University of Cambridge 
Department of Engineering 
Concepts for Retractable Roof Structures 
Dissertation submitted to the University of Cambridge 
for the Degree of Doctor of Philosophy 
Darwin College 
by 
Frank Vadstrup JENSEN 
v ,RRIDGE 
UNIV .RSITY 
UBRARY 
October 2004 
To Helle, Erik and Flemming 
Declaration 
The author declares that this dissertation is his own work and contains nothing which is 
the outcome of work done in collaboration with others, except for commonly understood 
and accepted ideas or as specified in the text and Acknowledgements. 
This dissertation has not been previously submitted, in part or in whole, to any Uni-
versity or Institution for any degree, diploma or other qualification. This report is 
presented in 158 pages and contains approximately 43.000 words and 124 figures. 
Acknowledgements 
The research presented in this dissertation has its roots in many places, much like the 
roots of a tree, and similarly to the tree it branches out into many interesting areas and 
results. However, the main focus, or trunk if you like, is the pursuit of novel retractable 
roof systems for large sporting venues. 
The first seed was planted by Dr Tim !bell while I was an exchange student at the 
University of Bath and for this I am deeply grateful. Though the seed had been 
planted it needed nutrition to grow - this was to be provided by my supervisor Professor 
Sergio Pellegrino at the Deployable Structures Laboratory, University of Cambridge, 
Department of Engineering. 
I would like to express my gratitude to Professor Sergio Pellegrino for his invaluable 
scientific guidance, encouragement and many ideas throughout the course of this re-
search. 
To my advisor Professor C. R. Calladine, FRS, I would like to extend my gratitude for 
his ideas, suggestions and scientific advice. 
At the Department of Engineering I would also like to thank the following: All the 
members of the Deployable Structures Laboratory for their advice and kind friendship. 
A special thanks must go to the following members or former members of the group; 
Tyge Schioler, Matthew Banter, Paul Ong, Dr Alan Watt, Jeffrey Yee, Dr Lin Tze Tan, 
Dr Wesley Wong, Dr Elizbar Kebadze, Dr Andrew Lennon and Richard Dietricht. For 
excellent technical support I would like to thank Alistair Ross, Steven Robinson, Jeremy 
Penfold, Roger Denston and Peter Knott. Importantly, my warmest thanks to Dr Lars 
Ekstrom, Dr Claus B. W. Pedersen and Dr Guido Morgenthal for their kind help and 
lasting friendship. 
I am also grateful for my fruitful collaboration with Dr Thomas Buhl, of the Technical 
University of Denmark, and for his continuing friendship. 
I gratefully acknowledge the financial support extended to me by the Cambridge Uni-
versity Engineering Department and Darwin College, which enabled me to take up a 
fellowship at the Faculty of Civil Engineering and Geosciences, Technical University of 
Delft, The Netherlands. There I was treated in the best possible manner by Professor 
Leo Wagemans, his staff and his students, and for this I am very grateful. 
I would also like to extend my thanks and gratitude to Dr Chris Williams at the 
University of Bath for allowing me to visit and for his very kind help and continuing 
ii 
advice. The same goes to Quintin Lake for his help, advice and friendship . And to my 
friends and former colleagues Dr Paul Shepherd and Dipl.Ing. Markus Balz for their 
friendship as well as their continued interest in my research. Also a heart felt thanks 
to all my other friends for enduring support and encouragements. 
I would also like to thank Mr Paul Meredith for his work cutting CFRP model parts 
at BNFL's test and rehearsal facility at Littlebrook. 
Many thanks should also go to the Happold Trust for financing my position as Design 
Teaching Assistant at the Cambridge University Engineering Department, and to the 
CUED Structures Group teaching staff and Professor fan Liddell, CBE of Buro Hap-
pold Consulting Engineers for their kind help and support on teaching issues. 
I am grateful for the financial support provided by S0ren Jensen Radgivende In-
geni0rfirma of Denmark and the interest and encouragements of those employed there. 
Finally, to those dearest to me, my parents Helle and Erik Jensen and my brother 
Flemming, I wish to express my love and give my most sincere thanks for all your 
love and support. A few further words of gratitude has been earned by Erik whom 
as a fellow engineer has not only been my father but also my tutor; without your 
nourishment I would not have come this far. 
Cambridge, October 2004 
Prank Vadstrup Jensen 
iii 
Abstract 
Over the last decade there has been a worldwide increase in the use of retractable roofs 
for stadia. This increase has been based on the flexibility and better economic perform-
ance offered by venues featuring retractable roofs compared to those with traditional 
fixed roofs. With this increased interest an evolution in retractable roof systems has fol-
lowed. This dissertation is concerned with the development of concepts for retractable 
roof systems. 
A review is carried out to establish the current state-of-the-art of retractable roof 
design. A second review of deployable structures is used to identify a suitable retract-
able structure for further development . 
The structure chosen is formed by a two-dimensional ring of pantographic bar elements 
interconnected through simple revolute hinges. A concept for retractable roofs is then 
proposed by covering the bar elements with rigid cover plates. To prevent the cover 
plates from inhibiting the motion of the structure a theorem governing the shape of these 
plate elements is developed through a geometrical study of the retractable mechanism. 
Applying the theorem it is found that retractable structures of any plan shape can be 
formed from plate elements only. To prove the concept a 1.3 meter diameter model is 
designed and built . 
To increase the structural efficiency of the proposed retractable roof concept it is invest-
igated if the original plan shape can be adapted to a spherical surface. The investigation 
reveals that it is not possible to adapt the mechanism but the shape of the rigid cover 
plates can be adapted to a spherical surface. Three novel retractable mechanisms are 
then developed to allow opening and closing of a structure formed by such spherical 
plate elements. 
Two mechanisms are based on a spherical motion for the plate elements. It is shown 
that the spherical structure can be opened and closed by simply rotating the individual 
plates about fixed points. Hence a simple structure is proposed where each plate is 
rotated individually in a synchronous motion. To eliminate the need for mechanical 
synchronisation of the motion, a mechanism based on a reciprocal arrangement of the 
plates is developed. The plate elements are interconnected through sliding connections 
allowing them mutually to support each other, hence forming a self-supporting structure 
in which the motion of all plates is synchronised. 
To simplify the structure further, an investigation into whether the plate elements 
can be interconnected solely through simple revolute joints is carried out. This is 
not found to be possible for a spherical motion. However, a spatial mechanism is 
iv 
developed in which the plate elements are interconnected through bars and spherical 
joints. Geometrical optimisation of the motion path and connection points is used to 
eliminate the internal strains that occur in the initial design of this structure so a single 
degree-of-freedom mechanism is obtained. 
The research presented in this dissertation has hence led to the development of a series 
of novel concepts for retractable roof systems. 
Keywords: Deployable structure, design, linkage, pantograph, retractable roof, spatial 
mechanism, spherical mechanism, stadia. 
V 
Contents 
Declaration 
Acknowledgements 
Abstract 
List of Figures 
List of Symbols and Abbreviations 
1 Introduction 
1.1 Stadia .. 
1.2 Retractable Roofs 
1.3 Deployable Structures 
1.4 Aims & Scope of Research . 
1.5 Outline of Thesis . ... . . 
2 Review of Literature and Previous Work 
2.1 Introduction . . . . . . . . . . . . . . . . . 
2.2 Retractable Roofs for Stadia ....... . 
2.2.1 Classification of Retractable Roofs 
2.2.2 Built Examples of Retractable Roofs 
2.2.3 Recent Proposals for Retractable Roofs 
2.3 Deployable Structures . . . . . . . . . . . . . . 
2.3.1 Space Applications ........... . 
2.3.2 Architectural and Civil Engineering Applications 
2.4 Retractable Pantographic Structures . . . . . . . 
2.4.1 Closed Loop Structures ..... .. . . . 
2.4.2 Structures formed by Angulated Elements 
2.4.3 Other Closed Loop Structures . 
2.4.4 Retractable Dome Structures . 
3 Retractable Bar and Plate Structures 
3.1 Introduction ... . .... . 
3.2 Retractable Bar Structures 
3.2.1 Radial Motion . . . 
3.2.2 Rotating Motion .. 
3.2.3 Non-circular Structures 
3.2.4 Additional Rotational Limits 
vi 
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3.3 Retractable Plate Structures . . . . . . . . . . 
3.3.1 Movement of Bar Linkage ..... .. . 
3.3.2 Application to Closed Loop Structures . 
3.3.3 Plate- only Structures . 
3.3.4 Periodicity of Boundary .. . 
3.4 Computer Assisted Design ..... . 
3.4.1 Optimisation of Plate Shapes 
3.5 Assemblies ........ . 
3.5.1 Planar Assemblies 
3.5.2 Stack Assemblies 
3.6 Discussion ........ . 
4 Design and Construction of Retractable Plate Structure 
4.1 Introduction . . . . . . 
4.2 Parts of the Model .. 
4.2.1 Plate Elements 
4.2.2 Hinges .. ... 
4.2.3 Actuator Assembly . 
4.2.4 Supports . . . . . . 
4.3 Structural Analysis . . . . . 
4.3.1 Finite Element Models . 
4.3.2 Physical Model .... 
4.4 Actuator Design . . . . . . . 
4.4.1 Virtual Work Analysis 
4.4.2 Gear Ratio 
4.5 Discussion . . . . . . . . . . . 
5 Spherical Retractable Structures: Preliminary Studies 
5.1 Introduction ...... . 
5.2 Spherical Geometry . .... . 
5.2.1 Spherical Excess ... . 
5.2.2 Spherical Trigonometry 
5.3 Pantographic Elements on a Spherical Surface . 
5.4 Plate Shape . . . . . . . . . . . . . 
5.4.1 Periodicity of Boundaries . 
5.4.2 Non-Symmetric Structures 
5.4.3 Physical Models 
5.5 Discussion ... ...... ... . 
6 Spherical Retractable Structures: Spherical Mechanisms 
6.1 Introduction . . . . . . . . . . 
6.2 Euler Pole ....... . ... . 
6.2.1 Compound Rotations . 
6.2.2 Spherical Trigonometry 
6.3 Varying the Location of the Euler Pole . 
6.3.1 Selection of Euler Pole . 
6.3.2 Physical Models . 
6.4 Relative Motion of Parts . . . . 
vii 
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6.5 Reciprocal Mechanism . . . . 
6.5.1 Small Physical Model 
6.5.2 Large Physical Model 
6.6 Discussion . . . . . . . . . . . 
7 Spherical Retractable Structures: Spatial Mechanism 
7.1 Introduction ..... . 
7.2 Spatial Mechanism .. 
7.2.1 Mobility Count 
7.3 Optimisation . . . . . 
7 .3.1 Element Orientation 
7.3.2 Connection Points 
7.4 Kinematic Model 
7.5 Discussion 
8 Conclusions 
8.1 Planar Structures . 
8.2 Spherical Structures 
8.3 Further Work 
Bibliography 
Vlll 
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List of Figures 
1.1 Coliseum in Rome (Escrig, 1996) . . . . . . . . . . . . . . . . . . . . . . 2 
1.2 Schematic of retractable roof over the Minute Maid Park, US (Post, 2000) 3 
1.3 Concept for deployable canopy (Mollaert et al., 2003) 4 
2.1 Methods for opening and closing (Ishii, 2000) . . . . . 10 
2.2 Methods for opening and closing of membrane structures by the Institute 
for Lightweight Structures (1971) . . . . . . . . . . . . . . . . . . . . . . 11 
2.3 Folding membrane roofs at (a) Montreal Olympic Stadium, and (b) Za-
ragoza Arena (Schlaich, 2000). Views from inside the stadia . . . . . . . 11 
2.4 Folding membrane roofs at (a) Pat·ken Stadium (Courtesy of CENO Tee), 
and (b) Toyota Stadium (Shibata, 2003) . . . . . . . . . . . . . . . . . . 12 
2.5 Miller Park (Courtesy of NBBJ Architects) . . . . . . . . . . . . . . . . 13 
2.6 Parallel retracting systems at (a) Ariake Colosseum (Ishii, 2000), and (b) 
Millennium Stadium (Courtesy of Atkins) . . . . . . . . . . . . . . . . . 14 
2.7 (a) Amsterdam Arena (Ishii, 2000), (b) Gelredome (Courtesy of ABT), 
and (c) Oita Stadium (Courtesy of Kisho Kurokawa Architects) . . . . . 15 
2.8 (a) Wembley Stadium (2004), and (b) Arizona Cardinals Stadium (Riberich, 
2003) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 
2.9 Four proposals for 2008 Beijing Olympic Stadium (a) B02; (b) B07; (c) 
B11; (d) B12 (Beijing Municipal Commission of Urban Planning, 2003) 17 
2.10 'HIMAT' retractable mast for space applications (Kitamura et al., 1990) 19 
2.11 Solid Surface Deployable Antenna (Guest & Pellegrino, 1996b) . . . . . 19 
2.12 (a) Umbrellas Medinah by Otto et al. (1995), and (b) Floating Concrete 
Pavilion by Calatrava (Tzonis, 1999) . . . . . . . 21 
2.13 Piiiero with his movable theater (Gantes, 2001) . . . 22 
2.14 Lazy-tong formed by three pantograph elements . . . 22 
2.15 Pantographic element consisting of two straight bars 23 
2.16 (a) Pantographic element consisting of two angulated elements, each 
formed by two bars, and (b) Multi-angulated element with three bars 24 
2.17 Radial movement of structure . . . . . . . . . . . . . . . . 25 
2.18 Retractable structure formed by multi-angulated elements 25 
2.19 Rotating movement of structure ............ . 
2.20 Cover elements by Kassabian et al. (1997) ........ . 
2.21 Reciprocal Plate Structure proposed by Chilton et al. . . 
2.22 Retractable systems formed by four-bar linkages (You, 2000) 
2.23 Swivel Diaphragm by Rodrigues & Chilton .... . 
2.24 Double-Chain mechanism by Wohlhart (2000) ... . 
2.25 Reciprocal dome proposed by Piiiero (Escrig, 1993) . 
lX 
26 
26 
27 
27 
28 
29 
29 
2.26 Iris Dome by Hoberman (Kassabian et al., 1999) . . . . . . . . . . . . . 30 
3.1 Circular structure with n;k = 8;3 (a) Closed Configuration, and (b) Open 
configuration . . . . . . . . . . . . . . . . . 33 
3.2 The opening ratio OR as a function of n;k . . . . . . . . . . . . . . . . . 34 
3.3 The stowage ratio SR as a function of n;k . . . . . . . . . . . . . . . . . 34 
3.4 The limits of rotation coincide with the intersections of two neighbouring 
circles of motion . . . . . . . . . . . . . . 35 
3.5 Definition of angles {3, "(, 6 and E . • • • • . . . . . • • • . . 36 
3.6 The rotation angle {3* as a function of n;k . . . . . . . . . 38 
3. 7 Sum of angles at internal hinges for structure with k = 3 . 38 
3.8 Structures formed from identical polygons and showing element with 
largest kink angle (a) An angulated element, (b) Similar rhombuses, and 
(c) Similar parallelograms . . . . . . . . . . . . . . . . . . . . . . . . . . 40 
3.9 Non-circular structures consisting of multi-angulated elements with vari-
able sum of kink angles . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 
3.10 Location of the fixed centre of rotation in (a) Intermediate configuration, 
and (b) Open configuration . . . . . . . . . . . . . . . . . . . . . . . 43 
3.11 Rotational limits of structure with hinges of finite size in (a) Closed 
configuration, and (b) Open configuration . . . . . . . . . . . . . 44 
3.12 Additional limit ( for the rotation angle {3* of circular structures 45 
3.13 Movement of four-bar linkage with two plates . . . . . . . . . . . 46 
3.14 Determining the boundary angle () . . . . . . . . . . . . . . . . . 47 
3.15 Boundary angle () for multi-angulated elements in (a) Closed configura-
tion, and (b) Open configuration . . . . . . . . . . . . . . . . . . . 48 
3.16 Wedge-shaped cover elements for circular structure . . . . . . . . . . . . 49 
3.17 Cover elements for a structure formed by projection onto a sphere . . . 51 
3.18 Wedge-shaped cover elements that fully cover the angulated elements 
and their hinges (a) k = 2, (b) k = 3, and (c) k = 4 . . . . . . . 52 
3.19 Plastic model of wedge-shaped plate structure with n; k = 12; 3 52 
3.20 Cardboard model of non-circular plate structure . . . . . . . . 53 
3.21 Periodic pattern of non-straight boundary . . . . . . . . . . . . 53 
3.22 Direction of initial movement and region defining possible boundary shapes 54 
3.23 Non-periodic end features . . . . . . . . . . . . . . . . . . . . . . . . 55 
3.24 Model of sixteen identical plates forming a perfect circular opening . . . 55 
3.25 Plastic model of plate structure formed from similar parallelograms . . . 56 
3.26 Top and bottom faces of model of non-circular structure where all plates 
and boundaries are unique . . . . . . . . . . . . . . . . . . . . . . . . . . 56 
3.27 Schematic of GTD application . . . . . . . . . . . . . . . . . . . . . . . 58 
3.28 Shape of cover elements after minimising the gaps and overlap between 
cover elements and maximising the size of central opening in the open 
configuration (Buhl et al., 2004) . . . . . . . . . . . . . . . . . . . . . . 59 
3.29 Single degree of freedom assembly of two rigidly interconnected structures 61 
3.30 (a) Two identical structures, (b) Single degree of freedom assembly of 
two structures with congruent rhombus, and (c) Branched pantographic 
elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 
3.31 Single degree of freedom assembly covered by rigid elements . . . . . . . 63 
X 
3.32 Movement in node structure and its linkages forming parts of three ring 
structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 
3.33 Cardboard model of assembly with node structure . . . . . . . . . . . . 64 
3.34 Two degree of freedom node structure (a) Intermediate configuration, 
(b) One linkage pair fully sheared, and (c) Both linkage pairs fully sheared 65 
3.35 Model of node assembly with two degrees of freedom 65 
3.36 Expandable circular plate structure. . . . . . . . . . . . . . . . . . . . 66 
3.37 Model of stacked assembly . . . . . . . . . . . . . . . . . . . . . . . . . 67 
3.38 Expandable free-form or "blob" structure (Jensen & Pellegrino, 2004) 67 
3.39 Stack assembly of two structures with three rigid connections 68 
3.40 Proposed use of plate structure to cover a sporting venue 69 
4.1 Hinge with a single rotational degree of freedom, scale 2:1 72 
4.2 Actuator assembly . . . . . . . . . . . . . . . . . . . . . . 73 
4.3 Switch and pin for reversing motion of model . . . . . . . 74 
4.4 Details of support cables in (a) Horizontal, and (b) Vertical configura-
tions, scale 1:1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 
4.5 Beam model, with n = 8, in its closed configuration. . . . . . . . . . . . 77 
4.6 Single plate element under self-weight (a) Mesh for shell elements, (b) De-
flections, and (c) Maximum principal stresses . . . . . . . . . . . . . . . 78 
4.7 Contours of deflections under self-weight for n = 9 plate model, held 
horizontal, with three support points . . . . . . . . . . . . . . . . . . . . 80 
4.8 Contours of deflections under self-weight for n = 9 plate model, held 
horizontal, with nine support points . . . . . . . . . . . . . . . . . . . . 82 
4.9 Contours of deflections under self-weight for n = 8 plate model, held 
horizontal, with eight support points . . . . . 83 
4.10 Physical model with n = 8, hung horizontally . . . . . . 84 
4.11 Physical model with n = 8, hung vertically . . . . . . . 84 
4.12 Vertical motion of hinge Ai hung from the fixed point P 87 
4.13 Vertical displacement and its derivative for rp = 400mm 89 
5.1 Spherical geometry (a) The sphere and its circles, and (b) Angle between 
great arcs and the angular length of these . . . . . . . . . . . . . . . 92 
5.2 Spherical geometry (a) A lune, and (b) A spherical triangle . . . . . 93 
5.3 Spherical geometry (a) Area of a triangle, and (b) Area of a polygon 94 
5.4 Pantographic element on a spherical surface . . . . . . . . . . . . . . 97 
5.5 Plots of LBP E for different nand lengths of AB . . . . . . . . . . . 99 
5.6 Spherical structure formed by multi-angulated elements (Kokawa, 2000) 100 
5.7 Spherical plates with straight boundaries in (a) Overlap free closed con-
figuration, and (b) Overlapping open configuration . . . . . . . . . . . . 101 
5.8 Spherical plates with kinked boundaries in (a) Overlap free open config-
uration, and (b) Overlap free closed configuration . . . 102 
5.9 Determining (a) Kink angle"'' and (b) Vertex angle a . . . . . 103 
5.10 Plate elements exhibiting periodicity . . . . . . . . . . . . . . . 104 
5.11 Plate elements with periodic boundary formed by a small circle 104 
5.12 Plate elements with periodic boundary forming a perfect circular opening 105 
5.13 Model made from plastic hemispheres . . . . . . 106 
5.14 Model made using rapid prototyping techniques . . . . . . . . . . . . . . 107 
6.1 Rotated plate element in (a) Two extreme configurations, labelled 0 and 
m, and (b) Intermediate configuration . . . . . . . . . . . . . . . . . . . 109 
6.2 The non-commutativity of vector rotations (a) Bx, By, Bz, and (b) By, Bx, 
Bz ....................................... 111 
6.3 Large three-dimensional rotation about OP . . . . . . . . . . . . . . . . 111 
6.4 A single rotation BE about PE moves the plate element from its closed 
to its open configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 
6.5 Determining the Euler pole PE and rotation BE from 6PEBoBm . . . . 113 
6.6 Determining PE and BE from 6PEBoBm and the initial position given 
by A ................. ..... ........ ........ 115 
6.7 Location of the Euler pole, PE , for varying positions A of the plate in 
the closed configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 
6.8 Compound rotations for -7f /2 :S A :S 7f /2 (a) Individual rotations, and 
(b) Solutions for e12 and e13 . . . . . . . . . . . . . . . . . . . . . . . . . 118 
6.9 Possible locations of PE for structures with n = 8 plates and increasing 
opening size L.POBm ..... .... ........... .. ...... 119 
6.10 Physical model with fixed points of rotation . . . . . . . . . . . . . . . . 120 
6.11 Computer generated images of how a retractable roof could be construc-
ted from spherical plates with fixed points of rotation . . . . . . . . . . 120 
6.12 Incremental rotations of apex Ar and An (a) Absolute, and (b) Relative 121 
6.13 Reciprocal retractable structures (a) Beam grillage, and (b) Swivel Dia-
phragm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 
6.14 Smaller model interconnected by pins running along curved paths . 123 
6.15 Model parts (a) Excavation from printer, and (b) Before polishing 124 
6.16 400 mm span model fabricated using 3D Ink-Jet printing . . . . . 125 
7.1 Spatial motion of n = 8 plate structure (a) Axes of rotation, and (b)-
(f) Opening of structure . . . . . . . . . . . . . . . . . . . . . 130 
7.2 Simulation results for (a) Internal strains, and (b) Rotations . . . . . 133 
7.3 Defining the location of point C using point B and angle cp . . . . . 134 
7.4 Convergence of tl(c/Jc, c/JE, c/Jc) using (a) fminsearch, or (b) fminunc . 135 
7.5 Simulation results for optimised values of cp (a) Internal strains, and 
(b) Rotations . . . . . . . . . . . . . . . . . . . . . . . . 136 
7.6 Convergence of 6. using (a) fminsearch, or (b) fminunc ... .... ... 137 
7.7 Two designs with finite sized spherical joints ... .. .... ...... 138 
7.8 Simulation results with lowest achieved strains (a) Internal strains, and 
(b) Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 
7.9 Simulation results for design with finite sized joints (a) Internal strains, 
and (b) Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 
7.10 Simulation results with four peaks in the strain function 6. (a) Internal 
strains, and (b) Rotations . . . . . . . . 140 
7.11 ProEngineer kinematic model . . . . . . 141 
7.12 Motion revealing conflicts between bars 143 
8.1 Rendering of novel concept retractable roof 147 
xii 
List of Symbols and 
Abbreviations 
Chapters 1-4 
Acen Centre of rotation for element A, p. 36 
Ai Hinge i of element A, p. 35 
Aj Internal hinge number j of element A, p. 37 
E Young's Modulus, p. 77 
G Weight of model, p. 87 
h Height of period, p. 54 
i Integer ::; k, p. 35 
j Integer 1 ::; j ::; k- 1, p. 37 
k Number of bars per element, p . 32 
Length of bar, p. 32 
lp 
L 
n 
0 
r 
r, z 
rp, zp 
R 
Length of hanger, p. 86 
Length of period, p. 47 
Number of elements per layer, p. 32 
Origin, p. 33 
Inner radius, p. 32 
Radius of joint, p. 43 
Coordinates in vertical plane, p. 87 
Coordinates of fixed point P, p. 87 
Outer radius, p. 32 
Actuator torque, p. 85 
Ball-bearing friction , p. 86 
Virtual work done by actuator, p. 85 
X Ill 
Wb Virtual work done by friction, p. 86 
W9 Virtual work done by gravity, p. 87 
a Kink angle, p. 32 
(3 Rotation angle, p. 36 
(3* Total rotation angle (f3open- f3closed), p. 35 
(3' Alternative total rotation angle (2a- (), p. 62 
1 Inner angle, p. 36 
8 Element angle (ka), p. 36 
E Limit angle (1r- a), p. 36 
( Additional limit angle, p. 45 
rJ Angle, p. 46 
() Boundary angle, p. 4 7 
v Poisson's ratio, p. 77 
Abbreviations 
ABS 
AWJ 
CFC 
CFRP 
EDM 
FA 
FIFA 
GTD 
OR 
SR 
UEFA 
uv 
Acrylonitrile-Butadiene-Styrene, p. 52 
Abrasive Water Jet Machine, p. 55 
Carbon Fibre reinforced Carbon, p. 16 
Carbon Fibre Reinforced Plastic, p. 71 
Wire Electrical Discharge Machine, p. 56 
British Football Association, p. 14 
Federation Internationale de Football Association, p. 8 
Computer application, p. 56 
Opening ratio (ropen/ Rclosed), p. 33 
Stowage ratio (Ropen/ropen), p. 33 
Union of European Football Associations, p. 8 
Ultra Violet, p. 11 
Chapters 5-8 
A 
E 
i 
I 
j 
L 
m 
NI 
n 
0 
P, P* 
r 
R 
R 
s 
x, y, z 
{3 
'T} 
if 
Area, p. 92 
Unit-vector, p. 110 
Spherical excess, p. 95 
Integer, p. 95 
Identity matrix, p. 112 
Number of joints per plate element, p. 129 
Length of period, p. 103 
Number of rotations between closed and open configurations, p. 109 
Relative mobility, p. 129 
Number of plate elements, p . 96 
Centre of sphere, p. 92 
Pole and antipodal, p. 92 
Euler pole, p. 110 
Radius of small circle, p. 92 
Radius of great circle and sphere, p. 92 
Rotation matrix, p. 112 
Spin matrix, p. 112 
Number of constraints imposed by joint i, p. 129 
Coordinates, p. 110 
Apex angle, p. 97 
Vertex angle, p. 94 
Vertex angle, p. 93 
Angular defect , p. 95 
Strain function (b. = J c~c + cfm + 4c ) , p. 132 
Strain, p. 132 
Internal angle of spherical angulated element, p. 97 
External angle ( 1r - {3), p. 94 
Pseudo-vector (Be), p. 110 
Rotation angle about axis i, p. 110 
XV 
BE Rotation angle about Euler pole PE, p. 110 
"' Kink angle, p. 102 
,\ Initial position angle, p. 115 
f1. Angle, p. 113 
~ Angle, p. 115 
1 Connection angle, p. 134 
w Modified pseudo-vector (2 tan(B /2)e) , p. 110 
Abbreviations 
ABS 
FDM 
PTFE 
Acrylonitrile-Butadiene-Styrene, p. 106 
Fused Deposition Modelling, p. 106 
Polytetrafluoroethylene, p. 126 
XVl 
I 
Chapter 1 
Introduction 
1.1 Stadia 
Throughout civilisation, mankind has assembled to view with awe the spectacular and 
heroic feats performed by their sporting fellows. The theatres for these performances 
have been known as stadia since the first were build together with western civilisation 
in ancient Greece. The Greek stadia were U-shaped foot racecourses and the 1 stadium 
(approximately 200 meters) length of the racecourse gave name to these venues ( Geraint 
& Sheard, 2001). The design of these first stadia was based on traditional open air 
theatres, excavated out of hillsides to form low uncovered seating tiers along both 
sides of the racecourse. They were of great civic importance in Greek life and could 
host up to 45,000 spectators. The sporting games held at the stadium in Olympia 
started what is still celebrated as the Olympic tradition. Centuries later the Romans 
dedicated the same importance to their stadia, the amphitheatre, though it no longer 
hosted peaceful athletic meetings. Instead they were the scenes for gory battles of life 
and death. The most famous of these is the Flavian Amphitheatre in Rome, better 
know as the Coliseum. From this particular venue many later stadia would inherit its 
structural form, with its good sightlines, structural stability and appropriate volumes 
of circulation space. 
With the fall of the Roman Empire, European societies concentrated their architectural 
efforts on religious buildings and no new stadia would be built in the world until the 
arrival of the industrial revolution fifteen centuries later. With the new social order that 
followed also came a renewed interest in mass spectator events. On June 23rd, 1894, the 
French Baron Pierre de Coubertin founded the International Olympic Committee with 
a ceremony held at the University of Sorbonne in Paris. Together, the revival of the 
Olympic tradition and the rapid growth in popular sports, like football and baseball, 
would form the catalyst and economic base for the construction of most stadia in the 
twentieth century and this is likely to continue in the twenty-first century also. Though 
present day society recognises the civic importance of the stadia and the events hosted 
there, it is increasingly unwilling to support financially the construction and running of 
these venues. Because of this, the long-term economic performance of the venues has 
now become the single most important planning and design issue for any new stadium. 
Ill 
1.1. STADIA 
Figure 1.1: Coliseum in Rome (Escrig, 1996) 
First generation modern stadia were not the grand monuments of the past. Instead 
they grew organically as attendance increased. First small shacks were erected for com-
fort, later cheaply build stands were built to improve sight-lines; both to attract the 
maximum number of spectators through the tills. To minimise the loss of revenue on 
rainy days many stands were fitted with simple corrugated steel sheet roofs to keep the 
rain out. The invention of TV and the broadcasting of live-sports from 1937 dramat-
ically changed the existing economic situation. Now spectators had the possibility of 
watching sporting events in the comfort of their own living room. This started a steady 
continuing decrease in attendance resulting in venues having to find additional streams 
of revenue (Sheard, 1998). 
This was mainly achieved by attaching a number of other facilities, such as shops, 
hotels, offices and fitness centres, to the venue itself (Roberts & Dickson, 1998). The 
stadia thus evolved from a simple sporting venue to a multi-facility complex aiming to 
secure constant revenues and long-term economic performance. 
Despite the multi-facility approach, many venues still struggle economically as their 
largest space, the sporting theatre itself, is largely unused. The home team only plays 
at home approximately once per week and revenues are therefore only generated on 
these few game days. To overcome this problem many stadia are today multi-purpose 
venues able to host a small range of sporting and non-sporting events such as football, 
athletics and concerts. To increase the range of events that can be hosted, during 
the last 40 years some stadia, mainly in the USA, have been built as fully covered 
arenas (Geraint & Sheard, 2001; Lenczner, 1998). This has only been possible as their 
main sporting tenants are able to play on artificial surfaces. This attractive solution 
has not been available to the majority of venues throughout the rest of the world as 
the predominant sport, football, requires a natural turf that cannot be grown in a fully 
covered arena. 
For these venues an alternative solution was also inspired by the Roman amphitheatre. 
Similarly to its structural section, its temporary roof formed by retractable canvas 
awnings, known as the velum, was to provide the inspiration for retractable roofs of 
modern day stadia (Escrig, 1996; Ishii, 2000; Sheard, 1998; Zablocki, 2002). 
2 
1.2. RETRACTABLE ROOFS 
1.2 Retractable Roofs 
A retractable roof is, unlike a demountable or temporary structure, a permanent struc-
ture capable of undergoing a geometric transformation or folding between two distinct 
configurations, usually referred to as the open and closed configurations. This ability to 
transform geometrically is what distinguishes retractable roofs from traditional static 
roofs. The transformation process of going from the open to the closed configuration 
is here referred to as closing or retracting and the reversed transformation as opening 
or expanding. 
A retractable roof can provide a variable amount of cover for the space below and thus 
gives an increasing number of stadia the ability both to grow natural turf and fully 
enclose the venue temporarily hence increasing the range of events that can be hosted. 
Retractable roofs have not only found their use in sport venues but also at ship yards, 
exhibition and recreational spaces and in cars. Retractable roofs used for stadia are 
typically stable and thus able to carry loads throughout their transformation but some, 
such as folding membranes, are only stable and able to carry load in one or both of 
their extreme configurations. 
Since large retractable roofs were first reintroduced in the 1950's a variety of retractable 
systems have been built and many more ideas and concepts have been proposed. The 
first systems were based on well-known crane technology, soon to be followed by folding 
membranes, inspired by umbrellas and tents, and telescopic systems. This evolution 
continues with various types of deployable structures being proposed as the next step 
in the evolution towards better structural and economical performance of retractable 
roofs (Ishii, 2000; Miura & Pellegrino, 2002). 
Figure 1.2: Schematic of retractable roof over the Minute Maid Park, US (Post, 2000) 
3 
I 
I 
j, 
1.3. DEPLOYABLE STRUCTURES 
1.3 Deployable Structures 
A large group of structures have the ability to transform themselves from a small, 
closed or stowed configuration to a much larger, open or deployed configuration; these 
are generally referred to as deployable structures though they might also be known as 
erectable, expandable, extendible, developable or unfurlable structures. As retractable 
roofs can also transform from their open to a closed configuration they are sometimes 
also classified as a type of deployable structure. 
The research subject of deployable structures is relatively young as it was pioneered 
only in the 1950s and '60s. Though the subject is young, many of its best known 
applications have been around for millennia, examples are the umbrella, the folding 
chair, and the velum of the Roman Coliseum, Figure 1.1. At present the two main 
applications areas of deployable structures are Aerospace and Architecture. 
From the first man made satellite, Sputnik, launched on October 4th, 1957, the scientists 
and engineers behind space programs throughout the world have been faced with tight 
weight and space restrictions for their space structures. Because of these restrictions 
all spacecraft structures, or parts thereof, larger than the space available in the launch 
rocket must be stowed and later deployed in space. This is done for antennas, reflectors, 
masts, solar panels and so on. Each one having its own unique shape and characteristics 
often resulting in equally unique stowage and deployment techniques. 
The Spanish architect Pinero was the first in modern times to explore systematically the 
possibilities and properties of deployable structures in architecture (during the middle 
ages prominent people such as Leonardo da Vinci had proposed deployable structures 
for trusses, bridges and machinery). Pinero proposed to use them for temporary cov-
erings of exhibition spaces or swimming pools. The architectural uses of deployable 
structures were initially reserved for these temporary or mobile structures but they are 
now being applied to a wider range of structures, such as retractable roof systems. 
Figure 1.3: Concept for deployable canopy (Mollaert et al., 2003) 
1.4 Aims & Scope of Research 
Though many different 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 
4 
1.5. OUTLINE OF THESIS 
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 structures. 
This dissertation aims to provide designers with viable alternatives to the existing 
solutions for large span retractable roof structures. The design of a full scale structure 
is beyond the scope of the research undertaken. Hence the basis for the development has 
been the assumption that most of the problems limiting the use of deployable structures 
are caused by the geometry of the structure and its mechanism. The research has 
therefore been focussed on eliminating these problems mainly by means of geometric 
studies. 
The problem has been approached by conducting a comprehensive review of the present 
state-of-the-art of large retractable roof structures and deployable structures proposed 
for retractable roofs. On the basis of this review it was decided to continue the devel-
opment of a retractable mechanism based on hinged bars. 
Through the development of this system a number of advances were made, notably a 
novel structure composed solely from plate elements forming a completely gap free roof 
surface in both open and closed configurations. However, a proof-of-concept model 
showed the potential structural stiffness limitations of flat two-dimensional systems. 
On the basis of this finding, three novel concepts with a spherical geometry have been 
developed. 
1. 5 Outline of Thesis 
In Chapter 2, a review of existing retractable roof technologies and deployable struc-
tures is presented. The first part is concerned with the current state-of-the-art of 
retractable roofs. The second part focusses on current technologies within the field of 
deployable structures and on their applications in civil engineering, and for retractable 
roof structures in particular. 
In Chapter 3 the concept of a retractable roof formed by hinged plates is developed. 
First a novel method for describing the motion of a particular retractable bar system 
is developed. Using this new method the kinematic consequences of introducing finite 
sized joints in the mechanism is investigated. A study of the motion of neighbouring 
elements within the overall structure reveals the possibility of replacing the bars of the 
structure with rigid plates, thus allowing the creation of retractable structures formed 
by plates. The method for designing such structures is then expanded to allow for a 
wide variety of possible shapes. 
In Chapter 4, a 1.3 metre span physical model of such a retractable structure is presen-
ted. The first part of the chapter presents the model as built while the latter part 
presents the analysis carried out in order to produce the model. From this study it 
is then concluded that this particular retractable structure is not well suited for large 
scale applications such as stadia roofs. A number of alternative uses are then proposed. 
In Chapter 5, a novel geometry for spherically shaped elements is revealed. Retractable 
5 
1.5. OUTLINE OF THESIS 
systems based on this new geometry seems better suited for applications on a large scale 
than geometries previously proposed by other authors. The derivation of the element 
shape is presented and linked to the geometry of the two-dimensional plate elements 
presented in Chapter 3. 
In Chapter 6 the motion of such spherically shaped elements on the sphere itself is 
investigated. It is found that it is possible to create retractable structures in which 
the individual elements only undergo a rotation about a fixed axis. The position of 
this axis thus becomes important, and it is shown that it is possible to position this 
axis within the boundaries of the elements themselves thus greatly simplifying the 
structure. By investigating the relative motion between neighbouring elements it is 
not found possible to link two of these elements together using only cylindrical hinges. 
Instead, it is proposed to use a sliding mechanism to create a reciprocal system with 
only a single internal mechanism capable of supporting itself efficiently. A physical 
model is built to demonstrate the feasibility of the concept. 
In Chapter 7 an alternative mechanism using the same spherical elements is proposed. 
The sliding mechanism and cylindrical hinges have been replaced by a series of bar 
linkages and spherical joints. Such a system is normally overconstrained and thus not 
capable of executing large motions without large strains occurring in the structure. 
However, it is shown that by carefully choosing the connection points it is possible to 
minimise the peak strains occurring in the structure, so they become very small, and 
thus allowing the structure to function as a mechanism. 
Chapter 8 concludes the study by comparing the proposed structures and discussing 
their possible application as large retractable roofs. A number of ideas for the direction 
of future work are also provided. 
Chapter 2 
Review of Literature and 
Previous Work 
2.1 Introduction 
The beginning of this chapter discusses the drivers which over the last decade have res-
ulted in a significant increase in the number of constructed and proposed stadia with 
retractable roofs. A classification of retractable roof systems is then presented and a 
number of commonly used systems are reviewed through built examples while state-
of-the-art systems are reviewed through recently proposed stadia and the competition 
entries for the 2008 Beijing Olympic Stadium. The second part of the chapter presents 
a brief review of current deployable structures technologies, both for space and archi-
tectural applications. For architectural and civil engineering applications, deployable 
systems using pantograph elements have become quite popular. As they can be used to 
form closed loop structures which can retract towards their perimeter, they have been 
proposed for use as retractable roofs for stadia. Hence the third part of the chapter 
presents the research that has been carried out on the use of closed loop retractable 
roofs made from pantograph elements. 
2.2 Retractable Roofs for Stadia 
Though modern retractable roofing systems have been used throughout the world since 
the 1930's, in one form or another, it is only in recent decades that they have become 
commonly used in stadium design. At the start of the new millennium there were 
approximately 25 large~span retractable roofs in the world, of which around 15 of those 
were stadia roofs and approximately half of the total were located in Japan (Ishii, 2000; 
Zablocki, 2002). There was at the same time an equal number of roofs at various stages 
of planning or construction, thus marking a sharp increase in the construction of large 
retractable roofs. 
Of the many factors contributing to this trend, economics is the main driving force, as 
demonstrated by Dean et al. (1998) using the retractable roof for the Victoria Dock-
7 
2.2. RETRACTABLE ROOFS FOR STADIA 
lands Stadium, later renamed the Colonial Stadium, in Melbourne, Australia, as an 
example (Sheldon & Dean, 2002). As nearly all sporting facilities around the world 
have been privatised and have had their public subsidies considerably reduced it has 
become both more important and more difficult to operate these facilities profitably. 
This is especially true for the larger national or regional stadium venues, which have 
increased construction costs and limited possibilities of public access to the facilities , 
unlike swimming pools, arenas and training facilities, which can generate additional rev-
enue through public use when no sporting events are hosted. This economic problem 
has led to most new stadia being built with a number of additional revenue generating 
facilities attached such as shops, hotels, cinemas and health clubs, turning them into 
what is known as multi-facility venues (Geraint & Sheard, 2001; Roberts & Dickson, 
1998). 
To improve their economic performance further many stadia have increased their host-
ing of non-sporting events, such as concerts and exhibitions, or sporting events for which 
the venue was not originally designed, such as boxing, tennis or speedway. These types 
of events are usually not as regular as the games played by the main tenant of the 
stadium. To achieve more regular use, an increasing number of stadia are home to 
more than one team and they are often from different sports. The best know example 
is the San Siro Stadium in Milan that hosts the city's two football teams Inter Milan 
and AC Milan. However, increasing the number of events hosted does cause a number 
of problems. 
A large number of non-sporting events require an enclosed space and thus most stadia 
are not capable of hosting events such as conferences and exhibitions. This led to the 
construction of a generation of enclosed stadia in the USA, between the late 1960s and 
the 1980s. A fully enclosed venue also has the advantage that cancellations due to 
adverse weather conditions will not occur. This strategy has not been followed in other 
parts of the world, where football is the most popular sport, as its governing body, 
Federation Internationale de Football Association (FIFA), still requires national and 
international games to be played "in the open-air" (Zablocki, 2002). 
Another inherent problem with increased use is that the playing surface slowly deteri-
01·ates with the increased activity. Many non-sporting events also require some sort of 
temporary covering of the turf and thereby further aggravates the problem. In addition 
to this, in many of the newer stadia with higher stands and larger overhanging roofs , 
built to accommodate more spectators, it has been found that the playing surface de-
teriorates more rapidly, as sun light and air movement have been reduced considerably. 
An often suggested solution to the above mentioned playing surface problems would 
be to install an artificial surface with synthetic grass. Such a surface would be able 
to withstand the use of both sports and pop concerts while eliminating many environ-
mental problems inside the stadium. This has already been tried in the USA, where 
the introduction of artificial surfaces allowed the construction of fully enclosed stadia. 
In Europe and the rest of the world the installation of artificial grass in football stadia 
has been prevented by FIFA, which requires national and international games to be 
played on natural grass. In a recent development, the European football organisation 
Union of European Football Associations (UEFA) has announced it is running anum-
ber of trials with artificial grass which has now been installed in several stadia on a 
8 
2.2. RETRACTABLE ROOFS FOR STADIA 
trial basis (UEFA, 2003). Two of these are high-profile national stadia, namely the 
Finnair Stadium, Helsinki, and the Amsterdam Arena, Amsterdam (Bisson, 2003). 
Another solution for the problem of poor conditions for natural grass has been imple-
mented at Gelredome, Arnheim, and the 2001 Sapporo Dome, Hokkaido. Here the turf 
is planted in a large tray, which can then be transported outside the stadium when not 
in use. This provides the turf with good growth conditions while allowing the stadium 
the maximum flexibility. At the Sapporo Dome the tray is lifted using air pressure and 
rolled on wheels, while at Gelredome the tray is mounted on a series of low friction 
rails. The approach taken at Gelredome has later proven troublesome according to the 
design engineers Nijsse & van Vliet (2003). 
In a 1995 British Sports Council publication on stadium design the problem of natural 
grass growth was said to be the limiting factor for further advances in stadium design 
in the United Kingdom. It was therefore found necessary either to find a method for 
growing turf beneath an enclosed roof, accept synthetic playing surfaces, or develop a 
cost-effective retracting roof system (Murray, 1995). The retractable roof solves the 
problem by reducing the size of the overhanging roof, thus allowing an increase in sun 
light and air movement and combining this with the flexibility of an enclosed venue. 
2.2.1 Classification of Retractable Roofs 
The first large retractable roof built for a sporting venue was the 1961 Pittsburgh Civic 
Arena, later renamed the Mellon Arena, USA (Ishii, 2001; Mellon Arena, 2004). It was 
a steel dome with four retractable steel truss panels making up half of the domed roof. 
In the open position these panels were then nested below the remaining fixed half of the 
roof in an overlapping arrangement. The dome was not self-supporting, rather it was 
hung from a large steel truss cantilevering over the dome. The dome was built using 
mechanical solutions borrowed from crane design. This is still the case for most large 
retractable roofs, as cranes provide tested and well-understood solutions for problems 
of tolerances, thermal expansion, and deflections - to name only a few among the many 
issues related to moving large and heavy structures along predefined paths. 
Since then other stadia have been constructed with retractable roofs. To ensure that the 
lessons learned through these pioneering projects were not forgotten the International 
Association for Shell and Spatial Structures set up Working Group No. 16 in 1993. 
The aim of the group was "to develop a State-of-the-Art report and Guidelines for the 
design and construction of retractable structures" and their conclusions were presented 
in their resulting publication Structural Design of Retractable Roof Structures (Ishii , 
2000) . Two types of structures were reviewed by the group: rigid frame structures and 
folding membrane structures. The emerging group of structures utilising expandable 
frames, in general referred to as deployable structures, was not reviewed as this type 
had yet to be realised on a large scale. 
Large retractable roofs are today an established building type and in Japan legal stand-
ards and design guidelines for new retractable roofs have been introduced through the 
Building Standard Law of Japan (Ishii, 2001). 
In Figure 2.1 existing methods for opening and closing a large retractable roof are 
2.2. RETRACTABLE ROOFS FOR STADIA 
shown. These methods can broadly be represented in a three-by-three matrix by clas-
sifying their movement as: (a) parallel, (b) circular or (c) vertical, and their method 
of stowing the retracted roof as: (i) overlapping, (ii) non-overlapping or (iii) folding, 
as shown in the figure. There are a number of systems that utilises a combination of 
these methods, as described below. 
(i) 
Overlapping 
systems 
(ii) 
Non-overlapping 
systems 
(iii) 
Folding 
systems 
(a) Parallel movement (b) Circular movement (c) Vertical movement 
~ ~ 
Figure 2.1: Methods for opening and closing (Ishii, 2000) 
2.2.2 Built Examples of Retractable Roofs 
Folding Membrane Systems 
Since the 1950s research has been carried out into the use of folding membranes as 
retractable roofs. The pioneering work was led by Prof Frei Otto at the Institute for 
Lightweight Structures, Stuttgart, Germany. Many schemes were developed and some 
of those were realised as retractable or temporary coverings for skating rinks, exhibitions 
and open-air theatres (Institute for Lightweight Structures, 1971; Mollaert, 1996). A 
schematic representation of many possible schemes that were considered is shown in 
Figure 2.2. 
The first application of a folding membrane roof to a stadium was the 1976 Montreal 
Olympic Stadium. The folding roof itself was however not completed until the mid-
eighties and has later been replaced by a fixed roof due to wind-induced failures of 
the retractable membrane. The membrane could be lowered from a tower leaning over 
the stadium. The membrane was guided along a series of cables running from the 
tower to the boundary of the fixed roof over the stands. Other cables fixed to the 
membrane would provide the tensioning of the membrane once the boundary was held 
in place (Ishii, 2000; Schlaich, 2000). 
10 
2.2. RETRACTABLE ROOFS FOR STADIA 
Construction system Type of movement Parallel Central Circular Peripheral 
<!) Bunching ~ ~~ ~ ~ s:: <ll Stationary H _,. .. ....._ 
supporting structure "S ~ ~ <!) ;8 Sliding 
<!) ~ ~ H Rolling ;:l """ u ;:l 
H ~ ~~ ~~ """ Moveable en bO Folding supporting structure .s 
""" '" H 0 ~ 0.. 0.. Rotating ;:l en 
Figure 2.2: Methods for opening and closing of membrane structures by the Institute for Light-
weight Structures (1971) 
The lessons learned from the very complicated Montreal roof were applied to another 
retractable membrane roof by the engineers Schlaich Bergermann and Partners. For 
the 1988 Zaragoza Arena, Spain, the existing bull fighting arena was covered with a 
partly retractable membrane roof. From a floating position at the centre of the opening 
in the fixed part of the roof, the folded membrane deploys along radial spokes, which 
also carry the load of the membrane (Ishii, 2000; Schlaich, 2000) . The material of 
the folding membrane was replaced after ten years of use due to Ultra Violet (UV) 
degradation of the fabric and developing tears along the folding lines (Monjo et al. , 
2002). 
(a) (b) 
Figure 2.3: Folding membrane roofs at (a) Montreal Olympic Stadium, and (b) Zaragoza 
Arena (Schlaich, 2000). Views from inside the stadia 
Two other stadia use folding membranes for their retractable roofs but unlike the 
11 
2.2. RETRACTABLE ROOFS FOR STADIA 
Montreal and Zaragoza roofs, the Toyota Stadium in Japan, built in 2001, and Parken 
Stadium in Denmark do not use a single large membrane to cover the pitch. Instead 
they are composed of air-inflateable membrane cushions attached between a number 
of large parallel trusses that run on two tracks either side of the opening in the fixed 
roof. In the open position the trusses are parked next to each other at o~e end of the 
pitch, with the cushions hanging deflated and folded between them. When the roof 
is closed the trusses move and the cushions are inflated and provide cover over the 
pitch. The system is simple in concept but in the case of the Toyota Stadium it was 
found necessary to also install smaller, folding trusses below the cushions to provide 
sufficient stiffness to the system (Shibata, 2003). In the case of the Par ken Stadium, 
the retractable roof was fitted in 2001 to the existing 1992 stadium roof to provide a 
venue for the 2001 Eurovision Song Contest. 
(a) (b) 
Figure 2.4: Folding membrane roofs at (a) Pm·ken Stadium (Courtesy of CENO Tee), and (b) 
Toyota Stadium (Shibata, 2003) 
Vertical Systems 
For large retractable roofs significant vertical movement is usually avoided, as gravita-
tional forces increases the power required by the mechanical systems that operates the 
roof. The impact of gravity effects can be minimised by constructing a very light roof, 
as was the case for the Montreal Olympic Stadium. A number of newer systems that 
are classified as having parallel movement do have gentle slopes in the path along which 
the roof is moved as it allows greater architectural freedom. Examples of this are the 
Toyota and Oita Stadia and Amsterdam Arena, see above and below in this section. 
Circular Systems 
The non-symmetrical layout of this type of retractable roof in its open configuration, 
and its ability to cover a triangular pitch has made it a popular choice for baseball 
venues in Japan and the USA. Figure 2.1(b) shows how the panels of the retractable 
roofs are pivoting about a fixed point allowing them to stowed by either overlapping 
or folding the panels. For stadia only overlapping systems using rigid panels have been 
constructed as the folding of large and non-symmetric panels is difficult (Geraint & 
12 
2.2. RETRACTABLE ROOFS FOR STADIA 
Shem·d, 2001; Ishii, 2000, 2001). Examples of circular, overlapping systems are the 
1993 Fukuoka Dome in Japan and the recent Miller Park, USA, and the Sendai Dome 
in Japan, both built in 2000 (Hewitt et al., 1998; WG16 lASS, 2001). 
Figure 2.5: Miller Park (Courtesy of NBBJ Architects) 
Parallel Systems 
The majority of large retractable roofs are based on parallel movement. The simplicity 
of this motion makes it an attractive option for most venues and many technical solu-
tions can be taken directly from large cranes such as those used in container terminals. 
The simplest variation of the parallel system is simply to move the entire rigid roof off 
the venue itself and put it next to it to achieve the open configuration. This however 
does not reduce the plan area covered by the roof and thus additional space has to be 
allocated to this purpose; hence this form of retractable roof is thus rarely used. An 
example of this is the 1991 Ariake Colosseum (Ishii, 2000), Figure 2.6(a), where the 
roof is composed of two parts which both span the entire seating bowl of the stadium 
and thus can be rolled off completely. 
Other variations of the parallel system have already been discussed above for the Toyota 
Stadium and Pm·ken Stadium, in Section 2.2.2. Similarly, the 1989 Toronto Sky Dome 
retracts three large rigid space frame panels using a telescopic system. In the open 
configuration, these panels overlap the fixed structure at one end of the venue (Ishii, 
2000). The most commonly used system, generally referred to as a bi-parting system, 
is also based on parallel movement (Griffins, 2003). 
Bi-parting Systems 
A retractable bi-parting roof is comprised of two large rigid panels, which when opened 
move in opposite directions. The panels are moved along two parallel rails mounted on 
two main trusses located along the edge of the fixed roof either along or perpendicular 
to the main axis of the venue, the first being the arrangement most frequently used. 
Hence this system covers only the large central opening between the fixed roof, that 
covers the seating areas on all four sides of the stadium. The two retractable panels 
can therefore be moved above or below the permanent roof when not in use. A typical 
example of this system is the 1999 Cardiff Millennium Stadium in Wales, shown in 
Figure 2.6(b). 
13 
2.2. RETRACTABLE ROOFS FOR STADIA 
Figure 2.6: Parallel retracting systems at (a) Ariake Colosseum (Ishii, 2000), and (b) Millen-
nium Stadium (Courtesy of Atkins) 
Other interesting examples of this system and stadia design in general are the 2001 
Oita Stadium, Japan, the 1996 Amsterdam Arena and the 1998 Gelredome, The Neth-
erlands, all shown in Figure 2. 7. The last two are interesting as they have retractable 
roof panels which span the longest direction of the opening, making them unusual. This 
solution has been found economical in both cases as it reduces the number of structural 
members. The Gelredome has two other interesting features: many retractable roofs 
are designed such that they cannot be moved when wind and/m· snow loading surpasses 
a pre-determined level. This is not the case for the Gelredome as the designers found 
that the internal fire load imposed greater requirements on the roof structure than the 
ultimate wind and snow loads. Thus the roof can be opened under all load conditions 
as in case of fire this permits the ventilation of the enclosed space and prevents the 
heat from building up (Nijsse & van Vliet, 2003). It is also noteworthy that it was 
found economically viable to build a multi-purpose venue featuring both a retractable 
roof and a movable pitch. For the Amsterdam Arena it is interesting that the venue, 
as one of the few in Europe with a retractable roof, has decided to install an artificial 
playing surface, as mentioned in Section 2.2. The Oita Stadium is interesting as it 
uses a spherical cap, which provides a circular plan for a football stadium. The two 
roof panels are not supported on two rails running along the central opening, instead 
they are supported on seven arches of which five span across the oval opening in the 
roof. The movement of the panels is not purely horizontal, instead it follows the curved 
profile of the fixed roof and hence requires additional energy to overcome gravitational 
forces during closing. This is also the case for the roof of the Amsterdam Arena. 
2.2.3 Recent Proposals for Retractable Roofs 
The British Football Association (FA) has commissioned a new stadium to replace 
the old Wembley Stadium, in London. This new venue, currently under construction, 
will, when completed in 2006, have a retractable roof unlike any other in the world. 
Engineered for letting sunlight onto the famous grass turf the retractable roof will allow 
the entire pitch to be bathed in sunlight during the full length of the FA Cup final in 
14 
2.2. RETRACTABLE ROOFS FOR STADIA 
Figure 2.7: (a) Amsterdam Arena (Ishii, 2000), (b) Gelredome (Courtesy of ABT), and (c) Oita 
Stadium (Courtesy of Kisho Kurokawa Architects) 
May. This was required by the FA as it avoids the presence of annoying contrasts 
between sunlit and shadow areas in TV pictures. Allowing the sunlight to reach all 
areas of the pitch could only be achieved by not covering the spectators on the South 
stand and hence a retractable roof system will be fitted over this. The retractable 
roof allows the stand to be covered when needed though it does not enclose the venue 
completely unlike other retractable roof systems (Barker, 2000; Manica, 2003). 
In the USA an American football team, the Arizona Cardinals, is also building a new 
multi-purpose venue. The Arizona Cardinals Stadium will feature both a movable pitch 
and a hi-parting retractable roof, originally planned as a telescopic system (Gee, 2002). 
This is part of the trend in America where synthetic playing surfaces are being replaced 
with natural ones again, and combining a retractable roof with a movable pitch is found 
by some designers as the best way of providing full multi-use and achieving the best 
economic performance despite the additional construction and running costs (Riberich, 
2003). 
(a) 
Figure 2.8: (a) Wembley Stadium (2004), and (b) Arizona Cardinals Stadium (Riberich, 2003) 
2.2. RETRACTABLE ROOFS FOR STADIA 
2008 Beijing Olympic Stadium 
The 29th Olympic Games, in 2008, will be hosted by Beijing, China, and as most 
other Olympic cities before it Beijing has commissioned a new stadium. The modern 
tradition for Olympic stadia is more than a hundred years old and nearly all stadia 
built for hosting the Games have been the pride and show piece of the host nation and 
thus often a state-of-the-art venue. It is therefore of particular interest to review the 
proposals for the architectural competition held for the 2008 Beijing Olympic Stadium 
by the Beijing Municipal Commission of Urban Planning (2003). Of the thirteen entries 
in the final round, all except one proposed a stadium with a retractable roof. Four 
schemes are presented below and shown in Figure 2.9: B02, B07, Bll and B12. 
Scheme B02 was proposed by a team led by Gerkan, Marg and Partners, Germany. 
The engineers were Schlaich Bergermann and Partners of Germany. The scheme pro-
posed a fixed tensegrity roof, see Section 2.3.2, covered with ETFE foil cushions. The 
retractable roof for the central circular opening was to consist of 16 cantilevering ele-
ments pivoting at the base about the edge of the fixed roof. Each element was to be 
fabricated as a single 63 meter long piece using Carbon Fibre reinforced Carbon (CFC) . 
For opening and closing the roof, a system consisting of three synchronised, structural 
cable systems was to be used (Balz, 2003) . 
Scheme B07 by Hiroshi Hara + Atelier, Japan proposed to use an American patented 
retractable roof system, called the Two Panel Spiral Rotating Shell Retractable Roof 
Geometry (Allen & Robbie, 1994). This system uses two large curved shells, which run 
, at ground level on two curved rails that form two overlapping circles. By running the 
two shells along the curved rails and thus rotating them, the roof is opened or closed 
as shown in Figure 2.9(b) . 
'-
Led by the Swiss architects Herzog & de Meuron Architekten, scheme Bll won the 
competition and this design is currently being taken forward. This structure will be 
67 meters high and made from steel box sections more than 10 meters deep, forming 
a structure that resembles the interwoven twigs of a birds' nest. On top of this birds' 
nest, a conventional bi-parting retractable roof, engineered by British engineers Arup, 
was proposed (Parrish, 2003). The retractable roof has later been removed from the 
project due to cost constraints. 
Scheme B12 by Tianjin Architects & Consulting Engineers, China, put forward an often 
proposed concept ~ that of using a balloon as the retractable roof. Under normal 
conditions the roof would be hovering gently above the venue, ready to be hoisted 
down and fastened to the fixed roof should the weather change. This proposal came 
joint-second in the competition. 
As can be seen from the examples above, there are many and varied design proposals 
for retractable stadium roofs. Current research within the field of deployable structures 
promises to deliver more structural concepts and solutions that in the future can be 
applied to large scale retractable roofs. 
16 
2.2. RETRACTABLE ROOFS FOR STADIA 
(c) (d) 
Figure 2.9: Four proposals for 2008 Beijing Olympic Stadium (a) B02; (b) B07; (c) Bll; (d) 
B12 (Beijing Municipal Commission of Urban Planning, 2003) 
2.3. DEPLOYABLE STRUCTURES 
2.3 Deployable Structures 
Currently deployable structures have two very different main applications, space explor-
ation and temporary, quick-to-erect earth-bound structures, each with its own pattic-
ular requirements. Hence solutions developed and found applicable for one application 
are rarely directly transferrable to the other. 
2.3.1 Space Applications 
The development of deployable structures for space applications has been governed by 
the tight space and weight requirements for structures that are to be launched into 
space and the extreme, zero-gravity environment of space. Therefore these structures 
have been developed to minimise their weight and volume in the stowed configuration. 
For space applications there are three main uses of deployable structures, according 
to Tibert (2002): 
• Masts, 
• Antennas, and 
• Solar panels 
Methods used for deploying these structures can be classified as 
• Coiled rods, 
• Flexible shells, 
• Inflatable/folded membranes, and 
• Structural mechanisms in the form of hinged, pinned, telescopic and sliding ele-
ments. 
It is beyond the scope of this dissertation to go into details regarding the specific 
applications and deployment methods for the above mentioned applications. For further 
information refer to the latest reviews by Jensen & Pellegrino (2001), Tibert (2002) 
and Pellegrino (1995) for masts, antennas and solar panels respectively. Figures 2.10 
and 2.11 show a few examples of deployable space structures. 
2.3.2 Architectural and Civil Engineering Applications 
The use of deployable structures for architectural applications has its roots in the an-
cient building types of tents, teepees and yurts. The architectural dream of temporary, 
adaptable and transportable structures has led to the development of this family of 
deployable structures though they have been found suitable for a number of other 
applications too. Hernandez et al. (1991) outlined, among other uses, seven suitable 
applications for deployable structures in architecture and civil engineering: 
18 
2.3. DEPLOYABLE STRUCTURES 
Figure 2.10: 'HIMAT' retractable mast for space applications (Kitamura et al., 1990) 
Figure 2.11 : Solid Surface Deployable Antenna (Guest & Pellegrino , 1996b) 
• Temporary shelters and enclosures suitable for storage, 
• All types of structures for inaccessible regions, 
• Large portable and rapidly erectable structures, 
• Retractable roofs, 
• Areas where !here are high risks and labour costs involved with erection, 
• Construction aids for traditional construction, and 
• As a fundamentally new way to think about erection of permanent structures 
A more exhaustive list is presented in Gantes (2001). Bulson (1991b), Hernandez 
et al. (1991), and more recently Miura & Pellegrino (2002) have proposed classification 
methodologies for deployable structures in general, which also include civil engineer-
ing applications. Escrig (1996) and Gantes (2001) have both proposed classification 
methods for deployable structures in architecture and civil engineering. 
Retractable roofs are often classified as a separate group of structures as they can be 
constructed using any of the following systems: 
• Tensegrity structures, 
• Membrane structures without an underlying bar or panel structure, 
• Plate structures consisting of hinged plates, and 
• Bar structures consisting of hinged bars 
I I 
I 
2.3. DEPLOYABLE STRUCTURES 
Tensegrity Structures 
This type of structure was invented by Fuller (1962). They consist solely of members 
either in tension or in compression, though many other interpretations of the phrase 
tensegrity have been put forward over the years (Motro, 1992) . Despite the efficiency 
using members in pure tension or compression this system is not widely used. It has 
found a niche as a structural system for circular domes where a series of spoke and 
wheel rings are interconnected to form a stable and efficient fixed roof. 
Membrane Structures 
Unlike tensegrity structures, membrane, or tensile structures, have been widely adop-
ted in both permanent and temporary architecture. The ability of the membrane to 
fold has made it the most widely used system for providing a deployable cover, either 
as an independent structure or as a cladding material for another deployable structural 
system. The best known deployable membrane structure is the umbrella. Frei Otto 
further developed the idea of the umbrella and created several retractable roofs based 
on this concept (Otto et al., 1995), see Figure 2.12. Many other types of deployable 
membrane structures have been built, see Section 2.2.2. The use of inflatable mem-
brane structures has also been researched and several small scale projects have been 
realised (Kronenburg, 2000) . 
Plate Structures 
Larger deployable structures, consisting of hinged rigid plates are not common in archi-
tecture as these systems are not as light and compact as other systems. Also a limited 
number of folding methods are available for irregular structural shapes. The best known 
examples of this type of deployable structure are the pivoting plates and beams used 
by the Spanish architect and engineer Calatrava in his transformable architecture, see 
Figure 2.12 for an example. 
Bar Structures 
Bar structures are the most used and researched type of deployable structure in civil 
engineering. The first to research and build deployable structures for architectural use 
was the Spanish architect Piiiero, during the 1960's. He proposed during his short life 
a number of novel structural systems for temporary enclosures, including retractable 
domes and travelling theatres and pavilions (Belda, 1996). One of his movable theatres 
is shown in Figure 2.13. Piiiero's structures were based on bars hinged together at 
their midpoints as well as at their ends. Other designers, such as Calatrava, have 
used other methods of interconnecting the bars. However, the system used by Piiiero, 
generally referred to as a pantographic element, has remained the most commonly used 
bar system. 
20 
2.4. RETRACTABLE PANTOGRAPHIC STRUCTURES 
(b) 
Figure 2.12: (a) Umbrellas Medinah by Otto et al. (1995) , and (b) Floating Concrete Pavilion 
by Calatrava (Tzonis, 1999) 
2.4 Retractable Pantographic Structures 
A pantograph element consists of two identical straight bars in two separate layers, 
shown as red and blue in this dissertation, connected through a revolute joint, also 
called a scissor hinge, allowing the two bars to pivot about an axis perpendicular to 
the element itself. When a series of such pantographic elements are interconnected at 
the ends using revolute joints, a two-dimensional transformable structure is formed. 
This is often referred to as a lazy-tong, the principle of which has been known for 
millennia. 
By arranging a series of such individually transformable structures in a grid, it was 
possible for Piiiero to form structures that could be expanded and retracted again. 
These structures originally expanded only in the plane, but by varying the lengths the 
bars in the individual pantographic elements and adjusting the overall arrangement 
accordingly it became possible to create structures that could expand into cylindrical 
or spherical shapes from their original bundled configuration (Belda, 1996; Escrig, 1993; 
Gantes, 2001; Miura & Pellegrino, 2002). 
As Piiiero's systems of pantographic elements were unstressed throughout their trans-
formation they had no stiffness in the directions of motion. Therefore, they formed 
unconstrained mechanisms and their hinges had to be locked, or the structure in some 
other way rigidified, once they had reached a configuration in which they were required 
to carry loads. During the transformation of the structure from one configuration to 
another, only a single point on the structure can remain in its original position. This 
prevents simple support conditions during the transformation, as is clearly visible in 
21 
2.4. RETRACTABLE PANTOGRAPHIC STRUCTURES 
Figure 2.13: Piiiero with his movable theater (Gantes, 2001) 
(b) 
Figure 2.14: Lazy-tong formed by three pantograph elements 
22 
2.4. RETRACTABLE PANTOGRAPHIC STRUCTURES 
Figure 2.13. A distinct advantage for this system over others, however, is the simplicity 
of its joints. 
After Piiiero's early death in 1972 his work was carried forward by others and they have 
improved his original design. Zeigler (1981, 1984, 1987, 1993) patented a system in which 
some of the bars must deform during the transformation to provide a locking effect in 
the extreme configurations eliminating the need for additional locks to be added to the 
structure. Escrig from the School of Architecture in Seville, Spain, and his collaborators 
have also further developed systems formed by pantographic elements. Over the last 
two decades they have greatly advanced the understanding of these structures as well 
as promoting their use. Many new shapes have been identified and several larger 
structures have been realised (Escrig & Valcarcel, 1993; Gantes, 2001; Sanchez et al., 
1996; Valcarcel & Escrig, 1996). Others have also used pantographic elements. 
2.4.1 Closed Loop Structures 
It is possible to form a closed loop ring structure from n pantographic elements, each 
subtending an angle a; the centre hinge has to be moved away from the mid-point 
of the bars, see Figure 2.15. Zanardo (1986) showed that such a pantographic ring is 
rigid and hence does not form a mechanism. You & Pellegrino (1997) illustrated this 
by deriving the relationship between the subtended angle a and the rotation angle {3, 
see also Pellegrino & You (1993) 
CE-AE 
tan a/2 = tan {3 /2 
AC 
(2.1) 
This relationship shows that any variation in the rotation angle {3 will result in change 
in the subtended angle a which is prohibited in a closed loop structure. Strictly, if 
CE = AE then a= 0 and independent of {3. 
\ 
\ 
\ c 
Figure 2.15: Pantographic element consisting of two straight bars 
23 
p 
2.4. RETRACTABLE PANTOGRAPHIC STRUCTURES 
2.4.2 Structures formed by Angulated Elements 
A considerable advance was made by Hoberman (1990), with his invention of the simple 
angulated element. Unlike the original pantograph element with its straight bars, this 
pantographic element consists of two identical angulated elements arranged in two sep-
arate layers as above. Each angulated element consists of two rigidly connected identical 
bars, each with the length l, that form a central kink of amplitude a. Connecting 
two such angulated elements through a scissor hinge at the centre, i.e. at node E 
in Figure 2.16(a), a pantographic element is formed that can be used for closed loop 
retractable structures. Writing the relationship between the subtended angle and the 
rotation angle (You & Pellegrino, 1997): 
\ \ 
\ 
\ c 
" 
' 
' 
' 
' 
..... 
..... 
..... 
..... 
..... 
(a) (b) 
Figure 2.16: (a) Pantographic element consisting of two angulated elements, each formed by 
two bars, and (b) Multi-angulated element with three bars 
CE-AE EF 
tan (a/2) = tan (/3/2) + 2= AC AC (2.2) 
If the conditions AE = C E and a = 2 arctan(EF / AF) are met then a becomes (i) 
constant for all values of {3, and so the pantographic element moves along fixed radial 
lines, and (ii) the angle between the radial lines has the required magnitude. 
A simple flat circular pantographic structure can therefore be formed and it will be 
capable of retracting towards its own perimeter. Such a structure is formed by two 
identical layers of identical angulated elements, one layer formed by elements arranged 
in a clockwise sense, when the elements are traced from the innermost to the outermost 
end, shown in red in Figure 2.17 and the other formed by elements arranged in a 
counter-clockwise sense shown in blue. 
As the structure expands along fixed radial lines each angulated element in the red 
layer also rotates clockwise as can be seen in Figure 2.17. The blue layer on the other 
hand rotates counter-clockwise. The rotations of the elements in the two layers are thus 
equal but opposite. 
24 
2.4. RETRACTABLE PANTOGRAPHIC STRUCTURES 
Figure 2.17: Radial movement of structure 
You & Pellegrino (1996, 1997) found that two or more such retractable structures can 
be joined through the scissor hinges at the element ends. Two angulated elements from 
layers that turn in the same sense of two such interconnected structures, with a scissor 
hinge in common were found to maintain a constant angle during the transformation of 
the structure and hence could be rigidly connected to each other, thus forming a single 
multi-angulated elements with more than one kink as shown in Figure 2.16(b). 
The retractable structure shown in Figure 2.18 consists of two layers each formed by 
12 identical multi-angulated elements made from 4 rigidly connected bars. At each 
cross-over point, there is a scissor hinge. Note that the structure also forms a pattern 
of identical rhombuses or four-bar linkages which are "sheared" when the structure 
transforms. 
Figure 2.18: Retractable structure formed by multi-angulated elements 
The same authors extended the proof based on in Equation 2.2 to include general-
ised angulated elements which allow for non-circular structures to be generated. Such 
structures form patterns of either rhombuses or parallelograms, as shown in Figure 3.8, 
depending on which type of generalised angulated elements was used for the struc-
ture (You & Pellegrino, 1996, 1997). 
Teall (1996) explored the structural behaviour of two circular retractable structures. 
The structures had a dome like shape, following Hoberman (1991), see Section 2.4.4. 
As the perimeter of this type of structure varies during the transformation, the support 
conditions must be designed to allow for this. Teall proposed to overcome this problem 
by mounting the structure on pairs of hinged columns, each creating a four-bar linkage, 
such that the support points would be able to move radially with the structure. 
A further investigations into the support conditions was carried out by Kassabian et al. 
25 
2.4. RETRACTABLE PANTOGRAPHIC STRUCTURES 
(1997). By studying the instantaneous centres of rotation for the elements, Kassabian 
found it was possible to find fixed points about which the elements rotate during their 
motion and thus can be fixed to. Such points, however, only exist for one layer of 
the structure. The solution requires a rigid body rotation to be imposed on the whole 
structure equal in magnitude to the original rotation of the elements which are to be 
fixed at a point. As the two layers rotate by equal but opposite amounts, imposing this 
rigid body rotation results in one layer of elements undergoing pure translations while 
the elements in the other layer undergo pure rotations about their own fixed point as 
shown in Figure 2.19 (Kassabian, 1997; Kassabian et al., 1997). See also Sections 3.2.2 
and 3.2.3. 
\ 
\ 
I 
I 
I 
Figure 2.19: Rotating movement of structure 
Kassabian et al. (1997) showed it to be possible to cover this type of structure using 
rigid cover elements. This allows the structure to perform as a retractable roof. To 
provide weatherproofing the cover elements creates a continuous, i.e. gap free, surface in 
· both the open and closed configurations of the structure. Using a kinematic approach, 
two possible designs, shown in Figure 2.20, were developed. 
Figure 2.20: Cover elements by Kassabian et al. (1997) 
Kassabian et al. (1999) presented an extensive review of retractable roof structures 
based on angulated, multi-angulated or cover elements and their possible uses. 
26 
2.4. RETRACTABLE PANTOGRAPHIC STRUCTURES 
2.4.3 Other Closed Loop Structures 
Several authors (Chilton et al., 1998; Escrig et al., 1996; Rodriguez & Chilton, 2003; 
Wohlhart, 2000; You, 2000), have presented work on closed loop retractable struc-
tures. Chilton et al. (1998) have developed the Retractable Reciprocal Plate Structure 
shown in Figure 2.21. The structure shown consists of 6 triangular rigid plate ele-
ments that each slide against each other as the structure is opened hence providing a 
continuous surface throughout the opening. There are no fixed points for the particu-
lar structure shown though the authors have proposed several other variations of this 
system that have fixed points which can be supported, see Section 6.5 . 
Figure 2.21: Reciprocal Plate Structure proposed by Chilton et al. 
You (2000) expanded and simplified the method for generating pantographic retract-
able structures with fixed support points . Instead of starting with a fixed geometry 
"for the angulated elements and then finding the fixed points as done by Kassabian 
et al. (1997), he prescribes the fixed points A1, B1 and C1 and connect these using 
three bars interconnected by scissor hinges, thereby creating a series of four linkages as 
shown in Figure 2.22(a). Rigidly connecting the bar pairs that share fix point creates 
angulated elements such as AoA1A2 in Figure 2.22(b). By adjusting the lengths of the 
bars and by adding additional hinged bars, structures similar to the earlier work on 
generalised angulatecl elements by You & Pellegrino (1997) can be formed as shown in 
Figure 2.22(d). You concluded that the solution for fixed points by Kassabian et al. 
(1997) is a special case of this more general approach and that "all multi-angulatecl 
structures consisting of parallelogram modules have fixed points." 
(a) (b) (c) (cl) 
Figure 2.22: Retractable systems formed by four-bar linkages (You, 2000) 
Extending this work, Rodriguez & Chilton (2003) proposed a novel retractable struc-
27 
2.4. RETRACTABLE PANTOGRAPHIC STRUCTURES 
ture shown in Figure 2.23(a-d) called the Swivel Diaphragm. As proposed by You 
(2000) it uses the fixed points of the structure, together with straight bars, to form 
a concentric series of parallelograms between angulated elements. This can be clearly 
seen in Figure 2.23(e) where the structure has been opened beyond its normal extreme 
configuration shown in Figure 2.23(d). 
By having the fixed points coincide with the scissor hinges it is guaranteed that the 
support points for the Swivel Diaphragm can be directly connected to its angulated 
elements. This is unlike the fixed points for the multi-angulated structures found 
by Kassabian et al., which are always located away from to the angulated elements. 
Furthermore the authors have shown that any one of the three scissor hinges on an 
angulated element can be fixed if the remaining two points are connected to the neigh-
bouring elements using bars. The angulated elements can be replaced by rigid plate 
elements, to form a continuous surface in both the open and closed positions as shown 
in Figure 2.23(a,d). Rodriguez & Chilton (2003) found this type of structure to have a 
number of advantages over retractable reciprocal structures and the retractable plate 
structures presented in Chapter 3. Rodriguez et al. (2004) have also developed meth-
ods for interconnecting several individual Swivel Diaphragms to form larger assemblies. 
Work on assemblies formed by retractable plate structures is presented in Section 3.5. 
(a) (b) (c) (d) (e) 
Figure 2.23: Swivel Diaphragm by Rodrigues & Chilton 
Wohlhart (2000) presents another type of plane retractable structure, also based on the 
motion of parallel four-bar linkages. By connecting two concentric rings of triangular 
plate elements, each connected to its three neighbouring triangles through scissor hinges 
at their apexes, he achieves a structure with a single internal degree of freedom, a so-
called double-chain mechanism. The structure is moved by rotating adjacent elements 
in opposite directions to one another. Interestingly, if the two rings are congruent the 
solutions found are those of Hoberman (1991) and You & Pellegrino (1997). 
2.4.4 Retractable Dome Structures 
Wohlhart (2000) expanded the solution of concentric rings to spherical structures that 
can expand along normals to the sphere. He also proposes spherical and conical chains 
but these are beyond the scope of this dissertation. Similar structures have been pro-
posed by Kovacs & Tarnai (2000) and Verheyen (1993). Several authors have proposed 
dome shaped structures that can retract towards their perimeter. 
2.4. RETRACTABLE PANTOGRAPHIC STRUCTURES 
Figure 2.24: Double-Chain mechanism by Wohlhart (2000) 
Piiiero proposed a diaphragm retractable dome where a number of spherical "wedge" 
shaped shells are each rotated about an axis normal to the sphere (Escrig, 1993). In the 
closed position the shells form a gap free spherical cap. When the shells are rotated an 
opening or aperture is created at the centre of the dome through a motion comparable 
to that of the shutter mechanism in a camera. The individual shells overlap in all 
positions other than the closed position. To create a mechanism with a single degree of 
freedom, pairs of adjacent shells are connected to each other through a revolute joint 
at the apex of one shell. This joint is then run along a path on the other shell as 
illustrated in Figure 2.25. 
Figure 2.25: Reciprocal dome proposed by Piiiero (Escrig, 1993) 
Hoberman (1991) found that angulated elements connected using scissor hinges will 
not only subtend a constant angle on a plan surface, but also on a conical surface. 
Hoberman proposed to use five concentric rings of angulated elements, each on a dif-
ferent conical surface. By connecting these rings to each other through scissor hinges 
with axes of rotation tangental to the circular plan of the rings, a retractable dome is 
formed. The resulting Iris Dome is shown in Figure 2.26. As seen from the figure the 
29 
,. 
2.4. RETRACTABLE PANTOGRAPHIC STRUCTURES 
dome can be clad by rigid plates attached to the individual angulated elements. When 
the dome is closed, a continuous surface is formed while in the open configuration the 
plates are stacked on top of each other. The Iris Dome uses considerably more elements 
and hinges than that proposed by Piiiero but as there are no sliding mechanisms the 
connections are simpler. 
Figure 2.26: Iris Dome by Hoberman (Kassabian et al., 1999) 
A dome structure constructed from pantograph elements which retracts towards its 
perimeter was presented by Escrig et al. (1996) . This structure is different from that 
proposed by Hoberman, as the pantographic elements are normal to the surface of the 
dome rather than parallel to the surface, as in the Iris Dome. 
You & Pellegrino (1997) found that the two-dimensional solution for the multi-angulated 
elements could be projected onto any three-dimensional surface if the axes of rotation 
for the scissor hinges remains perpendicular to the plane of the original two-dimensional 
solution. This significantly decreased the number of elements required, compared to the 
Iris Dome. The work of Teall (1996) was based on this finding. Rigid cover elements 
for such a projected structure will be presented in Section 3.3.2. 
The motion of the projected structure is purely horizontal, as all axes of rotation are 
vertical. The structure, therefore does not move on the surface of a sphere or dome; this 
is not because of the type of joints but because of the angular defect as will be shown in 
Section 5.3. Kokawa (2000, 2001) and Farrugia (2002) have proposed to overcome this 
problem by building additional freedoms into the joints. Figure 5.6 shows a spherical 
retractable dome proposed by Kokawa (2000). Kovacs (2000) has been successful in 
creating a bar structure that moves on the sphere using simple revolute joints with 
their axes normal to the sphere. However, this solution only covers part of the sphere 
and to form a retractable dome structure additional bars have been introduced - all 
connected through revolute joints to allow the angles between adjacent bars to change 
effectively creating a series of spherical four-bar linkages. This creates a large number of 
bars that must be connected through their joints, making this structure as complicated 
as that proposed by Hobermail. (1991). 
30 
Chapter 3 
Retractable Bar and Plate 
Structures 
3.1 Introduction 
This chapter extends the work of Kassabian et al. (1999) on covering bar structures with 
rigid cover elements. The first part is concerned with circular structures with n-fold 
symmetry. Expressions are obtained for the inner and outer radii for such retractable 
structures, in the open and closed configurations. Using these expressions, design 
· charts are determined, for which general conclusions are drawn regarding the radial 
displacements. 
The rotation undergone by the individual elements of these structures is then used to 
provide a simpler approach for describing their motion. This method provides a uniform 
approach, suitable for both circular and non-circular structures, which is easily adapted 
to take into account the limits associated with models with finite sized members and 
hinges. 
Using this new approach, the shape and properties of cover elements that do not affect 
the range of motion of the structure are determined. With this it becomes possible to 
cover the basic bar structure such that a continuous, gap free surface is generated in 
both the open and closed configurations of the structure. Furthermore, it is shown to 
be possible to construct structures comprised of two layer of plates, both of which form 
continuous surfaces in the extreme configurations of the structure. The positions of the 
scissor hinges are identical to those of an equivalent bar structure and their kinematical 
behaviour is therefore also identical. 
It is shown that it is possible to vary the shape of the individual plates, within certain 
constraints, without affecting the movement of the structure. This approach can be 
used to optimise, for example, the range of motion which such plate structures can 
undergo. The chapter concludes by showing methods for rigidly interconnecting such 
plate structures so they can form stacked or planar arrays. 
31 
3.2. RETRACTABLE BAR STRUCTURES 
3.2 Retractable Bar Structures 
In the following geometric studies the bars are considered to be lines of zero thickness 
and similarly the hinges are considered as having no finite size. 
A general retractable circular bar structure, such as that shown in Figure 3.1, consists 
of 2 x n identical angular or multi-angulated elements. Each layer, shown as red and 
blue, consists of n elements. The individual multi-angulated elements are comprised of 
k rigidly connected bars, hence the simple angular element, shown in Figure 2.16, can 
be seen as a special multi-angulated element with k = 2. Such a structure is therefore 
fully defined by the parameters n;k plus the length l of all the bars, which defines the 
overall size of the structure. 
The kink angle a between two consecutive bars in a simple angulated element is identical 
to the angle subtended by the element as shown in Figure 2.16(a). The kink angle 
can hence be found for circular structures using the n-fold symmetry of the struc-
ture (Hoberman, 1990) and You & Pellegrino (1997): 
3.2.1 Radial Motion 
27r 
a= -
n 
(3.1) 
. Consider a general configuration of the bar structure. We denote by r and R its 
inner and outer radii, respectively. We use the subscripts open and closed for the two 
extreme configurations of the structure. The extreme closed configuration is reached 
when all the innermost hinges coincide at the centre of the structure, hence rclosed = 0. 
The extreme open configuration is reached when the outermost rhombuses formed by 
the angulated elements have been sheared so two diagonally opposite scissor hinges 
coincide and further motion is inhibited. The two extreme configurations are shown in 
Figure 3.1. 
Using the maximum outer radius Ropen to define the size of the structure 
. a 
l = RopenSlll 2 (3.2) 
The following expressions for the radial positions can be derived using simple trigono-
metry (Jensen, 2001) 
Rc1osed = Ropen sin ( k ~) (3.3) 
Topen = Ropen COS ( (k- 1) ~) (3.4) 
To compare the effects of varying the parameters n;k two ratios are introduced. The 
opening ratio, OR, is defined as the radius of the central opening in the open position, 
32 
I I 
3.2. RETRACTABLE BAR STRUCTURES 
(a) (b) 
Figure 3.1: Circular structure with n;k = 8;3 (a) Closed Configuration, and (b) Open config-
uration 
ropen, divided by the outer radius of the structure in the closed position, Rclosed· 
OR= Topen 
Rclosed 
cos ( ( k - 1) ~) 
sin (k~) (3.5) 
Equation 3.5 is plotted as a function of n;k in Figure 3.2. OR < 1 indicates that there 
is an overlap between the two positions, which can be used when considering different 
methods for supporting the structure. Note that continuous lines have been plotted 
in Figures 3.2, 3.3, 3.6, and 3.12 though only whole numbers of n creates closed loop 
structures. 
The second ratio is the stowage ratio, SR. This is defined for the open position only 
and is defined as the radius of the opening, ropen, divided by the outer radius in the 
open position, Ropen· It indicates the size of the central opening compared with the 
maximum, outer size of the structure. 
The expression of S R is 
SR = -- = cos (k - 1 -r~ffi ( )a) 
Ropen 2 
(3.6) 
and is plotted for n;k in Figure 3.3. 
33 
3.2. RETRACTABLE BAR STRUCTURES 
2.5 
2 
~ 
.~ 1.5 
_.., 
(lj 
~ 
bD 
s:: 
·~ 1 Q) 
0.. 
0 
0.5 
0 50 20 15 10 9 8 7 6 5 3 
Elements, n 
Figure 3.2: The opening ratio OR as a function of n;k 
1 
0.8 
2;3 
0 0.6 
·~ 
.... 
Q) 
~ g 0.4 
_.., 
if) 
0.2 
0 3 
Elements, n 
Figure 3.3: The stowage ratio SR as a function of n;k 
34 
3.2. RETRACTABLE BAR STRUCTURES 
3.2.2 Rotating Motion 
As described in Section 2.4.2, Kassabian et al. (1999) found that if the structure is 
allowed to rotate while it opens and closes, i.e. the scissor hinges are not required to 
move on radial lines, the motion of one layer of elements could be described as a pure 
rotation and the motion of the other layer as a pure translation. As the positions of the 
scissor hinges for . the two layers coincide, of course, the motion and positions of both 
layers can be determined by only considering the rotation undergone by the elements 
of the layer that undergo a pure rotation. Hence, the motion of each element can be 
described by a circle with its centre at a fixed point, the location of which is described 
in Section 3.2.3. Note, the total rotation angle undergone by all elements in the red 
layer from the fully-closed to the fully-open configuration is denoted by {3* as shown in 
Figure 3.4. The radius r* of the circle describing the motion of the hinges, i.e. the kink 
points in the angulated elements, is 
* Ropen r =--
2 
Rewriting the bar length l with Equations 3.2 and 3. 7 
l = 2r* sin~ 
(3.7) 
(3.8) 
Figure 3.4(a) show that the circles of motion for two adjacent angulated elements 
intersect at the centre of the structure, 0, as all the angulated elements have their 
innermost joint at 0. Thus this intersection point represents the closed limit for the 
motion. Similarly the intersection point, P, represents the fully-open configuration, as 
shown in Figure 3 .4( c). Thus P is the open limit for the rotation. 
p 
(a) (b) (c) 
Figure 3.4: The limits of rotation coincide with the intersections of two neighbouring circles of 
motion 
Figure 3.5 shows the same angulated element as that emphasised in Figure 3.4. The 
angulated element consists of k bars and its hinges, Ai, are numbered such that starting 
with the innermost hinge they have the numbers 0, 1, 2, ... , k - 1, k. To describe the 
35 
3.2. RETRACTABLE BAR STRUCTURES 
rotation of the angulated element about its fixed point Aeen the four angles, fJ, /, 8 
and E are defined in Figure 3.5. Note that fJ + 1 + 8 + E = 27r. The outer angle or 
rotation angle, fJ, is the angle between the open limit P and the outermost hinge on the 
element, Ak. Similarly, the inner angle, 1, is defined as the angle between the closed 
limit, i.e. the origin 0 and the innermost hinge Ao. At the extreme configurations, 
open and closed, we denote the values of fJ and 1 by, fJopen and /open, fJctosed and /closed 
respectively. 
p 
········· ....... 
) 
__ .... --··· 
Figure 3.5: Definition of angles (3, {, o and E 
From Equation 3.8 the angle subtended by a single bar on the circle is equal to a, as 
sin ( a/2) = l j2r* (3.9) 
thus the element angle, 8, subtended by the multi-angulated element is 
8 = ka (3.10) 
In Section 3.2.3 it will be shown that for non-circular structures the kink angles are 
not identical for all kinks in the angulated element and it will therefore be useful if all 
angles are determined using summation. The element angle, 8, is then 
(3 .11) 
The limit angle, E, subtended by the intersection points of the two neighbouring circles 
of motion, 0 and P, is found by noting that .LAeen 0 Been = a and that the triangles 
6Aeen0Been and 6AeenP Been are identical, hence 
E=7r-a (3.12) 
36 
3.2. RETRACTABLE BAR STRUCTURES 
The total rotation of the angulated element between the open and closed configurations 
is /3* = f3closed - /3open. Clearly /3open = 0 and, since "/'closed = 0 
k k-1 
f3closed = 27f - ')'closed - 0 - E = 27f - L ai - ( 7f - a) = 7f - L ai 
i=1 
Hence, the rotation from the closed to the open configuration is 
k- 1 
!3* = !3closed - /3open = 7f - 2:.: ai 
i=1 
i=1 
(3.13) 
(3.14) 
The above equation clearly shows how the motion of the structure is governed by k and 
n, as a is solely a function of n. This simple expression is therefore an important tool for 
determining the motion of the circular retractable bar structure. In the Sections 3.2.3 
and 3.2.4 it is extended to cover non-circular structures and structures which have their 
motion limited by for example hinges of finite size. 
A plot of the rotation angle, /3*, for parameters n;k is shown in Figure 3.6. Note 
that a number of structures with different n;k parameters have identical rotations, in 
Figure 3.6 {3* = 60° is emphasised as an example. Structures with identical rotation 
angles also have identical radial displacements as can be seen in Figures 3.2 and 3.3. 
Above the rotation angle /3* was derived by considering the rotation of the angulated 
element as whole. The rotation angle can also be derived by considering the rotations 
at a single hinge. 
Consider the angulated elements in Figure 3.7(a). The hinges are numbered 0, 1, 2, ... , k-
1, k , as in Figure 3.5. Hence the internal hinges, and hence the kinks of the elements 
are numbered j = 1, . .. , k- 1. For each internal hinge the angles a, f3 and ')' are la-
belled with a subscript, see Figure 3.7(a), and so for j = 1 the sum of the angles is 
1r = a 1 + fJ1 + 1'1, for j = 2 the sum is 1r = a1 + a2 + fJ2 + 1'2 and more generally for 
hinge number j 
j 
1r = a1 + a2 + · · · + aj + /3j + "/'j = L ai + /3j + "/'j 
i= 1 
(3.15) 
Using this formulation the angle "/'j is identical for all internal hinges and thus will just 
be referred to as ')', while the angle /3j will always be the internal angle of a rhombus. 
This will prove useful in Section 3.3. 
Figure 3. 7(b) shows the angulated element when it has reached its closing limit. Since 
')'closed = 0, /3j,closed is then found from Equation 3.15 
j j 
7f = 2:.: ai + /3j ,closed + "/'closed :::} /3j,closed = 7f - 2:.: ai (3.16) 
i=1 i = 1 
37 
Tl 
3.2. RETRACTABLE BAR STRUCTURES 
180 
160 
Bars, k 
140 
~ 120 
~ 
~ 
CQ_ 
~ 100 ~ 
bO 
>:1 
<ll k=3: >:1 80 0 
·~ 
+" 0 p:; 60 
' 
' 
' 
40 
' 
' 
:k=4 
20 
0 50 20 15 12 10 9 7 6 5 4 3 
Elements, n 
Figure 3.6: The rotation angle /3* as a function of n;k 
\ I \ I 
\ I 
\ I 
\ I 
\ I 
\ I 
\ I 
\ \ I I fJ2,closerl 
\ \ I I \ I 
\ \ I I 
\ \ I I 1 
\\Ill 
(a) 
"*'0 (b) 
Figure 3.7: Sum of angles at internal hinges for structure with k = 3 
38 
3.2. RETRACTABLE BAR STRUCTURES 
Figure 3.7(b) shows also the same element when it has reached the opening limit. It 
can be seen that f3k-1 ,open = 0 at this limit. Using Equation 3.16 the limiting closing 
angle, f3k - l,closed, is determined for the same hinge 
k- 1 
!3k- l,closed = 7r- 2::: ai 
i=1 
(3.17) 
The rotation angle for the angulated element can now be determined using the limits 
found for hinge j = k - 1 
k- 1 
!3* = !3k- 1,closed- !3k-1 ,open = 7r- 2::: ai 
i= 1 
(3 .18) 
Note that Equations 3.17 and 3.18 are identical to the previously derived Equations 3.13 
and 3.14. 
From Equation 3.16 the angle f3j ,closed can be found for any internal hinges and then, 
using Equation 3.18, the angle f3j,open can be derived for any hinge 
k- 1 j 
f3j,open = f3j,closed - f3* = L ai - L ai = aj+l + aj+2 + · · · + ak- 1 (3.19) 
i=1 i= 1 
Note that, because all elements undergo the same rotation {3*, any single element can 
be considered when determining the rotation angle. 
3.2.3 Non-circular Structures 
The above derived equations can be applied to non-circular structures with variable 
kink angles; note that for non-circular structures a =F 21r jn. A simple technique pro-
posed by Hoberman (1990) for creating non-circular closed loop structures made from 
angulated elements, is to use a general polygon to define the open configuration of the 
structure. Hoberman's original technique made use of a series of similar rhombuses. 
This was later expanded by You & Pellegrino (1997) to multi-angulated elements and 
to allow the creation of retractable structures formed by similar parallelograms. 
Structures Consisting of Similar Rhombuses 
The simplest non-circular structure is formed using a general, n-sided polygon. A set of 
angulated elements, hence k = 2, is then formed by letting their single internal scissor 
hinge, i.e. j = k- 1, coincide with the vertices of the polygon, see Figure 3.8(a). The 
lengths of the individual bars are then equal to half the length of adjacent polygon 
sides. The two rigidly connected bars of an angulated element form a kink equal to 
the internal angle of the polygon vertex. The kink angle a of each angulated element 
is thus 1r minus the internal angle of the polygon. In this fully open configuration 
39 
3.2. RETRACTABLE BAR STRUCTURES 
the angulated elements coincide with the polygon and all n similar rhombuses have 
therefore an identical opening limit, /3open = 0. 
The chain of rhombuses formed by the angulated elements, see Figure 3.8(b), remains 
similar throughout the motion of the structure as their diagonals are reduced in length 
by proportional amounts and their diagonals remain at constant angles (You & Pel-
legrino, 1997). The rotation undergone by all angulated elements in a layer is therefore 
also identical. 
For circular structures the closing limit is reached when the innermost hinges all co-
incide with the origin of the structure. At this limit the innermost bars also coincide. 
For non-circular structures the closing limit is reached when the bars of a single pan-
tographic element coincide, i.e. the angle 'Y is equal to zero for this particular, limiting 
pantographic element. As the movements of all elements are linked, all other elements 
are then inhibited from reaching their own closing limit because the limiting panto-
graphic element cannot move any further. 
From Equations 3.16 and 3.18 it then follows that the limiting pantographic element 
must be that with the largest kink angle, i.e. the angulated elements coinciding with 
the smallest internal angle of the polygon, see Figure 3.8(a) 
(3.20) 
Interestingly, the above equation show that the rotation limits, and thus the structure's 
radial displacements are directly influenced by the smallest internal angle of the polygon 
used to form the structure. Therefore a structure formed from a regular polygon, where 
the internal angles are identical, can execute the largest motion. Also, as previously 
shown by Figures 3.2 and 3.3, the more internal angles there are in a polygon, the 
smaller their magnitude and thus structures with larger number of elements, n, are 
capable of executing larger rotations. 
Internal hinge 
(a) (c) 
Figure 3.8: Structures formed from identical polygons and showing element with largest kink 
angle (a) An angulated element, (b) Similar rhombuses, and (c) Similar parallelo-
grams 
40 
3.2. RETRACTABLE BAR STRUCTURES 
Structures Consisting of Similar Parallelograms 
The same arguments can be repeated for structures formed by similar parallelograms, 
see Figure 3.8(c), where both angulated elements of a pantographic element have 
identical kink angles. They, however, do not have the same bar lengths as the an-
gulated elements in one layer are proportionally larger than the elements in the other 
layer. In the open configuration all similar parallelograms have /3open = 0 as the par-
allelograms coincide with the defining polygon and hence are fully collapsed. Similarly 
to structures formed from similar rhombuses, the closing limit is governed by the two 
inner bars of a particular pantographic element coinciding. The two angulated elements 
will have /closed = 0 when their bars coincide. Therefore it is also found using Equa-
tions 3.16 and 3.18 that the movement of structures formed by similar parallelograms 
is limited by the angulated element with the largest kink angle, as already found in 
Equation 3.20. 
Note that for structures made from similar parallelograms the two layers of the structure 
do not rotate by equal amounts when the internal hinges are forced to move radially. 
Depending on the ratio between the longer and the shorter bar lengths in the paral-
lelogram one will rotate more than the other. However, the relative rotation between 
elements in different layers is not influenced and hence not the ability of either layer to 
rotate about fixed points. 
Structures Consisting of Multi-Angulated Elements 
For circular structures consisting of identical angulated or multi-angulated elements the 
rotation angle {3* can be found from Equation 3.18 by considering any of the identical 
elements. Considering the rotation angle for all the non-identical angulated elements 
composing a non-circular structure it was above found from Equation 3.18 that the 
limiting angulated element was that with the largest kink angle a . 
From Equation 3.18 it also follows that for a non-circular structure composed of non-
identical multi-angulated elements the motion of such a structure is not limited by the 
angulated element with the largest individual kink angle. Rather it is limited by the 
element with the largest sum of kink angles 
[
k-1 l /3* = 1r- Lai 
t= l max 
(3.21) 
If the number of bars in the individual multi-angulated elements is not identical through-
out the structure the motion of the structure is still limited by the particular element 
the largest sum of kink angles. This is shown in Figure 3.9 where four elements in the 
red layer, one of which only have two bars, all have a sum of kink angles equal to 60 
degrees. As all other elements have a lower sum of kink angles these four elements 
limits the motion of the structure as shown. 
Equation 3.14 has thus been extended to cover not only circular bar structures but also 
non-circular structures formed by non-identical multi-angulated elements. 
41 
3.2. RETRACTABLE BAR STRUCTURES 
Figure 3.9: Non-circular structures consisting of multi-angulated elements with variable sum of 
kink angles 
Location of Fixed Centres of Rotation 
As described in Section 2.4.2, Kassabian et al. (1997) found that if the structure is 
allowed to rotate while it opens and closes, i.e. the hinges are not required to move on 
radial lines, the motion of one layer of elements could be described as a pure rotation 
about fixed points. For a structure, of any plan shape, moving radially Kassabian found 
that the location of instantaneous centre of rotation Acen for the angulated element Ao-
Ak can be obtained from the angle f3k- 1 and the distance between the origin 0 and 
hinge Ak- 1· As shown in Figure 3.10(a) the centre is rotated by f3k - I/2 about the 
origin compared to the hinge Ak_ 1 and the radial distance is given by 
OA _ OAk-1 
cen - 2 COS (fJk-1/2) (3.22) 
Hence, in the open configuration, where {3 = 0, the centre is located on the radial 
line OAk- 1 and at exactly the half distance between the origin and the polygon vertex 
defining hinge Ak_ 1 . Thus, all the fixed centres of rotation are given at the vertices of 
a polygon that is half the size of that defining the angulated elements themselves, as 
shown in Figure 3.10(b). Note that the origin need not be at the centre of the polygon. 
Combining the polygon method for obtaining both the open configuration and the fixed 
centres of rotation with Equation 3.21, a simple method for determining the closed and 
any intermediate configurations is obtained. The shape of the individual angulated 
elements is given by the polygon and the closed configuration can easily be found 
by rotating the individual elements of a single layer by {3* about their individual fixed 
centres of rotation. The position of the elements of the second layer is then given by the 
connecting hinges on the rotated layer. Intermediate configurations for the structure 
can be found by rotating the elements by less than {3*. 
For non-circular structures, in particular where calculating the radial position of the ele-
ments during the motion of the structure is not straight forward, this method provides 
a much easier and simpler approach for finding various configurations for the structure. 
42 
3.2. RETRACTABLE BAR STRUCTURES 
' 
0 
(a) 
I 
I 
I 
(b) 
Figure 3.10: Location of the fixed centre of rotation in (a) Intermediate configuration, and 
(b) Open configuration 
3.2.4 Additional Rotational Limits 
As the bar structure is composed of two distinct layers, the only possible interferences 
to its movement are: between elements of the same layer, between elements and hinges, 
and between hinges. For structures generated from identical rhombuses, when all bars 
have the same length but different kink angles, interference will only occur between 
hinges. 
Assuming that the hinges are all circular in shape, with a radius of rj, then it can 
be seen from Figure 3.11 that such hinges will impose additional limits on the motion 
of the structure. ,Hence both the open and closed limits will be reached before the 
limits determined earlier, f3k-l,open = 0 and /closed = 0. The new limits are obtained 
by considering the minimum angle between two bars of length l attached to hinges of 
radius rj. Therefore, 
f3k-l ,open = /closed = 2 arcsin (?) (3.23) 
The reduce rotation angle is obtained by rewriting Equations 3.16, 3.18 and 3.19 for 
f3k - l,open = /closed =f. 0 
j j 
7f = 2:: ai + !3j,closed +/closed '* !3j,closed = 7f - 2:: ai - /closed (3.24) 
i = l i = l 
The rotation angle {3* is then, following Equation 3.21 
{3* = f3k-l,closed - f3k - l ,open = 7f- [I: ail - f3k-l,open -/closed 
t= l max 
(3.25) 
43 
I I 
3.2. RETRACTABLE BAR STRUCTURES 
\ I \ I 
\ I 
\ I 
\ I 
\ I 
\ I \ 
\ I \ I I 
\ \ I I 
\ \ I I 
/J2,closed \ \ I I 
Tclosed 
\ \ I I \ I 
\ \ I I 
\ \ I I 1 
,, ill 
(a) '~~'0 (b) ~0 
Figure 3.11: Rotational limits of structure with hinges of finite size in (a) Closed configuration, 
and (b) Open configuration 
and so 
fJj,open = fJj,closed - f3* = l:Yj+1 + l:Yj+2 + · · · + l:Yk- 1 + f3k - 1,open (3.26) 
Note that for j = k- 1, f3j,open = f3k-1,open as j + 1 > k- 1. 
By expressing any limits imposed on the motion of the structure in terms of a rota-
tional limit, as done in Equation 3.23, it is hence possible to subtract these from the 
uninhibited motion given by Equation 3.21. This allows the possible motion for almost 
all practical structures to be found using the resulting Equation 3.25. 
In structures not made from identical rhombuses often it is not two hinges that interfere 
but rather a hinge and an element as seen in Figure 3.8(b,c) at the bottom-left angulated 
elements of the structures shown. This, however, only occurs when the structure is near 
the fully-open or fully-closed configurations. Conservatively, the limits for f3open and 
')'closed for such structures can be estimated using Equation 3.23 with the shortest bar 
length in the structure. To find the exact limits of a complex shape one needs to identify 
the points of interference before any calculation is carried out. 
If a circular structure is considered, such as those considered in Section 3.2. 1, then 
defining the additional limit angle ( as ( = f3open +')'closed Equations 3.23 and 3.25 can 
be simplified to 
k- 1 
{3* = 7f - :z= ai - ( 
i= 1 
44 
(3.27) 
and 
3.2. RETRACTABLE BAR STRUCTURES 
( = 4 arcsin ( _r_j_ --1.,........,....) 
Ropen sin ( ~) (3.28) 
where Ropen is the maximum outer radius for hinge k- 1 for /3open = /closed = 0 as 
defined in Section 3.2.1. 
The additional limits for the rotation have the effect of changing the radial distances 
found in Section 3.2.1, further details can be found in Jensen & Pellegrino (2002). 
Plotting Equations 3.14 and 3.28 the rotation angle for structures with finite sized 
hinges can be readily found from Figure 3.12. In the figure the additional limit ( is 
plotted for different relative hinge sizes, given by the ratio rj/ Ropen· The reduced 
rotation angle is found by subtracting ( from the unreduced rotation angle given by 
Equation 3.14. 
180 
160 
140 ' 
' 
: k=2 
[k 1 l ~120 n- ~a; b.O . 
Q) 
.::Q. . 1=1 
V 
""d 100 
~ 
~ 
CQ 80 k = 3 u) 
Q) Limiting angle bb 
>:1 (for: ~ 60 
40 
20 
0 50 20 15 12 10 9 8 7 6 5 4 3 
Elements, n 
Figure 3.12: Additional limit ( for the rotation angle (3* of circular structures 
45 
3.3. RETRACTABLE PLATE STRUCTURES 
3.3 Retractable P late Structures 
By considering a retractable bar structure as a concentric ring of identical rhombuses or 
more generally as a ring of similar parallelograms it becomes possible to design plates 
that can be attached to the bar structure and provide a gap-free cover in both its open 
and closed configurations. This is done by considering what limitations are imposed 
on the movement of only a small part of the bar structure when covering plates are 
attached to it. 
3 .3.1 Movement of Bar Linkage 
Consider the four bar linkage, consisting of two parallel bars AkAk-1 and Bk-1Bk-2 
and a pair of parallel linking bars, shown in Figure 3.13. Bar Bk_ 1Bk- 2 is assumed to 
be fixed, so no rigid body motions are allowed and this leaves one internal mechanism 
which allows the linking bars to rotate and bar AkAk- 1 to translate, i.e. to shear the 
parallelogram AkAk- 1Bk-2Bk_1. The bottom-left bars define the angle, (3, as shown 
in Figure 3.13. Note that (3 can be defined at both Ak- 1 and Bk-1· 
Figure 3.13: Movement of four-bar linkage with two plates 
' Consider attaching a rigid plate to bars AkAk-1 and Bk-1Bk- 2 only. This rigid body 
eliminates the mechanism of the parallelogram. If a straight cut is made in the plate, 
at an angle () to the bars AkAk_ 1 and Bk-1Bk-2, as shown in Figure 3.13, then the 
mechanism is restored. The line of the cut is called the boundary line and () the boundary 
angle. So, now there are - in effect - two plates attached to the linkage and by not 
allowing these to overlap (3 is restricted to either increase or decrease depending on 
the boundary angle e. In the case illustrated in Figure 3.13 (3 decreases until the gap 
between the two plates has closed again and the movement of the linkage has reached 
its other limit. 
The two limits on the rotation angle are denoted by f3closed and f3open and, once they 
are known, () can be found from Figure 3.14 by considering the sum of the internal 
angles in L:.Ak,closectBk-1Ak,open 
2 ((3 {3 ) 1f - fJc iosed + f3open 1f = TJ + closed - open ::::::? TJ = 2 (3.29) 
46 
3.3. RETRACTABLE PLATE STRUCTURES 
Figure 3.14: Determining the boundary angle() 
In addition, by considering the parallel bars Ak,closedAk-l,closed and Bk-lBk-2 
7r = () + '17 + ,8closed =? '17 = 7r - () - ,8closed 
The boundary angle is thus related to the limiting ,B's by 
() = 7r - ,8closed - ,8open 
2 
(3.30) 
(3.31) 
This equation provides a simple method for finding suitable shapes for the plates used 
for covering retractable bar structures as will be shown in Section 3.3.2. 
As no length variables are present in the above equations, the position of the boundary 
line relative to the linkage does not affect the limits of the motion. The distance 
translated by bar AkAk-1, and its attached plate, parallel to the boundary line is 
L = 2l sin ( ,8closed ; ,8open) (3.32) 
where l is the length of the linking bars, i.e. AkBk-1 and Ak-lBk-2· 
3.3.2 Application to Closed Loop Structures 
Consider two adjacent multi-angulated elements, A0- A3 and B0- B3, which are part of 
a circular retractable structure with k = 3, as shown in Figure 3.15. Together with 
bars A1Bo, A2B1 and A3B2 these multi-angulated elements form two interconnected 
linkages, I, A1A2B1Bo, and II, A2A3B2B1, which are free to shear within the limits of 
motion for hinges A1 and A2, respectively. 
Consider attaching a rigid plate to the multi-angulated elements, A0- A3 and Bo- B3. 
This eliminates the single mechanism of the two interconnected linkages. To restore 
the mechanism the plate must be cut in two as was done for the four-bar linkage above. 
However, as there are now two linkages involved, any single straight cut must have a 
boundary angle () that satisfies Equation 3.31 for both linkages in order to not interfere 
with the motion of the bar structure. With ,6 defined at hinge Aj , with j = 1 and j = 2 
47 
3.3. RETRACTABLE PLATE STRUCTURES 
for linkages I and II respectively, the limits /3closed and /3open are replaced by /3j,closed 
and /3j,open respectively, 
e. _ 1r - /3j,closed - /3j,open 
J- 2 (3.33) 
(a) 
Figure 3.15: Boundary angle () for multi-angulated elements in (a) Closed configuration, and 
(b) Open configuration 
Figure 3.15 shows the angles at Aj, at the limits of the motion for the bar structure. As 
the hinges are all of identical size the two limits /closed and f3k - l,open are also identical 
following Equation 3.23. 
The boundary angle er that satisfies Equation 3.33 for I is found by considering the 
movement at A1. Equations 3.24, 3.26 and 3.33, noting that for a circular structure 
a1 = a2 and using /closed = f32,open, give 
7r - ( 7r- al -/closed) - ( a2 + f32,open) 
2 
al - a2 + /closed - f32,open 
2 
0 (3.34) 
The boundary line, given by er, is hence parallel to bars BoB1 and A1A2 as shown 
in Figure 3.15. The boundary angle eu that satisfies Equation 3.33 for II is similarly 
found by considering the movement at A2, 
48 
3.3. RETRACTABLE PLATE STRUCTURES 
7r - ( 7r - <XI - a2 - /closed) - fh,open 
2 
al + a2 + /closed - fh,open 
2 
(3 .35) 
The boundary line, given by On, is hence parallel to the boundary line given by Or 
as shown in Figure 3.15. It has been found that the two boundary lines are always 
parallel and hence a single straight, continuous boundary line can be defined without 
the motion of the interconnected linkages being inhibited and as a result a rigid plate 
attached to the multi-angulated elements can be cut by a single line, as for the simple 
angulated element. 
For the complete structure, n straight boundary lines can be determined. They will 
define n identical wedge-shaped covering elements, each subtending the angle a at the 
tip and their tips coinciding at the centre of the structure, as shown in Figure 3.16. 
The same result was first obtained by Kassabian et al. (1997), but through a different 
approach. 
Figure 3.16: Wedge-shaped cover elements for circular structure 
As the straight boundary line can be defined from any single linkage, it is possible to 
simplify the equations above. Equations 3.23 and 3.24 provides sufficient information to 
readily find the two limit angles for the parallelogram AkAk- lBk- lBk-2, and therefore, 
defining the boundary angle for Ak- l and using Equation 3.33, we obtain 
k-1 
O. = 7r - /3k-l,closed - /3k- l,open = ~"" . + /closed _ /3k-l,open 
k-l 2 2 ~a~ 2 2 
i=l 
(3.36) 
3.3. RETRACTABLE PLATE STRUCTURES 
7f - ( 7f - al - a2 - /closed) - f32,open 
2 
al + a2 + /closed - f32,open 
2 
(3.35) 
The boundary line, given by On, is hence parallel to the boundary line given by Or 
as shown in Figure 3.15 . It has been found that the two boundary lines are always 
parallel and hence a single straight, continuous boundary line can be defined without 
the motion of the interconnected linkages being inhibited and as a result a rigid plate 
attached to the multi-angulated elements can be cut by a single line, as for the simple 
angulated element. 
For the complete structure, n straight boundary lines can be determined. They will 
define n identical wedge-shaped covering elements, each subtending the angle a at the 
tip and their tips coinciding at the centre of the structure, as shown in Figure 3.16. 
The same result was first obtained by Kassabian et al. (1997), but through a different 
approach. 
Figure 3.16: Wedge-shaped cover elements for circular structure 
As the straight boundary line can be defined from any single linkage, it is possible to 
simplify the equations above. Equations 3.23 and 3.24 provides sufficient information to 
readily find the two limit angles for the parallelogram AkAk-lBk-lBk-2, and therefore, 
defining the boundary angle for Ak- 1 and using Equation 3.33, we obtain 
k-1 
0 = 7f - !3k-l,closed - f3k - 1,open = ~"" . + /closed - f3k - l ,open 
k- 1 2 2 ~at 2 2 (3 .36) 
i=l 
49 
3.3. RETRACTABLE PLATE STRUCTURES 
For identical motion limits, /closed = f3k - 1,open, Equation 3.36 gives simply 
1 k- 1 
fh 1 =- "'ai 
- 2D 
i = 1 
(3.37) 
Hence, in circular structures the boundary line is always parallel to a line from Ak to 
Ao. 
This is not the case for non-circular structures where one particular angulated or multi-
angulated element governs the motion limit of the whole structure. As (3* and f3k- 1,open 
are identical for all elements, () must also be identical for all elements, following Equa-
tions 3.25 and 3.33, 
() 7r - f3k-l,closed - f3k-1,open 
k- 1 = 2 
If f3k - 1,open = /closed = 0, 
7r- (3* 
()k-1 = 2 
7r- (3* 
2 - f3k - 1,
open (3.38) 
(3.39) 
This shows that the boundary angle is identical for all elements of circular and non-
circular structures formed using the polygon method, i.e. for structures consisting of 
similar rhombuses or parallelograms. Furthermore, the boundary angle is always the 
angle between the boundary line and bar AkAk_1. This implies that the apexes of all 
the cover elements will not necessarily coincide at the centre of the structure. 
This section has shown that it its possible to cover all types and shapes of retractable 
bar structures, formed by the polygon method, with rigid cover elements that form a 
continuous and gap free surface in both the opened and closed configurations. 
Cover Plates for Projected Structures 
As described in Section 2.4.4 it is possible to project a two-dimensional retractable bar 
structure onto a three-dimensional surface. It is also possible to cover such a structure 
with wedge shaped cover elements, as the two-dimensional and three-dimensional struc-
tures are kinematically identical. However, as the motion of the projected structure is 
purely horizontal, the determined boundaries of the cover elements must remain hori-
zontal in order to form a continuous surface in both the open and closed configurations. 
Between the boundaries, each cover element can have any three-dimensional shape, as 
long as interference with the motion of the underlying bar structure is prevented. An 
example of cover plates for a projected structure is shown in Figure 3.17. 
50 
3.3. RETRACTABLE PLATE STRUCTURES 
Figure 3.17: Cover elements for a structure formed by projection onto a sphere 
3.3.3 Plate- only Structures 
The shape of the covering elements shown in Figure 3.16 is such that they do not 
fully cover the multi-angulated elements to which they are attached. In this section 
it is shown that it is possible to construct a retractable plate structure by connecting, 
through hinges, two layers of rigid plate elements. 
In Section 3.3.1 it was shown that the boundary angle is independent of the position of 
the boundary line and it is therefore possible to perform a translation of the boundary 
line relative to the bar structure. By performing such a translation, the cover elements 
can be made to cover fully both the multi-angulated elements and their hinges, as 
shown in Figure 3.18. 
To allow the boundary line to pass between neighbouring elements it is often necessary 
to impose additional rotational limits. This is shown in Figure 3.15(b) where the 
boundary line for rhombus A2A3B2B1 cannot be arranged such that both the hinges 
A3 and B1 are fully covered. It is therefore necessary to increase either or both of 
the limits f3open and /closed; giving additional clearance between the hinges. If the 
limits are increased identically the boundary angle will remain unchanged, following 
Equation 3.36. 
As the boundary lines have been translated perpendicularly to themselves, they no 
longer meet at the centre, thus the cover elements will leave a small aperture at the 
centre of the structure in the closed configuration, as seen in Figure 3.18. 
For circular structures each wedge-shaped cover element subtends an angle a and it 
therefore becomes increasingly difficult to cover multi-angulated elements ask increases. 
No solutions have been found for k > 4 as the central aperture becomes disproportion-
ately large. Figure 3.18 shows solutions for k :S 4. The problem is treated in greater 
detail in Jensen (2001) . 
For structures with k :S 4 both the multi-angulated element and its hinges can be fully 
covered by the cover elements and hence a rigid cover element or a plate can be used 
to replace the multi-angulated element in the retractable structure. A plate structure 
51 
3.3. RETRACTABLE PLATE STRUCTURES 
(a) (b) (c) 
Figure 3.18: Wedge-shaped cover elements that fully cover the angulated elements and their 
hinges (a) k = 2, (b) k = 3, and (c) k = 4 
without bar elements can thus be constructed from two layers of n plates each, where 
the hinges connecting the plates coincide with the layout of the hinges in the equivalent 
n-element bar structure. 
Physical Models 
Models of plate structures have been constructed from identical wedge-shaped Acrylonitrile-
Butadiene-Styrene (ABS) plastic plates and more complex structures from cardboard. 
The connections were made using plastic snap rivets. Figure 3.19 shows an early cir-
cular model formed by plastic plates and Figure 3.20 shows a non-circular structure 
formed from cardboard plates. 
Figure 3.19: Plastic model of wedge-shaped plate structure with n ; k = 12; 3 
3.3. RETRACTABLE PLATE STRUCTURES 
Figure 3.20: Cardboard model of non-circular plate structure 
3.3.4 Periodicity of Boundary 
So far, only straight-edged cover elements have been considered, but non-straight shapes 
are also possible. Consider the linkage shown in Figure 3.21; for the cover plates to fit 
together without any gaps or overlaps in both extreme configurations, the boundaries 
of the two plates must fit together in both configurations. Hence, if the boundaries 
are shaped such that no gaps or overlaps occur in either extreme configuration, non-
straight features are allowed. These features must, however, repeat with period L, as 
shown in Figure 3.21. In general, common boundaries of neighbouring cover elements 
must be shaped such that all features have a periodic pattern. 
Figure 3.21: Periodic pattern of non-straight boundary 
There are two important restrictions to this general periodicity rule, as obviously a 
boundary that deviates significantly from the original straight line would inhibit the 
movement of the linkage. 
First, the movement at any time and of any point on either boundary is always per-
pendicular to the linking bars and this limits the maximum slope of any feature of the 
boundary. This condition is most severe in the two extreme configurations, where there 
is no gap between the plates. Hence, if the slope is at any point greater than the initial 
direction of movement of the boundary, all movement is inhibited. 
53 
3.3. RETRACTABLE PLATE STRUCTURES 
Second, any features of the boundary shape deviating from the original straight line 
need to lie within a region bounded by circular arcs passing through the points 0, 
P and Q, shown in Figure 3.22. Consider a point 0. Two circular arcs describe the 
movement of point 0; the upper-right arc represents the motion of 0 if it is attached to 
bar AkAk-1 and allowed to move relative to bar Bk- 1Bk-2i the lower-left arc represents 
the motion of 0 if it is attached to bar Bk- 1Bk- 2 and allowed to move relative to bar 
AkAk- 1· These arcs are identical to those describing the movements of bars AkAk- 1 
and Bk_ 1Bk- 2 respectively, hence they have radius land subtend the angle {3*. As the 
boundary is periodic, the region bounded by these arcs extends along the entire length 
of the boundary, as shown in Figure 3.22. If all features lie within this region the two 
plates will not interfere during the motion. 
' 
' 
' 
-.... -.... \ ---- Ak Ak-t 
. --- -Q·~; ...... ...... ......G!""~-~ ...... --~,q-------
. ...... 
.. ___ _ 
' 
' 
Figure 3.22: Direction of initial movement and region defining possible boundary shapes 
The length of the period L was found in Equation 3.32 and, since {3* = fJc1osed- f3open, 
it can be written as 
L = 2lsin (~*) (3.40) 
The maximum height of the bounded region, h, from the boundary line can be shown 
to be given by: 
h = l - l cos ( ~*) (3.41) 
As already found for the position of the boundary line, there are no conditions on the 
positioning of the features along a straight boundary line. Thus, the region defining 
the limits for the features can be moved along the boundary line. 
At either end of the periodic boundary it is possible to break the periodicity, as shown 
in Figure 3.23. This is because the movement of the four-bar linkage only allows the 
plates to translate by a single period and, importantly, always in the same direction. 
Therefore both ends of the boundary line need not be periodic at all, though they must 
not interfere with the motion. This feature can be applied to close the central aperture 
occurring in some structures and was first done by Kassabian et al. (1997), as shown 
in Figure 2.20. 
54 
3.3. RETRACTABLE PLATE STRUCTURES 
Figure 3.23: Non-periodic end features 
Physical Models 
A number of retractable plate structure models with non-straight boundaries have been 
built. Here only three will be presented though many others have been constructed. 
The first model consists of sixteen identical plates. It is based on an equivalent circular 
bar structure with n;k = 8;3 and the boundaries have been formed by circular arcs with 
a radius equal to the radius of the opening in the open configuration. Thus the model 
forms a perfect circular opening in the open configuration as shown in Figure 3.24. 
As all the plates are identical, they can all be cast using the same mould, from which 
thermoplastic plate elements were injection moulded. The plates were connected with 
plastic snap rivets. Figure 3.36 show the layers of the structure in red and blue colours. 
Figure 3.24: Model of sixteen identical plates forming a perfect circular opening 
The second model is based on the bar structure formed by similar parallelograms shown 
in Figure 3.8(c). One layer is formed by angulated elements and the other from plate 
elements, both cut in ABS plastic using an Abrasive Water Jet Machine (AWJ) and 
connected through plastic snap rivets. From the model it can be seen how the period 
of each boundary is different as each set of linking bars has a different length l. 
The third and final model presented is based on the non-circular bar structure shown 
in Figure 3.9. To show the wide range of possible boundary shapes, here all twenty-
six plate elements are unique, and thus the two layers are also different as seen in 
Figure 3.26. The periodic boundaries have been generated using a wide variety of 
shapes; straight lines, circular arcs, parabolic curves, sinusoidal curves and splines. 
55 
3.4. COMPUTER ASSISTED DESIGN 
Figure 3.25: Plastic model of plate structure formed from similar parallelograms 
All the boundaries are periodic except for the end features, which have been used to 
cover the central aperture in the closed configuration. Using a computer controlled Wire 
Electrical Discharge Machine (EDM) the plate elements were cut from aluminium-alloy 
plate and assembled with plastic snap rivets. 
) .. (· (/ I I 
.. ( 
. / '-
.. ! 
\ ,~ 
Figure 3.26: Top and bottom faces of model of non-circular structure where all plates and 
boundaries are unique 
3.4 Computer Assisted Design 
The model shown in Figure 3.26 was designed using a computer application written by 
the author in the programming language C++. The application has been called GTD 
after its functions of Generating, Transforming and Drawing bar and plate structures. 
GTD has been compiled for running on the MacOS 10.2 operating system. Figure 3.27 
56 
3.4. COMPUTER ASSISTED DESIGN 
shows the flow of the application from the input-file to the resulting drawings and 
output-files. 
The input-file contains the following information: 
• Number of sides in the polygon, n (in Figure 3.27 denoted A, B, ... , n), 
• Number of bars, k, for each element and coordinates for the polygon vertices, 
• Coordinates of the origin, {X origin, Yorigin}, 
• Radii of hinges, rj, 
• Additional rotational limits, f3open and "/closed, 
• Number of retraction steps to be drawn, S, and 
• Choice of radial or rotating motion, T, 
From the input-file GTD generates a hinge definition matrix containing the coordinates 
for all hinges in the structure in the configuration defined by f3open = 0. The application 
forms similar rhombuses, as they are best suited for forming plate structures because 
the elements of the two layers are equal in size. If similar parallelograms were used, then 
the elements in one layer would be larger than those in the other layer, see Figure 3.8. 
GTD retracts the structure by rotating the hinge positions given in the definition mat-
rix about fixed points. It then draws bars and boundaries from a constant element 
definition matrix. The rotation {3 is identical for all elements and hence the transform-
ation matrix is constant. However, each particular element and its hinges have its own 
centre of rotation. The centres of rotation are obtained by scaling the polygon defin-
ing the hinges Ak-1 by a factor of 0.5, as described in Section 3.2.3. If specified, the 
radial motion is generated by imposing an opposite rotation of magnitude fJ/2 about 
the origin. 
As the motion of the structure is limited by the angulated element with the largest 
sum of kink angles, the sum of kink angles is calculated for all elements and the largest 
sum obtained. 
The extreme configurations f3open and f3closed are found from Equations 3.23- 3.25 by 
considering only hinge k - 1 of the limiting angulated element. The limits for the 
motion, f3open and 'Yclosed, is set to be the higher of (i) a value specified in the input-file 
or (ii) the limit imposed by the hinges, found from Equation 3.23. 
Intermediate configurations are found by rotating the elements about their centres 
stepwise. 
Straight boundaries for plate elements can also be drawn. This is done by drawing 
a line from hinge k to k - 1 for each element. These lines are then all rotated by (} 
about hinge k of the elements, hence creating the boundary lines for a plate structure. 
These are then scaled by L/l so they obtain a length of a single period. Translating 
the boundary lines by a distance equal to rj, perpendicularly to themselves, the hinges 
k becomes positioned fully within the boundary of the plates. 
57 
n 
e ,{ X~-P Y~-1} 
k
8
, { X~-1' Y~-1} 
k ll { 11 1l } ' xk_,, Yk- 1 
{X origin , Yorigin } 
ri, f3 open , 'Yc!osed 
S,T 
[
k- 1 l ~ai ma.x 
7r- (3' 
e = ---- f3opcn 
2 
Offset by 7j 
f3opcn, 'Yc!oscct , {3', 0, L 
Read 
Generate mid-side hinge 
coordinates, k and k-2 
Generate hinge 
coordinates for k>2 
Generate centres of 
Determine maximum 
sum of kink angles 
Determine limits of 
extreme position f3cJosed 
Determine 0 and L 
3.4. COMPUT ER ASSISTED DESIGN 
{x~}={x~_2}=_!_[{x~-~}-{x~_1}l Yk Yk-2 2 [ Yk- l Yk-1 
{ x~_3} ={x~_2}+{ x~_2}-{x~_1} Yk-3 Yk-2 Yk-2 Yk-1 
{x~n}=~{x~_1 } 
Ycen Yk-1 
Rotate line A;_, A;_2 by 0 
and scale by L/ l to define 
a single boundary period 
Impose rigid body 
rotation {3/2 about 0 
Write for step s: 
• s 
{3 = {3 open + {3 X S 
Figure 3.27: Schematic of GTD application 
58 
3.4. COMPUTER ASSISTED DESIGN 
GTD generates an animation of the bar structure on screen for immediate visual in-
spection of the structure and its transformation. A text-file is generated with all hinge 
coordinates for f3open = 0 and other data. 
A graphical output-file in dxf-format is also generated. This allows the structure to 
be imported into most CAD and drawing applications for post-processing. Figure 3.9 
shows the drawing generated for a non-circular bar structure when the movement is 
plotted in four steps. 
3.4.1 Optimisation of Plate Shapes 
GTD does not detect if all hinges are positioned within the boundaries of the plate 
elements and thus the program still requires manual optimisation of the position of 
the boundaries, any periodic features and additional rotational limits. To overcome 
this problem an optimisation problem was formulated for cover elements by Buhl et al. 
(2004). It was formulated such that the gap and overlap areas of the plates were 
minimised for all possible configurations of the structure. A later formulation allowed 
for the maximisation of the central opening in the open configuration. 
The results obtained for the optimisation problem verify the rule of periodicity stated 
in Section 3.3.4 and the particular solution for the maximisation of the central opening, 
shown in Figure 3.28, is similar to the solution shown in Figure 3.24, which had been 
found by heuristic methods. More details are presented in Buhl (2002) and Buhl et al. 
(2004). 
Figure 3.28: Shape of cover elements after minimising the gaps and overlap between cover ele-
ments and maximising the size of central opening in the open configuration (Buhl 
et al., 2004) 
59 
3.5. ASSEMBLIES 
3.5 Assemblies 
For applications other than retractable roofs it is of interest that retractable bar and 
plate structures can be interconnected so they form larger assemblies. Such larger 
structures can be formed either by interconnecting the individual structures in their 
own flat plane, hence forming a two-dimensional planar assembly, or the structures can 
be rigidly connected to form a stack of such retractable structures. Assemblies of either 
type can be constructed so they possess only one internal degree of freedom and the 
motion of all individual structures in the assembly is hence synchronised. 
3.5.1 Planar Assemblies 
In Section 2.4.2 it was explained that the movement of a single layer in a bar structure 
can be a pure translation, and this was extended to structures formed by hinged plates 
in Section 3.3.3. If a single element in this layer is fixed, then no rigid body motions are 
allowed and all other plates in the structure will move in relation to this one element 
as the structure expands and retracts. 
For an identical neighbouring structure, the above is also true and hence a single element 
in this second structure can also be fixed without inhibiting its movement. The two 
neighbouring structures now both have a single element fixed and thus no relative 
motion occurs between these two elements during the motion of either structure. Hence 
these two elements can be rigidly connected to each other without interfering with the 
motion of either of the two individual structures. This, however, does not provide an 
assembly with a single degree of freedom. Instead two independent mechanisms are 
present in this assembly. 
To generate an assembly with only a single mechanism it is necessary to interconnect 
a minimum of two elements from each structure. This would allow the relative motion 
between two elements of the same structure to be imposed on the neighbouring structure 
and hence the motion of the two individual structures will become synchronised. Three 
methods for achieving single mechanism assemblies are presented below. 
Assemblies with Rigid Connections 
Consider the two identical bar structures I and II in Figure 3.29, each consisting of 
sixteen angulated elements. Say, the bottom, blue layer of each structure translates 
during the motion of each structure while the elements in the top, red layer undergo rigid 
body rotations about fixed points, as described in Section 2.4.2. The two structures are 
identical and the motion of element Ar in structure I is therefore identical to that of 
Au in structure II and similarly for elements Br, Bu. Hence the angulated elements Ar 
and Au can be rigidly connected to each other without inhibiting the motion of either 
structure if the connection is physically made in a way such that it does not interfere 
with any of the angulated elements. This is also possible for angulated elements Br 
and Bu. The bars connecting elements Ar and Er form four-bar linkages, as described 
in Section 3.3.1, and they are parallel and of identical length to the bars connecting 
60 
3.5. ASSEMBLIES 
elements An and Bn. Hence ArAn and ErEn can be considered to be two ri
gid bodies 
interconnected by four parallel bars and forming a single degree of freedom
 mechanism. 
Therefore, a relative movement between elements Ar and Er results in
 an identical 
relative movement between elements An and Err and hence the assembly
 is now a 
single degree of freedom system. Additional connections can be made b
etween other 
element pairs without changing this property. It is possible to use this me
thod also for 
non-identical structures. 
Figure 3.29: Single degree of freedom assembly of two rigidly interconnec
ted structures 
Assemblies with Congruent Linkages 
The method proposed above requires a third layer of elements to make th
e connections 
and it is therefore not ideal for some applications. However, by considerin
g each of the 
two bar structures as rings of identical linkages a simpler method of interco
nnecting the 
structures is proposed. From Figure 3.30(a) it can be seen that if both bar structures 
are in identical configurations, then rhombuses I and II are identical and t
heir four bars 
in the top, red layer are parallel to each other, as are the four bars in the
 bottom, blue 
layer. By positioning the two structures such that I and II are coincidin
g, the shared 
rhombus AoA1CoE1 is created in Figure 3.30(b). Noting linkage AoA1CoE1 is able 
to shear identically to both I and II, the internal mechanisms of both ri
ng structures 
have been preserved. However, their movement is now synchronous as al
l linkages are 
identical to AoA1 CoE1. This is illustrated in Figure 3.31. 
Around hinges Ao and Co, in Figure 3.30(b), a number of bars are found to be over-
lapping. Note Ao coincides with Eo in Figure 3.30(b). Bar AoA1 from the angulated 
element A2AoA1 overlaps with bar AoA1 from element A3AoA1 and simil
arly there are 
three other overlaps at EoB1, CoAl and CoB1. The angle A2AoA3 is cons
tant through-
out the motion as elements A2AoA1 and A3AoA1 are rigid. It is therefo
re possible to 
connect rigidly bars AoA1, AoA2 and AoA3 to form the branched element
, A, shown in 
Figure 3.30( c). 
Following Figure 3.30(c) it can be seen that it is now possible for the rotation about 
hinge Ao to be inhibited by E1 coinciding with A1 or A2, E2 coinciding w
ith A2 or A3 
or E3 coinciding with A1 or A3. The limits for the motion of E1 and E3
 are identical 
61 
3.5. ASSEMBLIES 
elements An and Bn. Hence A1An and B1Bn can be considered to be two rigid bodies 
interconnected by four parallel bars and forming a single degree of freedom mechanism. 
Therefore, a relative movement between elements AI and B1 results in an identical 
relative movement between elements An and Bn and hence the assembly is now a 
single degree of freedom system. Additional connections can be made between other 
element pairs without changing this property. It is possible to use this method also for 
non-identical structures. 
Figure 3.29: Single degree of freedom assembly of two rigidly interconnected structures 
Assemblies with Congruent Linkages 
The method proposed above requires a third layer of elements to make the connections 
and it is therefore not ideal for some applications. However, by considering each of the 
two bar structures as rings of identical linkages a simpler method of interconnecting the 
structures is proposed. From Figure 3.30(a) it can be seen that if both bar structures 
are in identical configurations, then rhombuses I and II are identical and their four bars 
in the top, red layer are parallel to each other, as are the four bars in the bottom, blue 
layer. By positioning the two structures such that I and II are coinciding, the shared 
rhombus AoA1CoB1 is created in Figure 3.30(b). Noting linkage AoA1CoB1 is able 
to shear identically to both I and II, the internal mechanisms of both ring structures 
have been preserved. However, their movement is now synchronous as all linkages are 
identical to AoA1CoB1. This is illustrated in Figure 3.31. 
Around hinges Ao and Go, in Figure 3.30(b), a number of bars are found to be over-
lapping. Note Ao coincides with Bo in Figure 3.30(b). Bar AoA1 from the angulated 
element A2AoA1 overlaps with bar AoA1 from element A3AoA1 and similarly there are 
three other overlaps at BoB1, CoAl and CoB1. The angle A2AoA3 is constant through-
out the motion as elements A2AoA1 and A3AoA1 are rigid. It is therefore possible to 
connect rigidly bars AoA1, AoA2 and AoA3 to form the branched element, A, shown in 
Figure 3.30(c). 
Following Figure 3.30(c) it can be seen that it is now possible for the rotation about 
hinge Ao to be inhibited by B1 coinciding with A1 or A2, B2 coinciding with A2 or A3 
or B 3 coinciding with A1 or A 3. The limits for the motion of B1 and B3 are identical 
61 
3.5. ASSEMBLIES 
AI A3 
Ez 
El 
Eo 
E3 
(a) (b) (c) 
Figure 3.30: (a) Two identical structures, (b) Single degree of freedom assembly of two struc-
tures with congruent rhombus, and (c) Branched pantographic elements 
and can be found to be identical to those derived for the simple angulated element in 
Section 3.2.2. For B2 the limit is found from 2a = (3 + 1 giving an alternative total 
rotation angle, denoted (3', and found for circular structures similarly to Equation 3.25 
(3' = 2a- ( (3.42) 
where the limit ( is determined from Equation 3.28. The possible motion for the 
assembly is then the lower value of either (3* or (3'. If n/2 = k + 1 for all structures in 
the assembly, then (3* = (3' as illustrated by the assembly shown in Figure 3.31 where 
both rings haven; k = 8; 3. Using the methods presented in Section 3.3.2 the assembly 
has been covered with rigid cover elements and can be constructed as a plate structure, 
as shown in Figure 3.31. The assembly shown can be further expanded by adding 
additional ring structures with one or more linkages congruent with the assembly. 
Non-identical structures formed by similar rhombuses or parallelograms can also be 
assembled using this method if they have a single identical linkage through which 
the structures can be joined. This, however, complicates the process of establishing 
the rotation angle (3' as the two kink angles at Ao and at Co are no longer required 
to be identical. The rotation angle {31, for non-circular structures formed by similar 
rhombuses, is then determined by the smallest sum of the kinks at either Ao or Go, 
(3.43) 
and fJ'c can be found similarly. 
62 
3.5. ASSEMBLIES 
Figure 3.31: Single degree of freedom assembly covered by rigid elements 
Assemblies with Node Structures 
Another method for generating planar assemblies also uses the idea of a congruent link-
age, i.e. a linkage that is part of two structures each with a single internal mechanism. 
Above, only structures formed by rings of angulated or multi-angulated elements were 
considered. However, other types of closed loop structures based on four-bar linkages 
exist and can be assembled using the method of congruent linkages. You (2000) pro-
posed a novel type of closed loop expandable structures, as described in Section 2.4.1, 
formed both by linkages and rigid elements as shown in Figures 2.22 . You showed that 
such a structure has a single internal degree of freedom and it is possible to design such 
structures so that the linkages are formed by three identical rhombuses, Figure 3.32. 
Each of these three linkages can form part of a ring structure. This is illustrated in 
Figure 3.32(b). The original three-linkage structure is called the node structure as it 
has the function of a central node for the ring structures. 
As all of the four individual closed loop structures are single degree of freedom mech-
anisms, the motion of the assembly is synchronised. The limit for the shearing of the 
linkages forming parts of the node structure is determined by the shape of the rigid 
element boundaries and, following Equation 3.39, (3* = 7f for f3open = /closed = 0 and 
() = 0. Figure 3.32 shows (3* ~ 7f. Hence, the rotation angle for the assembly is gov-
erned by the ring structures. A model of an assembly using a node structure is shown 
in Figure 3.33. The model is based on a three-linkage node structure and three bar 
structures with n; k = 6; 2 and is similar to that shown in Figure 3.32. 
63 
3.5. ASSEMBLIES 
(a) (b) (c) 
Figure 3.32: Movement in node structure and its linkages forming parts of three ring structures 
Figure 3.33: Cardboard model of assembly with node structure 
64 
3.5. ASSEMBLIES 
(a) (b) (c) 
Figure 3.32: Movement in node structure and its linkages forming parts of three ring structures 
Figure 3.33: Cardboard model of assembly with node structure 
64 
3.5. ASSEMBLIES 
Two Degree of Freedom Assemblies 
If a node structure is formed by four four-bar linkages at right angles it is possible to 
design an assembly with two distinct internal degrees of freedom. With the rigid node 
elements at right angles to each other, the linkages become collinear pairwise and hence 
each pair can be sheared without motion occurring in the other pair of linkages. This 
is shown in Figure 3.34 and it can be seen it is also possible for this type of structure to 
perform as nodes in an assembly of ring structures. Note that the straight bars in the 
linkages have been replaced in Figure 3.34(b) by kinked elements in the ring structure, 
thus allowing these to haven= 8. 
(a) (b) (c) 
Figure 3.34: Two degree of freedom node structure (a) Intermediate configuration, (b) One 
linkage pair fully sheared, and (c) Both linkage pairs fully sheared 
Figure 3.35: Model of node assembly with two degrees of freedom 
The model shown in Figure 3.35 consists of four plastic structures identical to that 
shown in Figure 3.24. The node structure is formed by cardboard fixed to the structures. 
65 
3.5. ASSEMBLIES 
3.5.2 Stack Assemblies 
It is possible to generate assemblies by vertically stacking and connecting individu-
ally expandable structures. It will be shown that this can be done for non-identical 
structures of almost any plan shape. 
Initially consider the circular plate structure, consisting of sixteen identical plate ele-
ments, shown in Figure 3.36. Expanding the structure in a radial motion the hinges 
move along radial lines and the two layers of plates translate and rotate. The rotations 
of the two layers of plates are equal but opposite, as previously described, i.e. the top, 
red layer rotates clockwise as the structure expands while the bottom, blue layer rotates 
counter-clockwise. Because of the opposite rotation, it is not possible to connect rigidly 
two identical structures stacked one above the other as the top, red, layer of the lower 
structure cannot be connected to the facing lower, blue, layer above. However, if the 
order of the two layers in either structure is swapped, corresponding plates will face 
each other and they can then be connected rigidly to each other, i.e. the order of the 
stack could be blue, red, red, blue where the two red layers are rigidly connected. 
Figure 3.36: Expandable circular plate structure. 
It is therefore possible to construct stacked assemblies of identical structures if the 
order of the layers is arranged such that layers that are to be connected have identical 
rotations, i.e. blue to blue and red to red. As the plates of the facing layers are identical 
the connections between the layers can be solid blocks with boundaries identical to those 
of the plates or any parts of such a solid, i.e. rods or walls. 
Physical Model 
The individual structures used for stacked assemblies do not have to be identical as long 
as they have an identical movement. An example of this is shown in Figure 3.37 where 
three plastic structures identical to that shown in Figure 3.36 have been connected 
to each other using foam board. To allow the structure to have a spherical profile in 
the closed configuration two of the plastic structures have had their outer boundaries 
trimmed so the outer diameter of these structures has been reduced and hence the 
outermost hinges have also been removed. Their motion remains unchanged as it is 
controlled by the boundary angle () which is unchanged. The connecting blocks of foam 
66 
3.5. ASSEMBLIES 
board have been cut using an AWJ machine such that their boundaries are identical 
to the boundaries of the plastic plates, including the periodic circular arcs along the 
boundary clearly visible in Figure 3.37. 
Figure 3.37: Model of stacked assembly 
Stacking Non-identical Structures 
As described above it is possible to change the shape of the outer boundary of the plate 
elements as the motion is independent of this. Also the periodic deviations along each 
boundary can be varied without influencing the motion. By considering the motion 
of two layers, to be rigidly connected, as rotations about fixed points, as described 
in Section 3.2.2, it is possible to stack non-circular and non-identical structures. This 
allows expandable free-form structures, such as that shown in Figure 3.38, to be formed. 
Figure 3.38: Expandable free-form or "blob" structure (Jensen & Pellegrino, 2004) 
One such assembly is shown in Figure 3.39. The assembly consists of the two structures 
I and II, structure I is shown with dashed lines. They consist of angulated elements 
formed using the polygon method. The polygons used for forming the elements are 
not congruent and the origins 01 and On do not coincide. Let the red elements, which 
are to be rigidly connected, rotate about fixed points and the blue elements translate. 
By scaling each polygon to half its original size about its origin the centres of rotation 
are found, as previously discussed in Section 3.2.3. Note the design is such that three 
centres of rotation for structure I coincide with three centres of rotation for structure II. 
67 
3.6. DISCUSSION 
These coinciding centres of rotation are denoted Acen , Been and Ccen. Let the rotation 
angle for both structures (3* be governed by the largest kink angle in the assembly at 
hinge Drr. Hence all red elements will undergo identical rotations about their respective 
centres of rotation. The two angulated elements with internal hinges Ar and Arr both 
rotate about Acen and the rotations they undergo are identical. Hence there is no 
relative movement between the two elements and they can be rigidly connected to each 
other. This is illustrated by the rigid plate .6.ArArrAcen in Figure 3.39. Similarly, 
elements Er and Err, with the coinciding centre of rotation Been and Cr, Crr with the 
centre Ccen, can be rigidly connected. 
Figure 3.39: Stack assembly of two structures with three rigid connections 
More generally, an element can be connected rigidly to another if their centres of ro-
tation coincide and they undergo identical rotations. By rigidly connecting elements 
together, their rotation angles must necessarily become identical and hence all struc-
tures in the assembly will have identical rotation angles. The condition of identical 
rotation angles is therefore always satisfied for a stack assembly. 
If the origins of two structures in an assembly coincide, their elements can be rigidly 
connected if their polygon vertices also coincide. This provides a simple and effective 
method for generating stack assemblies from non-circular structures formed using the 
polygon method, i.e. both structures of similar rhombuses and parallelograms can be 
assembled, see also Jensen & Pellegrino (2004) for further details. 
3.6 Discussion 
Extending the findings of Hoberman (1990), You & Pellegrino (1997) and Kassabian 
et al. (1999) a new uniform approach to describing the transformation of retractable 
bar structures has been developed. This new approach is based on the possibility of 
describing the motion of all parts of such structures by means of simple rotations. This 
allows the governing equations to be greatly simplified, and hence it has been possible 
to put forward simple design methods for previously unknown types of structures such 
as assemblies and plate structures. 
68 
3.6. DISCUSSION 
In Section 3.2.2 an expression for the rotation undergone through the transformation 
of the structure, Equation 3.14, has been obtained by considering the overall circular 
motion of the element. This expression was then shown to be identical to Equation 3.18, 
obtained by considering the relative rotations undergone at each of the scissor hinges. 
Hence, as shown in Section 3.2.4, it has been possible to link the local limits on the 
motion of the structure, due to physical properties such as finite hinge sizes, to the 
overall ability of both circular and non-circular structures to transform. 
Considering the shearing deformation of a four-bar linkage, a general condition, Equa-
tion 3.31, on the shape of the boundary between two rigid covering elements has been 
formulated. This equation guarantees that the plates do not restrict the motion of 
the linkage while resulting in a gap and overlap free surface in either extreme position 
of the linkage. As any bar structure is formed by a series of interconnected four-bar 
linkages it has thus been shown that it is possible to cover such bar structures with 
plates without inhibiting the motion of the structure. 
In fact, instead of covering a bar structure with plates, it has been found to be possible 
to remove the angulated elements and connect the plates directly, by means of scissor 
hinges at exactly the same locations as in the original bar structure. Thus, the kinematic 
behaviour of the expandable structure remains unchanged. In Section 3.3.4 it was shown 
that, as long as the plate boundaries have a certain periodic shape, they need not be 
straight. 
General methods for connecting expandable structures of any plan shape have been 
developed, leading to the possibility of creating plan or stacked assemblies composed 
0f individual expandable structures when certain conditions are satisfied. 
Figure 3.40 shows a novel concept for retractable stadium roof formed by hinged plates. 
The next step in the development is to investigate the structural and mechanical prop-
erties of such a roof. 
Figure 3.40: Proposed use of plate structure to cover a sporting venue 
69 
Chapter 4 
Design and Construction of 
Retractable Plate Structure 
4.1 Introduction 
In the previous chapter a method for constructing retractable roof structures formed 
by hinged plates was proposed. This chapter presents the design and analysis carried 
out to construct a 1.3 meter diameter model of such a retractable plate structure. 
The first part of the chapter presents the model, its individual parts in their final form 
and overall design considerations while the second and third parts are concerned with 
various aspects of the design process. This order of contents has been chosen as it 
reflects the design process for this model, where initial choices of material and design 
were subsequently verified and improved through analysis. 
The second part is concerned with the structural behaviour of the model. Numerical 
analysis was used to predict the structural behaviour under gravity loading, and allow 
sizing of the model parts. The model developed is capable of supporting itself in both 
a horizontal and a vertical position, i.e. spanning horizontally or hanging parallel to a 
wall. The final part of this chapter presents the virtual work analysis carried out to 
predict the required actuator torque for the model. This requirement is then used to 
design the motor and gears of the actuator such that the structure can be expanded 
and retracted. 
4.2 Parts of the Model 
The model was designed and constructed with the aim of gaining an increased un-
derstanding of the structural behaviour of hinged plate structures and to provide a 
mechanical concept model of exhibition standard, for demonstration purposes. It was 
decided to pursue a model that would be capable of supporting itself when arranged in 
both a horizontal and a vertical position. It was later found practical to only exhibit 
the model hanging in its vertical position as this would result in a simpler and more 
70 
4.2. PARTS OF THE MODEL 
elegant demonstration of the proposed new concept. 
The model is supported using thin steel cables, as a hanging system provides the 
simplest and least visually intrusive means of supporting the model. The overall size of 
the model was determined so that it can be transported in a small van in its assembled 
state, hence simplifying the hinges as they are not required to allow the model to be 
disassembled. 
The geometry of the model is similar to the plastic model presented in Figure 3.24 with 
its perfect circular opening in the extreme open configuration. Its main dimensions are 
listed in Table 4.1. Each layer contains eight identical plate elements, and each plate 
is connected through four hinges, hence n;k = 8;3. 
Radius: Diameter: 
Maximum open size, Ropen 857 mm 1714 mm 
Minimum closed size, Rclosed 652 mm 1304 mm 
Maximum opening, r open 368 mm 736 mm 
Minimum opening, r closed 42 mm 84mm 
Hinge radius, r1 16 mm 
r* 368 mm 
Table 4.1: Main dimensions of physical model 
4.2.1 Plate Elements 
The model was designed to only carry its own weight; to limit self-weight deflections 
a light, stiff material was to be used for the plate elements. However, the visual 
appearance was also taken into account in the material selection process. Carbon 
Fibre Reinforced Plastic (CFRP) was chosen, after considering materials such as wood, 
perspex and aluminium plates or honeycomb sandwich panels. The selection was made 
on the basis of its high stiffness-to-weight ratio, accurate machineability, low complexity 
of hinge connections and aesthetics. 
The individual plate elements were machined in pairs from 800 x 600 mm sheets of 
3 mm thick CFRP using diamond tipped tools. This was carried out at the BNFL's 
rehearsal and test facility at Littlebrook, UK, on a vacuum table and was computer 
controlled allowing for very high precision. 
As the retractable plate structures presented in Section 3.3.3 are overdetermined, i.e. a 
number of hinges could be removed without introducing any additional degrees of free-
dom, stresses can be introduced in the structure if the hinges are out of position. Hence, 
high precision in the manufacturing process was needed to limit the built-in stresses of 
the model. During final assembly limited stresses were introduced by hand to overcome 
the approximate 0.5 mm error at the last 2 connections assembled. 
71 
4.2. PARTS OF THE MODEL 
The thickness of 3 mm for the plate elements was determined using numerical analysis as 
described in Section 4.3.1. The weight of the plate elements and hence the overall model 
weight is approximately 13.5 kg. The overall weight influences the torque requirement 
for the actuator, as will be discussed in Section 4.4. 
4.2.2 Hinges 
During the retraction of the structure, the plates in the two layers rotate relative to 
each other, as described in Section 3.2.2, and hence the connections made between the 
layers must allow this rotation. However, for the structure to carry its own weight in 
both its horizontal and vertical positions the same connections must be able to transfer 
both in-plane and out-of-plane shear and out-of-plane bending between the connected 
plates, Section 4.3.1. The hinges were thus designed such that an aluminium pin is fully 
restrained in the lower of the two plates and connected through a ball-bearing to the 
upper plate. This permits rotation about the axis of the pin, while all other degrees of 
freedom are constrained. The use of ball bearings nearly eliminates the friction in the 
hinges and hence the required actuator torque is reduced, as described in Section 4.4.1. 
As shown in Figure 4.1 all three interfaces between plates, pin and ball-bearing were 
glued using DP490, an epoxy based structural adhesive from 3M. To provide extra 
contact surface for the adhesive and for ease of assembly, fianged bearings were used. 
The pin was shaped so it would act as a spacer between the layers and fit into holes 
in the plates identical to those used for the bearings, thus allowing all plates to be 
identical, see Figure 4.1. 
A minimum edge distance between the hinges and the boundary of the plates was 
included in the design, by using a hinge radius of Tj = 16 mm instead of the physical 
size of r = 5 mm when determining the limits for the movement of the structure and 
the plate boundaries as described in Chapter 3.2.4. 
CFRP plates 
DP490 adhesive 
10 mm 
Aluminium pin 
Ball bearing 
3.0mm 
1.5 mm 
3.0mm 
Figure 4.1: Hinge with a single rotational degree of freedom, scale 2:1 
4.2.3 Actuator Assembly 
An earlier model of a circular bar structure formed by multi-angulated beam elements 
had been retracted and expanded using a system of cables and springs (Kassabian, 
72 
4.2. PARTS OF THE MODEL 
1997), to demonstrate a solution that could be employed at large scale. A simpler 
actuation system was developed for the current plate model. Rather than imposing a 
change in the diagonal length of one or more of the rhombuses formed by the hinged 
elements of the structure, the new solution directly imposes a relative rotation between 
two elements in different layers, and by reversing the relative rotation it opens or closes 
the structure. Further details are given in Section 4.4. 
The relative rotation is imposed by a Maxon 12V electrical motor, (a) in Figure 4.2, 
through two gearing units and a friction clutch (b), all housed in the actuator assembly 
shown in Figure 4.2. The entire assembly is rigidly attached to the upper plate of 
the structure with four screws while the shaft for the wheel gear (c) , which can rotate 
relative to the rest of the assembly, is rigidly attached to the lower plate using adhesive 
and mechanical keys. The rotation of the worm gear (d) causes the assembly to rotate 
relative to the wheel gear (c) and hence a relative rotation between the upper and lower 
plates is imposed, driving the motion of the structure. 
(a) 12 V Motor with 
internal gearing 
(c) Wheel gear 
(b) Friction clutch (d) Worm gear 
Figure 4.2: Actuator assembly 
End stops are provided by two micro switches mounted with a spacing of a single period 
L along the boundary of the plate to which the actuator is attached. An adjustable pin 
mounted on the neighbouring plate is then able to activate one of the switches when 
the motion reaches either extreme, as shown in Figure 4.3. This reverses the current 
provided to the motor and the motion is hence also reversed . This simple control 
system provides the model with an autonomous actuation system. As a safeguard, 
an adjustable friction clutch, (b) in Figure 4.2, provides a slip mechanism preventing 
damage to both actuator and the structure in the event of failure of a switch or an 
object jamming the motion of the structure. 
73 
4.2. PARTS OF THE MODEL 
Figure 4.3: Switch and pin for reversing motion of model 
4.2.4 Supports 
Supports for the structure, when hung both vertically and horizontally, are provided 
by 2 mm diameter cables, as shown in Figures 4.10 and 4.11. These cables are attached 
to the enlarged hinges shown in Figure 4.4. By supporting the structure at the hinges 
both layers of plates can be supported at the same point. The choice of the number of 
supports and locations of these supports is treated in Sections 4.3.1 and 4.4.1. Using 
cables to support the model was preferred to other, more rigid support methods as it 
allows the supported hinges to move radially without the need for additional pivots, un-
like the rigid pinned columns as used by Hoberman (1990) and Teall (1996). Of course, 
this type of solution is only possible where it is feasible to provide fixed attachment 
points for the cable hangers. 
(a) 
2 mm Cable 
2 mm Cable 
Cable holder 
Cableshoe 
8 mm Pin 
Nut 
Spacer 
(b) 
Figure 4.4: Details of support cables in (a) Horizontal, and (b) Vertical configurations, scale 
1:1 
74 
4.3. STRUCTURAL ANALYSIS 
Hanging in the horizontal plane the model is supported at eight points, as shown in 
Figure 4.4(a). When hung in the vertical configuration the model is only supported at 
the two mid-height support points, Figures 4.4(b) and 4.11, and hence near the centre 
of gravity. Therefore only limited stability is provided to the model by gravitational 
forces . To increase stability in the vertical position the actuator has been placed as low 
as possible to increase the stability of the model. 
Above the model, the cables are attached to a ring which provides spacing between 
the attachment points for the cables and can be used both when hanging the model 
vertically and horizontally. The spacing between the attachment points for the cables 
is described in detail in Section 4.4.1. The ring can be supported by a tripod structure 
or hung from a suitable point in the exhibition space. 
4.3 Structural Analysis 
A finite element model was used to predict the internal forces in the model and its 
deformation when held both horizontal and vertical. The results were used for sizing 
the various parts of the model. To limit the scope of the analysis it was decided that 
only plates of constant thickness would be investigated. 
A previous study by Teall (1996) investigated the structural behaviour of a projected, 
dome shaped, three layer retractable bar structure with a span of 2 m. The third layer, 
identical to the bottom layer, was required to provide additional stiffness and reduce 
bending in the hinges. The study was based on comparing the structural behaviour of 
a physical model with that of a numerical model simulated in a finite element package. 
The main finding was that inaccuracies in manufacturing and assembly, particularly in 
the hinges, would allow the structure to deflect up to five times the value predicted 
under self-weight. When additional loads were imposed on this model, the observed 
deflections were twice those predicted and Teall suggested the likely cause to be that 
the stiffness of the hinges has been overestimated by the numerical model, partly due 
to slip in the hinges. 
Without substantially increasing the complexity of the finite element model, it seemed 
likely that similar difficulties in predicting the deformation behaviour would be en-
countered for the present model. Nonetheless, it did not seem worthwhile to pursue 
a complex study of the non-linear stiffness behaviour of the hinges for this concept 
model. From the small models built previously, it had been found necessary to intro-
duce a gap between the two layers of plates that make up the structure, in order to 
eliminate contact and friction between the plates. Hence, the main reason for estim-
ating the deformations of the structure was to establish the size of the gap needed to 
prevent such contacts and not the magnitude of the deflections themselves, though it 
was desirable that these should be kept visually small. 
4.3.1 Finite Element Models 
A finite element analysis was carried out in a number of stages. For each stage the 
complexity of the model was increased to allow more detailed and accurate modelling. 
75 
4.3. STRUCTURAL ANALYSIS 
The analysis was performed using Pro/Engineer and its structural analysis package 
Pro/Mechanica (Parametric Technology Corporation, 2001). This application would 
allow the same numerical model to be analysed in a number of different configurations 
as the mechanism of the plate structure could also be kinematically modelled and hence 
the structure expanded or retracted as required. By defining two support and loading 
conditions, a single numerical model could hence be used for all configurations - hung 
both vertically and horizontally - resulting in a reduced modelling time. To limit the 
scope of the analysis the structure should only be analysed in the three configurations, 
open, half-deployed and closed. 
Initially, a structure with 9 plates in each layer and four hinges in each layer, i.e. n;k = 
9;3, was chosen for the analysis. A structure with 9 plates in each layer would allow 
the structure to be supported symmetrically by three supports while using 4 hinges 
for each plate would demonstrate the possibility of replacing multi-angulated elements 
with plates. 
As a first step in modelling the full plate structure, a simple beam model of an equivalent 
bar structure with n;k = 9;3 was used to determine suitable supports and investigate 
methods for modelling the gap between the two layers of the structure. The next step 
was to model a single plate element supported at the position of its hinges. Thereby 
could meshing and local effects arising from concentrated loads at the support points, 
later to be connection points, be investigated. 
The full model consisting of 18 plates and their connections was then modelled with 3 
supports. This model displayed large deformations and the need for a large gap between 
the two layers to eliminate contact between plates of the two layers. Therefore, when 
the size of the model was increased by 40%, the number of supports was increased to 9 
in the model. To reduce the actuator torque required for the model to open and close, 
a final design with 8 plates in each layer and 8 supports was adopted and analysed. 
Bar Structure 
A first analysis was carried out on a simple beam model, similar to that shown in 
Figure 4.5 . This was done to find the best methods for modelling hinges and supports, 
as well as providing an easy-to-understand structural model of the complex load paths. 
It was decided to model the hinges as short beam elements rigidly connected to the 
lower layer; these short beams are connected at the upper end by a revolute joint to 
the upper layer, as shown in Figure 4.5. This is identical to the method used by Teall 
(1996) . 
The supports were modelled as translational restraints in a polar coordinate system. 
For example, in Figure 4.5 each support prevents both vertical translation and changes 
in the polar angle, i.e. the support acts as a pinned, radial roller support with four 
degrees of freedom. As the internal mechanism of the structure has been removed - by 
removing the rotational freedom at a single hinge - these support conditions provide a 
statically determinate structure with six restraints in space. 
A clear advantage was found in supporting the structure through the hinges as both 
76 
4.3. STRUCTURAL ANALYSIS 
Figure 4.5: Beam model, with n = 8, in its closed configuration 
layers can be supported at the same point reducing the deformation and internal forces 
in the structure. 
Teall (1996) had supported his structure at the outermost hinges, to provide the max-
imum span. Using the beam model it was found that deflections and forces in the 
hinges would be substantially reduced if the structure was supported further toward 
the centre, as shown in Figure 4.5. As this result is also valid for the case of a plate 
structure, all models were supported similarly, at the third hinge from the centre. 
Single Plate Element 
A single plate element under self-weight was analysed to investigate the convergence 
of the finite element analysis when supported at four individual points. Any problems 
arising from this type of support would likely also occur when modelling connections 
between individual plates using short beam elements. 
The plate was fully restrained at the four locations shown in Figure 4.6 and modelled us-
ing thin shell elements. A mesh of triangular and quadrilateral elements was generated 
by Pro/Mechanica and further subdivided by the application to obtain satisfactory 
convergence of the linear analysis performed. The unrefined mesh is shown in Fig-
ure 4.6(a). The thickness of the plate was initially set at 3 mm, and the quasi-isotropic 
CFRP laminate was modelled as isotropic with a Young's Modulus E = 38,000 N/mm2 
and a Poisson's ratio v = 0.4. 
The results exhibited, as expected, localised peak stresses and distortions at the sup-
ports. The stresses were small, though, with maximum principal stresses of 2 N/mm2 . 
As can be seen from Figure 4.6(c) the extent of the local stresses is very limited and 
contained within the physical size of the hinges. Hence, in reality, the physical model 
77 
4.3. STRUCTURAL ANALYSIS 
would not experience these peak stresses as they would be distributed within the size 
of the hinges. 
0.09 
0.05 
0.01 
-0.03 
-0.07 s 
-0.11 s 
-0.15 ~ 
-0.19 
-0.23 
-0.27 
-0.31 
(b) 
(a) 
1.73 
1.56 
1.38 
1.21 ~ 
1.o4 "'s 
0.86 s 
......... 0.69 z 
0.52 ~ 
0.35 
0.17 
0.00 
(c) 
Figure 4.6: Single plate element under self-weight (a) Mesh for shell elements, (b) Deflections, 
and (c) Maximum principal stresses 
Model with 9 Plates and 3 Supports 
Models of the complete structure were set up from plate models identical to that used 
for the single plate analysis. Plates in different layers were interconnected using beam 
elements, as in the beam model. Problems were not encountered with stability or 
convergence for any of the following models. 
Comparing early results for the structure held vertically and horizontally, it was found 
that the horizontal configuration would be governing the design, as the gravitational 
loading acts in the plane of the structure when it is held vertical and hence induces much 
smaller bending stresses. Therefore, only results from the horizontal configuration are 
presented below. 
For a plate model with a thickness of 3 mm and a gap between the two layers of 0.5 
mm the deflection contours are shown in Figure 4.7. From the contours it is clear that 
the two distinct layers of the model do not act as a single continuous plate. Instead 
78 
4.3. STRUCTURAL ANALYSIS 
the behaviour is that of a series of interconnected plates. This results in large bending 
moments having to be transferred between the individual plates, through the hinges. 
The largest bending moments in the hinges, 1700 Nmm, were found at the second hinge 
from the centre. At this connection the largest principal stresses of 118 N/mm
2 were 
also found for the plates. 
As seen in Figure 4. 7, the maximum downward deflection is approximately 16 mm in 
both the closed and half-deployed configurations. The structure is hence most effective 
when open, where it acts like a continuous ring beam due to the hinges of the plates 
being closer to a circle. This is unlike the closed configuration, where the behaviour is 
more like three cantilevering assemblies of plates . 
Increasing the thickness of the plates from 3 mm to 4 mm reduced the maximum 
deflection from 16 mm to 9 mm. However, importantly the paths of the contours were 
only changed marginally and hence it was concluded that there was little change in the 
way the structure carried the load, i.e. the relative stiffness between the hinges and the 
plates was largely unchanged. Changing the size of the gap between the two layers and 
the stiffness of the short beam elements used for modelling the hinges, within practical 
limits, had little influence on the overall structural behaviour of the model. Hence it 
was concluded that the most significant rotations and distortions at the hinges were 
occurring within the plates, near the joints. This lack of stiffness around the hinges 
cause three kinks to occur in the surface of the model in the closed configuration, see 
(a) in Figure4.7. 
From the contour lines in Figure 4. 7 it is possible to determine the gap needed to 
prevent contact between the two layers. The maximum gap is needed in the closed 
configuration and between the two plates at the supported hinges. At (b) in Figure 4.7 
it can be seen how the contour line for 5.0 mm upward deflection of the lower plate 
intersects with the contour line for 2.4 mm upwards deflection of the upper plate. Hence 
at the point of intersection a gap of minimum 2.6 mm between the two layers of plates 
is needed to avoid contact. For the entire structure a minimum gap of 3.0 mm was 
determined to be necessary. 
Model with 9 Plates and 9 Supports 
At this stage in it was decided to increase the size of the model by approximately 40%, 
to achieve a closed diameter of 1.3 m. This would not pose any problems for the model 
when exhibited vertically. However, in the horizontal configuration the deflections 
would be increased substantially and it was therefore decided to investigate the effect 
of increasing the number of supports. 
Supporting the structure at nine hinges, all plates were directly supported at one 
location. This resulted, for a thickness of 3 mm, in a reduction of the maximum 
deflections from 16 mm to 11 mm, despite the increased size. As can be seen from 
Figure 4.8 the deflections are more uniformly distributed than in the model with only 
three supports. As a result the minimum gap between the two layers of plates could 
be reduced to less than 1 mm. Note that the required gap between the layers is now 
governed by the open position. 
79 
4.3. STRUCTURAL ANALYSIS 
0JS~s::;;2t~...,.------ (b) Intersecting contour lines for 
upper and lower layers of plates 
7.5 
5.0 
2.4 
-0.1 
-2.7 s 
-5.3 ..[ 
-7.8 
-10.3 
-12.9 
-15.5 
Figure 4.7: Contours of deflections under self-weight for n 
with three support points 
9 plate model, held horizontal, 
80 
4.3. STRUCTURAL ANALYSIS 
It can also be seen that the hinge connections are now sufficiently stiff to ensure that 
at the connection points the slopes of the connected plates are nearly identical, unlike 
the previous model. It can hence be concluded that for any large span structure the 
stiffness requirements for the hinges are substantially lowered if all plates are directly 
supported. 
Model with 8 Plates and 8 Supports 
Based on the work carried out on the actuator assembly, presented in Section 4.4, it 
was decided to change the design to one with eight plates in each layer. An analysis 
was hence carried out on this updated but identically sized geometry. 
Figure 4.9 shows that using fewer and wider plates further reduces the expected de-
flections . The maximum downward deflections are 8.3 mm, 7.4 mm and 5.2 mm for 
the closed, half-deployed and open configurations, respectively. Equally, the necessary 
gap between the layers was also reduced to about 0.6 mm. However, because of the 
known issues with predicting the stiffness of the hinges both this final numerical model 
and the physical model presented in Section 4.3 .2 were built with a gap of 1.5 mm and 
hence a factor of 2.5 larger than necessary. 
The maximum principal stress was found to be 32 N /mm2 , at the connections; the 
stresses were much lower in regions without connections. Based on the maximum out-
of-plane bending moment in the connecting beam elements of 600 Nmm, a diameter of 
6 mm was found suitable for the Aluminium pin. 
4.3.2 Physical Model 
Based on the above results it was decided to build the n = 8 plate model using 3 mm 
thick CFRP as this would provide sufficient stiffness for the model. As no suitable 
miniature cylindrical bearings or double row ball-bearings were found, the model was 
built using single row ball-bearings as described in Section 4.2.2 resulting in a more 
flexible connection. 
After assembly the model was hung horizontally, as shown in Figure 4.10. Measuring the 
vertical positions of the supported hinges and the boundaries of the plates it was found 
that deflections were four to six times those predicted by the numerical model, hence 
exceeding the factor of 2.5 introduced in the design of the gap amplitude. This was 
partly due to the additional bending introduced in the model by the inclined supports, 
which induces in-plane compression of the model and hence additional bending in the 
connections. However, the main cause was out-of-plane rotation in the bearings. 
It was attempted to limit the deflections by attaching washers as spacers in the regions 
where contact or near-contact between the two layers was observed. It was also attemp-
ted to adjust the length of the support cables to redistribute the internal forces and 
deflections in the structure. However, the effects of these improvements were limited. 
Despite the contact and resulting friction between the two layers of plates the actuator 
was found capable of opening and closing the model when hung horizontally. 
81 
Figure 4.8: Contours of deflections under self-weight for n 
with nine support points 
82 
4.3. STRUCTURAL ANALYSIS 
3.0 
1.5 
0.0 
-1.5 
-3.0 s 
-4.5 ._[ 
-6.0 
-7.5 
-9.0 
-10.5 
9 plate model, held horizontal, 
4.3. STRUCTURAL ANALYSIS 
2.0 
0.9 
0.1 
-1.1 
-2.1 s 
-3.2 ._[ 
-4.2 
-5.2 
-6.3 
-7.3 
Figure 4.9: Contours of deflections under self-weight for n = 8 plate model, held horizontal, 
with eight support points 
83 
4.4. ACTUATOR DESIGN 
Figure 4.10: Physical model with n = 8, hung horizontally 
Having concluded that the structure was capable of carrying its own weight, but the 
resulting deformations were too large, the model was rotated and hung vertically, sup-
ported by four cables at two hinges, as described in Section 4.4. 
In this configuration, shown in Figure 4.11, the inclined supports were also imposing 
compression forces on the model and therefore a small-displacement snap-through buck-
ling in the out-of-plane direction was observed. Hence the model was not completely 
flush when hanging vertically as the central out-of-plane displacement was approxim-
ately 10 mm. There was, however, no contact between the two layers of plates. 
Figure 4.11: Physical model with n = 8, hung vertically 
4.4 Actuator Design 
In a previous study Kassabian (1997) had developed an actuation system for the 2 m 
span retractable model developed by Teall (1996). The actuation system was based on 
the concept of imposing a change in the diagonal length of one or more rhombus-shaped 
four-bar linkages that composed the structure. Such a change in length can be imposed 
using rigid bars, cables or - as in the Kassabian study - cables and springs. 
Kassabian used a single, continuous cable loop to retract the model, by shortening 
84 
4.4. ACTUATOR DESIGN 
all diagonals in a single concentric ring of rhombuses. Springs were mounted in the 
perpendicular, radial direction of another ring of rhombuses and were hence stretched 
as the model was retracted. Elegantly, the stored elastic energy of the springs could 
then be used to close the structure when the tension in the cable loop was relaxed. 
Using virtual work analysis the position of the cable loop was optimised such that 
the tension and hence the torque in the motor used for shortening the cable loop was 
minimised. The same analysis was also used to position the springs optimally. 
Imposing a change on the diagonal lengths requires additional diagonal members to be 
introduced in the structure. Such members were believed to be visually intrusive, and 
they would also increase the complexity of the model, hence it was decided to pursue 
an alternate actuation method. Noting that plates of different layers rotate relatively 
to one another, as described in Section 3.2.2, it was found that the model could also 
be actuated by simply imposing this relative rotation. As all plates rotate simultan-
eously, actuation is only required at a single connection point; both the expanding and 
retracting motions can be driven with a single rotational actuator. 
Using virtual work, the actuator torque required to move the structure slowly through 
a series of equilibrium configurations was investigated. 
4.4.1 Virtual Work Analysis 
The relative rotation of the two connected plates was defined, as in Section 3.2.2, by 
the rotation angle (3. The total rotation at the hinges of the model is (J* = 60 deg. To 
deter.mine the maximum torque required for the retraction and expansion of the model, 
consider a general configuration of the structure defined by fJ and a small change in 
the rotation angle d(J and its associate virtual work. 
The following forces were considered in the analysis: 
• Actuator torque, 
• Friction forces, 
• Gravity 
Due to symmetry and low accelerations in the structure all inertia forces were neglected. 
Actuator Torque 
The relative rotation between any two connected plates is identical and instantaneous 
for all hinges and hence the actual position of the actuator has no effect. The position 
of the actuator was hence governed by stability considerations, Section 4.2.4. Defining 
the actuator torque as Ta, the virtual work done by the actuator is then 
Wa =TaX d{J (4.1) 
85 
4.4. ACTUATOR DESIGN 
Friction Forces 
As explained in Section 4.3 the model was designed such that friction between the two 
layers of plates is avoided and hence the work done by friction between plates was 
assumed to be zero. 
Friction in the individual hinges is limited to the friction in the ball-bearings and was 
given by Tb = 470 x 10-6 Nm (SMB Bearings Ltd., 2002). Hence the total work done 
by friction is 
Wb = 4 X 8 X n X d,8 (4.2) 
Gravity 
The arrangement of the supports governs the motion of the centre of mass of the 
structure and hence the imposed gravitational forces . 
It is possible to support the model such that the supported hinges, which moves radially 
as the structure opens and closes, do not move in a horizontal plane perpendicular to 
the gravitational forces. An example of this is letting the supported hinges move along 
inclined paths as the structure opens and closes. As such inclined paths would lift 
and lower the entire structure, and hence its centre of mass, as it opens and closes, 
gravitational forces are imposed on the structure as it moves. A voiding gravitational 
forces arising from the lifting of the entire model, a design with n = 8 was chosen as 
this would allow the supported hinges to move in a horizontal plane when hung both 
vertically and horizontally. In the vertical configuration the four mid-height hinges, of 
which two are supported, all move on a horizontal plane while when hung horizontally 
all hinges of the structure move in the horizontal plane of the structure. 
However, by hanging the supported hinges from fixed points using cables the hinges 
must still vary their height z by a small amount when moving radially outwards in a 
vertical plane, as shown in Figure 4.12, independently of whether the structure hangs 
horizontally or vertically. Consider a hinge Ai of the plate structure that is connected by 
a hanger of length lp to a fixed support point P, as shown in Figure 4.12, of coordinates 
rp and zp. The radial coordinate r for the hinge is found by rewriting Equation 3.3 
using Figure 3.4 
r(f3) ~ 2r* sin ( ~ ( 7dooed + t "' + (3)) ~ 2r* sin C; {3) ( 4.3) 
where >. = /closed + 2a = 105° when the structure is supported at the third hinge, i.e. 
j = 2, and /closed = 150. 
From Figure 4.12 the height for the hinge Ai, and as the motion of all supported hinges 
86 
4.4. ACTUATOR DESIGN 
z 
p 
r 
0 r.:losed 
Figure 4.12: Vertical motion of hinge Ai hung from the fixed point P 
are identical the height of the structure's centre of mass, is 
z (r) = zp- .jz~- (r- rp) 2 (4.4) 
using Equation 4.3 to determine the radial coordinate r Equation 4.4 becomes 
z((J) = zp - l~- (2r*sin CA; f))- rp) 2 (4.5) 
The virtual work done by gravity is given by 
(4.6) 
where G is the total weight of the model and dz((J)jdf) is the derivative of Equation 4.5 
dz ((J) 
df) 
4 (r*) 2 sin2 (>'t.6) + 4r pr sin (>'t13 ) - r~ + l~ 
(4.7) 
In conclusion, the torque Ta required to impose a small change in the rotation angle f) 
can be found by equating Wa to Wb + W 9 . 
Results 
Determining the maximum torque required at any point during the motion, Ta,max, it 
was found that the friction in the bearings was negligible in comparison to the gravity-
induced forces and hence wb was set to zero. 
87 
4.4. ACTUATOR DESIGN 
With the maximum length of the supporting cables set at lp = 1500 mm, constrained 
by the height of the envisioned exhibition space, the required torque could be calculated 
as a function of the radial position, rp, of the fixed point. Table 4.2 shows the maximum 
required torque for closing and opening the model when G = 135 N. A single fixed point 
at the centre of the structure corresponds to r p = 0 mm while larger values corresponds 
to a ring of fixed points. The position rp = 557 mm is right above the mid-point of the 
path taken by Ai and the resulting symmetric pendulum motion produce the lowest 
possible value of Ta,max· A radial position of rp = 400 mm was chosen for the design 
of the structure. Equations 4.5 and 4.7 are plotted for rp = 400mm in Figure 4.13. 
Radial position, r p Omm 400 mm 557 mm 
Required torque, Ta,max 9.88 Nm 2.50 Nm 0.63 Nm 
Table 4.2: Maximum torque required for varying points of support 
4.4.2 Gear Ratio 
The gear ratio for the actuator was determined by the required maximum torque output, 
Ta,max, and the required duration of an opening and closing cycle. This was set at 1 
min. With a total rotation 2{3* = 120° per cycle the required rate of rotation for the 
actuator was found from 
120° 1 
3600 x 1 min = 3 r.p.m. (4.8) 
No small electrical motor was found capable of delivering the required torque at this 
speed, and therefore gear units had be included in the design. A motor with a built-in 
200:1 gear was chosen. This is capable of producing a continuous torque of 0.6 Nm 
at 12 V with a speed of 8 r.p.m. An additional gear was therefore required. A worm 
and wheel gear was chosen for the second gear unit as this would allow the motor 
to be attached parallel to the plates, rather than perpendicular to them, and hence 
substantially reducing the visual impact of the actuator assembly. A 30:1 worm gear 
reduced the speed to approximately 1/4 r.p.m. thus providing the required rate of 
opening and closing. The efficiency of the worm gear is 39% (HPC Gears Ltd., 2002), 
and hence the maximum torque of the actuator assembly is 
0.6 Nm x 0.39 x 30 = 7.0 Nm > 2.5 Nm (4.9) 
hence exceeding the required torque Ta,max by a factor of almost 3. The actuator has 
been found to be capable of retracting and expanding the constructed model when hung 
both vertically and horizontally. The complete actuator assembly is shown mounted 
on the model in Figure 4.2. 
88 
4.4. ACTUATOR DESIGN 
0.018 0.020 
0.016 
0.018 
0.014 dz({J) 
d,B 0.016 
0.012 
0.010 0.014 <::Q 
'lj 
s ......... s: 
"l 
';;l 0.008 z 0.012 'lj 
0.006 
0.010 
0.004 
0.008 0.002 
0.000 0.006 
0 10 20 30 40 50 60 
Rotation angle, ,B [deg] 
Figure 4.13: Vertical displacement and its derivative for Tp = 400mm 
89 
4.5. DISCUSSION 
4.5 Discussion 
A 1.3 metre diameter retractable plate structure has been constructed. It has been 
shown that such a structure, formed by hinged CFRP plates is capable of supporting 
its own weight in both a horizontal and a vertical configuration. A simple, effective 
method of autonomous actuation has also been developed. 
Several important issues were encountered during the design and construction process . 
Using finite element modelling it was shown that such plate structures are capable of 
spanning horizontally if sufficient bending strength and stiffness are provided at the 
connections between the plates. It was found that, due to the discontinuity of the two 
layers, it was necessary to provide support for all plates as this significantly reduced 
the required strength and stiffness of the hinges. The analysis showed that forces in 
the plates were concentrated around the hinges and it is therefore possible to reduce 
the thickness of the plates if they are reinforced around the hinges. This would reduce 
the overall weight and hence the resulting gravity loads on the structure. 
For the actuation of the model a new and innovative approach was taken. By using the 
relative rotation between plates of different layers it was found possible to design and 
construct a simple actuator with a minimum of moving parts and simple autonomous 
controls. 
When hung in its vertical position the model is an elegant verification of the concept 
of hinged plate structures. Its simple supports and actuation method allow the model 
to be easily transported and erected. 
Hanging the model horizontally large deflections were found. These were substantially 
larger than expected and were caused by the use of single row ball bearings, instead of 
double row or cylindrical bearings, in the hinges. However, the model did show that 
the structure is capable of both carrying itself and expanding and retracting when held 
horizontally. 
Thus it has been proven that such plate structures can be used for retractable roof 
structures, though they would necessarily have to be designed with greater structural 
stiffness, particularly in the hinge elements. The structural efficiency of such a stiff 
structure remains to be proven. 
90 
Chapter 5 
Spherical Retractable Structures: 
Preliminary Studies 
5.1 Introduction 
This chapter presents the background for two novel concepts for retractable structures 
that are presented · in Chapters 6 and 7. Both structures are formed from intercon-
nected curved, i.e. three-dimensional plate elements, unlike the structures presented in 
Chapters 3 and 4 which were based on fiat , two-dimensional plate elements. Forming 
retractable structures in three-dimensions will allow these to take advantage of in-plane 
forces, unlike planar structures that rely on bending strength, and hence such struc-
tures have the potential for spanning greater distances efficiently. This work has been 
focused on spherical shapes. 
The chapter begins by introducing some key concepts of spherical geometry and some 
relevant differences between the well-known Euclidean, two-dimensional geometry and 
the non-Euclidean, three-dimensional spherical geometry. Using these geometric tools 
it is shown that a ring structure formed by Hoberman's angulated elements is not 
mobile in non-Euclidean space, i.e. on the surface of a sphere. 
Though the basic mechanism of the plate structures presented in Chapter 3 cannot be 
adopted for non-Euclidean geometry, it is investigated if the wedge-shaped plates that 
were derived in Section 3.3.2 are capable of forming a gap and overlap free spherical 
surface in both an open and a closed configuration. This geometric study proves that 
the shape of wedge-shaped spherical plates must be modified in order to be capable of 
forming an overlap-free open configuration. The basic geometric shape for such modified 
spherical plate elements is then derived and extended to allow greater variation of the 
shapes of the elements. 
5.2. SPHERICAL GEOMETRY 
5.2 Spherical Geometry 
This introduction to Spherical Geometry and Spherical 1\·igonometry contains a brief 
review of the concepts and tools used by the author to investigate the geometric prop-
erties of spherical structures. The review provides all necessary information for the 
reader to follow the arguments presented later in this chapter and is based on Carne 
(2002); Clough-Smith (1978); Lenart (1996) and Wolfram Research (2004). 
The sphere in Figure 5.1(a) can be defined as the surface on which all points are at 
an identical distance from a single point, the centre of the sphere 0. A line from the 
centre of the sphere to the surface of the sphere is said to be normal to the surface. The 
length of this line is referred to as the radius of the sphere, R. The surface area of a 
sphere is A(sphere) = 47rR2 and hence a unit sphere has the surface area 41f . Two poles 
of a sphere are always located diametrically opposite, i.e. a line drawn between the two 
poles P and P* passes through the centre of the sphere. P* is called the antipodal of 
P. 
Small circle 
----+----
- I -
'0 
Antipodal, P* 
(a) (b) 
Figure 5.1: Spherical geometry (a) The sphere and its circles, and (b) Angle between great arcs 
and the angular length of these 
A great circle is a circle on the sphere with the same radius as the sphere, i.e. the 
Equator and all meridians are great circles. Furthermore, the plane defined by a great 
circle will always pass through the centre of the sphere. Circles with a smaller radius 
are know as small circles. The plane defined by a small circle does not pass through 
the centre of the sphere. Figure 5.1 (a) illustrates the concepts of poles, great and small 
circles. 
Let a great circle pass through two points A and B on the sphere. This great circle 
consists of two segments or two great arcs from A to B. The shorter of these two great 
arcs defines the shortest distance between the two points, also known as a geodesic, 
92 
5.2. SPHERICAL GEOMETRY 
which is the spherical equivalent to a straight line. The angle between two great arcs 
or circles is defined as the angle between their tangents at the point of intersection. 
In Figure 5.1(b) the angle L.BAC between the arcs AB and AC, at their point of 
intersection A, is equal to a. 
The angular length of a great arc is defined as the angle subtended by that arc at the 
centre of the sphere and hence the angular length of AB is L.AO B . The actual length 
of the arc is equal to the angular length times the radius R. Unless stated otherwise 
the angular length is always assumed in the following. 
A lune is a part of the surface bounded by two great arcs, which intersect at A and its 
antipodal A*. From Figure 5.2(a) it can be seen that the angles subtended at the poles 
or vertices of the lune are identical. The spherical triangle .6(ABC) in Figure 5.2(b) is 
similarly bounded by three great arcs, AB, AC and BC. The arcs intersect at the three 
vertices A, B and C. The angles between intersecting arcs, a, {3 and/, are referred to 
as vertex angles. 
(a) (b) 
Figure 5.2: Spherical geometry (a) A lune, and (b) A spherical triangle 
5.2.1 Spherical Excess 
Next it is shown that the sum of the angles of a spherical triangle is not constant, but is 
equal to 1r plus the area of the bounded triangle. This is another significant difference 
from Euclidean geometry for which the sum of angles is always equal to 1r . The area 
is therefore often referred to as the spherical excess, E, as it is the amount by which 
the sum of the angles in the spherical triangle exceeds the sum of the angles in a plane 
triangle. 
To find the area of a spherical triangle consider first the area of the lune AA* in 
Figure 5.2(a). From first principles the proportion of the total area of the sphere 
covered by the lune is identical to the vertex angle a divided by 27r. Hence the area of 
the lune is A(lune) = A(sphere) x a/27r = 47r x a/27r = 2a. 
93 
5.2. SPHERICAL GEOMETRY 
(a) (b) 
Figure 5.3: Spherical geometry (a) Area of a triangle, and (b) Area of a polygon 
Now consider the spherical triangle 6(ABC), bounded by three great circles as shown 
in Figure 5.3(a). The three circles intersect at A, B, C and also at the antipodals A* , 
B*, C*. Hence these circles divide the sphere into eight triangles. These triangles are 
denoted by their vertices, so they are 
6(ABC), 6(A* BC), 6(AB*C), 6(ABC*), 
6(A* B*C), 6(AB*C*), 6(A* BC*), 6(A* B*C*) 
Note, the area of the triangle A(6(ABC)) is identical to A(6(A* B*C*)). Similarly 
the areas of the other triangles are equal in pairs. Hence only the area of a single 
hemisphere, 21r, need be considered 
A(6(ABC)) + A(6(A*BC)) + A(6(AB*C)) + A(6(ABC*)) = 21r (5.1) 
The two triangles 6(ABC) and 6(A* BC) form a lune with the area 
Similarly, 
Adding these together 
A(6(ABC)) + A(6(A* BC))= 2a 
A(6(ABC)) + A(6(AB*C)) = 2{3 
A(6(ABC)) + A(6(ABC*)) = 2')' 
(5.2) 
(5.3) 
(5.4) 
3A(6(ABC)) + A(6(A* BC))+ A(6(AB*C)) + A(6(ABC*)) = 2(a + {3 + 1) (5.5) 
94 
5.2. SPHERICAL GEOMETRY 
From Equation 5.1 it is hence found 
2A(.6(ABC)) + 21r = 2(a + (3 + 1) (5.6) 
giving 
A(.6(ABC)) =a+ (3 + 1- 1r = E (5.7) 
This is called Gerard 's Spherical Excess Formula after the French mathematician and 
engineer Albm't Girard and, considering a triangle lying on a sphere of radius R, Equa-
tion 5. 7 gives 
a+ (3 + 1 = 1r + A(.6(ABC))/ R2 (5.8) 
This more clearly shows that the sum of angles is equal to 1r plus the area of the triangle 
and the well-known a+ fJ + 1 = 1r is obtained for the plane Euclidean triangle. 
Angular Defect 
More generally the area of ann-sided spherical polygon A(poly) , Figure 5.3(b), can be 
determined from the sum of its vertex angles f3i 
A(~~ly) = t f3i- (n- 2) 1f 
i = l 
(5.9) 
The area of the polygon is also known as the angular defect, o. From Figure 5.3(b) the 
external angle is defined as f/i = 1r- f3i· The angular defect can then be written, from 
Equation 5.9, as 
0 n 
R2 = 21f - L 'r/i 
i = l 
5.2.2 Spherical Trigonometry 
(5.10) 
Similarly to the trigonometric rules for a triangle in Euclidean geometry, a number 
of rules can be derived for the spherical triangle. Proofs are not given here but can 
be found in Carne (2002) and Clough-Smith (1978). Consider the spherical triangle 
.6ABC shown in Figure 5.2. 
The Spherical Cosine Rule I is 
cos(LBOC) = cos(LAOC) cos(LAOB) + sin(LAOC) sin(LAOB) cos a (5.11) 
95 
5.3. PANTOGRAPHIC ELEMENTS ON A SPHERICAL SURFACE 
The Spherical Cosine Rule II is 
cos a = -cos (3 cos 1 + sin (3 sin 1 cos( L:BOC) 
The Spherical Sine Rule is 
The Four-part Rule is 
sin(L:BOC) 
sin a 
sin(L:AOC) 
sin(J 
sin(L:AOB) 
sin/ 
cot(L:BOC) sin(L:AOC) =cot a sin 1 + cos(L:AOC) cos 1 
(5.12) 
(5 .13) 
(5 .14) 
If one or more of the six angles defining the spherical triangle, three at the vertices 
and three arc lengths, is a right-angle then given any two angles, any third can be 
found using what is known as Napier's Rules (Clough-Smith, 1978). These allow the 
relationship between the two given angles and the third unknown angle to be written 
directly without the use of the above equations. 
5.3 Pantographic Elements on a Spherical Surface 
Consider the spherical pantographic element shown in Figure 5.4. The element con-
sists of the two identical angulated elements AEB and A' EB'. Each element is com-
prised of two identical curved bars defined as great arcs, e.g. AE and BE. These two 
curved angulated elements are the spherical analogues of Hoberman's angulated ele-
ments presented in Section 2.4.2 for Euclidean geometry. The hinges are defined with 
axes of rotation normal to the sphere and hence all intersecting at the centre of the 
sphere. This allows the pantographic element to rotate about hinge E while main-
taining all hinges on the sphere. Similarly to the planar lazy-tong a number of these 
spherical pantographic elements can be connected while preserving the single internal 
mechanism of the pantograph. 
To form a closed loop ring structure from n identical spherical pantographic elements, 
each pantographic element must subtend the constant angle a defined by the two 
meridians P A and P B. The intersection point P of the meridians is hence the centre 
of the structure and a = 27r jn. A third meridian intersects the first two meridians at 
P and is defined such that it passes through hinge E. For the closed loop structure 
formed by these n pantographic elements to have an internal mechanism a must be 
constant when the angulated elements are rotated. Following symmetry the angles 
L:AP E and L:BP E must also remain constant. By considering the spherical triangles 
defined in Figure 5.4, next it will be shown that a ring structure formed by identical 
curved bars is rigid, and hence does not form a mechanism, as angle L:BP E does not 
remain constant when the hinges A and B are moved along meridians subtending the 
constant angle a . 
96 
5.3. PANTOGRAPHIC ELEMENTS ON A SPHERICAL SURFACE 
p 
B B' 
Figure 5.4: Pantographic element on a spherical surface 
The angulated element consists of identical bars AE and BE, here recall that L.AOE 
is the angular arc length of AE. The angles L.EAB and L.EBA are identical and 
denoted by (. By considering the closed configuration of the pantographic element, 
where A coincides with P, it can readily be seen that ( = a/2. Using Napier's Rules, 
the constant angular length of AB can be written from L.AEF and L.BEF as 
tan (L.AOB/2) = tan(L.AOE) cos(= tan(L.BOE) cos( (5.15) 
Consider L.PAB. As L.AOB and a are constant a change in L.POA will cause a change 
in L.PO B. By considering L.P AD and L.B AD this change can be found as follows. 
From L.PAD 
and from L.BAD 
tan(L.POD) = tan(L.POA) cos a 
sin(L.AOD) = sin(L.POA) sin a 
. sin(L.AOD) 
sm(L.ABD) = sin(L.AOB) 
cos(L.AOB) 
cos(L.DOB) = cos(L.AOD) 
(5.16) 
(5.17) 
(5.18) 
(5.19) 
The length of P B, L.PO B, can hence be found by adding together L.PO D and L.DO B. 
The angle L.BP E can then be found by considering L.BEC and L.P EC. Note, 
L.EBC = L.ABD + ( and hence for L.BEC 
97 
5.3. PANTOGRAPHIC ELEMENTS ON A SPHERICAL SURFACE 
tan(.LBOC) = tan(.LBOE) cos (.LABD + () 
sin(.LCOE) = sin(.LBOE) sin (.LABD + () 
(5.20) 
(5.21) 
From .LPOC = .LPOB- .LBOC the following expression is obtained for 6P EC 
tan(.LCOE) 
tan(.LBP E) = sin (.LPOB- .LBOC) (5.22) 
Hence it is possible to determine if .LBP E = a/2, and therefore constant for varying 
.LPOA. 
The limits for .LPOA corresponding to the open and closed configurations of a closed 
loop structure are found as follows. In the closed configuration A coincides with P 
and hence .LPOA is zero. In the open configuration AB is perpendicular toP E, since 
.LPO A = .LPO B, and hence 
sin(.LPOA) = sin (.LAOB/2) 
sin (a/2) (5.23) 
The length of AB gives the length of bars AE and BE and hence the size of the 
pantographic element. The length of the bars is then following Equation 5.15 
cot(.LAOE) = cot(.LBOE) =cot (.LAOB/2) cos (a/2) (5.24) 
In Figure 5.5 the angle .LBP E is plotted, using Equations 5.16-5.22, as a function of 
.LPOA for a number of particular designs given by n and five different lengths of AB 
ranging from a/5 to a. The figure shows that .LBP E is not constant and only equal to 
a/2 at the limits of .LPOA, i.e. in the fully opened or closed configurations. Hence, a 
closed loop structure formed by spherical pantographic elements is rigid and does not 
therefore form a mechanism. 
If ( is not equal to a/2, but still constant, then .LBP E = a /2 can be satisfied in 
the open and one other configuration. If ( is not required to be constant, i.e. the 
two bars of each angulated element are allowed to rotate relative to each other, then 
.LBP E = a/2 can be satisfied in all configurations. Of course, this structure is no 
longer a pantograph, but a series of spherical four-bar linkages. 
Another approach is to allow the hinge E to move away from the surface of the sphere, 
for example by rotating the angulated element about an axis through A and B. Thus, 
hinge E can be positioned so that it lies in a plane that is at an angle a/2 to the planes 
0 P A and 0 P B. As the pantograph element is opened or closed, hinge E will thus move 
away from and then back to the spherical surface. To allow this motion as well as a 
rotation about the normal, the hinges must allow additional rotations. For a closed loop 
structure all hinges have to allow rotations about all three axes, i.e. they are spherical 
joints, which would result in a system with additional internal degrees-of-freedom. 
98 
5.3. PANTOGRAPHIC ELEMENTS ON A SPHERICAL SURFACE 
~ 
.:Q. 
34.0 
33.0 
~ 32.0 
0... 
C!::l 
"' 31.0 
30.0 
"22.0 
21.0 
bo 
V 
.:Q. 
n =6 
0 10 20 30 40 50 60 70 80 90 
L.POA [deg] 
(a) 
n = lO 
~ 20.0 .. 
0... 
C!::l 
"' 19.0 
18.0 
0 10 20 30 40 50 60 70 80 90 
L.POA [deg] 
(c) 
~ 
.:Q. 
26.5 
25.5 
~ 24.5 
~ 
"' 23.5 
22.5 
19.0 
18.0 
bo 
V 
.:Q. 
~ 17.0 
~ 
"' 16.0 
15.0 
n =8 
0 10 20 30 40 50 60 70 80 90 
L.POA [deg] 
(b) 
n = 12 
0 10 20 30 40 50 60 70 80 90 
L.POA [deg] 
(d) 
Figure 5.5: Plots of L.BPE for different nand lengths of AB 
99 
5.4. PLATE SHAPE 
Instead of only allowing hinge E to move away from the spherical surface, hi
nges A 
and A' could also be allowed to move away from the surface. If the hinges A and A'
 are 
also allowed to move away from each other it is possible to arrange all five hinge
s of the 
antographic element on a spherical surface of higher curvature such that it su
btends 
~he angle a as before. Hence, when the pantographic element is opened, changes in 
angular defect and internal angles can be made compatible by changing the cu
rvature 
of the sphere. This, however, still require the hinges to allow rotation about a
ll three 
axes. 
Because the three hinges A, B and E always lie on a spherical surface, it then be
comes 
possible to use multi-angulated elements. The additional hinges of the multi-an
gulated 
element lie in the same plane as the original three hinges of the simple an
gulated 
element. The intersection of this plane and the sphere defines a circle of constant
 radius 
on which the hinges are then located similarly to the planar solution, regardless
 of the 
changes in curvature. Kokawa (2000, 2001) has shown it to be possible to construct 
such structures with only a single internal mechanism, as the additional conn
ections 
made by the multi-angulated elements limit the number of internal degrees-of-fr
eedom. 
One such structure is shown in Figure 5.6, and the change in curvature is clearly 
visible 
as it opens. Note, the lowest set of triangles form a supporting structure for the
 bottom 
edge. Kokawa's structure has also been covered with rigid cover elements whic
h form 
a gap and overlap free surface in the closed configuration. In the open config
uration 
there are considerable gaps and small overlaps of the cover plates, partly caused
 by the 
change in curvature between the two extreme configurations. 
Figure 5.6: Spherical structure formed by multi-angulated elements (Kokawa, 2000) 
5.4 Plate Shape 
Though it has been shown that spherical retractable structures cannot be for
med by 
pantograph elements it is still of interest to investigate if a series of wedge-
shaped 
curved plate elements would be capable of forming a gap and overlap free sur
face in 
two different configurations, open and closed, on the sphere. 
Consider a spherical cap on a unit sphere. If the cap is cut into n pieces by an 
equal 
number of meridians, the result is n identical spherical wedge-shaped plates as
 shown 
in Figure 5.7(a). Each plate subtends at its apex the constant angle a and the apexes 
of a ll plates coincide at the pole of the sphere in this closed configuration. Note 
that 
the two boundaries between adjacent plates, formed by the meridians, are great arcs 
while the third boundary is formed by a small circle. 
100 
5.4. PLATE SHAPE 
Instead of only allowing hinge E to move away from the spherical surface, hinges A 
and A' could also be allowed to move away from the surface. If the hinges A and A' are 
also allowed to move away from each other it is possible to arrange all five hinges of the 
pantographic element on a spherical surface of higher curvature such that it subtends 
the angle a as before. Hence, when the pantographic element is opened, changes in 
angular defect and internal angles can be made compatible by changing the curvature 
of the sphere. This, however, still require the hinges to allow rotation about all three 
axes. 
Because the three hinges A, B and E always lie on a spherical surface, it then becomes 
possible to use multi-angulated elements. The additional hinges of the multi-angulated 
element lie in the same plane as the original three hinges of the simple angulated 
element. The intersection of this plane and the sphere defines a circle of constant radius 
on which the hinges are then located similarly to the planar solution, regardless of the 
changes in curvature. Kokawa (2000, 2001) has shown it to be possible to construct 
such structures with only a single internal mechanism, as the additional connections 
made by the multi-angulated elements limit the number of internal degrees-of-freedom. 
One such structure is shown in Figure 5.6, and the change in curvature is clearly visible 
as it opens. Note, the lowest set of triangles form a supporting structure for the bottom 
edge. Kokawa's structure has also been covered with rigid cover elements which form 
a gap and overlap free surface in the closed configuration. In the open configuration 
there are considerable gaps and small overlaps of the cover plates, partly caused by the 
change in curvature between the two extreme configurations. 
Figure 5.6: Spherical structure formed by multi-angulated elements (Kokawa, 2000) 
5.4 Plate Shape 
Though it has been shown that spherical retractable structures cannot be formed by 
pantograph elements it is still of interest to investigate if a series of wedge-shaped 
curved plate elements would be capable of forming a gap and overlap free surface in 
two different configurations, open and closed, on the sphere. 
Consider a spherical cap on a unit sphere. If the cap is cut into n pieces by an equal 
number of meridians, the result is n identical spherical wedge-shaped plates as shown 
in Figure 5.7(a). Each plate subtends at its apex the constant angle a and the apexes 
of all plates coincide at the pole of the sphere in this closed configuration. Note that 
the two boundaries between adjacent plates, formed by the meridians, are great arcs 
while the third boundary is formed by a small circle. 
100 
5.4. PLATE SHAPE 
" \ 
\ 
I 
\ I 
(a) (b) 
Figure 5.7: Spherical plates with straight boundaries in (a) Overlap free closed configuration, 
and (b) Overlapping open configuration 
An open configuration is found by rearranging the plates such that parts of their left 
boundaries form the central symmetric opening shown in Figure 5.7(b). The opening 
is shaped as an equal, n sided polygon. Note how the plates are now overlapped in 
the region highlighted in red for plates I and II in the figure. To prove that this is 
always the case, and that simple wedge-shaped plates cannot form an overlap free open 
configuration, the angular defect of the opening is considered. 
First the open configuration is defined as a configuration where a central and symmetric 
opening is created by a continuous boundary formed by the plates. The apex of each 
plate must therefore ,be located on the boundary of the neighbouring plate, as is the 
case in Figure 5.7(b). The opening will hence be a symmetric n-sided spherical polygon 
with vertices located at the apexes of the plates. The area of this polygon is identical 
to the angular defect and hence J = A(poly) following Section 5.2.1. 
For no gap or overlap to occur between the plates, they must be arranged such that 
the left and right boundaries of two neighbouring plates coincide. Hence the polygon 
external angle rJ must be equal to the angle subtended by each plate, a. The total sum 
of external angles must then always be 2?T, as a= 2?Tjn. Following Equation 5.10 the 
angular defect must then be equal to zero. This is only the case in the closed configur-
ation as the area of the central polygon is larger than zero for all open configurations. 
Hence it is not possible to form an open configuration without overlaps using simple 
wedge-shaped plates where the boundaries are formed by straight arcs. 
The overlap can be eliminated by removing the overlapping part of plate II. Thus 
no overlap would occur in the open configuration and the left boundary of plate II 
would have a kink at the location of plate I's apex. Moving the plates to the closed 
configuration the structure now has a gap where the previously overlapping part of 
plate II has been removed. To close this gap, the right boundary of plate I must also 
be kinked such that it again fully coincides with the boundary of plate II. This is 
shown in Figure 5.8(b). Returning the plates to the open configuration, Figure 5.8(a), 
it is found that neither gaps or overlaps are present in this configuration also. It has 
thus been discovered that by making identical kinks in the two boundaries of a plate 
101 
5.4. PLATE SHAPE 
element it is possible to form a gap and overlap free surface in both an open and a 
closed configuration. Note that each piece of these kinked boundaries is still formed by 
a great arc. 
\ 
-----
(a) 
" I / 
' / 1(. 
" \ 
I 
(b) 
Figure 5.8: Spherical plates with kinked boundaries in (a) Overlap free open configuration, and 
(b) Overlap free closed configuration 
To determine the size of the necessary kink consider Figure 5. 9 (a). The kink angle, r;,, 
is defined as 
(5.25) 
using the previous definition 77 = 1r - (3 
(5 .26) 
or, from symmetry 
(5.27) 
The vertex angle (3 can be found by considering the spherical triangle 6P AB in Fig-
ure 5.9(b). Note that the boundary part AC is always perpendicular to a meridian PB 
at its midpoint B. Letting a minimum radius of the opening P B be the governing size 
and a/2 = 1r jn, then 
cos ((3/2) = cos(L.POB) sin (1rjn) (5.28) 
and the location of the kink is found from L.AOC = 2(L.AOB) and 
tan(L.AOB) = sin(L.POB) tan (1r jn) (5.29) 
102 
5.4. PLATE SHAPE 
The same triangle also produces the relationship 
cot(LPOA) = cot(LPOB) cos (1rjn) 
which will be used later on. 
/ '..,.,.-- ....... 
/ ,.. ...... ' " 
I / '- \ 
I ..- -;- - {3 \ 
/ 
" 
\ 
\ 
-----
(a) 
\ 
\ I 
I 
/ 
I 
\ 
I 
/ 
I 
-----
(b) 
Figure 5.9: Determining (a) Kink angle K, and (b) Vertex angle a 
(5.30) 
' 
" \ 
\ 
I 
/ 
'-V 
/ 
Hence it is possible to determine the shape of a kinked plate element capable of forming 
a gap and overlap free surface in an open and a closed configuration from only the 
immber of plates in the structure and the minimum radius of the wanted opening. 
Interestingly, the area A(L.(PAC)) = fJjn = K, from symmetry. 
5.4.1 Periodicity of Boundaries 
Though no specific motion path has yet been considered it can be seen that two adjacent 
plates must translate a distance L, equal to LAOC, and rotate through K relative to one 
another when moving between the two extreme, i.e. open and closed, configurations. 
Hence, if a boundary length is longer than L , additional kinks must be introduced to 
prevent overlap from occurring. From Figure 5.10 it can be seen that the kinks must 
then occur periodically and this kinked boundary exhibits similar characteristics to 
those found for the straight boundaries of planar plate elements, in Section 3.3.4. 
A common characteristic is the possibility of the boundary shape deviating from the 
original straight, or kinked form. Without a specific motion path only a single con-
dition must be satisfied by the boundary of the spherical plate element - the rule of 
periodicity. All shape features must repeat with a period of L and be rotated by K 
relative to the last period. 
Two other examples of periodic boundaries are shown in Figures 5.11 and 5.12. The first 
example is generated by observing that in both Euclidean and non-Euclidean geometry 
three points define both a plane and a circle. Thus, using the point of the apex and 
two other points along the boundary of a plate element, all with an internal spacing 
of L, and using these points as standard "kink points", a small circle can be defined. 
103 
5.4. PLATE SHAPE 
Figure 5.10: Plate elements exhibiting periodicity 
as shown in Figure 5.11. Using this small circle to define the periodic shape of the 
boundary a smooth continuous boundary is created. The radius r1 of this small circle 
can be determined by considering the plane defined by the small circle. 
/ ' 
' 
-----
' \ 
' I I 
'' I / 
'-./ 
/ 
I 
Figure 5.11: Plate elements with periodic boundary formed by a small circle 
The second example also uses a small circle. The small circle is defined by the apexes of 
the plates in the open position. Using this circle to define a periodic boundary formed 
by small arcs, a perfect circular opening is created, see Figure 5.12. The radius of the 
opening is found by considering the plane passing through apex A, pole P and the 
centre of the sphere 0. Hence, 
r2 = Rsin (L:POA) (5.31) 
104 
- ~ 
I 
I 
5.4. PLATE SHAPE 
' \ 
I 
/ ...... \ 
\ / ...... / ...... I I 
' 
/ ...... 
,I ,. ...... I / 
< 
......,. 
/ 
----- -----
Figure 5.12: Plate elements with periodic boundary forming a perfect circular opening 
5.4.2 Non-Symmetric Structures 
Non-symmetrical structures can also be formed if certain conditions are satisfied. To 
prevent gaps or overlaps in the closed configuration, clearly the following condition 
must be satisfied at the pole 
n 
Lai = 21r 
i=l 
(5.32) 
In the open configuration Equation 5.10 for the angular defect must be satisfied and 
Equation 5.25 must also be satisfied at the apex of each plate. Thus it is possible to 
write 
n n n 
2:: ~i = 2:: ai - 2:: '/]i = o (5.33) 
i=l i=l i=l 
This equation shows that there are an infinite number of possible solutions to this 
problem. As for the boundary period L for the symmetric structure the period for any 
particular boundary, Li in a non-symmetric structure is always equal to the length of 
AC for that particular boundary, i.e. the length of the opening polygon's side formed 
by that particular boundary. 
5.4.3 Physical Models 
The double-curvature of the spherical plate elements makes it more difficult to fabricate 
physical models of these structures. Since it is impossible to use fiat sheets of material, 
as in Section 3.3.3, new methods for making physical models had to be developed. 
105 
5.5. DISCUSSION 
Models Cut From Plastic Hemispheres 
Lemirt (1996) have developed a set of tools, including thin, transparent plastic hemi-
spheres, for teaching non-Euclidean geometry to students. Spherical plate elements 
were made by cutting pieces from such hemispheres. A model used for a preliminary 
study of kink angles is shown in Figure 5.13 and is identical to the structure illustrated 
in Figure 5.8. The straight meridians are visible in the closed configuration. The indi-
vidual elements are held together with tape and the structure is supported by another 
plastic hemisphere. 
Figure 5.13: Model made from plastic hemispheres 
Rapid Prototyping 
For geometrically more detailed models, which were difficult to draw and cut manually, 
a rapid prototyping technique was used. From three-dimensional computer models 
physical models were produced using Fused Deposition Modelling (FDM) (Stratasys, 
1999). This is a layered manufacturing method that extrudes a thin bead of plastic, one 
layer at a time. A thread of plastic is fed into an extrusion head, where it is heated into 
a semi-liquid state and extruded through a very small hole onto the previous layer of 
material. Support material is also laid down in a similar manner. The plastic used is a 
high strength ABS and hence the finished model is of high strength and high accuracy. 
The plates were produced individually and then mounted on a plastic hemisphere. 
Figure 5.14 shows a model with plate boundaries formed from small circles to provide 
continuous, smooth boundaries. To allow the plates, which have finite thickness, to fit 
together all boundaries are normal to the sphere. 
5.5 Discussion 
Since it has been shown in Chapter 3 that it is possible to create covered retractable 
bar structures with planar geometry, this chapter has investigated the possibility of 
adapting to spherical geometry both the simple pantographic element and the wedge-
shaped cover plates. 
106 
5.5. DISCUSSION 
Figure 5.14: Model made using rapid prototyping techniques 
Though a spherical surface is axi-symmetric and has a constant curvature, which is 
somewhat similar to a flat plane, it has been shown that the geometric conditions are 
not identical. It has been found that both the pantograph element and the shape of 
plate elements must be modified to allow their use in spherical geometry. Hence, it has 
been proved that it is impossible to construct a closed-loop retractable structure on a 
sphere using pantographic elements connected through simple scissor hinges. As the 
angle subtended by such an element is not constant during the transformation, it has 
been shown that it is necessary to allow additional rotations in the hinges to preserve 
its mechanism. 
Considering only the extreme configurations of a spherical, retractable plate structure 
it has been shown that simple wedge-shaped plate elements cannot form an overlap-free 
surface in any other configuration than the initial closed configuration. It has, however, 
been found possible to modify the shape of such spherical plate elements by forming 
kinks along the boundaries of the elements. This allows spherical structures consisting 
of such kin ked plates to create gap- and overlap-free surfaces both in an open and a 
closed configuration. It has been determined that the size and location of these kinks 
are, for symmetrical structures, only a function of the number of plates used and the 
size of the central opening created. 
Furthermore, it has been shown that this modified plate element, with kinked bound-
aries, exhibits similar periodic characteristics like those found for the two-dimensional 
plate elements derived in Section 3.3.4. However, as no motion paths have been pre-
scribed, no conditions have been defined for the range of possible periodic features. 
Further variations have been shown to be possible, as non-symmetric structures can 
also be formed. 
Finally, note that the plate structures studied in this chapter have only been considered 
in two extreme configurations. The next two chapters are concerned with mechanisms 
that will allow the plate elements presented in this chapter to form retractable roof 
structures. 
107 
Chapter 6 
Spherical Retractable Structures: 
Spherical Mechanisms 
6.1 Introduction 
Chapter 5 has presented a study of the shape of spherical plate elements that fit together 
on a spherical surface, in two different configurations. The present chapter is concerned 
with developing a spherical mechanism that will allow structures composed of such 
elements to open and close while moving on the spherical surface. 
In Euclidean geometry the motion of structures can be described in terms of translations 
and rotations. In non-Euclidean geometry, however, every motion can be described as 
a pure rotation and the work presented in this chapter is based on this concept. The 
first part of the chapter is concerned with this concept and how it can be used to derive 
a simple method for opening and closing a spherical retractable roof where each plate 
element is simply rotated about a fixed point. 
The second part investigates the parameters that govern the position of this fixed point . 
It is found that a geometric relationship between the two extreme configurations for a 
plate element governs the location of its fixed point. Thus, it is shown to be possible 
to design structures where the fixed points are suitably located within the boundaries 
of the plate elements, hence simplifying the structural design of the structure. 
The final part of this chapter describes the relative motion between neighbouring plates 
as they are rotated about their fixed points. Through the study of this relative motion 
it is shown that it is not possible to interconnect two neighbouring elements using only 
rigid bars and cylindrical hinges with only one axis of rotation. Hence an alternative 
system of constraints, which includes sliding mechanisms is developed. The ability of 
this system to form a self-supporting structure with only a single internal mechanism 
is proven by the construction of a half metre span physical model. 
108 
6.2. EULER POLE 
6.2 Euler Pole 
When deriving the spherical plate elements in Chapter 5, only the two extreme config-
urations were considered. To use these plates to form a spherical retractable structure 
it is necessary to consider also all intermediate configurations of these structures, in 
order to determine a suitable motion for the plate elements. Here, a suitable motion 
will be defined as a motion such that; (i) there is no overlap of the plates; (ii) the plates 
remain on the spherical surface at all times; and (iii) any hinges are required to have 
only a single axis of rotation. Initially, no structural requirements will be set and also 
no restrictions will be posed on the number of internal mechanisms of the structure; 
hence allowing also a structure consisting of a number of "free" plate elements to be 
considered initially. 
Consider the plate elements shown in Figure 6.l(a). In the closed configuration, labelled 
with the subscript 0 in the figure, it can be seen that the apex A of the element coincides 
with the pole of the sphere, P. It can also be seen that the first piece of the plate's 
left boundary AC coincide with the meridian PP' in this configuration. In Section 5.4 
it was found in the open configuration AC is perpendicular to a meridian at the mid-
point B, Figure 6.l(a) . Let this meridian be PP' also. Hence the relative location of 
the closed and open configurations has been defined. Following this, it is possible to 
describe a single motion or a set of motion increments that will move the plate from 
one extreme position to the other. In previous chapters only the extreme configurations 
were considered and hence denoted as open and closed. Here, however , intermediate 
configurations must also be identified and hence the points A , B and C will be denoted 
·as A0 , Bo, Go and Am, Bm, Cm for the closed and open configurations respectively, 
while intermediate positions will be denoted by A1, A2, . .. , Am-1 and similarly for B 
and C. Hence, m is the number of rotations undergone between the two extreme 
configurations. 
.... 
--
P,A0 _ 
..... .... 
/ / 
I //Eo 
I 
__ -7-
)_/ I 
/ 't.... 
I I ..... 
I ..... I ..... 
I I / 
\ 
" I / 
' / 
Q<... ...... -
(a) 
...... 
......,_.. 
_... P' 
" \ 
I 
I I I 
/ 
I 
-- p 
.... ..... .... 
/ / 
I / 
/ 
I .-- -/-
)_...- I 
't.... 
I ...._ 
..... 
I 
(b) 
Figure 6.1: Rotated plate element in (a) Two extreme configurations, labelled 0 and m, and 
(b) Intermediate configuration 
Consider a motion composed of two parts: first, a rotation of the plate element 
along PP' so that B1 coincides with Bm, as shown in Figure 6.1(b), and secondly 
109 
6.2. EULER POLE 
a rotation about the axis OB1 which aligns the element with its final configuration. 
The first rotation about the axis OQ perpendicular to the plane of the meridian is 
81 = - (L.POBm- L.POBo) and following Section 5.4 then 81 = L.AOB- L.POBm. 
Note, in Section 5.4 no subscripts were used when deriving the spherical plate elements. 
The second part of the motion is clearly a rotation of 7r/2 about the axis OB1 . Hence, 
it is possible to determine a motion for the plate element from the closed to the open 
position by dividing the motion into two separate parts. 
However, this incremental motion can be simplified using Euler's Theorem which states 
that:" The general displacement of a rigid body with one point fixed is a rotation about 
some axis which passes through that point" (Felippa, 2001). Hence, if the centre of 
a sphere is fixed, then any movement of a rigid body between any two positions or 
orientations on the sphere's surface can be described by a single rotation about a 
specific axis passing through the centre of the sphere. 
Therefore, following Euler's Theorem, the plate element does not need to undergo two 
separate rotations to move from the closed to the open position, as proposed above. 
Instead, the element can be moved between the two extreme positions by a simple 
rotation about an axis or an Euler pole as it is also known. As a single rotation 
is the simplest way of moving the plate element on the sphere, this is of particular 
interest . The location of the Euler pole, PE, and the rotation undergone by the plate 
element about this pole, BE, can be found using either compound rotations or spherical 
trigonometry. Both methods are used later and hence presented below. 
6.2.1 Compound Rotations 
The order in which large rotations are applied to a rigid body influences the final 
outcome as illustrated in Figure 6.2 and hence the addition of rotations cannot be 
analysed using simple vectors. A method to determine the combined or compound 
rotation is to use pseudo-vectors e and Rodrigues Formula. This method is outlined 
below and further information can be found in Crisfield (1997) and Felippa (2001) . 
In Figure 6.3, point A(vo), defined by the vector v0 , is rotated to A(v!) by the rotation 
e about the unit-axis e. The pseudo-vector for this rotation is defined as 
(6.1) 
For compound rotations it is useful to modify the pseudo-vector such that 
w =we= 2 tan (B /2) e (6.2) 
110 
6.2. EULER POLE 
X 
(a) 
X 
(b) 
Figure 6.2: The non-commutativity of vector rotations (a) Bx, By, Bz, and (b) By, Bx, Bz 
0 
Figure 6.3: Large three-dimensional rotation about OP 
111 
6.2. EULER POLE 
The rotation can then be written using the three-by-three rotation matrix R(w) 
ih = R(w)vo (6.3) 
where 
(6.4) 
and 
s(w) = w (6.5) 
If the point A(v0 ) undergoes two rotations fh and 02 about e1 and e2 respectively, then 
(6.6) 
hence 
(6.7) 
where w12 is the pseudo-vector for the rotation about the Euler pole for A(vo) to A(v2) 
and is given by 
(6.8) 
The unit-axis of the Euler pole is found through normalisation of w12, using Equation 6.2 
(6.9) 
The rotation angle can then be obtained through 
1 + 2cosB12 =trace (R(w12)) =trace (R(w2)R(w1)) (6.10) 
Hence, if the motion of a plate element can be defined as a series of individual rotations, 
then the Euler pole PE and its associated rotation BE can be determined using the 
equations above. For the two part motion described in Section 6.2, the Euler pole 
112 
6.2. EULER POLE 
and its associated rotation are thus found from Figure 6.1 using the right hand rule: 
el = OQ, fh = L.AOB- L.POBm and e2 = OBm, e2 = -7r/2. The resulting Euler 
pole and rotation are shown in Figure 6.4. 
..... 
..... 
...... 
/ 
I / 
I / 
-- -7-). ..... I 
/ '( 
'\ I I ' \ I ' I I 
I I 
\ / / I 
'\ 
,I / / 
<.. / 
- ----
Figure 6.4: A single rotation BE about PE moves the plate element from its closed to its open 
configuration 
6.2.2 Spherical Trigonometry 
Using the trigonometric functions provided in Section 5.2.2, the location of the Euler 
pole PE and rotation eE can also be determined from L:.PEBoBm, shown in Figure 6.5. 
I 
PI Ao 
P' 
Figure 6.5: Determining the Euler pole PE and rotation ()E from Lo.PEBoBm 
As the plate element undergoes a rigid body rotation eE about the axis OPE, the dis-
tance from any point on the plate to the axis must remain constant, i.e. L.PEOAo = 
L.PEOAm, L.PEOBo = L.PEOBm and so forth. Hence the angle L.PEBC is also con-
stant, giving L.PEBoCo = L.PEBmCm = J-t· From the isosceles triangle L:.PEBoBm it 
can be shown that L.PEBoBm = L.PEBmBo = J-t· 
113 
r 
6.3. VARYING THE LOCATION OF THE EULER POLE 
In the closed position AoBoCo is parallel to the meridian PP' and in the open position 
AmBmCm is perpendicular to the meridian. The angle p, can therefore be determined 
directly from the sum of angles at Em 
7f = 7f/2 + 2p, =? 1-L = 7r/4 (6.11) 
The rotation (}E can then be found from L:.PEBoBm 
( ) ( L.BoOBm) . cos BE/2 =cos 2 sm p, (6.12) 
where L.BoOBm is the angular length of the great arc BoBm and 
L.BoOBm = L.POBm- L.AOB (6.13) 
Both L.POBm and L.AOB were determined when the plate shape was found in Sec-
tion 5.4. The location of PE is then determined from 
(6.14) 
Using spherical trigonometry it is therefore possible to determine the position of PE 
without defining intermediate steps for the motion. This method is therefore simpler 
than that proposed in Section 6.2.1 using compound rotations where it is necessary to 
determine a possible motion for the plate before the Euler pole can be determined. 
6.3 Varying the Location of the Euler Pole 
In the previous section the Euler pole was found to be located outside the rotated plate 
element, Figure 6.4. If a simple rotation about an Euler pole is to be used as part of a 
mechanism for a retractable roof, it would be advantageous if the pole could be located 
within the boundaries of the plate element being rotated. 
Considering only the open and closed configurations of the plate element, the Euler 
pole can be determined and hence its position must be a function of the geometric 
relationship between those two configurations only. Therefore, to alter the position of 
the pole the relative position of the two configurations must be changed. However, there 
are a number of constraints that must be observed, which follow from the definition 
of the plate shape. In the closed configuration the apex A must coincide with P, thus 
making AB C parallel to a meridian PP'. In the open configuration the length L.PO B 
is defined and ABC must be perpendicular at B to a meridian PP". Hence only the 
angle A subtended by the meridians PP' and PP" can be varied. In Section 6.2 the 
two meridians, PP' and PP", were assumed to coincide and hence A was equal to zero. 
Figure 6.6 is identical to Figure 6.5, except that the two meridians PP' and PP" are 
no longer coincident. From L:.P BoBm 
114 
6.3. VARYING THE LOCATION OF THE EULER POLE 
P' 
Figure 6.6: Determining PE and eE from L.PEBoBm and the initial position given by A 
cos(LBoOBm) = cos(LPOBm) cos(LPOBo) + sin(LPOBm) sin(LPOBo) cos>. (6.15) 
The point E is defined as the intersection of meridian PP' and the great circle passing 
through Am and Bm, and is hence similar to point Bm which defines the intersection 
of the same great circle and the meridian PP". From E the triangles L.EPBm and 
L.EEoBm can be defined. For L.EP Em the following can be found 
cos( LP EBm) =sin).. cos(LPOBm) 
sin(L.EOEm) = sin>.sin(L.POEm) 
sin(LPEEm) 
sin(L.POE) = sin(LPOEm) 
sin(LPEBm) 
And from L.EEoBm, J.l can be determined as follows 
. ( _C)_ . ( / B ) _ sin(LPEEm) sin(L.EOEm) 
sm J.l ., - sm LEEo m - . ( /B O ) 
sm L o Em 
. ( c)_ . ( /EE E ) _ sin(LPEBm) sin (LPOE- LPOBo) 
sm J.l +"' - sm L m 0 - sin(LEoOBm) 
115 
( 6.16) 
(6.17) 
(6.18) 
(6.19) 
(6.20) 
(6.21) 
6.3. VARYING THE LOCATION OF THE EULER POLE 
Using Equations 6.14 and 6.12, respectively, can PE and ()E now be found. 
In Figure 6.7(a- e) the location of the Euler pole is shown for different values of>.. Inter-
estingly, the poles are all found to lie on a single great circle, as shown in Figure 6.7(f) . 
To understand this latent result, consider a compound rotation composed of three 
separate rotations, illustrated in Figure 6.8(a), instead of two rotations as described 
in Section 6.2.1. First the plate element is rotated ()1 = >. about the axis OP such 
that ABC coincides with the meridian PQ. Then the element is rotated ()2 = L.POBm 
about the axis OQ so that the plate apex A coincides with the final position of B, 
i.e. Bm . Finally, the element is rotated ()3 = L.AOB about an axis normal to the plane 
of OAmBm . Considering only the compound of the first two rotations, it can be seen 
from Figure 6.8(b) that for -1r /2 ::::; >. ::::; 1r /2 the solutions for e12 define a great arc 
from Q to the midpoint of P Bm . If - 7r ::::; >. ::::; 1r and the antipodal of e12 is also plotted, 
a complete great circle is created. Now plotting e13, also in Figure 6.8(b), it can be 
seen that the third rotation in the compound has only rotated the previous solution for 
e12 and hence these solutions must also lie on a great arc. 
6.3.1 Selection of Euler Pole 
For each point on the great circle defined by e13 in Figure 6.8(b) there is a corresponding 
value of >.. Hence it is possible to choose on the great circle a suitable location for the 
Euler pole, and thus find the corresponding closed position of the plate element in 
terms of>.. 
In Figure 6.9 four different plate shapes, each with a different opening size, have been 
plotted together with their possible locations of PE for -7r /2 ::::; >. ::::; 7f /2. All have n = 8 
plates, while the minimum size of the opening L.POBm is increased from 1r /16 to 7f /4. 
For an increase in the opening size the length of each boundary period L is increased 
and so is the kink angle"'· For L.POBm = 7r/4 it is not possible to find any location 
of PE inside the element. It is, however, possible to find a suitable Euler pole inside 
the plate elements if the opening size is reduced. It has therefore been shown that it 
is possible to move a plate element from its closed to its open position by rotating the 
element about a fixed point that lies within its own boundaries. 
6.3.2 Physical Models 
So far only a single plate element has been considered. To investigate the possibility of 
using this method for opening and closing a complete structure several physical models 
were built. One such model is shown in Figure 6.10. The model is a modification of 
that shown in Figure 5.14. Thin steel rods have been added to provide the fixed points 
of rotation for each plate. These rods are held in place by holes in the supporting 
plastic hemisphere and, similarly, at the centre of the sphere by holes in the cardboard 
base. Hence, these rods fix the axes of rotation normal to the sphere. 
Using this and other models it was found that the plates do not interfere with each 
other if their rotations are synchronised. When the structure is opened, a gap appears 
between the plates, as can be seen in the sequence in Figure 6.11. The relative motion 
116 
6.3. VARYING THE LOCATION OF THE EULER POLE 
\ 
\ 
\ ~-
\ \ / 
\ / 
\ 
\ 
' I 
' I, 
I \ 
I 
I / 
'- I / 
' I _,. ?-
- ?1p" 
/ I 
I 
- ...... , 
...... -/ 
(a) 
\ ~-
\ \ 
....-,& 
/ // 
/ I 
I I 
(c) 
p 
' \ 
\ 
' I 
' I, 
I \ 
' \ 
\ 
' I 
' I, 
I \ 
I 
I 
I I I 
\ I 
~--
Q I ' , 
...... / 
\ .~ -.. 
\.PE 
' 
\ 
~-
\ 
\. / 
/ 
I I 
I / 
'- I / 
' I _,. ?-
- ?1p" 
/ I 
- ...... , / 
...... -
(e) 
\ 
\ ~-
\ \ / 
\. / 
...... , 
(b) 
\ ~-
\. \ 
(d) 
I, 
I \ 
I I 
I I 
I / 
'- I / 
' I _,. ?-
- ?1p" 
/ I 
/ 
' \ 
\ 
' I 
' I, 
I \ 
I I 
I I 
\ 
\ 
' I 
' I, 
I \ 
I 
I 
...... , / 
...... _ 
(f) 
Figure 6.7: Location of the Euler pole, PE, for varying positions .>.. of the plate in the closed 
configuration 
117 
6.4. RELATIVE MOTION OF PARTS 
\ 
\ 
' I 
' 1\ 
I \ 
I I I 
I I / 
/ / 
(a) (b) 
Figure 6.8: Compound rotations for - 1f /2 ::; ,\ ::; 1f /2 (a) Individual rotations, and (b) Solutions 
for e12 and e13 
between neighbouring plates is described below in Section 6.4. In the concept model 
shown in Figure 6.11 the plate elements are not interconnected and hence do not form 
a single mechanism. In the figure, the individual plates are supported by a series of 
arches spanning the opening, as is the case for the Oita Stadium retractable roof, see 
Figure 2.7. 
6.4 Relative Motion of Parts 
The relative motion of two neighbouring spherical plate elements is similar to that of two 
adjacent planar plate elements that are connected through parallel bars as described in 
Section 3.3.1. Both types of plates move between extreme positions where the adjacent 
plates are in contact. In the intermediate positions there is a gap between the plates, 
in both cases. Hence, it might be possible to connect two adjacent spherical plates 
using parallel bars similarly to the planar structure, though they cannot form simple 
pantographic elements as this possibility was ruled out in Section 5.3. 
In the following the motion of the apexes has been considered though any other points 
on the plates could have been used instead. Figure 6.12(a) shows the circular motion of 
the apexes of two adjacent plates, AI and Au, as they are rotated incrementally about 
their fixed points. Denoting the fixed points e1 and eu, and their identical rotations BE, 
the pseudo-vectors for the rotation of the apexes can be obtained from Equation 6.2 
WI = 2 tan ( BE/2) el and WII = 2 tan ( BE/2) eu (6.22) 
Consider the same rotations if plate I is fixed and plate II is allowed to move relative to 
this by releasing eu. Figure 6.12(b) shows how eu then moves along a circular arc about 
118 
LP0Bn= 1f/16, £ = 9.24", K; = 0.91 " 
(a) 
LP0Bn=31f/16, £ = 25.92", K;=7.89" 
(c) 
6.4. RELATIVE MOTION OF PARTS 
LP0Bn=1f/8, £ = 18.01", K; = 3.59" 
(b) 
LP0Bn=1f/4, £ = 32.65", K;=13.60" 
(d) 
Figure 6.9: Possible locations of PE for structures with n = 8 plates and increasing opening 
size L.POBm 
119 
6.4. RELATIVE MOTION OF PARTS 
Figure 6.10: Physical model with fixed points of rotation 
Figure 6.11: Computer generated images of how a retractable roof could be constructed from 
spherical plates with fixed points of rotation 
120 
6.5. RECIPROCAL MECHANISM 
e1 hence keeping a constant length between the two points. This allows these points 
to be connected by a rigid bar through cylindrical hinges, as shown. To determine if 
An can similarly be connected to a fixed point, the instantaneous centre of rotation rh 
is plotted for An. This centre is the compound of the two rotations WI and wn and 
has been found using Equations 6.8 and 6.9. As can be seen from Figure 6.12(b) the 
instantaneous centre is not constant and hence there is no fixed point on the adjacent 
plate to which An can be connected. 
It was then investigated if, by imposing a rigid body rotation on the structure, a fixed 
point could be found for An. For ei to become constant the imposed rotation must 
be equal and opposite to w1, hence resulting in ei = en and thus not providing a new 
connection point. 
(a) (b) 
Figure 6.12: Incremental rotations of apex A1 and Au (a) Absolute, and (b) Relative 
Though only the motion of the apexes has been considered here, the same result can be 
found for any other point. It has thus been found that it is not possible to interconnect 
any point on plate I to plate II, other than the two fixed points. Therefore, other 
methods for interconnecting the plates are considered. 
6.5 Reciprocal Mechanism 
Chilton et al. (1998) proposed the planar retractable structure based on a transformable 
"reciprocal" frame, shown in Figure 6.13(a). A reciprocal framework consists of a three-
dimensional beam grillage in which the beams mutually support each other. Each beam 
is only supported externally at its outer end, while the inner end is supported on the 
adjacent beam in the closed-loop structure. As all beams rest on each other, the 
structure formed is structurally stable. By allowing the inner beam ends to slide along 
the length of the adjacent beam a retractable system can be formed. 
121 
6.5. RECIPROCAL MECHANISM 
Allowing the beams to rotate relative to each other by means of simple scissor hinges 
and letting the inner end of the beam slide along its supporting beam is not enough to 
form a retractable mechanism. Consider the structure shown in Figure 6.5(a). If the 
structure is to have only a single internal degree-of-freedom the rotations of all beams 
must be identical. However, simple trigonometry shows that if the distance between 
the external supports A and C is constant and L_BAC = L_DCE then both AB and 
BC must change lengths for 6ABC to maintain the constant sum of angles 1r. Hence 
two sliding connections and two hinges are needed for each beam in the structure. The 
sliding connection that allow changes in length of AB can be positioned at either A or 
B (Chilton et al., 1998). 
(a) (b) 
Figure 6.13: Reciprocal retractable structures (a) Beam grillage, and (b) Swivel Diaphragm 
A similar mechanism is proposed for the spherical plate structure. As for the planar 
mechanism, Figure 6.12(b) shows that the length of arc enAn must be increased if An 
is to lie on the arc e1A1 as the two apexes are rotated synchronically. The solution for 
the beam structure has been to add an additional sliding connection. This is however 
not necessary for the spherical plate structure. 
In the beam structure, the use of beam elements required hinge B to follow the straight 
beam CD, which on the sphere corresponds to the arc e1A1. This is not the case for a 
plate element, where non-straight paths can be accommodated within the boundaries 
of the plate. Hence it is possible for apex An to slide along its non-straight path 
shown in Figure 6.12(b), thus allowing the spherical plate elements to form a reciprocal 
retractable structure without a second sliding connection. 
The shape of the non-straight path is identical for all plates. Therefore, the motion of 
all plate apexes will occur synchronously along identical paths, following a rotation of 
each plate about its fixed point. Two smaller models have been constructed to show 
122 
6.5. RECIPROCAL MECHANISM 
the principles of such a reciprocal spherical structure. 
Note, for a planar structure consisting of plate elements where the length of AB is 
constant, the instantaneous centre of rotation for hinge B lies on the circular arc DEF, 
as shown in Figure 6.13(b). Points D and F on the neighbouring plate element follow 
the same path as the plate is rotated about C. It can be shown that the length of BF 
is constant and these two points can hence be connected, thus forming the four-bar 
linkage ABCF. This solution is that of Rodriguez & Chilton (2003), see Section 2.4.3. 
6.5.1 Small Physical Model 
The first smaller model is based on the design shown in Figures 6.11 and 6.12. The 
model, Figure 6.14, consists of a supporting plastic hemisphere, of spherical plate ele-
ments fabricated using FDM, of steel rods used for the fixed axes of rotation, and of 
sliding connections realised as follows. 
The sliding mechanism was modelled by attaching a drawing pin at the apex of each 
plate. The path of the apex lies outside the boundary of the plate and hence a curved 
slot in a piece of thin, clear plastic provided the path for the pin. This arrangement 
limits the motion of the plates, which therefore cannot be moved to their extreme 
positions, Figure 6.14. In this model, the top of the plastic sphere has been removed 
and the structure thus spans the gap; it is only supported at the fixed points, thus 
proving the concept of the reciprocal structure. 
Figure 6.14: Smaller model interconnected by pins running along curved paths 
6.5.2 Large Physical Model 
A larger and more detailed model was also fabricated but using a different prototyping 
technique. This 400 mm span model consists of eight spherical plate elements, eight 
curved columns and four curved beam elements, Figure 6.15(b). All parts were fabric-
ating in a single nine hour printing process at the Faculty of Architecture, University 
of Delft. 
The fabrication process used is based on 3D Ink-Jet printing of binder fluid which 
123 
~I 
6.5. RECIPROCAL MECHANISM 
fuse the elements formed from a plaster-based powder (Z Corporation, 2001). First, 
the printer spreads a thin layer of powder on top of a piston. Second, a layer of the 
part being created is then printed with a binder from an ink-jet print head. Next , 
the piston is lowered to make room for the next layer of powder and the process is 
repeated. As the part is taking shape it is surrounded and supported by loose powder. 
This is removed from the finished part using a vacuum cleaner and compressed air , 
Figure 6.15(a). After a twenty-four hour curing period the parts were infiltrated with a 
cyanoacrylate adhesive to increase strength and durability. This also allows the surface 
of the parts to be polished. 
(a) (b) 
Figure 6.15: Model parts (a) Excavation from printer , and (b) Before polishing 
The plate elements are 4 mm thick, and their boundaries are formed by small circles, 
see Section 5.4.1. The rotation point is located inside the plate element , simplifying the 
hinge connection. The sliding connection has been modified, compared to the previous 
model, to allow the structure to reach the two extreme configurations. The connection 
point is no longer located at the apex of the plate. Instead, it has been moved outside 
the element using four short beams. This allows the path of the connection point to 
lie within the boundaries of the neighbouring plate. The four beams were integrated 
into the plate shape and hence fabricated a.s a single part. The beams are arranged 
in pairs on the inside and the outside of the plate, to allow the neighbouring plate to 
slide between them. The four beams are used to hold a 2 mm diameter aluminium pin 
which connects the two plates through a curved slot. The slot was included in the 3D 
computer model, allowing it to be printed directly, which resulted in higher accuracy 
than if the slot had been cut after fabrication. 
The larger size of the model prevented the use of the smaller plastic hemispheres for 
support. The supporting structure was therefore made using the same fabrication 
method as for the plate elements, as this allowed a spherical support surface to be 
formed. The support structure is formed by four curved beam elements which when 
slotted together form a circular ring beam. The ring is carried by eight columns slotted 
into the beam elements. The ring wa.s prestressed using rubber bands to eliminate any 
124 
6.5. RECIPROCAL MECHANISM 
Figure 6.16: 400 mm span model fabricated using 3D Ink-Jet printing 
125 
6.6. DISCUSSION 
loss of stiffness due to the opening of any gaps. The connections between the spherical 
plates and the supporting structure was made with 5 mm diameter pins machined from 
a polytetrafluoroethylene (PTFE) rod. To reduce friction the contact surfaces were 
polished with fine sand paper and spraycoated with PTFE. 
The fabrication time and costs were governed by several factors: volume, surface area 
and orientation during printing. The parts of the model were designed to minimise 
the costs of the model. Without quantitative knowledge of the material properties, a 
thickness of 4 mm was decided upon for the plates. To provide sufficient stiffness to the 
structure while minimising material use the ring beam was formed as a hollow section. 
Further reductions in material use were achieved by introducing circular holes in the 
walls. 
Though the structure is relatively small, it exhibits some interesting structural prop-
erties. In the open and closed configurations the plates come in contact along their 
adjoining boundaries. Hence the structure performs more like a continuous shell in 
these configurations than a three-dimensionally connected series of plates. If this could 
be implemented on a larger scale, the structural efficiency of such a structure would 
be increased considerably as a retractable roof would be required to carry the ultimate 
loadings only in these extreme configurations. 
Another interesting feature of the model is that it normally tends to move toward 
the open configuration as in the closed configuration the potential energy is at its 
maximum. This enabled a very simple actuation system to be implemented on the 
model. By mounting a cable loop near the apexes of the plates it was found possible to 
control both the opening and the closing of the structure. As the model always wants 
to open itself the cable loop remains in tension. Therefore, by varying the length of 
the cable loop the model can be opened or closed. For a larger structure the required 
tension force and hence actuator effect could be lowered by running the cable around 
the opening a number of times, thus providing an efficient gearing. 
6.6 Discussion 
Based on the shape of spherical plate elements described in Chapter 5 this chapter has 
presented two simple mechanisms that allow the uninhibited movement of the plates 
between their extreme configurations. 
Using the theorem of Euler it has been found possible to determine a fixed point of 
rotation about which a spherical plate element can be moved between its two extreme 
configurations through a simple rotation on a spherical surface. The location of the 
fixed point has been found to be governed only by the relative position of the extreme 
configurations with respect to the central axis of the structure, see Section 6.3. Fur-
thermore, a single great arc has been shown to define all possible locations for the 
fixed point, as shown in Figure 6.9. The figure also shows that for some structures it 
is possible to choose the location of the fixed point such that it is located within the 
boundary of the rotated element thereby simplifying the structure. Based on this dis-
covery a novel type of retractable roof system has been proposed, where each individual 
spherical plate element is rotated about a fixed point. 
126 
6.6. DISCUSSION 
A geometric study of the relative motion of two adjacent plate elements rotated about 
fixed points has found that it is not possible to interconnect the plate elements using a 
rigid member and cylindrical hinges only. Instead, a single degree-of-freedom mechan-
ism based on a three-dimensional reciprocal system has been proposed. The mechanism 
connects the individual plate elements to each other through sliding connections, thus 
forming a self-supporting structure. Both novel concepts have been proved using phys-
ical models. 
This chapter has only been concerned with identifying possible mechanisms for which 
the motion occurs on a spherical surface as this permits the use of simple hinges with 
only a single axis of rotation. Since it has been shown that it is not possible to connect 
neighbouring plates using rigid bars if the motion is to occur on the spherical surface, 
the next chapter investigates the possibility of interconnecting the plates with rigid 
bars when the motion is not constrained to a spherical surface. 
127 
Chapter 7 
Spherical Retractable Structures: 
Spatial Mechanism 
7.1 Introduction 
In this chapter a third mechanism for spherical plate elements is presented. The mech-
anisms presented in the previous chapter were based on the use of cylindrical and 
sliding connections. The mechanism presented in this chapter uses spherical hinges 
only, i.e. hinges that allow rotation about all three axes. The motion is hence no longer 
constrained to the surface of the sphere and can thus be described as spatial. 
The first part of this chapter shows that a particular symmetric structure formed by 
spherical plate elements and interconnected by rigid bars through spherical hinges has 
zero internal degrees-of-freedom and is hence kinematically a "rigid" structure. This is 
shown to cause internal strains in the structure if it is forced open or closed, using a 
simple geometric model. 
The second part describes the optimisation process used to minimise the magnitude 
of the peak strain that occurs in the structure as it is forced to move. Modelling the 
opening of the structure through a number of identical steps the current spatial orient-
ation of the plate elements is optimised such that the strain energy in the mechanism 
is minimised. Using the same model the position of the connecting hinges is also op-
timised. The result of this optimisation process is that it is possible to form a hinged 
structure which has negligible internal strains during opening and closing, though it is 
overconstrained. 
The third part of this chapter presents the kinematic computer simulation carried out 
in order to verify the concept. The symmetry constraint imposed in the previous 
geometrical study is removed and using non-symmetric actuation it is found that the 
structure still exhibits a symmetric transformation. This shows that the structure has 
no other competing modes of deformation, which might allow it to start moving along 
an unexpected path. 
128 
7.2. SPATIAL MECHANISM 
7.2 Spatial Mechanism 
As shown in Chapters 5 and 6, the angular defect of the sphere does not allow a spherical 
mechanism based on simple cylindrical hinges to be formed from either spherical plate or 
pantographic elements. Kokawa (2000, 2001) overcame the problem, for pantographic 
elements, by introducing additional rotational freedoms in the connections, as described 
in Section 5.3. For the spherical plate elements presented in Chapter 5 to form a gap 
and overlap free surface in both extreme configurations, the plates must necessarily lie 
on the same spherical surface in these two configurations. This is not the case for the 
mechanism proposed by Kokawa and hence a different mechanism is proposed for the 
spherical plate elements. 
Consider the closed structure shown in Figure 7.1(a). This structure is composed 
of eight identical spherical plate elements, each fixed against translation through a 
spherical joint at point A. Each plate is hence capable of freely rotating about the 
three axes A1, A2 and A3. The axis A1 is normal to the sphere and the point A is 
chosen so that it coincides with PE and hence a rotation BE about the axis A1 will 
move the plate from its closed configuration to the open configuration as described 
in Chapter 5. The axis A2 is tangential to the sphere and horizontal in the closed 
configuration, while A3 is perpendicular to the plane defined by A1 and A2. The three 
axes are local and defined relative to the plate element . Following Section 6.2.1 any 
orientation in space of the plate element can be described by rotations about A1, A2 
and A3 . For the rotations about A2 and A3 equal to zero and that about A1 also equal 
to zero the closed configuration is obtained while for the rotation about A1 equal to BE 
and the rotations about A2 and A3 equal to zero the open configuration is obtained. 
7.2.1 Mobility Count 
A rigid body such as a plate element has six degrees-of-freedom in space. The three 
translational restraints of the spherical joint at A leave each plate element with three 
degrees-of-freedom, i.e. the ability to rotate freely about the fixed point. From this it 
is possible to determine how many connections must be made between adjacent plates 
in order for the structure to have only a single internal degree-of-freedom. This is done 
by determining the mobility of the structure. 
The number of relative degrees-of-freedom between the bodies in a structure or mech-
anism, M , is equal to number of rigid bodies, each with six degrees-of-freedom, minus 
the number of independent constraints in the system. As explained above a spherical 
joint imposes three constraints, while a simple cylindrical hinge imposes five . The num-
ber M is generally referred to as the relative mobility of a mechanism. For a structure 
consisting of n plate elements connected by j joints, where joint i imposes Ui constraints 
the mobility relative to an origin 0 is equal to (McCarthy, 1990) 
j 
Jvf = 6n- L.:ui 
i=l 
129 
(7.1) 
7.2. SPATIAL MECHANISM 
~ 
.._ 
' ( I ..... ... 
.... 
• 
(a) (b) 
(c) (d) 
(e) (f) 
Figure 7.1: Spatial motion of n = 8 plate structure (a) Axes of rotation, and (b)- (f) Opening 
of structure 
130 
7.3. OPTIMISATION 
For the structure shown in Figure 7.1(a) the relative mobility is hence 
M = 6 X 8 - 3 X 8 = 24 (7.2) 
To produce a structure with only a single internal degree-of-freedom it is therefore 
necessary to introduce a further 23 constraints. In Figure 7.1(b- f) the plate elements 
have been interconnected using 24 rigid bars. Each bar is connected to two adjacent 
plate elements through two spherical joints and the total number of spherical joints in 
the structure is thereby increased from 8 to 56. Determining the relative mobility of 
this mechanism 
M = 6 (8 + 24) - 3 X 56 = 24 (7.3) 
Hence, introducing the bars has apparently been without effect. Consider a single bar 
fixed in space by two spherical joints. The bar should have zero degrees-of-freedom as 
!VI = 6 - 3 x 2 = 0. It has however a single degree-of-freedom as the bar is capable 
of rotating about its own axis, i.e. !VI = 1. Hence an additional constraint must be 
introduced to prevent this rotation. 
This is also the case for the bars in Figure 7.1 and hence an additional 24 constraints 
are introduced in Equation 7.3 
!VI = 6 (8 + 8 X 3) - 3 (8 X 7) - 24 = 0 (7.4) 
Therefore, either this structure is not a mechanism, i.e. it is "rigid", or it is both stat-
ically indeterminate and kinematically indeterminate, in which case the mechanism is 
overconstrained (Pellegrino & Calladine, 1986). Removing a single bar should therefore 
give the structure a single degree-of-freedom but it would also remove the structural 
symmetry. Furthermore, the existence of a mechanism does not guarantee that there 
is a continuous motion between the two extreme configurations without any internal 
straining of the structure. Hence the fully symmetric structure was chosen for further 
investigation. 
7.3 Optimisation 
For both "rigid" and overconstrained mechanisms, internal strains occur in the struc-
ture if it forced to move. Therefore, an approach of minimising the peak strains occur-
ring in a particular structure as it is forced to move, is valid for both the structure's 
mechanism being rigid or overconstrained. Thus, it is not necessary to determine 
whether the structure is rigid or overconstrained for the purpose of forming a hinged 
structure which develops only negligible internal strains when moved. 
The magnitude of the strains developed in the structure as it is forced to move is 
determined by the overall shape of the structure and hence by altering the design of 
the structure, the strains can be influenced. It is therefore possible to minimise the 
131 
7.3. OPTIMISATION 
strains in the structure to prevent material failure and lower the energy required for 
opening and closing the structure. The technique used below is similar to that used for 
a solid surface deployable antenna by Guest & Pellegrino (1996a,b). 
To optimise the design the strains must be determined. These can be found using a 
variety of methods, and a non-linear finite element analysis would provide the most 
accurate results. However, instead of this complex analysis a simpler approach based 
on geometry is proposed. If the plate elements are modelled as rigid, all straining will 
occur in the connecting bars and the strains can then be found by considering the 
position of the connection points. By imposing perfect symmetry on the structure, and 
hence additional constraints, the problem is further simplified as only a single plate 
element and its connection points need be considered. 
The initial closed configuration is defined as unstrained and hence the strains for all 
other configurations can be found from the current distance between the connection 
points. The strain to be minimised, ~' is defined as the sum of the squares of the 
strains in the three bars connecting two adjacent plates, cBc, C:DE and cFG· Hence 
(7.5) 
7.3 .1 Element Orientation 
To find the current location of connection points B, C, D, E, F and G for any particular 
configuration let the orientation of the plate element be defined by rotations about the 
three axes A1, A2 and A3 . Then given, in the closed configuration, the position of 
connection point B(vo) the current position B(v3 ) for another configuration can be 
found using compound rotations as described in Section 6.2.1 
(7.6) 
where the three rotation matrices correspond to rotations fh, fh and 03 about the axes 
A1, A2 and A3 respectively. Note that the axes are local to the plate element and 
hence are also rotated, as can be seen in Figure 7.1 (b- f). A fourth rotation of 27f / n 
about the central, vertical axis of the structure produces the location of the connection 
points on the adjacent plate, allowing ~ to be determined for the current configuration. 
The strain is hence a function of the three rotations .6(81, 82, 03) if the locations of the 
connection points on the plate element are given. 
Because the fixed point A coincides with the Euler pole PE the rotation about A1 can be 
used to drive the opening of the structure using 0 ::::; 01 ::::; OE. This range of rotation is 
subdivided into s identical steps, defining the total number of configurations for which 
~ is to be evaluated. As 81 is predetermined for all configurations the optimisation 
problem then becomes a function of 02 and 03 only. 
The optimisation of ~(82, 83) has been solved using the MATLAB function fminunc 
from the optimisation tool box (The Math Works, Inc., 2002, 2003). This function 
finds a minimum using unconstrained nonlinear optimisation based on a Quasi-Newton 
132 
7.3. OPTIMISATION 
method. Gradients were not provided and hence MATLAB used its medium-scale al-
gorithm, which uses finite-difference to find the gradients. Convergence was achieved 
in approximately five iterations, and hence the formulation was found computationally 
efficient. The number of steps was found to have negligible effects on the convergence. 
To allow faster convergence, the values of fh and 83 obtained from the previous step 
were fed into a higher level optimisation loop. As the motion path of the plate elements 
is continuous the results from the previous step are in the vicinity of the results for the 
current step, and hence the last set of results provides a good initial estimate for the 
optimisation algorithm. For the first step the initial estimates were all set to zero. 
For a chosen particular design, it is thus possible to determine the spatial motion of 
the plate elements which minimises the strain in the bars at each opening step. The 
motion is given in terms of rotation about A 1 , A2 and A3 and allows the peak strain 
in the structure to be determined. 
Figure 7.2 shows the results for a simulation of the structure shown in Figure 7.1. 
This trial design was found using heuristic methods and with the AutoCAD drawing 
package (Autodesk, Inc., 2002) . Figure 7.2(a) shows the results for the strains in all 
three bars and the strain function .6. when the simulation is carried out in 50 steps. 
The peak strain was determined as EFG = -0.069%, i.e. close to the yield strain of 
steel. From the figure it can be seen that strains are not equal in all bars and they 
vary as the structure is opened. Figure 7.2(b) shows the optimised rotations fh and 83. 
As the rotations are not small they represent a significant deviation from the surface 
of the sphere. This is clearly visible in Figure 7.1 also. Note that both the strains and 
the rotations are equal to zero at both extremes, showing that these two configurations 
are unstrained and coincide with the surface of the sphere. 
0.10 10 
8 
0.05 
bO 6 <!) 
'cf2. ~ 
~ 0.00 ~ 4 0 
'@ ·~ H 
...., 
...., 
if1 ~ 2 
-0.05 
0 
-0.10 -2 
0 10 20 30 40 0 10 20 30 40 
()1 [deg] ()1 [deg] 
(a) (b) 
Figure 7.2: Simulation results for (a) Internal strains, and (b) Rotations 
133 
7.3. OPTIMISATION 
7.3.2 Connection Points 
In the previous section a particular design was considered. This section is concerned 
with optimising the design of the structure and hence the location of the connection 
points B, ... , G. The overall plate shape, location of the fixed point and its associated 
rotation are not included in the following optimisation and are hence held constant. 
Consider the structure in Figure 7.1. In the closed configuration the connection points 
lie on the surface of a unit sphere and hence each point is defined by only two spherical 
coordinates, longitude and latitude. Thus for the six connection points a total of twelve 
variables are to be optimised. However, by defining both the initial closed and the final 
open configuration as unstrained, it is possible to reduce this number. 
Consider Figure 7.3. In Section 5.4.1 it was found that two neighbouring plates move 
by a distance L and rotate by an angle "'relative to one another when moved from the 
closed to the open configuration, see Figure 5.10. Therefore, if plate 11 is fixed, plate I 
and hence point B must move by a distance L as shown. If bar BC is to be unstrained 
in the two extreme positions, then C must lie on a great arc which is perpendicular 
to the great arc BoBm, defined by the extreme positions. Hence, if B is defined then 
C can be determined from the connection angle c/Jc as shown. The total number of 
variables for the optimisation problem is thus reduced to nine as the same applies for 
connection points E and G. 
I 
I 
\ 
I 
/ 
I 
...... 
...... 
-----
\ 
\ I 
I / 
...... / 
' I ,.._./ 
/ 
...--
Figure 7.3: Defining the location of point C using point B and angle <jJ 
Optimisation Using Connection Angles Only 
Before attempting to solve the full optimisation problem a smaller problem was solved. 
By only optimising the location of the points C, E and G the design optimisation 
problem is limited to three variables and the strain function is thus tl(c/Jc, c/JE, c/Jc). 
For a particular design given by c/Jc,c/JE and c/Jc, the peak strain occurring in the struc-
ture as it is moved is found by optimising the problem fl(B2 , e3 ) as described in Sec-
134 
7.3. OPTIMISATION 
tion 7.3.1. Thus, for each loop of !:l(cpc,c/JE,c/JG) the problem !:l(fh,B3) is solved to 
allow the peak strain to be minimised for that particular design. 
To solve !:l(c/Jc, c/JE, c/Jc) MATLAB was used. Two different functions were tested for 
best performance, fminunc and fminsearch, both from the optimisation toolbox. From 
Figure 7.1(c) it can be seen that point C is located on the boundary of the plate and 
hence c/Jc should be constrained to prevent the point lying outside the boundary of the 
element. As both MATLAB functions use unconstrained algorithms, the constraints 
were imposed within the strain function itself. For the current problem it was found 
necessary to constrain c/Jc only. Figure 7.4 shows the convergence of the two func-
tions. Using 54 iterations and 302 function evaluations fminunc found a minimum of 
0.03866%, while fminsearch used 64 iterations and only 118 function evaluations to find 
the same minimum. Hence it was, in general, found for the current formulation that, 
although generally less efficient, the direct search method of fminsearch was superior 
to fminunc if given identical initial estimates for the variables. 
0.08 .-----~-----, 
If( 0 07 .. . ~. -111-
<1 
>:1 0.06 
.8 
..., 
u + 
>:1 0.05 <2 
* >:1 + · ~ 
0.04 H ..., - ~IIRIIRIIIIIRHRHIIiRIIRIIIHRI . 
(/) 
0.03 
0 50 
Iterations 
(a) 
100 
0.08 ..---~--~---. 
If( ~ 0.07 
<I 
.§ 0.06 
..., 
u 
>:1 
<2 
>:1 
·~ 
H 
..., 
(/) 
0. 0 5 -i+tt-· . . . . . . . . . . . . . . . . . . . . . 
0.04 . '"4illllf1AIIIilllfllllllllllf1Aiilllllfllllllll 
0.03 L----~--~-----' 
0 20 40 60 
Iterations 
(b) 
Figure 7.4: Convergence of 6.(4Jc, 4JE, 4Jc) using (a) fminsearch, or (b) fminunc 
The optimised locations for the connections were c/Jsc = 10.5°, c/JDE = 12.6387° and 
c/JFG = 20.1186° and the resulting strains and rotations are shown in Figure 7.5. When 
compared with the original results shown in Figure 7.5 for c/Jsc = 10.5°, c/JDE = 12.5006° 
and c/JFG = 19.8793° it can be seen that !:l has been correctly optimised as the peaks 
of !:l are now of the same magnitude. The location of the connection points has not 
been significantly changed, although the peak strain in bar FG has been lowered from 
EFG = 0.069% to EFG = 0.037%, i.e. almost halved. This has been achieved by finding 
a solution where the strain in the bars varies between tension and compression. Note 
that there are four configurations in which there are no strains in either bars and an 
additional configuration for which BC is unstrained. This indicates that in practice 
the structure will have some sort of snap-through behaviour. 
Parametric Study 
It was found through a limited parametric study that the length of bar BC governs the 
rotations about A2. As the bar is located the furthest away from the fixed point A, it 
135 
7.3. OPTIMISATION 
0.04 10 
8 . . . . . . . . . . . 
0.02 e2 : 
b:O 6 . . . . . . . . . . . . . . . . . . .,. .. . . .. (].) 
'i:J( ::2.. 
s:: 0.00 s:: 4 ·~ _g ...., 
1-< C<l ()3 ...., ...., 
U) 0 
~ 2 
-0.02 
0 
-0.04 -2 
0 10 20 30 40 0 10 20 30 40 
B1 [deg] B1 [deg] 
(a) (b) 
Figure 7.5: Simulation results for optimised values of cp (a) Internal strains, and (b) Rotations 
governs the extent of the deviation away from the spherical surface. It was observed 
that if the bar was shortened the plates would have to be rotated further away from 
the sphere in order to accommodate the length of the bar. This was found to be caused 
by the length of the bar being determined by both <Pc and L. If <Pc is set equal to zero 
the bar has a length of approximately L (it is a straight line and not a great arc hence 
the approximation to L) and hence the gap between adjacent plates is approximately 
L when the structure is opened halfway. As <Pc is increased the gap is reduced as this 
is approximately equal to the length of the bar minus the arc length subtended by </Jc. 
It was also found that the strains in all bars are generally reduced as <Pc is increased 
and hence <Pc was found to be constrained by the boundary of the plate element. 
Similarly it was observed that bars DE and FG governed the rotation about A3 . From 
Figure 7.5 it can be seen that the strains in these two bars are of opposite signs, though 
they were not of identical magnitude. It can also be seen that the strains decrease with 
an increased distance between a bar and the fixed point. 
Optimisation of All Points 
The optimisation of all or selected connection points is an optimisation problem of 
higher order than that for <P only. Convergence and solutions of the two MATLAB 
functions fminunc and fminsearch were therefore compared again. For a fifth order 
problem, i.e. five variables, convergence plots are shown in Figure 7.6. For this 
and other problems it was found that from identical initial estimates that fminsearch 
produced superior results in terms of the minimum found, here D. = 0.0029% compared 
with D. = 0.0157%. The difference was found to be large enough to justify the higher 
number of function evaluat ions required, 855 compared to 501, and hence fminsearch 
was used for subsequent simulations. Though this algorithm produced good results, 
restarting the algorithm at a previously found minimum would sometimes produce 
136 
7.3. OPTIMISATION 
another lower minimum hence suggesting that there exist local minima in the solution 
space. 
0.10 .----~--~----, 
~ 0.08 
<l 
-~ 0.06 
t) 
>::: 
.2 0.04 
>::: 
-~ 
b 0.02 
en 
... .. . . : .. . . .. . . ~ . . . .. . . . 
200 400 600 
Iterations 
(a) 
0.40 .-----~------, 
~ 
<l 0.30 
>::: 
.8 
1) 0.20 
>::: 
.2 
>::: 
'@ 0.10 
H 
...., 
en 
0.00 ~ll~lllll~lllll~lllll~llllll~lllll~lllll~lllll~lllll~lllli~lllll~lllll~llllltll __ _j 
0 50 100 
Iterations 
(b) 
Figure 7.6: Convergence of 6.. using (a) fminsearch, or (b) fminunc 
Instead of solving the problem with nine variables, it was found that it could be ad-
vantageous to only release a few points at a time and hence iterate towards a better 
solution by restarting the algorithm. Using this technique two issues were discovered. 
The problem had been formulated such that the strain function would become zero if 
two bars were to coincide, i.e. effectively reducing the number of bars to only two and 
resulting in the mechanism being no longer overconstrained. Based on this, constraints 
were introduced on the latitude of points B, D and E. Similarly, it was also found 
that point B would tend to move away from the constrained point C and hence lie 
outside the boundaries of the plate element. Constraints were thus imposed also on the 
longitudes of points B, D and F. 
By releasing the location of points B, D and F it was found necessary to update 
continuously the constraints for cf;c, cPE and cf;c, using spherical trigonometry. 
Connections of Finite Size 
With the aim of designing a physical model it was decided to introduce constraints 
that would represent spherical joints of finite size. Figure 7. 7 shows two designs where 
the extent of the connections is shown as circular holes in the plate elements. The 
radius of the joints is equal to 10% of the radius for the spherical surface. The size 
of the joints were chosen with the aim of constructing a small scale model, hence the 
relatively large size of joints. Inspection of the figure shows if joints interfere with each 
other or lie outside the boundary of the plate. In Figure 7.1 it was found that the bars 
would conflict with the plate elements and hence it is thought that a physical model 
would have plate elements defined on a larger spherical surface and attached to the 
connection points using rigid links normal to the surface. 
The solutions presented in Figure 7.7 were found by fixing the points Band D, resulting 
in an optimisation problem with five variables. As can be seen, B was located as close 
137 
(a) 
Fgo 
E 
G 
(b) 
7.4. KINEMATIC MODEL 
Figure 7.7: Two designs with finite sized spherical joints 
to the apex as possible so this would be properly supported. D was located such as to 
avoid interference with point C. As the location of D and the constraints for F were 
defined iteratively, early designs showed some interference, Figure 7.7(a). 
The peak strains for the design presented in Figure 7.7(a) were found to be 6. = 
0.00068%, Figure 7.8(a). For all practical purposes these strains are negligible and 
hence justify the use of the simple geometric model presented, instead of a more complex 
finite-element model. From Figure 7.8(b) it can be seen that the rotation about A2 has 
also been reduced, compared with Figure 7.5(b), as a result of an increased length of 
bar BC. 
Figure 7.7(b) shows a design where there is little interference between the spherical 
joints. This was achieved at a small increase in strains, though they were still negligible 
as shown in Figure 7.9. The larger strains were found to be caused by the increased 
distance of points F and G from the fixed point A. If no constraints were imposed on 
these points they would tend to coincide with A, hence eliminating the strain in bar 
FG and subsequently the mechanism. 
Figure 7.12 shows a design which is a compromise between the two shown in Figure 7.7. 
Unlike any of the results presented above, the strain function 6. has four peaks for this 
design, Figure 7.10. Though the first peak does not have the same height as the three 
others it shows that three peaks are not an inherent feature of this type of mechanism. 
7.4 Kinematic Model 
To verify the results presented above, a kinematic model was built using ProEngineer 
and its kinematic analysis package. The parts of the model, plate elements and bars 
were defined as rigid bodies. A fixed, rigid ring beam was used as a support. The 
138 
1.0 ,----~--~--~--....., 
~ 0.5 
1:!2. 
M 
'o 
,..., 
.i 
.@ 
..... 
..., 
if] -0.5 
-1.0 '------~--~--~---' 
0 10 20 30 40 
()1 [deg] 
(a) 
7.4. KINEMATIC MODEL 
4 
0 .......... . 
-2L---~-~~-~~---' 
0 10 20 30 40 
()1 [deg] 
(b) 
Figure 7.8: Simulation results with lowest achieved strains (a) Internal strains, and (b) Rota-
tions 
3.0 8 
2.0 6 
1:!2. 1.0 b:o <J) 
'? ::2.. 4 0 ,..., 
1':1 
>< 0.0 .s 
.i ..., Cll 2 .@ ..., 
-1.0 0 ..... 0::: ..., 
if] 
-2.0 0 
-3.0 
EFG 
-2 
0 10 20 30 40 0 10 20 30 40 
()1 [deg] ()1 [deg] 
(a) (b) 
Figure 7.9: Simulation results for design with finite sized joints (a) Internal strains, and (b) Ro-
tations 
plate elements were connected to this ring via spherical joints. Similarly the bars were 
connected to the plate elements through spherical joints. 
The design used was that optimised for <jJ only and hence with a peak strain of 0.037%. 
However, the kinematic simulation module available in ProEngineer is based on rigid 
bodies and hence no elastic deformation is modelled. This issue is solved in ProEngineer 
by allowing a definable misfit at the connections, i.e. the connection between two parts 
is modelled as a deformable element with a predefined maximum tolerance. Hence, 
139 
7.4. KINEMATIC MODEL 
3.0 .----~--~--~----. 8 
6 
bi5 
<I) 
::2.. 4 
~ 
.9 
..., 
ro 2 ..., 
0 
~ 
0 
-3.0 '-----~--~--~----' 
-2 
0 10 20 30 40 0 10 20 30 40 
el [deg] el [deg] 
(a) (b) 
Figure 7.10: Simulation results with four peaks in the strain function fl (a) Internal strains, 
and (b) Rotations 
to assemble the parts of the model, the program solves a numerical problem which 
may not necessarily result in a perfectly symmetric solution. Furthermore, for each 
configuration or step of the motion the program had to solve this numerical problem, 
i.e. reassembling the structure at each step. It was found that the program would have 
problems in obtaining solutions if the step size was too large, and hence s was set at 500 
though a high tolerance was still required to obtain convergence for all configurations. 
The original, non-optimised solution of Figure 7.2 could not successfully be modelled 
for all steps of opening. 
The geometry was based on a unit sphere and hence the length of bar FG was 0.398. 
For the kinematic analysis package a typical length of one was defined to allow the tol-
erance to be defined also. To obtain convergence in all configurations the tolerance was 
increased until the full motion could be modelled. This required the tolerance to be set 
at 0.010. For the bar FG this corresponded to a peak strain of 2 x 0.010/0.398 = 5.0%. 
Comparing this with the expected strain of 0.037% it is clear that the solution obtained 
by ProEngineer is less accurate than that found above. However, from the image se-
quence in Figure 7.11 it can be seen that the opening process is correctly modelled 
and the structure preserves symmetry throughout the opening, though symmetry is 
not imposed. 
In the geometric analysis of Section 7.3 symmetry was assumed and hence the structure 
was effectively actuated at eight locations thereby imposing additional constraints. The 
kinematic model was used to investigate if the removal of these additional constraints 
would result in additional degrees-of-freedom. This was done by using a single asym-
metrically placed actuator. To excite any additional mechanisms the actuation was not 
imposed as a rotation about an axis. Instead, the model was actuated by imposing an 
outward, horizontal translation of the apex of a single plate element . In Figure 7.11 the 
apex of the nearest plate element is moved horizontally in a direction perpendicular to 
140 
L l 
7.4. KINEMATIC MODEL 
Figure 7.11: ProEngineer kinematic model 
141 
7.5. DISCUSSION 
that defined by the nearest side of the octagonal ring beam. 
As the motion was only imposed at a single point and in a single constant direction no 
symmetry was imposed on the structure. With non-symmetric actuation and the high 
tolerances allowed, it was expected that some type of non-symmetric motion would 
occur. This did not happen, as shown in Figure 7.11, hence proving that the structure 
only possesses a single internal mechanism. This was confirmed by altering the direc-
tion of the imposed motion as these only achieved small motions before the maximum 
tolerance was reached at the connections and convergence failure occurred. 
7.5 Discussion 
Based on the spherical plate elements derived in Chapter 5, this chapter has developed 
a spatial mechanism that allows the plate elements to be interconnected through fixed 
connection points only, hence omitting the need for sliding connections. To allow 
the resulting spatial motion, connections between plate elements were made through 
spherical joints. 
The mechanism formed by the structure was found to be overconstrained by determ-
ining its mobility. Using a simple geometric model of the mechanism the magnitude of 
the internal strains occurring during a forced motion was determined. A limited para-
metric study based on the same geometrical model was used to determine the causes 
for these strains. The model was then used first to optimise the motion of the structure 
and then the location of the connection points on the plate elements. The optimisa-
tion process allowed the strains to be effectively eliminated, thereby creating a single 
degree-of-freedom for the structure. 
This result was verified using a numerical kinematic analysis. Unlike the geometric 
model, this analysis did not impose symmetry on the structure or additional constraints 
in the form of synchronous actuation. The kinematic model showed that the structure 
only possesses a single internal degree-of-freedom and hence forms a true mechanism. 
From the geometric study of the structure it was found that there are many possible 
designs with negligible small strains . It was also found to be possible to create such 
designs when joints of finite size were considered. However, the designs proposed can 
be further improved as collisions occur between the connecting bars as the structure 
opens and closes. This problem can be viewed in Figures 7.1 and 7.12. In the latter 
figure bar BC conflicts with DE in the final step shown, while DE collides with FG in 
the second step. Only limited attempts were made at resolving this issue and hence a 
satisfactory solution is still to be found. This might involve changes to the parameters 
excluded from the present study such as the number of plates, opening size and location 
of the fixed point. 
Compared to the mechanisms presented in Sections 2.4.4 and 6.5 the current structure 
has a number of advantages. Though the structure has been formed on a spherical 
surface the method of geometrically minimising the internal strain can be used for 
structures of other shapes as shown by Guest & Pellegrino (1996a,b) hence allowing 
greater freedom in the choice of shape. Compared to the flat solution, the main struc-
142 
7.5. DISCUSSION 
tural elements, the plate elements, are fixed directly with a more direct load path as a 
result. Though spherical joints are more complex than simple cylindrical hinges the use 
of these avoids the sliding connections employed in the previous chapter. The limited 
rotations about the two axes A2 and A3 allow spherical joints with limited rotations to 
be used. Furthermore, the use of spherical joints will allow the connecting bars to act 
in tension or compression only and hence increase their effectiveness, though friction 
in the connections will be considerable and might cause bending in the bars. The main 
structural members are the plate elements which can be sized accordingly. The cur-
rent design is thus structurally simpler than that proposed and constructed by Kokawa 
(2000, 2001) which also uses spherical joints. Importantly, the proposed structure also 
creates continuous, gap free surfaces in both the extreme configurations. 
Figure 7.12: Motion revealing conflicts between bars 
143 
Chapter 8 
Conclusions 
8.1 Planar Structures 
The aim of the work presented in this dissertation was to develop novel concepts for 
retractable roof systems, in particular large span structures such as covered stadia. The 
understanding of several previously known solutions for retractable structures has been 
advanced, and also a series of novel concepts for retractable roof structures has been 
proposed. 
Based on an initial review of both existing and proposed state-of-the-art concepts, a 
promising concept based on two layers of interconnected bar elements covered by rigid 
plates was identified and selected for further investigation. 
Extending earlier work, a new uniform approach for describing the transformation of 
these retractable bar structures has been developed. Describing the motion of all parts 
of the structure using rotations, rather than radial displacements, it was shown that 
the overall motion of the structure can be expressed simply in terms of the relative 
rotations undergone at each of the hinges in the structure. This, importantly, allows 
the constraints on the motion associated with members and hinges of finite size to be 
easily determined while also allowing structures of any planar shape to be designed 
much more easily than previously. 
Using the same approach, earlier work on covering bar structures with rigid plates, 
by Kassabian et al. (1999), has been extended significantly. By considering the motion 
of a simple four-bar linkage, forming part of the overall structure, a simple and elegant 
expression, Equation 3.31, was obtained for the shape of the individual cover plates 
such that they form a continuous, gap and overlap free cover for the structure in both 
an open and a closed configuration. Unlike the previous solution, which had been found 
for particular plan shapes by heuristic methods, this general expression allows solutions 
to be found for retractable plate structures of any plan shape. 
It has been shown that if certain constraints are satisfied the basic triangular wedge-
shaped solution for the cover plates can be modified to have non-straight boundaries 
of nearly any shape. Hence, a great degree of freedom in the design of both the overall 
plan shape and the shape of the individual cover plates has been shown to exist and 
144 
8.2. SPHERICAL STRUCTURES 
demonstrated through a number of physical models. 
A further advance that has been made is the design of plate elements that cover all the 
hinges of the underlying angulated elements to which they are attached. This allows 
the bar structure itself to be removed and two layers of plates to be connected directly, 
by means of hinges, hence forming a hinged plate structure. As the plates are connected 
at exactly the same locations as the original bar structure the kinematic behaviour of 
this hinged plate structure is identical to that of the original bar structure. 
To evaluate the structural properties of such a plate structure, a 1.3 m proof-of-concept 
model has been designed and constructed. Using finite element analysis the required 
thickness of the plates and the location and number of supports was determined for the 
model. With the model suspended horizontally the analysis showed that the optimal 
arrangement of the supports is that all plates are supported directly, thereby signific-
antly shortening the load paths in the structure. Comparing the analysis results with 
the physical model it was found that out-of-plane rotations in the bearings caused the 
maximum deflections of the physical model to be four to six times those predicted. 
However, the model did prove the concept of a retractable roof structure formed by 
hinged plates as the structure was, when held both horizontally and vertically, capable 
of supporting itself while opening and closing. 
8.2 Spherical Structures 
To increase their structural efficiency for longer spans, retractable structures of curved 
shape have been investigated. First, it was investigated whether the planar geometry 
of the plates could be adapted to a spherical surface. It was found that a structure 
composed of wedge-shaped spherical plates, with boundaries formed by great arcs would 
overlap in the extreme open configuration due to a change in the angular defect of 
the central opening. However, it has been discovered that by kinking or curving the 
boundaries of the spherical plates overlap can be avoided and a structure capable of 
forming a continuous gap and overlap free spherical surface, in both an open and a closed 
configuration, can be formed. Previous solutions for spherical retractable structures 
have all, as shown in Section 2.4.4, had some degree of overlap between their plates 
and hence this finding represents a significant improvement over existing solutions. It 
has been shown that it is possible to modify the basic shape of the spherical plates in 
a similar manner to the planar plates, if certain conditions are satisfied. This allows 
some freedom in the design of their final shape. 
By considering the motion of pantographic elements adapted to a spherical surface, i.e. 
by letting all rotation axes be normal to the spherical surface, it has been proved that 
a circular closed loop mechanism could not be formed by such pantographic elements. 
Hence, alternative mechanisms had to be considered for a spherical retractable plate 
structure and a total of three mechanisms have been developed, allowing structures 
formed by spherical plates to open and close. The motion of the first two mechanisms 
occur on the surface of the sphere while the third is spatial. 
When geologists model tectonic drift of the earth's continents, they describe the motion 
in terms of simple rotations about fixed axes running through the centre of the earth. 
145 
8.3. FURTHER WORK 
Using the same technique to describe the motion of the spherical plates that make up the 
retractable structure, it has been investigated how the spherical plates could be made 
to move from the closed configuration to the open configuration without interfering 
with one another. A simple solution where each plate rotates by equal amounts about 
symmetrically located fixed points was obtained. Based on this solution an original 
type of retractable dome structure was proposed. 
With the aim of generating a single degree-of-freedom mechanism for the structure, 
thus eliminating the need for external synchronisation of the rotation, compound ro-
tations were used to determine if neighbouring plates could be interconnected. It was 
shown not to be possible to use interconnecting bars and simple cylindrical hinges, but 
instead a self-supporting reciprocal mechanism, similar to that proposed by Piiiero, was 
used for the structure (Escrig, 1993). Unlike the concept proposed by Piiiero, shown 
in Figure 2.25, the current structure does not have any overlaps, and hence friction 
between neighbouring plates is removed. This makes it better suited for large scale ap-
plications. Other advantages are the possibilities of modifying the plate boundaries and 
the location of the fixed points about which the plates rotate. Small proof-of-concept 
models were constructed to demonstrate the ability of such a structure to open and 
close while supporting itself. 
Unlike any previous retractable roof structures, the third mechanism is based on the use 
of spherical joints, i.e. hinges which allow rotation about three axes. Interconnecting 
the spherical plates through bars and spherical joints, the plates are no longer con-
strained to a spherical surface, hence giving this new self-supporting structure a spatial 
motion. By initially forming an overconstrained mechanism, effectively a single degree-
of~ freedom mechanism was obtained through geometric design optimisation. This was 
achieved by minimising the internal strains occurring in the structure as it opens and 
closes. Connecting the plates through spherical joints increases the complexity of the 
structure but it also generates a unique mechanism which, unlike a spherical motion, 
might allow non-symmetric structures, capable of moving in three dimensions, to be 
formed. 
8.3 Further Work 
It has been shown that planar plate structures have limited efficiency as a cover for large 
horizontal spans. Hence, to further develop this type of retractable structure smaller 
scale applications should be sought. Many such applications have been proposed in the 
form of table tops, shading devices and small scale retractable roofs. For the planar 
plate structure to become a viable solution for these applications more detailed design 
studies are needed. This work should include determining viable methods for structural 
redundancy and tolerances depending on the application. This work might reveal the 
need for further expanding the geometric studies presented in this dissertation though 
it is likely that the requirements of most applications are already covered. 
Limited work has been carried out on atTays of interconnected, individually retractable 
plate structures. To develop these geometric shapes into practical applications further 
studies of the external support requirements, stiffness and actuation is needed. 
146 
I 
I 
I 
8.3. FURTHER WORK 
No structural analysis has been carried out for the proposed spherical retractable struc-
tures. Though proof-of-concept models have showed that the concepts are realisable 
on a small scale, they did not provide any data on the structural efficiency of these 
structures when applied on a much larger scale. Therefore the next step in the de-
velopment of these types of retractable roof systems should be the detailed analysis 
and verification of the structural efficiency of these doubly-curved structures. Specific 
areas of interest should be the forces in the connections and the exploitation of in-plane 
forces in order to reduce the out-of-plane bending of the spherical plates, especially in 
the two extreme configurations where load requirements are usually the highest and 
where adjacent plate elements are in direct contact. Structural redundancy, tolerances, 
thermal and non-symmetric loading should also be investigated. 
Further geometric studies are also needed for the proposed concept, using spherical 
joints in order to eliminate the collision of the bars during opening and closing. Altern-
ative applications for spherical retractable structures should also be investigated to-
gether with the possibility of applying a similar spatial mechanism to non-symmetrical 
structures. 
Figure 8.1: Rendering of novel concept retractable roof 
147 
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