bars that expand and retract from and towards an imaginary central point S

In addition, Kasabbian, You and Pellegrino found that in rings formed from angulated or multi-angulated bars the elements in one layer can be individually fixed in position at a point F from where they rotate during the deployment (Kassabian, You & Pellegrino, 1997) (Fig. 3.26, 3.27 and 3.28). As a result the inner and outer joints of the elements in that layer follow circular paths during the deployment without the need for additional constraints. S S q/2 q/2 q/2 q/2

q r* F

Fixed position support

r* F P

P B

P1 P2

Angulated bar Multi-angulated bar

Fig. 3.26. a) Fixed position point from where the angulated-scissor bar rotates, b) Fixed position point from where the multi-angulated-scissor bar rotates

CHAPTER 3

F F F

F F S F F F F F

F F S F

F F

1 Closed

F F F F S F F F F FF

3 Open

Fig. 3.27. One layer of elements of a dodecagonal ring formed from multi-angulated-scissor bars where each element rotates about a fixed point F

F F

F S F

F F

0 Closed

F F

5 Open

Fig. 3.28. Hexagonal ring formed from angulated-scissor bars where each element rotates about a fixed point F

Jensen found that the radius r* (fig. 3.29) of the circular path described by the joints of the angulated or multi-angulated elements can be found with the following equation (Jesen, 2004, p.35)( see also fig. 3.23) : r* = Ropen /2

(3.20)

Ropen= radius of the circle formed by the outer joints when the ring is fully open

Jensen also found that the total rotation angle b* depends on the number, k, of segments within the multi-angulated bar as follows (Jensen, 2004, p.37) (fig. 3.29): k-1 b* = p - a1

S i=1 50

(3.21)

CHAPTER 3

The angle b*is equivalent to the angle between the intersection points M and N of two adjacent circular motion paths, minus angle f between joints B, F and A (fig. 3.29).

P M

q k2

b* k3

r* F

f q

q B

k2 P1

q P2

Fig. 3.29. a) Multi-angulated-scissor bar at a closed position, b) Multi-angulated-scissor bar at a open position

Additional information of how to find the location of the points F within rings formed form multi-angulated bars can be found in the work of Kassabian, You, Pellegrino and Jensen, (Kassabian, You & Pellegrino, 1997,1999); (Jensen , 2004). Notice the differences between the patterns of movement and the displacements of the joints in the two cases of support (with no fixed points and with fixed points) studied in the present section. In rings similar to that one shown in figures 3.24 and 3.25, there are no fixed position supports. Instead, during the deployment, the joints slide along straight paths of movement used as constraints from and towards the centre of the ring. Hence, the displacement of the joints can be classified as pure translation. However, in order to induce movement into the ring it is necessary to push or pull the components simultaneously from opposite directions and provide additional elements (i.e. rails) that serve as paths for the joints. In contrast, in rings such as that shown in figures 3.28 and 3.29, all the elements in one layer can be connected to a fixed position support F from where they rotate during the deployment. According to Jensen, â&#x20AC;&#x153;the motion of one layer of elements could be described as pure rotation and the motion of the other layer as pure translationâ&#x20AC;? (Jensen,2004, p.35). However, notice that the pivotal hinges F are additional to the hinges that comprise the ring. Hence, extra elements (i.e. bars) are required in order to link these points to the angulated or multi-angulated bars. 51

CHAPTER 3

3.2.5.3 Examples of grid designs In two-dimensional schemes the concentric layers of elements work on the same plane of operation. As with other types of scissors, planar surfaces have low structural efficiency. For that reason, three-dimensional surfaces are always desirable for large assemblies of multi-angulated units. Three-dimensional shapes like spheres, or polyhedrons, are achievable by projecting two-dimensional grids onto sections of the required surface, (fig. 3.30 and 3.31). Base dome configuration Multi-angulated-bars

Fig. 3.30. Configuration of a dome with multi-angulated-scissor bars

Double curvature configurations are more complex in geometry since every segment k within the multi-angulated-scissor bar is cranked out of plane. Therefore, the bars are not only angled to configure a polygonal shape in plan, but also bent in order to obtain the curvature of the desired three-dimensional shape. In each particular case a prior analysis is necessary to calculate the two kink angles in every multi-angulated-scissor bar. Fig. 3.31 shows a deployable half sphere developed by Hoberman and named iris dome, due to its similarities to the retraction of the iris in the human eye. Note that the assembly increases in height when it is fully closed and in diameter when it is fully opened. Detailed analysis of the geometry and behaviour of these types of structure with angulated and multi-angulated-scissor bars can be found in the work of Hoberman, Kassabian, You, Pellegrino and Jensen (Hoberman, 1990); (Kassabian, You & Pellegrino, 1997, 1999); ( Jensen, 2004).

Fig. 3.31. Deployment process of dome with multi-angulated-scissor bars (the iris dome)

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3.2.6. INTERCONNECTING-SCISSOR BARS (ring grids) 3.2.6.1. Morphology In past research the author has experimented with grids formed from interconnected rings with angulated-scissor bars (Rodriguez, 2000). In contrast to grids with concentric rings designed by Hoberman, Kassabian, You, Pellegrino and Jensen, the grids in the authorâ&#x20AC;&#x2122;s past research are formed from interconnected rings one next to the other (fig. 3.34). As a result, new elements that serve to interconnect the rings are formed. These elements where named by the author: interconnecting-scissor bars. P4

End joint 1 2

A1 Interconnecting-scissor bar 1

w End joint 2

Intermediate pivot PP

Interconnecting-scissor bar 2

B P3

Fig. 3.32. Example of a interconnecting-scissor unit

Figures 3.32 and 3.33 illustrate examples of units with interconnecting-scissor bars. The grid proposed by the author where designed to deploy following straight lines and be supported upon rails, in a similar way to the rings described in section 3.2.4.2. 3.2.6.2. Examples of grid designs

Fig. 3.33. Examples of units with interconnecting-scissor bars (Rodriguez, 2000)

Using interconnecting-scissor bars it is possible to design grids combining different polygonal shapes (i.e. hexagons and triangles or squares and octagons). It is also possible to configure grids where some polygons retract whilst others contract during the process of movement (Rodriguez,2000).

CHAPTER 3

1 Closed

Interconnecting-scissor bar

Interconnecting-scissor bar 3

w =1208

4 Open

w =1208

Interconnecting-scissor unit Fig. 3.34. Grid formed from seven hexagonal rings with interconnecting-scissor bars

Other rings formed from angulated-scissor bars can be intersected with the grid to create three-dimensional polyhedral configurations similar to Hoberman’s expanding spheres (shown in fig. 3.20). For example, in the grid shown in figures 3.35, 3.36 and 3.37 formed from square rings with interconnecting and angulated-scissor bars, the central ring is intersected by other perpendicular square rings creating a expanding polyhedra or ’sphere’. When the grid is fully closed the polyhedron formed in the centre is a icosahedron, when the grid is fully open it changes into a cube.

CHAPTER 3

Base grid configuration Square ring with angulated-scissor bars

w =1808

w =908

w =908 Interconnecting-scissor unit

Fig. 3.35. a) Base grid configuration, b) Interconnecting-scissor unit

1Closed

3 Open

Fig. 3.36. Plan view of a grid formed from seven square rings using interconnecting-scissor bars with a central expanding sphere

Expanding sphere (Cube)

Expanding sphere (Icosahedron)

1Closed

Fig. 3.37. Grid formed by seven square rings using interconnecting-scissor bars with a central expanding sphere 55

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3.2.7. DEPLOYABLE PLATES BASED ON ANGULATED, MULTI-ANGULATED OR INTERCONNECTING- SCISSOR BARS 3.2.7.1. Morphology It can be seen in past sections that in structures using scissor-mechanisms the relationship between the joints and the way they are disposed is of great importance if one wishes to establish the type and direction of movement in the structure. The shape of the scissor bar is to a certain extent irrelevant. Hence, it can adopt different designs as long as the correlation between the joints is preserved and interference between the elements during motion is avoided. In structures formed from angulated, multiangulated or interconnecting-scissor bars, it is possible to replace the bars with other rigid elements such as plates. The plates can also be designed to cover the whole polygonal area of the ring. Options of plate designs are very broad, triangular shaped plates the most straight forward solution (fig. 3.38).

Ring with angulated-scissor bars used as base

Inner joint Intermediate joint S

Outer joint A

A1 w = 1208

Layer of plates 1 B

Layer of plates 2

Fig. 3.38.Hexagonal ring with triangular deployable plates based on angulated-scissor bars

0 closed

5 open

Fig. 3.39. Deployment process of a hexagonal ring with triangular deployable plates based on angulated-scissor bars 56

CHAPTER 3

Figures 3.38 and 3.39 show a ring comprising two layers of identical triangular plates based on a hexagonal ring with angulated-scissor bars. Figure 3.40 shows another configuration for the plates designed in past research by the author (Rodriguez, 2000). This example is based on the same hexagonal ring with angulated-scissor bars, however, it uses two types of triangular plates. Plate type 1 Plate type 2

1 Closed

4 Open

Fig. 3.40. Ring with two types of deployable plates based on a ring with angulated-scissor bars (Rodriguez, 2000)

Plates with irregular shapes are also possible. Jensen developed a method to design plates with non-straight boundaries (Jensen, 2004). In Jensenâ&#x20AC;&#x2122;s method the boundary of each plate is shaped with a periodic pattern, in such a way that no gaps or overlaps occur when the ring is fully open or closed (fig. 3.41). Jensen extracts one pantograph of the ring and analyses it as if joints P and B1 where fixed in position. The boundary line crosses AP and P1B1 at an angle u, which can be found using the following equation (Jensen,2004, p.47): u = p - bclosed - bopen 2

(3.22)

Inclination for the boundary line A

P1 u

bclosed P

1 Closed

B1 2

Bopen

3 Open

Fig. 3.41. Periodic pattern of a non-straight boundary (Jensen,2004, p. 53)

Figure 3.42 shows a design developed by the author of an octagonal ring with deployable plates based on multi-angulated-scissor bars. The plates in this example are shaped using Jensenâ&#x20AC;&#x2122;s method explained above for non-straight boundaries.

CHAPTER 3

Notice that the ring is designed in a way that when one layer of plates is fully open the other is fully closed.

1 Closed

4 Open

Fig. 3.42. Octagonal ring with two types of deployable plates with non-straight boundaries

3.2.7.2. Examples of grid designs Figure 3.43 illustrates a grid developed in past research by the author based on triangular rings with interconnecting-scissor bars (Rodriguez, 2000). In this example the interconnecting-scissor bars have been replaced by plates with three shapes with straight boundaries.

Plate type 1 1 Closed

2 Plate type 2

Plate type 3

4 Open

Fig. 3.43. Grid with triangular deployable plates based on interconnecting-scissor bars (Rodriguez,2000) 58

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3.3. HYBRID DEPLOYABLE LATTICE STRUCTURES WITH RIGID LINKS Structures resulting from combinations of various types of scissor-mechanisms or combinations between scissor-mechanisms and deployable reciprocal frames are classified in this thesis as hybrid deployable lattice structures with rigid links. In addition, angulated, multi-angulated and interconnecting-scissor bars based on irregular polygons or where the segments between the joints are different lengths or with viable kink angles w are also included within this group. Hybrid DLSRL can be divided into the following types: 1. Combinations of straight-scissor bars with central pivot and straight-scissor bars with an offset pivot. 2. Combinations of straight-scissor bars with central and offset pivot with angulated, multi-angulated and/or interconnecting- scissor bars 3. Irregular retractable ring structures with angulated, multi-angulated and/or interconnecting- scissor bars. 4. The â&#x20AC;&#x2DC;base structureâ&#x20AC;&#x2122; 5. Combinations with all types of scissor-mechanisms and deployable plates. Due to the vast range of designs possible with the combinations mentioned above, this section studies primarily the morphology and basic characteristics of each type of hybrid DLSRL. Conditions for fixed position supports and grid designs are only mentioned in general terms. 3.3.1. COMBINATIONS OF STRAIGHT-SCISSOR BARS WITH CENTRAL PIVOT AND STRAIGHT-SCISSOR BARS WITH AN OFFSET PIVOT 3.3.1.1. Morphology A very simple way to increase the size of a structure formed from straight-scissor bars with an offset pivot is to combine them with straight-scissor bars with central pivot. Elements with an offset pivot allow the structure to obtain curvatures when fully deployed, whilst elements with central pivot allow the structure to have a larger extensions but attain a fairly compacted form when fully closed. Figure 3.44 illustrates two deployable arches: arch (a) with twelve identical straight-scissor bars with an offset pivot and arch (b) with a combination of twelve straight-scissor bars with central pivot and twelve straight-scissor bars offset pivot.

CHAPTER 3

1 Closed

3 Straight-scissor bars with central pivot

Straight-scissor bars with an offset pivot

4 Open

Fig. 3.44. a)Deployable arch with straight-scissor bars with an offset pivot, b) Deployable arch with a combination of straight-scissor bars with central an offset pivot

The principals for fixed position supports and grid configurations described in sections 3.2.1 and 3.2.2 can also be applied to the type of combinations mentioned in this section. 3.3.2 COMBINATIONS OF STRAIGHT-SCISSOR BARS WITH CENTRAL AND/OR OFFSET PIVOT WITH ANGULATED, MULTI-ANGULATED AND/OR INTERCONNECTING-SCISSOR BARS 3.3.2.1. Morphology A simple variation of the rings with angulated-scissor bars consists of adding straightscissor bars in between the angulated units. With this solution the ring can reach a larger expansion when it is fully open and still be packed in a fairly small bundle of bars in its closed position. Figure 3.45 shows two rings: ring (a) is formed from eight identical angulated-scissor bars, whilst ring (b) is formed from a combination of eight angulatedscissor bars and eight straight-scissor bars with central pivot. In this example the angulated elements allow the structure to preserve the polygonal shape and the straight elements enable the ring to have a larger expansion. 60

CHAPTER 3

S S

3 Open

1 Closed

Angulated-scissor unit Straight-scissor unit

b) 1 Closed

3 Open

Fig. 3.45. a) Octagonal ring with identical angulated-scissor bars, b) Octagonal ring with a combination of angulated-scissor bars and straight-scissor bars with central pivot

3.3.3. IRREGULAR RETRACTABLE RING STRUCTURES WITH ANGULATED, MULTIANGULATED AND/OR INTERCONNECTING- SCISSOR BARS 3.3.3.1. Morphology The structures in sections 3.2.4 to 3.2.7 are formed from identical angulated, multiangulated or interconnecting- scissor bars or plates. In addition, the segments between the joints are of equal length and the kink angle w is the same for all the elements. However, Hoberman found that it is possible to generate a retractable ring structure based on irregular polygons and angulated-scissor units with variable kink angles w (Hoberman,1990). Figure 3.46 illustrates a method developed by Hoberman where the kink angles w in each angulated-scissor unit coincide with the angles of the desired irregular polygon. For example, the kink angles w between joints A,P and B and w1 between A1, P and B1 are equal to the angle w2 between the vertices N, P and G of the polygon. Hence, AP=(NP/2) and PB =(PG/2).The pantographs in these rings are formed by similar rhombuses with segments (a) of equal length .

CHAPTER 3

Pantograph with similar rhombus B1

P B1

w1 w a

B A1

w = w1 = w2 1 Closed

3 Open

Fig. 3.46. Deployable irregular polygon with similar rhombuses using Hobermanâ&#x20AC;&#x2122;s method

The absolute stages of deployment are reached when two or more inner or outer joints coincide at the centre or at the perimeter of the ring. In this method the following geometrical conditions are achieved (fig. 3.47 a) : AP = PB1 and PB = A1P , where w = w1

(3.23)

S A1

q Angulated-scissor bar

Angulated-scissor bar

B P

q Straight-scissor bar

Angulated-scissor bar

A w

B1 a)

P b)

w1 = 1808

Fig. 3.47. a) Irregular-scissor unit where w = w1, b) Irregular-scissor unit where w = w1

Hoberman also developed a second method to design rings based on regular polygons but with variable kink angles w and w1within the scissor unit (fig. 3.47b).This method consists of replacing one of the bars of the scissor-unit with a straight bar (fig. 3.47 and 3.48 b). The remaining angulated bar helps to preserve the polygonal shape of the ring whilst the straight bar helps the ring to expand further when fully open. Figure 3.48 shows three rings where the distances AP and PB are the same for all the elements. Ring (a) is formed from eight identical angulated-scissor bars, ring (b) is formed from sixteen identical angulated-scissor bars ring (b) is formed from eight identical straight bars (in blue) and eight identical angulated bars (in red and black) using Hobermanâ&#x20AC;&#x2122;s method. Notice that rings a) and c) use angulated bars with the same kink angle w and 62

CHAPTER 3

Angulated bar w = w1 = 908

w1 A1 B1 A

a) 1 Closed

3 Open

2 B Angulated bar w = w1 = 1358

A1 w1

B1 A

1 Closed

3 Open

2 B1

P w w1

Angulated bar w = 908

Straight bar w1= 1808 c)

1 Closed

3 Open

Fig. 3.48. a) Regular ring with angulated-scissor bars based on a square , b) regular ring with angulated-scissor bars based on a octagon, c)Irregular ring with angulated and straight bars based on a square using Hobermanâ&#x20AC;&#x2122;s method for variable kink angles

form squares when open. Ring c) uses twice as many elements than ring (a) and at the open position it is four times bigger. Ring (a) and (b) use angulated bars with different kink angle w. Ring (b) uses twice as many elements that ring (a) but when open it is almost 4.8 times bigger than ring (a). Hence, with Hoberanâ&#x20AC;&#x2122;s method for variable kink angles, it is possible to increase the size of a ring without changing its shape by introducing an equivalent number of straight bars to the exiting angulated bars. You and Pellegrino (You & Pellegrino, 1997) extend on this subject by proposing a method with similar parallelograms instead of the similar rhombuses used by Hoberman. In this method only the opposite bars in the pantograph are equal lengths and both angulated elements can have identical or variable kink angles w and w1 (fig. 3.50). You and Pellegrino named their method as: generalized angulated elements (GAE). They identified two types of GAE, type I compiles with the following conditions: AP = PB1 and PB = A1P , where in general w = w1

(3.24)

And type II: AP / PB1 = A1P / PB , and w = w1 63

(3.25) CHAPTER 3

Hoberman’s first method described at the beginning of this section, is classified by You and Pellegrino as a special case of GAE type II that complies with equations 3.23 and 3.25 simultaneously. In GAE the angle q is proportional to the kink angles w and w1 of both elements of the scissor unit as follows (You & Pellegrino, 1997)(fig. 3.49): q = 1808 - (w + w1)/2

(3.26)

A1 B

q qi

qii

b b B1

B2 b)

Fig. 3.49. a) Irregular angulated- scissor unit with GAE type I, b)Chain of irregular angulatedscissor units with GAE type II (after You and Pellegrino, 1997)

Figure 3.50 shows an irregular polygon equivalent to the one of figure 3.46, this example, however, uses You and Pellegrino’s method of similar parallelograms instead of similar rhombuses. Pantograph with similar parallelograms

w1 B1

A S

1 Closed

S a

A1 G

3 Open

Fig. 3.50. Deployable irregular polygon with similar parallelograms using You and Pellegrino’s method GAE type II

The methods of Hoberman, You and Pellegrino described above can be also applied to study structures with multi-angulated and interconnecting-scissor units following the same principals of similar rhombuses and similar parallelograms as described in this section. 64

CHAPTER 3

3.3.3.2 Examples of fixed position supports Figure 3.51 illustrates the displacements and the movement paths followed by the joints in the two examples of polygons with irregular angulated-scissor bars described in this section (fig. 3.46 and 3.50). Notice that in ring (a), with the similar rhombuses method, the inner and outer joints follow the same path, whilst in ring (b), with the similar parallelograms method, the inner and outer joint follow different but parallel paths.

Movement path followed by the joints

Fig. 3.51. a) Displacement of the joints during the deployment of a irregular polygon with similar rhombuses, b) Displacement of the joints during the deployment of a irregular polygon with similar parallelograms

Kassabian, You and Pellegrino (Kassabian, You & Pellegrino, 1997) found that it is also possible to have fixed points of rotation for the elements of one layer of the ring in a similar way as with rings using angulated and multi-angulated-scissor bars (see sections 3.2.4 and 3.2.5). When the ring is fully open (b =0) point F is located at exact half distance between the centre S and the vertex Popen of the polygon (fig. 3.52). The distance between the centre S and the fixed point F can be found using the following equation by Kasabian, You and Pellegrino: SF=

(3.27)

SPopen 2cos (b/2)

Popen

F S

1 Closed

3 Open

Fig. 3.52. Location of the fixed point of rotation in irregular rings with angulated-scissor bars using Kassabian, You and Pellegrinoâ&#x20AC;&#x2122;s method 65

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3.3.3.3. Examples of grid designs Three-dimensional assemblies are equally achievable with irregular-scissor units. Hoberman proposed a method to create three-dimensional structures using rings with variable kink angles w and w1 similar to the one shown in figure 3.48b. This method involves replacing every surface in a polyhedron by a deployable ring. The rings are interconnected at each vertex by special joints that allow every ring to operate on a different plane (fig. 3.53 and 3.54). Notice that to form the cuboctahedron of figure 3.54, only the square surfaces have been replaced by retractable rings and the triangular surfaces are formed by the edges of the squares but are not themselves rings.

Deployable ring where the angulated elements have variable kink angles (see fig. 3.48c) Base cuboctahedral configuration

Fig. 3.53. Base cuboctahedral configuration where a surface is replaced by a deployable ring where the angulated elements have variable kink angles using Hobermanâ&#x20AC;&#x2122;s method

Special joint to allow the rings to operate in different planes

Fig. 3.54. Deployable cuboctahedron using Hobermanâ&#x20AC;&#x2122;s method, where the square surfaces are replaced by rings using angulated elements with variable kink angles

CHAPTER 3

The great advantage when using the method illustrated above, compared with the method for expanding spheres shown in figure 3.20 is that the frame does not interfere with the space inside the structure during the deployment process. 3.3.4. THE 'base structure' 3.3.4.1. Morphology The 'base structure' is an alternative way to generate a closed loop with bar and pivots proposed by You (You, Z., 2000). Instead of finding fixed points to support the ring after having a defined geometry, as done by Kassabian, You and Pellegrino (Kassabian, You & Pellegrino,1997). You proposed a method that uses four-bar linkages. The four-bar linkage is a common type of assembly widely used in engineering to transmit forces and motion (fig.3.55). It comprises four bars (BA, AF ,FF1 and F1B) linked by four joints, (B, A, F and F1), where two of the joints (F and F1) are fixed in position. The link between the fixed points (FF1) is usually described as the frame, the link opposite to the frame (BA) is described as the coupler link, and the remaining links (AF and F1B) are described as the side links. Coupler link

Side

link

Frame

Fig. 3.55. Configuration of a basic four-bar linkage

In the 'base structure', You proposed that a closed loop be created with four-bar linkages forming a triangle (Fig. 3.56a). For such assembly to work, additional bars or other types of rigid elements need to be attached between links A and B (fig. 3.56b and 3.56c). Four-bar linkage

A B2

B2 F F2

F B

F2 A1

A1 B1

B F2

A1 B1 c)

Fig. 3.56. Configurations of the â&#x20AC;&#x2DC;base structureâ&#x20AC;&#x2122; proposed by You (You,2000)

CHAPTER 3

3.3.4.2. Examples of grid designs You found that the ''base structure'' can be expanded by adding more four-bar-linkages to the perimeter of the triangular loop as shown in figure 3.57.

A3 B2

A F

A5 A2

F1 F2

B3 B

A1 A4

B1 B4

Fig. 3.57. Method for expanding the â&#x20AC;&#x2DC;base structureâ&#x20AC;&#x2122; proposed by You (You, 2000)

The examples of the ''base structure'' shown by You (You,2000) comprise only triangular configurations with four-bar linkages. It is mentioned by You that variations are possible, however, in the literature available it is not clear how this variations can be achieved. The expansion of the 'base structure' described by You (Fig. 3.58) is also based on a loop with three internal fixed position supports. 3.3.5. COMBINATIONS WITH ALL TYPES OF SCISSOR-mechanisms AND DEPLOYABLE PLATES 3.3.5.1. Morphology A broad range of alternative retractable ring structures can be formed from combinations of various systems based on scissor-mechanisms, as long as they comply with the basic rules described in sections 3.2.1 to 3.3.3. Figure 3.58 shows an example proposed by the author of a combination using triangular retractable plates with straight-scissor bars with central pivot.

1 Closed

3 Open

Fig. 3.58. Octagonal ring with a combination of triangular deployable plates and straightscissor bars with central pivot 68

CHAPTER 3

Other multiple combinations of this type are possible, however, it is out of the scope of this chapter to study all of them in more detail. Ffixed position supports and grid configurations need to be accessed separately for each particular design. This can be achieved following the principals described in sections 3.2.1 to 3.3.3.

3.4. DEPLOYABLE RECIPROCAL FRAMES 3.4.1. DEPLOYABLE RECIPROCAL GRIDS 3.4.1.1. Morphology Reciprocal frame structures comprise rigid elements (i.e. bars) jointed together in such a way that they may support themselves mutually. As mentioned in chapter two section 2.3.3, this characteristic provides great potential for retractable systems. A simple way to transform a reciprocal frame into a deployable reciprocal ring is to replace the fixed connections between the bars with pivots. A deployable reciprocal ring of this type formed from four bars creates a assembly very similar to a pantograph (fig. 3.59).

A P1

Reciprocal bar

Hinge

B 1

2 Fig. 3.59. Deployable reciprocal frame with four bars

The main difference between pantographs with straight-scissor bars and deployable reciprocal rings with four bars is the way the bars are connected and how they support themselfs. In pantographs the bars operate in two layers(fig. 3.60a), one layer supports the other (on a horizontal position). In deployable reciprocal ring the bars are tilted to mutually support each other (fig. 3.60b). Width of the bar w P1

P1 A

B P a)

P b)

Depth of the bar d

Fig. 3.60. a) Pantograph with straight-scissor bars, b)Deployable reciprocal frame with four bars 69

CHAPTER 3

According to Ariza and Villate (Ariza, 2000), in a deployable reciprocal ring with four identical bars the relationship between the width (w) of the bar and the distance between joints A and P affects the mobility of the structure. The higher the value of AP/w the greater the mobility of the frame. (fig. 3.60). Options for fixed position supports are similar to those for straight-scissor with central or offset pivot bars described in sections 3.2.2.2 and 3.2.3.2. 3.4.1.2. Examples of grid configurations As with pantographic structures, series of deployable reciprocal rings with four bars can be interconnected to form planar or three-diemsional structures. Figure 3.61 shows a grid design by Ariza and Villate (Ariza, 2000) formed from square deployable reciprocal frames.

Fig. 3.61. Deployable reciprocal grid (after Ariza,2000)

3.4.2. DEPLOYABLE RECIPROCAL RINGS 3.4.2.1. Morphology As mentioned in section 2.3.3, deployable reciprocal rings where first proposed by Choo, Couliette and Chilton (Choo, Couliette & Chilton,1994) (fig. 3.62, 3.63 and 3.64). These structures are formed from identical adjoined bars with guided sliding joints. Each bar slides upon the preceding bar while fixed by an outer joint to an external support.

CHAPTER 3

Base polygon

Sliding inner joint

A2 A3

L1 u1

Reciprocal bar

Fixed in position support (outer joint)

Fig. 3.62. Deployable reciprocal ring with a hexagonal base

1 Closed

3 Open

Fig. 3.63. Deployment of a deployable reciprocal ring with a hexagonal base (Plan view)

Sliding joint Reciprocal bar

Base polygon Fixed in position support

b b 1 Closed

3 Open

Fig. 3.64. Deployment of a deployable reciprocal ring with a hexagonal base (Axonometric view)

CHAPTER 3

According to Popovic (Popovic, 1996, p. 6-18) deployable reciprocal frames fulfill the following conditions, where L1 an L2 are the projection in plan of FA and FA1, b = beam slope angle and d= depth of the bar: L2 = L1 o

sinu2 p Sinu1

b = sin-1 o

(3.28)

L1 =

d p L2 - L1

FF1

(3.29)

(3.30)

cosu1 sinu2 so p-cosu2t Sinu1 From equation 3.29 it possible to explore a hypothetical case where d=0 then b=0 and the ring operates only on a horizontal plane. However, in practical application, where d>0 the bars rotate in two planes: horizontal and vertical (fig. 3.64). According to Popovic, “as the beams come closer to the fully closed position, the rate of change of the vertical angle b increases”. In addition, “this effect is more pronounced for structures with a large dimension d” (Popovic,1996, p. 6-19). Figure 3.65 illustrates the relationship between the horizontal angle u1 and the vertical slope b for a hexagonal roof with perimeter length of 5m and three options for bar depths d= 500mm, d= 1000mm and

b Vertical angle (slope)

d=1500mm (Popovic,1996, p. 6-18). 50 45 40 35 30 25 20 15 10 5 0

d = 500mm d = 1000mm d = 1500mm

0 Open

40 Closed

u1 Horizontal angle Fig. 3.65. Relationship between horizontal angle u1 and vertical slope b (Popovic,1996, p. 6-18)

As can be seen in figure 3.65 that when the bars approach the closed position,the slope of the bars increases very rapidly. Since b is proportional to d, the greater the depth of the bars, the higher the vertical slope of the overall ring. Further information regarding the geometrical capabilities of this type of deployable reciprocal frame can be found in the work of Choo, Couliette, Chilton, Choo, Popovic and Wilkinson (Choo, Couliette & Chilton,1994); (Popovic, 1996); (Wilkinson,1997);(Chilton, Choo & Wilkinson, 1998).

CHAPTER 3

3.4.2.2. Fixed position supports Each bar in a deployable reciprocal ring rotates fixed in position relative to an outer joint. This outer joint should allow at least two degrees of freedom, so the bar can operate simultaneously on a horizontal and on a vertical plane during the deployment of the ring. At the same time, the inner joint of each bar slides in one direction along a track on the top of the neighbouring bar. This joint should be designed in such a way that the deployment of the ring can be controlled, as under the effects of gravity, the bars tend to slide very rapidly from the closed to the open position. 3.4.2.3. Examples of grid configurations Rings can be linked next to each other to form grids using interconnecting bars with a special intermediate joint (fig. 3.66). This joint should be designed so it connects the rings and at the same time allows each bar to rotate with two degrees of freedom. Figure 3.66 illustrates an example proposed by the author of a grid of this kind formed from seven interconnected hexagonal deployable reciprocal rings. Notice that this grid uses two types of interconnecting bars.

1 Closed

Interconnecting reciprocal bar type 2 Special intermediate joint Interconnecting reciprocal bar type 1

3 Open Fig. 3.66. Grid configuration formed from seven interconnected hexagonal deployable reciprocal rings

CHAPTER 3

3.4.3. DEPLOYABLE RECIPROCAL PLATES 3.4.3.1.Morphology As with angulated, multi-angulated and interconnecting-scissor bars (sections 3.2.4, 3.2.5 and 3.2.6), in deployable reciprocal rings the bars can be replaced by plates. In such structures each plate slides upon the preceding plate while fixed by an outer joint to an external support. Another alternative was proposed by Chilton, Choo and Wilkinson ( Chilton, Choo, & Wilkinson, 1998) (fig. 3.67). It consists of a ring formed from reciprocal plates where one vertex of each the plate (A) slides along the side of the adjacent plate (A1F) leaving no gaps during the deployment. As a result, the ring operates only on one plane.

B4 B3 w4

F2 A2 A3

w3 S

F1 A

A5 F5

B6 B1 Fig. 3.67. Hexagonal ring with deployable reciprocal plates using Chilton, Choo and Wilkinson method (Chilton, Choo, & Wilkinson, 1998)

1 Closed

6 Open

Fig. 3.68. Deployment process of an hexagonal ring with deployable reciprocal plates

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3.4.3.2.Options for supports Chilton, Choo and Wilkinson proposed two types of support for deployable reciprocal plates (Chilton, Choo, & Wilkinson, 1998) (fig. 3.69 and 3.70). In a ring with support type one (fig. 3.69), during the deployment the joint type F in each plate rotates about a support whilst the joint type A slides free. As a result, with this type of support there is no connection between the plates. In addition, in rings formed from regular triangular plates (fig. 3.69a) during the deployment gaps are formed between the plates. In rings that use irregular triangular plates (fig. 3.69b), during the deployment of the ring, the plates overlap each other. In a ring with support type two (fig. 3.70) both joints A and F in the plate slide along their corresponding adjacent plate. With this type of support there are no gaps between the plates during deployment. Despite of this, again no fixed in position support can be used for the plates. F

A A A

A F

Displacement of the joint during deployment

1 Closed

3 Open

F A A

A F

Displacement of the joint during deployment

1 Closed

F 2

3 Open

Fig. 3.69. a) Hexagonal ring with deployable reciprocal plates with type of support one and regular triangular plates, b) Hexagonal ring with deployable reciprocal plates with type of support one and irregular triangular plates (after Chilton, Choo, & Wilkinson, 1998)

Displacement of the joint during deployment

A F

1 Closed

3 Open

Fig. 3.70. Hexagonal ring with deployable reciprocal plates with type of support two and regular triangular plates, (after Chilton, Choo, & Wilkinson, 1998) 75

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3.5. DISCUSSION It can be seen in this chapter that within DLSRL there are many different systems. Some share similarities in terms of the morphology, options for fixed position supports and/or types of grid configurations. It can be argued that in terms of the morphology of the system, there are three very distinct families of structures: pantographic structures, deployable reciprocal frames and hybrid deployable structures. The majority of the systems that constitute hybrid deployable structures are the result of combinations between the other two families of structures. It is also noticeable that in terms of type of deployment there are two well defined groups: structures with linear deployment and structures with a radial deployment (or retractable rings). It was considered by the author that, in terms of geometry, retractable rings show a greater potential for further exploration in this research compared with structures exhibiting linear deployment, for the following reasons: 1. Morphology: Retractable rings tend to be less compactable than structures with linear deployment. This is due to the fact that in retractable rings when the inner joints meet at the centre, they restrict further deployment for other elements. As a result, in the fully closed position there is more free space between the rigid elements (i.e. bars or plates). Hence, the rigid elements of retractable rings are more flexible in terms of design, they can cover a larger area and their shape has more potential for variations. 2. Fixed position supports: in structures with linear deployment only one element (joint or bar) can be attached to a support. The rest of the elements present translation during the deployment of the structure. Constraints need to be included in some joints so the deployment is controlled and the structure follows desired movement paths. Retractable rings offer more options for supports. As illustrated in this chapter, some systems allow two or more joints to be fixed into position during the deployment of the structure. However, there are still aspects not fully resolved within the existing systems. For example, there are currently no retractable ring systems that use interconnected elements and where fixed position supports have only one degree of freedom and are placed on the perimeter, without interfering with the area covered by the structure. These and other aspects regarding fixed in position supports in retractable rings are studied further in chapter five. It can be argued that all the systems within deployable lattice structures with rigid links offer possibilities for grid configurations. Grids where interconnectivity between the elements is possible are arguably the best options. Interconnectivity allows the movement to be spread evenly throughout the structure and, in some cases, joints or bars can be omitted reducing the number of elements needed for the structure to operate. 76

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