thesis niels de temmerman

FACULTY OF ENGINEERING Department of Architectural Engineering Sciences

Design and Analysis of Deployable Bar Structures for Mobile Architectural Applications

Thesis submitted in fulfilment of the requirements for the award of

the degree of Doctor in de Ingenieurswetenschappen (Doctor in Engineering) by

Niels De Temmerman June 2007 Promotor: Prof. Marijke Mollaert

Members of the Jury:

Prof. Dirk Lefeber (President) Vrije Universiteit Brussel

Prof. Rik Pintelon (Vice-President)

Vrije Universiteit Brussel

Prof. Marijke Mollaert (Promotor) Vrije Universiteit Brussel

Prof. Ine Wouters (Secretary)

Vrije Universiteit Brussel

Prof. Sigrid Adriaenssens Vrije Universiteit Brussel

Prof. John Chilton

Lincoln School of Architecture

Prof. W.P. De Wilde Vrije Universiteit Brussel

Dr. Frank Jensen

rhus School of Architecture

Acknowledgements My interest in the exciting field of deployable structures came about through the process of writing my masters thesis on the subject, under the supervision of Prof. Marijke Mollaert. This has been the inspiration and drive to delve deeper into this rich and rewarding research topic of which this dissertation is the final result. I would like to express my gratitude to my supervisor Prof. Marijke Mollaert for sharing her vast research experience and for her invaluable scientific guidance. Also, I have great appreciation for her warm and kind personality and her con-tinuous encouragement throughout the course of this research. I would like to thank everyone who has contributed in making the past four years into an exciting and enriching experience: My most heartfelt sympathy goes out to my colleagues of the Department of Architectural Engineering, whom I thank for providing a kind and stimulating environment, and for their friendship and support: Maryse Koll, Tom Van Mele, Thomas Van der Velde, Lars De Laet, Lisa Wastiels, Anne Paduart, Caroline Henrotay, Michael de Bouw, Prof. Ine Wouters, Prof. Hendrik Hendrickx, Prof. Jos Depuydt, dr. Jonas Lindekens. At the department of Mechanical Engineering, I would like to thank Prof. Dirk Lefeber and Prof. Patrick Kool for their help on gaining an insight in the mobil-ity of mechanisms. Prof. Patrick De Wilde, Prof. Sigrid Adriaenssens and Wim Debacker from the department of Mechanics of Materials and Constructions, and Prof. Rik Pintelon from the Department of Fundamental Electricity and Instrumentation, I would like to thank for their scientific advice and sugges-tions. I gratefully acknowledge the financial support extended to me by IWT-Vlaanderen (Institute for the Promotion of Innovation through Science and Technology in Flanders).

dr. Frank Jensen and Prof. John Chilton I would like to thank for sharing their expertise on the subject and providing much valued comments and sugges-tions. Also, many thanks to Wouter Decorte, for the fruitful collaboration, his enthu-siasm on the subject and for sharing his excellent model-making skills. My parents, Eric and Monique, my sister Ilka and her husband Tom, and also Solange, deserve special thanks for their love and friendship and for their un-conditional support and encouragement. Above all, I wish to express my love and sincerest gratitude to Els, my partner and friend, for her continuous love and support. Without her I would never have come this far.

Vilvoorde, June 2007 Niels De Temmerman

IV

Abstract

Deployable structures have the ability to transform themselves from a small, closed or stowed configuration to a much larger, open or deployed configuration. Mobile deployable structures have the great advantage of speed and ease of erection and dismantling compared to conventional building forms. Deployable structures can be classified according to their structural system. In doing so, four main groups can be distinguished: spatial bar structures consisting of hinged bars, foldable plate structures consisting of hinged plates, tensegrity structures and membrane structures. Because of their wide applicability in the field of mobile architecture, their high degree of deployability and a reliable deployment, two sub-categories are studied in greater detail: scissor structures and foldable plate structures. Scissor structures are lattice expandable structures consisting of bars, which are linked by hinges, allowing them to be folded into a compact bundle. Foldable plate structures consist of plate elements which are connected by line joints allowing one rotational degree of freedom. A wide variety of singly curved as well as doubly curved structures are possible. Although many impressive architectural applications for these mechanisms have been proposed, due to the mechanical complexity of their systems during the folding and deployment process few have been constructed at full-scale. The aim of the work presented in this dissertation is to develop novel concepts for deployable bar structures and propose variations of existing concepts which will lead to viable solutions for mobile architectural applications. It is the intention to aid in the design of deployable bar structures by first explaining the essential principles behind them and subsequently applying these in several cases studies. Starting with the choice of a suitable geometry based on architecturally relevant parameters, followed by an assessment of the kinematics of the system, to end with a structural feasibility study, the complete design process has been demonstrated, exposing the strengths and weaknesses of the chosen configuration.

V

Contents Acknowledgements Abstract List of Figures List of Tables List of Symbols 1. Introduction 1

1.1 Deployable Structures 1 1.2 Aims and scope of research 4 1.3 Outline of thesis 5

2. Review of Literature 9 2.1 Introduction 9 2.2 Deployable structures based on pantographs 9 2.2.1 Translational units 10 2.2.2 Polar units 11 2.2.3 Deployability constraint 12 2.2.4 Structures based on translational and polar units 13 2.2.5 Angulated units 21 2.2.6 Closed loop structures based on angulated elements 22 2.3 Foldable plate structures 29

VI

3. Design of Scissor Structures 39

3.1 Introduction 39 3.2 Design of two-dimensional scissor linkages 40

3.2.1 Method 1: Geometric construction 42 3.2.2 Method 2: Geometric design 57 3.2.3 Interactive geometry 68

3.3 Three-dimensional structures 70 3.3.1 Linear structures 72 3.3.2 Plane grid structures 74 3.3.3 Single curvature grid structures 77 3.3.4 Double curvature grid structures 82

3.4 Conclusion 85

4. Design of Foldable Plate Structures 87 4.1 Introduction 87 4.2 Geometry of foldable plate structures 88 4.3 Geometric design 95 4.3.1 Regular structures 97 4.3.2 Right-angled structures 104 4.3.3 Circular structures 107 4.3.4 Alternative configurations 111 4.4 Conclusion 112

5. Introduction to the Case Studies 115 5.1 Introduction 116

5.2 Geometry 117 5.3 Structural analysis of the proposed concepts 120 5.3.1 General approach 120 5.3.2 Load cases 122 5.4 Conclusion 132

VII

6. Case Study 1: A Deployable Barrel Vault with Translational Units

on a Three-way Grid 135 6.1 Introduction 136 6.2 Description of the geometry 137 6.3 Geometric design 143 6.4 From mechanism to architectural envelope 149 6.4.1 Deployment and kinematic analysis 149 6.5 Structural analysis 159 6.5.1 Open structure (single curvature) 159 6.5.2 Closed structure (double curvature) 173 6.6 Conclusion 175

7. Case Study 2: A Deployable Barrel Vault with Polar and Translational Units on a Two-way Grid 177

7.1 Introduction 178 7.2 Description of the geometry 179 7.2.1 Open structure 179 7.2.2 Closed structure 183 7.2.3 Deployment 186 7.3 From mechanism to architectural envelope 188 7.3.1 Deployment and kinematic analysis 188 7.4 Structural analysis 199 7.4.1 Open structure (single curvature) 199 7.4.2 Closed structure (double curvature) 205 7.5 Conclusion 208

8. Case Study 3: A Deployable Bar Structure with Foldable Articulated Joints 211

8.1 Introduction 212 8.2 Description of the geometry 213 8.3 From plate structure to foldable bar structure 219 8.3.1 Deployment 223 8.3.2 Alternative geometry 229 8.3.3 Kinematic analysis 231

VIII

8.4 Structural analysis 235 8.4.1 Open structure (single curvature) 235 8.4.2 Closed structure (double curvature) 239 8.5 Conclusion 242

9. Case Study 4: A Deployable Tower with Angulated Units 245

9.1 Introduction 246 9.1.1 A concept for a deployable tower 247 9.2 Description of the geometry 249 9.3 Geometric Design 255 9.3.1 First approach: Design of the undeployed configuration 255 9.3.2 Second approach: Design of the deployed configuration 264 9.4 From mechanism to architectural structure 270 9.4.1 Mobility analysis 270 9.4.2 The erection process 272 9.4.3 Alternative configuration 275 9.4.4 Simplified concept: prismoid versus hyperboloid 277 9.5 Structural analysis 287 9.6 Conclusion 292

10. Conclusions 295 10.1 Novel concepts for deployable bar structures 296

10.1.1 Case study 1: A Deployable Barrel Vault with Translational Units on a Three-way Grid 296

10.2.2 Case study 2: A Deployable Barrel Vault with Polar and Translational Units on a Two-way Grid 297

10.2.3 Case study 3: A Deployable Bar Structure with Foldable Articulated Joints 298

10.2.4 Case study 4: A Deployable tower with Angulated Units 299 10.2 Comparative evaluation of the proposed concepts 300 10.2.1 Architectural evaluation 300 10.2.2 Kinematic evaluation 301 10.2.3 Structural evaluation 302 10.3 Further work 303

IX

References 305 List of Publications 313

305

List of Figures Figure 1.1: Mobile deployable bar structure ( Grupo Estran)..................................2 Figure 1.2: Classification of structural systems for deployable structures by

their morphological and kinematic characteristics [Hanaor, 2001] ..3 Figure 2.1 : Translational units ........................................................................................ 10 Figure 2.2: The simplest plane translational scissor linkage, called a lazy-tong

.............................................................................................................................. 11 Figure 2.3: A curved translational linkage in its deployed and undeployed

position .............................................................................................................. 11 Figure 2.4: Polar unit .......................................................................................................... 12 Figure 2.5: A polar linkage in its undeployed and deployed position .................. 12 Figure 2.6: The deployability constraint in terms of the semi-lengths a, b, c and

d of two adjoining scissor units in three consecutive deployment stages.................................................................................................................. 13

Figure 2.7: Piero demonstrates his prototype of a deployable shell [Robbin, 1996] .................................................................................................................. 14

Figure 2.8: Planar two-way grid with translational units and cylindrical barrel vault with polar units [Escrig, 1985] ........................................................ 15

Figure 2.9: Top view and side elevation of a two-way spherical grid with identical polar units [Escrig, 1987] ........................................................... 15

Figure 2.10: Top view and side elevation of a three-way spherical grid with polar units ......................................................................................................... 15

Figure 2.11: Top view and side elevation of a geodesic dome with polar units [Escrig, 1987] ................................................................................................... 16

Figure 2.12: Top view and side elevation of a lamella dome with identical polar units .................................................................................................................... 16

Figure 2.13: Deployable cover for a swimming pool in Seville designed by Escrig & Sanchez ( Performance SL)....................................................... 16

Figure 2.14: Bi-stable structure before, during and after deployment ............... 17 Figure 2.15: Collapsible dome and a single unit, as proposed by Zeigler [1976]

.............................................................................................................................. 17 Figure 2.16: Bi-stable structures: elliptical arch and geodesic dome [Gantes,

2004] .................................................................................................................. 18

306

Figure 2.17: Positive curvature structure with translational units in two deployment stages [Langbecker, 2001].................................................... 19

Figure 2.18: Negative curvature structure with translational units in two deployment stages [Langbecker, 2001].................................................... 19

Figure 2.19: Plane and spatial pantographic columns by Raskin [1998]............ 20 Figure 2.20: Pantographic slabs by Raskin [1998]..................................................... 20 Figure 2.21: Deployable ring structure [You & Pellegrino, 1993] ......................... 21 Figure 2.22: Angulated unit or hobermans unit ........................................................ 21 Figure 2.23: A radially deployable linkage consisting of angulated (or

hobermans) units in three stages of the deployment......................... 22 Figure 2.24: Multi-angulated element .......................................................................... 23 Figure 2.25: A radially deployable linkage consisting of multi-angulated

elements in three stages of the deployment.......................................... 23 Figure 2.26: Multi-angulated structure with cover elements in an intermediate

deployment position ...................................................................................... 24 Figure 2.27: Model of a non-circular structure where all boundaries and plates

are unique [Jensen, 2004]............................................................................ 24 Figure 2.28: Computer model of an expandable blob structure [Jensen &

Pellegrino, 2004] ............................................................................................. 25 Figure 2.29: Reciprocal plate structure......................................................................... 25 Figure 2.30: Swivel diaphragm in consecutive stages of deployment................. 26 Figure 2.31: Reciprocal dome proposed by Piero [Escrig, 1993]......................... 26 Figure 2.32: Iris dome by Hoberman [Kassabian et al, 1999]................................. 27 Figure 2.33: Retractable dome on Expo Hannover (courtesy of M. Mollaert)

Mechanical curtain Winter Olympics Salt Lake City 2002 [Hoberman, 2007]........................................................................................... 27

Figure 2.34: Retractable roof made from spherical plates with fixed points of rotation .............................................................................................................. 28

Figure 2.35: Novel retractable dome with spherical plates with modified boundaries......................................................................................................... 28

Figure 2.36: Basic layout of Fosters module [Foster, 1986] .................................. 29 Figure 2.37: Combination of different modules [Foster, 1986] ............................. 30 Figure 2.38: Simplest building with 90 apex angle [Foster, 1986] ..................... 31 Figure 2.39: Building formed by two 90-modules joined at their ends [Foster,

1986] .................................................................................................................. 32

307

Figure 2.40: Building with apex angle of 120 [Foster, 1986]............................... 32 Figure 2.41: Building formed by two 120-modules joined at their ends [Foster,

1986] .................................................................................................................. 33 Figure 2.42: Structure with 90-, 60- and 30-elements [Foster, 1986].......... 33 Figure 2.43: Temporary stage shell with 120 modules - Tension cables used to

provide bulkhead [Foster, 1986]................................................................. 34 Figure 2.44: Double curvature variable shape (hyperbolic type), plane pattern

[Tonon, 1993] ................................................................................................... 34 Figure 2.45: Fold pattern with different individual plate angles, but a constant

sum throughout the plate geometry, guaranteeing full foldability.35 Figure 2.46: Doubly curved folded shapes [Tonon, 1993] ....................................... 35 Figure 2.47: Linear and circular deployable double curvature folded shapes

[Tonon, 1993] ................................................................................................... 36 Figure 2.48: Folding aluminium sheet roof for covering the terrace of the pool

area of the International Center of Education and Development in Caracas, Venezuela [Hernandez & Stephens, 2000] ............................ 36

Figure 2.49: (a) Fold pattern; (b) Fold pattern with alternate rings to prevent relative rotation during deployment [Barker & Guest, 1998] ........... 37

Figure 3.1 : Translational and polar scissor unit ........................................................ 40 Figure 3.2: An ellipse as the graphic representation of the deployability

constraint for translational units, determining the locus of the intermediate hinge ......................................................................................... 41

Figure 3.3: A circle as the graphic representation of the deployability constraint for polar units, determining the locus of the intermediate hinge ................................................................................................................... 41

Figure 3.4: Circular arc as base curve, determined by the design values rH (rise) and S (span) .......................................................................................... 42

Figure 3.5: The arc is divided in equal angular portions. Circles intersecting the arc determine the loci of the intermediate hinge points ................... 44

Figure 3.6: The arc is divided in unequal angular portions. Variable circles intersecting the arc determine the loci of the intermediate hinge points.................................................................................................................. 46

Figure 3.7: An inner and outer arc determine the constant unit thickness....... 47

308

Figure 3.8: For the same span and rise, a pluricentred arc can offer increased headroom compared to a single-centred arc......................................... 48

Figure 3.9: Example of a pluricentred base curve consisting of three arc segments with decreasing radius............................................................... 49

Figure 3.10: Each arc segment (with a different radius) is divided in equal angular portions. Identical circles ensure a constant bar length..... 51

Figure 3.11: The original and double ellipse representing the deployability constraint intersection points M and M are the midpoints of the unit thickness t ................................................................................................ 53

Figure 3.12: Double ellipses impose the deployability constraint on a translational linkage with constant unit thickness.............................. 53

Figure 3.13: Two differently sized, but compatible ellipses representing the deployability constraint intersection points M and M are the midpoints of the unit thickness t1 and t2 ................................................. 55

Figure 3.14: Ellipses of different scale determine the location of the intermediate hinges on the base curve to form a translational linkage with varying unit thickness .......................................................... 56

Figure 3.15: The parameters used in the description of the geometry of the circular arc: rise (Hr) and span (S).............................................................. 57

Figure 3.16: Parameters needed for the geometric design of a polar linkage .. 60 Figure 3.17: Parameters for the geometric design of a translational linkage

with four units (U=4), of which two are shown.................................... 63 Figure 3.18: The relation between the original and the double ellipse in terms

of semi-axes a and b and the unit thickness t ...................................... 64 Figure 3.19: Translational linkage with U=2 fitted on a parabolic base curve. 67 Figure 3.20: Screenshot of interactive geometry file in Cabri Geometry II

[2007] software for designing arbitrarily curved translational linkages with constant unit thickness (base curve marked in black).............................................................................................................................. 69

Figure 3.21: Deployable landscape consisting of one arbitrarily curved translational linkage repeated in an orthogonal grid. Linkage designed using the interactive geometry tool (Aluminium, 4.5 m x 3 m, (photo: courtesy of Wouter Decorte).................................................. 69

Figure 3.22: Possible shapes for three-dimensional stress-free deployable structures, which can be designed using the tools presented .......... 70

309

Figure 3.23: Two-way grid with directions A and B.................................................. 71 Figure 3.24: Three-way grid with directions C, D and E .......................................... 71 Figure 3.25: Linear elements prismatic columns arches .................................. 72 Figure 3.26: Parallel linear structures connected by non-deployable elements

.............................................................................................................................. 73 Figure 3.27: Plane translational units on a two-way grid ...................................... 74 Figure 3.28: Plane translational units on a three-way grid.................................... 75 Figure 3.29: Plane translational units on a four-way grid..................................... 76 Figure 3.30: Plane and curved translational units on a two-way grid................ 77 Figure 3.31: Polar and translational units on a two-way grid............................... 78 Figure 3.32: Plane and curved translational units on a three-way grid ............. 79 Figure 3.33: Polar units on a three-way grid (variation 1) ..................................... 80 Figure 3.34: Polar and translational units on a three-way grid (variation 2) ... 81 Figure 3.35: Translational units on a two-way grid (synclastic shape)............... 82 Figure 3.36: Two variations for translational units on a two-way grid .............. 83 Figure 3.37: Translational units on a lamella grid ..................................................... 84 Figure 3.38: Polar units on a lamella grid .................................................................... 85 Figure 4.1: Typical foldable plate structure ................................................................. 88 Figure 4.2: Fold patterns of type A and B for the smallest possible regular

structure (p=5)................................................................................................. 89 Figure 4.3: Unfolded and fully folded configuration of patterns A and B (p=5)

.............................................................................................................................. 90 Figure 4.4: Elevation view of the compactly folded and fully deployed

configuration for a regular structure with five plates and an apex angle of 120.................................................................................................... 90

Figure 4.5: Right-angled fold pattern: altering one apex angle to 90 enables a compacter folded configuration (introduction of quadrangular plates near the sides)..................................................................................... 91

Figure 4.6: Elevation view of compactly folded and fully deployed configuration for a right-angled structure with five plates and an apex angle of 120 ......................................................................................... 91

Figure 4.7: Three stages of deployment for a basic regular foldable structure (p=7; =120): completely unfolded, erected position and fully compacted for transport............................................................................... 92

310

Figure 4.8: Plate element, compactly folded configuration and fully deployed configuration (front elevation) for the first three compactly foldable structures (p=5, p=7, p=9)........................................................................... 93

Figure 4.9: Side elevation of the fully deployed configuration of the first three compactly foldable structures (p=5, p=7, p=9)..................................... 94

Figure 4.10: For a chosen number of panels p the apex angle can be altered at will, affecting the width of the structure and the compactly folded state....................................................................................................... 95

Figure 4.11: Parameters used to characterise a foldable structure: length L, span S, width W, apex angle and the deployment angle ............ 96

Figure 4.12: A foldable plate and its parameters: length L, height H, H1, H2, apex angle , the deployment angle and angles , 1, 2 .............. 96

Figure 4.13: Perspective view and side elevation of the vertical projection of a plate linkage for empirically determining the relationship between 1 and p ............................................................................................................. 98

Figure 4.14: The relationship between the apex angle and the deployment angle for regular structures with p=5, p=7 and p=9....................... 99

Figure 4.15: Elevation view and perspective view of the deployment of a regular five-plate structure with =120..............................................100

Figure 4.16: The parameters associated with the polygonal contour of the flatly folded configurations with p=5, p=7 and p=9 and the expressions for the area in terms of the edge length Ledge................102

Figure 4.17: The relationship between the apex angle and the deployment angle for right-angled structures with p=5, p=7 and p=9...........105

Figure 4.18: Plate element, fold pattern, compactly folded configuration and fully deployed configuration (front elevation and side elevation) for three compactly foldable five-plate right-angled structures (drawn to scale)............................................................................................................106

Figure 4.19: Only for p=5 can any regular and any right-angled structure be interconnected along a common edge, regardless of the value for ............................................................................................................................107

Figure 4.20: Top view and perspective view of circular foldable structure .....108 Figure 4.21: Fold pattern and a single sector of a circular structure with q=8

............................................................................................................................108

311

Figure 4.22: Horizontal projection of a plate linkage for empirically determining the relationship between 2 and q ..................................108

Figure 4.23: Connecting a regular module with two half-domes leads to an alternative fully closed configuration with high plate uniformity110

Figure 4.24: Circular structure with q=6, q=8 and q=10 (top view) and its respective combination with a compatible regular structure (perspective view) .........................................................................................111

Figure 4.25: Some examples of alternative configurations ..................................112 Figure 5.1: Some of the concepts for mobile structures presented in the

following chapters........................................................................................115 Figure 5.2: Front elevation view of cases studies shows the mutual similarity of

the geometry. Case study 1, 2 and 3 are based on the same shape (semicircle with radius of 3 m).................................................................117

Figure 5.3: Overall geometry for the case studies: single curvature shape (open) and double curvature shape (closed) ......................................................118

Figure 5.4: Perspective view of the single and double curvature geometries.118 Figure 5.5: Perspective view and side elevation of case study 4 ........................120 Figure 5.6: Wind and snow action on the open and closed structure...............122 Figure 5.7: Schematic representation of considered wind loads on the closed

and.....................................................................................................................125 Figure 5.8: Schematic representation of snow loads on the closed and open

structures ........................................................................................................127 Figure 5.9 : Method of accumulate damage [Eurocode 3, 2007] .......................131 Figure 6.1: Deployable barrel vault with translational units on a triangular

grid: scissor structure and tensile surface ............................................135 Figure 6.2: Plan view and perspective view of the same double curvature

structure with translational units on a quadrangular grid ..............137 Figure 6.3: Translational scissor module with only single units..........................138 Figure 6.4: Translational scissor module with a double unit................................138 Figure 6.5: Plan view, perspective view and side elevation of a planar structure

with a triangulated grid..............................................................................138 Figure 6.6: Plan view, perspective view and side elevation of a barrel vault with

a triangulated grid........................................................................................139

312

Figure 6.7: Perspective view and plan view of three different triangular modules............................................................................................................139

Figure 6.8: OPEN structure: perspective view and plan view...............................140 Figure 6.9: CLOSED structure: perspective view and plan view (double scissor

marked in red)................................................................................................141 Figure 6.10: Front elevation, top view and perspective view of a portion of the

barrel vault with four modules in the span: the projected versions (marked in red) of the scissor units U1 and U2 determine the real curvature .........................................................................................................142

Figure 6.11: Developed view of units U1, U2 and U3: graphic representation of the deployability condition by means of ellipses................................143

Figure 6.12: An ellipsoid representing the geometric deployability condition in three dimensions...........................................................................................144

Figure 6.13: Vertical section view of the small and big ellipsoid, imposing the geometric deployability condition ...........................................................144

Figure 6.14: A scissor linkage fitted on a circular curve, with all relevant design parameters and the global coordinate system.....................................145

Figure 6.15: Developed view of the scissor linkage from Figure 6.14, showing a chain of double ellipses ..............................................................................146

Figure 6.16: Perspective view of the scissor linkage from Figure 6.15 .............147 Figure 6.17: Perspective view, front elevation and top view of the deployment

process of the barrel vault with translational units OPEN structure............................................................................................................................150

Figure 6.18: Proof-of-concept model of (half of the) closed structure (aluminium, scale 1/10) ..............................................................................151

Figure 6.19: Two double scissors in partially (left) and fully deployed (right) position ............................................................................................................151

Figure 6.20: Perspective view, front elevation and top view of the deployment process of the barrel vault with translational units CLOSED structure ..........................................................................................................152

Figure 6.21: From scissor mechanism to the equivalent hinged plate linkage for mobility analysis of the open structure (idem for closed structure) - minimal constraints .....................................................................................153

Figure 6.22: Fixing all lower nodes to the ground by pinned supports.............154

313

Figure 6.23: An active cable (marked in red) runs through the mechanism, connecting upper and lower nodes along its path. After deployment it is locked to stiffen the structure..........................................................154

Figure 6.24: Top view and perspective view of one scissor unit, its intermediate hinge and its end joints and their offset position relative to the theoretical plane ...........................................................................................156

Figure 6.25: Concept for an articulated joint, allowing the fins which accept the bars to rotate around a vertical axis, to cope with the angular distortion of the grid ...................................................................................156

Figure 6.26: Partially and undeployed state: as the structure is compactly folded, the imaginary intersection point of the centrelines travels on the vertical centreline through the joint.........................................157

Figure 6.27: Perspective view and top view of OPEN structure with integrated tensile surface................................................................................................158

Figure 6.28: Perspective view and top view of CLOSED structure with integrated tensile surface...........................................................................158

Figure 6.29: Top view and perspective view of the skeletal scissor structure (left) and the boundary geometry for the compatible membrane (right)................................................................................................................159

Figure 6.30: Views of the equilibrium form for the membrane...........................160 Figure 6.31: Typical stresses in the membrane range from 4 to 5.5 kN/m ......160 Figure 6.32: FEM-model of six bars attached to a node........................................162 Figure 6.33: An intermediate pivot hinge connects two scissor bars................162 Figure 6.34: Local coordinate system of a bar element (left) and global

coordinate system (right) ...........................................................................162 Figure 6.35: Typical pattern of load vectors for transverse wind + pre-stress of

the membrane................................................................................................163 Figure 6.36: Typical pattern of reaction forces under transverse wind.............163 Figure 6.37: Bending moments My under transverse wind ..................................163 Figure 6.38: Typical deformation under transverse wind: .....................................164 Figure 6.39: Perspective view of the resulting structure with rectangular

sections of 120x60mm................................................................................166 Figure 6.40: Reactions in the global coordinate system: the maximal reaction

force occurs under ULS 2 (pre-stress + snow + transverse wind) .167

314

Figure 6.41: The critically loaded bar is located at the top. Summary of the stresses occurring in the critically loaded bar (positive stresses indicate pressure, negative values mean tension) ..............................167

Figure 6.42: Axial forces, transverse forces and bending moments in the local coordinate system of the bars ..................................................................168

Figure 6.43: Maximal nodal displacements in the global coordinate system.169 Figure 6.44: Continuous cable zigzagging through the structure, connecting

upper and lower nodes and contributing to the structural performance ...................................................................................................170

Figure 6.45: Resulting structure after optimization, with cable elements ......170 Figure 6.46: Summary of the determining stresses and forces for the strength,

stability and stiffness of case study 1: OPEN structure....................171 Figure 6.47: Perspective view of case study 1: CLOSED structure: with sections

after structure design and total weight.................................................173 Figure 6.48: Summary of the determining parameters for the strength, stability

and stiffness of case study 1 _ CLOSED structure..............................174 Figure 6.49: Case study 1: Single curvature OPEN structure (barrel vault) .....175 Figure 6.50: Case study 1: Double curvature CLOSED structure ........................176 Figure 7.1: Deployable barrel vault with polar units on a quadrangular grid:

scissor structure and tensile surface ......................................................177 Figure 7.2: Plan view and perspective view of a planar structure with a

quadrangular grid .........................................................................................179 Figure 7.3: Plan view and perspective view of a barrel vault with quadrangular

grid ....................................................................................................................179 Figure 7.4: Series of polar linkages with 3, 4, 5 or 6 units in the span,............180 Figure 7.5: Geometric construction of the four-unit linkage...............................181 Figure 7.6: OPEN structure: perspective view and top view.................................181 Figure 7.7: Perspective view and developed view of units U1 (plane

translational) and U2, U3 (polar): graphic representation of the deployability condition by means of ellipses........................................182

Figure 7.8: Adding an end structure based on parallels and meridians to the main structure ...............................................................................................184

Figure 7.9: A lamella dome has a stress-free deployment ....................................184

315

Figure 7.10: The main structure is provided with half of an adapted lamella dome .................................................................................................................185

Figure 7.11: CLOSED structure: perspective view and plan view........................185 Figure 7.12: Perspective view, front elevation and top view of the deployment

process of the polar barrel vault OPEN structure............................186 Figure 7.13: Perspective view, front elevation and top view of the deployment

process of the polar barrel vault CLOSED structure .......................187 Figure 7.14: Proof-of-concept model (half of the structure) in three

deployment stages........................................................................................187 Figure 7.15: Deployment sequence of a polar linkage ...........................................189 Figure 7.16: Polar linkage in an intermediate deployment stage: 10

316

Figure 7.31: Improved result by inserting vertical cable ties ...............................200 Figure 7.32: Additional diagonal bars triangulate the grid...................................201 Figure 7.33: Double diagonal cross bars offer no real advantage structurally

............................................................................................................................202 Figure 7.34: Perspective view of case study 2 OPEN structure, with sections

after structure design and weight/m2.....................................................203 Figure 7.35: Summary of the determining parameters for the strength, stability

and stiffness for case study 2 OPEN structure ....................................204 Figure 7.36: Main structure and additional end structures with no additional

measures to improve structural performance......................................205 Figure 7.37: Perspective view of case study 2 CLOSED structure:, with resulting

sections after structure design and total weight................................206 Figure 7.38: Summary of the results for the structural analysis of case study 2

............................................................................................................................207 Figure 7.39: Case 2: OPEN structure............................................................................208 Figure 7.40: Case 2: CLOSED structure .......................................................................209 Figure 8.1: Foldable bar structure based on the geometry of foldable plate

structures ........................................................................................................211 Figure 8.2: Typical foldable plate structure ...............................................................213 Figure 8.3: Design parameters for a basic regular foldable plate structure. ...214 Figure 8.4: For a chosen number of panels p the apex angle can be altered

at will, only affecting the width of the structure...............................215 Figure 8.5: Graph showing the relation between the deployment angle and

the apex angle in the fully deployed configuration for p=5 ......216 Figure 8.6: The resulting regular geometry for the case study: two extreme

deployment states and the fold pattern ................................................217 Figure 8.7: Top view and a perspective view of a circular plate geometry with

six sectors arranged radially......................................................................217 Figure 8.8: The resulting circular geometry for the case study: two extreme

deployment states and the fold pattern ................................................218 Figure 8.9: A combination of a regular and a circular geometry........................218 Figure 8.10: Dimensions in plan view of the shapes...............................................219 Figure 8.11: A foldable plate structure (p=7) and its similar counterpart, a

foldable bar structure..................................................................................219

317

Figure 8.12: Pattern 1: double bars present ..............................................................220 Figure 8.13: Pattern 2: double bars removed ............................................................220 Figure 8.14: Pattern 3: double bars and diagonal bars removed, without

affecting the original kinematic behaviour ..........................................221 Figure 8.15: Foldable 3 D.O.F.-joint derived directly from the fold pattern,

therefore mimicking its kinematic behaviour ......................................221 Figure 8.16: Deployment sequence for the foldable joint: from the undeployed

to the fully deployed position ...................................................................222 Figure 8.17: The (regular) open structure complete with bars and joints:.......222 Figure 8.18: Detailed view of bars and three variations of foldable joints

occurring in the structure ..........................................................................223 Figure 8.19: Deployment sequence for the open structure perspective view,

front elevation and top view.....................................................................224 Figure 8.20: Proof-of-concept model of the regular structure (with scissors) in

four stages of the deployment..................................................................224 Figure 8.21: Deployment sequence for the dome structure perspective view,

front elevation and top view.....................................................................225 Figure 8.22: Proof-of-concept model of the foldable dome (with additional

scissor units) in six deployment stages..................................................225 Figure 8.23: Deployment sequence for the closed structure: 1 regular module +

2 semi-domes.................................................................................................226 Figure 8.24: Six stages in the deployment of the closed structure (top view)227 Figure 8.25: Kinematic joint allowing all necessary rotations (3 D.O.F.) and the

resulting bar structure Proof-of-concept model to verify the mobility ............................................................................................................228

Figure 8.26: Integration of the membrane beforehand by attaching it to the nodes Side elevation and perspective view of the undeployed and deployed position..........................................................................................229

Figure 8.27: Right-angled geometry with its own set of joints ..........................230 Figure 8.28: Deployment sequence of a concept model of a right-angled

structure with aluminium bars and resin connectors [De Temmerman, 2006a] ....................................................................................230

Figure 8.29: Several regular and right-angled structures connected together after deployment...........................................................................................231

Figure 8.30: The two loops and their common fold line........................................233

318

Figure 8.31: A foldable open structure with a compatible integrated scissor linkage one bar of each scissor unit doubles up as an edge the foldable bar structure..................................................................................234

Figure 8.32: Top view and perspective view of the finite element model of the foldable joint from Figure 8.15 (hinges are represented by dashed lines)..................................................................................................................235

Figure 8.33: Model with the middle bars in the rhombus-shaped modules still present..............................................................................................................236

Figure 8.34: Same model as in Figure 7.33, but with cross-bars........................236 Figure 8.35: Bars are grouped in pairs and joined by a fixed connection in their

apex angle.......................................................................................................237 Figure 8.36: Adding struts again only increases the weight, while the section

remains identical...........................................................................................237 Figure 8.37: Summary of the determining parameters for the strength, stability

and stiffness for case study 3 OPEN structure ....................................238 Figure 8.38: Resulting section and weight for the foldable dome .....................239 Figure 8.39: Perspective view of case study 3 CLOSED structure with sections

after structure design and total weight.................................................240 Figure 8.40: Summary of the determining parameters for the strength, stability

and stiffness for case study 3 CLOSED structure ................................241 Figure 8.41: Case 3 OPEN structure .............................................................................242 Figure 8.42: Case 3 Foldable DOME structure ..........................................................243 Figure 8.43: Case 3 CLOSED structure.........................................................................244 Figure 9.1: Design concept for a tensile surface structure with a deployable

central tower..................................................................................................245 Figure 9.2: Mobile structure with membrane surfaces arranged around a

demountable central tower ( The Nomad Concept).........................248 Figure 9.3: The top of the tower is accessible to visitors, allowing them to

enjoy the view................................................................................................249 Figure 9.4: Side elevation of the tower and canopy ...............................................250 Figure 9.5: Top view of the structure showing the three tensile surfaces

arranged radially around the central tower .........................................250 Figure 9.6: Dimensions of the tower and a single angulated bar .......................251

319

Figure 9.7: Comparison between a linkage with angulated SLEs and its polar equivalent........................................................................................................252

Figure 9.8: Imposed condition on the length of the semi-bars a and b (a

320

Figure 9.23: Three stages in the deployment of a hexagonal tower with 5 modules: elevation and top view .............................................................276

Figure 9.24: Hyperboloid geometry (as proposed in previous sections) angulated elements do not remain coplanar during deployment..278

Figure 9.25: Prismoid geometry (simplified alternative to the previously described geometry) - angulated elements remain coplanar during deployment .....................................................................................................279

Figure 9.26: Non-symmetrical identical angulated elements result in a fully compactable configuration: hyperboloid solution ..............................279

Figure 9.27: Symmetrical identical angulated elements cannot be fully compacted.......................................................................................................280

Figure 9.28: Symmetrical and non-identical angulated elements result in a fully compactable configuration: prismoid solution ..........................281

Figure 9.29: Symmetrical and non-identical angulated elements result in a fully compactable configuration: prismoid solution ..........................282

Figure 9.30: Three consecutive stages in the deployment of a prismoid geometry..........................................................................................................282

Figure 9.31: Three consecutive stages of the corresponding planar closed-loop structure ..........................................................................................................283

Figure 9.32: Perspective view of the deployment of a triangular tower ..........283 Figure 9.33: Top view and side elevation of the prismoid tower ........................284 Figure 9.34: Detailed view of the simplified hinge connecting four scissor bars

............................................................................................................................284 Figure 9.35: Triangular and quadrangular prismoid solution and their

respective equivalent hinged-plate structure, providing an insight in the kinematic behaviour .............................................................................285

Figure 9.36: Top view and perspective view of the structure with indication of the global coordinate system and the vector components of the wind action .....................................................................................................287

Figure 9.37: Side elevation of the equilibrium form of the membrane.............288 Figure 9.38: Top view of the equilibrium form of the membrane.......................288 Figure 9.39: Horizontal cable ties to improve structural performance .............289 Figure 9.40: Perspective view, top view and side elevation of deployable mast

............................................................................................................................290

321

Figure 9.41: Summary of the determining parameters for the strength, stability and stiffness of case study 4.....................................................................291

Figure 9.42: Case 4 A temporary canopy and its deployable tower with angulated units..............................................................................................292

Figure 10.1: Case study 1 (Chapter 6) Translational barrel vault....................296 Figure 10.2: Case study 2 (Chapter 7) Polar barrel vault...................................297 Figure 10.3: Case study 3 (Chapter 8) Deployable bar structure with foldable

joints .................................................................................................................298 Figure 10.4: Case study 4 (Chapter 9) Deployable mast ....................................299

322

List of Tables Table 4.1: The first eight values for in terms of p for compactly foldable

regular structures............................................................................................ 93 Table 4.2: Minimum and maximum possible apex angles for regular structures

with 5, 7 or 9 plates.....................................................................................100 Table 4.3: The span S and rise R for a given number of plates p of regular

foldable structures in terms of the plate length L..............................101 Table 4.4: The area of the compact configuration for (p=5, =90), (p=7,

=120) and (p=9, =135) in terms of the plate length L .............102 Table 4. 5: The area of the sectional profile of the deployed configuration for

(p=5, =90), (p=7, =120) and (p=9, =135) in terms of the plate length L ............................................................................................................103

Table 4. 6: The expansion ratio for (p=5, =90), (p=7, =120) and (p=9, =135) ............................................................................................................103

Table 4.7: Minimum and maximum possible apex angles for right-angled structures with 5, 7 or 9 plates, as can be read from the graph in Figure 9 ............................................................................................................105

Table 4.8: Values for and for a chosen q (circular structure), combined with a regular structure (p=5) ............................................................................110

Table 5.1: Values for the wind pressure w per zone................................................126 Table 5.2: The seven load cases used for calculations in EASY............................128 Table 9.1: Characteristics of the hyperboloid geometry ........................................286 Table 9.2: Characteristics of the prismoid geometry ..............................................286 Table 9.3: Load combinations for wind and snow ...................................................287

List of Symbols Chapter 2 Deployment angle p.10 a, b Semi-bars p.11 Unit angle p.11 a, b, c, d Semi-lengths p.12 Kink angle p.21 Angle p.21

Chapter 3 Deployment angle p.40 Unit angle p.40 a, b, c, d Semi-lengths p.40 M Intermediate hinge P.41 Hr Rise p.42 S Span p.42 t Unit thickness p.43 2 Total unit angle p.43 O Centrepoint p.43 M Centrepoint p.43 C Intermediate point p.43 M Centrepoint p.43 t1, t2 Unit thickness p.45 O1, O2,,O3 Centrepoint p.48 P, Q, S, T End nodes p.51 t Unit thickness p.54 K Intermediate hinge p.54 Quarter of total sector angle p.57

n Angle p.57 nP Base point of arc p.57

0P Apex point of arc p.57 O Centrepoint p.57

inR Internal radius p.57

U Number of units p.59 Sector angle p.59

eR External radius p.60

L Bar lenght p.60 E0, E1 Ellipse p.63

Chapter 4 P Basic plate element p.88 M Module p.88 p Number of plates p.88 Apex angle p.89 L Plate length p.95 W Module width p.95 Hr Rise p.95 S Span p.95 Deployment angle p.95 H Plate height p.96 H1 Horizontal projection of plate height p.96 H2 Vertical projection of plate height p.96 , 1, 2 Angle p.96 Ledge Edge length p.101 L Plate length p.102 Expansion ratio p.103 tp Thickness of a single plate element p.103 Tp Total thickness of the compactly folded configuration p.104 q Number of sectors p.107

Chapter 5 t Unit thickness p.117 Air density p.123

refv Reference velocity p.123

ALTc Altitude factor p.123

DIRc Direction factor p.123

TEMc Temporary factor p.123

refq Reference wind pressure p.123

w Total wind pressure p.123

we Pressure on the external surfaces p.123 wi Pressure on the internal surfaces p.123 Opening ratio p.123 AL,W Total area of openings at the leeward and wind

parallel sides p.124

AT Total area of openings at the windward, leeward and wind parallel sides

p.124

Cpi Internal pressure coefficient p.124 Cpe External pressurecoefficient p.124 Cpi,a Permeability p.124

ks Characteristic snow load on the ground p.127

tC Temperature coefficient p.127

eC Exposure coefficient p.127

i Form factor for the snow load p.127 G Permanent loads p.128 Q Mobile loads p.128 Safety factor p.128 D Damage p.130 ni Number of cycles p.130 Ni Critical amount of load cycles p.130

i Fluctuating stresses p.130 c Resistance against fatique p.130 i Shear stresses p.132

Chapter 6 M1 Plane module p.139 M2 Slightly curved module p.139 M3 Highly curved module p.139 U1, U2, U3 Linkage p.142 t Unit thickness p.144 a, b Semi axes p.144 U Number of units in the span p.145 R Radius of the circular arc p.145 2 Angle p.145 A , A Circular arc p.145

2a Distance between parallel arcs

p.145

P2, P0, 1P , P1, P2

Intersection point p.145

E0, E1 Ellipsoid p.146 Angle p.146 n Node p.152 2, 3 Angle p.155 fy Yield stress p.164 Smax Maximum stress p.168

Chapter 7 U Units p.180 O Centrepoint p.180 P, Q, R Intersection points p.180 h Unit height p.180 U1, U2, U3 Linkage p.181 S Span p.188 Hr Rise p.188 t Unit thickness p.188 a, b Semi-bar p.188 Deployment angle p.188

maxS Deployment angle for which the maximum span is reached

p.188

design Deployment angle in the fully deployed configuration

p.188

Smax Maximum span p.188 Sdesign Span of the deployed configuration p.188 Deployment ratio p.189

inR Internal radius p.189

eR External radius p.189 Unit angle p.189 Sector angle p.189 Se External span p.192

Chapter 8 p Number of plates p.214 Apex angle p.214

Deployment angle p.214 L Plate length p.214 S Span p.214 W Module width p.214 q The amount of sectors arranged radially p.216 m Number of modules p.231 R Degree of statical determinacy p.232 b Number of bars p.232 j Number of joints p.232 r Number of restraints p.232 Njoints Number of continuous joints p.233 Nlinks Total number of links p.233 Nloops Number of loops p.233

Chapter 9 a, b Semi-bar length p.253 Kink angle p.255 U Number of scissor units p.255 n Number of modules p.255 E Edge length p.255 Sector angle p.255 Deployment angle p.257 h Height of the undeployed position p.257 H Total height p.257 Deployment ratio p.258 R Radius p.259 h Unit height p.261 L Base length p.261

Chapter 1 Introduction

1

Chapter 1

Introduction

1.1 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 con-figuration. These are generally referred to as deployable structures though they might also be known as erectable, expandable, extendible, developable or un-furlable structures [Jensen, 2003]. Although the research subject of deployable structures is relatively young be-ing pioneered in the 1960s, the principle of transformable objects and spaces has been applied throughout history. Applications range from the Mongolian yurts, to the velum of the Roman Coliseum, from Da Vincis umbrella to the folding chair. At present day, the main application areas are the aerospace industry, requiring highly compactable, lightweight payload and architecture, requiring either mobile, lightweight temporary shelters or fixed-location re-tractable roofs for sports arenas. Mobile shelter systems are a type of building construction for which there is a vast range and diversity of forms and structural solutions. They are designed to provide weather protected enclosure for a wide range of human activities. The main applications are exhibition and recreational structures, temporary build-ings in remote construction sites, relocatable hangars and maintenance facili-ties and emergency shelters after natural disasters. Enclosure requirements are generally very simple, with the majority needing only a weather protecting membrane or skin supported by some form of erectable structure. In all appli-cations, both the envelope and structure need to be capable of being easily moved in the course of normal use, which very often requires the building sys-tem to be assembled at high speed, on unprepared sites [Burford & Gengnagel, 2004]. An example of an easily erectable temporary exhibition structure is shown in Figure 1.1.


2

Figure 1.1: Mobile deployable bar structure ( Grupo Estran)

Mobile deployable structures have the advantage of ease and speed of erec-tion compared to traditional building forms. Because they are reusable and easily transportable, they are of great use for temporary applications. However, the aspect of deployability is associated with a higher mechanical complexity and design cost compared to conventional systems. This increased cost has to be balanced by the structures potential to be suitable for the particular appli-cation. Deployable structures can be classified according to their structural system. In doing so, four main groups can be distinguished:

Spatial bar structures consisting of hinged bars Foldable plate structures consisting of hinged plates Tensegrity structures Membrane structures

It is noted that these deployable structural systems only constitute a portion of the possible applications in their respective field. The majority of spatial bar structures, plate structures, tensegrity structures and membrane structures is non-deployable and has a permanent location. What is referred to here are those specific applications which exhibit a certain ability to transform their shape, therefore adapting to changing circumstances and requirements. Hanaor [2001] has classified the aforementioned structural systems used in deployable structures by their morphological and kinematic characteristics (Figure 1.2). Because of their wide applicability in the field of mobile architec-ture, their high degree of deployability and a reliable deployment, two sub-categories will be studied in greater detail (marked in red in Figure 1.2):


3

Figure 1.2: Classification of structural systems for deployable structures by their morphological

and kinematic characteristics [Hanaor, 2001]


4

scissor structures and foldable plate structures. Scissor structures are expand-able structures consisting of bars linked together by scissor hinges allowing them to be folded into a compact bundle. Although many impressive architec-tural applications for these mechanisms have been proposed, due to the me-chanical complexity of their systems during the folding and deployment proc-ess, few have been constructed at full-scale [Asefi, 2006]. Foldable plate structures consist of rigid plate elements which are connected by continuous joints allowing one rotational degree of freedom. In their unde-ployed configuration they form a flat stack of plates, while a corrugated sur-face is formed in their fully deployed configuration. Singly curved as well as doubly curved surfaces are possible, characterised by a linear or radial deploy-ment.

1.2 Aims and scope of research Although many different deployable systems have been proposed, few have successfully found their way into the field of temporary constructions. A cause for this limited use can be found in the complexity of the design process. This entails detailed design of the connections which ensure the expansion of the structure during the deployment process. Therefore, not only the final de-ployed configuration is to be designed, but an insight is required in the mobil-ity of the mechanism, as a means to achieve that final erected state. Also, de-signing deployable structures requires a thorough understanding of the spe-cific configurations which will give rise to a fully deployable geometry. The aim of the work presented in this dissertation is to develop novel concepts for deployable bar structures and propose variations of existing concepts which will lead to architecturally as well as structurally viable solutions for mobile applications. It is the intention to aid in the design of deployable bar structures by first explaining the essential principles behind them and subse-quently applying these in several case studies. Starting with the choice of a suitable geometry, followed by an assessment of the kinematics of the system, to end with a structural feasibility study, the complete design process is dem-


5

onstrated. By doing so, the strengths and weaknesses of the chosen structural system and geometric configuration, are exposed. Ultimately, the designer is provided with the means for deciding on how to cover a space with a rapidly erectable, mobile architectural space enclosure, based on the geometry of foldable plate structures or employing a scissor system. A review of previous research concerning scissor structures and foldable plate structures is given, offering an insight in the wide variety of possible shapes and configurations. An understanding of their geometry is crucial, because it greatly influences the deployment behaviour of the structure. The design prin-ciples behind these structures and several construction methods are explained and novel geometric design methods are proposed, based on architectural pa-rameters such as the rise and span of the structure. These principles are then used in four case studies, which cover the key aspects of the design and are an application of novel proposed concepts for mobile deployable bar structures.

1.3 Outline of thesis In Chapter 2, previous work and a literature review of scissor structures and foldable plate structures is presented and the main researchers active in this field are discussed. The first part focuses on translational and polar scissor units employed in spatial structures and angulated elements applied in closed loop retractable structures. The second part is concerned with past develop-ments within the field of foldable plate structures and their possible configu-rations. In Chapter 3 the basic principles needed for the design of deployable scissor structures are clarified. As a simple means of obtaining a deployable scissor linkage, several construction methods for translational and polar arches are explained. A geometric design method is proposed, for which the derived equations are based on the rise and span of the deployed configuration. This method allows the design of polar linkages of circular curvature and transla-tional linkages of any curvature. It is shown how these can be used to obtain three-dimensional grid structures which are stress-free deployable.


6

Chapter 4 is concerned with the design of foldable plate structures. Some ba-sic single curvature or double curvature foldable configurations are identified which are compactly foldable for maximum transportability. The formulas needed for designing single curvature and double curvature configurations are derived. It is shown that a single plate element can be obtained from which domes and barrel vaults or combinations thereof can be composed. These de-sign principles are applied in Chapter 8, in which a concept for a deployable bar structure is proposed based on a foldable plate geometry. Chapter 5 serves as an introduction to the case studies which will bring into practice the design methods discussed in Chapters 3 and 4. The geometry for the case studies is presented as well as the general approach for the structural analysis and design. Also, the considered load combinations are discussed. In Chapter 6 case study 1 is designed, which is a novel type of single curvature deployable structure composed of translational units on a three-way grid. A geometric design approach is proposed which is then brought into practice for designing a translational triangulated barrel vault with a circular base curve. Also, based on this barrel vault, a fully closed double curvature shape is pro-posed as an alternative configuration. An insight is provided in the kinematic behaviour during and after the deployment. The concept is structurally ana-lysed according to the method specified in Chapter 5. In Chapter 7 a barrel vault with polar and translational units is designed. For the second case study a novel way of providing an open barrel vault with a compatible stress-free deployable end structure is proposed, making use of half of a slightly modified lamella dome. Analogous to case study 1, the kine-matics of the system are discussed and a structural analysis is performed. In Chapter 8 an innovative concept for a mobile shelter system, based on the kinematics of foldable plate structures, is proposed. For case study 3 a basic foldable barrel vault, as well as a foldable dome are designed, based on the principles presented in Chapter 4. By combining these two basic shapes a closed doubly curved foldable geometry is obtained. The transition from plate


7

structure to bar structures is discussed and a novel foldable articulated joint, serving as a connector for the bars, is proposed. The mobility of the mecha-nism is discussed and the concept is analysed structurally. Chapter 9 is concerned with the design of case study 4, which is a deployable tower with angulated scissor units. In the proposed concept the structure serves as a tower or truss-like mast for a temporary tensile surface structure and doubles up as an active element during the erection process. A compre-hensive geometric design method is proposed and the influence of the design parameters on the geometry and the deployment process are discussed. Finally, the kinematic behaviour is explained and the structural feasibility is checked. Chapter 10 concludes the study by discussing the proposed concepts in a comparative evaluation. Also, a number of suggestions for further work are provided.


8

Chapter 2 Review of Literature

9

Chapter 2

Review of Literature

2.1 Introduction In this chapter the main contributors to the field of deployable structures are discussed. A review is given of existing deployable scissor structures (or panto-graph structures) and foldable plate structures for architectural applications. The first part is concerned with an explanation of the characteristics of trans-lational and polar units, and the deployability condition they have to comply with when used in a scissor linkage, in order to guarantee deployability. Fur-ther, angulated elements, which are used to form closed loop structures, are discussed. These are characterised by a radial deployment, allowing the struc-ture to retract towards its perimeter. The second part discusses the application of foldable plate structures, includ-ing single and double curvature configurations. An explanation is given of the possible plate linkages which generate compactly foldable configurations. Also, the condition which foldable plate configurations have to satisfy in order to be compactly foldable is mentioned.

2.2 Deployable structures based on pantographs Scissor units, otherwise called scissor-like elements (SLEs) or pantographic elements, consist of two straight bars connected through a revolute joint, called the intermediate hinge, allowing the bars to pivot about an axis perpen-dicular to their common plane (Figure 2.1). By interconnecting such SLEs at their end nodes using revolute joints, a two-dimensional transformable linkage is formed, as shown in Figure 2.2. Altering the location of the intermediate hinge or the shape of the bars gives rise to three distinct basic unit types: translational, polar and angulated units.


10

2.2.1 Translational units

The upper and lower end nodes of a scissor unit are connected by unit lines. For a translational unit, these unit lines are parallel and remain so during de-ployment. In Figure 2.1 a plane and a curved translational unit are shown, the plane unit being the simplest translational unit having identical bars. When these units are linked, a well-known transformable single-degree-of-freedom mechanism is formed, called a lazy-tong, shown in Figure 2.2.

Figure 2.1 : Translational units

The curved unit named such because it is commonly used for curved linkages has bars of different length. When the latter is linked by its end nodes, a curved linkage is formed, pictured in Figure 2.3. By varying the deployment angle a linkage is transformed from its most compact configuration (a compact bundle) to its fully deployed position, as shown in Figure 2.2 and Figure 2.3.

Plane unit Curved unit

Unit line

End node

Intermediate hinge


11

Figure 2.2: The simplest plane translational scissor linkage, called a lazy-tong

Figure 2.3: A curved translational linkage in its deployed and undeployed position

2.2.2 Polar units

When in a plane translational unit the intermediate hinge is moved away from the centre of the bar, a polar unit is formed with unequal semi-bars a and b (Figure 2.4). It is this eccentricity of the intermediate hinge which generates curvature during deployment. The unit lines intersect at an angle . This angle varies strongly as the unit deploys and the intersection point moves closer to the unit as the curvature increases. In Figure 2.5 a polar linkage is shown in its undeployed and deployed configuration.


12

Figure 2.4: Polar unit

Figure 2.5: A polar linkage in its undeployed and deployed position

2.2.3 Deployability constraint

Crucial to the design of deployable scissor structures is the deployability con-straint. This is a formula derived by Escrig [1985] which states that in order to be deployable, the sum of the semi-lengths a and b of a scissor unit has to equal the sum of the semi-lengths c and d of the adjoining unit. This trans-lates theoretically into the ability of the bars to coincide in the compact state. Practically, this means that the scissor linkage is foldable into a compact bun-dle of bars. For the linkage in Figure 2.6, the deployability constraint is written as:

dcba +=+ (2.1)

a

b


13

Figure 2.6: The deployability constraint in terms of the semi-lengths a, b, c and d of two adjoin-

ing scissor units in three consecutive deployment stages

It should be noted that scissor linkages which do not comply with Equation 2.1 can still be partially foldable: one unit might be fully compacted, while the adjoining unit might still be partially deployed. However, since this disserta-tion is concerned with the design of compactly foldable scissor structures, the deployability constraint is treated as a minimum requirement.

2.2.4 Structures based on translational and polar units

In the early 1960s, Spanish architect Emilio Perez Piero [1961, 1962] pio-neered the use of scissor mechanism to make deployable structures. He was among the first in modern times to employ the principle of the pantograph for use in deployable architectural structures, such as his moveable theatre (Figure 2.7). This particular model consisted of rigid bars and wire cables, which would become tensioned to provide the structure with the necessary stabilisation. The members remain unstressed in the compact, bundled configuration and the deployed state, except for their own dead weight. Furthermore, the struc-ture is stress-free during the deployment, effectively behaving like a mecha-nism. Piero was very productive in the field of deployable scissor structures, until all this was brought to an end by his tragic death in 1972. Another Spanish architect became one of the most prolific researchers on the subject. Felix Escrig [1984, 1985] presented the geometric condition for de-ployability (Section 2.2.3) and demonstrated how three-dimensional structures

a

b

c

d


14

could be obtained by placing scissor units in multiple directions on a grid. Fur-ther, it was shown how curvature could be introduced in such a grid by vary-ing the location of the intermediate hinge of the scissor units.

Figure 2.7: Piero demonstrates his prototype of a deployable shell [Robbin, 1996]

Escrig has also investigated, in collaboration with J. Sanchez and J.P. Valcarcel, spherical two-way scissor structures based on the subdivision of the surface of a sphere. These two-way grids require measures, such as cross-bars or cables, to stabilise the structure in its deployed configuration, due to in-plane insta-bility caused by non-triangulation. A myriad of geometric models has been proposed by Escrig [1985,1987] based on two-way and three-way grids with no curvature, single curvature or double curvature. An example of each category is given in Figures 2.8 to 2.12.


15

Figure 2.8: Planar two-way grid with translational units and cylindrical barrel vault with polar

units [Escrig, 1985]

Figure 2.9: Top view and side elevation of a two-way spherical grid with identical polar units

[Escrig, 1987]

Figure 2.10: Top view and side elevation of a three-way spherical grid with polar units

[Escrig, 1987]


16

Figure 2.11: Top view and side elevation of a geodesic dome with polar units [Escrig, 1987]

Figure 2.12: Top view and side elevation of a lamella dome with identical polar units

[Escrig, 1987]

Besides constructing several models, Escrig has also designed a cover for a swimming pool in Seville. The design consists of two identical rhomboid grid structures with spherical curvature. The subdivision of the spherical surface is executed in such a way, that straight edges emerge, allowing several struc-tures to be mutually connected along these edges (Figure 2.13).

Figure 2.13: Deployable cover for a swimming pool in Seville designed by Escrig & Sanchez (

Performance SL)


17

Some of the proposed geometric configurations for three-dimensional grid structures demonstrate a snap-through effect during the deployment. This means that they do not deploy as mechanisms and are no longer stress-free during expansion (apart from their own dead weight). This snap-through ef-fect is caused by geometric incompatibilities between the member lengths associated with the way they are contained within the grid. Because they are in a stress-free state before and after deployment, but go through an interme-diate stage with deployment induced stresses, they are called bi-stable de-ployable structures. Figure 2.14 illustrates the snap-through effect on a square module with diagonal units. The diagonal units (marked in red) are subject to elastic deformation in the intermediate deployment stage, while the unde-ployed and fully deployed configuration are stress-free.

Figure 2.14: Bi-stable structure before, during and after deployment

Figure 2.15: Collapsible dome and a single unit, as proposed by Zeigler [1976]


18

Zeigler [1981, 1984] was the first to exploit this phenomenon as a self-locking effect, effectively making extra stabilisation after deployment (which is neces-sary for stress-free deployable structures) obsolete. He proposed, on these grounds, a partial triangulated spherical dome as shown in Figure 2.15. Charis Gantes [1996, 2001] has thoroughly investigated bi-stable deployable structures and has developed a geometric design approach for flat grids, curved grids and structures with arbitrary geometry. Also, he has researched the structural response during deployment, which is characterized by geomet-ric non-linearities. Simulation of the deployment process is, therefore, an im-portant part of the analysis requiring sophisticated finite element modelling. The material behavior, however, must remain linearly elastic, so that no resid-ual stresses reduce the load bearing capacity under service loads. Two of his proposals for bi-stable structures, an elliptical arch and a geodesic dome, are depicted in Figure 2.16.

Figure 2.16: Bi-stable structures: elliptical arch and geodesic dome [Gantes, 2004]

A geometric and kinematic analysis of single curvature and double curvature structures has been performed by Travis Langbecker [1999, 2001]. He has used translational units to design several models of positive (Figure 2.17) and nega-tive (Figure 2.18) curvature structures. By using compatible translational units and by keeping the structural thickness (unit thickness) constant throughout the whole structure, these configurations are always stress-free deployable.


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Figure 2.17: Positive curvature structure with translational units in two deployment stages

[Langbecker, 2001]

Figure 2.18: Negative curvature structure with translational units in two deployment stages

[Langbecker, 2001]

Pantographic deployable columns are linear deployable structures composed of translational or polar units and were researched by Raskin [1996, 1998]. His work focussed on pantographs behaving as mechanisms during deployment, which are to be stabilised in the deployed configuration by additional bound-ary conditions. First, plane linkages were investigated, which were subse-quently used to form prismatic columns (Figure 2.19). Expanding his findings, deployable pantographic slabs that can be packaged in different arrangements were proposed. Figure 2.20 shows two variations of such a deployable slab, consisting either of prismatic modules or an arrangement of prismatic col-umns.


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Figure 2.19: Plane and spatial pantographic columns by Raskin [1998]

Figure 2.20: Pantographic slabs by Raskin [1998]

Under the guidance of Dr. Sergio Pellegrino, a research group called the De-ployable Structures Laboratory, emerged at the Cambridge University in 1990 as a driving force in the field of deployable structure research. One of their proposals constituted a deployable pantographic ring structure developed as the edge beam of a deployable antenna. Together with Zhong You, the condi-tions for strain-free deployment of such a structure were derived [You & Pellegrino, 1993]. Structures of this type consist of translational linkages on the perimeter ring and inner ring, mutually connected by radially placed polar units. As an example, Figure 2.21 shows a structure based on a twelve-sided polygon.


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Figure 2.21: Deployable ring structure [You & Pellegrino, 1993]

2.2.5 Angulated units

Unlike common pantograph units with straight bars, angulated units consist of two rigidly connected semi-bars of length a that form a central kink of ampli-tude . Because they were invented by Hoberman [1990] they are commonly denoted as hobermans units. The major advantage is that, as opposed to polar units, angulated units subtend a constant angle during deployment (Figure 2.22). For this to occur, the bar geometry has to be such that = /2. This im-plies that angulated elements can be used for radially deploying closed loop structures, capable of retracting to their own perimeter, which is impossible to accomplish with translational or polar units, which demonstrate a linear de-ployment. (Figure 2.23) shows a circular linkage with angulated elements in its undeployed and deployed configuration.

Figure 2.22: Angulated unit or hobermans unit

a

a

= /2


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Figure 2.23: A radially deployable linkage consisting of angulated (or hobermans) units in three

stages of the deployment

The structure shown in Figure 2.23 is formed by two layers of identical angu-lated elements, of which one layer is formed by elements in clockwise direc-tion (marked in gray), while the other is arranged in counter-clockwise direc-tion (marked in red). As the structure deploys, each layer undergoes a rotation, equal in magnitude but opposite to each other.

2.2.6 Closed loop structures based on angulated elements

You & Pellegrino [1996, 1997] extended the previous concept to multi-angulated elements, which are elements with more than one kink angle, as can be seen in Figure 2.24. They found that two or more such retractable structures can be joined together through the scissor hinges at the element ends. Two angulated elements from layers that turn in the same direction of two such interconnected structures, were found to maintain a constant angle and could therefore be rigidly connected, thus forming a multi-angulated ele-ment. The deployment of such a structure, composed of two layers of twelve identical multi-angulated elements with three kinks, is depicted in Figure 2.25.


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Figure 2.24: Multi-angulated element

Figure 2.25: A radially deployable linkage consisting of multi-angulated elements in three stages

of the deployment

This concept was extended by You & Pellegrino [1996, 1997] to include gener-alised angulated elements (GAE) which allow non-circular structures to be ge

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