Design and Analysis of Deployable Bar Structures for Mobile

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

Århus School of Architecture

Acknowledgements My interest in the exciting field of deployable structures came about through the process of writing my master’s 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. José 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: Piñero 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 hoberman’s unit ........................................................ 21 Figure 2.23: A radially deployable linkage consisting of angulated (or

hoberman’s) 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 Piñero [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 Foster’s 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; β=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 β 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 β and the deployment angle θ ............ 96

Figure 4.12: A foldable plate and its parameters: length L, height H, H1, H2, apex angle β, the deployment angle θ and angles α, α1, α2 .............. 96

Figure 4.13: Perspective view and side elevation of the vertical projection of a plate linkage for empirically determining the relationship between α1 and p ............................................................................................................. 98

Figure 4.14: The relationship between the apex angle β and the deployment angle θ 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 β=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 β and the deployment angle θ 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 β............................................................................................................................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 α2 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 <<ψ ..190 Figure 7.17: Graph showing the relation between the deployment angle θ and

the span S for the polar linkage with U=4 ...........................................193 Figure 7.18: Deployment sequence of the polar linkage (U=4) ...........................193 Figure 7.19: Open barrel vault: scissor structure and equivalent hinged plate

structure ..........................................................................................................194 Figure 7.20: Geometry of a hinged plate module ....................................................194 Figure 7.21: Fixing all inner lower nodes to the ground by pinned supports..195 Figure 7.22: Closed barrel vault: scissor structure and equivalent hinged plate

structure ..........................................................................................................195 Figure 7.23: Geometry of the occurring plate modules .........................................195 Figure 7.24: Closed barrel vault: scissor structure and equivalent hinged plate

structure ..........................................................................................................196 Figure 7.25: Fixing all inner lower nodes to the ground by pinned supports..197 Figure 7.26: Joint connecting four bars (no rotation of the ‘fins’ of the joint

around a vertical axis, as is the case for the translational barrel vault in Section 6.4.1, Figure 6.24) .........................................................197

Figure 7.27: Top view and perspective view of one scissor unit and its intermediate and end joints ......................................................................198

Figure 7.28: Perspective view and top view of OPEN structure with integrated tensile surface................................................................................................198

Figure 7.29: Perspective view and top view of CLOSED structure with integrated tensile surface...........................................................................199

Figure 7.30: Result without any measures taken to improve structural performance ...................................................................................................200

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 β can be altered

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

the apex angle β 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 SLE’s and its polar equivalent........................................................................................................252

Figure 9.8: Imposed condition on the length of the semi-bars a and b (a<b), in order to make the linkage foldable along the vertical axis .............253

Figure 9.9: Initial unfolding of the compacted linkage to its polygonal form253 Figure 9.10: Six stages in the deployment of a hexagonal tower: elevation and

top view ...........................................................................................................254 Figure 9.11: Design parameters of a two-module tower with angulated SLE’s

(three states) ..................................................................................................256 Figure 9.12: Perspective view: design parameters of a two-module tower with

angulated SLE’s – Side elevation showing the non-coplanarity of the angulated elements (marked in red)................................................259

Figure 9.13: Illustration of the influence of the apex angle β on the geometry of a linkage with angulated SLE’s with two modules (n=2) in the undeployed (top) and fully deployed configuration (below) ...........267

Figure 9.14: Illustration of the influence of the apex angle β on the geometry of a linkage with angulated SLE’s with three modules (n=3) in the undeployed (top) and fully deployed configuration (below) ...........268

Figure 9.15: A schematic representation of the relative rotations of the quadrilaterals around imaginary fold axes during deployment......270

Figure 9.16: Kinematic joint connecting the angulated elements at their end nodes.................................................................................................................271

Figure 9.17: The kinematic joint and the axes of revolution for the seven rotational degrees of freedom ..................................................................271

Figure 9.18: The scissor linkage in its deployed state and its equivalent hinged plate structure for mobility analysis (left) – Fixing the structure by pinned supports (right)................................................................................272

Figure 9.19: Deployment of proof-of-concept model ............................................272 Figure 9.20: Deployment sequence (A, B and C) for the tower with the

membrane elements attached ..................................................................274 Figure 9.21: Design for a deployable hexagonal tower with angulated elements

............................................................................................................................275 Figure 9.22: Initial unfolding of the compacted linkage to its polygonal form

............................................................................................................................276

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 β 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, β=90°), (p=7,

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

Table 4. 6: The expansion ratio λ for (p=5, β=90°), (p=7, β=120°) and (p=9, β=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 β and θ 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 θ Deployment angle p.10

a, b Semi-bars p.11

γ Unit angle p.11

a, b, c, d Semi-lengths p.12

β Kink angle p.21

α Angle p.21

Chapter 3 θ Deployment angle p.40

γ 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

2δ 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


K Intermediate hinge p.54 ϕ Quarter of total sector angle p.57

nα 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

ω 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

β Apex angle p.89

L Plate length p.95

W Module width p.95

Hr Rise p.95

S Span p.95

θ 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

α, α1, α2 Angle p.96

Ledge Edge length p.101

L Plate length p.102

λ 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 ρ 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

µ 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

iµ Form factor for the snow load p.127

G Permanent loads p.128

Q Mobile loads p.128

γ Safety factor p.128

D Damage p.130

ni Number of cycles p.130

Ni Critical amount of load cycles p.130

iσ∆ Fluctuating stresses p.130

cσ∆ Resistance against fatique p.130

iτ∆ 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


a, b Semi axes p.144

U Number of units in the span p.145

R Radius of the circular arc p.145

α2 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

φ Angle p.146

n Node p.152

γ2, γ3 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


a, b Semi-bar p.188


maxSθ Deployment angle for which the maximum span is reached

p.188

designθ Deployment angle in the fully deployed configuration

p.188

Smax Maximum span p.188

Sdesign Span of the deployed configuration p.188 ψ Deployment ratio p.189

inR Internal radius p.189

eR External radius p.189

β Unit angle p.189

γ Sector angle p.189

Se External span p.192

Chapter 8 p Number of plates p.214

β Apex 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

β Kink angle p.255

U Number of scissor units p.255

n Number of modules p.255

E Edge length p.255

φ Sector angle p.255


h Height of the undeployed position p.257

H Total height p.257

ψ 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 1960’s, 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 Vinci’s 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 structure’s 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 (SLE’s) 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 SLE’s 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 θ 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.

θ

Plane unit Curved unit

Unit line

End node

Intermediate hinge

θ


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 γ. 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

θ

γ


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 1960’s, Spanish architect Emilio Perez Piñero [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. Piñero 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: Piñero 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.


19

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.


20

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.


21

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 β. Because they were invented by Hoberman [1990] they are commonly denoted as hoberman’s units. The major advantage is that, as opposed to polar units, angulated units subtend a constant angle γ during deployment (Figure 2.22). For this to occur, the bar geometry has to be such that α= γ/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 hoberman’s unit

θ

γ

β

α

a

a

α = γ/2


22

Figure 2.23: A radially deployable linkage consisting of angulated (or hoberman’s) 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.


23

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 generated. Depending on which type of GAE is used, such structures form pat-terns of either rhombuses of parallelograms. By providing this type of structure with cover elements, Kassabian et al. [1997, 1999] has shown it possible to employ them as a retractable roof (Figure 2.26).

α

α = γ/2 γ/2 γ/2 γ/2

α α α


24

The cover elements provide, both in the open and closed position a gap-free, weatherproof surface.

Figure 2.26: Multi-angulated structure with cover elements in an intermediate deployment posi-

tion

Jensen [2004] has found that, instead of covering a bar structure with plates, it is 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. He has developed general methods and the conditions for con-necting expandable structures of any plan shape (Figure 2.27), leading to the possibility of creating plane or stacked assemblies composed of individual ex-pandable structures. This has led to the development of transformable free-form or ‘blob’-structures, as shown in Figure 2.28 [Jensen & Pellegrino, 2004].

Figure 2.27: Model of a non-circular structure where all boundaries and plates are unique [Jen-

sen, 2004]


25

Figure 2.28: Computer model of an expandable blob structure [Jensen & Pellegrino, 2004]

Several other types of closed loop structures have been developed by Escrig et al. [1996], Chilton et al. [1998], Wohlhart [2000], You [2000] and Rodriguez & Chilton [2003]. Retractable reciprocal plate structures have been developed by Chilton et al. [1998], of which an example is shown in Figure 2.29. The struc-ture shown consists of six triangular rigid plate elements which each slide against each other as the structure is retracted, hence providing a continuous surface throughout the deployment process.

Figure 2.29: Reciprocal plate structure

Rodriguez & Chilton [2003] have proposed a novel retractable structure called the swivel diaphragm. It forms a ring of congruent parallelograms between angulated elements by using the fixed points of the structure together with straight bars. In Figure 2.30 a swivel diaphragm is shown in several stages of the deployment. As opposed to the multi-angulated elements proposed by Kassabian et al., the support points can always be directly connected to the angulated elements, which allows the angulated elements to swivel around the fixed points. The angulated elements can be replaced by rigid plate ele-ments to form a continuous surface in both the open and closed position. Rod-


26

riguez et al. [2004] has also developed methods for interconnecting several individual swivel diaphragms to form larger retractable assemblies.

Figure 2.30: Swivel diaphragm in consecutive stages of deployment

Several researchers have proposed dome shaped structures that can retract towards their perimeter. Piñero pioneered a diaphragm retractable dome with a number of wedge-shaped plates which are able to rotate about an axis nor-mal to the sphere [Escrig, 1993]. When the plates are rotated, an aperture is created at the centre of the dome. During opening and closing, all plates show an overlap, except in the fully closed position where the plates form a gap-free spherical cap. To create a single-degree-of-freedom mechanism, pairs of adjacent plates are mutually connected through a revolute joint at the apex of one plate. This joint is then run along a certain path on the other plate, as il-lustrated in Figure 2.31.

Figure 2.31: Reciprocal dome proposed by Piñero [Escrig, 1993]

Angulated elements connected by scissor hinges not only subtend a constant angle in a plane surface, but also on a conical surface. This was discovered by Hoberman [1991], who has proposed the Iris Dome shown in Figure 2.32. The structures consists of five concentric rings of angulated elements, connected with axes of rotation tangential to the circular plan of the rings, and thus a


27

retractable dome is formed. The dome uses rigid plates as cladding material, 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 upon each other. Hoberman exhibited a model for a dome, which was continuously retracted by an actuator, at the Expo 2000 in Hannover. In 2002 he designed and built a semi-circular retractable mechanical curtain for cere-monial purposes at the Winter Olympics in Salt Lake City. Both structures are shown in Figure 2.33.

Figure 2.32: Iris dome by Hoberman [Kassabian et al, 1999]

Figure 2.33: Retractable dome on Expo Hannover (courtesy of M. Mollaert) – Mechanical curtain

Winter Olympics Salt Lake City 2002 [Hoberman, 2007]

Extending his work on two-dimensional retractable plate structures, Jensen [2004] has proposed an elegant solution for a retractable dome structure, us-ing only plate elements, instead of a combination of multi-angulated elements and cover plates. One of his proposals is a novel type of retractable dome, de-scribed as a self-supporting reciprocal mechanism, similar to that proposed by Piñero [Escrig, 1993]. Unlike the concept developed by Piñero, the current structure does not have any overlaps, and hence friction between neighbour-


28

ing plates is removed, which makes it better suited for large scale applications. Other advantages are the possibilities of modifying the plate boundaries and the location of the fixed points about which the plates rotate. Figure 2.34 shows a retractable dome with plates having fixed points of rotation. The plates provide a gap-free surface in the open and closed position. Figure 2.35 shows a retractable dome with modified boundaries.

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

[Jensen, 2004]

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

[Jensen, 2004]


29

2.3 Foldable plate structures A family of foldable, portable structures, based on the Yoshimura buckle pat-tern for axially compressed cylindrical shells, has been presented by Foster and Krishnakumar [1986/87]. These structures have considerable shape flexibility (multiple degrees of freedom), but once erected they possess significant stiff-ness.

Figure 2.36: Basic layout of Foster’s module [Foster, 1986]

Foldable plate structures consist of a series of triangular plates, connected at their edges by continuous joints, allowing each plate to rotate relative to its neighbouring plate. The plates can fold into a flat stack and unfold into a pre-determined three-dimensional configuration, with a corrugated surface. De-termining the actual shape of the deployed configuration, is an origami-like pattern formed by intersecting mountain folds and valley folds. The foldable plate linkage consists of a repetition of the basic plate element which is of triangular shape and must possess one angle of at least 90°, in order to gener-


30

ate a configuration which is compactly foldable into a flat stack of plates. All elements within sections (modules) of the structures must have the same shape and equal apex angles. A module, as Figure 2.36 shows, is essentially a strip of plates joined in such a way that opposite sides are parallel when fully deployed. The module pictured contains four plates: three full and two halves (which are counted as one full plate). Separate modules, which must each be capable of being compactly foldable, are joined together by continuous hinges along the parallel sides to form the complete structure. The apex angle of the triangular plate is the main parameter which determines the fold pattern, and therefore the geometry of the final deployed shape. As can be seen from Figure 2.36, some apex angles give rise to folded configurations, which are not fully compacted, i.e. there is a gap in the folded configuration. This can be avoided when the apex angle is a sub-multiple of 360°. The simplest foldable shape is a four-plate linkage with an apex angle of 90°. Now if all plates are isosceles triangles, the compact folded configuration has a square shape but when all plates are right-angled triangles, the compactly folded configuration is of rectangular shape.

Figure 2.37: Combination of different modules [Foster, 1986]


31

When two modules, one with isosceles triangles and one with right-angled triangles, are connected, their flatly packed shape is an overlapping square and rectangle, as shown in Figure 2.37. This shows that not all elements in a struc-ture have to be identical, but, on the other hand, the usefulness and the ele-ment uniformity of such a mixed structure is decreased. There is a maximum number of plates associated with each apex angle which will give a foldable configuration. Using a higher number of plates would make folding of the con-figuration physically impossible, due to plate overlap. For instance, a 90° apex angle corresponds with a maximum of four full plates, a 108° apex angle with five plates, a 120° apex angle with six plates, and so on.

Figure 2.38: Simplest building with 90° apex angle [Foster, 1986]

Consider the minimal foldable configuration: a four-plate linkage with an apex angle of 90°, shown in Figure 2.38. This linkage is the most compactly foldable


32

configuration, as it folds into a compact square shape. Despite its advanta-geous folding properties, a major shortcoming of practical use is that clear headroom is quite low. A possible solution is to make a longitudinal field joint between two modules, as depicted in Figure 2.39. Of course, the joined sub-structures will have to be disassembled in order to be foldable.

Figure 2.39: Building formed by two 90°-modules joined at their ends [Foster, 1986]

A plate linkage with appealing characteristics is the one with an apex angle of 120°. From Figure 2.40 it can be seen that the width of the collapsed configu-ration is identical to the plate length. If, for example, the length is made 2.4 m, the headroom is 2.08 m and this would make it suitable for human habita-tion, as opposed to the 90°-elements, where the headroom would only be 1.7 m. As Figure 2.41 shows, by connecting two modules by means of a field joint, the resulting deployed structure would be employable as a temporary field service hangar for light aircraft and similar applications [Gantes, 2001].

Figure 2.40: Building with apex angle of 120° [Foster, 1986]


33

Figure 2.41: Building formed by two 120°-modules joined at their ends [Foster, 1986]

Employing isosceles triangular plates leads to skewed deployed configurations, which can equally be combined along their parallel, straight edges. Figure 2.42 shows a structure with 90°-, 60°- and 30°-elements.

Figure 2.42: Structure with 90°-, 60°- and 30°-elements [Foster, 1986]

The previously discussed structures are all characterised by a linear deploy-ment. As the structure deploys, it undergoes a linear expansion in the longitu-dinal direction and a variation of the curvature in the transverse direction. The 120°-structure can also be circularly deployed, by holding the bottom ele-ments together and only deploying the middle section. The example in Figure 2.43 could be used for a temporary stage shell.


34

Figure 2.43: Temporary stage shell with 120° modules - Tension cables used to provide bulkhead

[Foster, 1986]

Due to the fact that these structures are basically a mechanism, a number of constraints have to be considered to make them statically determinate. Figure 2.43 shows how tension cables can be integrated in the span of a 120°-structure to provide bulkhead. Further expanding this concept, Tonon [1991, 1993] has studied the geometry of single and double curvature foldable plate structures, such as domes, conics, paraboloids and hyperboloids. To obtain these shapes, variations of the apex angle and/or the plate dimensions are imposed in subsequent modules. Although they are compactly foldable into a compact stack of plates, some fold patterns cannot be developed in a plane, as opposed to the configurations previously described by Foster.

Figure 2.44: Double curvature variable shape (hyperbolic type), plane pattern [Tonon, 1993]


35

As can be seen from Figure 2.44, which has a variable double curvature (hy-perboloid shape), such a pattern cannot be developed in a plane. However, once the corresponding edges are connected by a field joint in a partially de-ployed state, the structure can be further folded until it reaches its fully com-pacted shape. Tonon has formulated the condition, which the plate geometry has to satisfy, in order to guarantee full foldability: the sum of the individual base angles of two neighbouring plate elements has to be constant through-out the plate geometry. This is illustrated by Figure 2.45, from which it can be seen that the size of the individual plate angles varies, although their sum re-mains constant throughout the pattern. It is noted that Tonon uses the base angle of the triangular plates to describe the fold pattern, as opposed to the apex angle used by Foster.

Figure 2.45: Fold pattern with different individual plate angles, but a constant sum throughout

the plate geometry, guaranteeing full foldability.

A selection of doubly curved paper models is shown in Figure 2.46 and Figure 2.47.

Figure 2.46: Doubly curved folded shapes [Tonon, 1993]

Plates with variable base

angles

Constant sum


36

Figure 2.47: Linear and circular deployable double curvature folded shapes [Tonon, 1993]

Using a plane foldable geometry, Hernandez & Stephens [2000] have proposed a folding aluminium sheet roof for covering the terrace of a pool area. The fold pattern consists of trapezoidal plate elements which give rise to a plane cor-rugated surface (Figure 2.48). Because no curvature is introduced by folding, the roof retracts on a supporting steel structure consisting of seven parallel trusses, to provide the necessary headroom. The retraction, during which the sheets are supported by wheels running on a rail, is realised by a motor driven system of wire cables. Special attention has been given to the joint design, providing a waterproof sealing between consecutive plates.

Figure 2.48: Folding aluminium sheet roof for covering the terrace of the pool area of the Inter-

national Center of Education and Development in Caracas, Venezuela [Hernandez & Stephens,

2000]

Another interesting idea, although not of immediate use for an architectural application, is brought forward by Guest and Pellegrino [1994], treating the folding of triangulated cylinders, later expanded by Barker & Guest [1998]. Such triangulated cylinders consist of identical triangular panels, placed on a


37

helical strip of the cylinder. All cylinders made of isosceles triangles fold down to prisms and are packaged as a compact stack of plates. They are strain-free in their compacted and fully deployed configuration. However, in intermediate folding positions, some deformation of the surface is required. The concept has been tested and applied to inflatable, thin walled metal cylinders.

Figure 2.49: (a) Fold pattern; (b) Fold pattern with alternate rings to prevent relative rotation

during deployment [Barker & Guest, 1998]


38

Chapter 3 – Design of Scissor Structures

39

Chapter 3

Design of Scissor Structures

3.1 Introduction This chapter is concerned with explaining the basic principles needed for the design of deployable scissor structures, composed of translational or polar scissor units, of which the characteristics have been discussed in Sections 2.2.1 and 2.2.2. Retractable closed loop structures consisting of angulated elements (Section 2.2.5) are not discussed in this chapter. As can be seen from Figure 2.14, such structures are not foldable into a compact bundle of bars in their undeployed position. In the context of highly compactable, easily transportable single cur-vature or double curvature grid structures, they offer therefore no added value over translational or polar units. However, a specific design approach will be proposed to employ angulated elements in a different setting: a linear deploy-able tower for case study 4, where they contribute substantially to the con-cept. It will be shown how translational and polar units can be connected to form the simplest of scissor mechanisms: a two-dimensional linkage, which can be designed by either of two methods: pure geometric construction or a novel geometric design method, using equations to obtain the complete geometry in its deployed configuration. Further, it is shown how these two-dimensional linkages can be combined to form stress-free deployable three-dimensional grid structures of single or dou-ble curvature, which will form the basis of the design of case studies 1 and 2. In Section 2.2.4 (Figure 2.14) the difference between stress-free deployable and bi-stable structures has been explained. Because the aim of this disserta-tion is to provide the means for designing easily deployable scissor structures, the focus is on stress-free deployable structures only. Finally, the grid struc-


40

tures known from literature (Section 2.2.4) which are guaranteed stress-free deployable, are identified and classified according to their grid type (two- or three-way) and their curvature (single or double).

3.2 Design of two-dimensional scissor linkages Two methods for obtaining a deployable scissor linkage will be presented. Whether it be pure geometric construction, or a geometric design method em-ploying a series of equations, both approaches are based on the deployability constraint (Section 2.2.3). The two-dimensional and three-dimensional struc-tures of which the design is explained in this chapter, are based on the trans-lational or the polar unit (Figure 3.1).

Figure 3.1 : Translational and polar scissor unit

It should be noted that scissor linkages which do not comply with equation 3.1 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. The deployability constraint can be made more tangible by using its graphic representation. Consider the scissor unit with semi-lengths a and b in Figure 3.2. When a compatible unit is to be linked with the first unit, the location of

θ

t θ

γ

t


41

the intermediate hinges has to be determined, in such a way that semi-lengths c and d satisfy the deployability constraint. The locus of all valid intermediate hinges M that comply with the deployability constraint is an ellipse, with the common end nodes of both units as its foci. This method of representation is valid for all general scissor linkages.

Figure 3.2: An ellipse as the graphic representation of the deployability constraint for transla-

tional units, determining the locus of the intermediate hinge

When polar units are used, however, the locus of valid intermediate hinges can be represented by a circle, as shown in Figure 3.3.

Figure 3.3: A circle as the graphic representation of the deployability constraint for polar units,

determining the locus of the intermediate hinge

a

b

c

d

M

a

b

c

d

M


42

In the following section the deployability constraint and its geometric repre-sentation will be used to design two-dimensional deployable linkages com-posed of translational and polar units.

3.2.1 Method 1: Geometric construction

Apart from a plane (or rectilinear) curve, the most common curve for a scissor linkage is a circular arc (Figure 3.4). From an architectural point of view, it makes sense to describe this base curve in terms of a given rise (Hr) and span (S). Once the geometry of the base curve is determined, it can be fitted with a series of compatible scissor units, each obeying the geometric constraint (or deployability constraint). The construction of polar and translational units will now be discussed in greater detail.

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

rH

S

O


43

Polar linkages

Method 1: the base curve contains the intersection points of the scissor bars (intermediate hinges). Linkages can be constructed with either a constant or a variable unit thickness t. The unit thickness is defined as the distance between the internal and external end nodes of the scissor units, as shown in Figure 3.1. Constant unit thickness (Figure 3.5) A polar linkage with a constant unit thickness has bars of identical length. All intermediate points C of the scissor units lie on the base curve. Construction:

• A: the base curve is divided in equal angular portions by the polar unit lines, which all intersect in centre O. Segment MC is tangent to the base curve, perpendicular to OC, which is the bisector of a segment. Point M is the centre of the circle which represents the geometric constraint for deployability. This circle intersects the base curve in the intermediate point C.

• B: now a scissor unit can be drawn, in such a way that MC is the bi-sector of unit angle 2δ. Now the unit thickness t is determined.

• C: the line through MC intersects the next unit line in point M’. A cir-cle can now be drawn with centre M’ and M’C as radius. The intersec-tion point of this circle with the base curve is the intermediate point of the next unit.

• D: proceeding this way leads to a complete linkage


44

A B

C D Figure 3.5: The arc is divided in equal angular portions. Circles intersecting the arc determine

the loci of the intermediate hinge points

t C

δ

δ M

M


45

Variable unit thickness (Figure 3.6) A polar linkage with a variable unit thickness has different units comprising of bars of variable length. All intermediate points C of the scissor units lie on the base curve. Construction:

• A: the base curve is divided in unequal angular portions by the polar unit lines, which all intersect in centre O. Point C is arbitrarily placed on the base curve between two consecutive unit polar lines. Segment MC is tangent to the base curve, perpendicular to OC. Point M is the centre of the circle which represents the geometric constraint for de-ployability. This circle intersects the base curve in the intermediate point C.

• B: now a scissor unit can be drawn, in such a way such that MC is the bisector of unit angle 2δ. Now the unit thickness t1 is determined.

• C: the line through MC is intersected with the next unit line in point M’. A (differently sized) circle can now be drawn with centre M’ and M’C as radius. The intersection point of this circle with the base curve is the intermediate point of the next unit. The extended scissor bars of the first unit intersect with the unit line through M’ by which unit thickness t2 is obtained.



46

A B

C D Figure 3.6: The arc is divided in unequal angular portions. Variable circles intersecting the arc

determine the loci of the intermediate hinge points

t1 C δ

δ

M

M t2


47

Method 2: an inner and outer curve, with a constant distance t (unit thickness) between them, contain the inner and outer end nodes of the polar units. This is a simple method for constructing a linkage with constant unit thickness. All units have identical bars.

A B

C D

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

t C


48

Constant unit thickness (Figure 3.7) Construction:

• A: the inner curve is divided in equal angular portions by the polar unit lines, which all intersect in centre O.

• B: an outer arc is offset a distance t from the inner base curve, by which the unit thickness is determined and kept constant.

• C: the intersection points of the polar unit lines with the inner and outer curve determine the position of the inner and outer end nodes of the units. Constructing the units is simply a matter of connecting the appropriate nodes


A pluricentred polar linkage

Using a pluricentred arc as base curve for the 2D-linkage (as opposed to the previously discussed single-centred arc) can increase the overall headroom, while maintaining the same span and rise. Figure 3.8a shows a single-centred arc with centre O1. The segmented arc in Figure 3.8b has an identical span and rise, but consists of three arcs: one arc with centre O2 and two arcs with cen-tre O3 and a decreased radius. Figure 3.8c shows the difference in headroom between the single-centred and the pluricentred arc .

a b c

Figure 3.8: For the same span and rise, a pluricentred arc can offer increased headroom com-

pared to a single-centred arc

O1 O2

O3


49

Figure 3.9: Example of a pluricentred base curve consisting of three arc segments with decreas-

ing radius

Constant unit thickness (Figure 3.10) In a polar pluricentred linkage with a constant unit thickness all bars are iden-tical. The base curve is a concatenation of arc segments, each of which has its own centre. Because of the varying curvature of the base curve, there is a dif-ference in unit geometry per segment, although all bars are identical. This means that the location of the intermediate hinge of the units differs per segment. The geometry is determined by the end nodes lying on the inner and outer curve. Construction:

• A: the first segment of the inner base curve with centre O1 is divided in equal angular portions by the polar unit lines, which all intersect in centre O1. A second arc segment with centre O2 has a smaller radius


50

• B: an outer curve is drawn, with an offset distance t from the inner curve

• C: the intersection points of the polar unit lines with the inner and outer curve determine the position of the inner and outer end nodes of the units. One bar of the first unit is constructed by connecting the appropriate end nodes. A circle, which represents the geometric con-straint is drawn with centre M and radius MM’.

• D: now the same circle is drawn, but with centre M’. The intersection point with the outer curve is the centre for the next circle. This is re-peated until the end of the complete base curve is reached. The repe-tition of the circles ensures a constant bar length.

• E, F: analogous to phase D, the second chain of bars is now con-structed, effectively completing the scissor linkage

A B

R2 R1

t

M

M

O1

O2


51

C D

E F

Figure 3.10: Each arc segment (with a different radius) is divided in equal angular portions.

Identical circles ensure a constant bar length

Translational linkages

As opposed to polar linkages, which are commonly based on a circular curve, translational linkage are easily based on any arbitrary curvature. Therefore, the general method for base curves with arbitrary curvature will be explained, by which all possible curves are covered. Linkages with variable or constant unit thickness are possible. Method: the base curve contains the midpoints M, M’ of the segments PQ and ST representing the unit thickness t, shown in Figure 3.11. Points P, Q and S, T are also the foci of the ellipses representing the deployability constraint.


52

Points M and M’ are found by intersecting a double ellipse (twice the size of the original ellipse) with the base curve. Once these intersection points are found, completing the scissor units is simply a matter of appropriately con-necting the end nodes by segments PT and QS. Constant unit thickness (Figure 3.12) A translational linkage with a constant unit thickness has bars of variable length. All midpoints M lie on the base curve. Construction:

• A: a vertical segment of length t is placed twice on the base curve, a chosen distance removed from each other, in such a way that mid-points M and M’ are located on the base curve. This determines the geometry of the first translational unit. The small ellipse (deployability constraint) can be drawn and subsequently the double ellipse is de-rived.

• B: now the double ellipse and the unit thickness t are placed repeat-edly on the base curve, each time using the intersection point from the previous ellipse.

• C, D: now the end nodes are appropriately connected to form the complete linkage.

Figure 3.11: The original and double ellipse representing the deployability constraint – intersec-

tion points M and M’ are the midpoints of the unit thickness t

M t

M

M

M

Q

P

S

T


53

A

B


54

C

D

Figure 3.12: Double ellipses impose the deployability constraint on a translational linkage with

constant unit thickness

Variable unit thickness (Figure 3.13 and Figure 3.14) A translational linkage with a variable unit thickness {t1, t2,…, tn} has bars of variable length. The intersection points K of the (small) ellipses with the base curve determine the location of the intermediate hinges. This approach is somewhat laborious because for each new unit a separate ellipse has to be drawn to determine the location of the next intermediate hinge. Because of the varying unit thickness, the method of the double ellipses cannot be used. Construction:

• A: a vertical segment of length t1 is placed on the base curve. The lo-cation of the intermediate hinge K of the first unit is arbitrarily cho-sen on the base curve. P, Q and K are determined. Now the ellipse can be drawn. Lines through PK and QK are intersected by an arbitrary


55

vertical unit line. These intersection points S and T determine the unit thickness t2. Now the second ellipse can be drawn.

• B: the intersection point of the second ellipse and the base curve is determined and the previous steps are repeated

• C, D: now the end nodes are appropriately connected to form the complete linkage.

Figure 3.13: Two differently sized, but compatible ellipses representing the deployability con-

straint – intersection points M and M’ are the midpoints of the unit thickness t1 and t2

K

t1

M

M

Q

P

S

T

t2


56

A

B

C


57

D

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

3.2.2 Method 2: Geometric design

Parameterisation of the base curve based on the rise and the span

This section is concerned with determining the geometry of polar and transla-tional linkages based on architecturally relevant design parameters: the rise

rH and the span S of the circular base curve, as shown in Figure 3.15. The

equations for a geometric design approach will be presented. Design values: rH , S

Unknowns: inR ,O , 0P , nP , nα ,ϕ


58

Figure 3.15: The parameters used in the description of the geometry of the circular arc: rise (Hr)

and span (S)

First ninR αϕ ,, are found through the following relationships:

=⇔= −

SH

SH rr 2tan

2tan 1ϕϕ (3.1)

( ) ( )ϕϕ2sin2

122sin SRRS

inin

=⇔= (3.2)

ϕπα 22−=n (3.3)

The general polar equation of the circular arc is given by Eqn (3.4), with

nn απαα −≤≤ :

==

αα

sincos

in

in

RyRx

(3.4)

nαinR


59

By joining Eqns (3.1), (3.2) and (3.4) we can now write the equation in terms of rH and S:

=

=

−

−

α

α

sin2tan2sin2

1

cos2tan2sin2

1

1

1

SH

Sy

SH

Sx

r

r

(3.5)

Point ),0(0 inRP is the midpoint of the curve and )sin,cos( ninninn RRP αα is

an endpoint. Now the base curve has been derived from two design parameters S and rH ,

only two more parameters are to be given a value, in order to fully determine the complete linkage: the number of units U and the unit thickness t . This is shown for both a polar and a translational linkage in the following section.

Geometric design of a polar linkage

The base curve can now be fitted with a chosen number of polar units U. Therefore, the sector angle ω has to be divided into equal angular portions γ ,

as shown in Figure 3.16. From Figure 3.15 and Figure 3.16, we know that

γω U= (3.6)

and

ϕω 4= (3.7)

Therefore,

Uϕγ 4

= (3.8)


60

With the inner radius (radius of the base curve) inR obtained from Eqn (3.2), a unit thickness t is chosen, which determines an outer curve with radius eR :

tRR ine += (3.9)

Because the scissor units from Figure 3.16 are polar units, each bar length L is divided into two unequal semi-bars a and b . Now the geometry of the linkage can be totally derived by finding values for semi-bar lengths a , b and the deployment angle θ . From Figure 3.16 and from Sastre [1996] we can find the following relations:

baL += (3.10)

With γ known from Eqn (3.8) and inR , eR known from Eqns (3.2) and (3.9) we

can use the cosine rule to obtain L : γcos2222

einein RRRRL −+= (3.11)

Figure 3.16: Parameters needed for the geometric design of a polar linkage

Re Rin

tS

γ ω

m

n

θ a

b

γ

θ


61

Through similarity of triangles is

nm

RR

e

in = (3.12)

and

ba

nm= (3.13)

Equating (3.12) and (3.13) results in

inee

in RbRaba

RR

=⇔= (3.14)

Substituting Eqn (3.10) in Eqn (3.14) gives

ein

e

RRRLb+

= (3.15)

and ein

in

RRRLa+

= (3.16)

Also,

2sin θam = (3.17)

and 2

sin γinRm = (3.18)

Finally, by equating (3.17) and (3.18), an expression for the deployment an-gleθ is obtained:

= −

2sinsin2 1 γθ

aRin (3.19)

The complete geometry has now been derived from design parameters Hr, S, U and t.


62

Geometric design of translational linkages

This section is concerned with the geometric design of a translational linkage with constant unit thickness on a base curve. As opposed to the approach pre-viously described for the polar linkage, a translational linkage is not character-ised by the angular portions of the sector angle of the arc. This is because the unit lines for a translational unit are parallel and not radially arranged as is the case with polar units. Therefore, another approach is used for subdividing the base curve, which will involve solving a system of equations. First, a circu-lar arc is used as base curve, but the approach is easily extended to other curves as well. Let Hr and S determine a circular arc as shown in Figure 3.15. Again, a semicir-cle or an arc segment can be derived, determined by the angle nα .

The additional design parameters that will fully determine the geometry of the linkage are the number of units U, the unit thickness t. The number of desired units U will determine how many intersection points between the unit lines and the base curve will have to be calculated. To explain the method, a linkage with U=4 is used. Due to symmetry, only two units are shown. As shown in Figure 3.17 the location of P0 and P2 is known: P0 is the midpoint and P2 the endpoint of the curve. Essentially, the only solution needed is the position of point P1 on the base curve.


63

Figure 3.17: Parameters for the geometric design of a translational linkage with four units

(U=4), of which two are shown

Analogous to the geometric construction method previously discussed, the double ellipse is used for imposing the geometric constraint and determining the intersection points P0, P1 and P2 with the base curve (the original, small ellipses in Figure 3.17 are merely there to illustrate the geometric constraint of the units, but serve no further purpose in the calculation). Figure 3.18 shows the relation between the semi-axes of the double ellipse and the unit thick-ness. Their relation is given by

222 abt −= (3.20)

P0

P1

P2

t

E0 E1

α1 α2

Pe

Pin


64

For the given design parameters (U and t) there is only one solution for which the array of double ellipses in Figure 3.17, fits exactly on the curve, i.e. the position of P0, P1 and P2 is exactly as pictured.

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

The general polar equation for a circular arc is given by

αcosRx = αsinRy = (3.21)

The general parametric equation for an ellipse, with πθ 20 ≤≤ , is given by

θcosax = θsinby = (3.22)

For brevity, the radius of the circular arc is denoted by R, but as usual it can be expressed in terms of the rise and span using Eqn (3.2). The point P0(x0, y0) has coordinates (0, R) and P2 is determined by 2α , which in turn can be expressed

in terms of Hr and S.

t

t

a

b


65

The coordinates of P1 are determined by the unknown angle 1α . P1(x1, y1) is

located on the base curve and has coordinates:

11 cosαRx =

11 sinαRy = (3.23)

P1 also lies on ellipse E0, with centre P0(x0, y0):

001 cos xax += θ

001 sin yby += θ (3.24)

Equating (3.23) and (3.24) gives:

001 coscos xaR += θα (3.25)

001 sinsin ybR += θα (3.26)

Analogously, we can write the coordinates for point P2(x2, y2), lying on the base curve:

22 cosαRx =

22 sinαRy = (3.27)

and on ellipse E1, with centre P1(x1, y1):

112 cos xax += θ

112 sin yby += θ (3.28)

Equating (3.27) and (3.28) gives:

112 coscos xaR += θα (3.29)

112 sinsin ybR += θα (3.30)


66

Eqns (3.25), (3.26), (3.29), (3.30) and (3.20) form a system of five equations in five unknowns. With R, 2α and t as input parameters the system can be solved for 1α , a, b, θ0 and θ1

−=

+=+=+=

=

222112

112

01

01

sinsinsincoscoscos

sinsincoscos

abt

RbRRaR

RbRaR

αθααθα

θαθα

(3.31)

Two different cases can be distinguished:

• 02 =α : this evidently means that P2 lies on the X-axis and coincides

with the endpoint of the circular arc. Therefore, a semi-circle is used as the base curve.

• 2

0 2πα << : in this case P2 lies somewhere on the arc and therefore

only a user-defined portion of the circular arc is used for carrying the linkage.

Now that a solution for 1α has been found, the coordinates of P1 is given by

Eqn (3.23). The y-coordinate of the external end nodes Pe of the scissor units are easily found by adding an amount 2/t to the y-coordinates of P0, P1 and P2. Analogously, by subtracting an amount 2/t the internal nodes Pin are found. Calculating higher numbers of units is simply a matter of adding the appropri-ate equations to the system. Per additional angle α (extra intersection points P on the base curve) to be calculated, two equations are added. Generally, for n units (U=n) and R, αn and t as design parameters, the system with 2n+1 equa-tions for 2n+1 unknowns can be written as


67

−=

+=

+=

+==

−−

−−

22211

11

01

01

sinsinsincoscoscos

...sinsincoscos

abt

RbRRaR

RbRaR

nnn

nnn

αθααθα

θαθα

(3.32)

which gives solutions for a, b,{ }11 ...,, −nαα and { }10 ...,, −nθθ .

The approach is not limited to circular arcs only. Any equation for a curve can be used, such as the parametric equation for a parabola. Eqn (3.33) stands for an inverted (open towards negative y-values) parabola with parameter v. Figure 3.19 shows such a parabola, of which the focus lies in the origin, as base curve for a four-unit linkage, of which two are considered due to symme-try.

vqx 2= 2vqy −= (3.33)

Figure 3.19: Translational linkage with U=2 fitted on a parabolic base curve

P1

P2

P0

q

o

Y

X


68

The base curve is determined by assigning a value to q. Analogous to the ap-proach for circular arcs, values for U and t are chosen. The endpoint of the base curve P2 is found by determining v2. Now the coordinates of P1 can be calculated from a system of five equations in five unknowns:

−=

−=−

+=+=−

=

222

211

22

112

02

1

01

sin

2cos2sin

cos2

abt

vqbvq

vqavqqbvq

avq

θ

θθ

θ

(3.34)

which gives solutions for v1, a, b, θ0 and θ1. The value obtained for v1 will suf-fice for determining the position on the parabola of P1, by using Eqn (3.33).

3.2.3 Interactive geometry

The geometric construction of scissor linkages can be automated by using a program called Cabri Geometry II [2007]. This allows the creation of interac-tive drawings responding in real-time to changes made by the user. For exam-ple, the geometry of a translational linkage with constant unit thickness has been drawn up using the method of the ellipses (Figure 3.20) After altering the geometry at will, all necessary values for the bar lengths and angles can be read. This tool has been used by Wouter Decorte [2007], a designer of kinetic art, for the formfinding of a ‘deployable landscape’. It consists of translational units with constant unit thickness on an arbitrary base curve. Figure 3.21 shows three consecutive deployment stages.


69

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)

Figure 3.21: Deployable landscape consisting of one arbitrarily curved translational linkage re-

peated in an orthogonal grid. Linkage designed using the interactive geometry tool (Aluminium,

4.5 m x 3 m, (photo: courtesy of Wouter Decorte)


70

3.3 Three-dimensional structures In this section a description will be given of possible configurations for three-dimensional scissor structures composed of translational and polar units. These can be designed by utilising the methods discussed in this chapter. There is a myriad of possible configurations for such structures, all obeying the deploy-ability constraint, some of which have been reviewed in Section 2.2.4. How-ever, only part of those will be stress-free deployable, meaning that the scissor members will be in a stress-free state – the stresses induced by selfweight left aside - before, during and after deployment, effectively behaving like a mechanism. Two-dimensional scissor linkages obeying the deployability con-straint will always be stress-free deployable, but it is when they are placed into a three-dimensional grid that additional effects can come into play.

Linear/Flat Single curvature Double curvature Figure 3.22: Possible shapes for three-dimensional stress-free deployable structures, which can

be designed using the tools presented

Although, theoretically, the configurations which will be discussed are linkages (or mechanisms) they will be referred to as structures, because ultimately they will be used as such in an architectural environment and become stabilised in order to be able to carry loads. They are classified according to their overall curvature (plane, single curvature or double curvature), the grid directions


71

(two-, three-, or four-way grid) and the type(s) of units used (translational, polar or both). The characteristic shapes which can be designed using the methods previously described are shown in Figure 3.22. In Figure 3.23 and Figure 3.24 a two-way and a three-way grid are shown with their respective directions, which will be used to contain the plane or curved translational and polar linkages.

Figure 3.23: Two-way grid with directions A and B

Figure 3.24: Three-way grid with directions C, D and E


72

3.3.1 Linear structures

The most elementary three-dimensional shape is the linear structure, which is often used as a prismatic column, masts or beam. It consists of translational or polar units, or a combination thereof. As these linear structure deploy, their length increases while their width decreases (Figure 3.25, left). Also, combina-tions of scissor units and non-deployable elements are possible, in which case the non-deployable elements give the structure a certain initial width, which is invariable throughout the deployment (Figure 3.25., right).

Square and triangular beam Square and trapezoidal arch

Triangular column Quadrangular column

Figure 3.25: Linear elements – prismatic columns – arches

(left): linear elements consisting solely of scissor units

(right): Parallel scissor units connected by non-deployable elements


73

This concept can be extended to the structures shown in Figure 3.26. Here, the non-deployable elements are of such length, that a space enclosure is formed. By laterally connecting arches of increasing size by non-deployable elements, numerous variations become possible with single or double curvature (Figure 3.26, bottom right). Also, three-dimensional linear structures can be placed parallelly and become mutually connected (Figure 3.26, top left).

Parallel triangular 3D-arches Parallel 2D- arches

Polar and translational linkages Scaled 2D-arches

Figure 3.26: Parallel linear structures connected by non-deployable elements


74

3.3.2 Plane grid structures

These are structures with zero curvature (zeroclastic) and consist of plane (rectilinear) translational linkages placed on either a two-, three- or four-way grid.

Two-way grid

• Translational units This is the simplest of geometries. Plane translational units are placed in direc-tions A and B of a two-way grid with square of rhombus-shaped cells (Figure 3.27).

Plan view Perspective view

Elevation view

Figure 3.27: Plane translational units on a two-way grid


75

Three-way grid

• Translational units By placing plane translational linkages in three directions C, D and E of a three-way grid, a plane grid is formed with equilateral grid cells (Figure 3.28).


Elevation view

Figure 3.28: Plane translational units on a three-way grid


76

Four-way grid

• Translational units By adding two more directions diagonally to the already present orthogonal directions A and B of the grid in Figure 3.29, a four-way grid structure is formed with isosceles grid cells. Per grid cell, the four units contained in the diagonal directions are basically identical to those in directions A and B, ex-cept that they are shortened, to fit inside the grid cells. It should be noted that for the grid to be stress-free deployable, all units have to be plane. Using curved units on the diagonals would render the geometry bi-stable.


Elevation view

Figure 3.29: Plane translational units on a four-way grid


77

3.3.3 Single curvature grid structures

Cylindrical grids are monoclastic shapes, often referred to as barrel vaults. They are obtained by curving one direction of a two-way grid, or two, respec-tively three directions of a three-way grid.

Two-way grid

• Translational units A combination of plane and curved translational units is used to form a trans-lational barrel vault with an orthogonal grid. Grid direction A is kept plane, while curvature is introduced in direction B (Figure 3.30).


Elevation view

Figure 3.30: Plane and curved translational units on a two-way grid


78

• Polar and translational units

Rows of polar units are now contained in curved direction B, while direction A contains plane translational units. This structure is referred to as a polar barrel vault (Figure 3.31).


Elevation view

Figure 3.31: Polar and translational units on a two-way grid


79

Three-way grid

• Translational units Plane and curved translational units are placed in a three-way grid. Direction C contains plane translational units, while D and E contain curved units (Figure 3.32). The result is a translational barrel vault with isosceles grid cells. During deployment the angles inside the grid cells vary slightly, a phenomenon called angular distortion. This will be explained in greater detail in Section 6.4.1.


Elevation view

Figure 3.32: Plane and curved translational units on a three-way grid


80

• Polar and translational units (variation 1)

Now direction C, which contains polar units, is curved and placed in the direc-tion of the span (transverse direction). Directions D and E are also curved, but run diagonally over the span and contain translational units (Figure 3.33). The rows of translational units warp out of their common plane during deploy-ment. In order for this structure to be deployable, the bars of each transla-tional unit have to be able to move apart by sliding along their intermediate pivot hinge.


Elevation view

Figure 3.33: Polar units on a three-way grid (variation 1)


81

• Polar and translational units (variation 2)

Direction C contains translational units and is placed perpendicular to the span (longitudinal direction). Directions D and E are also curved, but run di-agonally over the span and contain polar units (Figure 3.34). The rows of translational units in direction C warp out of their common plane during de-ployment. In order for this structure to be deployable, the bars of each transla-tional unit have to be able to move apart by sliding along their intermediate pivot hinge.


Elevation view

Figure 3.34: Polar and translational units on a three-way grid (variation 2)


82

3.3.4 Double curvature grid structures

Two-way grid

• Translational units Translational units are placed in a two-way grid. A linkage of any curvature can be repeatedly placed in the grid in one direction (A). When the same link-age is also placed in the perpendicular direction (B), a synclastic shape is ob-tained (Figure 3.35). When the linkage in direction B is inverted, an anticlastic shape is obtained (Figure 3.36a). As long as a constant unit thickness is used throughout the structure, any two arbitrarily curved linkages can be combined in a grid (Figure 3.36b). Using translational units of constant unit thickness in a two-way grid is a very powerful method of creating structures with positive or negative Gaussian curvature [Langbecker, 1999, 2001].


Elevation view

Figure 3.35: Translational units on a two-way grid (synclastic shape)


83

Variations

Anticlastic shape (a) Arbitrary curvature (b)

Figure 3.36: Two variations for translational units on a two-way grid


84

Lamella grid

Spherical lamella grids have rhombus-shaped cells which are arranged circu-larly around a pole of the structure. They are deployed radially, either from the edge toward the centre or vice versa, in which case an opening exists in the middle.

• Translational units Curved translational units with constant unit thickness are placed in radial direction, to form a translational lamella dome (Figure 3.37).


Elevation view

Figure 3.37: Translational units on a lamella grid


85

• Polar units

Identical polar units are arranged radially to form a polar lamella dome (Figure 3.38).

3.4 Conclusion In this chapter the basic principles behind the design of deployable scissor structures composed of translational or polar scissor units have been ex-plained. It has been demonstrated how two-dimensional scissor linkages, with either translational or polar units, can be obtained by pure geometric construction, based on the geometric deployability constraint (Section 2.2.3). Several meth-ods for obtaining polar linkages based on a circular base curve and transla-tional linkages based on any curve have been discussed.


Elevation view

Figure 3.38: Polar units on a lamella grid


86

A geometric design method, for which the equations have been derived, has been proposed, based on the key design parameters for a space enclosure: the rise and span. It has been shown how based on these parameters, a base curve is determined, which is subsequently translated into a scissor geometry in its deployed configuration. Two-dimensional scissor linkages which comply with the deployability con-straint are always stress-free deployable, since the bi-stable deployment is caused by geometric incompatibilities associated with a three-dimensional configuration. For three-dimensional grid structures, the deployability con-straint is therefore a necessary, but not a sufficient condition for stress-free deployability. It has been found that, whether or not a certain configuration will be stress-free deployable, fully depends on the specific combination of the grid type (two-, or three-way), unit type (translational and/or polar) and cur-vature (plane, single or double). To assist the designer in the choice of a suit-able configuration, the stress-free deployable configurations known from lit-erature have been identified and discussed. It was found that designing single curvature structures (plane grids, barrel vaults) is quite straightforward when translational (plane or curved) and/or polar units are used on a two-way or three-way grid, provided that a constant unit thickness is imposed throughout the structure. But when it comes to us-ing these units in doubly curved deployable grids (domes, saddle shapes, arbi-trarily curved) some care must be taken in preserving the stress-free deploy-ability. As opposed to polar units, for which the only valid stress-free deploy-able geometry is a lamella dome, translational units can be used for a myriad of arbitrarily curved double curvature grids, provided that a constant unit thickness is used throughout the structure and the sums of the semi-lengths of the scissor bars is constant. This has lead to the conclusion that transla-tional units are a powerful means for designing stress-free deployable bar structures with positive or negative curvature.

Chapter 4 – Design of Foldable Plate Structures

87

Chapter 4

Design of Foldable Plate Structures

4.1 Introduction This chapter extends the work of Foster [1986/87] and Tonon [1991, 1993] who have proposed a variety of geometric shapes for foldable plate structures such as beams, barrel vaults and some doubly curved surfaces, as discussed in Section 2.3. However, not all configurations are foldable: some are merely de-mountable while others are constructed from smaller foldable sub-structures that have to be joined together afterwards [Hanaor, 2001]. Since the focus of this research is on rapidly erectable structures, only fully foldable configura-tions are discussed. An insight in the basic fold patterns and the impact they have on the geome-try of the final deployed configuration is offered. The individual plate geometry as well as the complete fold pattern in both the flatly folded and the final de-ployed configuration are thoroughly described. An advance is made in the design process by developing a geometric design method for which the necessary equations are derived. These equations are used to geometrically describe regular structures, which are single curvature structures (barrel vaults) characterised by a linear deployment. Additional pa-rameters, such as the expansion ratio, as a measure for the increase in size between the compacted, stowed configuration and the fully deployed configu-ration, are introduced. The geometric design method is then extended to right-angled structures – a variation of the regular structures – and circular structures. The latter are foldable domes characterised by a circular deployment. It is shown that it is possible, with appropriately chosen design values, to connect single and dou-ble curvature modules to form alternative foldable shapes, with maximum plate uniformity.


88

Figure 4.1: Typical foldable plate structure

4.2 Geometry of foldable plate structures Foldable plate structures consist of a series of triangular plates, connected at their edges by continuous joints, allowing each plate to rotate relative to its neighbouring plate. The plates can fold into a flat stack and unfold into a pre-determined three-dimensional configuration, as the one shown in Figure 4.1. The origami-like fold pattern consists of intersecting mountain folds (continu-ous lines) and valley folds (dashed lines), as shown in Figure 4.2. A fold pattern consists of a repetition of the basic plate element marked P, which is of trian-gular shape and must possess one angle of at least 90°. An array of several interconnected plates in the directrix direction is called a module, marked M. Now, the fold pattern can be expanded by repeating the module M any num-ber of times in the generatrix direction. In this case both patterns consist of two modules (m=2), which leads to the smallest symmetrical and usable solu-tion. Foldable plate structures can be characterised by the number of plates (p) they have in the span, which is found by adding up the number of whole and half elements in one module. The possible values for p are all odd integers equal to or greater than five. The minimal configuration (p=5) with two mod-ules (m=2) is depicted in Figure 4.2. Even values for p would make the struc-ture asymmetrical and are therefore discarded. Another important parameter is


89

the apex angle β at which the mountain fold lines intersect. The apex angle strongly influences the way curvature is introduced during deployment. So together, the three main design parameters p, m and β of the fold pattern, determine the overall morphology of the structure and its behaviour during deployment.

Figure 4.2: Fold patterns of type A and B for the smallest possible regular structure (p=5)

The apex angle can range from 90° to 180° and can be identical or variable throughout the structure. When all β’s are identical the structure is called regular, while changing one of the apex angles makes it irregular. Countless variations of shapes are possible but generally speaking the more exotic the fold pattern becomes, the less useful the resulting structure will be as a fold-able space enclosure. Figure 4.5 and Figure 4.6 show that minor changes, such as altering the outer most apex angle of pattern A to 90° (the minimum), can give it an interesting quality: the fully folded configuration is much compacter that the one of pattern A and B, shown in Figure 4.2, although all patterns have p=5 and share the same dimensions. Another consequence is that the triangular plate located at the base becomes quadrangular which makes in-corporating an entrance in the side panel easier than would be the case with triangular panels as in Figure 4.3 and Figure 4.4. Also, this so called right-angled structure has increased headroom near the sides when compared to its regular counterpart.

β


90

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

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°


91

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)

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°

A specific class of foldable structures which are of interest are the regular structures with their apex angle being a multiple or sub-multiple of 360°. In their most compact, fully folded configuration there is only one corresponding apex angle β for each specific number of plates p for which the edges of the end plates meet to form a closed circle. When designing a mobile structure, looking for the most compact folded configuration for a regular structure can only improve transportability. The example shown in Figure 4.7 is a regular structure with seven panels in the span (p=7), which will fold to its most com-pact form, only when the apex angle is 120°.


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Figure 4.7: Three stages of deployment for a basic regular foldable structure (p=7; β=120°):

completely unfolded, erected position and fully compacted for transport

A greater apex angle would cause the folded configuration to be less compact and show a gap, while a smaller apex angle would cause an overlap, therefore making the configuration unable to fold. Eqn (4.1) returns the apex angle β (degrees) for a given number of plates p and the first eight pairs of p and β are shown in Table 4.1.

( )( )1

3−−

=ppπβ (4.1)


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p β [°]

5 90

7 120

9 135

11 144

13 150

15 154,3

17 157,5

19 160

Table 4.1: The first eight values for β in terms of p for compactly foldable regular structures

The first three configurations from Table 4.1 (p=5, p=7, p=9) are pictured in Figure 4.8 and Figure 4.9. p=5; β=90° p=7; β=120° p=9; β=135°

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)


94

p=5; β=90° p=7; β=120° p=9; β=135°

Figure 4.9: Side elevation of the fully deployed configuration of the first three compactly fold-

able structures (p=5, p=7, p=9)

It is interesting to note what influence the apex angle has on the overall ge-ometry. Figure 4.10 shows two structures with similar fold patterns and the same number of panels (p=5). Although they have a different apex angle, their projection (or silhouette) in the erected position is identical. This means that for a certain number of elements the width of the resulting structure can be dramatically increased, simply by increasing the apex angle. As a result, mate-rial can be used more economically, since the structure with an increased apex angle will reach more width with the same amount of connections. This ex-plains an important characteristic concerning the design of such structures. From an architectural point of view, the span and the rise of a structure can be treated as key design parameters, as they determine the silhouette of the structure in vertical projection. This implies that for a certain chosen number of plates and a chosen rise and span, a myriad of possible configurations exist, all equal in projection, but with different actual plate dimensions. Since these plate dimensions, together with the apex angle β, determine whether or not the configuration will be compactly foldable, it is important to choose the ap-propriate plate geometry that will lead to the required deployed shape.


95

Figure 4.10: For a chosen number of panels p the apex angle β can be altered at will, affecting

the width of the structure and the compactly folded state

4.3 Geometric design In order to design these structures, a parameterisation of a single module and the structure as a whole is needed. This provides us with a description of all relevant design parameters such as the apex angle (β), bar length (L), the span (S), rise (Hr), width (W) and the number of plates (p) in one module. In order to understand the way the various design parameters affect the deployment be-haviour and the geometry of the various configurations, we need to establish their mutual relationships. A significant parameter is the deployment angle θ, measured between a triangular face and the vertical axis, as shown in Figure 4.11. The deployment angle θ determines to what degree a structure is un-folded with values ranging from 0° (fully compacted configuration) to 90° (flat, completely unfolded position). In between these extremes there is a unique value for θ that corresponds with the fully erected position, i.e. a semi-circular shape for the regular structure when viewed in elevation. For an ir-regular structure such as the one depicted in Figure 4.5 we are looking for the configuration whereby the side plates are standing perfectly vertical.


96

Figure 4.11: Parameters used to characterise a foldable structure: length L, span S, width W,

apex angle β and the deployment angle θ

Figure 4.12: A foldable plate and its parameters: length L, height H, H1, H2, apex angle β, the

deployment angle θ and angles α, α1, α2

The following relationships can be derived from Figure 4.12:

α

α2

α1 θ

β/2

L

H

H2

H1

Hr


97

2βπα −

= (4.2)

αtan2LH = (4.3)

θcos1 HH = (4.4)

11 tan2

αLH = (4.5)

θsin2 HH = (4.6)

22 tan2

αLH = (4.7)

Substituting (4.3) in (4.4) and (4.6), and equating (4.4) to (4.5) and (4.6) to (4.7) gives:

θαα costantan 1 = (4.8)

θαα sintantan 2 = (4.9)

Equations (4.8) and (4.9) will be used to determine the relationship between the apex angle, the number of plates and the deployment angle for regular, right-angled and circular structures.

4.3.1 Regular structures

The geometric relationship between the apex angle, the number of plates and the deployment angle can now be derived. There is a unique value for the de-ployment angle that corresponds with the fully erected position of the regular structure (although in reality the deployment angle can range from 0° to 90°, when θ is mentioned hereafter, it is this unique value for the fully deployed configuration that is being referred to). As Figure 4.1, Figure 4.3 and Figure 4.4 show, in this deployed position the bottom most (half) panels touch the ground along their bottom edge and in elevation view the silhouette of the structure is a perfect semi-circle. It can be described as a barrel vault with cylindrical curvature.


98

Eqn (4.8) allows α1 to be written in terms of α and θ: ( )θαα costantan 1

1−= (4.10)

Figure 4.13 represents the projection in the vertical plane of a regular plate linkage with p plates. Empirically, it can be observed that α1 is a sub-multiple of π/2. This results in the following equation:

( )2

1 1πα =−p (4.11)

Figure 4.13: Perspective view and side elevation of the vertical projection of a plate linkage for

empirically determining the relationship between α1 and p

Substituting (4.2) and (4.11) in (4.10) gives:

( )2

cos2

tantan1 1 πθβπ=

−

− −p (4.12)

Finally, Eqn (4.13) gives the value of θ in terms of p and β for regular struc-tures:

( )

−

= −

2tan

12tancos 1 βπθ

p (4.13)

The relationship between these parameters for geometries with p=5, p=7 and p=9 is plotted in the graph of Figure 4.14. For example, looking at the regular structures of Figure 4.10 we can see that, while both configurations have p=5,

α1


99

the one on the left has β=120° and the one on the right has β=90°. From the graph we can read the appropriate deployment angle θ for both structures: 44.16° and 65.53° respectively. Looking at the graph from Figure 4.14 we can draw the following conclusions:

• The higher the number of plates, the blunter the apex angle can be. There is, for a given number of plates, a minimum and maximum value for the apex angle. These values are given in Table 4.2. Lower or higher values generate configurations that cannot be made into a foldable structure

• For a fixed number of plates, an increased apex angle means a de-creased deployment angle

• For a fixed apex angle, increasing the number of plates also increases the deployment angle

Relationship between the apex angle β and the deployment angle θ for REGULAR STRUCTURES

p=5 p=7 p=9

0

10

20

30

40

50

60

70

80

90

90 95 100 105 110 115 120 125 130 135 140 145 150 155Apex angle β [deg]

Dep

loym

ent a

ngle

θ [

deg]

Figure 4.14: The relationship between the apex angle β and the deployment angle θ for regular

structures with p=5, p=7 and p=9


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Number of plates p βmin[°] βmax [°]

5 90 135

7 120 150

9 135 157.5

Table 4.2: Minimum and maximum possible apex angles for regular structures with 5, 7 or 9

plates

Figure 4.15: Elevation view and perspective view of the deployment of a regular five-plate

structure with β=120°

Crucial to the design of space enclosures are the span and rise, which are in-fluenced by the number of plates and their length. The relationship between the number of plates p, the plate length L, the span S (defined between the outer points) and the rise Hr for regular structures is expressed by Eqns (4.14) and (4.15). Table 4.3 shows the span S and rise Hr for a given number of plates p for regular foldable structures in terms of the plate length L.

−

=1

tanp

SL π (4.14)


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Because in projection the profile is a semi-circle, the rise is given by

2SH r = (4.15)

p S (x L) Hr (x L)

5 1.41 0.71 7 2.00 1.00 9 2.61 1.31

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.

An expression can be found for the relationship between the width w of a sin-gle module and the plate length L, the apex angle β, the deployment angle θ for both regular and right-angled structures. From Figure 4.11 it can be seen that the width w of a single deployed module is equal to H2. By substituting Eqn (4.9) in (4.7) we obtain the following expression for w:

θα sintan2Lw = (4.16)

Substituting Eqn (4.2) in (4.16) gives an expression for w in terms of β and θ:

θβπ sin2

tan2

−=

Lw (4.17)

Or in other form:

θβ sin2

cot2Lw = (4.18)

Eqn (4.19) gives the total width W in terms of the number of modules m. wmW = (4.19)

An important characteristic in transportable structures is the compactness of the flatly folded plate linkage in its stowed condition. The space required will be the volume determined by the area of the polygonal footprint of the folded configuration and the total thickness of the stack of plates, determined by the individual plate thickness. For the fully foldable configurations from Figure 4.8, the area of the polygon is expressed in terms of the edge length Ledge.. Figure 4.16 shows the relationship


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between the apex angle β, the plate length L and the edge length Ledge which allows the following relationship to be written:

2sin2 β

LLedge = (4.20)

Now that the edge length Ledge of the polygon is known, the area can be ex-pressed. Figure 4.16 shows the expressions for the area of a polygon in terms of its edge length [Mathworld, 2007]. p=5; β=90° p=7; β=120° p=9; β=135°

2

2edgeL

23

23

edgeL ( ) 2212 edgeL+

Figure 4.16: The parameters associated with the polygonal contour of the flatly folded configu-

rations with p=5, p=7 and p=9 and the expressions for the area in terms of the edge length Ledge

Using Eqn (4.20) and the expressions from Figure 4.16, the area of the com-pact configuration is expressed in terms of Ledge for the three basic fully fold-able configurations (Table 4.4).

Configuration Areacompact

p=5, β=90° 0.5 2L

p=7, β=120° 0.87 2L

p=9, β=135° 1.41 2L Table 4.4: The area of the compact configuration for (p=5, β=90°), (p=7, β=120°) and (p=9,

β=135°) in terms of the plate length L

β

L

Ledge


103

The area of the sectional profile of the fully folded configuration gives us in-formation on the space the fully deployed configuration will occupy. The span S has been defined as the distance between the lower extremities of the circu-lar profile (Figure 4.11). The area of the sectional profile equals the area of a semi-circle with radius Hr. Using the expressions for Hr in terms of the plate length L from Table 4.3, we can now express the area of the sectional profile for the deployed configurations, for which the results are given in Table 4. 5.

Configuration Areadeployed

p=5, β=90° 0.79 2L

p=7, β=120° 1.57 2L

p=9, β=135° 2.7 2L Table 4. 5: The area of the sectional profile of the deployed configuration for (p=5, β=90°), (p=7,

β=120°) and (p=9, β=135°) in terms of the plate length L

Now, the ration between the compacted shape and the fully deployed configu-ration can be expressed as the expansion ratio λ:

deployed

compact

AreaArea

=λ (4.21)

The values for λ are given in Table 4. 6. It can be seen that the smallest con-figuration demonstrates the largest expansion.

Configuration Expansion ratio λ

p=5, β=90° 0.63 p=7, β=120° 0.55 p=9, β=135° 0.52

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

The plate thickness has a great impact on the compactness of the stowed con-figuration. As can be observed from Figure 4.3, Figure 4.7 and Figure 4.15, the geometry of these structures is such, that each module (as defined in Figure 4.2) in its flatly folded configuration consists of two overlapping layers of plates. When tp represents the thickness of a single plate element, then the


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total thickness Tp of the compactly folded configuration with m modules is given by:

pp tmT 2= (4.22)

The expressions for the area and the thickness of the compactly folded con-figuration allow to determine the volume in the stowed position, simply by multiplying Areacompact by the total plate thickness Tp. Also, the volume occupied by the fully deployed configuration can be determined by multiplying the area of the sectional profile Areadeployed by the total width W from Eqn (4.19).

4.3.2 Right-angled structures

Analogously, when p and β are known, the deployment angle for a right-angled structure such as the one in Figure 4.5 can be calculated by solving Eqn (4.23) for θ:

( )22

tancostan22

cotcostan3 11 πβθβθ =

+

− −−p (4.23)

The approach used for deriving Eqn (4.23) is identical to the empirical method used for regular structures and will therefore not be repeated. When looking at the graph from Figure 4.17, it is clear that altering the first and last apex angles to 90° generates a completely different behaviour as compared to regular structures:

• The higher the number of plates, the less sharp the apex angle can be. There is, for a given number of plates, a minimum and maximum value for the apex angle, which is given in Table 4.7. Lower values generate configurations that cannot be made into a foldable structure

• For a fixed number of plates, an increased apex angle means an in-creased deployment angle

• For a fixed apex angle, increasing the number of plates also increases the deployment angle


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Relation between apex angle β and deployment angle θ for RIGHT-ANGLED STRUCTURES

p=5

p=7

p=9

65

70

75

80

85

90

90 100 110 120 130 140 150 160 170 180

Apex angle β [deg]

Dep

loym

ent a

ngle

θ [d

eg]

Figure 4.17: The relationship between the apex angle β and the deployment angle θ for right-

angled structures with p=5, p=7 and p=9

Number of plates p βmin[°] βmax [°]

5 90 180

7 120 180

9 135 180

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

Although the apex angle can range from βmin to 180°, the highest values will generate quite useless structures since their structural thickness converges to zero. As opposed to the regular structures from Figure 4.8, there is no unique optimal value for the apex angle β for which a right-angled structure with p=5 will be compactly foldable without overlap or gap. All valid values for β (from the graph in Figure 4.17) will lead to compactly foldable five-plate con-figurations. This can be seen in Figure 4.18, which shows three configurations for p=5, each of which has a different apex angle. No matter which apex angle


106

is chosen (within bounds imposed by Eqn (4.23) and Figure 4.17), all resulting configurations are compactly foldable. p=5; β=90°; θ=65.5° p=5; β=120°; θ=68.1° p=5; β=135°; θ=71.5°

Figure 4.18: Plate element, fold pattern, compactly folded configuration and fully deployed con-

figuration (front elevation and side elevation) for three compactly foldable five-plate right-

angled structures (drawn to scale)

To form an alternative configuration, a five-plate configuration of a right-angled structure can be connected to a regular structure, when a type B pat-tern is used (Figure 4.2). When five plates are used, such patterns will always have an identical vertical edge, for any value for β. By connecting along this common edge, linear arrays of regular and right-angled structures are possible, as shown in Figure 4.19.


107

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 β

4.3.3 Circular structures

Formulas for the geometric design in terms of the apex angle and the number of plates have been proposed. Not only single curvature structures with linear deployment are possible: dome-like structures with circular deployment can also be used as foldable space enclosures. As opposed to regular structures, their fold pattern cannot be developed in a plane. To flatten the shape, radial incisions have to be made. The foldable dome from Figure 4.20 and Figure 4.21 consists of a radial pattern of sectors joined together along their common edge. When this configuration is cut along one radius, it can be folded into a compact stack of plates by rotating its sectors around a vertical axis. These configurations are characterised by the number of sectors q that make up a full circle in plan view: in this case q=8. Empirically, it has been found that its value can be any even integer equal to or greater than six (due to apex angle restrictions for βmin). Values below six would give rise to non-foldable solu-tions. An interesting application is to connect half of a dome to an array of regular structures, making a fully closed space enclosure, as shown in Figure 4.23. It would make the design process and fabrication easier when a single plate size could be used for both the half domes and the regular structure. Therefore, once the number of sectors q is chosen, the resulting apex angle β for the circular structure is also used for the regular structure, leading to a geometry with uniform plate elements.

Identical vertical edge


108

Figure 4.20: Top view and perspective view of circular foldable structure

Fold pattern developed in a plane Sector module

Figure 4.21: Fold pattern and a single sector of a circular structure with q=8

Let q be the number of sectors in the circular structure, which can be freely chosen. Again, the geometric relationships derived from Figure 4.12 are used.

Figure 4.22: Horizontal projection of a plate linkage for empirically determining the relationship

between α2 and q

α2


109

Eqn (4.9) allows α2 to be written in terms of α and θ:

( )θαα sintantan 12

−= (4.24)

Figure 4.22 represents the horizontal projection of half a circular plate linkage. Empirically, it can be observed that α2 is a sub-multiple of π. This results in the following equation:

πα =2q (4.25)

Substituting Eqn (4.2) and Eqn (4.24) in Eqn (4.25) gives the relationship be-tween q, β and θ for circular structures:

πβπθ =

−−

2tansintan 1q (4.26)

Or in other form

= −

2tantansin 1 βπθ

q (4.27)

When a number of units q is chosen, solving Eqns (4.13) and (4.27) simultane-ously will return a unique value for β and the corresponding deployment angle θ. Table 4.8 gives for a chosen q the appropriate β and θ. For example, the combined regular-circular structure (p=5, q=8) from Figure 4.23 has β=119.3° and θ=45 throughout the entire structure.


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For p=5

q β [°] θ [°]

6 109.2 54.3

8 119.3 45.0

10 124.5 38.1

12 127.5 32.9

14 129,4 28.9

16 130.6 25.7

Table 4.8: Values for β and θ for a chosen q (circular structure), combined with a regular struc-

ture (p=5)

Figure 4.23: Connecting a regular module with two half-domes leads to an alternative fully

closed configuration with high plate uniformity

Figure 4.24 shows the first three configurations (q=6, q=8 and q=10) for cir-cular structures which are calculated with Eqns (4.13) and (4.27). They are combined with a compatible regular structure (p=5) with identical plate ele-ments. Although Eqn (4.13), when used separately, allows the calculation of regular structures with any number of plates (p>5), the combined calculation for the fully closed configuration using Eqns (4.13) and (4.27) is only specifi-cally applicable to regular structures with five plates, combined with a circular structure with any q equal to, or greater than six. Empirically, it has been found that regular structures with a higher number of plates (e.g. p=7) cannot be mutually connected with a seven-plate circular counterpart, if all plates remain identical. With uniform plates, no common edge for connecting the


111

structures was found. At this stage, it is unclear whether an irregular geometry with variable apex angles can be found which would solve this problem. q=6; β=109.21°; θ=54.34°

q=8; β=119.28°; θ=44.9°

q=10; β=124.47°; θ=38.11°

Figure 4.24: Circular structure with q=6, q=8 and q=10 (top view) and its respective combina-

tion with a compatible regular structure (perspective view)

4.3.4 Alternative configurations

By using the previously discussed regular, right-angled and circular plate structures as building blocks, certain alternative configurations can be com-posed. Figure 4.25 shows a few possible variations.


112

Figure 4.25: Some examples of alternative configurations

4.4 Conclusion In this chapter the findings of Foster [1986/87] and Tonon [1991, 1993], who have proposed a number of geometric shapes for foldable plate structures have been extended. Some foldable configurations are merely demountable while others are constructed from smaller foldable sub-structures that have to be joined together afterwards [Hanaor, 2001], thus compromising the trans-portability and the speed of erection. Therefore, a geometric design method has been developed which entails the design of fully foldable and rapidly erectable configurations with single or double curvature. A comprehensive study of the design parameters (deployment angle, apex an-gle, plate length, number of plates) has provided a thorough insight in the ge-ometry of individual plates and the fold patterns they constitute, as well as


113

the impact on the compactly folded shape and the final deployed configura-tion. An advance has been made in the design process by developing a geometric design method for which the necessary equations have been derived. These equations are used to geometrically describe regular structures, which are sin-gle curvature structures (barrel vaults) characterised by a linear deployment. By introducing the deployment angle θ, it was made possible to derive equa-tions - expressed in terms of the apex angle β and the number of plates p - which explain the relationship between the plate geometry and the fully de-ployed configuration. By analysing these formulas and the resulting graphs, an understanding has been provided on what minimum and maximum values for β and p give rise to compactly, fully foldable configurations. Additional characteristics, such as the expansion ratio – a measure for the in-crease in size between the compact and the deployed state - have been stud-ied and expressed in terms of the key parameters, i.e. the plate length and the apex angle. It has been shown that, by slightly altering the regular plate geometry, an in-teresting variation arises: a right-angled structure. This variation provides, in its deployed configuration, increased headroom near the sides of the structure and has quadrangular side panels, providing the option of a larger entrance space. It has also been found that, for a five-plate geometry, this type of structure is always compactly foldable. The design method has been extended to include right-angled structures, as well as circular structures. The latter are foldable domes characterised by a circular deployment. Further, it has been proven possible that, by solving the equations for dome-like configurations and regular configurations simultaneously, a single plate element is found which can be used for both regular (single curvature) and circular (double curvature) shapes with five plates. Because of the plate uni-formity, it was shown possible to combine these shapes into alternative double curvature shapes.


114

The current geometric design method is based on basic foldable shapes with a circular sectional profile and with high plate uniformity. The approach can be extended to include single and double curvature shapes with variable curva-ture. Although the method provided allows the design of regular foldable structures with any number of plates, the equations for circular structures, as well as those for the combined shapes, are currently valid for the smallest configuration (p=5). Further study is required to include configurations with a higher number of plates.

Chapter 5 – Introduction to the Case Studies

115

Chapter 5

Introduction to the Case Studies

Figure 5.1: Some of the concepts for mobile structures presented in the following chapters


116

5.1 Introduction This chapter serves as an introduction to the concepts proposed in the follow-ing chapters, in which a number of case studies will be discussed, bringing into practice the findings and methodologies presented in the previous chapters. All concepts presented are mobile structures for architectural applications that use either scissor-hinged bars or bars connected by foldable joints combined with a tensile surface for weather protection (Figure 5.1). An advance is made in the field of mobile structures by proposing novel concepts for deployable bar structures or by making use of existing concepts in an alternative, novel way. Four case studies, in which these concepts are put into practice, are chosen in such a way that a variety of structural systems is used: bars, cables and mem-branes in single and double curvature shapes, built from translational and po-lar scissor units on a triangular (case study 1) or quadrangular grid (case study 2), or based on the geometry of foldable plate structures (case study 3), or angulated (hoberman’s) units connected in a linear way (case study 4). The geometry of the case studies is chosen in such a way that mutual comparison becomes possible. Per case study, a general description is given and the overall geometry, together with the dimensions, is discussed. As part of a feasibility study in which the proposed concepts are evaluated architecturally, kinematically and structurally, this chapter elaborates on the approach taken for the structural analysis of the case studies. It is explained how a simplified approach is used for the preliminary design and what con-secutive steps are taken in the structural analysis. By using Eurocode 1 [2007], the wind and snow loads are calculated and it is shown how these will effect the different load zones of the structure.


117

5.2 Geometry The first three case studies are concepts for small, easily erectable mobile shelters of semi-cylindrical shape (barrel vault). Deriving all three geometries from the same basic shape - a semicircle with a radius of 3 m - makes com-parison in terms of architectural and structural qualities more convenient (Figure 5.2). The length of the barrel vaults is approximately 10 m, which does not exceed the maximum length/width ratio for barrel vaults of 2:1. In case several structures are connected in the longitudinal direction, and therefore exceeding the length/width ratio, diaphragm walls or some kind of stiffening cable arrangement should be introduced. Also, the structural thickness t is chosen to be identical for each case. These barrel vaults are open structures with no back nor front, as shown in Figure 5.3. Then, using the same structural elements from the single curvature structures, alternative configurations are proposed, to form double curvature shapes with a fully closed surface. The sin-gle and double curvature shapes will be referred to as open, respectively closed structures.

Case study 1 Case study 2 Case study 3

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)

t t t


118

Figure 5.3: Overall geometry for the case studies: single curvature shape (open) and double cur-

vature shape (closed)

The first case study consists of translational scissor units placed on a three-way grid (Figure 5.4). Scissor structures with translational units can be easily made into single or double curvature structures with a quadrangular grid. Two-way grids have quadrangular grid cells which make them susceptible to skewing (angular distortion of the grid cells). The proposed concept is a varia-tion: a single curvature shape with translational units on a three-way grid (tri-angular grid cells). This makes triangulation of the grid cells, to counter the skewing effect, obsolete.

Figure 5.4: Perspective view of the single and double curvature geometries

for cases studies 1, 2 and 3


119

Polar scissor arches, combined with translational units on a two-way grid are used for the second case study (Figure 5.4). Using polar scissor arches is an effective way of introducing curvature in a quadrangular scissor grid, there-fore, these polar barrel vault structures are relatively common. Identical polar units can also be used to make a deployable dome-like shape with rhombus-shaped grid cells, otherwise called a lamella dome. As a variation, two halves of this dome are added to the open structure to turn it into a closed, double curvature shape. The third case study has its geometry and kinematic behaviour based on that of foldable plate structures. Instead of plates, bars are used as structural com-ponents to hold up a membrane surface (Figure 5.4). The reason for using a continuous membrane instead of separate plates is explained in Section 8.1. The bars are connected by foldable joints, which behaves like a miniature fold-able plate mechanism. Optionally, scissor units can be added to influence the kinematic behaviour. The fourth and last case study is a deployable mast of approximately 8.5 m high which, when deployed, holds up three circularly arranged membrane canopies, each measuring 10 m x 5 m. It is made from scissor modules, stacked upon each other to form a vertical linear deployable truss-like structure. Dur-ing deployment the mast - to which the membrane elements are attached - expands vertically. By doing so the canopy becomes gradually tensioned until maximal deployment is reached. The three scissor modules which make up the structure consist of angulated scissor elements. Angulated scissor units were not discussed in the context of the design of deployable scissor arches to form single curvature barrel vaults, for reasons specified in Section 3.1. However, in this particular case, as they are connected in a linear manner to expand verti-cally, they demonstrate a particular deployment behaviour which is used to the advantage of the proposed concept (Section 9.1).


120

Figure 5.5: Perspective view and side elevation of case study 4

5.3 Structural analysis of the proposed concepts

5.3.1 General approach

Since all considered case studies are mobile architectural shelters, a canopy serving as a climatological protection is an integral part of the design. Because this tensile surface is subject to wind and snow loading and it is physically attached to the bar structure, its behaviour under load is to be determined first. Subsequently, the actions of the membrane are transferred to the bar structure which allows the members to be sized as part of a preliminary struc-tural design. So first, the formfinding of the tensile surface is done with EASY-software [Technet, 2007] which uses the force-density method [Mollaert, Forster, 2004]. All membranes are attached to the nodes of the bar structure. These nodes become the boundary points for the membrane geometry that is used as input for the formfinding process. A moderate level of pre-tension is introduced in the membrane (1 kN/m in warp and weft direction). For mobile structures as the ones presented here this seems reasonable, as it provides a fair balance between the tension in the membrane and the resulting forces transferred to the bar structure. Also, avoiding the need for highly sophisticated material for introducing the pre-tension is in accordance with the low-tech nature of the structures. After the equilibrium form has been calculated, snow and wind


121

loads are applied, of which the latter are determined according to the Euro-code 1 [2007]. Seven load cases are used for the structural analysis of the ten-sile surface which returns the reaction forces of the membrane on the bound-ary points. These vectors are then transferred to a FE-model (Finite Element Model) of the bar structure which is used for a preliminary structural design performed in ROBOT [Robobat, 2007]. This simplified approach is chosen because these software packages are readily available and perform their respective tasks (tensile surface design and struc-tural analysis and design) well and because this is a preliminary design to test the feasibility of the proposed concepts, this method seems well-suited for the purpose. The three loads (transverse wind, longitudinal wind and snow) cannot be separately applied in EASY and the subsequent load vectors (action of the membrane on the boundary points) combined into several load cases, which will be imposed on the bar structure in ROBOT. After all, this would mean that the load vectors, obtained from the non-linear calculations on the membrane, would become superimposed, which leads to false results. Instead, the loads are combined into several load cases (pre-stress combined with wind and/or snow) which are applied in EASY, and the non-linear response of the mem-brane is measured. Besides, combining the separate load vectors obtained from EASY into load cases in ROBOT would falsely take the pre-tension several times into account, because it would be an integral part of each calculated load case in EASY. Therefore each possible load case has to be manually de-termined and applied on the membrane model in EASY, after which the reac-tion forces are transferred to the FE-model of the bar structure. There, the load cases are combined with the self-weight of the bar structure. While the wind and snow load are live loads, the pre-tension in the membrane is treated as a permanent load, with all safety factors applied accordingly. It should be noted that this method is a simplification in the sense that it does not take into account the effect the bar structure has on the membrane, since its displacements under load would alter the boundary conditions of the mem-brane. If a more profound and accurate analysis would be required, an inte-grated, more detailed model of both the tensile surface and the skeletal struc-ture - which takes into account the mutual response - would be a better


122

route. Or, in the current approach, going back and forth between programmes, each time using the obtained results from one calculation as new boundary conditions for the next, in an iterative way, would lead to more accuracy, but with a huge effort and little return for this study. In summary, these are the steps to be taken in the structural analysis: - Formfinding of the tensile surface: calculation of the equilibrium form of the membrane under pre-stress - Statical analysis of the tensile surface: determining the reaction forces of the membrane on the boundary points under different load cases - Structure design of the skeletal structure: the bar structure is dimensioned with the action of the membrane and the selfweight of the structure applied

Figure 5.6: Wind and snow action on the open and closed structure

5.3.2 Load cases

Three different live loads are considered: longitudinal wind, transverse wind and a snow load (Figure 5.6). For this simplified approach, only two wind di-rections are used, for mutual comparison of the different configurations. In a more profound analysis, the 45° wind direction should also be included, since its effect, especially on the open structure, could prove significant. The closed


123

structure as well as the open structure are divided into three zones or load areas, allowing a differentiated wind load application: pressure or suction. Whenever assumptions are made or simplifications are done, the most unfa-vourable value is chosen. The reference wind pressure is expressed as [Eurocode 1]:

refref vq 2

2ρ

=

(5.1)

with the air density ρ =1.25 kg/m3 and the reference velocity refv determined

from:

0,... refALTTEMDIRref vcccv = (5.2)

For Belgium 0,refv =26.2 m/s, the altitude factor ALTc =1, the direction fac-

tor DIRc =1 and for the temporary factor TEMc a value of 0.8 is chosen which

corresponds with a one month exposure (November). This gives a value for

refq = 0.275 kN/m2.

The total wind pressure acting on the surfaces is obtained from:

ie www −= (5.3)

with we the pressure on the external surfaces and wi on the internal surfaces, given by:

( ) peeerefe CZCqw ..= (5.4)

( ) piierefi CZCqw ..= (5.5)

As terrain category, the most severe is chosen: category I, flat country. The external pressure coefficient is determined by the geometry of the structure, which means the surface is divided into several zones, each with their own Cpe. The internal pressure depends on Cpi and is calculated by taking the influence of openings in the surface into account, by means of a factor µ (opening ratio), which is given by:


124

T

WL

AA ,=µ

(5.6)

Here, AL,W stands for the total area of openings at the leeward and wind paral-lel sides and AT stands for the total area of openings at the windward, leeward and wind parallel sides. The closed structure is assumed to have an opening (entrance) on both the front and back (windward and leeward side for the lon-gitudinal wind) which will allow internal wind pressure to be generated. By calculating the µ’s for both closed and open structures, values for Cpi of -0.5 (suction under transverse wind) and 0.14 (pressure under longitudinal wind) are obtained. The permeability Cpi,a of the membrane has not been taken into account. Figure 5.7 shows the different load zones in plan view for both the open and closed structures. The wind pressure and suction are shown sche-matically in a transverse and longitudinal section view. Table 5.1 gives a sum-mary of all obtained values for the external and internal wind pressure and exposure coefficients and the resulting total pressure per load zone and wind direction. Although the schematic representations of the sectional profiles are shown as semi-circles, the loads are applied to the actual non-smooth shape of the membrane model in EASY.


125

Plan view Section view

CLOSED structure Transverse wind

OPEN structure Transverse wind

CLOSED structure Longitudinal wind

OPEN structure Longitudinal wind

Figure 5.7: Schematic representation of considered wind loads on the closed and

open structures

A B C

D E F

I

H

G

L K J


126

Transverse wind

Zone Cpe

we [kN/m2]

Cpi

wi [kN/m2]

w [kN/m2]

A 0.8 0.4 -0.5 -0.25 0.65 B -1.2 -0.6 -0.5 -0.25 -0.35

CLO

SED

C -0.4 -0.2 -0.5 -0.25 0.05

D 0.8 0.4 -0.5 -0.25 0.65 E -1.2 -0.6 -0.5 -0.25 -0.35

OPE

N

F -0.4 -0.2 -0.5 -0.25 0.05

Longitudinal wind

Zone Cpe we [kN/m2]

Cpi

wi

[kN/m2] w [kN/m2]

G 0.7 0.4 0.14 0.1 0.3 H -0.2 -0.12 0.14 0.1 -0.22

CLO

SED

I -0.3 -0.18 0.14 0.1 -0.3

J -0.6 -0.35 0.14 0.1 -0.45 K -0.3 -0.2 0.14 0.1 -0.3

OPE

N

L -0.2 -0.12 0.14 0.1 -0.22

Table 5.1: Values for the wind pressure w per zone

for the closed and open structures


127

Transverse section view Longitudinal section view

CLOSED structure

OPEN structure

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

The snow load is determined with the following formula [Eurocode 1]:

ktei sCCs ...µ= (5.7)

with ks the characteristic snow load on the ground in kN/m2, the temperature coefficient tC , the exposure coefficient eC and the coefficient iµ (form factor

for the snow load). All factors tC , eC and iµ are given the value 1 and for the

characteristic snow load a value of 0.5 kN/m2 is chosen. This leads to a snow load of 0.5 kN/m2, which is a typical value. Because surfaces with an inclina-tion of 60º and above are assumed to be without snow, the snow load is only applied to the zones at the top of the canopy, as shown in Figure 5.8. Based on the three mobile loads– transverse wind, longitudinal wind and snow - seven load cases are compiled, according to the prescribed combination fac-


128

tors, with the most severe conditions chosen (weather conditions statistically occurring once every 50 years). It is assumed that the two wind loads, each having a distinct direction, cannot occur simultaneously. With G the perma-nent loads and Q the mobile loads and γ their respective safety factors, the total load can be written in its general form:

S = Σi γG Gk,i + γQ Qk,1 + Σj γQ Ψ0,1 Qk,j

(5.8)

For the ultimate limit state (ULS) γG=1.35 and γQ=1.50. These values become equal to 1 when the service limit state (SLS) is considered. In ULS the strength (determination of sections) and stability (buckling analysis) of the structure are verified, while SLS is used for verification of the stiffness (displacements). Table 5.2 shows the resulting seven load cases for both ULS and SLS. For ex-ample, ULS 1 and SLS 1 are a combination of pre-stress, transverse wind and snow and are given by: ULS 1=1,35 × pre-stress + 1.5 × transverse wind + 1.5 × 0.6 × snow SLS 1=1× pre-stress + 1 × transverse wind + 1 × 0.6 × snow

Load case Permanent load

Main solicitation Additional solicita-tion

ULS/SLS 1 Pre-stress transverse wind snow ULS/SLS 2 Pre-stress snow transverse wind ULS/SLS 3 Pre-stress longitudinal wind snow ULS/SLS 4 Pre-stress snow longitudinal wind ULS/SLS 5 Pre-stress snow ULS/SLS 6 Pre-stress transverse wind ULS/SLS 7 Pre-stress longitudinal wind

Table 5.2: The seven load cases used for calculations in EASY

These load cases are used to determine the actions of the membrane on the boundary points in EASY. Therefore, the self-weight of the bar structure can-not be included at this stage. These load vectors will be transferred to the FE-


129

model in ROBOT, to analyse and design the bar structure under the considered load cases. With all load cases determined, a structural analysis of each case study will be performed, following the earlier described approach. At this stage the self-weight of the bar structure (with a safety factor of 1.35 applied) is included in the load cases. The structural analysis of the first case study – the three-way grid barrel vault with translational scissor units – will be discussed in detail. Of the remaining case studies, which are calculated analogously, a concise sum-mary of the results will be given. Structural analysis in the following chapters will show that either buckling or strength is the dimensioning ULS criterion. Although (static) stiffness stan-dards are not stringent for this kind of foldable structures, other SLS criteria, such as fatigue should be considered. Fatigue is defined as a progressive, localised structural damage that occurs when a material or a component is subjected to cyclic or fluctuating strains at nominal stresses. The values of stresses when fatigue occurs are less than the static yield strength of the material. This means that structural failure can happen at stresses below ultimate tensile stress (ULS). Although this phenomenon has not been included in the structural feasibility study, the designer should be aware of its possible impact on the structural performance. Since fatigue is caused by dynamic (and/or moving) loads, an obvious cause in the case of deployable structures could be the repeated folding and unfolding of the structure. However, as this will amount to a relative low number of cy-cles compared to the lifespan of the construction, another phenomenon is likely to influence the structural performance, namely fluctuating wind loads (on the erected configuration). Therefore the calculation methodology used in the European standards [Eurocode 3] will be explained.


130

In case of an applied load with variable amplitude, the accumulative damage D has to remain smaller than 1. D is expressed by the Palmgren-Miner equation [Eyrolles, 1999]:

∑=i

i

NnD (5.9)

with ni the number of cycles of the extent of fluctuating stresses iσ∆ during

a reference period and Ni the critical amount of load cycles during the same reference period and under the extent of fluctuating stresses iM σγ ∆ which

the structure or the material can withstand. Figure 5.9 shows the consecutive steps to determine all the necessary parame-ters [Eurocode 3, 2007]:

a) Based on existing knowledge of similar constructions the typical so-licitation sequence can be determined. Herein a realistic fluctuation of wind pressure is difficult to define. A conservative way to solve this is-sue is to consider a constant amplitude of the design wind pressure.

b) A stress history can be set up for a particular structural detail. A stress-time graph can also be determined; either by measuring on similar constructions or using dynamic calculations of the response of the construction.

c) Stress histories may be evaluated by using one of the following known techniques to determine the extent of stresses and their number of cycles: i. Rain drop method

ii. Reservoir method d) The range of stress variations are determined by drawing them, in

combination with the number of cycles, in a descending way. e) Using the Wohler or S-N curves, the number of cycles Ni can be read

of. For typical structural details these curves already exist and can be plotted in families. Each construction detail family is defined by a ref-erence value of the resistance against fatigue cσ∆ at a constant am-

plitude and a number of cycles of 2.106 cycles. f) The accumulative damage caused by the different stress variations can

be calculated using the Palmgren-Miner equation


131

a b c d e f

Figure 5.9 : Method of accumulate damage [Eurocode 3, 2007]


132

A similar way of working can be followed to determine the damage created by varying shear stresses iτ∆ .

Further experimental research has to be conducted to see if fatigue is a di-mensioning criterion. The determination of the resistance against fatigue of the specific joints used in the presented cases requires experimental data, which can not be given at this stage of research. Furthermore, a more realistic stress history due to fluctuating wind loads requires wind tunnel testing.

5.4 Conclusion In this chapter the concepts have been introduced which will be presented in the subsequent chapters. These case studies use either scissor-hinged bars or bars connected by foldable joints combined with a tensile surface for weather protection. The proposed concepts contribute to the field of mobile deployable bar structures by being either novel, or by making use of existing ideas in a new way. To facilitate mutual comparison, an overall shape has been determined on which the single curvature geometry from case studies 1, 2 and 3 is based: a barrel vault shape with a radius of 3 m and an approximate length of 10 m. A fourth concept has been introduced, which uses an 8.5 m high linear deploy-able truss-like structure to support its architectural envelope. The general approach for the structural analysis and the steps it involves have been clarified in this chapter. It has been explained how the considered load cases [Eurocode 1, 2007], for both ULS and SLS, have been compiled and ap-plied to the numerical model of the membrane. Subsequently, the obtained reaction forces on the boundary points are transferred to the corresponding nodes of the FE-model of the primary structure. This simplified approach seems suited for a preliminary design within the framework of a feasibility study.


133

It has been noted that this approach does not take into account the mutual response between the bar structure and the membrane. Therefore, it is sug-gested that, for a more profound analysis, an integrated model of both the primary structure and the tensile surface should be used. Also, the joints which are currently excluded from the structural design, will require detailed analy-sis. It is unclear at this stage whether strength will be the governing design criterion, or fatigue, caused by fluctuating wind loads. This fatigue calculation falls out of the scope of this dissertation, but requires further study. Therefore, the consecutive steps to determine the needed parameters for the calculation have been discussed [Eurocode 3, 2007], [Eyrolles, 1999].


134

Chapter 6 – Case study 1

135

Chapter 6

Case study 1: A Deployable Barrel Vault with Trans-lational Units on a Three-way Grid

Figure 6.1: Deployable barrel vault with translational units on a triangular grid: scissor structure

and tensile surface


136

6.1 Introduction The present chapter is concerned with the development of a new concept for a mobile deployable shelter. As described in Section 5.2, a semi-circular single curvature shape (barrel vault) is designed, consisting of plane and curved translational units on a three-way grid. Using translational scissor units is a very powerful method for developing positive and negative curvature surfaces [Langbecker, 1999]. This statement applies to two-way scissor grids (Sections 3.3.3 and 3.3.4) but is no longer valid for three-way (or triangulated) scissor grids. Langbecker [1999] states that for a curved three-way scissor geometry to be stress-free foldable, double scissors units have to incorporated, as shown in Figure 6.4. In this chapter it is shown, however, that it is possible to obtain a single cur-vature triangulated grid without the use of double scissor units, which leads to the development of a novel scissor geometry. A novel geometric design method is developed based on equations nased on the rise and span of the base curve and the number of units in the span. Also, it is investigated how the semi-cylindrical shape can be equipped with suitable double curvature end structures to obtain a fully closed architectural space. A simple, yet elegant solution is proposed which does not alter the original deployment behaviour. The implications of the scissor geometry and the effect of the deployment on the joints is discussed. It is shown that the phenomenon of angular distortion, as a consequence of the deployment behaviour, can be dealt with by a custom joint design. By introducing the concept of an equivalent hinged-plate model, an interest-ing way of gaining insight in the kinematic behaviour of the deployed configu-rations is proposed. To test the structural feasibility of the concept, both the open barrel vault and the closed double curvature configuration are structur-


137

ally analysed, using the method explained in Section 5.3.1 and the results are discussed.

6.2 Description of the geometry Consider the saddle-shaped (anticlastic) structure depicted in Figure 6.2. As can be seen in both the plan and perspective view, it consists of identical translational scissor linkages placed in two directions of an orthogonal grid. Using translational units on an orthogonal grid, while maintaining a constant unit thickness throughout the structure, is a powerful method for obtaining a wide variety of singly or doubly curved structures (Sections 3.3.3 and 3.3.4).

Figure 6.2: Plan view and perspective view of the same double curvature structure with transla-

tional units on a quadrangular grid

But this statement is no longer valid for three-way grids with single or double curvature. If a triangulated grid is to be designed, with single or double curva-ture, the introduction of double scissors is needed, in order to obtain a stress-free deployable configuration [Langbecker, 1999]. Figure 6.3 shows a scissor module composed of single scissor units, while Figure 6.4 shows a module with a double unit. The effect of the integration of a double unit is that the module is no longer triangular, but quadrangular. This non-triangulation of the grid can lead to in-plane instability, resulting in swaying or skewing of the structure. Although double scissor units are inevitable for triangulated double curvature structures, it will be shown that single curvature grids can be de-signed with only single units.


138

Figure 6.3: Translational scissor module with

only single units

Figure 6.4: Translational scissor module with a

double unit

Consider Figure 6.5 which represents a planar translational scissor grid. The two-dimensional scissor linkages A, B and C contain plane units with constant unit height throughout the grid with equilateral triangular cells.

Figure 6.5: Plan view, perspective view and side elevation of a planar structure with a triangu-

lated grid

Now, by introducing curvature in direction B and C a cylindrical shape is ob-tained, as shown in Figure 6.6: - direction A (longitudinal direction) contains parallel rows of identical plane units - direction B and C run diagonally over the span and contain rows of non-identical curved translational units

A B C

Double unit Single unit


139

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

grid

Curving these scissor units has a particular consequence: for the scissor mod-ule to have its unit height unaltered while the units become curved, the equi-lateral grid cells have to turn into isosceles ones, i.e. their apex angle increases while their projected area decreases. Figure 6.7 shows a plane module M1 and its curved derivatives M2 and M3, which are both used to compose the barrel vault from Figure 6.6.

M1: plane M2: slightly curved M3: highly curved

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

γ1 γ2

γ3

A B C


140

In Figure 6.7 three different scissor modules are shown in their deployed posi-tion (not one single unit in consecutive deployment stages).

So the only way to make a deployable barrel vault with translational scissor units on a triangular grid – without the use of double scissor units - is to use isosceles scissor modules of constant unit height (Figure 6.7). But a double scissor can also prove itself useful. The barrel vault described earlier is an open structure, which means the membrane canopy forms only a roof, while the front and back side remain open (Figure 6.8). However, by using a novel way of providing the open barrel vault with double curvature ‘end structures’, the open structure can be fully closed. This is done by adding two modules – which are identical to those used in the rest of the structure – to both the front and back, as shown in Figure 6.9.

Figure 6.8: OPEN structure: perspective view and plan view

For the doubly curved end structures, the integration of double scissors cannot be avoided. In order for the structure to be able to deploy, each added module is provided with its own separate double scissor unit. The effect this has on the deployment is discussed in Section 6.4.1.

9.4 m

6 m


141

Figure 6.9: CLOSED structure: perspective view and plan view (double scissor marked in red)

Now that the composition of the grid has been discussed, a closer look is drawn to the structure itself: as shown in Figure 6.10 it has a circular curva-ture when viewed in front elevation and consists of four modules in the span. As can be seen in Figure 6.10, the curved scissor units – which make up the modules – are not coplanar, nor do they lie in the direction of the span. There-fore, the curvature of the structure - which is of importance to the design - is different from that of the scissor linkages B and C, because these find them-selves at an angle with the span direction. The eventual curvature is determined, not by the actual scissor units, but by their projected counterparts, marked in red in Figure 6.10, with the projection plane lying in the direction of the span.

6 m

10.6 m


142

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

All scissor units obey the geometric deployability condition (Section 3.3), which is graphically represented by ellipses on the linkage U1, U2 and U3, de-veloped in a common plane, as shown in Figure 6.11. Now there are two ways to design such a barrel vault, depending on how much control over the eventual curvature is desired. The first approach is based purely on geometric constructions and involves no numeric computation. It is simply a matter of constructing a two-dimensional linkage such as the one shown in Figure 6.11. Subsequently, the scissor units are rotated in 3D until they form closed triangular scissor modules, as depicted in Figure 6.10, which are repeated to form a complete grid. As mentioned ear-lier, the definitive curvature can be evaluated by projecting the units on the plane lying in the direction of the span.

U1

U2 U3


143

Top view of units U1, U2 and U3: rotated until coplanar

Figure 6.11: Developed view of units U1, U2 and U3: graphic representation of the deployability

condition by means of ellipses

It is evident that, when a designer wants total control over the final curvature, another, more precise approach is to be used. When, for example, a circular curve with a certain rise and span for the barrel vault is desired, numeric com-putation has to be used to find a geometry that will satisfy this requirement. The second approach will now be discussed in greater detail.

6.3 Geometric design Using the rise and span as initial design parameters, a circular arc is deter-mined, which represents the actual curvature of the barrel vault. The goal of the numeric approach is to find, based on a number of design parameters, a series of scissor modules which will fit on the desired base curve. For this ex-

U2

U3

U1

U1 U2 U3


144

ample a circular arc is chosen – to facilitate comparison between the barrel vaults from case study 1, 2 and 3 - but any curve, in its parametric form, can be used. For this approach, the method of the ellipses for imposing the geo-metric deployability condition onto two-dimensional linkages (Chapter 3) is extended to three dimensions. By simply revolving the ellipses around their vertical axis, an ellipsoid is obtained with three semi-axes, of which two are identical, as shown in Figure 6.12. Where the ellipse determines the locus of all valid positions for the intermediate hinges in two dimensions (Figure 6.13), the ellipsoid fulfils the same role in three dimensions. Instead of the small ellipsoid, marked in red in Figure 6.13, an ellipsoid of dou-ble size is used to determine the locus of the intersection points between the unit lines of a scissor unit and the curve. The small unit has unit thickness (t) which is also half the distance between the foci of the double ellipsoid. The relation between the design parameter t and the length of the semi axes a and b is expressed by Eqn (6.1):

222 abt −= (6.1)

Figure 6.12: An ellipsoid representing the

geometric deployability condition in three

dimensions

Figure 6.13: Vertical section view of the small

and big ellipsoid, imposing the geometric de-

ployability condition

K

θ

φ

K

t

t

b

a


145

When the following four design parameters are given a value, the geometry is fully determined:

• U: the number of units in the span • R: the radius of the circular arc • t: the unit thickness

α2: the angle which determines the position of point P2 and whether the curve

is a semi-circle (α2=0) or an arc segment (2

0 2πα ≤≤ ) - see Section 3.2.2.

Now consider Figure 6.14 which shows a circular arc A and a parallel circular

arc A′with a distance of 2a

between them. The location of endpoint 2P of the

arc is determined by α2.

Figure 6.14: A scissor linkage fitted on a circular curve, with all relevant design parameters and

the global coordinate system

A solution has to be found for 1P′ (x1, y1, z1). Its position on the circular arc is

such that it determines a translational SLE (scissor-like element) which is

P0

A

A’

P1’

P2

P0’

R

α2


146

compatible with the plane SLE, as depicted in the developed view of the link-age in Figure 6.15. This is the intersection point of ellipsoid E0 with the circle arc A′ , determined by the angle α1 (which determines the position of P1 on the circular arc). Figure 6.15: Developed view of the scissor linkage from Figure 6.14, showing a chain of double

ellipses

The general form of the parametric equation for an ellipsoid with centre (x0 ,y0, z0), semi-axes a and b and the axis of revolution parallel to the Z-axis (see Figure 6.12), with πθ 20 ≤≤ ; πφ ≤≤0 becomes:

0

0

0

cossinsinsincos

zbzyayxax

+=+=+=

φφθφθ

(6.2)

The general equation of a circular arc parallel to the ZX-plane (y=cte) with centre around the origin with πα ≤≤0 is

αα

sincos

RzRx

==

(6.3)

Plane

unit Compatible

curved units

Plane

unit

P0

P0

P1

P2

E0

E1


147

From Figure 6.14 the following relations between ellipsoids E0, E1 and points

0P , 1P′ , 2P are derived:

The location of endpoint 2P of the arc A depends on α2 with 2

0 2πα ≤≤

22

22

sincos

αα

RzRx

==

(6.4)

Figure 6.16: Perspective view of the scissor linkage from Figure 6.15

The coordinates for point 1P′ on A′ are given by

11

1

11

sin2

cos

α

α

Rz

ay

Rx

=

=

=

(6.5)


148

We can now write the parametric equation for ellipsoid E0 with centre 0P (0, 0,

R):

Rbzayax

+===

0

00

00

cossinsinsincos

φφθφθ

(6.6)

1P′ (x1, y1, z1) is the intersection between E0 and the circular arc A′ , which is

expressed by joining Eqns (6.5) and (6.6) in:

10

00

100

sincos2

sinsin

cossincos

αφ

φθ

αφθ

RRb

aa

Ra

=+

=

=

(6.7)

(6.8)

(6.9)

1P′ is also the centre for ellipsoid E1 (size identical to E0) which intersects with circular arc A in point 2P . We can write the parametric equation for ellipsoid E1 with centre 1P′ :

11

111

111

cossinsinsincos

zbzyayxax

+=+=+=

φφθφθ

(6.10)

Joining Eqns (6.4), (6.5) and (6.10) leads to the following relations:

211

11

2111

sinsincos

02

sinsin

coscossincos

ααφ

φθ

ααφθ

RRb

aa

RRa

=+

=+

=+

(6.11)

(6.12)

(6.13)

Equations (6.1), (6.7), (6.8), (6.9), (6.11), (6.12) and (6.13) lead to a system of seven equations in seven unknowns which is solved numerically. By assigning values to R, α2 and t a solution is found for { }10101 ,,,,,, φφθθαba


149

−=

=+

=+

=+=+

=

=

222211

11

2111

10

00

100

sinsincos

02

sinsin

coscossincossincos2

sinsin

cossincos

abt

RRb

aa

RRaRRb

aa

Ra

ααφ

φθ

ααφθαφ

φθ

αφθ

(6.14)

The previous parameterisation determines a geometry of four units (U=4) in the span. More units can be calculated by adding the appropriate equations for the ellipsoids in an analogous manner, but as the number of units rises, the complexity of the system of equations increases.

6.4 From mechanism to architectural envelope

6.4.1 Deployment and kinematic analysis

A two-dimensional scissor linkage has a single rotational degree of freedom (D.O.F.). When such linkages are placed on a grid, this rotational D.O.F. is, de-pending on the grid geometry, either preserved or removed. All this depends on the type of scissor units used (translational, polar, angulated), the type of grid they form (two-way, three-way, four-way) and the curvature (plane, single or double). An insight in the mobility of the mechanism is needed to understand to what degree constraints have to be added after deployment to turn it into a load bearing structure. During deployment the rotational degree of freedom is used to expand the scissor mechanism. Figure 6.17 shows the different stages in the deployment. As is typical for translational scissor units, the mechanism expands gradually in two directions in the horizontal plane, while maintaining its overall height during deployment. The deployment itself can be performed by pulling the four


150

lower corner nodes away from the centre. When on even terrain, wheels at the corners could aid in the deployment. Alternatively, small lifting equipment - which would usually be already present for loading and unloading during transport – can be used to lift the mechanism at a central node.

Figure 6.17: Perspective view, front elevation and top view of the deployment process of the

barrel vault with translational units – OPEN structure


151

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

As mentioned before, the custom end structures which turn the open structure into a closed one, have double scissor units. There are two of these, one for each side, as shown in Figure 6.19. In the partly deployed position there is a gap in the structure that will close during deployment. That is the reason for providing each module with its separate double scissor. In the fully deployed state these modules meet and the double scissors coincide. The deployment process of the closed structure is depicted in Figure 6.20. A proof-of-concept model of the closed structure has been constructed, which has shown that the kinematic behaviour is as desired.

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

Two double units,

side by side

Two double units

theoretically coincide


152


barrel vault with translational units – CLOSED structure

A novel way of gaining an insight in the mobility of the mechanism and the restraints which are needed to turn it into a structure in the deployed configu-ration, is to represent the configuration by an equivalent hinged plate struc-ture. This structure shares the same kinematic properties as the scissor geome-try would have when its rotational degree of freedom (scissor action) is re-moved. Removal of this rotational D.O.F. is a minimum requirement for a scis-sor mechanism to act as a structure, otherwise it is allowed to collapse and return to its initial undeployed state. Since the triangulated geometry with translational units is a single-D.O.F.-mechanism, it is sufficient to block the movement of a single unit in order to remove the rotational D.O.F., e.g. by fix-ing the distance between two end nodes n1 and n2 (as shown in Figure 6.21) by means of a bar or cable element. Alternatively, two lower end nodes n3 and n4


153

at opposite sides of the structure can be kept at the same distance in the same way (Figure 6.22). The reason the equivalent hinged plate model is a valid representation of the scissor grid with the rotational D.O.F. removed, is illustrated in Figure 6.21. A scissor unit with its D.O.F. removed can be replaced by a rigid plate of which the vertices correspond with the end nodes of the scissor. The original parallel unit lines (imaginary lines connecting the upper and lower nodes of the scissor units) now become fold lines acting as continuous joints for the plates. Each module or grid cell consists of three plates, which is a rigid body. And because the complete configuration consists of such three-plate modules, the grid has no in-plane mobility (no skewing or swaying). This means we can form an idea of the extra restraints needed in the deployed configuration, apart from con-straining the scissor action, to turn the mechanism into a structure. A mini-mum of seven translations need to be constrained (including constraining the scissor action), as Figure 6.21 shows. In practice however, it is suggested to fix all nodes touching the ground by pinned supports (Figure 6.22). Standard solu-tions for fixing mobile, lightweight structures the ground can be used. An overview of recoverable lightweight anchors is given by Llorens [2006].

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

n1

n2

▲

●▲


154

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

Undeployed state: cable at full length Fully deployed state: cable shortened and

locked

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

Crucial to any deployable structure are the joints. Every unit consists of two bars connected by a revolute joint (intermediate hinge). At their ends, the bars are connected by another revolute joint. In reality, the members of the scissor units and the joints have discrete dimensions, unlike the theoretical geometric line models which have zero thickness. In theory, both bars of a scissor unit lie in a common plane. As opposed to the theoretical one-dimensional coplanar scissors, the physical bars are not in the same plane. A scissor unit has an imaginary centreline, which separates the two scissor bars lying on either side of that axis (Figure 6.24). The size of the joint is influenced by the number of bars it has to connect and by the dimensions of the bar. The wider the section of the scissor bars be-comes, and the higher the number of bars that to be connected in the joint,

n3

n4


155

the larger the radius of the joint becomes, in order to accommodate all ele-ments without interference during deployment. Some joint solutions have been proposed by Escrig [1984]. The joints will have to allow every possible movement, imposed by the grid geometry, to ensure a stress-free deployment. It has been shown that, in the deployed position, the grid cells are isosceles triangles. But during deployment of the mechanism proposed here, the apex angles inside the scissor modules increase in size. Take the angles γ2 and γ3 from Figure 6.7: when the maximal deployment is reached, γ2=63º and γ3=95º. This angular distortion has to be allowed by the specially constructed joint, which adds some complexity to the design. A joint is pro-posed, of which the ‘fins’ can freely rotate around the cylindrical hub. Figure 6.25 shows a top view of the designed joint connecting six bars. The bars are provided with wedge-shaped end pieces, which allow them to be as compactly arranged as possible, without interfering with one another during deployment. The centrelines of all units connected by a certain joint have a single intersec-tion point G, lying on a vertical axis through the joint, as Figure 6.26 shows. In the fully deployed position, this intersection point of centrelines of the scissor units lies on the vertical axis through the joint. As the structure is compacted towards its undeployed state, point G moves further upward until all centre-lines become parallel in the undeployed position.


156

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

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


157

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

The bars are connected to a hinge point which is not at the intersection point of bar centrelines (different from the scissor unit centreline from Figure 6.24). This eccentricity has an effect on the structural performance of the structure in the sense that this geometric imperfection induces a 2nd order effect, i.e. bending in the bars. The hollow, cylindrical hub around the fins of the joint rotate, can accept an active cable (Figure 6.23), guided by a pulley system, for stiffening the struc-ture after deployment (or for raising the membrane and bringing it under pre-tension. In the deployed state, after the structure is appropriately fixed to its supports, the cable can also be used to influence the stiffness by introducing more or less tension. After deployment, the membrane is raised toward the inner end nodes of the bars, after which a basic level of pre-tension is introduced. In Figure 6.27 and Figure 6.28 the resulting mobile shelters (open and closed) are shown together with the area they cover. Alternatively, instead of attaching the membrane to the inner nodes of the structure, the tensile cover could function as an outer layer, attached to the external nodes, providing weather protection for the structure as well.

Joint

centreline

Unit centre-

line

G


158

Perspective view Top view

Covered area

60 m2

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


Covered area

58 m2

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


159

6.5 Structural analysis

6.5.1 Open structure (single curvature)

Formfinding of the tensile surface

Before any load can be applied to the membrane surface, an equilibrium form under pure pre-tension is to be found first. The boundary geometry used for generating the membrane in the formfinding process in EASY® consists of a pattern of rhombi and triangles. A pre-tension of 1 kN/m in both directions is introduced in the net with a mesh length of 0.2 m. A force of 2kN is intro-duced in the boundary cables which have a stiffness of 12000 kN, while the membrane is given a basic stiffness of 400 kN/m in both directions. A typical PVC-coated polyester fabric is chosen for all case-studies. In Figure 6.29 a top view and perspective view of the skeletal structure and the matching boundary geometry for the membrane are shown. After formfinding, the equilibrium form for the tensile surface is found, as depicted in Figure 6.30. Typical values for stresses in the membrane vary from 4 to 5.5 kN/m (Figure 6.31).

Figure 6.29: Top view and perspective view of the skeletal scissor structure (left) and the bound-

ary geometry for the compatible membrane (right)


160

Figure 6.30: Views of the equilibrium form for the membrane

as a result of the formfinding process

Figure 6.31: Typical stresses in the membrane range from 4 to 5.5 kN/m


161

Statical analysis under load combinations

With all load combinations (Section 5.2.2) applied, the actions of the mem-brane on the outer boundary points are determined. In accordance with the hypothesis, the pre-tension is treated as a permanent load and is therefore multiplied with its appropriate safety factor. In the case of the ultimate limit state (ULS) the pre-tension is increased to 1.35 kN/m and the force in the boundary cable is raised to 2.7 kN. For the service limit state calculations (SLS) the membrane pre-tension and the force in the boundary cable are left at their initial values of 1kN/m and 2 kN respectively. Now the seven load combina-tions are applied and the resulting forces in the boundary points are calcu-lated.

Structural design of the scissor structure

The resulting forces are applied to the corresponding nodes of the FE-model of the scissor structure in ROBOT [2007]. The lower nodes touching the ground are fixed with pinned supports. This removes the mobility from the mechanism and enables it to act as a structure and transfer loads. In accordance with the physical model, the joints which connect the scissor bars consist of separate elements that share one rotational degree of freedom around an axis through their common point and perpendicular to their common plane. Figure 6.32 shows six bars (black lines) attached at their ends (P) to six node elements (PQ) and the attributed rotational degrees of freedom: the bars of the scissor units are allowed to rotate around their local Y-axis (Figure 6.34) at their ends (P) as well as at their intermediate pivot hinge (RS). For modelling purposes, each physical bar is represented in the FE-model by two (half-length) separate lines, joined together with a fixed connection (R and S): 3 translations and 3 rotations constrained.


162

Figure 6.32: FEM-model of six bars attached

to a node

Figure 6.33: An intermediate pivot hinge

connects two scissor bars

The line elements that make up the nodes rotate around their local Z-axis. The global coordinate system and the local coordinate system of a bar are shown in Figure 6.34.

When, for example, load case 6, which is transverse wind is considered (a full description of the load cases is given in Section 5.3.2), the distribution of the load vectors and their relative size is shown graphically in Figure 6.35. The resulting reaction forces in the supports are depicted in Figure 6.36. The load vectors from Figure 6.35 represent the action of the membrane on the nodes of the structure, as a result of transverse wind load and priestess and the re-sultant load vectors are pointed inward. The reaction forces in the support points of the structure are shown in Figure 6.36. As can be seen from the schematic, the action on the middle section induces a reaction force pointed

LOCAL Coordinate System GLOBAL Coordinate System

Figure 6.34: Local coordinate system of a bar element (left) and global coordinate system

(right)

P

Q

R

S


163

upward and outward, while the reaction forces in the sections near the open ends are pointed inward and downward.

Front elevation Side elevation

Figure 6.35: Typical pattern of load vectors for transverse wind + pre-stress of the membrane

Front elevation Side elevation

Figure 6.36: Typical pattern of reaction forces under transverse wind

Moments around the local Y-axes of the bars are shown in Figure 6.37, which is a typical diagram for a scissor arch. Here the maximal and minimal values for My are 2.21 kNm and -2.49 kNm respectively.

Figure 6.37: Bending moments My under transverse wind


164

A typical deformation pattern (not to scale) under transverse wind load is shown in Figure 6.38.

Figure 6.38: Typical deformation under transverse wind:

undeformed [grey] and deformed [red] configuration – no scale

Ultimate Limit State (ULS)

With all load cases and combinations applied, a structural analysis is per-formed to examine the structure’s strength (section design) and stability (buckling analysis), in accordance with Eurocode 3 [2007] in the deployed con-figuration. The extreme values for the reactions, forces and stresses will be discussed. When a load case is mentioned, it is assumed that the selfweight of the structure is included in the combination, with its appropriate safety factor applied, depending on whether ULS or SLS is considered. Aluminium is chosen as the material for the bars which are tubular and have a rectangular section. Because the joints will not be designed in this study, their influence on the structure design is made as small as possible. Therefore, their size is limited and they are awarded a very large section together with a high stiffness and yield stress (fy). The structure design is performed by means of an iterative static analysis with ‘minimum weight’ as the optimization criterion. When three possible tubular sections for the bars are suggested – round, square and


165

rectangular - the algorithm evaluates the latter as being the most structural efficient. Rectangular sections not only seem to be appropriate from a struc-tural point of view, but they are preferred anyway over round or square sec-tions because of the higher degree of compactness they provide in the unde-ployed position. A buckling analysis is performed with all bars having a buck-ling length coefficient of 1. After optimization, the proposed section is rectan-gular: TREC 120x60x3.2 mm. This leads to a total weight of 856 kg, which gives, for a total covered surface of 63 m2, a weight-per-square-meter of 13.6 kg. Figure 6.39 shows the optimized structure with the resulting weight/m2 and the material characteristics. Because the joints are not structurally ana-lyzed and designed, their weight is not taken into account. Therefore, the structural performance is only a measure of the main members. Also, the weight the membrane is neglected and no analysis is performed in the par-tially deployed state.


166

Section hxbxt [mm] Total weight [kg] Weight/m2 [kg/m2]

TREC 120x60x3.2 mm 856 13.6

Material E-modulus [GPa] fy [MPa]

Aluminium 75 180

Figure 6.39: Perspective view of the resulting structure with rectangular sections of

120x60mm

In ROBOT positive stresses indicate pressure, negative values mean tension. The reaction forces in the global coordinate system are mentioned in Figure 6.40. Induced by load combination ULS 2 (pre-stress + snow + transverse wind), the largest reaction force is 52 kN in the Z-direction. Marked in red is the support in which Fz max occurs.


167

REACTIONS

FX (kN) FY (kN) FZ (kN)

MAX 13,36 44,50 52,03

Case ULS 5 ULS 1 ULS 2

MIN -13.44 -38.58 -40.57

Case ULS 5 ULS 1 ULS 2

Figure 6.40: Reactions in the global coordinate system: the maximal reaction force occurs

under ULS 2 (pre-stress + snow + transverse wind)

STRESSES • Section TREC 120×60×3.2 mm

Bar S max [MPa] S max(My) [MPa] S max(Mz) [MPa]

154 157.8 115.08 14.67

Fx/Ax [MPa] S min [MPa] S min(My) [MPa] S min(Mz) [MPa]

30.54 -96.74 -115.08 -14.67

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)


168

The governing load combination for the design for strength is ULS 5 (pre-stress + snow). In the critically loaded bar a total stress level of 157.8 MPa occurs, part of which is caused by My, inducing a bending stress of 115.8 MPa (Figure 6.41). The bending stress around the local Z-axis is a fraction of Smax (My). When the stability is checked, ULS 2 (pre-stress + snow + transverse wind) is the governing load combination, which induces an axial force Fx of 45.22 kN (Figure 6.42). The bar, which is most susceptible to buckling, is located near the bottom of the structure. During buckling analysis in ROBOT, second order effects are taken into account.

FORCES

Bar FX (kN) FY (kN) FZ (kN)

540 45.22 -0.17 0.04

Load Comb. MX (kNm) MY (kNm) MZ (kNm)

ULS 2 -0.02 0.04 -0.41

Figure 6.42: Axial forces, transverse forces and bending moments in the local coordinate

system of the bars


169

Service limit state (SLS)

The stiffness of the structure is analysed under service limit state by checking the displacements of the nodes. The nodes which are subject to the largest displacements are marked in Figure 6.43: 2.0 cm in the X-direction (SLS 2), 1.0 cm in the Y-direction (SLS 5) and -1.4 cm along the Z-axis. Because this is a concept for a mobile structure, the same strict requirements as for permanent buildings do not apply. These values seem perfectly acceptable for a structure of this type and the displacements will not degrade the serviceability of the structure. The maximum deflection occurring in the structure is approx. 1/100.

DISPLACEMENTS

UX (cm) UY (cm) UZ (cm)

2.0 1.0 -1.4

Load Comb. SLS 2 SLS 5 SLS 5

Figure 6.43: Maximal nodal displacements in the global coordinate system

The obtained section of 120×60×3.2 mm causes the total weight to mount up to 856 kg which makes the structure bulky and heavy for transport. The sec-tion can be reduced by connecting the lower and upper nodes with a continu-ous cable running over pulleys. Figure 6.44 shows the cable and its path through the structure in both the undeployed and deployed state. After de-ployment, the cable is locked from which moment it can contribute to the overall structural performance. The effect is a decrease in section of the scissor


170

bars and an increase of the structural stiffness. The steel cable with a section of 28 mm2 has been assigned a calculation strength of 1500 MPa (safety factor of 1.5 applied). For practical reasons, the continuous cable is modelled as dis-crete cable fragments, each of which connects an upper and lower node. Al-though this is an approximation of the real situation, the results of the analy-sis indicate a noticeable positive effect on the structural behaviour (Figure 6.46).

Figure 6.44: Continuous cable zigzagging through the structure, connecting upper and lower

nodes and contributing to the structural performance

Scissor bars

Tension cable

Section hxbxt, d Total weight/section [kg] Weight/m2 [kg/m2]

TREC 60x40x3.2mm 459

Cable 6 mm 6

Total weight: 465 7.75

Figure 6.45: Resulting structure after optimization, with cable elements

Cable tie


171

RESULTS for case study 1: OPEN structure (with cable elements)

STRESS [MPa]

Section [mm] S max

S max

(My)

S max

(Mz)

S max

Fx/Ax

Load

comb.

TREC 60x40x3.2 75.7 55.0 9.2 11.5 ULS 5

Cable d=6 -856.7 - - -856.7 ULS 5

FORCE

Section [mm] Fx [kN] My [kNm] Mz [kNm] Load comb.

TREC 60x40x3.2 16.4 0.2 0.2 ULS 5

Cable d=6 -25.7 - - ULS 5

REACTIONS [kN]

FX Load comb. FY Load comb. FZ Load comb.

-6.34 ULS 6 12.06 ULS 6 7.95 ULS 5

DISPLACEMENTS [cm]

Ux Load comb. Uy Load comb. Uz Load comb.

3.3 SLS 6 0.9 SLS 5 -2.4 SLS 6

Figure 6.46: Summary of the determining stresses and forces for the strength, stability and

stiffness of case study 1: OPEN structure

Ux

Uy

Uz


172

When considering Figure 6.40 (no cable) and Figure 6.46 (with cable), a no-ticeable difference in values for Fy and Fz can be observed. This is due to the fact that only the highest peak value occurring in the structure, under a cer-tain load case, is represented. The sums of all values of Fy (and equally for Fz) are identical for both structures, only the structure without cables demon-strates higher peak values. The effect of incorporating a cable is an alteration of the stiffness, resulting in a redistribution of forces and a decrease in peak values.


173

6.5.2 Closed structure (double curvature)

Section hxbxt, d[mm] Total weight/section [kg] Weight/m2 [kg/m2]

TREC 60x30x3.2 420

Cable 6 6


Figure 6.47: Perspective view of case study 1: CLOSED structure: with sections after structure

design and total weight

The closed structure is provided with the same cable elements as in the open structure. The added modules lead to a slight decrease in section width (30 mm as opposed to 40 mm for the open structure). The weight/m2 for both structures is approximately 7.5 kg/m2.


174

RESULTS for Case study 1: CLOSED structure (with cable elements)

STRESS [MPa]

Section [mm] S max

S max

(My)

S max

(Mz)

S max

Fx/Ax

Load

comb.

TREC 60x30x3.2 54.5 48.3 4.8 1.5 ULS 2

Cable d=6 -949.9 - - 928.6 ULS 2

FORCE


TREC 60x30x3.2 9.7 0.03 -0.05 ULS 2

Cable d=6 -26.8 - - ULS 2

REACTIONS [kN]


-7.8 ULS 6 8.3 ULS 2 6.3 ULS 2

DISPLACEMENTS [cm]


3.1 SLS 1 3.7 SLS 1 -3.7 SLS 1

Figure 6.48: Summary of the determining parameters for the strength, stability and stiffness

of case study 1 _ CLOSED structure

Ux

Uy,

Uz


175

6.6 Conclusion It is known from literature that using translational scissor units on a two-way grid allows the design of stress-free deployable positive and negative curva-ture surfaces, but this does not apply to three-way (triangulated) scissor grids. For non-flat three-way grids to be stress-free deployable, the integration of double scissors is required [Langbecker, 1999].

Figure 6.49: Case study 1: Single curvature OPEN structure (barrel vault)

In this chapter an advance has been made by developing a stress-free deploy-able scissor geometry of single curvature with translational units on a three-way grid. It has been shown that the curved triangulated grid can be solely composed of single scissor units, therefore making the integration of double scissor units obsolete. By avoiding double scissors, the number of connections is kept to a minimum. Also, the inherent triangulation of the grid provides in-plane stability. As a consequence, the deployed configuration does not require additional cross-bracing of the grid cells to prevent the structure from skew-ing. A geometric design method has been developed and the necessary equations have been derived, based on architecturally relevant design parameters. Al-though this design method allows the design of barrel vaults with a semi-circular section, this approach can easily be extended to other base curves as well. To test the feasibility of the concept, a barrel vault with four units in the span has been designed, of which two variations have been presented: an open, single curvature structure (barrel vault) (Figure 6.49) and a closed double curvature structure (Figure 6.50). In case of the double curvature structure, it


176

has been found indeed inevitable, as opposed to the single curvature structure, to incorporate double scissors to guarantee stress-free deployability.

Figure 6.50: Case study 1: Double curvature CLOSED structure

An explanation has been given for the angular distortion, a phenomenon asso-ciated with the deployment of the proposed geometry. A design for a joint has been proposed, which takes this effect into account. By introducing the concept of an equivalent hinged-plate model, an interest-ing way of gaining insight in the kinematic behaviour of the mechanism has been offered. It has been shown that the proposed geometry is a single-degree-of-freedom mechanism. As a result of the structural analysis, it has been found that, due to high in-plane bending stress in the scissor members, the strength is the governing de-sign criterion. By adding a continuous cable between the upper and lower nodes, a significant increase in structural performance and a reduction in weight has been achieved. The open and closed structure, both covering approx. 65 m2, achieve a weight ratio of approximately 7.5 kg/m2. It is noted that the joints are not included in the structural design and therefore no statement is done concerning their weight.


177

Chapter 7

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

Figure 7.1: Deployable barrel vault with polar units on a quadrangular grid: scissor structure and

tensile surface


178

7.1 Introduction In this chapter the design of the second case study, a deployable barrel vault with polar and translational scissor units on an orthogonal two-way grid, is described. Using polar units is an effective way of introducing single curvature in an orthogonal grid. A number of examples have been presented in literature, such as single curvature shelters and scissor arches (Section 2.2.4). This case study is an illustration of how a basic stress-free deployable single curvature structure can be obtained by putting the construction methods and the geometric design methods, as described in Section 3.2, into practice. A structure with four units in the span is proposed, similar in shape and dimen-sions to case study 1. What sets this concept apart is the innovative way the open structure is fully closed by adding doubly curved end structures, which are compatibly deploy-able with the original structure. Previous solutions, such as the elegant pro-posal by Escrig [2006], also provide a doubly curved closed surface, but are, however, not stress-free deployable. Therefore, it is shown how an existing geometry, the lamella dome (Section 3.3.4) is adapted to fit the required pur-pose. Based on the geometric design method proposed in Section 3.3.2, the equa-tions, which allow the study of the geometry in several stages of the deploy-ment, are derived. These are then used to predict the maximum span the con-figuration will reach during deployment. Similar to case study 1, the mobility of the system is discussed by means of an equivalent hinged-plate model providing an insight in the stability of the de-ployed configuration. As quadrangular grids are prone to skewing due to a lack of in-plane stability, measures are taken such as the introduction of cross-cables or cross-bars in several configurations, which are structurally analysed and discussed.


179

7.2 Description of the geometry

7.2.1 Open structure

Consider Figure 7.2 which represents a plan view and perspective view of a planar translational scissor grid. The two-dimensional scissor linkages A and B contain plane units with constant unit height throughout the grid with square cells.

Figure 7.2: Plan view and perspective view of a planar structure with a quadrangular grid

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

As mentioned in chapter 3, a polar unit is simply obtained by moving the in-termediate hinge of a plane translational unit away from the middle of the bar. This eccentricity of the revolute joint creates curvature when the unit be-comes deployed. Now, by introducing curvature in direction B, a cylindrical shape is obtained as shown in Figure 7.3: - direction A (longitudinal direction) contains parallel rows of identical plane units

A

B

B A

A

B

B A


180

- direction B runs in the direction of the span and contains rows of identical polar units Figure 7.4 shows a series of polar linkages with 3, 4, 5 and 6 units in the span. From Figure 7.4, the linkage with four units in the span (U=4) is chosen, for reasons explained in Section 5.1: all case-studies (with single curvature shape) are based on the same circle arc with a radius of 3 m and share the same structural thickness of 1.25 m. This facilitates mutual comparison.

3 units 4 units

5 units 6 units

Figure 7.4: Series of polar linkages with 3, 4, 5 or 6 units in the span,

based on the same circular arc

The geometric construction is based on a chosen circular arc, marked in red in Figure 7.5, which is divided into four segments (U=4) by radial lines through centre O. The intersection points on the arc, such as point Q, determine the location of the inner end nodes of the scissor units. Now a line is drawn through Q, tangent to the circle arc. P and R are the intersection points which mark the outer end nodes of the units, by which the unit height (h) is immedi-ately determined. By symmetry, the complete geometry can now be con-structed.


181

Figure 7.5: Geometric construction of the four-unit linkage

Subsequently, a curved four-by-four grid is composed by connecting parallel placed polar arches (direction B) by plane translational linkages (direction A). All bars, both translational and polar, are of identical length, the only differ-ence between the translational and polar units being the location of the in-termediate hinge, relative to the middle of the bars. The resulting configura-tion is shown in Figure 7.6.

Figure 7.6: OPEN structure: perspective view and top view

Evidently, all scissor units obey the geometric deployability constraint (Section 3.2), which is graphically represented by ellipses on the linkage U1, U2 and U3, developed in a common plane, as shown in Figure 7.7.

10.9 m

8.5 m

6 m

P

Q

R

O h


182

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

The translational unit U1 belongs row A in Figure 7.3, while the polar units U2 and U3 belong to the perpendicular direction B. When this linkage is devel-oped in a common plane, the ellipses which impose the deployability con-straint can be drawn, as shown in Figure 7.7. It should be noted that, although polar units are used here, an ellipse is required to impose the deployability

U2

U3

U1

U1

U2

U3


183

constraint. This is because the polar units from the transverse direction are to be linked with the translational units from the longitudinal grid direction. Us-ing circles is only valid for polar units, while the ellipse is valid for both unit types. Therefore, mixing polar and translational units requires the general method, hence, an ellipse is to be used.

7.2.2 Closed structure

As an alternative to the open structure, a closed double curvature structure is proposed. For this, half of the open structure (called ‘main structure’) is kept and provided with compatibly deployable ‘end structures’. Escrig [2006] has proposed an elegant solution for a deployable dome which consists of polar units placed on a grid of parallels and meridians. Figure 7.8 shows a single ‘slice’ of such a dome added to part of the open structure, in three consecutive deployment stages. However elegant this solution may be, there is a slight snap-through phenomenon during deployment and, as can be seen in Figure 7.8, severe angular distortion in the end nodes, to allow the de-formation of the grid cells. This proposal is discarded because a stress-free deployable solution is to be found. Consider the polar structure from Figure 7.9 (Section 3.3.4) called a lamella dome. It consists of identical polar scissor units placed on a spherical grid with six rhombus-shaped grid cells arranged radially, as can be seen in the top view. There is no snap-through phenomenon during deployment, so it is fully compatible with the stress-free deployment of the main structure. A minor adaptation to the lower scissor units is needed to ensure a completely closed surface. The dashed lines in Figure 7.10 represent the unaltered lower units of the lamella dome, of which the end nodes would float above ground level, if all units were to be identical. This is due to the fact that the lower lamella-units meet the ground plane at an angle, as opposed to the lower units of the main structure, which are normal to the ground plane. For the simple reason of the longer distance these have to bridge to actually touch the ground, this requires them to be elongated. The width of the compact bundle of bars is not


184

influenced by the longer bars, however, they protrude at the top and at the bottom, therefore increasing the height of the undeployed shape.

Full dome (meridians and parallels) Adding a single row to the main structure

During deployment there is a snap-through phenomenon and angular distortion of the

grid cells

Figure 7.8: Adding an ‘end structure’ based on parallels and meridians to the main structure

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


185

The resulting closed structure is shown in Figure 7.11.

Figure 7.11: CLOSED structure: perspective view and plan view

Top view of full lamella

dome with six modules

arranged radially

In order to form a gapless enclosure, the lower units

(red) of the lamella modules have to be elongated until

they touch the ground

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

6 m

15.2 m

11.5 m


186

7.2.3 Deployment

The deployment process of the open and closed structure is depicted in Figure 7.12 and Figure 7.13.


polar barrel vault – OPEN structure


187


polar barrel vault – CLOSED structure

Figure 7.14: Proof-of-concept model (half of the structure) in three deployment stages


188

7.3 From mechanism to architectural envelope

7.3.1 Deployment and kinematic analysis

The polar linkage with four units in the span (U =4) can easily be constructed as explained in Section 3.2.1. Alternatively, the parametric design approach, described in Section 3.2.2 can be used. Either way, the geometry can be fully derived from four design parameters: the span S and rise rH of the base

curve, the number of units U and the unit thickness t . Ultimately, values for the semi-bar lengths a and b and the deployment angle designθ are obtained

which fully determine the geometry of the linkage in its deployed position. Furthermore, these three parameters suffice to study the deployment behav-iour of the polar linkage, as will be shown later. An expression for the span S as a function of the deployment angleθ , including the design constants a , b and U has been derived and is used to determine the value maxSθ for which

the maximum span is reached during the deployment. Now polar linkages have a peculiar property: during deployment, as the de-ployment angle θ increases from 0 (compact configuration) to designθ (de-

ployment angle in the fully deployed configuration), the span S will not merely increase. Asθ increases, a maximum span (Smax) is reached, only to de-crease again slightly until designθ is reached (Sdesign). Figure 7.15 shows a typical

deployment pattern for a polar linkage.


189

1 2 3

4 (Smax=maximum span) 5 (Sdesign)

Figure 7.15: Deployment sequence of a polar linkage

It is useful to know, for practical reasons, what the precise behaviour of the linkage will be during deployment, in terms of variation of the span, and at what point this maximum is reached. In order to locate this point in the deployment, an expression has to be found for the span, as a function of the constants a , b and U and the variable θ . With designθθ ≤≤0 the deployment ratio ψ can be defined as:

designθθψ = with 10 ≤≤ψ (7.1)

where 0=ψ stands for the undeployed configuration and 1=ψ for the fully

deployed configuration. The constants U , a and b are known, while θ , t , inR , eR , β , γ and S are

unknown and vary throughout the deployment. We can derive a few relations from Figure 7.16. Using the cosine rule we can write the following expression:

)cos(2222 θπ −−+= babat (7.2)


190

which gives for t :

θcos222 babat ++= (7.3)

Figure 7.16: Polar linkage in an intermediate deployment stage: 10 <<ψ

From Figure 7.16 the following can be readily written:

tRR ine += (7.4)

inee

i RbRaba

RR

=⇔= (7.5)

Thus, by substituting Eqn (7.4) in Eqn (7.5) an expression for inR is obtained:

abatRin −

= (7.6)

Re Rin

t

β

γ 2φ

φ

a

b

θ

S

mt η

Se


191

Further, we know that βγ U= (7.7)

and from Figure 7.16 we can see that

ϕγ 4= (7.8)

From Eqns (7.7) and (7.8) we can find ϕ

4βϕ U

= (7.9)

Also,

2sinθam = (7.10)

and 2

sin βinRm = (7.11)

from which an expression for β is found:

= −

2sinsin2 1 θβ

inRa

(7.12)

From Section 3.4.2 (Eqn (3.3)) we know that

)2sin(2 ϕinRS = (7.13)

Finally, substituting (7.3), (7.6), (7.9) and (7.12) in (7.13) results in an expres-sion for the span S as a function of U , a , b and θ

= −

2sinsinsin2 1 θ

inin R

aURS (7.14)

with θcos222 babaab

aRin ++−

= (7.15)

Alternatively, an expression can be found for the external span Se as defined in Figure 7.16. It can be observed that

ηcos2tSSe += (7.16)


192

and

ϕπη 22−= (7.17)

Substituting (7.17) in (7.16) gives

−+= ϕπ 2

2cos2tSSe (7.18)

Or in other form: ϕ2sin2tSSe += (7.19)

Now, the value for the deployment angle θ(Smax) at which the structure will have reached it maximum span Smax, can be found. Using f(x) for Eqn (7.14), then θ(Smax) is found by

0)( =′ θf (7.20)

When applied to the polar linkage from which the barrel vault is built, the key stages in the deployment sequence are obtained, as shown in Figure 7.18, marked A, B and C. The relation between the deployment angle and the span is represented by the graph in Figure 7.17. In the undeployed configuration (A) both θ and S are evidently equal to 0. Between stages (A) and (B) the correla-tion betweenθ and S is virtually linear. The maximum span is reached in stage (B). Then the linkage further deploys until stage (C) is reached, for which it is designed, where a slight decrease in span occurs. Theoretically, the deploy-ment can be continued until the extremities of the linkage meet to complete a circle and the span is reduced to 0, but as an architectural space enclosure this configuration is useless. Alternatively, the external span Se can be used (Eqn 7.19) to describe the maximum span the structure will reach at its extremities during the erection process. This could prove especially useful on sites with limited dimensions, to check whether Se remains within the limits allowed.


193

Relation between the deployment angle (θ) and the span (S) of the polar linkage with U=4

-400

-200

0

200

400

600

800

0 30 60 90 120 150 180

Deployment angle (θ)

Spa

n (S

)

Figure 7.17: Graph showing the relation between the deployment angle θ and the span S for the

polar linkage with U=4

Stage A represents the undeployed position: both θ and S are equal to 0. The maximum span of approximately 7 m is reached for θ=111.6°. Stage C is the final deployed position for which the linkage is designed and has θ=135° for a span of 6 m, which are evidently the chosen design values.

A: θ=0°, S=0 B: θ=111.6°, S=701.7 cm C: θ=135°, S=600 cm

Figure 7.18: Deployment sequence of the polar linkage (U=4)

Similar to the approach used for the translational barrel vault, a hinged plate model is made, based on the grid geometry. This eliminates the rotational de-gree of freedom of the scissors and allows to study the kinematic properties of

Smax

Sdesign

θ(Smax)

θdesign

A B C


194

the grid, in order to determine to what extent constraints have to be added to turn the mechanism into a structure. Figure 7.19 shows the equivalent hinged plate structure, consisting of plates that are mutually connected by a line joint, allowing one rotational degree of freedom. This model represents the mobility of the grid.

Figure 7.19: Open barrel vault: scissor structure and equivalent hinged plate structure

The complete structure is built from one module, pictured in Figure 7.20. Be-cause the fold lines that represent the line joints do not share a single inter-section point, this module has no mobility. By deduction, the complete struc-ture has no mobility. Therefore, there is no need for additional constraints, other than the one needed to eliminate the rotational degree of freedom of the original scissor mechanism. However, similar to the translational barrel vault (Section 6.4.1, Figure 6.21), all inner nodes touching the ground are fixed by pinned supports, as shown in Figure 7.21.

Figure 7.20: Geometry of a hinged plate module


195

Figure 7.21: Fixing all inner lower nodes to the ground by pinned supports

Figure 7.22: Closed barrel vault: scissor structure and equivalent hinged plate structure

Figure 7.23: Geometry of the occurring plate modules


196

3 modules undeformed 2 deformed (red) and 1 undeformed

(black) modules

Figure 7.24: Closed barrel vault: scissor structure and equivalent hinged plate structure

Figure 7.22 and Figure 7.23 show the individual modules from which equiva-lent hinged plate structures can be built for the closed configuration. Because the middle part of the structure (half of the previously discussed open structure) has no mobility, the degrees of freedom of the complete structure are those of the added ‘end structures’ alone. In Figure 7.24 the undeformed and deformed state of an end structure is shown. When one module is kept immobile, the two other can be simultaneously deformed. This leads to the conclusion that there are two degrees of freedom per end structure, which gives a total of four for the complete geometry. Again, in the deployed posi-tion all inner nodes touching the ground are pinned to the ground, effectively removing all mobility, as shown in Figure 7.25. The resulting open and closed structure, with the tensile surface attached, are shown in Figure 7.28 and Figure 7.29, with a covered area of 66 m2 and 60 m2 respectively. The joint solution is similar to that of the translational barrel vault, as de-scribed in Section 6.4.1. However, this is a simpler version because no rotation of the plate elements (or ‘fins’) of the joint has to be allowed to cope with an-gular distortion of the grid.


197

Figure 7.25: Fixing all inner lower nodes to the ground by pinned supports

Figure 7.26: Joint connecting four bars (no rotation of the ‘fins’ of the joint around a vertical

axis, as is the case for the translational barrel vault in Section 6.4.1, Figure 6.24)


198

Figure 7.27: Top view and perspective view of one scissor unit and its intermediate and end

joints


Covered area

66 m2

Figure 7.28: Perspective view and top view of OPEN structure with integrated tensile surface


199


Covered area

60 m2

Figure 7.29: Perspective view and top view of CLOSED structure with integrated tensile surface



Analogous to case 1, the structural analysis will now be discussed. Without taking any measures to improve structural performance, the calculated section for the polar barrel vault is 150×100×3.2 mm, which, for a covered surface of 66 m2, translates into a weight-per-square-metre of 15.8 kg/m2 (Figure 7.30). With steel cable elements (d=6 mm, design strength=1500 MPa) inserted be-tween upper and lower nodes (cfr. Section 6.5.1, Figure 6.45), the section of the scissor bars is decreased to 120×80×3.2 mm, leading to an improved 12.7 kg/m2, as shown in Figure 7.31.


200

Element

type Section [mm]

Weight [kg]

Scissor bar

150×100×3.2 1046

T-cable

Cross bar

Cross ca-ble

Total weight 1046 Weight/m2 15.8

Figure 7.30: Result without any measures taken to improve structural performance

Element type

Section [mm]

Weight [kg]

Scissor bar 120×80×3.2 828 T-cable d=6 7

Cross bar Cross ca-

ble

Total weight 835

Weight/m2 12.7

Figure 7.31: Improved result by inserting vertical cable ties

Triangulation of the quadrangular grid can enhance the structural stability. Therefore, diagonal struts are added inside each quadrangular scissor module, as shown in Figure 7.32. These crossbars can be added after deployment by fixing them to the nodes, which requires some extra manual labour. Alterna-tively, these crossbars could be equipped with an intermediate hinge which enables them to be integrated into the structure beforehand and allows them


201

to deploy compatibly with the scissor modules. However, these hinged bars need to be locked at their intermediate hinge in order to fulfil their structural role. Also, they are the longest bars in the structure which makes them sus-ceptible to buckling. The outcome of the analysis of this configuration is a sec-tion of 100×50×2.5 mm for the scissor bars, while the crossbars have a tubu-lar round section of 70×2.5 mm, both made from aluminium (Young’s modulus: 75 GPa, design strength: 180 MPa). The section of the scissor bars decreases to 100×50×2.5 mm and although 100 kg is added through the crossbars, the weight-covered area ratio drops to 8.9 kg/m2 because of the reduced section of the scissor bars.

Element type

Section [mm]

Weight [kg]


Cross bar 70×2.5 101

Cross ca-ble

Total weight 589

Weight/m2 8.9

Figure 7.32: Additional diagonal bars triangulate the grid

By adding another series of diagonal crossbars, as shown in Figure 7.33, their section is slightly decreased to 60×2.5 mm and the section of the scissor bars is further decreased to 80×40×2.5 mm, leading to a weight ratio of 8.5 kg/m2. This is too little a difference to be called an improvement, especially since a lot more connections have to be made.


202

Due to the added complexity of the locking mechanisms in case of the foldable crossbars or the additional on-site assembly in case of non-folding bars, an-other option is chosen. Steel cables can instead be used to triangulate the scissor grid to become tensioned by deployment of the structure. As a result, the section of the scissor bars, and therefore the weight increases in compari-son to the versions with cross-bars, but nonetheless this option is chosen be-cause of its lower complexity. The resulting structure weighs 9.9 kg/ m2 with a section for the scissor bars of 100×50×3.2 mm. The resulting geometry and the weight is given in Figure 7.34. It is noted that the membrane, if warp and weft directions are placed diagonally over the grid, can simulate the effect of cross-bracing with cables. In Figure 7.35 a summary is given of the resulting stresses, forces and displacements of the analysis of the open structure. The governing load case is ULS 6 (pre-stress + transverse wind). The nodal dis-placements are small and are comparable to those for the translational barrel vault from Chapter 6. The bar deflections (not shown) do not exceed 1/300 of the bar length.

Element type

Section [mm]

Weight [kg]


Cross bar 60×2.5 173 Cross ca-

ble

Total weight 560

Weight/m2 8.5

Figure 7.33: Double diagonal cross bars offer no real advantage structurally


203

100×50×3.2 mm

Cable d=6 mm

Section hxbxt, dxt [mm] Total weight/section

[kg]

Weight/m2 [kg/m2]

TREC 100x50x3.2 616

Cable d=6 7

Diagonal cable d=6 30


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

design and weight/m2

Diagonal cable


204

RESULTS for Case study 2 OPEN structure (with cable elements)

STRESS [MPa]

Section [mm] S max

S max

(My)

S max

(Mz)

S max

Fx/Ax

Load

comb.

TREC 100x50x3.2 129.3 98.5 11.3 27.9 ULS 6

Cable d=6 -1153.7 - - -1153.7 ULS 6

Cross Cable d=6 -401.31 - - -401.31 ULS 2

FORCE


TREC 100x50x3.2 41.13 -1.85 -0.14 ULS 6

Cable d=6 -32.53 - - ULS 6

Cross Cable d=6 -11.3 - - ULS 6

REACTIONS [kN]


-16.7 ULS 6 -24.5 ULS 6 11.1 ULS 5

DISPLACEMENTS [cm]


-1.2 SLS 6 -0.9 SLS 5 -2.1 SLS 6


for case study 2 OPEN structure

Uy

Uz

Ux


205


After structural analysis and design of the closed structure, without any addi-tional measures taken to improve structural performance, a section of 150×100×3.2 mm for the bars is obtained. This leads to a total weight of 1060 kg, which gives a massive 17.6 kg/m2 for a covered area of 60 m2 (Figure 7.36). When now diagonal steel cables (d=6 mm, design strength=1500 MPa) are added to cross-brace the quadrangular scissor modules, the structural per-formance is enhanced. The section is reduced to 120×60×3.2 mm which gives, for a total weight of 798 kg and a covered area of 60 m2, a ratio of 13.3 kg/m2.

Element type

Section [mm] Weight

[kg]

Structure bar

150×100×3.2 1060

Cable d=6 Total weight 1060 Weight/m2 17.6

Figure 7.36: Main structure and additional end structures with no additional measures to im-

prove structural performance

Form Figure 7.38 it can be seen that ULS 3 (pre-stress + longitudinal wind + snow) is the governing load case. Again, displacements and deflections (not shown) are within limits and are comparable to those of previous calculations.


206

120×60×3.2 mm

Cable d=6 mm

Section hxbxt, dxt [mm] Total weight/section

[kg]

Weight/m2 [kg/m2]

TREC 120x60x3.2 755

Cable d=6 16

Diagonal cable d=6 27


Figure 7.37: Perspective view of case study 2 CLOSED structure:, with resulting sections after

structure design and total weight

Diagonal cable


207

RESULTS for Case study 2 _ CLOSED structure (with cable elements)

STRESS [MPa]

Section [mm] S max

S max

(My)

S max

(Mz)

S max

Fx/Ax

Load

comb.

TREC 120x60x3.2 161.1 150.9 2.7 8.14 ULS 3

Cable d=6 -523.8 - - -523.8 ULS 4

FORCE [kN]

Section [mm] Fx My Mz Load comb.

TREC 120x60x3.2 12.5 -0.05 -0.07 ULS 3

Cable d=6 -14.77 - - ULS 4

REACTIONS [kN]


-9.1 ULS 6 -7.77 ULS 7 10.43 ULS 4

DISPLACEMENTS [cm]


1 SLS 6 1.8 SLS 3 -2.2 SLS 3

Figure 7.38: Summary of the results for the structural analysis of case study 2

CLOSED structure

Ux

Uy Uz


208

7.5 Conclusion This chapter was concerned with the design of a deployable barrel vault with polar and translational scissor units on an orthogonal two-way grid. First, an open barrel vault with four units in the span has been designed (Figure 7.39), similar in shape and dimensions to case study 1, as described in Section 5.2. For this case study, the construction methods and the geometric design meth-ods, as described in Section 3.2, have been put into practice, to obtain a basic, single curvature stress-free foldable solution.

Figure 7.39: Case 2: OPEN structure

Despite the simplicity of the design, providing the barrel vault with doubly curved end structures to form a fully closed envelope, while these added sub-structures are required to be stress-free foldable, has proven troublesome. A clever solution proposed by Escrig [2006] consisting of polar scissor units, placed in parallels and meridians, is characterised by an unwanted bi-stable deployment. An innovative solution was found in the lamella dome (Section 3.3.4). It has been shown that, with a slight modification to the bottom most scissors, half of such a dome can be connected to a polar barrel vault and demonstrate a compatible, stress-free deployment (Figure 7.40). Based on the geometric design method proposed in Section 3.3.2, the equa-tions, which allow the study of the evolution of the span throughout the de-ployment, have been derived. These were then used to predict the maximum span the configuration reaches during deployment. This parameter has pro-


209

vided information on the minimum space required for deployment and can prove useful during erection on a site with limited dimensions.

Figure 7.40: Case 2: CLOSED structure

The mobility of the system has been discussed by means of an equivalent hinged-plate model, a method proposed in Section 6.4.1. It was found that the open barrel vault is a one-degree-of-freedom mechanism, while each added end structure has two D.O.F., leading to a total of five D.O.F. for the fully closed, doubly curved structure. It was suggested to fix all lower nodes to the ground by pinned supports, effectively eliminating all mobility from the struc-ture. To prevent the quadrangular grids from skewing due to a lack of in-plane sta-bility, several configurations were provided with either cross-cables or cross-bars. From the structural analysis it was found that the configuration with double cross-bars provided the lightest solution (8.5 kg/m2). However, on prac-tical grounds, the slightly heavier solution with the cross-cables was opted (9.9 kg/m2). Although this structure has larger sections for the scissor mem-bers, the connection of four extra cables to a joint, as opposed to four extra bars, is preferred because of simpler connection and a more compact unde-ployed configuration. The structural analysis of the fully closed configuration has led to a structure with a weight ratio of 13.3 kg/m2. The increase in section is attributed to the longer bars of the doubly curved end structures and the resulting higher buckling sensitivity. A uniform section has been chosen for all bars in the structure and, hence, an increase in weight was observed.


210


211

Chapter 8

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

Figure 8.1: Foldable bar structure based on the geometry of foldable plate structures


212

8.1 Introduction In this chapter a novel concept for a mobile shelter system is developed, based on the geometry and kinematic behaviour of foldable plate structures, of which the architectural application has been described in Section 2.3. Evaluated by Hanaor [2001], plates are found to require an substantial thick-ness because they are subject to compression and bending. The resulting over-all weight may be higher than a structure surfaced with a membrane. Storage efficiency is low due to the plates’ dimensions and thickness and waterproof-ing of joints between individual plates poses a problem. Therefore, as an archi-tectural envelope for the third case study, a textile membrane is proposed, as it provides a flexible, lightweight, translucent and continuous shelter surface. The membrane is held up and tensioned within the foldable bar structure, act-ing as the primary load bearing structure. As this case study is based on a foldable plate structure, the geometric design method proposed in Section 4.3 is used to design a single curvature shape (barrel vault, as shown in Figure 8.1), similar to the shape of case study 1 and 2. Also, a double curvature configuration (foldable dome) consisting of the same plate elements is designed. Further, it is shown how these basic shapes can be connected to form new, alternative configurations [De Temmerman, 2006b]. Next, it is explained how the transition is made from plate geometry to bar structure. An innovative foldable articulated joint, serving as a connector for the bars, is proposed [De Temmerman, 2006a]. It is demonstrated that the bar-joint system preserves the kinematics of the plate structure it is derived from, even when certain bars are discarded from the structure. An insight in the mobility is offered [Foster1986/87], since these mechanisms typically possess multiple degrees of freedom [Kool, 2006], which are to be removed after deployment to stabilise the structure. The structural feasibility of the concept is assessed in a preliminary structural analysis, where several


213

configurations are tested and mutually compared. Key aspects regarding the architectural use are discussed.

Figure 8.2: Typical foldable plate structure

8.2 Description of the geometry As described in Chapter 4 foldable plate structures consist of triangular plates, hinged together along their edges by continuous joints which provides them with one rotational degree of freedom. These hinges allow the structure to fold according to a pattern of mountain and valley folds. By unfolding the initially flattened plate linkage, a three-dimensionally expanding corrugated surface arises, as shown in Figure 8.2. In Chapter 4 an overview is given of some typical configurations and the equations needed for parametric design are presented. For clarity, the key design parameters will be concisely explained. Although in the end a foldable bar structure is to be designed, the underlying geometry is no different from that of the plate structure it is derived from. Therefore the geometry will be described in terms of plates and all previously determined design parameters and equations remain valid. In Figure 8.3 the parameters are shown which completely determine the geometry of a foldable plate structure. This particular shape is called regular. The key parameters which determine the overall geometry are: the number of


214

plates (p) in the span, the apex angle of the plates (β), and the deployment angle (θ) for the fully deployed configuration. What is called the fully deployed configuration corresponds with the position for which the edges of the plates nearest to the ground are perfectly horizontal, i.e. they touch the ground (Figure 8.2). In elevation view the silhouette of the structure is then a perfect semi-circle. The impact additional parameters such as the plate length (L), the span (S) and the module width (W) have, is limited to a variation of the dimensions of the structure.

Fully deployed position

Fold pattern

Figure 8.3: Design parameters for a basic regular foldable plate structure.


215

As stated in the introduction to the case studies (Chapter 5) a circular arc with a radius of 3m is used as base curve. This two-dimensional profile can be seen as a projection in the vertical plane of a three-dimensional plate geometry with five plates in the span (p=5). Taking a projected profile as a starting point, implicates that there is an endless collection of plate geometries which will fit this requirement. This is illustrated in Figure 8.4: the two pictured fully deployed configurations are both valid solutions for p=5, but have a different apex angle β and a different width W.

Figure 8.4: For a chosen number of panels p the apex angle β can be altered at will, only

affecting the width of the structure

However, not all values for the apex angle give rise to a valid, foldable configuration which in addition can be folded into a compact configuration. It has been shown that the minimal apex angle for a five-plate linkage which is still compactly foldable is βmin=90° while the maximum value is βmax=135°. This can also be seen in the graph from which, for a certain chosen β, the appropriate deployment angle θ can be derived for linkages with p=5 (the graph is drawn up using Eqn (4.13) from Section 4.3.1).


216

Relation between the apex angle (β) and the deployment angle (θ) in the fully deployed configuration for p=5

0

10

20

30

40

50

60

70

90 95 100 105 110 115 120 125 130 135Apex angle (β) [°]

Dep

loym

ent a

ngle

(θ) [

°]

Figure 8.5: Graph showing the relation between the deployment angle θ and the apex angle β in

the fully deployed configuration for p=5

Because the 90° solution achieves - for a certain number of plates and con-nections - the greatest expansion (and therefore the greatest width), this seems a logical choice, but there are other factors that play a part in choosing a suitable geometry. The equations for designing circular structures have been proposed as well. It has also been shown that a common module geometry can be found which can be used in both regular and circular configurations, leading to a higher uniformity of plate elements. The amount of sectors arranged radially in the circular structure (q) can be freely chosen. Figure 8.7 shows such a sector of which there are six in this particular example, which will be the amount used for this case study. With chosen values p=5 and q=6, solving Eqn (4.13) and Eqn (4.27) simultaneously results in an apex angle β=109.2° and an accompa-nying deployment angle θ=54.3° (marked in red in the graph from Figure 8.5). This means that the original value for β has risen from 90° to 109.2°, which translates in a reduced width in the deployed configuration. But the benefit is


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that a plate geometry is obtained which can be used to form both regular and circular configurations with the same plate elements and that these can be mutually connected to form new alternative configurations, as shown in Figure 8.9.

Fully deployed Compactly folded

Fold pattern

Figure 8.6: The resulting regular geometry for the case study: two extreme deployment states

and the fold pattern

Figure 8.7: Top view and a perspective view of a circular plate geometry with six sectors ar-

ranged radially


218

Fully deployed Compactly folded

Fold pattern

Figure 8.8: The resulting circular geometry for the case study: two extreme deployment states

and the fold pattern

Figure 8.9: A combination of a regular and a circular geometry

a

c

b

c

a b

c


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In Figure 8.10 the dimensions are given for three obtained geometries, which all share the same interior height at their highest point: 4.25 m.

Open structure Closed structure Dome

Figure 8.10: Dimensions in plan view of the shapes

8.3 From plate structure to foldable bar structure Now that the geometry has been fully determined, the transition from foldable plates to foldable bars can be made. The goal is to devise a foldable bar struc-ture with a kinematic behaviour identical to that of its similar counterpart, the foldable plate structure. So instead of plate elements as cladding components, a continuous membrane supported by a skeletal structure will be used to form the architectural envelope.

Figure 8.11: A foldable plate structure (p=7) and its similar counterpart, a foldable bar structure

A first way of obtaining the bar structure is to ‘cut away’ the middle sections of the plates until only thin borders remain, which represent the bars. Where

8.5 m

10.4 m

8.5 m

10.8 m

8.5 m


220

there were originally two neighbouring plates meeting edge to edge, there are now two parallel bars, one on each side of the fold line. These double bars, however, lead to an inefficient use of material (Figure 8.12).

Figure 8.12: Pattern 1: double bars present

Therefore, as many superfluous bars as possible are removed, without affecting the original kinematics of the system. Of each pair of double bars, one is con-sequently discarded, as shown in Figure 8.13.

Figure 8.13: Pattern 2: double bars removed

Additionally, provided that the bars are appropriately connected, some of the middle bars (valley between two triangles) can be left out, without altering the kinematic behaviour, provided that the two remaining V-shaped legs are joined by a fixed connection at the apex. The reason for discarding these bars, is to be able to incorporate a membrane, hung inside from the nodes. If the bars would remain in place, the membrane would be unable to reach its anti-clastic shape inside the V-shaped folded rhombuses. The resulting bar pattern


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is shown in Figure 8.14, which is the type that will be used for the open struc-ture.

Figure 8.14: Pattern 3: double bars and diagonal bars removed, without affecting the original

kinematic behaviour

By connecting the bars by means of a custom foldable joint, shown in Figure 8.15 the bar-node system demonstrates the same kinematic behaviour as the plate structure it is originally derived from. It consists of six parts connected by hinges in such a way that the folding and unfolding is unhindered by the thickness of the bars (Figure 8.16). In this way a lightweight articulated bar structure is obtained which behaves like a foldable mechanism.

Figure 8.15: Foldable 3 D.O.F.-joint derived directly from the fold pattern, therefore mimicking

its kinematic behaviour


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Figure 8.16: Deployment sequence for the foldable joint: from the undeployed to the fully de-

ployed position

By appropriately connecting bars and foldable joints according to the fold pat-tern previously discussed, a foldable barrel vault is formed, as shown in Figure 8.17. All joints are identical, except for those at ground level, which are slightly modified versions. The dome-shaped structure from Figure 8.8 and the combined geometry shown in Figure 8.9 can both be built in a similar manner, composed from the same elements.

Figure 8.17: The (regular) open structure complete with bars and joints:

perspective and top view – side and front elevation


223

Figure 8.18: Detailed view of bars and three variations of foldable joints occurring in the struc-

ture

8.3.1 Deployment

The deployment process of the open structure is shown in Figure 8.19. During deployment almost all the expansion occurs in the longitudinal direction (per-pendicular to the span). In the transverse direction (parallel to the span) the variation of the geometry is not as significant. In the undeployed position, the outermost nodes, marked in Figure 8.19, touch the ground. These remain in contact with the ground throughout the deployment and can therefore be used to attach wheels to facilitate the deployment on even terrain.

1

2

3


224

Figure 8.19: Deployment sequence for the open structure – perspective view, front elevation and

top view

Figure 8.20: Proof-of-concept model of the regular structure (with scissors) in four stages of

the deployment

The foldable dome is deployed in a circular manner, by rotating all sector around a vertical axis through the highest point. The deployment process of the circular structure is shown in Figure 8.21.


225

Figure 8.21: Deployment sequence for the dome structure – perspective view, front elevation

and top view

Figure 8.22: Proof-of-concept model of the foldable dome (with additional scissor units) in six

deployment stages


226

The deployment of the combined structure is similar to that of its regular and circular sub-structure. First the regular structure is deployed while the circular modules are kept compacted, as shown in Figure 8.23 and Figure 8.24 (stage A, B and C). Then, the circular modules are deployed until the structure becomes a fully closed envelope (stage D, E and F).

Figure 8.23: Deployment sequence for the closed structure: 1 regular module + 2 semi-domes

A B

C

D

E

F


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Top view

A B C D

E F

Figure 8.24: Six stages in the deployment of the closed structure (top view)

An alternative way of translating the fold pattern into a bar structure has been looked at. As opposed to the previous method, a bar is placed exactly on every fold line of the pattern. The bars are connected by custom kinematic joints which allow all necessary rotations during deployment. Some bars are grouped into triangles to make connections simpler. The resulting geometry is shown in Figure 8.25. A model has been constructed to evaluate the mobility of the mechanism which behaves in the desired way. However, fully folding or un-folding proves difficult because obstruction occurs in the joints, and therefore this solution cannot be folded as flat as the configuration with the foldable joint. Consequently, this proposition will not be further investigated.


228

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

Ultimately the foldable bar structure has to provide shelter by means of a ten-sile surface. Two approaches are proposed, depending on whether the mem-brane is integrated in advance or not. Because the deployment of the structure is characterised as folding, a membrane can well be integrated beforehand by attaching it to the nodes. Then, it is unfolded along with the structure and brought under tension as the structure reaches its fully deployed position. In Figure 8.26 the integrated membrane is shown in both the undeployed and deployed position. Another option is to deploy the bar structure first, after which the membrane can be pulled up to the nodes by cables, until it becomes sufficiently tensioned.


229

Figure 8.26: Integration of the membrane beforehand by attaching it to the nodes – Side eleva-

tion and perspective view of the undeployed and deployed position

8.3.2 Alternative geometry

As an alternative configuration, a right-angled structure as described in Chap-ter 4 has been designed using the same approach. Such a geometry, shown in Figure 8.27, can be very compactly folded, provided that the design parameters are appropriately chosen. Also, the vertical sides are quadrangular, which could make it easier to provide access to the structure. As a downside, the higher variation of the apex angles, when compared to the regular structure, translates in three different foldable joints, with another two variations at ground level. But the main concern is the relatively low structural thickness, due to the gentler corrugation of the surface. This is because a right-angled structure has a larger deployment angle θ and it is this increased deployment range that gives the ‘roof’ its higher slenderness in the deployed configuration, making it susceptible to sagging or even snapping through. A model (Figure


230

8.28) with aluminium bars and resin connectors (not to scale) has been built to test the deployment behaviour, which proves to be as required.

A B C

Figure 8.27: Right-angled geometry with its own set of joints

Figure 8.28: Deployment sequence of a concept model of a right-angled structure with

aluminium bars and resin connectors [De Temmerman, 2006a]

Right-angled and regular structures with a type B pattern (Section 4.2) and with p=5 have identical edges in the vertical plane. This implies that they can be linked together along that edge to form a chain of structures. However, the

A

B

C


231

difference in shape between the regular and the right-angled structure in the undeployed configuration prevents a simultaneous deployment. Therefore, they are to be coupled after deployment (Figure 8.29).

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

8.3.3 Kinematic analysis

The devised articulated bar system and the plate system it is derived from, are mechanisms with multiple degrees of freedom. The mobility of foldable plate linkages can be determined from the layout of the plates. In chapter 4, the difference between a type A and type B pattern has been explained. For all patterns similar to pattern A the formula for the total number of degrees of freedom is derived by Foster [1986/87]:

22 ++ mp (8.1)

where p stands for the number of plates in one module and m for the number of modules in the pattern. Analogous to Eqn (8.1), a similar formula can now be derived for patterns of type B:

42 ++ mp (8.2)


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From these two expressions it is immediately apparent that patterns of type B posess, for an identical number of plates and modules, a higher mobility. Because we want to limit the mobility of the mechanism, in order to gain control over the deployment, the patterns used in this case study are therefore chosen to be of type A.

Applying Eqn (8.1) to the open structure (Figure 8.19) with p=5 and m=6, the mechanism has 19 D.O.F. Now it is necessary to determine what constraints have to be added in order to turn the linkage into a structure. Pinning eight lower nodes of the linkage to the ground adds 24 constraints (8×3 translations removed). This ensures stability for the structure which is now 5 times overconstrained. Now, for comparison, the number of constraints for the articulated bar linkage from Figure 8.25 is determined. Eqn (8.3) for pin-jointed truss systems gives the degree of statical determinacy (R) in terms of the number of bars (b), the number of joints (j) and The number of restraints (r) on the structure [Callad-ine, 1978]:

)3( rjbR −−= (8.3)

When the pattern for the open structure is built up using the structural system shown in Figure 8.25, the number of bars b=53 and the number of joints j=24. With again 8 pinned supports to constrain the structure (8×3 constraints), the statical determinacy R=-5. This means the structure is 5 times overconstrained and therefore stable. These identical results demonstrate the similarity between a foldable plate system and the articulated bar structure in terms of kinematic behaviour. It can be asked to what extent the predicted degree of mobility represents the actual kinematics of the system. Often, formulas such as Kützbach-Grübler [Hiller, 1991] for predicting the mobility of a system based on counting the number of joints and links, fail due to singularities occurring in mechanisms with high symmetry. Therefore Kool [2006] has devised a method for deter-


233

mining the mobility of foldable plate structures which uses a joint matrix for each kinematic loop in the structure. The number of kinematic loops can be easily determined with Eqn (8.4), attributed to Euler.

1int +−= linkssjoloops NNN (8.4)

Here, Njoints is the number of continuous joints between plate elements, while Nlinks is the total number of plates in the configuration. In short, once the num-ber of loops has been established, the mobility can be predicted with a struc-ture matrix. The columns of the joint matrix are the line vectors representing the hinges between the plates. The joint matrices can be assembled into a lar-ger structure matrix. The folding patterns can be calculated by using a svd-composition of this matrix and the zero singular values which are indicative for the rank-deficiency of the matrix correspond to the number of possible patterns. For a detailed explanation of the method please refer to [Kool, 2006]. Applying this approach to the foldable plate configuration from Figure 8.7, the number of loops is found using Eqn (8.4): Nloops=11–10+1=2. These two loops are shown in Figure 8.30.

Figure 8.30: The two loops and their common fold line

By applying the mentioned approach, five degrees of freedom are found for the foldable plate configuration from Figure 8.30: two in each loop and one angle around the common fold line. Practically, this means that, when the an-


234

gle around the fold line is determined (deployment angle θ), another angle in each loop can be chosen to place the bottom edges on the ground. Finally, a second angle in each loop can be chosen to force the side edges in a vertical plane in order to be able to mutually connect several modules. Possibly, this high degree of mobility of the mechanism can cause undesired movement. Therefore, incorporating a scissor linkage – a one D.O.F-mechanism – could be a solution. Figure 8.30 and Figure 8.31 show the deployment of a dome and a regular structure, both with an integrated compatible scissor link-age.

A foldable dome with a compatible integrated scissor linkage – only the upper nodes of the

scissors are connected to the dome

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


235



The bars and foldable joints are modelled in ROBOT as shown in Figure 8.32. The foldable articulated joints are entered as six separate entities, each con-sisting of fixed bar elements. The bars, in turn, are connected to these joint elements by fixed connections. By releasing the rotations, represented by the axes in Figure 8.32, the joints have the same kinematic behaviour as the actual joint model, shown in Figure 8.15.

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)

The load combinations, specified in Chapter 5, are applied to the model as nodal forces. These load vectors are distributed over the six elements of the nodes. The joints elements are excluded from the design process. Therefore, their size is limited and they are awarded a very large section together with a high stiffness and design strength. Again, some variations of the configurations have been structurally analysed. First, a configuration where the middle bars of each rhombus-shaped module are still present, has been calculated. All lower nodes are fixed to the ground by pinned supports. The resulting section and the weight ratio is shown in Figure 8.33. As material for the bars, aluminium is chosen (Young’s modulus: 75 GPa, design strength: 180 MPa). This results in a tubular round section of 88×2.5 mm, with a weight ratio of 5 kg/m2.

Joint

element

Bars


236

Element type Section

[mm]

Weight

[kg]

Structure bar 88×2.5 375

Strut

Total weight 375

Weight/m2 5

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

When cross-bars are added to the top layer of the system, the section remains unaltered. Instead, the added bars only increase the weight which has mounted to 403 kg, leading to a weight ratio of 5.4 kg/m2 (Figure 8.34).


[mm]

Weight

[kg]


Strut 55×2.5 28

Total weight 403

Weight/m2 5.4

Figure 8.34: Same model as in Figure 7.33, but with cross-bars

The configuration from Figure 8.35 is the one which is presented in the previ-ous sections. It has no middle bars in the rhombus-shaped modules and the bars are joined, in pairs, at their apex angle by a fixed connection. This is also shown in Figure 8.14, Figure 8.15 and Figure 8.18. The result is again a round tubular section of 88×2.5 mm, but with fewer bars, which gives a total weight of 296 kg, resulting in a weight ratio of 3.9 kg/m2, which is the lightest solu-tion of all proposals.


237


[mm]

Weight

[kg]


Strut

Total weight 296

Weight/m2 3.9

Figure 8.35: Bars are grouped in pairs and joined by a fixed connection in their apex angle

When struts are added, the section isn’t decreased. Adding struts only adds to the weight, which mounts up to 335 kg, resulting in a weight ratio of 4.5 kg/m2 (Figure 8.36).


[mm]

Weight

[kg]


Strut 76×2.5 39

Total weight 335

Weight/m2 4.5

Figure 8.36: Adding struts again only increases the weight, while the section remains identical

Figure 8.37 shows a summary of the results of the structural analysis of the proposed configuration. For the strength, the governing load case is ULS 5 (pre-stress + snow). The stresses are quite low and buckling sensitivity is the determining phenomenon here. Also, the snow action (in SLS 5) is the govern-ing load for the displacements, which stay within acceptable bounds. Deflec-tions (not shown here) do not exceed 1/1000.


238

RESULTS for Case study 3 _ OPEN structure

STRESS [MPa]

Section [mm]

S max S max

(My)

S max

(Mz)

S max

Fx/Ax

Load

comb.

TRON 88x2.5 41.5 7.1 8.7 25.9 ULS 5

FORCE


TRON 88x2.5 23.65 0.01 0.06 ULS 5

REACTIONS [kN]


-14.22 ULS 6 9.08 ULS 6 -19.6 ULS 5

DISPLACEMENTS [cm]


0.6 SLS 5 1.3 SLS 5 -1.6 SLS 5


for case study 3 OPEN structure

Ux

Uy

Uz


239



[mm]

Weight

[kg]


Total weight 208

Weight/m2 4.5

Description:

Foldable dome: bar structure without

middle bars and no struts

Figure 8.38: Resulting section and weight for the foldable dome

First, the proposed dome structure is analysed (Figure 8.38). The same results as for the open structure are obtained: tubular round aluminium sections measuring 88×2.5 mm. The weight is 208 kg leading to a weight ratio of 4.5 kg/m2.


240

Section hxbxt, d[mm] Total weight/section [kg] Weight/m2 [kg/m2]

TRON 88x2.5 375

Total weight: 375 5

Figure 8.39: Perspective view of case study 3 CLOSED structure with sections after structure

design and total weight

Next, the closed structure is analysed (Figure 8.39). Again, the same sections are obtained and a comparable result in terms of weight ratio is obtained: 5 kg/m2. Figure 8.40 gives an overview of the strength, stability and displace-ments. ULS 5 (pres-stress + snow) is the governing load case for the strength and stability. Displacements are still acceptable without harming the service-ability of the structure, but have risen in comparison with the open structure. Also, deflections (not shown) have risen (compared to the open structure) to 1/600.


241

RESULTS for Case study 3 CLOSED structure

STRESS [MPa]

Section [mm]

S max S max

(My)

S max

(Mz)

S max

Fx/Ax

Load

comb.

TRON 88x2.5 81 68.4 5.9 12.59 ULS 5

FORCE [kN]

Section [mm] Fx My Mz Load comb.

TRON 88x2.5 29.7 0.03 0.37 ULS 5

REACTIONS [kN]


-15.43 ULS 6 -11.36 ULS 7 30.29 ULS 5

DISPLACEMENTS [cm]


0.6 SLS 6 1.3 SLS 7 -3.5 SLS 5


for case study 3 CLOSED structure

Uz

Ux Uy


242

Transverse forces and bending moments are high in the elements which con-stitute the joints (bodies of the foldable articulated joint) and are low in the bars. This structural behaviour is comparable with that of a truss with fixed nodes (no releases for bars in the nodes). The fact that the transverse forces and the shear stresses are high in the nodes requires further, detailed analysis in which the joints are structurally designed. The governing phenomenon is buckling of the bars. The relatively great length of the bars – especially the inverted V-shapes at the front and back of the open structure – makes them susceptible to buckling. This explains the relatively low stresses in the bars, which leads to the conclusion that the stability is determining for the design, and not the strength.

8.5 Conclusion In this chapter a novel concept for a rapidly erectable shelter system, based on the geometry and kinematic behaviour of foldable plate structures, has been proposed. First a single curvature foldable plate geometry (barrel vault – Figure 8.41) has been designed using the geometric design method proposed in Sec-tion 4.3.1. A five-plate geometry was chosen, since its sectional profile is simi-lar to that of case study 1 and 2, as explained in Section 5.2. Also, a double curvature shape (dome – Figure 8.42) has been designed using the same ap-proach.

Figure 8.41: Case 3 OPEN structure


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Due to the plate uniformity, achieved by using the equations derived in Sec-tion 4.3, it has been shown possible for both these shapes to be combined into a closed doubly curved foldable geometry (Figure 8.43). It has been shown that the transition from plate structure to bar structures can be made, while preserving the kinematic behaviour of the original plate system. To simplify the structure, it was found that some excess bars could be discarded from the configuration, while still maintaining the same deployment behaviour.

Figure 8.42: Case 3 Foldable DOME structure

For connecting the bars, an innovative foldable articulated joint has been de-veloped. Through a proof-of-concept model, it has been proven that the bar-joint system preserves the kinematic properties of the plate structure it is de-rived from. By counting the number of loops, it was found that a five-plate module has five D.O.F.’s. To ensure a controlled deployment, it has been proposed to stra-tegically incorporate scissor mechanisms, turning the configuration in a sin-gle-D.O.F. mechanism.


244

Figure 8.43: Case 3 CLOSED structure

From the structural analysis it was found that transverse forces and bending moments were high in the joints and low in the bars. Buckling of the bars has been recognised as the governing phenomenon. Therefore, other configura-tions with a higher number of units (for the same span), leading to shorter bars, should be investigated. Currently, all three configurations (joints ex-cluded) have achieved a weight ratio of approximately 5 kg/m2. While the proposed concept seems promising and the architectural and kine-matic feasibility have been demonstrated, more profound and detailed struc-tural analysis is needed on an integrated model consisting of the bar structure, the joints and the tensile surface.


245

Chapter 9

Case study 4: A Deployable Tower with Angulated Units

Figure 9.1: Design concept for a tensile surface structure with a deployable central tower


246

9.1 Introduction This chapter is concerned with the design of the fourth and final case study, which is an innovative concept for a deployable hyperboloid tower with angu-lated scissor elements. Its purpose is two-fold: serving as a mast for a tensile surface structure while acting as an active element during the erection proc-ess. Angulated scissor elements have been extensively investigated and this has yielded a wide range of concepts and applications in the field of deployable scissor structures [Hoberman, 1991], [You & Pellegrino, 1996, 1997], [Rodri-guez & Chilton, 2003], [Jensen, 2004]. Although primarily intended for ra-dially deployable closed loop structures, it is shown in this chapter that angu-lated elements can also prove valuable for use in a linear three-dimensional scissor geometry. It is explained how angulated elements offer, for the pro-posed application, an advantage over polar units in terms of deployment be-haviour and a reduction of the number of connections. A comprehensive geometric design method is proposed for which the equa-tions, expressed in terms of relevant design parameters, are derived, enabling the design of any hyperboloid shape. Two different approaches are introduced, offering the designer a choice between designing the undeployed or the de-ployed configuration. As the deployment is an integral part of the design, an insight in the relation-ship between the geometry of the structure and its subsequent kinematic be-haviour is offered. The mobility of the system is assessed through use of the equivalent hinged-plate model, as introduced for case study 1 (Section 6.4.1) and case study 2 (Section 7.3.1). An innovative design for a joint is proposed, allowing all necessary rotations between subsequent elements. Finally, the tower is structurally analysed under wind and snow action and conclusions are drawn on the structural feasibility of the proposed design.


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9.1.1 A concept for a deployable tower

This case study is quite different from the previous three. Although a deploy-able bar structure and a tensile surface are once again involved, the way it provides an architectural enclosure is what sets it apart. For the barrel vaults of case studies 1, 2 and 3 the bar structure and the membrane surface follow the same curvature (monoclastic in the case of the open structure and syn-clastic for the closed structures). In this case on the other hand, the scissor structure is a central vertical linear element, used to hold up the anticlastic membrane canopies at one of their high points. The question was raised whether it would be possible to design such a deployable tower for a tempo-rary tensile structure and to use it as an active element during the erection process. In addition, the pantographic tower allows visitors to access several platforms to enjoy the views, under or above the different membrane elements (Figure 9.1).

The proposed concept is based on a design by The Nomad Concept [2007], a company active in the field of tensile surface structure design. Figure 9.2 shows the original (undeployable) tower (or mast), consisting of several mod-ules which are assembled and dismantled on-site by stacking them vertically, for which a lifting device is needed. After assembly the membrane would have to be attached to the top, after which the pre-tension in the membrane can be introduced.


248

Figure 9.2: Mobile structure with membrane surfaces arranged around a demountable central

tower (© The Nomad Concept)

By making the tower deployable, all connections could be made on ground level, while the mechanism is in its undeployed, compact state, therefore eliminating the need for additional lifting equipment. After connections be-tween the membrane elements and the tower have been made, the mechanism is deployed until the required height is reached and the membrane elements become tensioned. The tower could be deployed to such an extent that a suffi-cient amount of pre-tension is introduced in the membrane, ensuring the abil-ity to withstand external loads. Since the tower is basically a mechanism addi-tional bracing is needed after full deployment to turn it into a load-bearing structure.


249

Figure 9.3: The top of the tower is accessible to visitors, allowing them to enjoy the view

(© The Nomad Concept)

9.2 Description of the geometry The deployable tower is horizontally divided in several modules, which are closed-loop configurations of identical hoberman’s units or otherwise called angulated SLE‘s. Figure 9.4 shows an example of a tower with triangular mod-ules, of which three are stacked vertically. Because every bar in the structure is identical, the sums of the semi-lengths are evidently constant, therefore, the geometric deployability constraint is automatically fulfilled. The dimensions of the structure are shown in Figure 9.4, Figure 9.5 and Figure 9.6. The tensile surfaces are identical and measure 10 m along their longest diagonal. The top of the second module, at which the membrane elements are attached, is located at 5.2 metres above ground level. The other high point of the membranes is held 4 m above ground by additional masts.


250

Figure 9.4: Side elevation of the tower and canopy

Figure 9.5: Top view of the structure showing the three tensile surfaces arranged radially around

the central tower

10 m

4 m 5.2 m


251

Figure 9.6: Dimensions of the tower and a single angulated bar

It could be argued that a tower with a broad base and a narrow top can equally be built with polar units with decreasing size as they are located nearer to the top. In Figure 9.7 two linkages – one with angulated elements, another with polar units - are shown, with identical height and width, but with varying number of units U and different bar lengths. Using the angulated elements offers an advantage: while the linkage with angulated elements is built from only 3 SLE’s with 11 hinges and nodes, the equivalent polar mecha-nism needs 8 units with 26 connections to reach a similar deployed geometry. The effect that the angulated elements have on the modules is that, during deployment, the top of a module becomes narrower than its base. The radius of the top of a certain module becomes equal to the radius of the base of the next, higher located module. This means that the narrowing effect is enhanced and passed on through the mechanism, from module to module, from bottom to top.


252

Linkage with angulated SLE’s: Equivalent linkage with polar SLE’s:

Number of SLE’s: n=3

Number of hinges and end nodes: 11 Number of SLE’s: n=8

Number of hinges and end nodes: 26

Figure 9.7: Comparison between a linkage with angulated SLE’s and its polar equivalent

The dimensions of the individual bars of the scissor units are such, that the horizontal projection of b is equal to a, as shown in Figure 9.8. The imaginary vertical axes connecting the end nodes of the bars can act as fold lines, used to further flatten the linkage. Therefore, the modules are ‘cut open’ along one


253

fold line, after which the whole can be flatly folded for easier transport. Such a fold sequence is shown in Figure 9.9. This way of further compacting is pre-sented as an option and could be ignored, provided that the dimensions in the undeployed state are kept reasonable.

Figure 9.8: Imposed condition on the length of the semi-bars a and b (a<b), in order to make the

linkage foldable along the vertical axis

Step 1 (Compacted for transport) Step 2

Step 3 Step 4 (ready to deploy)

Figure 9.9: Initial unfolding of the compacted linkage to its polygonal form

a b


254

The deployment sequence of the tower is presented in Figure 9.10, showing a top view and a side elevation for each stage. The maximum deployment is reached when the upper end nodes of the top module meet in one point.

Step 5 (Undeployed) Step 6 Step 7

Step 8 Step 9 Step 10 (Fully deployed)

Figure 9.10: Six stages in the deployment of a hexagonal tower: elevation and top view


255

9.3 Geometric design Two different design approaches are presented here, depending on what the determining design criteria are:

• The first method – from unit to linkage – allows to design the mecha-nism in its compact, undeployed configuration, after which it is de-ployed into the desired position. This offers control over the geometry of the scissor elements themselves, rather than the overall final de-ployed result. This is a straightforward, linear design approach that al-lows each subsequent parameter to be derived from the previous one.

• The second approach – from linkage to unit – allows the overall ge-ometry of the deployed configuration to be determined from a set of design values. Finding solutions for the remaining parameters (geome-try of the SLE’s) relies on numerical calculations.

In this section the two approaches and all relevant design parameters are discussed. All equations are valid for any n-sided polygon (n>3) as basic shape for the tower. For simplicity, the parameterisation is applied to a two-module mechanism, but the approach is easily extended to more modules.

9.3.1 First approach: Design of the undeployed configuration (Unit linkage)

The tower is designed by means of parameters which determine the geometry of the linkage in the undeployed state (Figure 9.11b) and remain constant throughout the deployment: semi-bar lengths a and b, the kink angle β of the angulated SLE’s, the number of scissor units U in one module and the total number of modules n stacked vertically. The number of units U – with the minimum obviously being three – in the closed loop determines the sector an-gle φ (Figure 9.11e). The edge length E (a parameter specific to the unde-ployed state) of the base polygon determines the length of the semi-bars a and b, with a being half the length of E and b’s horizontal projection is equal to a (Figure 9.11d). (As mentioned earlier, this extra requirement makes it pos-sible to further compact the whole linkage in its folded configuration).


256

(d) Elevation view of angulated element

(a) Intermediate state

(0≤ψ≤1); (0≤θ1≤θ2≤θmax)

(e) Top view showing the edge length E

and the sector angle φ

(b) Undeployed state

(ψ=0); (θ1=θ2=0) (c) Fully deployed state

(ψ=1); (0≤θ1≤θ2=θmax)

Figure 9.11: Design parameters of a two-module tower with angulated SLE’s (three states)

Lb

La

Lb La

La

Lb

γ

β

θ1

θmax

φ

β

θ1

θ2

E


257

The design parameters a, b, β, U and n remain constant throughout the de-ployment which is achieved by letting the deployment angle θ vary over a cer-tain interval. As the module deploys, its width decreases while its height in-creases.. E, β, U and n completely determine the geometry of the linkage in the unde-ployed configuration. The sector angle φ is a submultiple of 2π and depends on the number of units U, corresponding with the number of sides of the base polygon:

Uπϕ 2

= (9.1)

The shortest semi-bar a measures half of the length of the edge of the base polygon E:

2Ea = (9.2)

The length of semi-bar b is given by:

( ) ββπ coscosaab −=

−= (9.3)

In the undeployed position the height of one module equals:

βtanahundeployed −= (9.4)

while the total height of all modules is:

undeployedundeployed hnH = (9.5)

Now that all necessary design parameters are given a value, we can unfold the linkage by increasing the deployment angle θ. Figure 9.11a shows two mod-ules (n=2) in an intermediate deployment position in which the bottom and top module both have their separate deployment angle θ1 and θ2 respectively.


258

We can define to what degree a linkage is deployed by introducing the de-ployment ratio ψ, with 0≤ψ≤1:

maxθθψ = (9.6)

The uppermost (nth) module will have reached its maximal deployment before any other module. Therefore its deployment angle θn shall be used for deter-mining the deployment of the whole structure. When θmax is reached, the up-permost end nodes meet in one point, rendering further deployment physically impossible. We can express θmax in terms of the kink angle β, via definition of an extra angle γ or it can be found directly from the drawing in Figure 9.11:

max2θπγ −= (9.7)

βπγ −= (9.8) which gives for θmax:

2maxπβθ −= (9.9)

There are three distinct deployment stages: undeployed (ψ=0), intermediate (0≤ψ≤1) or fully deployed stage (ψ=1). These will now be discussed in greater detail.


259

Intermediate state (0≤ψ≤1); (0≤θ1≤θ2≤θmax)

Figure 9.12: Perspective view: design parameters of a two-module tower with angulated SLE’s

– Side elevation showing the non-coplanarity of the angulated elements (marked in red)

The general, intermediate configuration (Figure 9.11a) is used to draw up the geometric relations between the design parameters. The undeployed and fully deployed states can be seen as special cases of the intermediate one. From Figure 9.12 a number of geometric relations can be drawn:

( )βθθ tancossin2 111 −= ah (9.10)

M2

M1

M0

L0

L1

L2

H2

H1

h1

h2

R2

R1

R0

p0

p1

p2

p0

p1

p2


260


10 cos2 θaL = (9.12)

21 cos2 θaL = (9.13) ( )βθθ tansincos2 111 += aL (9.14) ( )βθθ tansincos2 222 += aL (9.15)

ϕsin2 11 RL = (9.16) ϕsin2 22 RL = (9.17)

2cos00

ϕRM = (9.18)

2cos11

ϕRM = (9.19)

2cos22

ϕRM = (9.20)

21 HHH += (9.21)

( )2102

11 MMhH −−= (9.22)

( )2212

22 MMhH −−= (9.23)

a) The intermediate position (0≤ψ≤1)

(0≤θ1≤θ2≤θmax) The linkage is designed from top to bottom, by assigning a value to θ2 with 0≤θ2≤ θmax. Together with the previously determined design parameters n, U, β, a, b and φ, all dimensions can be derived by subsequently solving Eqn (9.24) to (9.31). The parameters used in this section are pictured in Figure 9.12.

For the top module:

( )βθθ tancossin2 222 −= ah (9.24) ( )βθθ tansincos2 222 += aL (9.25)

ϕsin22

2LR = (9.26)


261

2cos22

ϕRM = (9.27)

21 cos2 θaL = (9.28)

ϕsin21

1LR = (9.29)

2cos11

ϕRM = (9.30)

( )2212

22 MMhH −−= (9.31)

For the bottom module: θ1 can be calculated from this equation:

( )βθθ tansincos2 111 += aL (9.32) Alternatively, θ1 can be expressed in terms of L1, a and β:

+−−

= −2

2421

21

11 2

sinsec4coscoscos

a

aLaLa ββββθ (9.33)

The unit height h1, base length L0 , radius R0 and projected height H1 are given by:


10 cos2 θaL = (9.35)

ϕsin20

0LR = (9.36)

2cos00

ϕRM = (9.37)

( )2102

11 MMhH −−= (9.38)

When the total linkage consists of more modules (n≥2), these can be calcu-lated by repeating Eqns (9.32) to (9.38) for each extra module.


262

b) Fully deployed position (ψ=1)

(0≤θ1≤θ2=θmax) With θ2= θmax Eqns (9.24) to (9.31) become: For the top module:


Knowing that

2max2πβθθ −== (9.40)

Eqn (9.39) becomes ( )βββ tansincos22 +−= ah (9.41)

Alternatively, a simpler form can be found:

ββ

cos1cos2

2+

−= ah (9.42)

Because the upper end nodes meet in one point:

0222 === MRL (9.43) Also,

21 cos2 θaL = (9.44)

ϕsin21

1LR = (9.45)

2cos11

ϕRM = (9.46)

With M2=0 Eqn (9.31) becomes 2

12

22 MhH −= (9.47)


263

For the bottom module: The calculation is identical to that for 0≤ψ≤1, idem as Eqns (9.32) to (9.38): θ1 can be calculated from this equation:

( )βθθ tansincos2 111 += aL (9.48) Alternatively, θ1 can be expressed in terms of L1, a and β:

+−−

= −2

2421

21

11 2

sinsec4coscoscos

a

aLaLa ββββθ (9.49)

The unit height, base length, radius and projected height are given by:


10 cos2 θaL = (9.51)

ϕsin20

0LR = (9.52)

2cos00

ϕRM = (9.53)

( )2102

11 MMhH −−= (9.54)

When the total linkage consists of more modules (n>2), these can be calcu-lated by repeating the last set of Eqns (9.48) to (9.54) the appropriate number of times.

c) Undeployed position (ψ=0)

Evidently, in the undeployed position both modules are equal. Deployment an-gles θ1=θ2=0 in which case

L0=L1=L2=E (9.55) R0=R1=R2 (9.56)

h1=h2= βtanahundeployed −= (9.57)


264

undeployedtotal hnH = (9.58)

This position is after all the one from which the mechanism was designed.

9.3.2 Second approach: Design in the deployed configuration (Linkage unit)

By following the reverse approach, it is possible to impose overall dimensions on the linkage, after which the remaining parameters such as the bar length and the kink angle are derived. The governing equations for a simple one-module linkage are derived first, after which the geometry for two modules will be simultaneously derived. For determining the global geometry n, U, R0, R1, H1 are chosen and can be awarded any value (Figure 9.12 ). Depending on the value of R1, this can be an intermediate position (R1>0) or an fully deployed position (R1=0), in which case the top nodes coincide. The sector angle φ is found from

Uπϕ 2

= (9.59)

while L0 and M0 are given by

ϕsin2 00 RL = (9.60) and

2cos00

ϕRM = (9.61)

L1 and M1 are given by

ϕsin2 11 RL = (9.62)

2cos11

ϕRM = (9.63)


265

which gives for h1:

( )2102

11 MMHh −+= (9.64)

A system of four equations (9.10), (9.12), (9.14) and (9.3) in four unknowns is used to calculate the remaining parameters.

( )

( )

−=

+==

−=

β

βθθθ

βθθ

cos

tansincos2cos2

tancossin2

111

10

111

ab

aLaL

ah

(9.65)

Solving this system numerically gives two sets of solutions { a , b , β , 1θ } and { a′ , b′ , β ′ , 1θ ′} which means that two different geometries meet the imposed

design requirements. It is up to the designer to make a choice on which one to use. It is likely that the most extreme configurations, although strictly a cor-rect solution, will not be of great use as a deployable structure with a large deployment range. We can use a similar approach for simultaneously calculating two modules. Again we impose dimensions on the top and bottom extremities of the link-ages. Choose n, U, R0, R2. The first parameters are calculated from:

Uπϕ 2

= (9.66)

ϕsin2 00 RL = (9.67)

2cos00

ϕRM = (9.68)

ϕsin2 22 RL = (9.69)

2cos22

ϕRM = (9.70)


266

Now, the remainder of the parameters can be found from a system of twelve equations (9.71) in twelve unknowns. Solving this system numerically gives two sets of solutions { a , b , β , 1θ , 2θ , 1h , 2h , 1H , 2H , 1R , 1L , 1M } and { a′ , b′ , β ′ , 1θ ′ , 2θ ′ , 1h , 2h , 1H , 2H , 1R , 1L , 1M }:

( )( )

( )( )

( )

( )

+=

−−=

−−=

=

=+=+=

==

−=−=

−=

21

221

222

210

211

11

11

222

111

21

10

222

111

2cos

sin2tansincos2

tansincos2cos2cos2

tancossin2tancossin2

cos

HHHMMhH

MMhH

RM

RLaLaLaLaL

ahah

ab

ϕϕ

βθθβθθ

θθ

βθθβθθ

β

(9.71)

Again, a choice can be made between the two obtained geometries. Compared to the calculation for one module, the complexity has risen. For each addi-tional module to be calculated (simultaneously) an extra set of six equations is needed.

a) Influence of parameters

Minimal changes in the design values can have a significant effect on the overall geometry. The two parameters with the strongest impact on the ge-ometry are the kink angle and the number of modules in the linkage. Figure 9.13 shows the undeployed and fully deployed position for six different con-


267

figurations with various combinations of two or three modules with values for β of 135°, 150° and 165°. All configurations have the same edge length.

n=2

β=135° n=2

β=150° n=2

β=165°

Figure 9.13: Illustration of the influence of the apex angle β on the geometry of a linkage with

angulated SLE’s with two modules (n=2) in the undeployed (top) and fully deployed configura-

tion (below)


268

n=3 β=135°

n=3 β=150°

n=3 β=165°

Figure 9.14: Illustration of the influence of the apex angle β on the geometry of a linkage with

angulated SLE’s with three modules (n=3) in the undeployed (top) and fully deployed configura-

tion (below)


269

We can draw the following conclusions from Figure 9.13 and Figure 9.14:

• n=cte, β≠cte: a fixed number of modules, but different kink angles. As β increases, the overall height of the deployed configuration also in-creases, while the radius decreases. The biggest impact however is no-ticeable in the undeployed configuration. By increasing β from 135° to 165°, the height in the stacked position is reduced to a third. So blunter kink angles lead to linkages which are more compact – easier transportable - in their undeployed state.

• n≠cte, β=cte: Since all configurations are maximally deployed, in-

creasing the number of modules actually means adding more modules at the bottom of the linkage, each of which will be less deployed than the previous one. As a consequence – although it is obvious that add-ing more modules leads to an increased height in both the undeployed and deployed position – the contribution of each subsequently added module becomes less.

The top module in the linkage is the determining factor for the deployment range. Units with sharp kink angles tend to quickly reach their maximal de-ployment, therefore halting the deployment of the remaining modules. So if a substantial expansion in height is desired, it would be a better option to choose a blunt kink angle in combination with a higher number of modules: the blunt kink angle makes the undeployed configuration more compact and increases the deployment interval (0 to θmax). A choice will have to be made concerning the optimal number of modules that will suit the design, taking all relevant parameters into consideration.


270

9.4 From mechanism to architectural structure

9.4.1 Mobility analysis

Figure 9.15 shows a schematic representation of an undeployed and an inter-mediate deployment position of the same linkage. As the deployment pro-gresses, the angulated SLE’s of each module tilt inward at the top. The dotted lines are imaginary fold lines around which mobility has to be allowed in order to complete the deployment. Through connection of the end nodes, each scis-sor unit can be represented by a trapezoid, of which the contour changes con-stantly during deployment. Between quadrilaterals ABDC and CDFE and be-tween CDFE and EFHG there is a relative rotation which causes them not to remain coplanar. The joints connecting the end nodes of the units will have to take into account all aspects of this mobility. In Figure 9.16 a proposal for such a joint is pictured, showing the seven rotational degrees of freedom needed for the deployment, as well as for the linkage to be compactly folded (Figure 9.17).

Figure 9.15: A schematic representation of the relative rotations of the quadrilaterals around

imaginary fold axes during deployment

C

A

B

D

E F

G H


271

Figure 9.16: Kinematic joint connecting the angulated elements at their end nodes

Figure 9.17: The kinematic joint and the axes of revolution for the seven rotational degrees of

freedom

In order for the mechanism to be usable as a structure, the mobility will have to be constrained. Analogous to the previous case studies, an equivalent hinged plate model is presented. Figure 9.18 represents the linkage with the rotational degree of freedom of the scissor linkage removed. After removal of this D.O.F. the remaining mobility determines to what extent constraints have to be added. Due to triangulation of the modules, there is no additional mobil-ity which means it is basically a single D.O.F.-mechanism. Therefore, it is suffi-cient to constrain the movement of the rotational degree of freedom of the


272

scissor units. As usual, fixing two appropriately chosen nodes is enough to re-move the rotational D.O.F from the scissor linkage. But for using the tower as a load bearing structure, all three lower nodes have to be fixed to the ground by pinned supports.

Figure 9.18: The scissor linkage in its deployed state and its equivalent hinged plate structure for

mobility analysis (left) – Fixing the structure by pinned supports (right)

9.4.2 The erection process

Figure 9.19: Deployment of proof-of-concept model

A proof-of-concept model has been constructed to verify the deployment be-haviour (Figure 9.19). A short description is given of how the erection process could be executed, as shown in Figure 9.20:


273

• A: the tower is in its undeployed form. The membrane elements are attached to the nodes of the mechanism and fixed by their low points to the ground

• B: As the tower gradually deploys, the membranes are raised. When

sufficient height is achieved, the additional masts are inserted and gradually put in their right location. Then, the cables fixing the secon-dary masts to the ground are brought under tension.

• C: Finally, the tower is slightly deployed further to add pre-tension in

the membrane. Then, the tower is fixed to the ground by pinned sup-ports and additional horizontal ties (cables or struts) can be inserted at the appropriate level.


274

Figure 9.20: Deployment sequence (A, B and C) for the tower with the membrane elements at-

tached

After deployment horizontal ties are added to enhance structural stiffness (see structure analysis). Several solutions are possible: cable ties could be used,


275

which are already present before deployment and are shortened as the struc-ture deploys and becomes narrower. Struts could be added afterwards to brace the structure. An active cable can run over appropriately chosen nodes along a path and can be shortened to aid in the deployment. This needs to be further investigated.

9.4.3 Alternative configuration

As mentioned earlier, deployable towers with a base polygon of more than three sides are possible. Shown here is an alternative configuration with hex-agonal modules (Figure 9.21). The upper two modules are identical, but mir-rored versions of the two modules located just below. Figure 9.22 shows how the six-unit modules can be compacted for transport and Figure 9.23 shows three stages in the deployment.

Figure 9.21: Design for a deployable hexagonal tower with angulated elements


276

Step 3 Step 4 Step 5 (ready to deploy)

Figure 9.22: Initial unfolding of the compacted linkage to its polygonal form

Step 6 (Undeployed) Step 7 Step 8

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

top view


277

9.4.4 Simplified concept: prismoid versus hyperboloid

The previously described geometry is an exact translation of the static hyper-boloid geometry of the tower, as originally proposed by The Nomad Concept [2007], into a deployable version. In its undeployed state the scissor linkage has a prismatic shape and all angulated elements per vertical row (or lateral face of the prism) are coplanar. During deployment the shape gradually changes into a hyperboloid, which means that the angulated elements per ver-tical row are no longer coplanar, i.e. they experience relative rotation, as can be seen in the triangular example of Figure 9.15. As a consequence, the articu-lated hinges (Figure 9.17) will have to allow an extra rotational D.O.F around the horizontal axes between modules to cope with this movement which adds to the complexity of the joint design. The described deployment behaviour is caused by the particular geometry of the angulated elements which - as explained in Figure 9.8 and Figure 9.9 - is such that in its undeployed state the linkage can be further compacted by folding along the vertical fold axes. This means that the angulated elements consist of two differently sized semi-bars a and b, turning the angulated ele-ments non-symmetrical. The overall geometry of this solution shall be referred to as hyperboloid.


278

Undeployed Partially deployed

Figure 9.24: Hyperboloid geometry (as proposed in previous sections) – angulated elements do

not remain coplanar during deployment

Now, an alternative concept is proposed, which is similar in setup to the hy-perboloid version, but has simplified joints for interconnecting the modules. If the angulated elements within a vertical row can be kept coplanar, then the hinges between modules would not have to allow an extra rotational D.O.F. around the horizontal axes between modules, effectively decreasing the me-chanical complexity. Also, the end nodes of the angulated elements remain collinear, as shown in the triangular example of Figure 9.25. The effect on the overall shape is that it resembles a prism before, during and after deployment. More precisely, such a shape is known in geometry as a prismoid [Mathworld, 2007].

Module edges

Non-symmetrical

angulated elements

Axis of rota-

tion between

modules


279

Undeployed Partially deployed

Figure 9.25: Prismoid geometry (simplified alternative to the previously described geometry) -

angulated elements remain coplanar during deployment

The particular geometry of Figure 9.25 can only be achieved if symmetrical angulated elements are used. Figure 9.26 and Figure 9.27 show the difference between non-symmetrical and symmetrical elements. As described earlier, non-symmetrical elements have differently sized semi-bars a and b and are used for the hyperboloid solution.

Non-symmetrical and identical angulated elements

Figure 9.26: Non-symmetrical identical angulated elements result in a fully compactable con-

figuration: hyperboloid solution

The symmetrical elements used for the prismoid solution consist of two identi-cal semi-bars a, as Figure 9.27 shows.

Symmetrical

angulated elements

End nodes are

collinear

a b


280

Symmetrical and identical angulated elements

Figure 9.27: Symmetrical identical angulated elements cannot be fully compacted

In both cases the angulated elements are identical throughout the entire structure. An angulated scissor linkage built from non-symmetrical identical elements is fully compacted in its undeployed position, as Figure 9.26 shows. But a linkage consisting of symmetrical identical elements (Figure 9.27) is not fully compactable: when the bottom most SLE is fully compacted, the above SLE is still partially deployed. To overcome this issue, an extra condition is imposed on the scissor geometry, as Figure 9.28 shows. The angulated elements become smaller near the top and their geometry is such that their lower end nodes and intermediate hinge are collinear in the undeployed condition. This configuration ensures the high-est degree of compactness.

a a


281

Symmetrical and non-identical angulated elements

Figure 9.28: Symmetrical and non-identical angulated elements

result in a fully compactable configuration: prismoid solution

The relationship between the lengths of semi-bars of consecutive angulated elements is depicted in Figure 9.29. The length of the semi-bar a1 can be ex-pressed in terms of a0 as follows:

( )βπ −= cos01 aa (9.72)

With the kink angle β and the length of the semi-bar of the bottom most angulated element a0 chosen as design parameters, the length semi-bar of the nth element can be written as:

( )( )nn aa βπ −= cos0 (9.73)

a0

a0

a1

a1

Lower end nodes and intermediate

hinge are collinear


282

Figure 9.29: Symmetrical and non-identical angulated elements

result in a fully compactable configuration: prismoid solution

To demonstrate what happens during deployment, three consecutive stages - undeployed, partially deployed and fully deployed - are shown in Figure 9.30. During the course of the deployment, a constant angle (marked by the red lines), is subtended by each vertical linkage. This characteristic is precisely what makes the design of radially deployable closed loop structures possible. To illustrate this, Figure 9.31 shows the corresponding closed loop structure which uses the same vertical linkage, but arranged radially in a common plane.

Undeployed Intermediate Fully deployed

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

a0

a0

a1

a1 β

β

a1


283

Corresponding closed loop structures

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

Figure 9.32 shows a three-dimensional model of a triangular prismoid tower in three deployment stages, while Figure 9.33 shows a top view and a side eleva-tion.

Figure 9.32: Perspective view of the deployment of a triangular tower


284

Figure 9.33: Top view and side elevation of the prismoid tower

The simplified solution for the articulated hinge - which connects four bars at once - is shown in Figure 9.34.

Figure 9.34: Detailed view of the simplified hinge connecting four scissor bars

Note that the rotational D.O.F. around the vertical axis - marked with the dashed line in Figure 9.34 - is not necessary during the deployment of the tower. Its purpose is to allow the structure to be further compacted in the un-deployed position, by means of folding, analogous to the method described in Figure 9.9 and Figure 9.22.


285

In Figure 9.18, an equivalent hinged-plate structure for the hyperboloid ge-ometry was introduced, which has shown that the only D.O.F. in the system is the rotational D.O.F. of the scissors. When the same method is applied to the prismoid solution, it can be seen that this holds no longer true. Figure 9.35 shows a triangular and a quadrangular geometry and their equivalent hinged-plate structure. The triangular shape cannot be folded along the vertical axes between neighbouring plates, while the quadrangular solution can be flatly folded. It can be concluded that the prismoid solution is – apart from the tri-angular geometry – a multiple D.O.F.-mechanism. To turn the mechanism into a structure, and therefore removing all D.O.F.’s, all lower nodes are fixed to the ground by pinned supports.

Figure 9.35: Triangular and quadrangular prismoid solution and their respective equivalent

hinged-plate structure, providing an insight in the kinematic behaviour

Table 9.1 and Table 9.2 contain a comparison between the hyperboloid and the prismoid solution in terms of the geometry of the angulated elements, the D.O.F. during deployment and the mechanical complexity of the hinges be-tween modules.


286

Hyperboloid geometry - Identical, non-symmetrical angulated elements - Deployed shape is hyperboloid: angulated elements per vertical row do not remain coplanar - End nodes of angulated elements are not collinear during deployment: 1 D.O.F.-mechanism - Articulated hinges between modules require extra D.O.F.: increased mechani-cal complexity

Table 9.1: Characteristics of the hyperboloid geometry

Prismoid geometry - Non-identical, symmetrical angulated elements - Deployed shape is prismoid: angulated elements per vertical row are copla-nar - End nodes of angulated elements remain collinear during deployment: multi-ple D.O.F. (except for triangular geometry) - Articulated hinges between modules do not need an extra D.O.F.: decreased mechanical complexity

Table 9.2: Characteristics of the prismoid geometry


287


For this case study, a nominal wind load of 0.6 kN/m2 is combined with a snow load of 0.5 kN/m2, applied on the whole surface, so no load zones are defined. For this analysis, only a transverse direction is considered for the wind load. The alternative wind direction at 60° was not included in the analysis. The re-sulting load vectors are calculated automatically by EASY, the snow load is treated analogously. The considered load combinations are given in Table 9.3.

Load combination Permanent Main solicitation Secondary ULS/SLS 1 pre-stress + snow ULS/SLS 2 pre-stress + snow + wind ULS/SLS 3 pre-stress + wind ULS/SLS 4 pre-stress + wind + snow

Table 9.3: Load combinations for wind and snow

Figure 9.36: Top view and perspective view of the structure with indication of the global co-

ordinate system and the vector components of the wind action


288

The approach discussed in Chapter 5 is used for determining the actions of the membrane on the structure under wind and snow load. Shown in Figure 9.36 are the load vectors in the global coordinate system. Figure 9.37 and Figure 9.38 show the equilibrium form of the membrane calculated in EASY-software.

Figure 9.37: Side elevation of the equilibrium form of the membrane

Figure 9.38: Top view of the equilibrium form of the membrane

First, a finite element model of the tower with additional cable ties (steel ca-ble d=8 mm, design strength = 1500 MPa) is analysed under the specified load


289

combinations. The resulting section is 120×60×3.2 mm, with a weight of 187 kg (Figure 9.39). For a height of the tower of 8.43 m, the weight ratio (the weight of the membrane itself is neglected as in the previous case studies) is 23 kg/m.

Element type Section [mm] Weight

[kg]

Structure bar 120×60×3.2 187

Tension cable d=8 7

Total weight 194

Weight/m 23

Figure 9.39: Horizontal cable ties to improve structural performance

When the horizontal cable ties are replaced by aluminium bars, the structural performance increases. Figure 9.40 shows the results for which ULS 4 (pre-stress + wind + snow) is the governing load combination (Figure 9.41). The section is decreased to 90x50x3.2 mm and the additional struts are assigned a tubular round section of 42x2.6 mm. The total weight drops to 160 kg which results in a weight ratio of 2.2kg/m2. Although ULS 4 is the determining load combination for the strength as well as the stability, the greatest displace-ments are a result of SLS 2 (pre-stress + snow + wind). The displacements are


290

of an acceptable magnitude, especially when the nature of the structure is considered. The deflections (not shown) do not exceed 1/200.

Section hxbxt, d[mm] Total

weight/section

[kg]

Weight/m [kg/m]

TREC 90x50x3.2 144

TRON 42x2.6 16

Total weight: 160 19

Figure 9.40: Perspective view, top view and side elevation of deployable mast


291

RESULTS for Case study 4

STRESS [MPa]

Section [mm] S max

S max

(My)

S max

(Mz)

S max

Fx/Ax

Load

comb.

TREC 90x50 x3.2 85 13.5 22.8 62.1 ULS 4

TRON 42x2.6 -129.3 -5.7 0 -123.4 ULS 4

FORCE


TREC 90x50 x3.2 53.3 0.3 -0.32 ULS 4

TRON 42x2.6 -40.4 0 0 ULS 4

REACTIONS [kN]


27 ULS 1 78.5 ULS 4 59.45 ULS 4

DISPLACEMENTS [cm]


0.3 SLS 4 -4.5 SLS 2 -1.1 SLS 4


of case study 4

Ux

Uy

Uz


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9.6 Conclusion

In this chapter, a novel idea has been put forward for a deployable hyperboloid tower, used for the deployment of a membrane canopy, without the need for additional lifting equipment. The two-fold purpose of the tower, namely hold-ing up the membrane elements in the deployed position and serving as an ac-tive element during the erection process, has been demonstrated. It has been found that the proposed linear structure offers an advantage over existing solutions: using angulated elements instead of polar units for the same deployed geometry, has lead to a significant reduction of the number of scissor members and connections.

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

A geometric design method has been proposed for which the equations were derived and expressed in terms of relevant design parameters, such as the shape of the base polygon, the bar length or the kink angle of the angulated element. Although the equations were derived for a two-module geometry, the design method is equally applicable to configurations with a higher number of modules. However, it was found that the number of equations to be solved simultaneously was equally raised, therefore significantly increasing the com-plexity of the calculation. Also, the method has been devised such, that by


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simply changing the kink angle to 180°, a polygonal tower with translational units is obtained. It has been found that the deployment behaviour is heavily influenced by the kink angle of the angulated element: the blunter the kink angle, the larger the expansion range is before the fully deployed position is reached. The influence of these parameters on the geometry and the deployment process has been discussed. It has been shown that the hyperboloid shape of the tower causes the angu-lated units, per vertical row, not to remain co-planar during deployment. A novel articulated joint which allows this relative rotation has been proposed and found to work well. Further, an alternative shape, called a prismoid geometry, has been proposed. This has proven to be a simpler solution compared to the hyperboloid geome-try in terms of kinematic behaviour, therefore allowing the use of greatly sim-plified joints. Through the use of an equivalent hinged-plate model, the mobil-ity of both the hyperboloid and prismoid geometry has been assessed. It was found that the hyperboloid configuration is always, regardless of the polygonal shape, a one-degree-of-freedom mechanism. The prismoid solution, on the other hand, is always, apart from the triangular geometry, a multiple-D.O.F.- mechanism. Finally, the structural feasibility has been checked under wind and snow ac-tion. The resulting structure has a weight/height ratio of 19 kg/m, excluding the weight of the joints, which have not been structurally designed. This fourth case study has made innovative use of angulated elements in an original application. Although the concept has been proven to work, more de-tailed analysis, including structural design of the joints, is needed.


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Chapter 10 - Conclusions

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Chapter 10

Conclusions The aim of the work presented in this dissertation was to develop novel con-cepts for deployable bar structures and propose variations of existing con-cepts, leading to architecturally and structurally viable solutions for mobile applications. Equally, it was the objective to provide the designer 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 struc-tures or employing a scissor system. By presenting a review of previous research on scissor structures and foldable plate structures, an insight is given in the wide variety of possible shapes and configurations. The design principles behind scissor structures have been ex-plained and it has been shown that using geometric constructions is a straightforward way of designing deployable scissor linkages of any curvature. The design of scissor structures, as well as foldable plate structures, has been advanced by proposing novel geometric design methods based on parameters which are relevant to the designer, such as the rise and span of the structure. These principles were then used in four case studies, which cover the key as-pects of the design, including a kinematic and preliminary structural analysis of novel concepts for deployable bar structures.


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10.1 Novel concepts for deployable bar structures 10.1.1 Case study 1: A Deployable Barrel Vault with Translational Units on a Three-way Grid

(a) OPEN structure

(b) CLOSED structure

Figure 10.1: Case study 1 (Chapter 6) – Translational barrel vault

With the first case study, an advance has been made by developing a stress-free deployable scissor structure of single curvature with translational units on a three-way grid. It has been shown that the curved triangulated grid can be uniquely composed of single scissor units, therefore avoiding the integration of double scissor units. By doing so, the number of connections could be kept to a minimum and the grid demonstrated an inherent triangulation, therefore providing in-plane stability. It has been shown that, as a consequence, the deployed configuration does not require additional cross-bracing of the grid cells to prevent the structure from swaying.


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The geometric design method developed for this case study is not limited to circular base curves only, but it has been devised such, that it can be extended to structures of any curvature. It has been proven possible to provide the open barrel vault with suitable end structures, providing a fully closed envelope and greatly enhancing the architectural applicability of the concept. 10.1.2 Case study 2: A Deployable Barrel Vault with Polar and Translational Units on a Two-way Grid

(a) OPEN structure

(b) CLOSED structure

Figure 10.2: Case study 2 (Chapter 7) – Polar barrel vault

For the second case study, a deployable barrel vault with polar and transla-tional scissor units on an orthogonal two-way grid, has been proposed. This case study is an illustration of how a basic stress-free deployable single curva-ture structure can be obtained by putting the construction methods and the


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proposed geometric design methods into practice. It has been shown that the geometric design equations can be used for studying the configuration in vari-ous stages of the deployment, allowing to determine the maximum span reached during the erection process. An innovative way of rendering the open structure fully closed by providing it with doubly curved end structures has been proposed. This was proven possible by adapting the geometry of a lamella dome in such a way, that when attached to the open structure, a fully closed stress-free deployable configuration is formed. 10.1.3 Case study 3: A Deployable Bar Structure with Foldable Articulated Joints

(a) OPEN structure

(b) DOME structure

(c) CLOSED structure (combined)

Figure 10.3: Case study 3 (Chapter 8) – Deployable bar structure with foldable joints


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Case study 3 has successfully shown that a novel type of deployable bar struc-ture can be designed, based on the geometry and kinematic behaviour of fold-able plate structures. The versatility of the design method proposed in Chapter 4, has been demonstrated by producing a single curvature (barrel vault) and a double curvature (dome) space enclosure with high element uniformity. As a consequence, it was shown possible for both shapes to be combined into a closed doubly curved foldable geometry, effectively increasing the range of possible applications. For connecting the bars, an innovative foldable articu-lated joint has been developed, which has proven to preserve the kinematic properties of the plate structure it is derived from. 10.1.4 Case study 4: A Deployable Tower with Angulated Units

Figure 10.4: Case study 4 (Chapter 9) – Deployable mast


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For case study 4, a novel idea has been put forward for a deployable hyperbol-oid tower, as part of a temporary tensile surface structure. It has been shown possible for the tower to be employed as an active element during the erection process, therefore eliminating the need for additional lifting equipment. By using angulated elements instead of polar units for the same deployed geome-try, it was shown that a significant reduction of the number of scissor mem-bers and connections could be realised. A novel geometric design method has been proposed allowing the design of a tower with any polygonal shape. Al-though the equations were derived for a two-module geometry, the design method is equally applicable to configurations with a higher number of mod-ules. The method has been devised such, that by simply changing the kink an-gle to 180°, a polygonal tower with translational units can be obtained. Also, it was shown possible to design an alternative geometry with greatly simplified connections, by imposing a specific condition on the angulated elements.

10.2 Comparative evaluation of the proposed concepts 10.2.1 Architectural evaluation Easily transportable structures require a compact shape in their undeployed configuration and a subsequent high volume expansion during the erection process, which provides a basis for comparison. Although the first three case studies (open barrel vaults and doubly curved closed structures) reach a similar overall shape and volume in the deployed configuration, the way they achieve that final deployed state differs. It was found that the scissor structures of case study 1 and 2, being a tight bundle of bars, demonstrate the highest vol-ume increase between the undeployed and fully deployed state, where all structures reach a volume of approximately 140 m3. Case study 1 was found to demonstrates the greatest expansion, having an initial volume of only 0.6 m3 for both the open and closed structure. A slightly higher initial volume was found for case study 2, with 1 m3 for the open structure and 1.8 m3 for the closed structure.


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The lowest volume expansion was, not surprisingly, found with case study 3, having an initial volume of 17 m3. Its characteristic folding movement leads to a significant expansion in the longitudinal direction only , while in the trans-verse direction almost no expansion occurs. For case study 4, which has a linear, vertical expansion, it was shown that its geometry is designed such, that in the undeployed configuration it can be fur-ther compacted by releasing one row of joints. The tower expands to a final height of 8.5 m, 5 times its initial height of 1.7 m. Apart from the structural system employed, another factor has been found to influence the compactness in the undeployed configuration. The higher the amount of members to be connected in a single joint, the larger the radius of that hub becomes, in order to accommodate all elements without hindering the deployment. This explains the aforementioned difference in size of the un-deployed configurations of case studies 1 and 2. 10.2.2 Kinematic evaluation By constructing several 1/10 scale proof-of-concept models of the proposed designs, the validity of the concepts has been successfully demonstrated. These models have significantly aided in understanding the deployment behav-iour and have given an idea of the connections needed to guarantee a com-plete and stress-free deployment. By introducing the concept of an equivalent hinged-plate model, an interesting way of gaining insight in the kinematic behaviour of the mechanisms has been offered. Case study 1 has been found to be a single-D.O.F.-mechanism. It was shown that the geometry is subject to angular distortion, an effect associated with the introduction of curvature in translational units during the deployment. As a consequence, the joints are designed such, that this in-plane rotation is al-lowed.


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Case study 2 is also a single-D.O.F.-mechanism, but by closing the open barrel vault with doubly curved end structures, the mobility of this particular geome-try was augmented to 4 D.O.F. It has been shown that both the single curvature and double curvature struc-ture from case study 3 are multiple D.O.F.-mechanisms. By appropriately in-corporating scissor linkages, it has been shown that the mobility can be re-duced to 1 D.O.F., allowing for a more controlled deployment The deployable tower from case study 4 has been found to possess a single D.O.F. during deployment, due to its hyperboloid geometry. Its simplified counterpart, the prismoid tower, has 1 D.O.F. in case of a triangular tower, but multiple D.O.F.’s for any other polygonal shape. It has been shown that this extra mobility is introduced by intersecting fold lines inherent to the prismoid geometry. In order to remove all mobility from the mechanisms in the fully deployed po-sition and to effectively employ them as a structure, it has been suggested to fix all lower nodes to the ground by pinned supports. 10.2.3 Structural evaluation For case studies 1, 2 and 4 it has been found from the preliminary structural analysis that, due to high in-plane bending stress in the scissor members, the strength is the governing design criterion. Compared to case study 3, these structures have significantly shorter bars, leading to a decrease in buckling sensitivity. By adding a continuous cable between nodes, a positive effect on the structural performance and a reduction in weight has been achieved. For case study 3 however, it has been found that the governing design criterion is buckling of the bars. due to their relatively great length compared to the span of the structure. The open and closed configuration from case study 1, both covering 65 m2, achieve a weight ratio of approximately 7.5 kg/m2. Case study 2 is a heavier


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solution, reaching 10 kg/m2 and 13.3 kg/m2 for its open and closed structure respectively, while case study 3 is by far the lightest solution of the three bar-rel vaults with 4 kg/m2 for single curvature structure, 4.5 kg/m2 for the dome and 5 kg/m2 for the combined shape. The 8.5 m high tower from the fourth case study achieves a weight-per-metre ratio of 19 kg/m. It is noted that the joints have not been included in the structural design and therefore no state-ment has been done concerning their weight. It is unclear at this stage what the influence is of fatigue, caused by fluctuat-ing wind loads, on the erected structure. Although a detailed calculation was out of the scope of this dissertation, the consecutive steps to determine all the necessary parameters needed for the calculation have been discussed. Further work The simplified approach used for the structural analysis was sufficient within the framework of a preliminary design to test the feasibility of the concepts, however, profound and detailed analysis is needed using a more accurate ap-proach, preferably by using an integrated model which takes the mutual re-sponse of the tensile surface and the bar structure into account. Since the joints, which were currently excluded from the analysis, constitute a major part of the design, a detailed study in which they are structurally designed and analysed is needed. For appreciating the full effect of fluctuating wind on the structure, experi-mental data is needed, such as the resistance of the joints against fatigue or wind tunnel testing to obtain a more realistic stress history due to fluctuating wind loads. Also, the effects of thermal influences. tolerances, friction and geometric imperfections on the structural performance of the structure re-quires further study. The current case studies were designed as small-scale shelters. Therefore, de-termining the maximum span which can be reached with a certain configura-tion could be the subject of further study. Also, configurations with a higher number of units (for the same span), leading to shorter bars, should be investi-


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gated. The expected beneficial effect of shorter bars on the buckling sensitivity compared with the effect of the introduction of more joints. By building a prototype, including the real size design of the joints, the ease of erection, dismantling and transportation could be verified. Also, this would allow to further investigate some aspects concerning the integration of the membrane, the adjustment of the pretension and the detailing of the connec-tions.

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List of Publications De Temmerman, N., Mollaert, M. [2003], Construction Teaching Methods, Proc. of ENHSA 2003 Second Workshop of Construction Teachers at Les Grands Ateliers, Lyon, pp. 46-52, EAAE-ENHSA, Thessaloniki. De Temmerman, N., Van Mele, T., Mollaert, M. [2004], Kinetic Structures: Architectural Organisms as a Design Concept, Proc. of the IASS 2004 Int. Symposium on Shell and Spatial Structures from Models to Realization, pp. 410-411, IASS Montpellier, (paper on cd-rom). De Temmerman, N., Mollaert, M., Van Mele, T., De Laet, L. [2006], A Concept for a Foldable Mobile Shelter System, Proc. of the IASS-APCS 2006 Int. Symposium on the New Olympics & New Shell and Spatial Structures, pp. 240-241, IASS-APCS, Bejing, (paper on cd-rom). De Temmerman, N., Mollaert, M., Van Mele, T., De Laet, L. [2006], Development of a Foldable Mobile Shelter System, Adaptables 2006: Proc. of the International Conference on Adaptability in Design and Construction, vol. 2, pp. 13-17, Eindhoven University of Technology, Eindhoven. De Temmerman, N., Mollaert, M., Van Mele, T., De Laet, L. [to be published], Design and Analysis of a Foldable Mobile Shelter System, Special Issue on Adaptive Structures of the International Journal of Space Structures, Multiscience Publishing, Brentwood. Mollaert, M., De Temmerman, N., Van Mele, T., Daerden, F., Block, P. [2003], Adaptable Tensioned Coverings, Proc. of the IASS-APCS 2003 Int. Symposium on New Perspectives for Shell and Spatial Structures, pp. 204-205, IASS-APCS Taipei, (paper on cd-rom).

Mollaert, M., De Temmerman, N. [2003], (At) Large: To Scale: working with bigger scale models has its advantages, Fabric Architecture, pp. 12-14, July-August 2003. Mollaert, M., De Temmerman, N., Van Mele, T. [to be published], Deployable and Retractable Tensile Structures, Proc. of the 12th Int. Workshop on the

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Design and Practical Realisation of Arch. Membrane Structures, Technical University Berlin, (paper on cd-rom). Mollaert, M., De Temmerman, N., Van Mele, T. [2006], The Development of a Tensile Surface for a Non-Static Application, Proc. Of the 11th int. Workshop on the design and Practical Realisation of Arch. Membrane Structures, Issue: Textile Roofs 2006, published by: Technical University Berlin. Mollaert, M., De Temmerman, N., Van Mele, T. [2006], Variations in Form and Stress Behaviour of a V-shaped Membrane in a Foldable Structure, Proc. of the HPSM 2006 Third International Conference on High Performance Structures and Materials, pp. 41-50, Witpress, Oostende. Van Mele, T., Mollaert, M., De Temmerman, N., De Laet, L. [2006], Design of Scissor Structures for Retractable Roofs, Adaptables 2006: Proc. of the International Conference on Adaptability in Design and Construction, vol. 2, pp. 68-73, Eindhoven University of Technology, Eindhoven.

Design and Analysis of Deployable Bar Structures for Mobile

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