-
FACULTY OF ENGINEERING Department of Architectural Engineering
Sciences
Design and Analysis of Deployable Bar Structures for Mobile
Architectural Applications
Thesis submitted in fulfilment of the requirements for the award
of
the degree of Doctor in de Ingenieurswetenschappen (Doctor in
Engineering) by
Niels De Temmerman June 2007 Promotor: Prof. Marijke
Mollaert
-
Members of the Jury:
Prof. Dirk Lefeber (President) Vrije Universiteit Brussel
Prof. Rik Pintelon (Vice-President)
Vrije Universiteit Brussel
Prof. Marijke Mollaert (Promotor) Vrije Universiteit Brussel
Prof. Ine Wouters (Secretary)
Vrije Universiteit Brussel
Prof. Sigrid Adriaenssens Vrije Universiteit Brussel
Prof. John Chilton
Lincoln School of Architecture
Prof. W.P. De Wilde Vrije Universiteit Brussel
Dr. Frank Jensen
rhus School of Architecture
-
Acknowledgements My interest in the exciting field of deployable
structures came about through the process of writing my masters
thesis on the subject, under the supervision of Prof. Marijke
Mollaert. This has been the inspiration and drive to delve deeper
into this rich and rewarding research topic of which this
dissertation is the final result. I would like to express my
gratitude to my supervisor Prof. Marijke Mollaert for sharing her
vast research experience and for her invaluable scientific
guidance. Also, I have great appreciation for her warm and kind
personality and her con-tinuous encouragement throughout the course
of this research. I would like to thank everyone who has
contributed in making the past four years into an exciting and
enriching experience: My most heartfelt sympathy goes out to my
colleagues of the Department of Architectural Engineering, whom I
thank for providing a kind and stimulating environment, and for
their friendship and support: Maryse Koll, Tom Van Mele, Thomas Van
der Velde, Lars De Laet, Lisa Wastiels, Anne Paduart, Caroline
Henrotay, Michael de Bouw, Prof. Ine Wouters, Prof. Hendrik
Hendrickx, Prof. Jos Depuydt, dr. Jonas Lindekens. At the
department of Mechanical Engineering, I would like to thank Prof.
Dirk Lefeber and Prof. Patrick Kool for their help on gaining an
insight in the mobil-ity of mechanisms. Prof. Patrick De Wilde,
Prof. Sigrid Adriaenssens and Wim Debacker from the department of
Mechanics of Materials and Constructions, and Prof. Rik Pintelon
from the Department of Fundamental Electricity and Instrumentation,
I would like to thank for their scientific advice and sugges-tions.
I gratefully acknowledge the financial support extended to me by
IWT-Vlaanderen (Institute for the Promotion of Innovation through
Science and Technology in Flanders).
-
dr. Frank Jensen and Prof. John Chilton I would like to thank
for sharing their expertise on the subject and providing much
valued comments and sugges-tions. Also, many thanks to Wouter
Decorte, for the fruitful collaboration, his enthu-siasm on the
subject and for sharing his excellent model-making skills. My
parents, Eric and Monique, my sister Ilka and her husband Tom, and
also Solange, deserve special thanks for their love and friendship
and for their un-conditional support and encouragement. Above all,
I wish to express my love and sincerest gratitude to Els, my
partner and friend, for her continuous love and support. Without
her I would never have come this far.
Vilvoorde, June 2007 Niels De Temmerman
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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
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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
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VIII
8.4 Structural analysis 235 8.4.1 Open structure (single
curvature) 235 8.4.2 Closed structure (double curvature) 239 8.5
Conclusion 242
9. Case Study 4: A Deployable Tower with Angulated Units 245
9.1 Introduction 246 9.1.1 A concept for a deployable tower 247
9.2 Description of the geometry 249 9.3 Geometric Design 255 9.3.1
First approach: Design of the undeployed configuration 255 9.3.2
Second approach: Design of the deployed configuration 264 9.4 From
mechanism to architectural structure 270 9.4.1 Mobility analysis
270 9.4.2 The erection process 272 9.4.3 Alternative configuration
275 9.4.4 Simplified concept: prismoid versus hyperboloid 277 9.5
Structural analysis 287 9.6 Conclusion 292
10. Conclusions 295 10.1 Novel concepts for deployable bar
structures 296
10.1.1 Case study 1: A Deployable Barrel Vault with
Translational Units on a Three-way Grid 296
10.2.2 Case study 2: A Deployable Barrel Vault with Polar and
Translational Units on a Two-way Grid 297
10.2.3 Case study 3: A Deployable Bar Structure with Foldable
Articulated Joints 298
10.2.4 Case study 4: A Deployable tower with Angulated Units 299
10.2 Comparative evaluation of the proposed concepts 300 10.2.1
Architectural evaluation 300 10.2.2 Kinematic evaluation 301 10.2.3
Structural evaluation 302 10.3 Further work 303
-
IX
References 305 List of Publications 313
-
305
List of Figures Figure 1.1: Mobile deployable bar structure (
Grupo Estran)..................................2 Figure 1.2:
Classification of structural systems for deployable structures
by
their morphological and kinematic characteristics [Hanaor, 2001]
..3 Figure 2.1 : Translational units
........................................................................................
10 Figure 2.2: The simplest plane translational scissor linkage,
called a lazy-tong
..............................................................................................................................
11 Figure 2.3: A curved translational linkage in its deployed and
undeployed
position
..............................................................................................................
11 Figure 2.4: Polar unit
..........................................................................................................
12 Figure 2.5: A polar linkage in its undeployed and deployed
position .................. 12 Figure 2.6: The deployability
constraint in terms of the semi-lengths a, b, c and
d of two adjoining scissor units in three consecutive deployment
stages..................................................................................................................
13
Figure 2.7: Piero demonstrates his prototype of a deployable
shell [Robbin, 1996]
..................................................................................................................
14
Figure 2.8: Planar two-way grid with translational units and
cylindrical barrel vault with polar units [Escrig, 1985]
........................................................ 15
Figure 2.9: Top view and side elevation of a two-way spherical
grid with identical polar units [Escrig, 1987]
........................................................... 15
Figure 2.10: Top view and side elevation of a three-way
spherical grid with polar units
.........................................................................................................
15
Figure 2.11: Top view and side elevation of a geodesic dome with
polar units [Escrig, 1987]
...................................................................................................
16
Figure 2.12: Top view and side elevation of a lamella dome with
identical polar units
....................................................................................................................
16
Figure 2.13: Deployable cover for a swimming pool in Seville
designed by Escrig & Sanchez ( Performance
SL)....................................................... 16
Figure 2.14: Bi-stable structure before, during and after
deployment ............... 17 Figure 2.15: Collapsible dome and a
single unit, as proposed by Zeigler [1976]
..............................................................................................................................
17 Figure 2.16: Bi-stable structures: elliptical arch and geodesic
dome [Gantes,
2004]
..................................................................................................................
18
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306
Figure 2.17: Positive curvature structure with translational
units in two deployment stages [Langbecker,
2001].................................................... 19
Figure 2.18: Negative curvature structure with translational
units in two deployment stages [Langbecker,
2001].................................................... 19
Figure 2.19: Plane and spatial pantographic columns by Raskin
[1998]............ 20 Figure 2.20: Pantographic slabs by Raskin
[1998]..................................................... 20
Figure 2.21: Deployable ring structure [You & Pellegrino, 1993]
......................... 21 Figure 2.22: Angulated unit or
hobermans unit
........................................................ 21 Figure
2.23: A radially deployable linkage consisting of angulated (or
hobermans) units in three stages of the
deployment......................... 22 Figure 2.24: Multi-angulated
element
..........................................................................
23 Figure 2.25: A radially deployable linkage consisting of
multi-angulated
elements in three stages of the
deployment.......................................... 23 Figure
2.26: Multi-angulated structure with cover elements in an
intermediate
deployment position
......................................................................................
24 Figure 2.27: Model of a non-circular structure where all
boundaries and plates
are unique [Jensen,
2004]............................................................................
24 Figure 2.28: Computer model of an expandable blob structure
[Jensen &
Pellegrino, 2004]
.............................................................................................
25 Figure 2.29: Reciprocal plate
structure.........................................................................
25 Figure 2.30: Swivel diaphragm in consecutive stages of
deployment................. 26 Figure 2.31: Reciprocal dome
proposed by Piero [Escrig, 1993]......................... 26 Figure
2.32: Iris dome by Hoberman [Kassabian et al,
1999]................................. 27 Figure 2.33: Retractable
dome on Expo Hannover (courtesy of M. Mollaert)
Mechanical curtain Winter Olympics Salt Lake City 2002
[Hoberman,
2007]...........................................................................................
27
Figure 2.34: Retractable roof made from spherical plates with
fixed points of rotation
..............................................................................................................
28
Figure 2.35: Novel retractable dome with spherical plates with
modified
boundaries.........................................................................................................
28
Figure 2.36: Basic layout of Fosters module [Foster, 1986]
.................................. 29 Figure 2.37: Combination of
different modules [Foster, 1986] ............................. 30
Figure 2.38: Simplest building with 90 apex angle [Foster, 1986]
..................... 31 Figure 2.39: Building formed by two
90-modules joined at their ends [Foster,
1986]
..................................................................................................................
32
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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
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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
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309
Figure 3.23: Two-way grid with directions A and
B.................................................. 71 Figure 3.24:
Three-way grid with directions C, D and E
.......................................... 71 Figure 3.25: Linear
elements prismatic columns arches
.................................. 72 Figure 3.26: Parallel linear
structures connected by non-deployable elements
..............................................................................................................................
73 Figure 3.27: Plane translational units on a two-way grid
...................................... 74 Figure 3.28: Plane
translational units on a three-way
grid.................................... 75 Figure 3.29: Plane
translational units on a four-way
grid..................................... 76 Figure 3.30: Plane and
curved translational units on a two-way grid................ 77
Figure 3.31: Polar and translational units on a two-way
grid............................... 78 Figure 3.32: Plane and
curved translational units on a three-way grid ............. 79
Figure 3.33: Polar units on a three-way grid (variation 1)
..................................... 80 Figure 3.34: Polar and
translational units on a three-way grid (variation 2) ... 81 Figure
3.35: Translational units on a two-way grid (synclastic
shape)............... 82 Figure 3.36: Two variations for
translational units on a two-way grid .............. 83 Figure
3.37: Translational units on a lamella grid
..................................................... 84 Figure
3.38: Polar units on a lamella grid
....................................................................
85 Figure 4.1: Typical foldable plate structure
.................................................................
88 Figure 4.2: Fold patterns of type A and B for the smallest
possible regular
structure
(p=5).................................................................................................
89 Figure 4.3: Unfolded and fully folded configuration of patterns
A and B (p=5)
..............................................................................................................................
90 Figure 4.4: Elevation view of the compactly folded and fully
deployed
configuration for a regular structure with five plates and an
apex angle of
120....................................................................................................
90
Figure 4.5: Right-angled fold pattern: altering one apex angle
to 90 enables a compacter folded configuration (introduction of
quadrangular plates near the
sides).....................................................................................
91
Figure 4.6: Elevation view of compactly folded and fully
deployed configuration for a right-angled structure with five
plates and an apex angle of 120
.........................................................................................
91
Figure 4.7: Three stages of deployment for a basic regular
foldable structure (p=7; =120): completely unfolded, erected
position and fully compacted for
transport...............................................................................
92
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310
Figure 4.8: Plate element, compactly folded configuration and
fully deployed configuration (front elevation) for the first three
compactly foldable structures (p=5, p=7,
p=9)...........................................................................
93
Figure 4.9: Side elevation of the fully deployed configuration
of the first three compactly foldable structures (p=5, p=7,
p=9)..................................... 94
Figure 4.10: For a chosen number of panels p the apex angle can
be altered at will, affecting the width of the structure and the
compactly folded
state.......................................................................................................
95
Figure 4.11: Parameters used to characterise a foldable
structure: length L, span S, width W, apex angle and the deployment
angle ............ 96
Figure 4.12: A foldable plate and its parameters: length L,
height H, H1, H2, apex angle , the deployment angle and angles , 1,
2 .............. 96
Figure 4.13: Perspective view and side elevation of the vertical
projection of a plate linkage for empirically determining the
relationship between 1 and p
.............................................................................................................
98
Figure 4.14: The relationship between the apex angle and the
deployment angle for regular structures with p=5, p=7 and
p=9....................... 99
Figure 4.15: Elevation view and perspective view of the
deployment of a regular five-plate structure with
=120..............................................100
Figure 4.16: The parameters associated with the polygonal
contour of the flatly folded configurations with p=5, p=7 and p=9
and the expressions for the area in terms of the edge length
Ledge................102
Figure 4.17: The relationship between the apex angle and the
deployment angle for right-angled structures with p=5, p=7 and
p=9...........105
Figure 4.18: Plate element, fold pattern, compactly folded
configuration and fully deployed configuration (front elevation and
side elevation) for three compactly foldable five-plate
right-angled structures (drawn to
scale)............................................................................................................106
Figure 4.19: Only for p=5 can any regular and any right-angled
structure be interconnected along a common edge, regardless of the
value for
............................................................................................................................107
Figure 4.20: Top view and perspective view of circular foldable
structure .....108 Figure 4.21: Fold pattern and a single sector of
a circular structure with q=8
............................................................................................................................108
-
311
Figure 4.22: Horizontal projection of a plate linkage for
empirically determining the relationship between 2 and q
..................................108
Figure 4.23: Connecting a regular module with two half-domes
leads to an alternative fully closed configuration with high plate
uniformity110
Figure 4.24: Circular structure with q=6, q=8 and q=10 (top
view) and its respective combination with a compatible regular
structure (perspective view)
.........................................................................................111
Figure 4.25: Some examples of alternative configurations
..................................112 Figure 5.1: Some of the
concepts for mobile structures presented in the
following
chapters........................................................................................115
Figure 5.2: Front elevation view of cases studies shows the mutual
similarity of
the geometry. Case study 1, 2 and 3 are based on the same shape
(semicircle with radius of 3
m).................................................................117
Figure 5.3: Overall geometry for the case studies: single
curvature shape (open) and double curvature shape (closed)
......................................................118
Figure 5.4: Perspective view of the single and double curvature
geometries.118 Figure 5.5: Perspective view and side elevation of
case study 4 ........................120 Figure 5.6: Wind and snow
action on the open and closed structure...............122 Figure
5.7: Schematic representation of considered wind loads on the
closed
and.....................................................................................................................125
Figure 5.8: Schematic representation of snow loads on the closed
and open
structures
........................................................................................................127
Figure 5.9 : Method of accumulate damage [Eurocode 3, 2007]
.......................131 Figure 6.1: Deployable barrel vault with
translational units on a triangular
grid: scissor structure and tensile surface
............................................135 Figure 6.2: Plan
view and perspective view of the same double curvature
structure with translational units on a quadrangular grid
..............137 Figure 6.3: Translational scissor module with
only single units..........................138 Figure 6.4:
Translational scissor module with a double
unit................................138 Figure 6.5: Plan view,
perspective view and side elevation of a planar structure
with a triangulated
grid..............................................................................138
Figure 6.6: Plan view, perspective view and side elevation of a
barrel vault with
a triangulated
grid........................................................................................139
-
312
Figure 6.7: Perspective view and plan view of three different
triangular
modules............................................................................................................139
Figure 6.8: OPEN structure: perspective view and plan
view...............................140 Figure 6.9: CLOSED
structure: perspective view and plan view (double scissor
marked in
red)................................................................................................141
Figure 6.10: Front elevation, top view and perspective view of a
portion of the
barrel vault with four modules in the span: the projected
versions (marked in red) of the scissor units U1 and U2 determine
the real curvature
.........................................................................................................142
Figure 6.11: Developed view of units U1, U2 and U3: graphic
representation of the deployability condition by means of
ellipses................................143
Figure 6.12: An ellipsoid representing the geometric
deployability condition in three
dimensions...........................................................................................144
Figure 6.13: Vertical section view of the small and big
ellipsoid, imposing the geometric deployability condition
...........................................................144
Figure 6.14: A scissor linkage fitted on a circular curve, with
all relevant design parameters and the global coordinate
system.....................................145
Figure 6.15: Developed view of the scissor linkage from Figure
6.14, showing a chain of double ellipses
..............................................................................146
Figure 6.16: Perspective view of the scissor linkage from Figure
6.15 .............147 Figure 6.17: Perspective view, front
elevation and top view of the deployment
process of the barrel vault with translational units OPEN
structure............................................................................................................................150
Figure 6.18: Proof-of-concept model of (half of the) closed
structure (aluminium, scale 1/10)
..............................................................................151
Figure 6.19: Two double scissors in partially (left) and fully
deployed (right) position
............................................................................................................151
Figure 6.20: Perspective view, front elevation and top view of
the deployment process of the barrel vault with translational units
CLOSED structure
..........................................................................................................152
Figure 6.21: From scissor mechanism to the equivalent hinged
plate linkage for mobility analysis of the open structure (idem for
closed structure) - minimal constraints
.....................................................................................153
Figure 6.22: Fixing all lower nodes to the ground by pinned
supports.............154
-
313
Figure 6.23: An active cable (marked in red) runs through the
mechanism, connecting upper and lower nodes along its path. After
deployment it is locked to stiffen the
structure..........................................................154
Figure 6.24: Top view and perspective view of one scissor unit,
its intermediate hinge and its end joints and their offset position
relative to the theoretical plane
...........................................................................................156
Figure 6.25: Concept for an articulated joint, allowing the fins
which accept the bars to rotate around a vertical axis, to cope
with the angular distortion of the grid
...................................................................................156
Figure 6.26: Partially and undeployed state: as the structure is
compactly folded, the imaginary intersection point of the
centrelines travels on the vertical centreline through the
joint.........................................157
Figure 6.27: Perspective view and top view of OPEN structure
with integrated tensile
surface................................................................................................158
Figure 6.28: Perspective view and top view of CLOSED structure
with integrated tensile
surface...........................................................................158
Figure 6.29: Top view and perspective view of the skeletal
scissor structure (left) and the boundary geometry for the
compatible membrane
(right)................................................................................................................159
Figure 6.30: Views of the equilibrium form for the
membrane...........................160 Figure 6.31: Typical
stresses in the membrane range from 4 to 5.5 kN/m ......160 Figure
6.32: FEM-model of six bars attached to a
node........................................162 Figure 6.33: An
intermediate pivot hinge connects two scissor
bars................162 Figure 6.34: Local coordinate system of a
bar element (left) and global
coordinate system (right)
...........................................................................162
Figure 6.35: Typical pattern of load vectors for transverse wind +
pre-stress of
the
membrane................................................................................................163
Figure 6.36: Typical pattern of reaction forces under transverse
wind.............163 Figure 6.37: Bending moments My under
transverse wind ..................................163 Figure 6.38:
Typical deformation under transverse wind:
.....................................164 Figure 6.39: Perspective
view of the resulting structure with rectangular
sections of
120x60mm................................................................................166
Figure 6.40: Reactions in the global coordinate system: the maximal
reaction
force occurs under ULS 2 (pre-stress + snow + transverse wind)
.167
-
314
Figure 6.41: The critically loaded bar is located at the top.
Summary of the stresses occurring in the critically loaded bar
(positive stresses indicate pressure, negative values mean tension)
..............................167
Figure 6.42: Axial forces, transverse forces and bending moments
in the local coordinate system of the bars
..................................................................168
Figure 6.43: Maximal nodal displacements in the global
coordinate system.169 Figure 6.44: Continuous cable zigzagging
through the structure, connecting
upper and lower nodes and contributing to the structural
performance
...................................................................................................170
Figure 6.45: Resulting structure after optimization, with cable
elements ......170 Figure 6.46: Summary of the determining stresses
and forces for the strength,
stability and stiffness of case study 1: OPEN
structure....................171 Figure 6.47: Perspective view of
case study 1: CLOSED structure: with sections
after structure design and total
weight.................................................173 Figure
6.48: Summary of the determining parameters for the strength,
stability
and stiffness of case study 1 _ CLOSED
structure..............................174 Figure 6.49: Case study
1: Single curvature OPEN structure (barrel vault) .....175 Figure
6.50: Case study 1: Double curvature CLOSED structure
........................176 Figure 7.1: Deployable barrel vault
with polar units on a quadrangular grid:
scissor structure and tensile surface
......................................................177 Figure
7.2: Plan view and perspective view of a planar structure with
a
quadrangular grid
.........................................................................................179
Figure 7.3: Plan view and perspective view of a barrel vault with
quadrangular
grid
....................................................................................................................179
Figure 7.4: Series of polar linkages with 3, 4, 5 or 6 units in the
span,............180 Figure 7.5: Geometric construction of the
four-unit linkage...............................181 Figure 7.6:
OPEN structure: perspective view and top
view.................................181 Figure 7.7: Perspective
view and developed view of units U1 (plane
translational) and U2, U3 (polar): graphic representation of the
deployability condition by means of
ellipses........................................182
Figure 7.8: Adding an end structure based on parallels and
meridians to the main structure
...............................................................................................184
Figure 7.9: A lamella dome has a stress-free deployment
....................................184
-
315
Figure 7.10: The main structure is provided with half of an
adapted lamella dome
.................................................................................................................185
Figure 7.11: CLOSED structure: perspective view and plan
view........................185 Figure 7.12: Perspective view,
front elevation and top view of the deployment
process of the polar barrel vault OPEN
structure............................186 Figure 7.13: Perspective
view, front elevation and top view of the deployment
process of the polar barrel vault CLOSED structure
.......................187 Figure 7.14: Proof-of-concept model
(half of the structure) in three
deployment
stages........................................................................................187
Figure 7.15: Deployment sequence of a polar linkage
...........................................189 Figure 7.16: Polar
linkage in an intermediate deployment stage: 10
-
316
Figure 7.31: Improved result by inserting vertical cable ties
...............................200 Figure 7.32: Additional diagonal
bars triangulate the grid...................................201
Figure 7.33: Double diagonal cross bars offer no real advantage
structurally
............................................................................................................................202
Figure 7.34: Perspective view of case study 2 OPEN structure, with
sections
after structure design and
weight/m2.....................................................203
Figure 7.35: Summary of the determining parameters for the
strength, stability
and stiffness for case study 2 OPEN structure
....................................204 Figure 7.36: Main structure
and additional end structures with no additional
measures to improve structural
performance......................................205 Figure 7.37:
Perspective view of case study 2 CLOSED structure:, with
resulting
sections after structure design and total
weight................................206 Figure 7.38: Summary of
the results for the structural analysis of case study 2
............................................................................................................................207
Figure 7.39: Case 2: OPEN
structure............................................................................208
Figure 7.40: Case 2: CLOSED structure
.......................................................................209
Figure 8.1: Foldable bar structure based on the geometry of
foldable plate
structures
........................................................................................................211
Figure 8.2: Typical foldable plate structure
...............................................................213
Figure 8.3: Design parameters for a basic regular foldable plate
structure. ...214 Figure 8.4: For a chosen number of panels p the
apex angle can be altered
at will, only affecting the width of the
structure...............................215 Figure 8.5: Graph
showing the relation between the deployment angle and
the apex angle in the fully deployed configuration for p=5
......216 Figure 8.6: The resulting regular geometry for the case
study: two extreme
deployment states and the fold pattern
................................................217 Figure 8.7: Top
view and a perspective view of a circular plate geometry with
six sectors arranged
radially......................................................................217
Figure 8.8: The resulting circular geometry for the case study: two
extreme
deployment states and the fold pattern
................................................218 Figure 8.9: A
combination of a regular and a circular
geometry........................218 Figure 8.10: Dimensions in plan
view of the
shapes...............................................219 Figure
8.11: A foldable plate structure (p=7) and its similar counterpart,
a
foldable bar
structure..................................................................................219
-
317
Figure 8.12: Pattern 1: double bars present
..............................................................220
Figure 8.13: Pattern 2: double bars removed
............................................................220
Figure 8.14: Pattern 3: double bars and diagonal bars removed,
without
affecting the original kinematic behaviour
..........................................221 Figure 8.15: Foldable
3 D.O.F.-joint derived directly from the fold pattern,
therefore mimicking its kinematic behaviour
......................................221 Figure 8.16: Deployment
sequence for the foldable joint: from the undeployed
to the fully deployed position
...................................................................222
Figure 8.17: The (regular) open structure complete with bars and
joints:.......222 Figure 8.18: Detailed view of bars and three
variations of foldable joints
occurring in the structure
..........................................................................223
Figure 8.19: Deployment sequence for the open structure perspective
view,
front elevation and top
view.....................................................................224
Figure 8.20: Proof-of-concept model of the regular structure (with
scissors) in
four stages of the
deployment..................................................................224
Figure 8.21: Deployment sequence for the dome structure perspective
view,
front elevation and top
view.....................................................................225
Figure 8.22: Proof-of-concept model of the foldable dome (with
additional
scissor units) in six deployment
stages..................................................225 Figure
8.23: Deployment sequence for the closed structure: 1 regular
module +
2
semi-domes.................................................................................................226
Figure 8.24: Six stages in the deployment of the closed structure
(top view)227 Figure 8.25: Kinematic joint allowing all necessary
rotations (3 D.O.F.) and the
resulting bar structure Proof-of-concept model to verify the
mobility
............................................................................................................228
Figure 8.26: Integration of the membrane beforehand by attaching
it to the nodes Side elevation and perspective view of the
undeployed and deployed
position..........................................................................................229
Figure 8.27: Right-angled geometry with its own set of joints
..........................230 Figure 8.28: Deployment sequence of a
concept model of a right-angled
structure with aluminium bars and resin connectors [De
Temmerman, 2006a]
....................................................................................230
Figure 8.29: Several regular and right-angled structures
connected together after
deployment...........................................................................................231
Figure 8.30: The two loops and their common fold
line........................................233
-
318
Figure 8.31: A foldable open structure with a compatible
integrated scissor linkage one bar of each scissor unit doubles up
as an edge the foldable bar
structure..................................................................................234
Figure 8.32: Top view and perspective view of the finite element
model of the foldable joint from Figure 8.15 (hinges are
represented by dashed
lines)..................................................................................................................235
Figure 8.33: Model with the middle bars in the rhombus-shaped
modules still
present..............................................................................................................236
Figure 8.34: Same model as in Figure 7.33, but with
cross-bars........................236 Figure 8.35: Bars are grouped
in pairs and joined by a fixed connection in their
apex
angle.......................................................................................................237
Figure 8.36: Adding struts again only increases the weight, while
the section
remains
identical...........................................................................................237
Figure 8.37: Summary of the determining parameters for the
strength, stability
and stiffness for case study 3 OPEN structure
....................................238 Figure 8.38: Resulting
section and weight for the foldable dome .....................239
Figure 8.39: Perspective view of case study 3 CLOSED structure with
sections
after structure design and total
weight.................................................240 Figure
8.40: Summary of the determining parameters for the strength,
stability
and stiffness for case study 3 CLOSED structure
................................241 Figure 8.41: Case 3 OPEN
structure
.............................................................................242
Figure 8.42: Case 3 Foldable DOME structure
..........................................................243
Figure 8.43: Case 3 CLOSED
structure.........................................................................244
Figure 9.1: Design concept for a tensile surface structure with a
deployable
central
tower..................................................................................................245
Figure 9.2: Mobile structure with membrane surfaces arranged around
a
demountable central tower ( The Nomad
Concept).........................248 Figure 9.3: The top of the
tower is accessible to visitors, allowing them to
enjoy the
view................................................................................................249
Figure 9.4: Side elevation of the tower and canopy
...............................................250 Figure 9.5: Top
view of the structure showing the three tensile surfaces
arranged radially around the central tower
.........................................250 Figure 9.6: Dimensions
of the tower and a single angulated bar
.......................251
-
319
Figure 9.7: Comparison between a linkage with angulated SLEs and
its polar
equivalent........................................................................................................252
Figure 9.8: Imposed condition on the length of the semi-bars a
and b (a
-
320
Figure 9.23: Three stages in the deployment of a hexagonal tower
with 5 modules: elevation and top view
.............................................................276
Figure 9.24: Hyperboloid geometry (as proposed in previous
sections) angulated elements do not remain coplanar during
deployment..278
Figure 9.25: Prismoid geometry (simplified alternative to the
previously described geometry) - angulated elements remain coplanar
during deployment
.....................................................................................................279
Figure 9.26: Non-symmetrical identical angulated elements result
in a fully compactable configuration: hyperboloid solution
..............................279
Figure 9.27: Symmetrical identical angulated elements cannot be
fully
compacted.......................................................................................................280
Figure 9.28: Symmetrical and non-identical angulated elements
result in a fully compactable configuration: prismoid solution
..........................281
Figure 9.29: Symmetrical and non-identical angulated elements
result in a fully compactable configuration: prismoid solution
..........................282
Figure 9.30: Three consecutive stages in the deployment of a
prismoid
geometry..........................................................................................................282
Figure 9.31: Three consecutive stages of the corresponding
planar closed-loop structure
..........................................................................................................283
Figure 9.32: Perspective view of the deployment of a triangular
tower ..........283 Figure 9.33: Top view and side elevation of the
prismoid tower ........................284 Figure 9.34: Detailed
view of the simplified hinge connecting four scissor bars
............................................................................................................................284
Figure 9.35: Triangular and quadrangular prismoid solution and
their
respective equivalent hinged-plate structure, providing an
insight in the kinematic behaviour
.............................................................................285
Figure 9.36: Top view and perspective view of the structure with
indication of the global coordinate system and the vector
components of the wind action
.....................................................................................................287
Figure 9.37: Side elevation of the equilibrium form of the
membrane.............288 Figure 9.38: Top view of the equilibrium
form of the membrane.......................288 Figure 9.39:
Horizontal cable ties to improve structural performance
.............289 Figure 9.40: Perspective view, top view and side
elevation of deployable mast
............................................................................................................................290
-
321
Figure 9.41: Summary of the determining parameters for the
strength, stability and stiffness of case study
4.....................................................................291
Figure 9.42: Case 4 A temporary canopy and its deployable tower
with angulated
units..............................................................................................292
Figure 10.1: Case study 1 (Chapter 6) Translational barrel
vault....................296 Figure 10.2: Case study 2 (Chapter 7)
Polar barrel vault...................................297 Figure
10.3: Case study 3 (Chapter 8) Deployable bar structure with
foldable
joints
.................................................................................................................298
Figure 10.4: Case study 4 (Chapter 9) Deployable mast
....................................299
-
322
List of Tables Table 4.1: The first eight values for in terms of
p for compactly foldable
regular
structures............................................................................................
93 Table 4.2: Minimum and maximum possible apex angles for regular
structures
with 5, 7 or 9
plates.....................................................................................100
Table 4.3: The span S and rise R for a given number of plates p of
regular
foldable structures in terms of the plate length
L..............................101 Table 4.4: The area of the
compact configuration for (p=5, =90), (p=7,
=120) and (p=9, =135) in terms of the plate length L
.............102 Table 4. 5: The area of the sectional profile of
the deployed configuration for
(p=5, =90), (p=7, =120) and (p=9, =135) in terms of the plate
length L
............................................................................................................103
Table 4. 6: The expansion ratio for (p=5, =90), (p=7, =120) and
(p=9, =135)
............................................................................................................103
Table 4.7: Minimum and maximum possible apex angles for
right-angled structures with 5, 7 or 9 plates, as can be read from
the graph in Figure 9
............................................................................................................105
Table 4.8: Values for and for a chosen q (circular structure),
combined with a regular structure (p=5)
............................................................................110
Table 5.1: Values for the wind pressure w per
zone................................................126 Table 5.2:
The seven load cases used for calculations in
EASY............................128 Table 9.1: Characteristics of
the hyperboloid geometry
........................................286 Table 9.2:
Characteristics of the prismoid geometry
..............................................286 Table 9.3: Load
combinations for wind and snow
...................................................287
-
List of Symbols Chapter 2 Deployment angle p.10 a, b Semi-bars
p.11 Unit angle p.11 a, b, c, d Semi-lengths p.12 Kink angle p.21
Angle p.21
Chapter 3 Deployment angle p.40 Unit angle p.40 a, b, c, d
Semi-lengths p.40 M Intermediate hinge P.41 Hr Rise p.42 S Span
p.42 t Unit thickness p.43 2 Total unit angle p.43 O Centrepoint
p.43 M Centrepoint p.43 C Intermediate point p.43 M Centrepoint
p.43 t1, t2 Unit thickness p.45 O1, O2,,O3 Centrepoint p.48 P, Q,
S, T End nodes p.51 t Unit thickness p.54 K Intermediate hinge p.54
Quarter of total sector angle p.57
n Angle p.57 nP Base point of arc p.57
0P Apex point of arc p.57 O Centrepoint p.57
inR Internal radius p.57
-
U Number of units p.59 Sector angle p.59
eR External radius p.60
L Bar lenght p.60 E0, E1 Ellipse p.63
Chapter 4 P Basic plate element p.88 M Module p.88 p Number of
plates p.88 Apex angle p.89 L Plate length p.95 W Module width p.95
Hr Rise p.95 S Span p.95 Deployment angle p.95 H Plate height p.96
H1 Horizontal projection of plate height p.96 H2 Vertical
projection of plate height p.96 , 1, 2 Angle p.96 Ledge Edge length
p.101 L Plate length p.102 Expansion ratio p.103 tp Thickness of a
single plate element p.103 Tp Total thickness of the compactly
folded configuration p.104 q Number of sectors p.107
Chapter 5 t Unit thickness p.117 Air density p.123
refv Reference velocity p.123
ALTc Altitude factor p.123
DIRc Direction factor p.123
TEMc Temporary factor p.123
refq Reference wind pressure p.123
w Total wind pressure p.123
-
we Pressure on the external surfaces p.123 wi Pressure on the
internal surfaces p.123 Opening ratio p.123 AL,W Total area of
openings at the leeward and wind
parallel sides p.124
AT Total area of openings at the windward, leeward and wind
parallel sides
p.124
Cpi Internal pressure coefficient p.124 Cpe External
pressurecoefficient p.124 Cpi,a Permeability p.124
ks Characteristic snow load on the ground p.127
tC Temperature coefficient p.127
eC Exposure coefficient p.127
i Form factor for the snow load p.127 G Permanent loads p.128 Q
Mobile loads p.128 Safety factor p.128 D Damage p.130 ni Number of
cycles p.130 Ni Critical amount of load cycles p.130
i Fluctuating stresses p.130 c Resistance against fatique p.130
i Shear stresses p.132
Chapter 6 M1 Plane module p.139 M2 Slightly curved module p.139
M3 Highly curved module p.139 U1, U2, U3 Linkage p.142 t Unit
thickness p.144 a, b Semi axes p.144 U Number of units in the span
p.145 R Radius of the circular arc p.145 2 Angle p.145 A , A
Circular arc p.145
2a Distance between parallel arcs
p.145
-
P2, P0, 1P , P1, P2
Intersection point p.145
E0, E1 Ellipsoid p.146 Angle p.146 n Node p.152 2, 3 Angle p.155
fy Yield stress p.164 Smax Maximum stress p.168
Chapter 7 U Units p.180 O Centrepoint p.180 P, Q, R Intersection
points p.180 h Unit height p.180 U1, U2, U3 Linkage p.181 S Span
p.188 Hr Rise p.188 t Unit thickness p.188 a, b Semi-bar p.188
Deployment angle p.188
maxS Deployment angle for which the maximum span is reached
p.188
design Deployment angle in the fully deployed configuration
p.188
Smax Maximum span p.188 Sdesign Span of the deployed
configuration p.188 Deployment ratio p.189
inR Internal radius p.189
eR External radius p.189 Unit angle p.189 Sector angle p.189 Se
External span p.192
Chapter 8 p Number of plates p.214 Apex angle p.214
-
Deployment angle p.214 L Plate length p.214 S Span p.214 W
Module width p.214 q The amount of sectors arranged radially p.216
m Number of modules p.231 R Degree of statical determinacy p.232 b
Number of bars p.232 j Number of joints p.232 r Number of
restraints p.232 Njoints Number of continuous joints p.233 Nlinks
Total number of links p.233 Nloops Number of loops p.233
Chapter 9 a, b Semi-bar length p.253 Kink angle p.255 U Number
of scissor units p.255 n Number of modules p.255 E Edge length
p.255 Sector angle p.255 Deployment angle p.257 h Height of the
undeployed position p.257 H Total height p.257 Deployment ratio
p.258 R Radius p.259 h Unit height p.261 L Base length p.261
-
Chapter 1 Introduction
1
Chapter 1
Introduction
1.1 Deployable structures A large group of structures have the
ability to transform themselves from a small, closed or stowed
configuration to a much larger, open or deployed con-figuration.
These are generally referred to as deployable structures though
they might also be known as erectable, expandable, extendible,
developable or un-furlable structures [Jensen, 2003]. Although the
research subject of deployable structures is relatively young
be-ing pioneered in the 1960s, the principle of transformable
objects and spaces has been applied throughout history.
Applications range from the Mongolian yurts, to the velum of the
Roman Coliseum, from Da Vincis umbrella to the folding chair. At
present day, the main application areas are the aerospace industry,
requiring highly compactable, lightweight payload and architecture,
requiring either mobile, lightweight temporary shelters or
fixed-location re-tractable roofs for sports arenas. Mobile shelter
systems are a type of building construction for which there is a
vast range and diversity of forms and structural solutions. They
are designed to provide weather protected enclosure for a wide
range of human activities. The main applications are exhibition and
recreational structures, temporary build-ings in remote
construction sites, relocatable hangars and maintenance facili-ties
and emergency shelters after natural disasters. Enclosure
requirements are generally very simple, with the majority needing
only a weather protecting membrane or skin supported by some form
of erectable structure. In all appli-cations, both the envelope and
structure need to be capable of being easily moved in the course of
normal use, which very often requires the building sys-tem to be
assembled at high speed, on unprepared sites [Burford &
Gengnagel, 2004]. An example of an easily erectable temporary
exhibition structure is shown in Figure 1.1.
-
Chapter 1 Introduction
2
Figure 1.1: Mobile deployable bar structure ( Grupo Estran)
Mobile deployable structures have the advantage of ease and
speed of erec-tion compared to traditional building forms. Because
they are reusable and easily transportable, they are of great use
for temporary applications. However, the aspect of deployability is
associated with a higher mechanical complexity and design cost
compared to conventional systems. This increased cost has to be
balanced by the structures potential to be suitable for the
particular appli-cation. Deployable structures can be classified
according to their structural system. In doing so, four main groups
can be distinguished:
Spatial bar structures consisting of hinged bars Foldable plate
structures consisting of hinged plates Tensegrity structures
Membrane structures
It is noted that these deployable structural systems only
constitute a portion of the possible applications in their
respective field. The majority of spatial bar structures, plate
structures, tensegrity structures and membrane structures is
non-deployable and has a permanent location. What is referred to
here are those specific applications which exhibit a certain
ability to transform their shape, therefore adapting to changing
circumstances and requirements. Hanaor [2001] has classified the
aforementioned structural systems used in deployable structures by
their morphological and kinematic characteristics (Figure 1.2).
Because of their wide applicability in the field of mobile
architec-ture, their high degree of deployability and a reliable
deployment, two sub-categories will be studied in greater detail
(marked in red in Figure 1.2):
-
Chapter 1 Introduction
3
Figure 1.2: Classification of structural systems for deployable
structures by their morphological
and kinematic characteristics [Hanaor, 2001]
-
Chapter 1 Introduction
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-
-
Chapter 1 Introduction
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.
-
Chapter 1 Introduction
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
-
Chapter 1 Introduction
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.
-
Chapter 1 Introduction
8
-
Chapter 2 Review of Literature
9
Chapter 2
Review of Literature
2.1 Introduction In this chapter the main contributors to the
field of deployable structures are discussed. A review is given of
existing deployable scissor structures (or panto-graph structures)
and foldable plate structures for architectural applications. The
first part is concerned with an explanation of the characteristics
of trans-lational and polar units, and the deployability condition
they have to comply with when used in a scissor linkage, in order
to guarantee deployability. Fur-ther, angulated elements, which are
used to form closed loop structures, are discussed. These are
characterised by a radial deployment, allowing the struc-ture to
retract towards its perimeter. The second part discusses the
application of foldable plate structures, includ-ing single and
double curvature configurations. An explanation is given of the
possible plate linkages which generate compactly foldable
configurations. Also, the condition which foldable plate
configurations have to satisfy in order to be compactly foldable is
mentioned.
2.2 Deployable structures based on pantographs Scissor units,
otherwise called scissor-like elements (SLEs) or pantographic
elements, consist of two straight bars connected through a revolute
joint, called the intermediate hinge, allowing the bars to pivot
about an axis perpen-dicular to their common plane (Figure 2.1). By
interconnecting such SLEs at their end nodes using revolute joints,
a two-dimensional transformable linkage is formed, as shown in
Figure 2.2. Altering the location of the intermediate hinge or the
shape of the bars gives rise to three distinct basic unit types:
translational, polar and angulated units.
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2.2.1 Translational units
The upper and lower end nodes of a scissor unit are connected by
unit lines. For a translational unit, these unit lines are parallel
and remain so during de-ployment. In Figure 2.1 a plane and a
curved translational unit are shown, the plane unit being the
simplest translational unit having identical bars. When these units
are linked, a well-known transformable single-degree-of-freedom
mechanism is formed, called a lazy-tong, shown in Figure 2.2.
Figure 2.1 : Translational units
The curved unit named such because it is commonly used for
curved linkages has bars of different length. When the latter is
linked by its end nodes, a curved linkage is formed, pictured in
Figure 2.3. By varying the deployment angle a linkage is
transformed from its most compact configuration (a compact bundle)
to its fully deployed position, as shown in Figure 2.2 and Figure
2.3.
Plane unit Curved unit
Unit line
End node
Intermediate hinge
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Figure 2.2: The simplest plane translational scissor linkage,
called a lazy-tong
Figure 2.3: A curved translational linkage in its deployed and
undeployed position
2.2.2 Polar units
When in a plane translational unit the intermediate hinge is
moved away from the centre of the bar, a polar unit is formed with
unequal semi-bars a and b (Figure 2.4). It is this eccentricity of
the intermediate hinge which generates curvature during deployment.
The unit lines intersect at an angle . This angle varies strongly
as the unit deploys and the intersection point moves closer to the
unit as the curvature increases. In Figure 2.5 a polar linkage is
shown in its undeployed and deployed configuration.
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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
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Figure 2.6: The deployability constraint in terms of the
semi-lengths a, b, c and d of two adjoin-
ing scissor units in three consecutive deployment stages
It should be noted that scissor linkages which do not comply
with Equation 2.1 can still be partially foldable: one unit might
be fully compacted, while the adjoining unit might still be
partially deployed. However, since this disserta-tion is concerned
with the design of compactly foldable scissor structures, the
deployability constraint is treated as a minimum requirement.
2.2.4 Structures based on translational and polar units
In the early 1960s, Spanish architect Emilio Perez Piero [1961,
1962] pio-neered the use of scissor mechanism to make deployable
structures. He was among the first in modern times to employ the
principle of the pantograph for use in deployable architectural
structures, such as his moveable theatre (Figure 2.7). This
particular model consisted of rigid bars and wire cables, which
would become tensioned to provide the structure with the necessary
stabilisation. The members remain unstressed in the compact,
bundled configuration and the deployed state, except for their own
dead weight. Furthermore, the struc-ture is stress-free during the
deployment, effectively behaving like a mecha-nism. Piero was very
productive in the field of deployable scissor structures, until all
this was brought to an end by his tragic death in 1972. Another
Spanish architect became one of the most prolific researchers on
the subject. Felix Escrig [1984, 1985] presented the geometric
condition for de-ployability (Section 2.2.3) and demonstrated how
three-dimensional structures
a
b
c
d
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14
could be obtained by placing scissor units in multiple
directions on a grid. Fur-ther, it was shown how curvature could be
introduced in such a grid by vary-ing the location of the
intermediate hinge of the scissor units.
Figure 2.7: Piero demonstrates his prototype of a deployable
shell [Robbin, 1996]
Escrig has also investigated, in collaboration with J. Sanchez
and J.P. Valcarcel, spherical two-way scissor structures based on
the subdivision of the surface of a sphere. These two-way grids
require measures, such as cross-bars or cables, to stabilise the
structure in its deployed configuration, due to in-plane
insta-bility caused by non-triangulation. A myriad of geometric
models has been proposed by Escrig [1985,1987] based on two-way and
three-way grids with no curvature, single curvature or double
curvature. An example of each category is given in Figures 2.8 to
2.12.
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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]
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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)
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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]
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Zeigler [1981, 1984] was the first to exploit this phenomenon as
a self-locking effect, effectively making extra stabilisation after
deployment (which is neces-sary for stress-free deployable
structures) obsolete. He proposed, on these grounds, a partial
triangulated spherical dome as shown in Figure 2.15. Charis Gantes
[1996, 2001] has thoroughly investigated bi-stable deployable
structures and has developed a geometric design approach for flat
grids, curved grids and structures with arbitrary geometry. Also,
he has researched the structural response during deployment, which
is characterized by geomet-ric non-linearities. Simulation of the
deployment process is, therefore, an im-portant part of the
analysis requiring sophisticated finite element modelling. The
material behavior, however, must remain linearly elastic, so that
no resid-ual stresses reduce the load bearing capacity under
service loads. Two of his proposals for bi-stable structures, an
elliptical arch and a geodesic dome, are depicted in Figure
2.16.
Figure 2.16: Bi-stable structures: elliptical arch and geodesic
dome [Gantes, 2004]
A geometric and kinematic analysis of single curvature and
double curvature structures has been performed by Travis Langbecker
[1999, 2001]. He has used translational units to design several
models of positive (Figure 2.17) and nega-tive (Figure 2.18)
curvature structures. By using compatible translational units and
by keeping the structural thickness (unit thickness) constant
throughout the whole structure, these configurations are always
stress-free deployable.
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Figure 2.17: Positive curvature structure with translational
units in two deployment stages
[Langbecker, 2001]
Figure 2.18: Negative curvature structure with translational
units in two deployment stages
[Langbecker, 2001]
Pantographic deployable columns are linear deployable structures
composed of translational or polar units and were researched by
Raskin [1996, 1998]. His work focussed on pantographs behaving as
mechanisms during deployment, which are to be stabilised in the
deployed configuration by additional bound-ary conditions. First,
plane linkages were investigated, which were subse-quently used to
form prismatic columns (Figure 2.19). Expanding his findings,
deployable pantographic slabs that can be packaged in different
arrangements were proposed. Figure 2.20 shows two variations of
such a deployable slab, consisting either of prismatic modules or
an arrangement of prismatic col-umns.
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Figure 2.19: Plane and spatial pantographic columns by Raskin
[1998]
Figure 2.20: Pantographic slabs by Raskin [1998]
Under the guidance of Dr. Sergio Pellegrino, a research group
called the De-ployable Structures Laboratory, emerged at the
Cambridge University in 1990 as a driving force in the field of
deployable structure research. One of their proposals constituted a
deployable pantographic ring structure developed as the edge beam
of a deployable antenna. Together with Zhong You, the condi-tions
for strain-free deployment of such a structure were derived [You
& Pellegrino, 1993]. Structures of this type consist of
translational linkages on the perimeter ring and inner ring,
mutually connected by radially placed polar units. As an example,
Figure 2.21 shows a structure based on a twelve-sided polygon.
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Chapter 2 Review of Literature
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Figure 2.21: Deployable ring structure [You & Pellegrino,
1993]
2.2.5 Angulated units
Unlike common pantograph units with straight bars, angulated
units consist of two rigidly connected semi-bars of length a that
form a central kink of ampli-tude . Because they were invented by
Hoberman [1990] they are commonly denoted as hobermans units. The
major advantage is that, as opposed to polar units, angulated units
subtend a constant angle during deployment (Figure 2.22). For this
to occur, the bar geometry has to be such that = /2. This im-plies
that angulated elements can be used for radially deploying closed
loop structures, capable of retracting to their own perimeter,
which is impossible to accomplish with translational or polar
units, which demonstrate a linear de-ployment. (Figure 2.23) shows
a circular linkage with angulated elements in its undeployed and
deployed configuration.
Figure 2.22: Angulated unit or hobermans unit
a
a
= /2
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Figure 2.23: A radially deployable linkage consisting of
angulated (or hobermans) units in three
stages of the deployment
The structure shown in Figure 2.23 is formed by two layers of
identical angu-lated elements, of which one layer is formed by
elements in clockwise direc-tion (marked in gray), while the other
is arranged in counter-clockwise direc-tion (marked in red). As the
structure deploys, each layer undergoes a rotation, equal in
magnitude but opposite to each other.
2.2.6 Closed loop structures based on angulated elements
You & Pellegrino [1996, 1997] extended the previous concept
to multi-angulated elements, which are elements with more than one
kink angle, as can be seen in Figure 2.24. They found that two or
more such retractable structures can be joined together through the
scissor hinges at the element ends. Two angulated elements from
layers that turn in the same direction of two such interconnected
structures, were found to maintain a constant angle and could
therefore be rigidly connected, thus forming a multi-angulated
ele-ment. The deployment of such a structure, composed of two
layers of twelve identical multi-angulated elements with three
kinks, is depicted in Figure 2.25.
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Figure 2.24: Multi-angulated element
Figure 2.25: A radially deployable linkage consisting of
multi-angulated elements in three stages
of the deployment
This concept was extended by You & Pellegrino [1996, 1997]
to include gener-alised angulated elements (GAE) which allow
non-circular structures to be ge