DEPLOYABLE TENSAIRITY STRUCTURES - Vrije Universiteit Brussel

DEPLOYABLE TENSAIRITY STRUCTURES
development, design and analysis
Lars De Laet

DEPLOYABLE TENSAIRITY STRUCTURES
Development, design and analysis

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© 2011 Lars De Laet
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FACULTY OF ENGINEERING
Department of Architectural Engineering Sciences
DEPLOYABLE TENSAIRITY STRUCTURES
Development, design and analysis
Thesis submitted in fulfilment of the requirements for the award of the degree of
Doctor in Engineering (Doctor in de Ingenieurswetenschappen) by
Lars De Laet
March 2011
Advisors: prof. dr. Marijke Mollaert
dr. Rolf H. Luchsinger

ii
This research is funded by the Research Foundation - Flanders (FWO)

Members of the jury
prof. dr. ir. Marijke Mollaert (advisor)
Vrije Universiteit Brussel, ARCH
dr. Rolf H. Luchsinger (advisor)
EMPA - Center for Synergetic Structures, Zurich, Switzerland
prof. dr. ir. Dirk Lefeber (chairman)
Vrije Universiteit Brussel, MECH
prof. dr. ir. Rik Pintelon (vice-chairman)
Vrije Universiteit Brussel, ELEC
prof. dr. ir. arch. Ine Wouters (secretary)
Vrije Universiteit Brussel, ARCH
prof. dr. ir. René Motro
Laboratoire de Mécanique et Génie Civil - Université de Montpellier II, France
dr. ir. Frank Jensen
Søren Jensen Consulting Engineering & Aarhus School of Architecture, Denmark
iii

Acknowledgements
There are many people I would like to thank for their contribution to this
PhD research. Thanks to their support, encouragement, guidance, feedback,
expertise, time, love, humor ... was this PhD research an interesting, enriching,
unique and unforgettable project of which this dissertation is the final result.
I would like to thank my advisor Prof. Marijke Mollaert for her scientific
guidance, enthusiasm and continuous encouragement throughout the course
of this research. Thank you, Marijke, for believing in me as a researcher and
for all the opportunities you offered me. The freedom you gave me to explore
and develop new ideas - at home and abroad - enriched me tremendously.
Thank you for your kind and warm personality.
I am very grateful to my co-advisor dr. Rolf Luchsinger for his invaluable
scientific guidance throughout my many visits and stays at the EMPA-Center
for Synergetic Structures. Thank you Rolf, for teaching me a great deal about
Tensairity structures and showing me the importance of doing research in the
most thorough and insightful way possible.
I am honored that René Motro, Frank Jensen, Dirk Lefeber, Rik Pintelon and
Ine Wouters accepted to be with Marijke and Rolf member of the jury of this
work. Thank you for your scientific advice and for providing much valued
comments and suggestions.
I would like to thank the Research Foundation Flanders (FWO) for funding
this research, as well my two-month stay at the EMPA-Center for Synergetic
Structures in Zürich. I am also very grateful for the additional support of
the department of Architectural Engineering Sciences of the Vrije Universiteit
Brussel.
I would like to thank Daniel Debondt, Gabriël Van den Nest, Frans Boulpaep,
v

Patrick Vanroose, Tinneke Van Thienen and Jan Roekens for their contribution
to the design, manufacturing and assembling of the final prototype. Thank
you Jan, for assisting me twice in conducting the experiments at EMPA-CSS
and for your endless patience.
During my many visits at the EMPA-CSS, I had the pleasure to work with an
international group of young researchers. Thank you, Rolf, Markus, Cédric,
René, Louis, Mircea, Roberto, Fanis, Joep and Antje for making me feel more
than welcome in Zürich and considering me each time as part of the team. I
enjoyed the many group meetings, interesting discussions and collaborations.
Thank you for supporting me in conducting the experiments and numerical
calculations. I am also grateful to the EMPA-CSS for allowing me to use their
equipment to conduct experiments and materials to fabricate scale models. My
trips to Zürich during the last year would never have been possible without the
unrestrained hospitality of Philippe Block. Thank you Philippe, for allowing
me to stay in style in Switzerland and offering me your amazing apartment.
My most heartfelt sympathy goes out to my colleagues of the department of
Architectural Engineering Sciences, whom I thank for providing a kind and
stimulating environment, and for their friendship and support. Thank you
all, for the relax discussions over lunch, for the (sometimes too) many tasteful
beers after work in spring and summer and for being friends as much as
colleagues.
My friends and family deserve special thanks for always being there and
encouraging me. Thank you Kenny and Lisa, for your support, humor and
friendship. Thank you mama, papa, Jens, Katrien, Brigitte, Jos and Kelly, for
your love and your unconditional support and encouragement.
I wish to express my love and gratitude to Romy, my partner and friend.
Thank you, for everything you did for me, especially those last couple of
months. Thank you for your immense and unconditional support and above
all, thank you for your love. Thank you Yano, for making worries fade away
with your beautiful smile, for your love and showing what life is about.
vi
Lars De Laet
Brussels, Belgium
February 2011

Abstract
A Tensairity structure has most of the properties of a simple air-inflated beam,
but can bear to hundred times more load. This makes Tensairity structures
very suitable for temporary and mobile applications, where lightweight solutions
and small transport volume are a requirement. However, the standard
Tensairity structure, which is comprised of several interacting components
such as an airbeam, cables and struts, can not be compacted to a small
package without being disassembled. By replacing the standard compression
and tension element with a mechanism, a deployable Tensairity structure is
achieved that needs - besides changing the internal pressure of the airbeam -
no additional handlings to compact or erect the structure.
The development of such a deployable Tensairity structure is investigated in
this research. Insight is gained in the structural and kinematic behaviour of
this type of Tensairity structures by means of experimental and numerical
investigations on small and large scale models.
The first part of the dissertation focuses on the development of an appropriate
mechanism for the deployable Tensairity structure. The exploration and
analysis of ideas for deployable systems is presented by means of experiments
on various scale models. By doing this, the boundary conditions and
requirements that have to be taken into account in the design are observed and
presented. These boundary conditions are imposed by the application where
the structure is designed for and by the structural concept Tensairity. With
this knowledge, a solution for a deployable Tensairity structure, the ‘foldable
truss system’, is being improved with regard to its structural and kinematic
behaviour. As a result, an easily foldable proposal for the deployable Tensairity
structure is obtained.
The second part investigates the structural behaviour of a Tensairity beam
vii

y means of experiments on scale models and numerical investigations and
identifies the influence of several design parameters. More precise, the contribution
of hinges and cables to the structural behaviour is investigated, as
well as the influence of the internal pressure, section of the strut and shape
of the airbeam. Among other things is learned from the investigations that
hinges decrease the structure’s stiffness and that a decent structural behaviour
requires that all hinges are connected with a cable.
A prototype of a deployable Tensairity beam is designed, fabricated, experimentally
tested and evaluated in the third part. The detailing and manufacturing
of the prototype is presented, as well how the experiments are conducted.
The general behaviour of the deployable Tensairity beam, such as its stiffness
and maximal load, is first discussed. Then, various configurations are studied
experimentally and numerically to analyse the influence on the structural
behaviour of the cables and hinges. Also here, the necessity of connecting
hinges of the upper and lower strut with each other is shown. During the
study, the experimental results are compared with the outcome of numerical
calculations on the finite element model of the prototype.
Finally, the deployable Tensairity beam is compared with other (non-deployable)
Tensairity prototypes. The deployable structure shows to be a factor two less
stiff than the same prototype with continuous strut. This difference can be
attributed to the presence of hinges in the strut of the Tensairity structure.
The proposal for the deployable Tensairity beam, containing thus the cable
configuration that allows complete folding, is with regard to its structural
behaviour insufficient. The main reason is that not all hinges are connected
with a cable. As a consequence, the proposal for the deployable Tensairity
structure should be adapted taken the formulated improvements into account.
Nevertheless, new insights in the structural behaviour of Tensairity structures
are created with this research. They form a solid base for further research on
deployable Tensairity structures and bring us one step closer to the realisation
of a deployable Tensairity beam.
viii

Samenvatting
Het constructief concept Tensairity bezit de meeste eigenschappen van een
opblaasbare balk, maar kan tot honderd keer meer belasting dragen. Hierdoor
zijn Tensairity constructies zeer geschikt voor tijdelijke en mobiele toepassingen,
waar het lage gewicht en het compact volume bij transport een vereiste
zijn. Echter, de standaard Tensairity constructie, die opgebouwd is uit verschillende
samenwerkende delen zoals een opblaasbare balk, kabels en staven,
kan niet opgeborgen worden tot een klein volume zonder de constructie te
demonteren. Door middel van het vervangen van de standaard druk- en trek
elementen door een mechanisme, wordt een opplooibare Tensairity constructie
gerealiseerd die - behalve het aanpassen van de interne druk - geen bijkomende
handelingen vereist om de constructie op te vouwen of te ontplooien.
De ontwikkeling van een dergelijke opplooibare Tensairity constructie is het
onderwerp van dit onderzoek. Inzichten in het constructief en kinematisch
gedrag van dit type Tensairity constructies worden verworven door middel
van experimenteel en numeriek onderzoek op kleine en grote schaalmodellen.
Het eerste deel van dit onderzoek richt zich op de ontwikkeling van een
geschikt mechanisme voor de opplooibare Tensairity constructie. De studie en
analyse van ideeën voor opplooibare systemen is weergegeven door middel
van experimenten op verscheidene schaalmodellen. De randvoorwaarden en
vereisten die in rekening moeten gebracht worden bij het ontwerpen, worden
op deze manier weergegeven. Deze randvoorwaarden worden opgelegd door
de toepassing waarvoor de constructie ontworpen is en door het constructief
concept Tensairity zelf. Een oplossing voor een opplooibare Tensairity constructie
wordt met deze kennis verbeterd op het vlak van diens constructief
en kinematisch gedrag. Het resultaat is een eenvoudig opplooibaar voorstel
voor de opplooibare Tensairity constructie.
ix

Het tweede deel onderzoekt het constructief gedrag van een opplooibare
Tensairity constructie door middel van experimenten op schaalmodellen en
numerieke simulaties, en identificeert de invloed van verscheidene ontwerpparameters.
Meer precies wordt de bijdrage van scharnieren en kabels op
het constructief gedrag bestudeerd, alsook de invloed van de interne druk,
de sectie van het druk-en trekelement en de vorm van opblaasbare balk. Uit
deze studie wordt onder andere geleerd dat de stijfheid van de constructie
verminderd wordt door de aanwezige scharnieren en dat een degelijk constructief
gedrag vereist dat alle scharnieren met kabels verbonden worden.
Een prototype van een opplooibare Tensairity balk is ontworpen, vervaardigd,
experimenteel onderzocht en geëvalueerd in het derde deel. De details en
vervaardiging van dit prototype worden er weergegeven, alsook hoe de experimenten
zijn uitgevoerd. Het algemeen constructief gedrag van de opplooibare
Tensairity balk, zoals diens stijfheid en maximale belasting, worden
eerst besproken. Daaropvolgend worden verschillende configuraties
bestudeerd om de invloed van de kabels en scharnieren op het gedrag van de
constructie te onderzoeken. Ook hier wordt de noodzaak duidelijk gemaakt
om de scharnieren van de druk-en trekelementen door middel van een kabel
te verbinden. De experimenteel verworven resultaten worden doorheen de
hele studie vergeleken met de uitslag van numerieke berekeningen van een
eindige elementen model van het prototype.
De opplooibare Tensairity balk wordt vergeleken met andere (niet-opplooibare)
Tensairity prototypes. De opplooibare configuratie is een factor twee minder
stijf dan hetzelfde prototype met continue drukstaven. Dit verschil kan
worden toegewezen aan de aanwezigheid van scharnieren. Het voorstel voor
een opplooibare Tensairity constructie, bestaande uit de kabelconfiguratie die
het volledig toevouwen van de structuur toelaat, is wat betreft het draagvermogen
onvoldoende. De voornaamste reden is dat niet alle scharnieren
verbonden zijn met een kabel. Dit heeft als gevolg dat het voorstel voor een
opplooibare Tensairity constructie aangepast moet worden, rekening houdend
met de geformuleerde voorstellen tot verbetering.
Dit onderzoek heeft bijgedragen tot het creëeren van nieuwe inzichten met
betrekking tot het constructief gedrag van opplooibare Tensairity structuren.
Deze inzichten vormen een solide basis voor verder onderzoek naar opplooibare
Tensairity structuren en brengen ons een stap dichter bij het realiseren
van een opplooibare Tensairity structuur.
x

Contents
Introduction 1
1 Introduction 3
1.1 Lightweight structures . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Tensairity structures . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Research goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Literature review & Basic principles 9
2.1 Inflatable structures . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Tensairity structures . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Deployable mechanisms for Tensairity structures . . . . . . . . . 26
I Design of Deployable Tensairity Structures 33
3 Mechanisms for deployable Tensairity structures 35
3.1 Goal and boundary conditions . . . . . . . . . . . . . . . . . . . 35
3.2 Folding segmented compression elements . . . . . . . . . . . . . 37
3.3 Adapting the foldable truss system . . . . . . . . . . . . . . . . . 45
3.4 Redesign of mechanism . . . . . . . . . . . . . . . . . . . . . . . 52
xi

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
II Structural behaviour of the Deployable Tensairity Beam 63
4 Experiments on scale models 65
4.1 Experimental set-up . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Observations and results . . . . . . . . . . . . . . . . . . . . . . . 68
4.3 Physical and numerical model . . . . . . . . . . . . . . . . . . . . 72
4.4 Cable configurations . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.5 Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5 Numerical Investigations 91
5.1 Finite element analysis of Tensairity structures . . . . . . . . . . 91
5.2 Modeling the Tensairity beams . . . . . . . . . . . . . . . . . . . 92
5.3 Solving the models . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6 Analysis of the structural behaviour 103
6.1 Hull section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.2 Dimensioning the struts . . . . . . . . . . . . . . . . . . . . . . . 112
6.3 Introducing hinges . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.4 Cable configuration . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
III Prototype of a Deployable Tensairity Beam 153
7 Prototype and experimental set-up 155
xii

7.1 Prototype of Tensairity beam . . . . . . . . . . . . . . . . . . . . 155
7.2 The experimental set-up . . . . . . . . . . . . . . . . . . . . . . . 169
7.3 Postprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
7.4 Goal of the experimental investigation . . . . . . . . . . . . . . . 174
7.5 Observations and improvements . . . . . . . . . . . . . . . . . . 175
7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
8 Experimental investigation of the prototype 179
8.1 Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
8.2 General structural behaviour . . . . . . . . . . . . . . . . . . . . 179
8.3 Influence of cables . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
8.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
9 Evaluation of the prototype 197
9.1 Comparison with other Tensairity prototypes . . . . . . . . . . . 197
9.2 Deployable versus non-deployable Tensairity beam . . . . . . . 199
9.3 Concluding remarks and suggestions . . . . . . . . . . . . . . . 203
Conclusions 205
10 Conclusions 207
10.1 Design of deployable Tensairity structures . . . . . . . . . . . . . 208
10.2 Structural behaviour of the deployable Tensairity beam . . . . . 208
10.3 Evaluation of the prototype . . . . . . . . . . . . . . . . . . . . . 211
Bibliography 213
List of publications 219
xiii

xiv

Introduction
1

Introduction
1.1 LIGHTWEIGHT STRUCTURES
1
Architectural and structural engineering have always been subject of the
search for lighter, more efficient and more performing structural systems.
Different types of structures have been developed with the goal of maximizing
structural efficiency and thus reducing the weight of the structure and minimizing
the use of resources. This maximization of the structural efficiency is a
major goal of lightweight engineering.
Various types of lightweight structures exist. They often thank their structural
efficiency to the concept whereby the structure’s material is used as optimal
as possible, ie. by means of loading the structural components only in tension
or compression and this preferably until the material’s yield limit. This can
be achieved by physically separating tension and compression within one
structure, as in the case with tensegrity structures. Another option is using
structures that are only loaded in compression, such as thin compression-only
shells, or in tension, such as membrane structures (figure 1.1).
Figure 1.1: Left: tensegrity structure, middle: compression-only shell,
right: membrane structure
Membrane structures are well known lightweight solutions for covering large
3

4
CHAPTER 1 INTRODUCTION
areas or as component (cladding) for temporary constructions. Two main
categories of tensile structures can be distinguished: the mechanically stressed
and the pressurized membranes. In this latter category the membrane stress is
caused by a pressure load (from a medium, like air or water). A well-known
application of pressurized membranes in civil engineering are the inflatable
structures, such as air-inflated tubes, also called airbeams, and air-inflated
cushions.
Inflatable structures have been used by engineers and architects for several
decades. These structures offer lightweight solutions with a relatively high
structural efficiency and provide several unique features, such as collapsibility,
translucency, and a variety of shapes that can be produced. The assembling
and dismantling of inflatable structures is fast by simple inflation and deflation,
and the transport and storage volume are minimal.
In spite of these exceptional properties, one of the major drawbacks of inflatable
components is their limited load bearing capacity. After all, the external
loading is only carried by the pretension in the membrane of the inflated
component (caused by the internal overpressure). When the air-inflated beam,
also called airbeam, is loaded distributed, compression will occur at one side
of the beam and tension at the other, the same way as occurs in a beam
made of rigid material. The compressive forces are then taken by a decrease
in membrane stress, until this stress is zero. From this point on, additional
compression causes wrinkling of the membrane, which has a large decrease of
the structure’s load bearing capacity as result. Thus the initial prestress in the
membrane of the inflated component is related with the maximum amount of
compression the structure can bear.
As a consequence, significant loads can only be carried with high pressures in
the structure. This leads to very high fabric tensions that can only be supported
by expensive fabrics. Moreover, with increasing pressure, the air tightness,
pressure management and safety become issues. High pressure inflatable
structural components are therefore generally not suitable and desirable for
civil applications.
1.2 TENSAIRITY STRUCTURES
The problem of the limited load bearing capacity of pneumatic structures can
be overcome by avoiding the membrane to take up the external loading. This
is achieved by combining the inflatable component, such as an airbeam, with
traditional building elements such as cables and struts. The strut is applied as

1.2 TENSAIRITY STRUCTURES
compression element at one side of the airbeam, the cable is positioned at the
opposite tension side 1 . Figure 1.2 shows this.
The tension and compression element are thus separated by the air-inflated
beam, which - when inflated to a low pressure 2 - pretensions the tension
element and stabilizes the compression element against out-of-plane buckling.
Indeed, because the compression element is connected continuously
to the prestressed membrane, a movement out-of-plane in one direction of
the element is counterbalanced by the tension of the membrane in the other
direction.
Thus, by combining the air-inflated element with cables and struts, a synergetic
structure emerges whereby the components with different properties
complement each other to a new entity by means of constructive interaction
(Luchsinger and Crettol, 2007). This structural concept is called Tensairity, a
combination of tension, air and integrity, that reflects the relationship with
tensegrity (Luchsinger et al., 2004a).
ession element
compression element
cable
cable
airbeam
Figure 1.2: A basic Tensairity structure is constituted of a compres-
sion and tension element, separated by a low pressurized airbeam
(Luchsinger et al., 2004b).
Due to this constructive interaction of the different elements it is constituted
of, the Tensairity beam has a much higher load bearing capacity than an
airbeam: the Tensairity beam is ten to one hundred times stronger than a
simple airbeam with the same dimensions and pressure 3 . When comparing
the Tensairity beam with a steel profile (HEB-profile) and truss, all designed
for the same load, the Tensairity beam is a factor 6 lighter than the steel profile
and a factor 2 lighter than the truss (Luchsinger et al., 2004a).
Thus, a Tensairity structure has more or less the low weight and compact trans-
1 In the case of a cylindrical airbeam is the cable winded around the airbeam in such a way that
it is positioned at mid span at the tension side
2 Order of magnitude between 100 and 500 mbar (10 - 50 kN
m 2 )
3 From equation 2.5 on page 18.
5

6
CHAPTER 1 INTRODUCTION
port volume of an air-inflated beam, but a much higher load bearing capacity.
These are properties interesting for adaptable or temporary applications, such
as mobile shelters or retractable roofs.
However, while the air beam is easily deployed by inflation, the basic Tensairity
beam needs to be disassembled before it can be packed together. After all,
the compression element of the Tensairity structure possesses some bending
stiffness and must be tightly connected with the hull (in order to maintain its
buckling-free behaviour). Thus, the basic Tensairity beam cannot be folded or
rolled together when deflated.
By replacing the standard compression and tension element with a mechanism,
a deployable Tensairity structure is achieved that needs - besides changing the
internal pressure of the airbeam - no additional handlings to compact or erect
the structure. The development of such a deployable Tensairity structure is
investigated in this research 4 .
1.3 RESEARCH GOAL
The main goal of this research is to gain insights in the structural and kinematic
behaviour of deployable Tensairity structures and develop a first prototype of
such a structure.
Given the lack of general knowledge on deployable Tensairity structures, this
investigation is pursued by means of experimental and numerical investigations
on small and large scale models. Many issues regarding the design of a
deployable system and its structural behaviour need to be investigated. These
different issues are discussed in this dissertation in separate parts.
1.4 OUTLINE OF THESIS
The first part of the dissertation focuses on the development of a mechanism
for the deployable Tensairity structure. The second part investigates the
structural behaviour of a Tensairity beam by means of experiments on scale
models and numerical investigations and identifies the influence of several
design parameters. As a result, a prototype of a deployable Tensairity beam is
designed, fabricated, experimentally tested and evaluated in the third part.
4 Remark that in this research no distinction is made between the terms ‘foldable’ and
‘deployable’: unfolding carries here the same meaning as deploying, just like folding and
compacting. This is in contrast with other research whereby folding only occurs according a
folding line (Jensen, 2004; De Temmerman, 2007)

1.4 OUTLINE OF THESIS
In chapter 2, a literature review describes the context of this research. The
basic principles of the inflatable and Tensairity technology are presented for a
good understanding. A state-of-the-art with regard to deployable Tensairity
structures is given as starting point for this research.
part I - Design of deployable Tensairity structures
Chapter 3 focuses on the development of an appropriate mechanism for the
deployable Tensairity structure. The exploration and analysis of ideas for
deployable systems is presented by means of experiments on various scale
models. By improving the ‘foldable truss system’ with regard to its structural
and kinematic behaviour, an easily foldable proposal for the deployable
Tensairity structure is obtained and proposed.
part II - Structural behaviour of the deployable Tensairity beam
Chapter 4 investigates the structural behaviour of deployable Tensairity beams
by means of experiments on scale models. The focus in the experiments
is gaining qualitative insight in the load bearing behaviour of deployable
Tensairity structures.
In chapter 5, the most significant aspects regarding the finite element calculation
of (deployable) Tensairity structures are discussed. The modelling of the
deployable Tensairity beams, as well how they are solved, is presented.
In chapter 6, the influence on the structural behaviour of several design parameters
and configurations are investigated numerically. The main parameters
whose influence is monitored are the struts’ and hull section, the hinges and
the cable configuration.
part III - Prototype of a deployable Tensairity beam
Chapter 7 presents the prototype of the deployable Tensairity beam. The
components it is constituted of are discussed in detail, as well the assembling
and the way this prototype is investigated.
Chapter 8 discusses the results of the experimental investigation of the prototype
and compares it with the outcome of numerical calculations on the
finite element model of the prototype. As well the general behaviour of the
deployable Tensairity beam as the influence of several parameters, such as
cables and hinges, is discussed.
Chapter 9 evaluates the prototype by comparing experimental and numerical
results with those of other (non-deployable) Tensairity structures. The
7

8
CHAPTER 1 INTRODUCTION
discussion of the results leads to the proposal of improvements.
Finally, chapter 10 compiles all those results and presents the conclusions. The
main findings concerning the design and structural behaviour of deployable
Tensairity structures and the general contributions to the field are discussed.

Literature review & Basic principles
2
First, the context of this research is described by means of a brief literature
review. Then, the basic principles of the inflatable and Tensairity technology
are presented for a good understanding. A state-of-the-art with regard to
deployable Tensairity structures is finally given as starting point for this
research.
2.1 INFLATABLE STRUCTURES
2.1.1 HISTORICAL CONTEXT
Inflatable structures are not new. The first technologically relevant realization
of an inflatable device dates back to 1783, when the Montgolfier brothers were
the first to venture the sky with their hot air balloon. The fascination of the sky
in those days lead to further technological improvements and inventions such
as hydrogen balloons and zeppelins (Topham, 2002). The idea of transposing
the zeppelin technology to architecture tracks back to the English engineer F.
W. Lanchester. His patent of a pneumatic system for campaign hospitals was
approved in England in 1918, but was never constructed due to the lack of
adequate membrane materials or appeal to possible clients (Herzog, 1976).
On solid ground, pneumatic structures had a first breakthrough during the
World War II. This mobile war demanded the invention of easily transportable
and easy-to-store devices. The properties of lightness, durability and mobility
which resulted in air balloons proved to be useful on the ground. As a result,
inflatable structures were applied as decoys to distract the opponent and as
emergency and radar shelters. In order to protect the radars from extreme
weather conditions, thin non-metallic coverings were developed by Walter
9

10
CHAPTER 2 LITERATURE REVIEW & BASIC PRINCIPLES
Bird. These spherical pneumatic domes, the so called radomes, were applied
on many sites all over the world (figure 2.1).
Figure 2.1: On solid ground, pneumatic structures had a first
breakthrough during the World War II. Left: Walter Bird on top of a
radome (1948), middle and right: inflatable decoys. (Topham, 2002)
In the 1960’s, civil and commercial applications were established and empirical
knowledge was gained with the development of pressurized airhouses as
covering for warehouses, swimming pools and sport facilities. At the same
time, the first academic investigations were undertaken by Frei Otto about the
process of form finding. Through the IASS Pneumatic Colloquium (University
of Stuttgart, 1967) and several publications and designs, Otto broadened
the landscape, not only of pneumatics, but of tension structures in general.
Pneumatics was also part of the repertoire of Richard Buckminster Fuller. His
proposal of a pneumatic dome to cover New York (1962) is a famous example
of Utopian pneumatic architecture (Herzog, 1976).
Also during the 1960’s, a new generation of architects debuted which disagreed
with the principles of Le Corbusier’s modernistic architecture. Radical
architecture groups embraced inflatable forms as reaction to the traditional
architecture (Topham, 2002). The Paris group Utopie formulated critics to the
inertia of the post-war European society and its architecture, urbanism and
daily life. As reaction, they reinterpreted the aesthetic of pneumatic structures
to express ephemerality and mobility (Dessauce, 1999) (figure 2.2).
Figure 2.2: Group Utopie: radical architecture groups embraced
inflatable forms as reaction to the traditional architecture (Topham, 2002)

2.1 INFLATABLE STRUCTURES
The inherent portability of pneumatic structures soon inspired their use in
temporary and itinerant exhibitions. Several pioneering pneumatic buildings
using air-supported roofs and air-inflated elements were shown at the Expo ‘70
in Osaka, which was the heyday of pneumatic architecture (figure 2.3). Inspired
by the pavilions, the structural concept of the airhouse was applied in
large sport arenas and coverings for tennis courts and swimming pools in the
next decade.
Figure 2.3: Expo ’70 in Osaka was the heyday of pneumatic architecture.
Fuji Pavilion is shown. (Topham, 2002)
However, since those days, no substantial progress in pneumatic architecture
has been seen (Forster, 1994; Hamilton et al., 1994; Luchsinger et al., 2004c).
A widespread exploitation of inflatable structures was setback due to (1) poor
structural performance of the structure and the membrane material, (2) the
complex and expensive pressure management, and (3) the absence of guide
lines and design tools. The use of air-supported roofs decreased because of
their high maintenance cost, vulnerability to strong wind- and snow loads
and the inefficient use of volume by their spherical or cylindrical shape. For
air-inflated elements, such as airbeams 1 , a major drawback was their limited
load bearing capacity, especially for large structures in architecture and civil
engineering. Substantial loads could only be carried with very high pressures
in the airbeam. This leads to very high fabric tensions demanding for high
strength and expensive fabrics.
However today, an increased interest towards ‘inflatables’ can be identified.
After all, in general they offer lightweight solutions with an optimal use of
material. Because of their limited load bearing capacity with low pressure,
pneumatics are nowadays applied as complementary elements to other stiff
structural systems. Two categories can be distinguished here.
One category deals with inflatables that are applied as secondary structural
1 Airbeam is the short name for an air-inflated beam, like a cylindrical inflated tube.
11

12
CHAPTER 2 LITERATURE REVIEW & BASIC PRINCIPLES
elements. An example of such pneumatic components are the air-inflated
cushions. They are used as cladding and bracing of the primary structure and
transfer wind and snow loads to this structure. In some recent large buildings,
air-inflated cushions have shown a good performance as complementary
elements to other stiff structural elements. The enormous domes built to house
the Eden Project in Cornwall are an outstanding combination of rigid and
pneumatic construction methods. Panels of inflated ETFE-foil - a lightweight,
highly transparent foil - are held in place by a self-supporting network of
tubular steel frames. Glass proved far too heavy to be supported by the
lightweight skeleton, whereas an air-inflated cushion weighs less than one
percent of a pane of glass of equivalent size (Liu et al., 2006). Other contemporary
projects promoting the use of cushions as a cladding and covering system
are the Allianz Arena in Munich (June 2005) and the National Aquatics Center
in Beijing (January 2008) (figure 2.4).
Figure 2.4: Application of air-inflated cushions as cladding. Left: Eden
project, middle: Allianz Arena, right: National Aquatics Center.
In the second category, the pneumatics are applied as a part of a structural
element whereby different parts interact with each other. The structural
concept Tensairity is such a combination of parts: cables, struts and an airbeam
interact with each other. The result is a structure with more or less the low
weight and compact transport volume of an air-inflated beam, but a much
higher load bearing capacity. This structural system Tensairity is discussed in
detail in section 2.2.
2.1.2 INTRODUCTORY CONCEPTS
A membrane that is tensioned by means of a pressure is called a pneumatic
structure. The word ‘pneumatic’ comes from the Greek ‘pneuma’, meaning
‘breath of air’ 2 . The pressure is a distributed load perpendicular to the surface
of the membrane and positions the membrane to a form of equilibrium where
2 However, the medium that tensions the membrane can be a liquid or gas.

2.1 INFLATABLE STRUCTURES
it is stable in both position and form. Due to the tension in the membrane, it
is capable of resisting a certain amount of external loading.
If a pneumatically stressed membrane does not form a closed cavity, it is called
an open pneumatic structure (Herzog, 1976). Examples of such structures
are sails, parachutes, kites. In the case of closed pneumatic structures, the
membrane encloses a pressurized, airtight volume. In this research, only this
latter group is taken into account when mentioning pneumatic structures.
As building application, two different types of closed pneumatic structures can
be distinguished, as illustrated in figure 2.5. In the first category is the space to
be utilized and enclosed by the membrane in overpressure. This category
is called ‘air-supported’ structures, because the single layer of membrane
dividing the interior from the exterior is supported by the air. Often, these
structures are called airhouses. Of course, the supporting medium must be air
and the internal overpressure must be in a range acceptable and comfortable
for the users. Typical overpressures here are in the range of 1 to 10 mbar 3 . In the
second category, the area in overpressure is fully enclosed by a membrane and
is not aimed to be utilized. These structures are called ‘air-inflated’. Another
pressurizing medium than air can be applied here. Depending whether the
horizontal components from the membrane stress are taken by a supporting
structure, the distinction here can be made between ‘airbeams’ and ‘cushions’.
After all, the ‘lens’-shape of a cushion can only be achieved when the horizontal
reaction forces are supported by a primary structure, illustrated on the right
of figure 2.5. The range of internal pressure in these structures depends on
the geometry and the external loads that have be supported, but are typical at
least a factor 10 larger than in the case of air-supported structures. The reason
for this is explained next.
a b
Figure 2.5: Two different types of closed pneumatic structures can be
distinguished. Left: air-supported; middle and right: air-inflated ((a)
airbeam and (b) cushion).
3 1 mbar = 100 N
m 2 = ±1 × 10 −3 atm
13

14
CHAPTER 2 LITERATURE REVIEW & BASIC PRINCIPLES
2.1.3 BASIC PRINCIPLES
For a membrane in equilibrium, the following relation can be derived between
the external loading and the membrane tensions:
nxx
Rx
+ nyy
Ry
= p (2.1)
Rx and Ry represent the radius of curvature [m] in two perpendicular directions
in the plane of the membrane. Usually, the largest and smallest radii of
curvature of the surface are chosen. nxx and nyy are the membrane tension
in the corresponding directions [ N
m ]. p is the load per area and represents in
the case of pneumatic constructions (without additional external loading) the
pressure, expressed in N
m2 (or (m)bar).
If above mentioned equation is applied for a sphere, whereby nxx = nyy and
Rx = Ry, one obtains
n = p.R
2
(2.2)
The membrane tension is thus proportionally to the radius and internal pressure.
As a result, to obtain the same membrane tension, a lower pressure can
be applied in a pneumatic structure with a larger radius. This is the reason
why the internal pressure of air-supported structures (with a larger radius) is
relative small in comparison with airbeams.
For cylindrical structures, the radii in longitudinal and radial direction differ.
After all, the surface of the cylinder is singly curved and the radius in longitudinal
direction can be regarded as infinite. With Rxx = ∞, the membrane stress
in radial direction becomes nyy = p.Ry. When the cylinder is terminated with
two half spheres as end caps, the membrane stresses can be calculated as:
nradial = p.R (2.3)
nlongitudinal = p.R
2
(2.4)
The load-carrying mechanism of an air-inflated beam, also called airbeam
in this research, is quite similar to that of a pretensioned beam (Schodek,
2000). After all, the internal pressure introduces uniformly longitudinal tensile
stresses, as illustrated on the left of figure 2.6. When loading, these stresses
interact with the external loading. This loading causes compression stresses
along the upper surface and tension stresses along the lower surface, the same

2.1 INFLATABLE STRUCTURES
way as occurs in a beam made of rigid material. As a consequence, tensile
stresses originally present along the upper surface are reduced and those along
the lower surface increased (figure 2.6). Clearly, the internal pressure must be
such that no compressive stresses, which would manifest as wrinkles, are
developed along the top surface. After all, once wrinkles are developed, the
airbeam deflects considerably. The moment necessary to initiate wrinkling
equals M = 1
2 .π.p.R3 , the maximum bending moment the inflated beam can
withstand is π.p.R 3 , with p the internal pressure and R the radius (Comer and
Levy, 1963; Main et al., 1994).
Figure 2.6: The load-carrying mechanism of an airbeam is quite similar
to that of a prestressed beam (Schodek, 2000). Left: the inflated airbeam
causes pretension in the membrane, right: under loading - the prestress
decreases on the compression side and increases on the tension side.
2.1.4 BUILT EXAMPLES OF AIR-INFLATED STRUCTURES
Most of the current research and development conducted on air-inflated structures
can be traced to space, military, commercial, and recreational applications.
Examples include air ships, weather balloons, inflatable antennas,
temporary shelters, pneumatic muscles and actuators, inflatable boats, temporary
bridging, and energy absorbers such as automotive air bags (Cavallaro
and Sadegh, 2006). This brief overview focuses on the current application of
airbeams in construction.
A distinction can be made between the application of inflatable airbeams in
temporary and permanent construction. As mentioned before, air-inflated
components are in permanent large scale constructions mostly used as complementary
elements to other stiff structural systems. Often is this in the form
of cushions as cladding of the primary structure (Robinson-Gayle et al., 2001;
Oñate and Kröplin, 2007). After all, they offer a lightweight alternative to glass
with another aesthetical and prestigious appearance. Examples are illustrated
in figure 2.4.
With regard to airbeams for temporary constructions, the distinction can be
made between employment on ground or in space. The inflatable technology
15

16
CHAPTER 2 LITERATURE REVIEW & BASIC PRINCIPLES
proves to be useful for ephemeral and mobile constructions. After all, they
provide a combination of features no other type of structure has, such as rapid
and self-erecting deployment, enhanced mobility, large deployed-to-packaged
volume ratios, fail-safe collapse, and possible rigidification.
In the domain of aerospace, investigations are focusing on the development of
large space-based inflatable structures for a variety of applications. Such applications
include lunar habits, radar antennas, solar arrays, sunshields, telescope
reflectors, etc. Reinforced air-supported structures serve as lunar habits, while
airbeams are applied for the other applications. The inflated airbeams can
remain pressurized to achieve stiffness for short missions; however, for longer
missions they can be permanently rigidized to keep their configuration when
space debris penetrates the structure and to avoid the influence of pressure
decrease on structural accuracy (Natori et al., 1995; Cadogan et al., 1999; Salama
et al., 2000; Darooka and Jensen, 2001; Benaroya et al., 2002; Hublitz et al., 2004;
L’Garde, 2010).
With regard to the application of airbeams on the ground, no substantial
progress in the airbeam technology can be seen in scientific literature. Besides
the new structural concept Tensairity, which will be discussed more in detail
in section 2.2, a project worth mentioning is the ‘Airtecture Exhibition Hall’,
developed by Festo (figure 2.7). The supporting structure of this hall is mainly
constituted of inflated elements. The structure contains inflated columns,
inflated wall components and airbeams as roof structure. The airbeams have a
slenderness of approx. 8 and are pressurized till 500 mbar. The hall measures
40 m x 20 m x 7,5 m in inflated state and can fit in a standardized container
to be transported (Wagner, 1997; Schock, 1997). This design for temporary
structures has not been developed further.
Figure 2.7: ‘Airtecture Exhibition Hall’, developed by Festo (Tensinet
Database, 2010).
Most inflatable structures applied nowadays rely on the standard airbeam

2.1 INFLATABLE STRUCTURES
concept. They are developed for commercial purposes, such as coverings for
events, or for military and emergency applications, such as rapidly deployable
shelters (Inflate, 2010; Buildair, 2010) (figure 2.8). This latter one is usually
composed of a frame of airbeams, covered with a membrane, as illustrated in
figure 2.9. These tents typically have a span of 4-6 m, a height of ± 3 m and
varying length. Their weight per area measures 3-5 kg
m 2 . The internal pressure
of the grid of airbeams varies from producer and ranges from 250 mbar -3 bar
(Losberger, 2010; Eurovinil, 2010; HDT, 2010) 4 . For structures with a larger
span and height, the necessary internal pressure quickly increases, leading to
very high membrane forces and the need for expensive high tech fibres.
Figure 2.8: Application of inflatables for events (Inflate, 2010).
Figure 2.9: ‘Application of inflatables as structure for rapid deployable
emergency and/or military shelters (Losberger, 2010).
4 The Belgian Army applies the Losberger-rds inflatable tents from the typeTDR.
17

18
CHAPTER 2 LITERATURE REVIEW & BASIC PRINCIPLES
2.2 TENSAIRITY STRUCTURES
2.2.1 PRINCIPLE
As mentioned in section 2.1.3, a standard airbeam reaches its load limit for
practical applications when wrinkling occurs at the compression side. After all,
when the prestress becomes zero, the membrane cannot take any compressive
forces. By connecting a stiff element on this compressed side, the load bearing
behaviour is increased because the compressive forces are now partly taken
by this stiff element. When this element is additionally connected with cables
that are spiralled around the airbeam, the structural behaviour is even more
optimised. The result is thus a structure composed of several parts interacting
with each other. This combination of an airbeam and traditional building
elements such as cables and struts is called a Tensairity structure, illustrated
in figure 2.10.
The tension and compression elements are thus separated by the air inflated
beam, which - when inflated - pretensions the tension element and stabilizes
the compression element against buckling. After all, the compression element
is connected continuously to the prestressed membrane which has as result that
a movement out-of-plane in one direction of the element is counterbalanced by
the tension of the membrane in the other direction. The compression element
is thus stabilized by the airbeam and not prone to buckle. As a result, as
well the material of the cable and the compression element can be used to its
yield limit and thus in its most efficient way. Because the airbeam only has
a stabilizing function, the Tensairity beam can operate with low air pressure.
Typical pressures are in order of 100 to 500 mbar.
Because of the interaction of the different elements it is constituted of, the Tensairity
beam has a higher load bearing capacity than an airbeam. Luchsinger
et al. (2004a) calculated the ratio of the load bearing capacity of a Tensairity
beam at a given pressure to a standard airbeam at the same pressure as being
qa,Tensairity
qa,airtube
= 4
.γ2
π3 (2.5)
with qa the load per area and γ the slenderness. Thus, depending on the
slenderness (from 10 to 30 taken into account), the Tensairity beam is ten to
one hundred times stronger than a simple airbeam with the same dimensions
and pressure. When comparing the Tensairity beam with a steel profile (HEBprofile)
and truss, all designed for the same load, the Tensairity beam is a factor

2.2 TENSAIRITY STRUCTURES
6 lighter than the steel profile and a factor 2 lighter than the truss (Luchsinger
et al., 2004a). Thus, a Tensairity structure has more or less the low weight and
compact transport volume of an air-inflated beam, but a much higher load
bearing capacity.
compression element
cable
ession element
cable
airbeam
Figure 2.10: The basic cylindrical Tensairity beam (Luchsinger et al.,
2004a).
2.2.2 BASIC MODEL
Luchsinger et al. (2004a) describes the interaction of the cables, struts and
membrane by means of a simple analytical model for cylindrical Tensairity
beams (figure 2.11). He derives in this model the maximal force T in the
tension and compression element for a homogeneous distributed load q as
T = q.L.γ
8
(2.6)
with γ being the slenderness and L the length of the beam. The slenderness is
determined as the ratio length over diameter (thus thickness): γ = L
2.R .
2 . R
0
p
Figure 2.11: Clarification for the analytical model of the basic cylindrical
Tensairity beam (Luchsinger et al., 2004a).
The compression element is supported and stabilized by the air-inflated beam.
As a result, the interaction between the compression element and the airbeam
can be modeled as a beam on elastic foundation. Luchsinger et al. (2004a)
calculated the spring constant k of this elastic foundation as being k = π.p,
whereby p represents the internal pressure. The critical buckling load of this
q
T
L
19

20
CHAPTER 2 LITERATURE REVIEW & BASIC PRINCIPLES
compression element becomes then
P = 2. π.p.E.I (2.7)
with E the modulus of elasticity of the compression element and I its (area)
moment of inertia.
The buckling load of the compression element is a function of the square root
of the internal pressure, the struts material and its geometry. Note that it is
independent of the length of the beam. This implies that a proper choice of
the bending stiffness of the compression element in relation with the applied
internal pressure can lead to the compression element being loaded to its yield
limit. This is called buckling free compression and an import feature of the
Tensairity technology.
By comparing the load-bearing behaviour of standard airbeams with Tensairity
beams, the influence of the struts and cables becomes visible. The Tensairity
beam is - depending on the slenderness - ten to hundred times stronger than
a standard airbeam with the same dimensions and geometry Luchsinger et al.
(2004a,b). Thus to withstand the same load, the pressure in an airbeam needs
to be ten to hundred times larger than in a Tensairity beam. It is obvious that
in relation to the Tensairity beam, the load bearing capacity of simple airbeams
is very limited.
2.2.3 SHAPE OF THE AIRBEAM
The standard Tensairity beam has a cylindrical shape. Pedretti et al. (2004)
investigated by means of finite element calculations various beam shapes,
illustrated in figure 2.12. The cigar-shaped geometry (b)proves to be more
efficient than the cylindrical form. The spindle shape (c-d), where the tube
end converges to a point, is even stiffer. After all, the shape of the strut is an
arch and transfers thus more optimal forces to the supports. As a result, it has
lower deflections and contributes thus to the stiffness of the Tensairity beam.
Another advantage of the spindle shape is that, compared to a cylindrical
airbeam, the amount of expensive fabric (and thus weight) can be reduced
by up to 33 percent. The volume of the compressed air can be reduced by
up to 47 percent (Luchsinger and Crettol, 2006b). A third asset of the spindle
configuration is that the cable becomes a straight line. As a consequence, a
tension rod or compression element can be applied here. This is an interesting
property for roofs whereby the load case can be negative and compression has
to be taken at the lower side of the beam.

2.2 TENSAIRITY STRUCTURES
Given these advantages, many Tensairity applications are based on spindleshaped
airbeams (Luchsinger et al., 2004b; Pedretti, 2005; Luchsinger and
Crettol, 2006b).
a
b
Figure 2.12: Various forms of Tensairity beams: (a) cylinder, (b)
cigar-shaped, (c) symmetric spindleshaped and (d) asymmetric spindle-
shaped (Luchsinger et al., 2004b).
2.2.4 SPINDLE SHAPED TENSAIRITY STRUCTURES
The behaviour of a spindle shaped Tensairity beam under central point load is
investigated by Luchsinger and Crettol (2006b). As well the load-displacement
behaviour, the general deformation as the forces in the strut are noted and
discussed in the paper.
In these experiments can be seen that the behaviour of the membrane of
the airbeam is reflected in the deformation behaviour of the spindle. The
fabric adapts in a first load-cycle to the new loading, resulting in a residual
displacement. The load-displacement response to subsequent loadings is from
that point on identical. Thus, when investigating these fabric structures, the
displacements should be noted in the second or third load cycle. Figure 2.13
illustrates this.
The results show that the displacement is a non-linear function of the load because
the structure slightly softens with increasing load. The largest deflections
are detected at the upper strut, where the load is applied. The displacement of
the lower strut is smaller, approximately half of the order of the displacement
of the upper strut. This reduction in thickness of the airbeam indicates the
load transfer between upper and lower strut by means of the tensioned fabric.
In addition, the upper strut deforms in a similar way to arches: when the load
acts in the middle of the upper strut, the beam moves slightly upwards at the
end regions. This is illustrated in figure 2.13.
Luchsinger also determines the compression and tension forces in the struts
as function of the applied load. He compares the analytical assumed values
with numerically and experimentally derived values and good agreement is
c
d
21

22
CHAPTER 2 LITERATURE REVIEW & BASIC PRINCIPLES
Load [kN]
1.2
1
0.8
0.6
0.4
0.2
0
−0.2
0
10
p = 150 mbar
20 30 40 50
Displacement [mm]
x [m]
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
p = 150 mbar
F = 0 kN
F = 1 kN
−2.5 −2 −1.5 −1 −0.5 0
z [m]
0.5 1 1.5 2 2.5
Figure 2.13: Left: load-displacement response of the spindle, Right:
experimental deformation of the spindle under central load of 1 kN.
(Luchsinger and Crettol, 2006b).
observed. If high loads at low pressures are not taken into account 5 , there
can be stated that these forces in the struts are independent of the pressure.
The equation whereby the struts’ forces are determined is an adaption of
equation 2.6, taking the distributed load applied over a short distance l 6 into
account: the compression force in the strut and tension force in cable equals
T = Q.γ l
.(2 − ) (2.8)
8 L
where Q is the total applied central load, γ the slenderness and L the length of
the spindle.
The role of the internal pressure on the structure’s behaviour is also determined
in this research. From the experiments can be concluded that the displacements
decrease with increasing pressure, as was also found in the numerical results.
Finally, Luchsinger also presents the influence of the connection of the strut
with the hull. By means of finite element calculations is shown that a connection
whereby the strut can move relative to the membrane (in longitudinal
and vertical direction) causes displacements almost twice as large as in the case
of a fixed connection. An example of such a gliding connection is when the
compression element is connected to the airbeam by means of a pocket (similar
as it is done eg. with the poles in a camping tent). Thus, a tight connection of
the compression element with the fabric can significantly stiffen the Tensairity
structure.
5 The analytical model does not take second order effects in account that occur for high loads
at low pressures (large displacements).
6 l=L for a distributed load and 0 for a point load.

2.2 TENSAIRITY STRUCTURES
Where Luchsinger and Crettol (2006b) investigated and described the spindle
shaped Tensairity beam under point load, is this done for homogeneous load
in Luchsinger and Teutsch (2009). By means of two coupled equations, the
deflections of the upper and lower strut are derived analytically. The inflated
body of the Tensairity structure is modeled as an elastic foundation with shear
stiffness, while the bending stiffness of the struts is neglected. The deflection at
mid span is found to be the sum of two terms. One term is due to the elasticity
of the struts, i.e. the extensioning or shortening of the strut caused by the axial
force. The other term is due to the deformation of the inflated body which
depends on the air pressure. For smaller pressure values, the deformation
of the upper strut is dominated by the deformation of the elastic foundation,
while for higher pressure values, the elasticity of the strut is dominant.
Another development in the domain of Tensairity beams is the application
of a web inside the beam. A web is a connecting element between upper
and lower strut that becomes prestressed due to the outwards movement of
the struts, caused by inflation. This web is typically constituted of cables
or a membrane. This web gives an additional support for the compression
element increasing the value of the spring constant of the elastic foundation.
This improves the stabilization of the compression element and can further
reduce the necessary internal pressure. Breuer et al. (2007) investigated the
application of a membrane web in an inflatable wing for kites. From the
theoretical weight study in this research can be concluded that applying a web
in the Tensairity structure increases the structure’s stiffness and thus efficiency.
Wever et al. (2010) investigated the influence of internal fabric webs on the
structural behaviour of a spindle shaped Tensairity column and showed an
increase of both the axial stiffness and the buckling load.
An airbeam, a Tensairity structure with and without web and a truss are
compared in the theoretical weight study conducted by Breuer et al. (2007).
This is done by means of calculating their weight for a given slenderness and
load. All structures measure three meter, are optimised and are all constituted
of the same materials. The slenderness is varied by varying the parameters
specific for each structure. From this study can be concluded that the weight
of a Tensairity structure is comparable to the weight of an optimized truss.
The web Tensairity structure is lighter than the normal Tensairity structure
due to a reduction of the air pressure, because less pressure is needed for the
same stabilization of the struts. The airbeam is the heaviest configuration for
slenderness values above 10. Figure 2.14 shows the four investigated cases
and the results. More on this can be found in Breuer et al. (2007).
23

24
CHAPTER 2 LITERATURE REVIEW & BASIC PRINCIPLES
weigth (kg)
a b
c
0.09
0.08
0.07
0.06
0.05
0.04
Theoretical weight comparison ( L = 3 m q = 250 N/m)
0.03
Tensairity web spindle
0.02
Tensairity spindle
0.01
truss spindle
air beam cylinder
0
0 10 20 30 40 50
slenderness (-)
60 70
Figure 2.14: Upper: the four investigated cases (with an illustration of
their section), lower: their weight for a given slenderness and under a
given load (Breuer et al., 2007).
As it is clear from this literature review, the structural concept Tensairity is very
new (Luchsinger et al., 2004a; Pedretti et al., 2004). As a result, the research for
this hybrid structure is still in its infancy (Luchsinger, 2009). All aspects of its
structural behaviour are not yet fully understood and many issues are waiting
to be tested, verified and modified.
Among other things, the influence of the connection between upper and
lower strut on the structural behaviour of a five meter Tensairity beam is
currently being investigated. Various prototypes with different connection
between upper and lower strut are tested experimentally. The research in this
dissertation is part of these ongoing investigations.
2.2.5 TENSAIRITY APPLICATIONS AND REALIZED PROJECTS
Tensairity structures are not only applied as beams for roofs or bridges.
Research is also being conducted on the application of this technology in
columns, arches, kites and actuators (Plagianakos et al., 2009; Wever et al.,
2010; Gauthier et al., 2009; Luchsinger, 2009; Breuer and Luchsinger, 2009;
Luchsinger and Crettol, 2006a). It is beyond the scope of this research to
present al these applications in detail.
d

2.2 TENSAIRITY STRUCTURES
Figure 2.15: Patents for several Tensairity applications. Left: arch,
middle: actuator, right: advertising column (Pedretti and Prospec-
tiveConcepts, 2004a; Luchsinger and ProspectiveConcepts, 2005; Fuchs
et al., 2005)
When applying Tensairity structures in civil constructions like roofs or bridges,
the safety is an important issue. Air leakages as a result of a damaged hull lead
to a loss of air and a decrease in internal pressure. This can cause a collapse
of the structure. However, because of the presence of the cable-strut structure
in a Tensairity beam, such an unstable situation can be avoided. After all,
the struts in a Tensairity structure have bending stiffness and can be designed
such that the dead load of the structure is carried by the struts without the
need of stabilization of the airbeam. The air pressure is needed for sustaining
additional live loads (Luchsinger and Crettol, 2006a).
In a lightweight Tensairity structure as the roof over the parking garage
in Montreux, the live load is roughly ten times higher than the dead load
(Luchsinger and Crettol, 2007). As a consequence, the structure is designed to
take its dead load; the live load are taken by the interaction of the structure’s
components. In structures like the roof in Montreux, high live load events are
rare, as well the occurrence at the same time of a complete air loss and high
live loads.
Furthermore, all larger Tensairity structures are connected to an external fan,
which is activated when the pressure needs to be increased. This fan can easily
compensate undesired air loss caused eg. gunshot of vandals (Luchsinger and
Crettol, 2007). In addition, the condition of Tensairity structures can be easily
checked by a simple pressure sensor or even by eye.
For reasons of illustration, two realized Tensairity projects are presented briefly.
The first application, shown on the left of figure 2.16 is a roof over a parking
garage in Montreux, Switzerland. Twelve spindle shaped inflated beams with
span up to 28 m build up the structure. As a special feature, the Tensairity
structures are illuminated during night. The roof was completed by the end
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CHAPTER 2 LITERATURE REVIEW & BASIC PRINCIPLES
of 2004 (Detail, 2005). The second construction, illustrated on the right of
figure 2.16 is a skier bridge in the French Alps with 52 m span, supported by
two asymmetric spindle shaped Tensairity girders. During the winter season,
a ski slope runs over the bridge. The thick layer of snow covering the desk
of the bridge during winter leads to high loads. The bridge was completed in
2005 (Pedretti and Luscher, 2007).
Figure 2.16: Left: roof over parking in Montreux, Switzerland - Luscher
Architectes SA & Airlight Ltd, right: skier bridge in the French Alps
- Charpente Concept SA, Barbayer Architect & Airlight Ltd (Airlight,
2010).
2.3 DEPLOYABLE MECHANISMS FOR TENSAIRITY STRUCTURES
To comply with the structural principles of Tensairity, the compression element
should be attached for stabilizing reasons continuously to the airbeam.
Otherwise, local buckling can occur. As a consequence, no relative movement
between the struts and the membrane is allowed, which implies that the folding
of the mechanism should be compatible with the inextensible membrane. This
is discussed more in detail in section 3.2 on page 37.
Because of this constraint, many investigated kinematic systems cannot be
used as deployable alternative for the linear compression element. To give an
example, solutions that erect in one dimension like e.g. telescopic compression
elements require relative movement between strut and membrane and can thus
not be applied. Also mechanisms whereby the membrane has to (over)stretch
are not a solution, such as is the case with mechanisms comprised of scissor like
elements. It is not in the scope of this literature review to name all investigated
mechanisms that were rejected because of incompatibility with the folding of
the membrane. Instead, a review is given from the mechanisms that finally
led to a proposal for the deployable Tensairity structure.

2.3.1 EXPLORATIVE MODELS
2.3 DEPLOYABLE MECHANISMS FOR TENSAIRITY STRUCTURES
The first developments regarding deployable Tensairity structures are conducted
by Crettol and Luchsinger (2006). Three categories of deployable
compression elements, based on the type of mechanism, are evaluated by
means of experiments on small scale models. This section briefly presents
these explorative experiments.
The compression elements which allow folding of the Tensairity beam because
of their low bending stiffness when the Tensairity beam is deflated are called
flexible compression elements, of which two are discussed here briefly (the ‘hose’
and ‘chain’) (figure 2.18). The insight gained with these flexible systems leads
to the investigation of two discretized compression elements (the ‘segmented
compression pipes’ and ‘separate wooden segments’) (figure 2.19). As a result
of this exploration, the hinged compression element is developed by Crettol and
Luchsinger (2006) (figure 2.20).
These three categories of deployable compression elements are evaluated by
means of experiments on small scale models. An air beam made from PVC
coated polyester fabric of two meter length and 0,2m diameter is used for
the experiments. The internal pressure of the beam is kept constant at 30 kN
m 2
(300 mbar). The investigated proposals are placed in a pocket attached at the
upper side along the beams length (figure 2.17). Remark that the compression
elements are for a first investigation only placed at the compression side and
not as well at the lower side of the beam. A description of the different cases
is given below, as well as a brief discussion of the test results.
Figure 2.17: The two meter air beam with the various investigated
compression elements (Crettol and Luchsinger, 2006).
From the experiments can be concluded that the flexible compression elements,
such as the ‘hose’ and ‘chain’, allow very well the folding of the hybrid structure
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CHAPTER 2 LITERATURE REVIEW & BASIC PRINCIPLES
(figure 2.18). The technology of pressurized compression elements has been
patented (Pedretti and ProspectiveConcepts, 2004b). However, the maximal
load these structures can bear and their stiffness is not sufficient for civil
engineering applications. In addition, the complexity of the solution is too
high. Therefore, solutions with a better load bearing behaviour and higher
stiffness are investigated, like mechanisms constituted of stiff elements: the
discretized systems.
Figure 2.18: The pressurized hose and chain as flexible compression
element (Crettol and Luchsinger, 2006).
These discretized systems are - unlike the flexible ones - constituted of stiff elements
(figure 2.19). Therefore, rolling or folding them together in deflated state
is not so straightforward and a compact configuration is achieved less easy.
When the compression element will be applied at the upper and lower side
of the air beam, folding these discretized systems will be even more difficult.
When looking at their structural behaviour however, a large improvement
is achieved when comparing with the flexible compression elements. The
results show a higher maximal load and stiffness. Applying stiff segments in
the compression element is from a structural point of view a better choice.
This technology of separate segments as compression element for a deployable
Tensairity beam has been patented by Luchsinger et al. (2006).
However, the structural behaviour of these discretized systems can still be
improved by connecting the different segments. Because of the lack of this
connection, the systems are poor in bearing the shear forces and high local
loads. In addition, a good connection in inflated state can decrease the gaps
between the segments and thus increase the stiffness of the structure.
These suggested improvements lead to the design of the ‘composite compression
element’. This compression element is constituted of wooden segments
that are connected by a fabric/tape, attached at the lower side of the elements,
illustrated in figure 2.20. Because of this fabric, the composite compression
element is able to bear shear forces and the tension forces resulting from
bending stresses at large deflections. Aluminum plates are placed on the

2.3 DEPLOYABLE MECHANISMS FOR TENSAIRITY STRUCTURES
Figure 2.19: The segmented compression pipes and separate wooden
segments as compression element (Crettol and Luchsinger, 2006).
upper side of the elements to bear the compressive forces. The compression
element can thus be folded to one side and is bending stiff when loaded
on the upper side. The stiffness of the structure is better than that of the
Tensairity beam with separate wooden segments. After all, due to the presence
of the tape, immediate contact is achieved between the segments in deployed
configuration.
Summary
Figure 2.20: The composite compression element (Crettol and
Luchsinger, 2006).
Table 2.1 summarizes roughly and as indication the ‘experimental results’ of
the investigated proposals mentioned before. The load at the deflection of 5 cm
is given for those structures that did not fail yet at a smaller deflection. For
those structures, the maximal load and reason of failure is given. The results
from tests on the air beam without compression element and a Tensairity beam
with wooden continuous compression element are included as reference.
Summarizing the research, there can be stated that there is an evolution in the
development of these foldable Tensairity structures. First, flexible solutions
for a compression element are investigated. They show characteristics very
appropriate for folding the Tensairity beam. However, their stiffness is too low
and their complexity too high for applications in civil engineering. Secondly,
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CHAPTER 2 LITERATURE REVIEW & BASIC PRINCIPLES
Table 2.1: Indication of load bearing behaviour of the investigated
proposals
compression element maximal load deflection reason of limit
air beam without compression element 18 kg - buckling
hose 30 kg 5 cm -
chain 30 kg - buckling
segmented compression pipes 45 kg - buckling
separate wooden segments 48 kg - buckling
composite compression element 60 kg 5 cm -
Tensairity with continuous compr. element 70 kg 3 cm material failure
there is then observed that applying stiff segments in the foldable compression
elements increases the stiffness and maximal load considerably. By connecting
the various segments with a hinge, a mechanism with satisfying properties is
developed.
More details and discussions of the research by Crettol and Luchsinger can be
found in De Laet et al. (2008).
2.3.2 FOLDABLE TRUSS
Santiago Calatrava developed in 1981 in his PhD-dissertation ‘Zur Faltbarkeit
von Fachwerken’ (‘On the folding of trusses’) novel deployable structures
by introducing hinges in trusses and by investigating their kinematics (Calatrava
Valls, 1981). One of his deployable structures is a conventional truss
where the horizontal tension and compression bars of each triangle are divided
in two and reconnected with an intermediate hinge (figure 2.21). This way, the
truss becomes a mechanism. After all, for a truss to be statically determinate,
the number of equations must equal the number of unknowns, or: 2 × j = n + 3
with j the number of joints and n the number of bars (members). As can be
seen from figure 2.21, this is not the case here: 2 × 12 > 16 + 3 and the truss
is as a consequence geometrical unstable. To stabilize the system in deployed
configuration, Calatrava applied vertical bars and a locking mechanism at the
intermediate hinges.
A certain analogue can be detected between Tensairity structures and regular
trusses: the airbeam fulfils the role of the diagonal struts. This reasoning
brought Luchsinger to the idea of adapting Calatrava’s foldable truss system

2.3 DEPLOYABLE MECHANISMS FOR TENSAIRITY STRUCTURES
Figure 2.21: Three steps in the unfolding sequence of the foldable plane
truss (Calatrava Valls, 1981).
for a deployable Tensairity structure (Luchsinger et al., 2007). He adjusted the
system for applying it in a Tensairity structure by replacing the vertical bars
with (pre-tensioned) cables, as illustrated in figure 2.22. The diagonals can
be included or excluded. The linear compression and tension bars are in the
deployable Tensairity structure continuously attached with the hull, and this
way, the truss is stable when the air beam is fully inflated.
Figure 2.22: Adaption of the foldable truss system for applying it in a
Tensairity structure, by Luchsinger et al. (2007)
This is a promising concept as alternative for a deployable Tensairity structure.
The structure can be folded and unfolded without disassembling. In addition,
the mechanism is present at tension and compression side, which allows the
structure to withstand as well downwards loading as upwards loading (which
is the case for wind loadings). An alternative for only downwards loading has
also been developed. In this case, the lower strut is replaced by cables that are
attached to the upper strut by means of a web (Luchsinger et al., 2009).
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CHAPTER 2 LITERATURE REVIEW & BASIC PRINCIPLES

PART I
Design of Deployable Tensairity Structures
33

3
Mechanisms for deployable Tensairity structures
The standard Tensairity beam can not fold because of its continuous compression
element. Therefore, the struts need to be replaced by a suitable
deployment mechanism. This chapter presents the exploration and analysis
of ideas for deployable systems by means of experiments on various scale
models. By investigating these models, insight is gained to improve the
existing foldable truss system. This system and the adaptations necessary
to improve it are presented. But first, before exploring several solutions,
the design requirements and boundary conditions that have to be taken into
account are discussed.
3.1 GOAL AND BOUNDARY CONDITIONS
There is chosen in this research to design for civil engineering structures,
like bridges or roofs. This choice imposes several restrictions on the design of
deployable Tensairity structures. It is the challenge of this research to find a
suitable solution that complies with these boundary conditions.
Before addressing the various requirements, the definition of ‘deployable
Tensairity structure’ in this research is clarified. After all, two categories of
applying Tensairity elements in a deployable structure can be distinguished.
The first one, on the left in figure 3.1, can be understood as a deployment
of Tensairity structures. Here, the deployment does not occur at the level of
the Tensairity elements, but at the connection of multiple Tensairity-elements.
This kind of deployment can be seen as the replacement of bars or plates of
conventional structures by Tensairity-beams or cushions. The second category,
shown on the right of figure 3.1, is called in this research a ‘deployable
Tensairity structure’. Here, the deployment of the continuously attached struts
elements is the main focus. This latter category is investigated in this research.
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CHAPTER 3 MECHANISMS FOR DEPLOYABLE TENSAIRITY STRUCTURES
Figure 3.1: Two categories of applying Tensairity elements in a
deployable structure can be detected. Left: a deployment of Tensairity
structures; right: a deployable Tensairity structure. The right category is
the subject of this research.
The unfolding of the deployable Tensairity beam has to be actuated by simple
increase of the inner pressure of the air beam. Once this structure is inflated
and thus completely unfolded, it should behave by the rules of the structural
concept of Tensairity. The difference in structural behaviour between the
deployable and standard Tensairity beam in inflated state should be as minimal
as possible. On the other hand, once the deployable Tensairity structure is
deflated, it should be able to be folded, rolled or packed together to a compact
configuration without disassembling the different components it is constituted
of. No additional operations (e.g. fixations with fasteners, mechanisms, ... )
on the structure are allowed in this research.
Since the deployable Tensairity structure will be applied in the context of civil
engineering, requirements are also made on the type of solution. Specific for
lightweight structures is that - due to wind suction - the load case can be
upwards. Therefore, to be able to resist such an inversion of the load, the cable
at the lower side of the beam is replaced with a bending stiff strut. When the
structure is not loaded and operational, it should also withstand transportation
and (un)folding. As a consequence, a durable and simple solution has to be
proposed. In addition, the structure should fold and unfold for a large amount
of times without significant damages to the membrane. A simple solution
(with the less degree of freedom as possible) guarantees a straightforward, lowtech
and thus robust solution. At the same time, the elegance of a Tensairity
beam in general and the lightweight appearance of the struts in relation to the
air beam should be preserved. Therefore, the visual presence of the mechanism
has to be as minimal as possible.
Above mentioned design constraints are all chosen to be imposed in this
research. In addition to that, and in order to meet the requirements of the
structural principle Tensairity, some other boundary conditions have to be
taken into account. The most important is that the compression element for

3.2 FOLDING SEGMENTED COMPRESSION ELEMENTS
stabilizing reasons should be attached along its length to the membrane as
much and as tight as possible. This requirement has a large influence on the
design of the mechanism, since no relative movement between the struts and
the membrane is allowed. As a consequence, the folding of the membrane
should be compatible with the proposed foldable compression element. After
all, the fabric has to be regarded as an inextensible material and can therefore
not stretch when e.g. it has to bend around a hinge.
3.2 FOLDING SEGMENTED COMPRESSION ELEMENTS
From the explorative models discussed in section 2.3.1 is concluded that stiff
elements connected with hinges are - with regard to the structural behaviour
- a good solution for the foldable compression element of the deployable
Tensairity beam. This section investigates the folding of these systems by
means of scale models.
3.2.1 INTERACTION BETWEEN MECHANISM AND MEMBRANE
To comply with the structural principles of Tensairity, the compression element
should be attached for stabilizing reasons along its length to the membrane. As
a consequence, no relative movement between the struts and the membrane is
allowed, which implies that the folding of the mechanism should be compatible
with the inextensible membrane. Because of this constraint, many kinematic
systems can not be used as deployable compression element. Solutions that
erect in one dimension like e.g. telescopic compression elements can thus not
be applied. Also mechanisms whereby the membrane has to (over)stretch are
not a solution.
Overstretching occurs where the distance between two points of the membrane
has to increase to comply with the kinematics of the mechanism. Consider
two hinged struts which are coplanar with and attached to a membrane, as
illustrated in figure 3.2. The axis of rotation of the hinge is perpendicular
to the membrane, and it is obvious that - when rotating - the struts would
move in such a way that the membrane has to stretch, which is not possible
and/or allowable. Thus, the axis of rotation can only lie in the plane of the
membrane 1 . However, fulfilling this condition is not sufficient. In addition,
the membrane should be coplanar with the center line of the hinge. When this
is not the case, the membrane will be stretched or wrinkled, as illustrated in
1 When the membrane is curved, as is the case for a cylindrical airbeam, the axis of rotation
should be tangential to the membrane at the position of the hinge.
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CHAPTER 3 MECHANISMS FOR DEPLOYABLE TENSAIRITY STRUCTURES
figure 3.3. Stretching damages the membrane and wrinkling can obstruct the
folding. Thus, a lot of attention has to be paid to the relative position of the
center of rotation of the hinges with regard to the membrane.
Figure 3.2: The folding of the mechanism should be compatible with
the membrane.
Figure 3.3: The membrane shoud be coplanar with the center line of the
hinge (upper). When this is not the case, wrinkling (middle) or stretching
(lower) of the membrane occurs.
Another example, illustrating the consequences of the inextensible membrane

3.2 FOLDING SEGMENTED COMPRESSION ELEMENTS
attached to the strut is illustrated in figure 3.4. Here, a cylindrical air beam
is used as inflated component of the Tensairity beam. The upper and lower
strut are positioned with a distance of 2 × R from each other. This distance
can due to the membrane only be increased maximally to a value of π × R,
thus approximately by fifty percent. As a consequence, there are limitations
in applying mechanisms which move the upper and lower strut further from
each other. Examples of such systems are concertina-type solutions (like an accordion)
and (translational) scissor mechanisms in the airbeam for decreasing
the degree of freedom of the folding process.
2.R
Figure 3.4: The folding of the mechanism should be compatible with
the membrane. Left: inflated airbeam, right: most compact position due
to stretching of membrane.
Now the design consequences of the interaction between hinged stiff segments
and membrane are understood, various systems of hinged folding can be
developed and investigated.
3.2.2 HINGED FOLDING
The compression element, comprised of stiff elements connected with hinges,
should be designed such that the Tensairity beam can fold undisturbed. In
addition, as mentioned before, the kinematic system should be robust and
simple. Therefore, in this research only hinges with just one axis of rotation
are applied. With the x-axis being the longitudinal direction of the strut, the
y- and z- axes are considered (figure 3.5). Three of the investigated models
are discussed here briefly, two with hinges folding around the y-axis, called
the ‘loop folding’ and ‘foldable truss’ and one around the z-axis, called ‘out of
plane folding’.
Out of plane folding
Figure 3.6 shows the model whereby the hinges rotate around the z-axis (and
thus allow movement ‘out-of-the-plane’). Since the external loading acts in
the xz-plane, the hinged struts can take up bending moments around the yaxis
and act as a continuous strut. Another advantage of this system is the
straightforward and neat folding of the membrane when deflated, because the
π.R
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CHAPTER 3 MECHANISMS FOR DEPLOYABLE TENSAIRITY STRUCTURES
Figure 3.5: The coordinate system: the x-axis is the longitudinal direction
of the strut, the loading acts in the xz-plane.
beam can be folded like a flat membrane as illustrated in figure 3.6.
However, this solution is not suitable from a structural point of view. After
all, previous research on Tensairity structures with continuous compression
elements showed that structural failure is mostly caused by global out-of-plane
buckling. Introducing hinges that allow rotation out of the plane weaken the
structure considerably. In addition, when bending moments occur in the
upper strut under external loading, these hinges will also be bent. This is not
favourable, since the smallest deformation of the kinematic system can cause
the system to be jammed.
Loop folding
Figure 3.6: The out-of-plane folding.
The second model is constituted of hinges that allow rotation around the yaxis
(see fig. 3.5) and is very similar to the ‘composite compression element’
discussed in section 2.3.1: the segments are connected at their lower side to
each other by a fabric/tape. The compression element can thus be folded to
one side and is bending stiff when loaded on the upper side (figure 3.7). Note

3.2 FOLDING SEGMENTED COMPRESSION ELEMENTS
that when the separate segments are fully fixed to the air inflated membrane
(glued, stitched, ..), the membrane takes up the role of the textile hinge and an
additional connection is thus unnecessary.
Figure 3.7: Tensairity beam with ‘loop-folding’ compression element
(Crettol, EMPA).
This system can be folded in different ways, as illustrated in figure 3.8. One
way of folding is called the loop-folding (figure 3.8a). This implicates that all
segments have different lengths, which is not desirable from a manufacturing
point of view. Note the space between the different loops in figure 3.8a for
‘accommodating’ the membrane. Another way of folding is called the closedloop
folding (figure 3.8b). Here, the structure is folded until one loop is
reached. The advantage of this proposal is that every segment has the same
length. On the other hand, the radius of the packaging becomes very large in
the case of long beams. A solution for folding the Tensairity beam where all
segments can have the same length and where the folding is independent of
the length of the Tensairity beam is called the ‘spiral folding mechanism’. This
mechanism is a modified ‘closed-loop’ folding system: the hinge between the
segments is inclined. This way, the mechanism will have the shape of a spiral
in folded position (figure 3.8c). The stability of this packaging is guaranteed by
using tapered segments and a fabric hinge at the starting point of the tapering
(figure 3.9). More on the design of this ‘closed-loop’ folding system can also
be found in De Laet et al. (2009).
Because of the specificity of the hinges, this model folds only in one direction.
The hinge is always positioned at the interface between the inflated beam
and the compression element, because the membrane of the airbeam can
not overstretch. The ‘segmented compression element’ can therefore only
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CHAPTER 3 MECHANISMS FOR DEPLOYABLE TENSAIRITY STRUCTURES
Figure 3.8: Different folding possibilities: a. loop-folding, b. closed-loop
folding, c. spiral folding mechanism.
Figure 3.9: The stability of the ‘closed-loop’ folding system can be
guaranteed by using tapered segments and a fabric hinge at the starting
point of the tapering.
be positioned at one side of the Tensairity beam. This concept is thus not
suitable for applications where the direction of the loading can change (e.g.
wind pressure and suction), because this implies that the Tensairity beam
should have compression elements at both sides of the beam, which is not the
case with this solution for a deployable Tensairity structure. Therefore, the
folding concept is not developed further in this research.
Foldable truss
The third model is the foldable truss system, already introduced in chapter 2.
Figure 3.10 shows the struts’ mechanism (Luchsinger et al., 2007). The foldable

3.2 FOLDING SEGMENTED COMPRESSION ELEMENTS
truss can be seen as a conventional truss where the horizontal tension and
compression bars are divided in two and reconnected with an intermediate
hinge. This way, the truss becomes a mechanism. Figure 3.11 shows the
deployment without airbeam of the foldable truss. Figure 3.12 shows the
deployment sequence at inflation. The diagonals can be included or excluded.
The compression and tension bars are in the deployable Tensairity structure
continuously attached with the hull, and this way, the mechanism is stable
when the air beam is fully inflated.
Figure 3.10: The foldable truss system by Luchsinger et al. (2007).
Figure 3.11: The deployment of the foldable truss system without air
beam.
As can be seen, this foldable solution allows the application of a compression
element at the upper and lower side of the air beam, which is a requirement
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CHAPTER 3 MECHANISMS FOR DEPLOYABLE TENSAIRITY STRUCTURES
Figure 3.12: The deployment sequence of the foldable truss (Luchsinger
et al., 2007).
for the design. Because the upper and lower strut move to each other when
folding, the upper and lower strut can be directly connected with cables. No
stretching of the circumferential membrane will occur because of this decrease
in distance between upper and lower strut. Note that some local stretching at
the hinges that rotate inwards occurs.
The hinges in this solution fold down in the direction of the loading. There
was expected that this would have a big influence on the structure’s stiffness.
However, the explorative models show that the air beam counteracts - although
not completely - the downwards movement of the hinges, which is beneficial
for the structural behaviour. It is subject of further research (presented in the
next section) to improve the design of the mechanism in general and the hinges
in specific such that the stiffness is increased.

3.2.3 SUMMARY
3.3 ADAPTING THE FOLDABLE TRUSS SYSTEM
The deployable Tensairity beam, designed for being applied in the field of
civil engineering, has to be robust and simple. Therefore, the compression
element is chosen to be linear to accommodate and transfer the compressive
forces the most optimal. This linear element is preferably constituted of
stiff segments connected with a hinge and should have a straightforward
deployment. Investigation of explorative models showed that the interaction
between the membrane and the struts, which are fully connected together, is
a major boundary condition that has to be taken into account in the design of
the mechanism.
Several things are learned from the explorative models. Folding out of the
plane seems a solution that results in a structural sound beam because the
hinges do not fold in the direction of the loading. However, out-of-plane
buckling will then occur at smaller loads. In addition, the hinges are loaded
in an unfavourable way, which should be avoided because deformation of the
kinematic system will happen. Solutions whereby the hinges rotate only in the
direction opposite to the loading have a good structural behaviour. However,
mechanisms constituted only of these hinges just fold in one direction. Therefore,
they are not fully applicable where upper and lower strut are required.
The foldable truss mechanism shows characteristics, structural as well kinematic,
that are all suitable for the deployable Tensairity beam. Off course,
the system needs to be investigated further and adapted to be applied for a
deployable Tensairity beam. This will be discussed in the next sections.
3.3 ADAPTING THE FOLDABLE TRUSS SYSTEM
The foldable truss system is identified as a suitable mechanism for the
deployable Tensairity structure. Now, the configuration and detailing is
adjusted to improve the deployment and structural behaviour.
3.3.1 CONFIGURATION
The prototype of the foldable truss, developed by Luchsinger et al. (2007),
shows a straightforward deployment and a good load-bearing behaviour,
especially when cables (or diagonal bars) are used to connect the upper and
lower strut.
However, there are also issues to be solved and optimized. The bars are
positioned adjacent to each other with the type of hinge used (figure 3.13). As
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CHAPTER 3 MECHANISMS FOR DEPLOYABLE TENSAIRITY STRUCTURES
a result, the hinges are loaded eccentric, which should be avoided. Another
disadvantage of the present system is that it can not be fully folded, because
the segments of upper and lower strut interfere with each other. In addition,
the interaction between the mechanism and the membrane is such that when
folding the strut, the membrane gets jammed between the struts. Cutting
of the membrane can occur this way. Of course, this should be prevented.
Figure 3.13 shows the issues to be solved with a new configuration.
Figure 3.13: Issues to be solved in a new configuration: (a) eccentric
loading of the hinges, (b) membrane is jammed between bars and (c) the
system can not be fully folded because of interfering of the upper and
lower strut.
The hinge mechanism of the ‘foldable truss’-model is improved by adapting
the initial configuration. The mechanism is changed in such a way that the
membrane is positioned between the bars (instead of being cut by the bars)
and that the bars lie in the same plane (instead of two parallel planes). With
this configuration, the hinges are loaded centrically.
The length of the bars with this configuration need to be derived from some
basic equations in order to fold correctly. From figure 3.14 can be seen
L1 = 2A + 3B + 4D (3.1)
L2 = 2B + 5C + 6D (3.2)
B = C + t (3.3)
with t the thickness of the strut and C the distance (clearance) chosen by the
designer to accommodate the membrane. In all cases has to be checked that
A > D and preferably that A > D + C.
The main issues of the previous model seem to be solved with this new
configuration of the foldable truss. However, new problems are arisen. The

A A
D
C
B
B
D D D
C C C
C
D
B
C
B B
C
3.3 ADAPTING THE FOLDABLE TRUSS SYSTEM
L 1
L 2
D D membrane
Figure 3.14: Left: the annotation of the bars of the new configuration,
right: the membrane is accomodated between the bars.
foldable truss is now constituted of almost twice as many hinges. As a result,
more segments can rotate relative to each other which makes the packing of the
structure to a compact configuration more complex. This way, some elements
tend to deploy in other directions than they should, which hinders the compact
folding.
A solution for this issue is investigated in the next sections, first by decreasing
the degree of freedom of the deployable Tensairity truss, then by designing
the hinges such that a more straightforward deployment is reached.
3.3.2 DEGREE OF FREEDOM
The decrease in degree of freedom of the truss is investigated with the goal to
improve and simplify the folding process of the deployable Tensairity beam.
First, there is looked at the effect on the kinematic behaviour of introducing
the diagonals (figure 3.15). The decrease in degree of freedom achieved by
this is however not enough to have a major influence on the folding process.
Mainly weight is added here.
The diagonals of the deployable truss are replaced by a scissor mechanism,
illustrated in figure 3.16. The resulting structure is a foldable truss with one
degree of freedom, as intended. However, because the scissor mechanism lies
in the same plane as the bars, the configuration can not completely fold to a
47

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CHAPTER 3 MECHANISMS FOR DEPLOYABLE TENSAIRITY STRUCTURES
Figure 3.15: Diagonals are introduced to decrease the DOF.
closed position. In addition, there is expected to have buckling in these slender
diagonals when the Tensairity beam will be loaded.
Figure 3.16: The diagonals of the deployable truss are replaced in this
proposals by a scissor mechanism.
More models are manufactured in the attempt to decrease the degree of
freedom when folding. Sliders are applied to guide the intermediate hinges,
cables transfer the rotations of the hinges, etc. The conclusion is always the
same: to ease and simplify the packing of the deployable Tensairity structure,

3.3 ADAPTING THE FOLDABLE TRUSS SYSTEM
more material, details, fragility and especially complexity needs to be added
to the mechanism. This is in contradiction with the purpose of designing a
lightweight, robust and simple deployable Tensairity beam. Therefore, a more
straightforward packing of the structure is pursued by means of the geometry
and configuration of the hinges.
3.3.3 HINGES
From the closed configuration of the mechanism, illustrated in 3.14, can be seen
that there are two types of hinges, characterized by their range of deployment:
some hinges allow the bars to rotate 180 degrees, some 90 degrees. To increase
the control of the deployment and to create a better load-bearing behaviour,
‘stops’ are introduced in the hinges. This way, the range whereby the bars can
rotate around the hinges are limited. Several solutions for achieving the ‘90
degree-stops’ are modeled and are shown in figure 3.17.
In two proposals for these stops, extra elements are added around the rotating
segments, such that the bars are hindered and can not rotate further than a
predefined angle. In another solution, a stop is being made by the geometry of
the contact surface of the two parts of the hinge. This stopping mechanism has
already been applied for the ‘loop’-folding in section 3.2.2 and had satisfying
effects.
Figure 3.17: To increase the control of the deployment and to create a
better load-bearing behaviour, ‘stops’ are introduced in the hinges.
Up to now, the proposals of hinges have always been limited to being assem-
49

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CHAPTER 3 MECHANISMS FOR DEPLOYABLE TENSAIRITY STRUCTURES
bled out of flat laser cut shapes. Because this is a limiting factor regarding the
evaluation of the hinges, the new proposal is fabricated by means of computer
numerical controlled (CNC) machine tools. Figure 3.18 shows the aluminum
hinge where a square tube can be placed around. This proposal is redesigned
to fit in a regular aluminum strut for a Tensairity structure, also illustrated in
figure 3.18.
Figure 3.18: Two proposals for hinges in aluminum fabricated by means
of CNC machine tools.
All solutions show an increase of control when packing the Tensairity beam to a
compact configuration. After all, the hinges can now only rotate in a predefined
interval. However, from the small scale laser cut models is seen that the least
play in these ‘stops’ is sufficient for cancelling their effect. Attention should
thus be paid to the materialization of the idea. The fabrication of the hinges in
aluminum by means of CNC allows to better evaluate the solution.
3.3.4 FROM STRAIGHT TO CURVED
Up to now, the deployable Tensairity structure is composed of a cylindrical air
beam and a truss with the shape of a trapezium. However, research showed
that a Tensairity beam with a spindle shape and thus a curved upper and lower
strut, is six times stiffer than a cylindrical Tensairity beam (Pedretti et al., 2004).
From now on, this spindle shape will be applied for the deployable Tensairity
structure. This does not have a great influence on the foldable configuration
or the design of the hinges, as can be seen in figure 3.19.

3.3 ADAPTING THE FOLDABLE TRUSS SYSTEM
Figure 3.19: A spindle shape will be applied for the deployable Tensairity
structure for reasons of increase in stiffness.
51

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CHAPTER 3 MECHANISMS FOR DEPLOYABLE TENSAIRITY STRUCTURES
3.4 REDESIGN OF MECHANISM
The influence of hinges in the struts of a deployable Tensairity beam are
investigated experimentally and numerically. These studies are presented
further in this dissertation and show that the position of the hinge can have an
influence on the load bearing behaviour of the structure: the maximum load
the structure can bear decreases when the hinge is positioned too close to the
ends of the beam. After all, the compressed strut will buckle outwards at this
position. This is discussed more in detail in section 4.5 and section 6.3.
3.4.1 COMPACTING THE CONFIGURATION
Moving the hinges towards the middle improves thus the structural behaviour
of the deployable Tensairity beam. For this reason it is chosen to redesign the
configuration of the mechanism. The opportunity is taken at the same time to
improve the kinematic behaviour of the mechanism. Indeed, after additional
evaluations and explorative experiments, some issues open to improvement
are detected. In the current configuration, the upper strut has to fold around
the lower strut and vice versa. Even with the limitation of the rotation of the
hinges to 90 degrees, this folding is in some cases still tough going. Another
disadvantage of the current foldable truss is the large amount of small bars.
There should be evaluated whether a new system is possible whereby there
is less variation in bar length. At the same time is investigated if the folded
configuration can be conceived more compact.
In the new proposal, the segments of the upper strut are ‘moved upwards’ in
their folded configuration, as illustrated in 3.20. This way, the upper strut does
not has to fold anymore around the lower strut and vice versa. This means
that only two layers of membrane have to be accommodated between the bars
and hinges instead of a bunch of membrane. As a result, the small bars in
the configuration are not necessary anymore and the hinges do not have to
stop rotating after 90 degrees, but can close completely. This simplifies the
design of the hinges and decreases their fragility. In addition, by shortening
the upper middle struts and thus ‘moving’ the hinges in folded configuration
upwards, the segments of the upper strut at the ends become longer, which
means that the distance between the supports and the first hinge in the upper
strut increases, which is aimed for as discussed before. Finally, another pluspoint
of this solution is that the size of the folded configuration is decreased.
Notice that asymmetrical upper and lower strut (with regard to the position
of hinges) are required to have the supports at the same side of the folded

A
C
package.
A
B
3.4 REDESIGN OF MECHANISM
Figure 3.20: In the new proposal (annotated as C), the segments of the
upper strut are ‘moved upwards’ in their folded configuration.
As mentioned, a compact configuration is one of the objectives of this ‘redesign’.
Therefore is chosen for symmetry between upper and lower strut with
regard to the amount of segments folding inwards. This way, a configuration
with parallel segments in folded configuration and segments with‘straightforward’
length is achieved. With L being the length of a strut, the lower segments all
measure L
L
6 , the middle upper segments have a length of 12 and the two side
segments are L
4 long.
3.4.2 HINGES
As already mentioned, all hinges are now allowed to rotate 180 degrees, which
simplifies their design. However, they are not all the same. Two types of
hinges exist, the hinges at the supports not taken into consideration. Their
difference lies in the direction of rotation: one type rotates towards (or ‘in’ )
the air beam, the other outwards (or ‘away’). As explained in section 3.2.1, the
membrane should be coplanar with the center line of the hinge. Otherwise,
the membrane will be stretched or wrinkled.
C
53

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CHAPTER 3 MECHANISMS FOR DEPLOYABLE TENSAIRITY STRUCTURES
In the case of the deployable Tensairity beam, the membrane lies on the same
side of the struts for the hinges that rotate towards or away from the air beam.
This means that for both cases, the axis of rotation has to be the same and
collinear with the membrane. For the hinge that rotates outwards, no problem
arises. However, for the hinge that rotates inwards, these conditions lead to the
design whereby the hinge is constituted of two axes of rotation, as illustrated
in figure 3.21.
Figure 3.21: The two hinges, rotating 180 degrees. Left: the hinge
rotating ‘away’ from the air beam. Right: the hinge rotating ‘inwards’.
Because the axis of rotation has to be collinear with the membrane, this
latter hinge is constituted of two axes of rotation.
These hinges are first fabricated in wood as test case, later in aluminum. For
reasons of manufacturing and cost, several proposals were made. A hinge
out of one piece (solid) proved too complex and expensive to manufacture
(figures 3.22). Therefore, the design is adapted to allow manufacturing out of
flat pieces (subdivided), shown in figure 3.23.
3.4.3 DIAGONAL CABLES
The mechanism is constituted of hinges that move outwards and inwards.
As a consequence, the distance between points of the upper and lower strut
changes during deployment. As will be discussed further in this dissertation,
cables connecting the upper and lower strut may increase the stiffness of the
deployable Tensairity beam under loading. Off course, when points move
away from each other, cables can not be applied here. Therefore, the distance
between hinges before, during and after folding is calculated.
Because the mechanism has many degrees of freedom, it is unpredictable what
the exact intermediate configurations are during the folding and unfolding.
Therefore, there is assumed that angles between the struts rotate with the
same aspect ratio from open till closed position, thus in relative values from

Figure 3.22: The designs of the aluminum solid hinges
Figure 3.23: The subdivided hinges.
3.4 REDESIGN OF MECHANISM
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CHAPTER 3 MECHANISMS FOR DEPLOYABLE TENSAIRITY STRUCTURES
0 percent to 100 percent closed. Because this is an assumption, the distances
during deployment may be incorrect. After all, it is possible that one hinge is
already completely folded, while the other has not rotated yet. This does not
pose a problem however, since it are especially the distances in the fully open
and closed position that are relevant for this research.
The position of each point is first calculated during deployment. Then the
distances between the hinges are plotted. Figure 3.24 presents the annotation
of the bars and angles. The coordinates of the points are given as:
x0 = 0 y0 = 0 (3.4)
x1 = L1. cos(α) y1 = L1. sin(α) (3.5)
x2 = x1 + L2. cos(α + γ − π) y2 = y1 − L2. sin(α + γ) (3.6)
x3 = x2 + L2. cos(α + γ − ε) y3 = y2 + L2. sin(α + γ − ε) (3.7)
x4 = x3 + L2. cos(π − α − γ + ε) y4 = y3 − L2. sin(π − α − γ + ε) (3.8)
xa = L3 ya = 0 (3.9)
xb = xa + L4. cos(π − β) yb = ya − L4. sin(π − β) (3.10)
xc = xb + L5. cos(β − δ) yc = yb + L5. sin(β − δ) (3.11)
For clarity, the angles mentioned above are the angles at a certain deployment.
Thus when the structure is folded by 50%, α = α50% = αopen
2 . By means of above
coordinates, the distances between the various points on opposite struts are
calculated and plotted, see figure 3.25.
y
x
α
L 3
a
β
L 1
Figure 3.24: The annotation of the bars and angles for calculating the
distances between the hinges during deployment.
From the graph can be seen that the distance between the upper and lower
hinges varies. All links from the upper strut to hinge b increase in length
during folding. Also the length of 1 − c and 3 − c increases. However, these
links shorten again after a certain amount of folding. It can be that when the
folding is not uniform, these links do not increase in length. The links whereby
1
γ
L 4
L 2
δ
b
ε
2
L 2
3
φ
L 5
L 2
4
c

Distance between hinge (1,2,3,4) and (a,b,c) [mm]
70
60
50
40
30
20
10
a
1
4c
0
1 2 3 4 5 6 7 8 9 10
γ − folding : from open ( value 1) to closed position (value 10) [/]
2
b
3
1b
3b
4b
2b
1c
3a
3c
4a
2c
2a
4
1a
3.4 REDESIGN OF MECHANISM
Figure 3.25: Plot of the distances between the various points on opposite
struts for a 2 m spindle with slenderness 10.
the distance increases during folding can thus not be connected with a cable.
Figure 3.26 shows all cable configuration allowing (upper) and obstructing
(lower) the folding.
3.4.4 FOLDING THE HULL
Some points of the upper strut move away from the lower strut when folding,
as shown above. This implies that the outer hull has to be designed such
that it allows for this increase in length. As can be seen in figure 3.25, the
links whose length increases most are 1 − b and 3 − b. The increase of both
lengths is identical. It is thus sufficient to design the membrane such that it can
c
57

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CHAPTER 3 MECHANISMS FOR DEPLOYABLE TENSAIRITY STRUCTURES
allow folding
obstruct folding
Figure 3.26: Cables connecting the upper and lower strut that can be
used in this configuration, because their length decreases when folding.
accommodate the increase in length between 1−b. After all, due to the spindle
shape, the amount of outer hull between 1 − b is smaller than 3 − b. Thus, if
the membrane will accommodate the outwards movement between 1 − b, no
problem will arise between 3 − b. In figure 3.27 can be seen that stretching of
the membrane occurs between 1 − b.
Figure 3.27: Stretching of the membrane occurs between 1 − b.
To avoid this stretching of the membrane and thus obstruction of the folding,
the outer hull should be dimensioned such that enough membrane is available
between 1 − b. In the case of a standard spindle where the section is circular,
the hull is determined by the shape of the struts and can thus not be changed.
As a consequence, stretching of the membrane will occur and the standard
spindle can not be folded completely with this configuration. In the case of a
web spindle, the radii of the outer hull along the length of the spindle beam
can be chosen. After all, the struts are kept in position by means of cables
between the upper and lower strut.
To calculate the distance along the hull between point 1 and b, the segment
of the spindle is considered as a cone. The error made by this approximation
is low due to the small curvature of the spindle in the longitudinal direction.
Consider radii R1 and Rb as given, as well the height of the spindle h1 and
hb 2 . Points 1 and b are positioned a distance l from each other, this is thus the
height of the cone (figure 3.28).
2 Remark that the height of the spindle, thus the distance between the upper and lower strut

1
h 1
R 1
l
b
h b
3.4 REDESIGN OF MECHANISM
Figure 3.28: Stretching of the membrane occurs between 1 − b.
2πRb
α 1
2πR1 1
s
Figure 3.29: The cone can be flattened and from this, the distance s can
be calculated.
The cone can be flattened and the geodesic line between points 1 and b becomes
a straight line, as illustrated in figure 3.29. From this figure, the distance s can
be calculated as:
s =
α
β
R b
αb
l’
b
l
L
l ′ 2 + L 2 − 2.L.l ′ cos β (3.12)
When the distance s does not match the length necessary for a complete folding,
the radii need to be changed. As will be discussed in section 6.1, the radii of
every section along the spindle’s length are dependent: once a radius at a
certain position along the length is chosen (R1 for example), all other radii are
along the spindle’s length, is always determined and known by design.
h
R
l
ϕ
59

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CHAPTER 3 MECHANISMS FOR DEPLOYABLE TENSAIRITY STRUCTURES
determined. Thus by means of iterative calculation, the appropriate radii can
be determined for achieving a distance s between the points 1 and b. This
means that the cutting pattern of the hull of the air beam can not be chosen,
but is determined by the geometry and kinematics of the foldable compression
element.
The cutting pattern is redesigned, taking the above mentioned design rules
into account. It is applied in a two meter long Tensairity beam by way of
evaluation. The Tensairity beam can completely be folded, which indicates
that the cutting pattern of the airbeam allows the outwards movement of the
points 1-b. However, as already observed before, the folding of the structure
and thus the packing of the membrane to a dense bunch does not go very
smoothly. This is because the membrane contains a certain amount of bending
stiffness. In addition, the outer hull of the membrane tends to move between
the struts, which can obstruct folding. With this new cutting pattern for the
model, this difficult folding is more obvious than before, since a membrane
with larger bending stiffness (coating on both sides) is applied and because
more membrane is used in the airbeam with the new cutting pattern.
Therefore, research is conducted on the linear folding of the cylindrical and
spindle shaped membrane according a predefined folding pattern (figure 3.30).
After all, folding a membrane along lines goes much easier than folding it to
a random bunch. An additional advantage is that the folding occurs in a
controlled way, i.e. the folding pattern dictates how the membane moves.
However, all solutions investigated imply mechanisms with many hinges
or acquire a complex mechanism which would result in complicated and
fragile hinges. Although the folding of the adapted foldable truss does not
go smoothly, it is still experienced to be the most appropriate solution so far.
Therefore is chosen to develop and investigate this system further. Figure 3.31
illustrates de model for the deployable Tensairity beam.
3.5 CONCLUSIONS
Several models were designed and investigated in this research to gain insight
in the development of an appropriate mechanism for the deployable Tensairity
structure. These explorations resulted in the ‘redesign’ of the foldable truss
system with regard to the structural and kinematical behaviour.
The explorations show the necessity of having in mind the application where
the structure is designed for. After all, many different solutions exist for a

3.5 CONCLUSIONS
Figure 3.30: Research is conducted on the linear folding of the cylindrical
and spindle shaped membrane according a predefined folding pattern.
Figure 3.31: Folding sequence of the proposal for a deployable Tensairity
beam.
deployable Tensairity structure, however, they are not all suitable for the same
use. In this research is chosen to design for an application in the field of
civil engineering (like a roof or bridge structure), and therefore is aimed for
a solution that is as strong, robust, low-tech and straightforward as possible.
Besides the boundary conditions dictated by the application purpose and/or
designer, there are also some requirements and consequences inherent to the
structural concept of Tensairity. The study cases show that the continuous
connection of the membrane with the compression element is the most crucial
and imperative requirement. The folding of the membrane should thus be
compatible with the proposed foldable compression element.
The investigation of the structural behaviour of the various models indicates
the ‘hinged stiff segments’ as being an appropriate mechanism as compression
61

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CHAPTER 3 MECHANISMS FOR DEPLOYABLE TENSAIRITY STRUCTURES
element. The folding of this system occurs preferably in the same plane as the
loading, because otherwise deformation of the kinematic system is risked. As
a result, the hinges fold towards or away from the air beam. When designing
the hinges, the inextensibility of the membrane has to be taken into account.
The centre line of the hinge has to lie in the same plane as the membrane,
otherwise stretching or wrinkling will occur.
From various studied models with hinged elements, the ‘foldable truss system’
proves to be the most appropriate. Evaluations of the system lead to the
improvement with regard to the structural and kinematic behaviour. The
system’s load bearing capacities are ameliorated by changing the longitudinal
shape from cylindrical to spindle, by decreasing the amounts of hinges and
segments and by positioning hinges on compression side towards the middle.
The kinematic behaviour is improved by adapting the configuration in such
a way that the deployment occurs more straightforward. This is achieved
with a redesign of the foldable truss which has as result that the segments of
upper and lower strut do not have to ‘fold’ into each other. In addition, less
hinges and less complicated joints that are compatible with the folding of the
membrane are necessary.
The result of this chapter is a proposal for a compression and tension element
(both capable of taking compression) for a deployable Tensairity structure,
achieved by means of investigations on scale models. However, before developing
further the deployable Tensairity structure and building a prototype,
the influence of several design parameters (such as the configuration of cables
or the amount and position of hinges) on its structural behaviour needs to be
understood. This is investigated in the next part.

PART II
Structural behaviour of the
Deployable Tensairity Beam
63

Experiments on scale models
4
Before simulating or developing a large scale model for a deployable Tensairity
beam, it is necessary to gain some first insights in the structural behaviour of
this complex synergetic structure. This chapter investigates the structural
behaviour of these beams by means of experiments and numerical investigations
on small scale models. The focus in this chapter is gaining qualitative
insight in the load bearing behaviour of Tensairity structures in general and
in the influence of the different parameters on the load-bearing behaviour of
the deployable Tensairity structure in specific. After all, no experiments have
been conducted yet on deployable Tensairity beams.
Experiments are conducted on a two meter deployable cylindrical and spindle
Tensairity beam. The experimental set-up and the models are first described in
more detail. Then the influence on the structural behaviour of the parameters
such as the amount of hinges, the presence of internal cables connecting the
upper and lower strut of the Tensairity beam, the configuration of these cables,
the shape of the beam etc. are discussed. Finally, some conclusions are made
that will provide the basis for a proposal for a deployable Tensairity structure.
4.1 EXPERIMENTAL SET-UP
4.1.1 SMALL SCALE MODELS
Two deployable Tensairity beams, a cylindrical and spindle shaped are investigated
experimentally. The studied models are two meter long and the
maximum height in the middle of the beams is 0,25m. The air beams are
composed of two layers. An outer PU coated polyester fabric 1 carries the
1 ‘Riverjacket 200’ from Rivertex
65

66
CHAPTER 4 EXPERIMENTS ON SCALE MODELS
tension forces due to the air pressure and surrounds two internal bags made
of thin PU foil that keep the structure airtight. These internal bags are
positioned adjacent to each other in the longitudinal direction. This way,
elements connecting the compression and tension side of the beam can be
placed between the two bags. This is illustrated in figure 4.1. These two bags
are also called air chambers.
Figure 4.1: Left: Longitudinal view of the investigated cylindrical and
spindle shaped deployable Tensairity beams (isostatically supported),
right: section of the beams
The internal pressure of these air chambers is kept constant at 100 mbar
during the experiments by means of a compressed air regulator. The internal
overpressure is monitored continuously by means of a water column. Cables
and bars used are made of steel and the bars have a rectangular cross section
(10 mm height × 6 mm wide). The cables connecting the upper and lower strut
have a diameter of 2 mm. The compression and tension elements are placed
in a pocket. Holes are made in these pockets to connect upper and lower side
of the beam by cables or struts.
Various configurations of the cylindrical Tensairity beam are tested in order to
reveal the influence of the different parameters on the load bearing behaviour,
such as the amount of hinges and the presence of pretensioned cables that
connect the upper and lower hinges. The investigated configurations are
illustrated in figure 4.2. Only the struts and the connection between them are
illustrated. The hull is not shown but is the same for all configurations. Full
lines are used to indicate struts, the dotted line represents cable elements and
black dots illustrate the hinges.
Configurations 1, 15 and 16 are used to investigate the influence of the amount
of hinges on the load bearing behaviour. The results of configurations 2 to 14
are used to study the influence of vertical and diagonal cables and diagonal
bars on the load-bearing behaviour of the deployable Tensairity beam.
h = 0.25 m
outer hull
pocket
strut
inner layer
cable

1
2
3
4
4.1.2 SET-UP
5
6
7
8
9
10
11
12
4.1 EXPERIMENTAL SET-UP
Figure 4.2: Investigated configurations: they differ in amount of hinges
and/or cable configuration.
The two meter long statically determined deployable Tensairity beam is supported
in its endpoints by a stiff steel frame. The loads are applied manually in
the upper hinges. Depending on the load case, 50 N, 30 N or 10 N is applied in
each load step for respectively distributed, asymmetrical and point load. The
deflections of the beams are measured in ten points at each load step by means
of analogue clock gauges, distributed evenly at the upper and lower side of
the beam. Figure 4.3 illustrates the test set-up. Figure 4.4 gives an illustration
of the data gained for one configuration and one load case. The displacement
of one case is plotted throughout loading.
Figure 4.3: Set-up for load-deflection experiments on scale model.
There must be remarked that the results presented in this chapter are quantita-
13
14
15
16
67

68
CHAPTER 4 EXPERIMENTS ON SCALE MODELS
displacement [mm]
0
5
10
15
20
25
30
-1 0
position along x axis [m]
1
Figure 4.4: Illustration of the data gained for one configuration and one
load case. The displacement is noted after each load step of 10 N per
node.
tively not exact as it would be for large scale models. After all, the scale models
can contain more easily manufacture imperfections and approximations due
to their size, such as using a pocket for the connection with the strut. In
addition, it is difficult to quantify these influences on the test results. Also
the procedure for conducting the experiments is very basic; think about
applying loads in discrete steps and reading the deflections from analogue
clock gauges. However, it is not the purpose of this chapter to retrieve very
accurate quantitative data, but to gain insight in the structural behaviour by
comparing the results qualitatively.
4.2 OBSERVATIONS AND RESULTS
This section presents the observations and conclusions drawn from the experimental
investigations. The influence of various parameters on the load
bearing behaviour is discussed.
4.2.1 APPLYING THE LOAD
The load-displacement response of the deployable Tensairity beam is monitored
when loading and unloading the structure several times. Figure 4.5
(left) shows the applied load as function of the time. The load is applied and
released in three cycles. The response of the deployable Tensairity structure,
illustrated on the right of figure 4.5, is very similar to the behaviour of a spindle
shaped Tensairity beam with continuous compression element under a point
load, investigated experimentally by Luchsinger and Crettol (2006c). The xaxis
of the load-displacement response represents the average displacement of
the five points on the upper strut, the y-axis shows the applied load.

Load [N]
350
300
250
200
150
100
50
0
0 50 100
time [min]
150 200
Load [N]
350
300
250
200
150
100
50
4.2 OBSERVATIONS AND RESULTS
0
0 10 20 30 40 50
Average deflection [mm]
Figure 4.5: Left: The applied load as function of the time. Right: The
load-displacement response of the spindle.
The graph shows that the Tensairity structure adapts to the new load condition
during the first load cycle. This can be seen from the residual displacement
of the unloaded structure after the first cycle. The displacements under the
second and third load cycle are almost identical. The deflections of the second
or third cycle are noted and used in the next sections. The graph also shows that
the structure has another load-displacement path during loading than during
unloading. This phenomenon, called hysteresis, indicates energy dissipation.
It has to be attributed to the air beam, since the steel chords of the Tensairity
girder have an elastic behaviour. It can result from the fabric. The details
behind the hysteresis are not in the scope of this research. It is in any case an
interesting aspect regarding the dynamic properties and damping behaviour
of Tensairity girders. The residual displacements as well the hysteresis are
known behaviours of fabrics and can also be seen in the biaxial testing of
fabrics.
4.2.2 INFLUENCE OF POINT(S) OF APPLICATION AND LOAD CASE
It is evident that the displacement of the beam along its length is dependent
of the load case. In addition, in the case of a Tensairity beam is the deflection
of the upper strut also dependent whether the load is applied on the upper or
lower strut of the beam. Figure 4.6 shows the deformed shape of a deployable
Tensairity beam loaded at the upper strut and at the lower strut. A larger
displacement arises at the side where the load is applied, which indicates that
load is being taken up by the stressed hull.
To be able to compare the various configurations from figure 4.2, the average
displacement of the upper strut is taken into account (unless mentioned
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CHAPTER 4 EXPERIMENTS ON SCALE MODELS
F
Figure 4.6: Different deflection when the load is applied on the upper or
lower side of the beam.
differently). This is the average value of the displacement of several evenly
distributed points on the upper strut. In the case of these explorative models
are the values taken into account of the five analogue clock gauges at the upper
strut.
In the experiments, the load will always be applied at the upper strut of
the beam. The displacements under three different load cases, illustrated in
figure 4.7, were registered to verify whether the conclusions derived from the
comparison of the various configurations are dependent of the applied load
cases. The results are incorporated in the following sections.
4.2.3 INTERNAL PRESSURE
Figure 4.7: The three applied load cases.
The internal pressure of the beam is varied during the experiments. The deflections
of various configurations were measured with an internal pressure of 75,
100 and 125 mbars. The load-deflection behaviour of case 1 under the different
pressures is illustrated in figure 4.8 (left). From the results can be concluded
that the stiffness of the structure is increased with increasing pressure, such as it
is the case for Tensairity beams with continuous compression element (Breuer
et al., 2007; Luchsinger and Crettol, 2006a,b). After all, a higher pressure and
thus a more pretensioned membrane leads to a more constant spacing between
tension and compression element. Moreover, the friction between the pocket
and the compression element increases with higher pressure values, which
results in a stiffer structure.
F

load [N]
7
6
5
4
3
2
1
075 mbar
100 mbar
125 mbar
0
0 10 20 30 40 50
deflection [mm]
deflection [mm]
0
10
20
30
40
4.2 OBSERVATIONS AND RESULTS
50
075 mbar
100 mbar
125 mbar
60
-1 0
position along beam [m]
1
Figure 4.8: Influence of the internal pressure on the stiffness of the
deployable Tensairity beam (case 1). Left: load-displacement curves;
right: the displaced shape of the upper strut at 300 N.
4.2.4 INFLUENCE OF NUMBER OF HINGES
To study the influence of the number of hinges on the load bearing behaviour
of the structure, three cases with different amount of hinges in the upper
and lower strut are investigated. Their load-displacement response under
a distributed load and with an internal pressure of 100 mbar is illustrated
in figure 4.9 (left). The displacement of the compression chord at mid span
is given on the x-axis, the y-axis represents the applied load. The (scaled)
displacement of the upper chord at 300 N (60 N per node) is given on the
right of figure 4.9. From the results can be seen that the cases 1 and 15 have
similar deflections, despite the different number of hinges. The presence and
influence of the middle hinge on the stiffness is thus much greater than that of
other hinges. To conclude more on the influence of the amount of hinges on
the structural behaviour of the Tensairity beam, more investigations need to
be done. This will be done by means of simulations in chapter 6.
4.2.5 INFLUENCE OF CABLES
The compression and tension element of a Tensairity beam can be connected
by means of a fabric web or cables. This way, the section of the airbeam is no
circle anymore, but seems constituted of two circle segments. Figure 4.1(right)
shows this. The connecting element, a fabric web or cable, becomes stressed
when the airbeam is inflated. Previous research showed that using a fabric web
improves the structural behaviour of the Tensairity beam (Breuer et al., 2007;
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load [N]
7
6
5
4
3
2
1
case 1
0
0 10 20 30 40
displacement [mm]
case 15
case 1 case 15 case 16 case 1 case 15 case 16
displacement [mm]
0
10
20
30
40
case 16
-1 0
position along beam [m]
1
Figure 4.9: Influence of the amount of hinges on the stiffness of the
structure. The internal pressure measures 100 mbar. Left: load-
displacement graph, right: deformed shape of the upper strut under
300 N (60 N per node).
Wever et al., 2010) (see p. 23). Since using a continuous web would obstruct
the folding of the Tensairity beam, cables are used to connect the upper and
lower hinges.
The influence of this connection between upper and lower strut on the load
bearing behaviour of a deployable Tensairity structure is investigated. Before
discussing the experimental results of the various cable configurations, the
influence of cables on the load-bearing behaviour of a deployable Tensairity
beam is first investigated and understood by means of a physical and numerical
model.
4.3 PHYSICAL AND NUMERICAL MODEL
Since the structural behaviour of deployable Tensairity beams has never been
studied before, the experimental investigations from the previous section help
to gain general insight in these structures. However, it is difficult from these
experimental results to derive how the structure really ‘works’. Therefore, a
physical model (with its simplifications and approximations) is developed in
the next section for a basic understanding of the effect of interactions between

4.3 PHYSICAL AND NUMERICAL MODEL
load, pressure, membrane, compression element and cables. Conclusions
derived from this simplified model are verified by means of a numerical
model 2 .
Many work on the basic understanding of regular Tensairity beams has already
been conducted by Luchsinger et al. (2004a). This section focuses on the loaddisplacement
behaviour of the deployable Tensairity beam: its stiffness and
maximal load. More investigations with regard to the hinges, the effect of the
section of the struts and geometry of the section of the hull are presented in
chapter 6.
As seen before in this chapter, the investigated deployable Tensairity beam is
constituted of an airbeam, an upper and lower strut and cables connecting the
hinges of upper and lower strut. The shape of the section of the airbeam is
constituted of two circle segments. Figure 4.10 illustrates a longitudinal and
sectional view of the deployable Tensairity beam (case 10).
1 a 2 b 3 c 4 d 5
Figure 4.10: A longitudinal and sectional view of the deployable
Tensairity beam.
4.3.1 INFLATION
When inflating the airbeam, the overpressure pushes the upper and lower
struts outwards and the hull tends to become a circle. This action will
be counterbalanced by the cables that connect the compression and tension
element. As a result, the cables become tensioned and experience thus a
tensile force. The value of this force can easily be calculated.
In chapter 2 (p. 14) is seen that the radial membrane tension nradial [ N
m ] is the
product of the radius of the hull and the internal overpressure: nradial = p × R.
As a result, the cable force F in one cable equals
F = 2 × p × R × l × sinα (4.1)
with α being the angle between the membrane (tangential) and the horizontal
(indicated in figure 4.10) and l the distance between two adjacent cables. This
2 Details on the development of finite element models in this research can be found in chapter 5.
h
R
α
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CHAPTER 4 EXPERIMENTS ON SCALE MODELS
normal force F [N] can also be called the pretension in the cable due to inflation
(Fpre).
The deployable Tensairity beam with vertical and diagonal cables connecting
upper and lower strut (as shown in figure 4.10) is investigated by means of
finite element calculations. The finite element model, illustrated in figure 4.11,
is inflated with an internal overpressure of 100 mbar. From the figure can be
seen that the membrane is not fully closed at the ends and only modeled until
the first cable. Otherwise, convergence was an issue. This approximation has
as result that the cables closest to the ends (cable 1 and 5) experience half of the
calculated and expected pretension ( 1
2 Fpre). The cable pretension derived from
the numerical calculations is 149 N for the middle cables and 74,5 N for cables
1 and 5. All forces introduced by inflation are taken by the vertical cables.
The diagonals are not pretensioned under inflation because of their angulated
position.
The pretension in the vertical cables is also calculated with equation 4.1. With
using the same parameters as in the finite element model 3 , one obtains the
value of 147 N, which is a good approximation of the numerical value.
Figure 4.11: The deployable Tensairity beam is also investigated
numerically. The airbeam is only modeled until the first and last cable
for reasons of convergence.
4.3.2 LOADING
The cable pretension is decreased by loading the deployable Tensairity beam
(downwards). When the amount of external load taken by one cable is equal
to the pretension in this cable (Fpre), it becomes slack. This means that the
cable has from that point on zero stiffness and cannot support any additional
loading. As a consequence, the cable does not contribute anymore to the
3 An angle α between the hull and the horizontal of 10 ◦ , a height h between upper and lower
strut of 0.25 m, a radius R of 0.127 m, length l between adjacent cables of 0.333 m, and internal
pressure p of 10 kN
m 2 .

4.3 PHYSICAL AND NUMERICAL MODEL
structural behaviour and one can expect the stiffness of the Tensairity beam to
change at the value whereby the cables become slack.
The deployable Tensairity beam from figure 4.10 is loaded with a point load
in each upper hinge. If Fext is the total amount applied load, then in each
hinge a load of 1
5 Fext is applied. From standard trusses, one knows that the
first verticals (cable 1 and 5) experience a compression force of 1
2 Fext. This
means that these cables become slack when Fpre = 1
2 Fext or Fext = 2Fpre. Since
the cables 1 and 5 are tensioned with a normal force of 74,5 N, the maximal
load this deployable Tensairity beam can bear before changing stiffness is thus
149 N.
This is also investigated by means of finite element calculations on the model
presented in figure 4.11. The displacement of all upper hinges is noted and
the average value in relation to the applied load is illustrated in figure 4.12.
From the curve can clearly be seen that the stiffness changes at a total load of
approximately 150 N, which corresponds with the analytical derived value.
Figure 4.13, plotting the tension in the cables throughout loading, shows that
cables 1 and 5 indeed reach zero tension at this value. The graph also shows
that the diagonal cables are tensioned under loading, as is also the case for a
truss with the same configuration of diagonals. This holds true until cable 1
and 5 become slack. From that point on, the diagonals are compressed.
load [N]
250
200
150
100
50
0
truss
case 10
airbeam
0 0.2 0.4 0.6 0.8 1
x 10 −5
average displacement [m]
Figure 4.12: Average displacement of the upper strut of case 10 in relation
to the applied load. (pressure is 100 mbar, five point loads in upper
hinges). (Numerical results).
Case 10 has also been investigated numerically under various pressures.
Figure 4.14 shows the load-displacement graph of the case under 50, 100
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CHAPTER 4 EXPERIMENTS ON SCALE MODELS
cable tension [N]
160
140
120
100
80
60
40
20
cable 3
cable 2 & 4
cable 1 & 5
0
0 50 100 150 200 250 300 350
load [N]
cable tension [N]
160
140
120
100
80
60
40
20
cable a & d
cable b & c
0
0 50 100 150 200 250 300 350
load [N]
Figure 4.13: The tension in the cables of case 10 in relation to the applied
load. (Numerical results).
load [N]
350
300
250
200
150
100
50
200 mbar
100 mbar
050 mbar
0 0.5 1 1.5 2 x 10 −5
0
average displacement [m]
Figure 4.14: The load-displacement graph of the case 11 under 50, 100
and 200 mbar (loaded with five point loads in the hinges) (Numerical
results).
and 200 mbar. As long as all cables are pretensioned, all curves have the
same stiffness. Because the pretension in the cable is dependent of the internal
pressure (equation 4.1), the case with the lowest internal pressure experiences
as first a slack cable and thus another stiffness.
When all cables are pretensioned and thus able to take compressive forces,
the deployable Tensairity beam has the same stiffness as a truss (with the
same configuration of diagonals and with the same sections). This can be seen
in figure 4.12. Once a cable does not contribute anymore to the structural

4.4 CABLE CONFIGURATIONS
behaviour, the bar structure becomes a ‘mechanism’. This is illustrated in
figure 4.15. However, the deployable Tensairity beam does not collapse
immediately since it is still supported by the airbeam. This is why the stiffness
of the beam is similar to the stiffness of an airbeam after the first cables are
slack (figure 4.12).
Figure 4.15: Once a cable does not contribute anymore to the structural
behaviour, the bar structure becomes a ‘mechanism’.
Thus, a relation between the structure’s load-displacement behaviour and the
contribution of pretensioned cables to this structural behaviour is shown. The
cables of the deployable Tensairity beam are pretensioned when the airbeam is
inflated. When both diagonal and vertical cables are present, only the vertical
cables become tensioned. These tensioned cables are able to take compressive
forces, by the same amount as their initial pretension. This has as result
that these cables avoid the hinges to deflect under compression. Or in other
words, the pretensioned cables ‘block’ the hinges. Once the external load has
reached the value whereby the value of the pretension becomes zero in at least
one cable, the hinge is not blocked or supported anymore by this cable. The
hinge will experience larger displacements and the stiffness of the deployable
Tensairity beam decreases.
Now the contribution of cables to the structural behaviour is understood,
the influence of the various cable configurations on the structure’s loaddisplacement
is studied.
4.4 CABLE CONFIGURATIONS
The deployable Tensairity beams with various cable configurations are investigated
numerically and experimentally. Note that the experimental and
numerical results are not compared quantitative, but qualitative. After all, the
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CHAPTER 4 EXPERIMENTS ON SCALE MODELS
values do not correspond due to imperfections in the two meter prototype
(movement of the struts in the pockets), approximations in the finite element
model (perfect connection between membrane, struts and cables), and a slight
different geometry of the section of the airbeam in the FEA (a higher value for
α is chosen to achieve higher cable tensions for reasons of clarity). The next
sections explain the effect of diagonal and vertical cable configurations on the
structural behaviour.
4.4.1 VERTICAL CABLES
The load-displacement behaviour of case 6, containing only vertical cables,
is investigated numerically. The value of the pretension of the cables after
inflation is identical as in case 10. This is because in case 10 only the vertical
cables were pretensioned during inflation.
However, the load-displacement behaviour of these two cases is different.
After all, no diagonals are present in case 6 to contribute to the behaviour under
loading. This has as result that the structure experiences a larger deformation.
Figure 4.16 illustrates the deformation of the structure under point loads on
the five upper hinges. The total load is F, thus each point load has a value of
F
5 . Because a part of the external loading is taken by the deformation of the
airbeam, cables 1 and 5 typically become slack when4 Fpre = 1
3Fext or Fext = 3Fpre.
The load at which the cable - pretensioned by 74,5 N - becomes slack is 223,5 N,
as can be seen in figure 4.17. This figure shows the load-displacement of case
6 and the cable tensions during loading.
4.4.2 DIAGONAL CABLES
Cases 2, 3, 4 and 5 are investigated experimentally and numerically to determine
the effect of the configuration of the diagonal cables on the loadbearing
behaviour. Figure 4.18 shows the load-displacement of the cases
under distributed load, derived from the finite element analysis. It is clear
that the configuration of the diagonal cables has an influence on the stiffness
of the structure. Cases 4 and 5 have similar load-displacement behaviour and
are less stiff than cases 2 and 3.
Before explaining the behaviour of the different cases, it is useful to discuss
the action of the external loading on the cables. After all, depending on the
configuration of the cable, it will experience compression or tension when
4 F There is assumed that cable 1 bears 5 + 1 2 F 5 + 1 6 F 5 = 1 3 F. The first term comes from the point
load on cable 1, the second term from the point load on cable 2 and the third on cable 3. The
numerical results (verified at different pressures) confirm this assumption.

load [N]
350
300
250
200
150
100
50
4.4 CABLE CONFIGURATIONS
Figure 4.16: Deformation of case 6 under loading, caused by the absence
of diagonal cables.
case 6 − p = 100 mbar
0
0 0.01 0.02 0.03 0.04 0.05
average displacement [m]
cable tension [N]
160
140
120
100
80
60
40
20
cable c
cable b
cable a
0
0 100 200
load [N]
300 400
Figure 4.17: The load-displacement of case 6 and the cable tensions
during loading. (Numerical results).
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CHAPTER 4 EXPERIMENTS ON SCALE MODELS
case 2
case 3
case 4
case 5
load [N]
350
300
250
200
150
100
case 2
case 3
50
case 4
case 5
0
0 0.02 0.04 0.06 0.08
average displacement [m]
Figure 4.18: The load-displacement of cases 2, 3, 4 and 5 under five point
loads and with an internal pressure of 100 mbar. (Numerical results).
loading and this has an influence on the load-bearing behaviour, as mentioned
in section 4.3. Whether the pretensioned cable is tensioned or compressed under
external loading is determined by means of the finite element calculations
and by observing the tension in the cables throughout the experiments: a loose
cable indicates that the cable is ‘compressed’. Both methods have the same
result and correspond to the signs of the normal forces in a conventional truss
with similar diagonal configurations under distributed load. The sign of the
forces are illustrated in figure 4.19. A ‘+’ means that the pretensioned cable
experiences tension during loading, a ‘–’ indicates that this cable is compressed.
- + + - + - - +
case 2
- - - -
case 3
+
tension
-
compression
+
case 4
+
case 5
Figure 4.19: The tension (+) and compression (–) forces from the cases
with diagonal cables indicated. (Isostatically supported)
The lower stiffness of cases 4 and 5 is not caused by the action of the external
loading on the cables, but has to be attributed to the cable configuration, and
more precise to the deformation of the first and last ‘mesh’ of the truss. After
all, this has the shape of a parallelogram and deforms much easier than the
triangles in cases 2 and 3. Figure 4.20 illustrates this. This can also be seen
+
+

4.4 CABLE CONFIGURATIONS
in figure 4.21 (left), where the displacement of the first hinge of each case is
plotted throughout loading: the hinge of cases 4 and 5 experience a larger
displacement. Thus in cases 4 and 5, the stiffness of the structure is not really
defined by a prestressed cable that becomes slack at a certain load, but by a
geometrical deformation that is made possible by the cable configuration. In
fact, the cables of case 5 did not become slack during the loading.
This behaviour can also be seen in the experimental results. Figure 4.21 (right)
plots the displacement of the first hinge of the four cases. Also here, the hinge
of cases 4 and 5 experience a larger displacement than cases 2 and 3.
load [N]
350
300
250
200
150
Figure 4.20: The unstable shape of the first mesh of cases 4 and 5 causes
a deformation of the truss mechanism. Case 5 is shown.
numerical − p = 100 mbar
100
case 2
case 3
50
case 4
case 5
0
0 0.02 0.04 0.06 0.08
displacement [m]
load [N]
350
300
250
200
150
experimental − p = 100 mbar
100
case 2
case 3
50
case 4
case 5
0
0 0.005 0.01 0.015 0.02
displacement [m]
Figure 4.21: The first hinge of cases 4 and 5 deflects more than cases 2
and 3 because of the unstable shape of the first mesh. Left: numerical
results, right: experimental results.
When having a closer look at the load-displacement behaviour of cases 2
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CHAPTER 4 EXPERIMENTS ON SCALE MODELS
and 3 (figure 4.18), one can see that case 2 is initial stiffer than case 3, but
changes stiffness at a lower load. The higher stiffness of case 2 can also here be
attributed to the configuration of the cables: they are positioned such that all
meshes in the ‘truss’ are triangles. In case 3, the second and fourth meshes are
parallelograms and deform easier, hence the lower stiffness. Because case 2
possess cables that are tensioned when the structure is loaded, the pretension
in the compressed cables reaches zero value at a lower load. This can be seen
in the graph by the change in stiffness of case 2 at a lower load (100 N).
To summarize, the investigation of the diagonal cables revealed that the
displacement behaviour of the structure is strongly influenced by the shape of
the first mesh of the truss. The shape of a parallelogram should be avoided,
since large displacements will occur at the level of the first hinge. The diagonal
cables are pretensioned under inflation, when no vertical cables are present.
Diagonal cables versus bars
case 11
case 12
case 13
case 14
load [N]
350
300
250
200
150
100
case 11
case 12
50
case 13
case 14
0
0 5 10 15 20 25
average displacement [mm]
Figure 4.22: The average displacement in relation with the applied load
of the cases with diagonal bars. The cases are uniformly distributed
loaded, each load step represents 50 N. (Experimental results).
Experiments are also conducted on cases with diagonal stiff bars instead of
cables. Cases 11, 12, 13 and 14 are the analogue of respectively cases 2, 3, 4
and 5. Note that only case 11 is deployable. Figure 4.22 shows the average
displacement in relation to the applied load. From the graph can clearly be
seen that two cases (13 and 14) have much larger displacement than the other
two configurations. As already mentioned in the previous section, this is
because of the shape of the first mesh of the strut, formed by the outer struts

4.4 CABLE CONFIGURATIONS
and the first diagonal (illustrated on the right of figure 4.21). After all, this
parallelogram cannot retain its shape as in the case of a triangle and thus
experiences larger deformation.
When comparing the configurations with diagonal bars and cables, interesting
similarities can be detected. Figure 4.23 shows the deflected configurations
under the same distributed load. In the case of configurations 10 and 14, all
connecting elements are under tension. As a result and as can be expected,
the deflected shape is identical. The figure also shows clearly the larger
deformation at the location of the first hinge of cases 13 and 14 due to the
presence of the unstable parallelogram. For cases 2 and 3, it is obvious that
their counterpart with diagonals, able to take compression, is much stiffer.
displacement [mm]
displacement [mm]
0
10
20
30
40
0 1 2 3 4 5 6
x position along axis [mm]
0
10
20
30
case 2 case 11 case 3 case 12
40
0 1 2 3 4 5 6
x position along axis [mm]
displacement [mm]
0
10
20
30
40
0 1 2 3 4 5 6
x position along axis [mm]
case 4 case 13 case 5 case 14
displacement [mm]
0
10
20
30
40
0 1 2 3 4 5 6
x position along axis [mm]
Figure 4.23: Comparison of configurations with diagonal cables and
struts. (Experimental results).
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CHAPTER 4 EXPERIMENTS ON SCALE MODELS
4.4.3 DIAGONAL AND VERTICAL CABLES
Cases 7, 8, 9 and 10 contain vertical cables in every hinge combined with the
diagonal cables of resp. cases 2, 3, 4 and 5. Their load-displacement behaviour
is investigated numerically and plotted in figure 4.24.
case 7
case 8
case 9
case 10
load [N]
350
300
250
200
150
p = 100 mbar
100
case 7
case 8
50
case 9
case 10
0
0 0.005 0.01 0.015 0.02
average displacement [m]
Figure 4.24: Load-displacement behaviour of cases 7, 8, 9 and 10 at
pressure of 100 mbar and loaded distributed (point load in each upper
hinge). (Numerical results).
From the graphs can be concluded that cases 9 and 10 are stiffer than cases
7 and 8. This is remarkable, because this is the opposite of their analogue
cases without vertical cables (figure 4.18). The explanation of this different
behaviour can be found in the pretension of the cables. After all, as seen in
section 4.3, the diagonal cables are not pretensioned due to inflation when
vertical cables are also connecting the hinges. This means that under loading,
only the tensioned diagonal cables contribute to the stiffness. The diagonals
under compression can be considered as non-existing. Figure 4.25 illustrates
the cases without the compressed diagonals and their deformed shape.
From this figure becomes clear that the presence of a mesh with the shape of a
quadrangle (here mainly a rectangle or parallelogram) weakens the structure.
In the cases 7 and 8 is this mesh near the end supports, which results in a large
decrease of stiffness. In fact, case 8 has identical load-bearing behaviour as
case 6. Cases 9 and 10 are stiffer, because they have triangular meshes near
the end supports. Case 10 is the stiffest case, since it contains only triangular
meshes under loading. This results in a stiffness identical to that of a truss
with the same configuration, as illustrated in figure 4.12.

7
8
9
10
4.4 CABLE CONFIGURATIONS
Figure 4.25: The diagonals that are compressed during loading can be
considered as non-existing. The cases without the compressed diagonals
and their deformed shape.
This contribution of the vertical and diagonal cables to the stiffness of the
deployable Tensairity beam is also investigated experimentally. Figure 4.26
shows the stiffness and the deflection of cases 10 (vertical + diagonal cables),
case 6 (only vertical cables) and case 5 (only diagonals). As can be expected,
the case with both diagonal and vertical cables has a larger stiffness than the
other two cases.
load step
7
6
5
4
3
case 5 case 10 case 11
2
1
case 5
case 10
case 11
0
0 5 10 15 20 25
average displacement [mm]
displacement [mm]
0
5
10
15
20
25
30
0 1 2 3 4 5 6
postition along x axis [mm]
Figure 4.26: The stiffness and deflection of the cases 5, 6 and 10.
(Experimental results).
4.4.4 SUMMARY
The numerical and experimental investigations show clearly the effect of the
configuration of cables on the load-displacement behaviour. In brief, there can
be concluded that the shape of the ‘mesh’ - formed by the pretensioned cables
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CHAPTER 4 EXPERIMENTS ON SCALE MODELS
and the cables that will become tensioned under loading - has a large influence
on the structure’s stiffness. If this mesh is a rectangle or parallelogram, it can
easily be deformed under external loading and causes larger displacements of
the structure. The more this quadrangle mesh is located at the end supports (as
in the case of 4 and 5) the higher the impact of this mesh on the deformation.
To give an overview and to compare, the load-displacement of cases 2 to 10 is
plotted in figure 4.27.
4.5 SHAPE
load [N]
350
300
250
200
150
distributed load − p = 100 mbar
100
case 2
case 3
case 4
case 5
case 6
50
case 7
case 8
case 9
case 10
0
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
average displacement [m]
Figure 4.27: Load-displacement of cases 2 to 10. (Numerical results).
Until now, the experiments are all conducted on the foldable truss system
with cylindrical airbeam. However, as already mentioned in the previous part,
research showed that the spindle shaped Tensairity beam has a better loadbearing
behaviour than the cylindrical one (Pedretti et al., 2004). Therefore, a
two meter deployable spindle is investigated experimentally. The upper and
lower strut of the spindle, having a section of 6 mm width by 10 mm height,
contain five evenly distributed hinges. No connecting cables between upper
and lower hinges are applied here. Figure 4.28 shows a picture of the investigated
spindle, as well its load-displacement behaviour and deflection. For
reasons of comparison, the stiffness and deflection of a cylindrical deployable
Tensairity beam are included. From the graphs can clearly be seen that the
spindle shaped beam has a larger stiffness than the cylindrical.

load step
7
6
5
4
3
2
1
0
cylinder
spindle
0 10 20 30 40
average displacement [mm]
displacement [mm]
0
10
20
30
40
4.6 CONCLUSIONS
0 1 2 3 4 5 6
postition along x axis [mm]
Figure 4.28: The investigated spindle and its load-displacement be-
haviour and deflection under distributed load. Each load step represents
50 N.
During the experiments on the spindle shaped structure, some problems were
encountered. Since the upper strut is now loaded as an arch under distributed
loading, the bending moment forces the struts near the supports to bend
outwards. Because the investigated model possesses a hinge in this zone and
because this hinge is not well supported by the membrane, the struts ‘buckled’
outwards. After all, due to the smaller radius of the hull at the ends of the
spindle beam, the membrane stress is at this place lower. This implies that the
strut and hinge, positioned in a pocket, are less supported by the membrane
and can thus more easily buckle outwards. This ‘buckling’ of the strut can be
seen on the deformed shape of the strut in figure 4.28 (right), indicated with an
arrow. Another observation during the experiments is that under distributed
loading, the spindle had the tendency to rotate. After all, the centre of gravity
lies on the same line as the supports, which has as consequence that the smallest
eccentricity out of the plane of the loading is sufficient to rotate the structure.
From a structural point of view, the spindle shaped deployable Tensairity beam
proves to be more efficient than the cylindrical. However, the specificities of
the shape have to be taken into account when developing further.
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4.6 CONCLUSIONS
Before simulating or developing a large scale model for a deployable Tensairity
beam, it is necessary to gain some first insights in the structural behaviour
of this complex synergetic structure by means of small scale experimental and
numerical investigations. Therefore, this chapter presents the experiments on
two meter long cylindrical and spindle shaped deployable Tensairity beams
with various configurations.
The experiments show us that the location of applying the load has an influence
on the outcome of the experiment. Because of the compressibility of the
airbeam, the results are different when the load is applied at the top or bottom
of the beam. This is unlike conventional structural elements such as wooden,
concrete or steel beams. Also different from these conventional elements is
that the displacements of the loaded structure should be noted in the second
or third load cycle. After all, the structure first adapts to the new load condition,
which results in a residual displacement of the unloaded structure after the
first cycle.
To detect the influence of several parameters - such as the internal pressure,
the amount of hinges in the struts and the configuration of cables - on the
structural behaviour of the beam, various configurations were investigated.
The experimental results show that the internal pressure influences the stiffness
of the structure, as is also the case for Tensairity beams with continuous
compression element. A higher internal pressure results in a stiffer structure.
Regarding the influence of the hinges in the strut, no conclusions can yet be
derived. This is because the amount of hinges was not varied enough in the
experiments.
Numerical simulations and analytical approximations revealed the contribution
of pretensioned cables to the structure’s load-displacement behaviour. As
long as the cables connecting upper and lower hinges are pretensioned due to
inflation, they can take compressive forces. This means that these cables can
support (or ‘block’) the hinges. Once the pretension in a cable is zero, the hinge
is not supported anymore and the structure experiences larger displacement.
This has as result that the stiffness of the deployable Tensairity beam changes
when one cable becomes slack.
Several cable configurations connecting the upper and lower strut of the
airbeam are investigated experimentally and numerically. The study showed
that the shape of the ‘mesh’ - formed by the pretensioned cables and the

4.6 CONCLUSIONS
cables that will become tensioned under loading - has a large influence on the
structure’s stiffness. If this mesh is a quadrangle, it can easily be deformed
under external loading and cause large displacements of the structure. When
these deformations occur, the deployable Tensairity beam is supported by
the airbeam and has stiffness similar to that one of an airbeam. When all
cables are prestressed and thus able to take compressive forces, the deployable
Tensairity beam has the same stiffness as a truss (with the same configuration of
diagonals and with the same sections). The stiffest configuration is constituted
of diagonal cables that are tensioned during loading and vertical cables.
This configuration contains, as long as the verticals are pretensioned, only
triangular stable meshes.
There can be concluded that the configuration influences the structure’s stiffness.
Vertical cables have more influence on the stiffness than diagonals, also
for asymmetrical load cases. In the case of the cylindrical foldable truss, it is
beneficial for the stiffness to have the first mesh of the truss being a triangle
and not a parallelogram. In the latter case, the first hinge will experience large
displacements. Applying struts as connecting element increases the stiffness
when they are positioned such that they will be loaded in compression. If not,
the stiffness will be identical to the configuration with cables and only weight
will be added.
In addition to the cylindrical beam, a spindle shaped deployable Tensairity
structure is investigated. The results show an increase in stiffness in comparison
with the cylindrical beam (with similar distance between upper and
lower strut in its middle). The observations during the experiments reveal
also some consequences inherent to this more optimal shape. Because of the
bending moment of the curved strut under distributed load, the position of the
hinges has an influence of the structure’s behaviour and should be investigated
thoroughly.
Now the first qualitative insight of the structural behaviour of a deployable
Tensairity beam is gained with these scale models, it is time to investigate more
configurations on a larger structure by means of simulations in finite element
analysis. This is done in the next two chapters: the finite element modeling is
considered in chapter 5, the results are presented and discussed in chapter 6.
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Numerical Investigations
5
Conducting experiments on Tensairity structures is a very instructive way of
gaining insight in this new structural concept. However, these experiments
are time consuming and thus not always feasible when many different configurations
have to be investigated. Besides, there are also often manufacturing
imperfections involved with these experiments which can cause difficulties in
comparing experimental results from different models. This can be avoided
by investigating the structure numerically by means of finite element models
which simulate the structural behaviour. It is thus essential to ‘build’ the
finite element model as accurate as possible in order to have reliable results.
Therefore, it is important to know how the structure is modeled, which
materials are used, which assumptions are made, etc. This chapter discusses
all these aspects regarding the finite element analysis of (deployable) Tensairity
structures.
5.1 FINITE ELEMENT ANALYSIS OF TENSAIRITY STRUCTURES
The finite element analysis (FEA) method is a computational technique whereby
approximate solutions of a variety of “real-world” engineering problems can
be calculated. This is also the case for the structural behaviour of Tensairity
structures.
As far as the finite element analysis of Tensairity structures is concerned,
some numerical research is published. Pedretti et al. (2004) introduces the
simulation of cylindrical Tensairity structures by means of the commercial
finite element code ‘Ansys’. He discusses the modeling of the compression
element, the cables and the membrane, as well as the interaction and contact
between these materials. Pedretti also addresses convergence issues as a
result of the membrane-element (able to resist only tension) and how to avoid
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this. Luchsinger and Crettol (2006b) compare the numerical, experimental
and analytical results of a spindle shaped Tensairity beam. The used finite
element model is an improvement of the one Pedretti presented, more precise
by using a more specific shell-element 1 instead of a membrane-element for the
membrane. Although Luchsinger models the membrane as linear isotropic
material, the experimental non-linear load displacement behaviour of the
structure is well reproduced by the numerical results. Plagianakos et al. (2009)
and Wever et al. (2010) used in the finite element model of a spindle-shaped
Tensairity column the same element types as Luchsinger, but modeled the
membrane with linear orthotropic properties. Also here, good agreement
between the numerical and experimental results is observed.
The finite element models in this research use the same material types and
parameters as in the research of Plagianakos. The material properties of the
membrane however are derived from mean values of biaxial testing measurements
conducted on the same fabric as the one that will be used for
the prototype in Part III. Besides, these models will primarily be used for a
qualitative comparison. This means that the value derived from a simulation
on itself is not the main focus (quantitative), but that comparing different
models should give us insight in the structural behaviour of the Tensairity
structure.
There are three main steps in a typical finite element analysis: modeling
the structure (also known as preprocessing), solving the finite model and
reviewing the results (postprocessing). Reviewing the results is subject of
the next chapter. modeling and solving the Tensairity structures in FEA is
discussed in the next sections.
5.2 MODELING THE TENSAIRITY BEAMS
The purpose of a finite element analysis is to recreate numerically the behavior
of an actual engineering system. In other words, the analysis must be an
accurate numerical model of a physical prototype. However, small details that
are unimportant to the analysis should not be included in the finite element
model, since they will only make the model more complicated than necessary
and increase the calculation time.
1 The applied shell-element (with membrane option activated) is a 4-node element with
translational degrees of freedom. The element is well-suited for linear and large strain nonlinear
applications. It accounts for load stiffness effects of distributed pressures.

5.2 MODELING THE TENSAIRITY BEAMS
To gain insight in the structural behaviour of deployable Tensairity structures,
various cases are modeled and compared. These cases however differ often
only in one parameter from each other, for example the amount of hinges, the
section of the strut, etc. This means that with exception of these parameters, the
modeling explained here holds true for all cases. modeling Tensairity beams
comprises creating and specifying the geometry, all the nodes, elements, material
properties, real constants (thickness and cross-sectional area), boundary
conditions, and other features that are used to represent the physical system.
5.2.1 CREATING THE MODEL GEOMETRY
The spindle shaped Tensairity beam has a length of five meter and a maximal
diameter in the middle of 50 cm, representing a slenderness of ten. Figure 5.1
shows the Tensairity beam.
Figure 5.1: The Tensairity beam in the finite element software Ansys.
The model is built up from the bottom, beginning by defining the lowest-order
model entities, keypoints. By connecting these keypoints, lines, areas and
volumes can then be defined. First, the keypoints are positioned to construct
the spindle shape and the struts. Their coordinates are calculated by means of
an arc of a circle with radius of 12,625 m and subtending angle of 0,3984 rad.
Figure 5.2 shows this. The keypoints are then connected with lines. The upper
line will be used to model the upper strut, the lower line for the lower strut.
In the case of a spindle with one circular air chamber (and thus no cables
connecting upper and lower strut), the upper line is rotated around the axis
connecting the ends of the arc. As a result, an area is constructed representing
the hull of the membrane, see figure 5.2. In the case of an airbeam with
cables connecting upper and lower strut, the section of the airbeam is no circle
anymore. Therefore, arcs are applied which determine the shape of the hull.
The spindle with the two hull shapes is shown in figure 5.3.
From the figure 5.2 can be seen that the hull of the airbeam ends in a point
at the supports. This should be avoided, since converging issues occur due
to local phenomenon. Besides, end pieces like this are hard to manufacture
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CHAPTER 5 NUMERICAL INVESTIGATIONS
Figure 5.2: Some snapshots during the modeling of the Tensairity beam
with circular cross section.
Figure 5.3: The spindle with the two hull shapes: a circular section and
a section composed of two circle segments.
and will normally not be used in practice. The ends are usually finished by
means of round end caps, as illustrated in figure 5.4. To reduce the modeling
complexity and ‘gain’ calculation time, there is chosen not to model the ends
of the membrane hull, but to apply an equivalent tangential line loading along
the ends. Figure 5.5 represents this. Numerical investigation of both cases
shows that their structural behaviour is identical and the approximation with
an equivalent line load justified is.
Many of the investigated finite element models should also contain hinges.
These hinges are modeled by positioning two keypoints at the same location.
The line (strut) at one side of the coincident points will contain a different
one than the strut at the other side. The nodes have coupled degree of freedom,
which means that these nodes are forced to take the same translational

5.2 MODELING THE TENSAIRITY BEAMS
Figure 5.4: The ends are usually finished by means of round end caps.
Figure 5.5: The ends of the membrane hull are replaced by the action of
an equivalent tangential line loading.
displacement. Of course, they have different rotational degree of freedom
(around the three main axes) in order to ‘create’ the hinge.
The Tensairity structure is in all investigated models statically determined:
both endpoints allow rotation around the z-axis (axis perpendicular to the
plane), whereby one endpoint also has a translational degree of freedom in the
x-direction (longitudinal direction).
5.2.2 MESHING THE MODEL
Analytical solutions give at any point within a system the calculated value
(like a displacement). In contrast, numerical calculations approximate the
exact solutions only at discrete points, called nodes. Therefore, the model
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needs to be discretized, also known as ‘meshed’. This process divides the
model into a number of nodes and small subregions, called elements.
Analytical solutions show the exact behaviour of a system at any point within
that system. In contrast, numerical calculations approximate the exact solutions
only at discrete points, called nodes. Therefore, the model needs to be
discretized, also known as ‘meshed’. This process divides the model into a
number of nodes and small subregions, called elements.
The size of these subregions and thus the mesh density is very important in
finite element calculations. If the mesh is too coarse, the approximate solution
can contain errors. A mesh which is too fine on the other hand, results in a
excessively long calculation time. To avoid this, an appropriate mesh density
should be used.
The question that rises is of course what an appropriate mesh density is. To
find this out, a first analysis is performed with what seems a ‘reasonable’
mesh. Then, the analysis is reanalyzed using twice as many elements and the
results of these two analysis are compared. The mesh is most likely adequate
if the two meshes give nearly the same results. If the results are substantially
different, further mesh refinement is required until nearly identical results for
succeeding meshes are obtained. It is of course also useful and wise to compare
the numerical result with a known accurate analytical result, if possible and
existing.
In order to study the effect of mesh refinement for a Tensairity structure, four
different mesh densities were applied on the fabric hull: 40 × 6, 60 × 12, 120
× 24 and 240 × 48 elements along the axial and radial direction respectively.
Figure 5.6 illustrates the cases. The central membrane hoop stress after inflation
and the central displacement under distributed load (200 N
m ) were noted and are
given in table 5.1. As can be seen from the table, all mesh densities produce
nearly identical results regarding the middle displacement. The analytical
value, derived from the analytical model in Luchsinger and Teutsch (2009),
corresponds with the calculated values. The hoop stress on the other hand
is affected by the mesh density. A larger element size produces a coarser
approximation. This is because the hoop stress in the centre of the beam is the
maximal hoop stress of the hull. The larger the element size, more areas with
a lower stress are comprised in this element, which results in a lower mean
value. When comparing the results with the analytical value of the hoop stress
for a cylindrical airbeam2 , it is clear that a denser mesh size approximates
2 The analytical value for the hoop stress of a cylindrical airbeam is nhoop = p.r, with n hoop being

5.2 MODELING THE TENSAIRITY BEAMS
Figure 5.6: Four different mesh densities were applied on the fabric hull
to study the effect of mesh refinement for a Tensairity structure.
better the theoretical value.
There can be concluded that the simulation of the structural behaviour is
rather insensitive to mesh refinement of the hull, in contrast with the membrane
stress. Because in this research, the focus will mainly be on the loaddisplacement
behaviour of the structure, a ‘coarser’ mesh size is sufficient to
obtain reliable results. The mesh with 60 × 12 elements is implemented in all
investigated finite element models. Regarding the struts, a non-uniform mesh
was applied being finer near the ends of the beam to avoid approximation
errors at the ends due to the coarseness of the mesh.
Table 5.1: Results reveal very little influence of the investigated mesh
densities on the value of the middle displacement of the beam.
40 × 6 60 × 12 120 × 24 240 × 48 analytical
middle displacement [mm] 2,92 2,96 2,98 2,98 2,9
hoop stress [ N ] 3202 3570 3688 3700 3750
m
After creating nodes and elements by means of meshing, the nodes of struts
and hull are merged together. This implies a perfect connection between these
components. Luchsinger and Crettol (2006b) investigated by means of finite
element analysis for a spindle shaped Tensairity beam under central point load
the influence on the structural behaviour of an imperfect connection between
the membrane stress [ N m ], p the air pressure and r the radius.
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struts and hull. Results show that the displacement at the compression side of
the model with imperfect connection is almost a factor of two larger compared
to the model with perfect connection. On the tension side is this effect
even more pronounced. A tight connection of the struts and hull can thus
significantly stiffen the Tensairity spindle. Physical models will therefore be
manufactured with a good connection between the elements. Regarding the
numerical model, a tight connection between hull and struts is thus assumed
in all cases, modeled by means of merging the coincident nodes of membrane
and strut.
Meshing does not only discretizes the structural components, but also assigns
the element type, element real constants and material properties to the elements.
5.2.3 ASSIGNING ELEMENTS AND MATERIALS
The appropriate element attributes need to be defined and assigned to the respective
elements. The element attributes comprises the element type, the real
constants and the material properties. The element type determines, among
other things, the degree of freedom set and whether the element lies in 2D or
3D space. The element’s geometric properties, such as thickness and crosssectional
area, are categorized under the real constant set. Material properties
include for instance the Young’s and shear modulus, poisson coefficient and
the density.
The finite element model of a Tensairity structure in ansys consists of shell
elements for the membrane, beam elements for the struts, and link elements
for cables when needed.
The fabric hull is modeled by four-node shell elements (shell 181) whose
bending stiffness is not taken into account in the solution by means of a feature
provided in the element properties. The 0.85 millimeter thick membrane,
which is a coated woven fabric, is modeled with linear elastic orthotropic
properties. In fact, the membrane should be modeled as non-linear, but this
would increase the complexity of the finite element model considerably. The
approximation with a linear model however shows fairly good agreement with
experimental results ((Luchsinger and Crettol, 2006b; Plagianakos et al., 2009)).
The struts are represented by linear two-node Timoshenko beam elements
(beam 188), where shear deformation effects are included. The sections have
a full rectangular shape. Some investigated models have cables connecting
the upper and lower strut . These cables are modeled by two-node three

5.3 SOLVING THE MODELS
dimensional elements (link 10) with the tension-only option activated, which
means that stiffness is removed if the element goes into compression. The
material properties for fabric (PVC-coated polyester fabric), struts (Aluminum)
and cables (Steel) are listed in table 5.2.
Table 5.2: In-plane properties of materials used in the finite element
models.
element type E11 [GPa] E22 [Gpa] G12 [GPa] ν12 size [mm]
membrane shell 181 1,06 0,53 2 × 10 −2 0,313 t = 0,85
strut beam 188 69 69 26,2 0,330 b × h = 30 × 10
cable link 10 100 - - 0,300 φ = 5
5.2.4 SCRIPTING THE GENERATION OF THE FINITE ELEMENT MODEL
When building the finite element model in ansys, there can be communicated
with the software by means of the graphical user interface or by means of
commands. These commands can be written in a text file, which is then called
a ‘script’, that can be loaded entirely in the software. The successive commands
are then executed automatically.
The advantage of producing and using these scripts, is that the model does
not has to be generated all over again manually when performing identical
or comparable analysis. Loading the file is sufficient to run the prescribed
list of commands. If the model is slightly different, the script can be altered
quickly where necessary. Script files are also used to parameterize the variables
involved in the construction and analysis of each case so that different designs
can easily be obtained by changing these parameters. Examples of such a
parameters in this research are the load, the pressure and the amount of hinges.
Off course, writing such a parameterized script is time consuming, but permits
to gain a lot of effort and time once it is ‘built’.
All finite element models in this research are built by means of scripted files.
5.3 SOLVING THE MODELS
Now the finite element model is entirely generated, it is almost ready to
be solved. But the loads have to be applied first and the analysis type and
options have to be specified. Then the finite element solution can be initiated.
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5.3.1 APPLYING LOADS
The Tensairity structure is loaded in two successive steps. The internal
overpressure is first applied in the airbeam, then the upper strut is loaded
with a distributed line load. The overpressure is modeled by means of an
outwards surface load on the membrane and is applied in twenty substeps.
Substeps are used in non-linear analysis to apply the loads gradually so that an
accurate solution can be obtained and for convergence purposes. The pressure
forces acts as so called ‘following’ forces, which means that they remain normal
to the surface during displacements under inflation. As already mentioned,
the end caps were replaced by an equivalent tangential line loading along the
ends, with respect to the value of the internal pressure.
After inflation, the distributed line load is applied. This load consists of
vertical point loads on every node of the compression strut. In the case of
an asymmetric load, only the nodes at one side of the strut are loaded. The
point loads on every node are calculated such that the load per length remains
constant. Especially at the ends of the struts has this to be taken into account,
where the nodes are closer to each other because of a finer meshing. Also
the distributed load is applied in discrete substeps, up to the point where the
geometrical non-linear solution diverges.
5.3.2 OBTAINING THE SOLUTION
The combination of compression elements, membranes and cables in Tensairity
make the finite element model by no means trivial. Converging to a solution
and obtaining reliable results is a critical issue in such a nonlinear model.
Therefore it is essential to specify the principles that have to be taken into
account in the solving. In all finite element models in this research, the
structural static analysis is specified to take geometric nonlinearities and the
effects of stress stiffening into account. Geometric nonlinearities refer to the
large deflection effects which have to be considered during the solving. Stress
stiffening is the out-of-plane stiffening of a structure due to its in-plane-stress
(e.g. the out-of-plane stiffness of a cable or membrane is dependent on its inplane-stress).
This coupling is called stress stiffening and is taken into account
in all analyses.
The solving can start once the options are specified. Due to the tension-only
behavior of the membrane and cables, the problem is nonlinear and Newton-
Raphson 3 iterative method is utilized to solve the nonlinear FE equations.
3 The Newton-Raphson method works by iterating the equation [Kt].{u} = {Fa} − {Fnr}, where

5.4 DISCUSSION
As already mentioned before, converging to a solution is a critical issue in such
a nonlinear model. Once the system matrix has a singularity, the solving is
terminated and no convergence can occur. Singularities in the solution process
may be caused by cables, membranes, etc. When for example wrinkling in a
membrane occurs, the structure of the matrix experiences a singularity. In
reality however, the structure still has approximately its full load bearing
capacity when the first wrinkles appear. The load at the last converging
step is thus not the real maximal load. It is therefore with this analysis very
hard to find a reliable value for the maximal load the structure can bear. As
a consequence, this research does not focus on this value, but investigates
the load-displacement behaviour of the various models. If one would like
to calculate the maximal load the structure can bear – mostly this load will
initiate global buckling – finite element investigations should be conducted
with ‘explicit’ finite element analysis tools, like eg. ‘Abaqus/Explicit’. The
software used in this research, Ansys, calculates according to the rules of
implicit analysis methods 4 .
5.4 DISCUSSION
Conducting experiments is very time consuming, especially when the influence
of many parameters on the structural behaviour needs to be investigated.
Therefore, the behaviour of structures can be simulated by means of finite
element calculations. However, before conducting these investigations, it is
important to determine how the structure is investigated and which assumptions
and approximations are made.
In this chapter, the modeling and solving of deployable Tensairity structures
in the finite element software ‘Ansys’ are briefly described.
The development of the spindle shaped Tensairity beam is described step by
step. To reduce the modeling complexity and gain calculation time, there is
chosen not to model the ends of the membrane hull, but to apply an equivalent
tangential line loading along the ends. Also, investigations revealed that an
appropriate mesh density is chosen, which makes the results more reliable. The
mesh density is preferably chosen to be not too dense, since this increases the
[Kt] is the tangential stiffness matrix, {Fa} is the applied load vector and {Fnr} is the internal load
vector, until the residual, {Fa} − {Fnr}, falls within a certain convergence criterion. The Newton-
Raphson method increments the load a finite amount at each substep and keeps that load fixed
throughout the equilibrium iterations (Moaveni, 2008).
4 Implicit: u t+∆t = f (u t+∆t , u t ) - iterative, convergence not guaranteed, ∆t is not limited;
Explicit: u t+∆t = f (u t ) - solved directly, no need to converge, ∆t is limited.
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calculation time. The hinges are modeled by means of two coincident points
that share identical translational degrees of freedom, but different rotational
degree of freedom. Furthermore, this chapter describes which materials are
applied in the research.
The Tensairity structure is loaded in two successive steps. The internal
overpressure is first applied in the airbeam, then the upper strut is loaded
with a distributed line load. By applying the load in discrete substeps, large
deformations between successive steps are prevented to avoid divergence of
the calculations. Divergence occurs when a singularity in the model occurs,
for example wrinkling of the membrane. As a consequence, with these finite
element calculations, it is not possible to calculate the realistic maximal load.
Many simulations are conducted in this research to gain insight in the general
behaviour of Tensairity structures and to monitor the influence of specific
design parameters on the structural behaviour of the deployable beam. These
results are presented in the next chapter.

Analysis of the structural behaviour
6
The design of the deployable Tensairity beam does not only rely on the
foldability constraint. Decisions should also be made from a structural point
of view. It is therefore important to investigate and understand the influence
of the various design parameters on the structural behaviour of the deployable
Tensairity beam. A first study on this is already conducted in chapter 4 by
means of experiments on scale models. The influence of cables and the shape
of the airbeam on the structural behaviour of the deployable Tensairity beam
were discussed.
In this chapter, more parameters and configurations with regard to the deployable
Tensairity beam are studied to understand their influence on its structural
behaviour. The main parameters whose influence will be monitored by means
of numerical investigations are the hinges, the cable configuration, the struts’
and hull section. Figure 6.1 illustrates the different investigated elements in
this chapter. These elements are all interacting with each other in the Tensairity
beam: the hull section has an influence on the cable tension, the cables and
hinges have an influence on the section of the strut, etc. Therefore is chosen for
reasons of clarity to discuss each element separately in a section. The insights
gained will then be brought together at the end of this chapter, discussing the
effect of the cables, hinges, hull and strut section on each other. These insights
will lead to the design of the prototype of the deployable Tensairity beam,
presented in the next chapters.
6.1 HULL SECTION
The section of an inflatable volume always tends to become a circle. This
is because the pressure acts perpendicular to the surface of the outer airtight
hull. This is also the case for a Tensairity structure with no connection between
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cables
hinges
strut section
hull section
Figure 6.1: When two parts of the hull are connected by means of a
tension element (e.g. cable, membrane) shorter than the diameter, the
section will not be a circle anymore, but a combination of two circles.
the upper and lower strut: the standard spindle shaped beam is an object of
revolution and has thus a circular section (with the section plane perpendicular
on the longitudinal axis of the structure). This standard spindle is also called
‘O-spindle’ in this research. When two parts of the hull are connected by
means of a tension element (e.g. cable, membrane) shorter than the diameter,
the section will not be a circle anymore, but a combination of two circles.
Figure 6.2 illustrates this.
By connecting the upper and lower strut of a Tensairity beam with such a tension
element, this element becomes stressed when the airbeam is inflated and
contributes to the structural behaviour of the beam (Breuer et al., 2007; Wever
et al., 2010). After all, this connecting element gives an additional support for
the compression element, increasing the value of the spring constant of the
elastic foundation. This improves the stabilization of the compression element
and the structure’s stiffness. This section investigates what the radius of the
airbeam along the spindle length should be to pretension the connecting web
in such a way that it contributes as much as possible to the stiffness of the
Tensairity beam.
The elements connecting the upper and lower strut of the Tensairity beam can
be cables or membrane and is generally called ‘web’ in this section, no matter
how many elements, the material or the configuration. This web is stressed

inflating
6.1 HULL SECTION
Figure 6.2: When two parts of the hull are connected by means of a
tension element (e.g. cable, membrane) shorter than the diameter, the
section will not be a circle anymore, but a combination of two circles.
due to the action of the inflated hull on this web: by inflating the airbeam,
the hull tends to give the section a circular shape and thus moving the struts
outwards. This way, the web becomes stressed and counteracts this outwards
movement of the struts.
The stress in the hull and the angle between the web and the hull determine
the stress in the web. The stress in the hull depends on its radius and internal
overpressure (see equation 2.3 on page 14). The angle between the web and
the hull is - for a fixed height h - determined by the radius of the hull: the larger
the radius, the larger the angle. Figure 6.3 shows the action from the stressed
hull on the web. The horizontal components of this action, represented by nh,
counterbalance each other. The vertical components, represented by nv, are
superposed and are thus the tension force that act on the web.
Two cases are investigated and compared. In the first model, the ratio h
R is kept
constant along the spindle’s length, whereby height h represents the distance
between the upper and lower strut and R the radius of the hull at one side
of the struts. The second case is modeled in such a way that the vertical stress
in the web (membrane or cables) due to the inflation is constant throughout
the length of the beam. The geometry of the middle section of the beam
of the two cases are chosen identical for comparison reasons. This means
that the stress in the web of the second case is identical to the middle web
105

106
CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR
stress of the first model. Figure 6.4 illustrates the outer hull of the two cases,
figures 6.5 and 6.6 show sections of each case. The shape of the struts - which
defines the ‘longitudinal shape’ of the structure - will not be modified here.
The struts cover a span of 5 m and are in the middle of the beam 0,5 m apart
from each other.
6.1.1 CONSTANT H R -RATIO
The shape of the outer hull is determined by choosing the h
R-ratio. After all, at
each position along the length of the beam is the height h between the upper
and lower strut defined. The radius can then be easily derived from the above
mentioned relation. Figure 6.5 shows the hull of the Tensairity beam at two
positions along the beams length. Because h
R is constant, the angle is the same
for both sections. However, since the stress of the hull is larger in the case of
a larger radius, the tension force F on the web is larger in the case of a larger
radius. This means that in the case of a spindle shaped Tensairity beam with
constant h
R-ratio, the web stress varies with the height (and thus the length) of
the spindle and thus from large in the middle towards smaller at the ends.
6.1.2 CONSTANT STRESS
The spindle shaped Tensairity beam is modeled in such a way that the vertical
stress in the web – membrane or cables – due to the inflation is constant
throughout the length of the beam. Since the pressure is the same in the
whole spindle airbeam, the ratio h
R has to decrease towards the ends. Once the
wanted stress in the web is decided (or once the radius at a certain point of the
spindle (and thus height of the web) is chosen), the radii at other web heights
can be calculated easily. Figure 6.6 shows sections of the Tensairity beam at
two different locations along the beams length. The height and radius are thus
chosen in such a way that the stress in the web is equal in both sections.
Suppose that the radius Ra at a certain height of web ha is given. Then, in order
to have everywhere the same web stress, the radius Rb at spindle height hb can
be calculated by the following expression:
Rb =
R 2 a − h2 a
4
+ h2
b
4
(6.1)
To calculate the normal force F in the web at a certain internal pressure p, the
following expression can be used:
F = 2p.
(R2 − h2
) (6.2)
4

hull
n
α
h R
web
F
n v
n h
n
6.1 HULL SECTION
Figure 6.3: The action of the hull (stress n) on the web. This results in
the tension force F on the web.
Figure 6.4: The outer hull (and a zoom of the ends) of the two
investigated cases. Upper: constant stress, lower: constant h
R -ratio.
107

108
CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR
6.1.3 NORMAL FORCE IN WEB
Membrane as web
The configurations with constant h
R-ratio and constant stress are both numerically
investigated in a spindle shaped beam without hinges in the struts. The
y-component of the stress in the membrane connecting the upper and lower
strut is illustrated in figure 6.7 and visualizes very well the above mentioned
difference between the cases. After inflation and loading, the web is equally
stressed along the spindle length in the ‘constant stress’-case, while this is
not true for the ‘constant h
R-ratio’-case where the web has smaller membrane
stresses at smaller spindle heights.
Cables as web
It is obvious that the Tensairity beam constituted of continuous struts and a
membrane as web is not deployable. Therefore, the two hull configurations
are also investigated with a Tensairity beam containing hinges in the struts
and cables connecting the upper and lower strut. The beam is illustrated
in figure 6.8. The forces acting in the cables throughout the loading are
represented in figure 6.9.
The cases are being inflated to 150 mbar in the first loadstep of the finite element
model. This is shown on the left of the dotted line in the graphs of figure 6.9 and
represented by an increasing cable tension. The cable numbers are indicated
and correspond with the cable numbers in figure 6.8. The cable forces under
distributed loading represent very well the influence of the geometry of the
two cases. The forces in the constant ‘ h
R-ratio case’ are higher for the cables
positioned at a larger height of the spindle. After all, as illustrated in figure 6.5
is the vertical tensile component then larger. The cable forces of the ‘constant
stress’-case are all nearly identical, as expected. Notice that the force in cable 3
is identical for both cases, which was expected since both configurations have
the same hull section in the middle of the beam because they were designed
for an identical web stress at that position.
6.1.4 DISPLACEMENTS AND STIFFNESS
The deformation of the upper strut of the Tensairity beam from figure 6.8 is
monitored during loading. This is done by measuring the displacement of
88 points, distributed evenly over the strut. The average displacement of the
upper strut of the two hull configurations is plotted on the left of figure 6.10.

n a
h a
F a
n a
R a
n b
h b
F b
6.1 HULL SECTION
Figure 6.5: The hull of the spindle shaped beam with constant h
R -ratio.
n a
h a
Fa Fa = Fb Fb n a n b n b
R a
Figure 6.6: The hull of the spindle shaped Tensairity beam is modeled
in such a way that the stress in the web is constant along the length of
the beam.
h b
R b
n b
R b
109

110
CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR
cable tension [N]
1500
1000
500
Figure 6.7: The difference between the two cases can be seen in the
y-component of the stress of the web. Upper: after inflation, lower: after
loading.
1 2 3
Figure 6.8: The Tensairity beam with hinges and cables connecting
upper strut is used in the cases of constant stress and constant h
R -ratio.
The outer hull is not shown.
constant stress
2
1
3
0
0 0.5 1
loadstep
1.5 2
cable tension [N]
1500
1000
500
constant
h
R -ratio.
3
2
0
0 0.5 1
loadstep
1.5 2
Figure 6.9: The cable forces of the two configurations along the loading
show very well the influence of their difference in geometry. The cases
are inflated to 150 mbar and loaded with a distributed load of 5000 N .
1

6.1 HULL SECTION
The average stiffness of both cases is identical until a certain load. At higher
loads is the case with constant web stress more stiff than the case with constant
h
R -ratio.
As discussed in chapter 4 can this behaviour be attributed to the influence of the
pretensioned cables. When all cables in both cases are still pretensioned, their
load-displacement behaviour is identical because the cables support all hinges.
From a certain load on, the less pretensioned cables from the h
R -configuration
(which are the cables at the sides) will become slack and allow the hinge to
displace more, hence the lower stiffness. Figure 6.10 (right) illustrates the
displaced shape of the two cases.
load [N]
7000
6000
5000
4000
3000
2000
1000
constant stress−case
h/R − case
0
0 0.002 0.004 0.006 0.008 0.01
average displacement [m]
constant h
R -ratio
constant stress
Figure 6.10: Left: the average displacement of the upper strut of the
two hull configurations, right: the displaced shape of the two cases at
5000 N. (Displacements are scaled with factor 10. Also displacements
due to inflation, hence the outwards movement of the strut. The outer
hull is not shown).
This investigation showed the influence of the shape of the section of the
airbeam on the forces in the elements connecting the upper and lower strut of
the Tensairity beam. The ‘constant stress’ configuration proves to be the most
optimal, since all cables are tensioned by the same amount. After all, in the
case with a constant h
R-ratio are the cables towards the ends less pretensioned
under inflation. As a result, these cables will become slack under a smaller
load and large displacements will occur. Therefore is chosen to apply the
‘constant stress’-configuration for further numerical models and the physical
prototype.
111

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CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR
6.2 DIMENSIONING THE STRUTS
In this section, the influence of the dimensions of the struts on the structural
behaviour of a deployable Tensairity beam is investigated. After all, the
deployable structure differs from a standard Tensairity beam by containing
hinges and cables that connect the upper and lower strut. No investigations
are conducted yet on the influence of these additional elements on the strut’s
behaviour.
The upper and lower strut of a uniformly vertical loaded standard Tensairity
beam are respectively in compression and tension. The strength, stability and
stiffness have to be considered when dimensioning the strut in compression
while only the strength requirement has to be taken into account for the tension
element. In this research, upper and lower struts are chosen to be identical.
This way, the structure can also bear negative load cases, such as wind suction.
Therefore, only the dimensioning of the compression element is discussed here.
First, the strength and stability requirement are considered. Then the influence
of its dimensions on the global stiffness of the structure is investigated.
6.2.1 STRENGTH AND STABILITY
The strut has to be dimensioned to withstand the compressive forces. They
can be determined by means of finite element models or by analytical approximation.
The compressive force for a cylindrical Tensairity beam can be
approximated by
N = q.L.γ
, (6.3)
8
with q the applied distributed load [ N
m ], L the length of the Tensairity beam [m]
and γ the slenderness [/], defined as the beams length divided by its height
(Luchsinger et al., 2004a). This value approximates also well the axial forces in
the middle of the struts of a spindle shaped Tensairity beam (Luchsinger and
Crettol, 2006b). The compressive forces in the middle of the strut are thus a
linear function of the applied load, but independent of the pressure.
A compressed structural element is prone to buckling. The compression
element of a Tensairity beam is tightly connected to the inflated hull along
its length. This stressed membrane acts as a continuous elastic support for the
strut. The buckling load for such a beam on elastic foundation is P = 2 √ k.E.I
with k the spring constant of the elastic foundation [ N
m 2 ], E the modulus of

6.2 DIMENSIONING THE STRUTS
elasticity [ N
m 2 ] and I the moment of inertia 1 [m 4 ]. In Tensairity structures,
the spring constant depends on the overpressure p of the airbeam and is
determined by Luchsinger et al. (2004a) as p×π. Hence, the maximum buckling
load for a Tensairity beam is
Pbuckling = 2 p.π.E.I (6.4)
This buckling load increases with increasing square root of the pressure and
the moment of inertia of the beam. Notice that it is independent of the length
of the beam. When dimensioning the compression strut, the moment of inertia
for a given material and overpressure can be chosen such that the buckling
load is higher than the yield load. This way, the compression element can be
loaded to its yield limit and the yield load becomes then the limiting factor of
the compression element. This is called buckling free compression (Luchsinger
et al., 2004a). The compressive force is thus
P = σyield.A (6.5)
with σyield the yield stress and A the cross sectional area of the compression
element.
6.2.2 INFLUENCE OF CABLES
Because of the constructive separation of tension and compression, no bending
occurs in the struts of a standard spindle shaped Tensairity beam under
homogeneous load. As a result, the bending stiffness of the struts is in these
cases of little importance and can be neglected when investigating the load
displacement behaviour. The deflection at mid span of such a Tensairity beam
is proposed analytically by Luchsinger and Teutsch (2009). They find that the
deflection is the sum of two terms, one due to the elasticity of the struts and
one due to the deformation of the inflated body which depends on the air
pressure. The bending stiffness of the struts is neglected in this model of a
standard spindle shaped Tensairity beam and comparisons with finite element
results show that this is an allowable approximation.
However, when cables connect the upper and lower strut of the Tensairity
beam, additional forces are introduced. These point loads introduce large
bending moments in the struts and the bending stiffness can not be neglected
anymore. Figure 6.11 illustrates the moment diagrams of four cases. Two
cases have a membrane connecting the upper and lower strut, the other two
have cables as web. Each web type has a case without hinges and one with
1 With moment of inertia is meant in this research the ‘area moment of inertia.’
113

114
CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR
five hinges. From the figure can be seen that the cases with a membrane as
connection have very little bending moments in the struts. After all, the forces
of the web are introduced homogeneous over the struts. This is in contrast
with the cases with cables, because point loads occur which introduce bending
moments in the struts. Since hinges do not transfer bending moments, the cases
with and without hinges have different bending moments.
cables
membrane
with hinges
without hinges
Figure 6.11: The bending moment of four Tensairity beams under
homogeneous load. Two cases have a membrane connecting the upper
and lower strut, the other two have cables as connection. Each connection
type has a case without hinges and one with five hinges.
The bending moments in the struts, introduced by the cables, have an influence
on the load-displacement behaviour of the deployable Tensairity beam. It is
clear that the struts should be dimensioned taking the bending moments into
account. If not, the struts will deform considerable and the Tensairity beam
will experience larger displacements than the same configuration without
cables. This is in contradiction with the purpose of adding cables, namely
increasing the stiffness of the deployable Tensairity beam by blocking the
hinges as discussed in chapter 4and as will be presented in section 6.3.
6.2.3 BENDING STIFFNESS
The influence of the bending stiffness of the struts on the load displacement
behaviour of the deployable Tensairity structure is studied numerically. A
statically determined five meter Tensairity beam is inflated by an overpressure
of 150 mbar (15 kN
m 2 ) and loaded with a uniform distributed load. The shape of
the hull is chosen in such a way that the pretension in the cables throughout
the length is constant. The upper and lower struts are connected by means
of vertical cables which are attached to the strut at the location of the five
upper and lower hinges, see figure 6.12. The length of the cables is equal to
the distance of the hinges that are connected by that cable which means that
there is no prestress present in the cables before inflation. Inflating the airbeam

introduces a tensile force in the cables.
6.2 DIMENSIONING THE STRUTS
The structural response of several Tensairity beams is simulated and the
average stiffness of each case is noted. The cases differ only in one thing,
the cross-section of the struts. All struts have identical cross-sectional area
(of 600 mm 2 ), but different bending stiffness. Since the Young’s modulus is
constant for all cases (all struts are made of aluminum), only the moment of
inertia is varied. The dimensions of the cross sections of the cases can be
found in table 6.1. It is important to note that out-of-plane buckling was not
taken into account in these simulations (by blocking the struts to move outof-plane).
After all, the focus in this study is on the influence of the struts’
bending stiffness on the load-displacement behaviour of the beam. The results
also hold true for strut sections with the same bending stiffness, but other
profile geometry (e.g. hollow rectangular sections).
Figure 6.12: The configuration of the investigated Tensairity beam. The
dots represent hinges, the vertical lines are cables.
Table 6.1: The investigated struts’ cross sections.
case 1 = A case 2 case 3 case 4 case 5 case 6
B [mm] 50,00 45,00 40,00 35,00 32,00 30,00
H [mm] 12,00 13,33 15,00 17,14 18,75 20,00
Iz × 10 3 [mm 4 ] 7,20 8,88 11,25 14,69 17,58 20,00
case 7 case 8 case 9 case 10 case 11 case 12
B [mm] 28,00 26,00 24,00 22,00 20,00 18,00
H [mm] 21,43 23,08 25,00 27,27 30,00 33,33
Iz × 10 3 [mm 4 ] 22,96 26,63 31,25 37,19 45,00 55,55
case 13 case 14 case 15 case 16 case 17
B [mm] 16,00 14,00 12,00 11,00 10,00
H [mm] 37,50 42,86 50,0 54,55 60,0
Iz × 10 3 [mm 4 ] 70,31 91,84 125,00 148,76 180,00
115

116
CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR
The average stiffness versus the moment of inertia of all the cases is plotted in
figure 6.13. The curve, which results from connecting the average stiffness of
the cases, can be divided in three parts. In the first part, a small rise of Iz stiffens
the structure considerably. Increasing the moment of inertia in the third part
of the graph, even by a large amount, has very little influence on the stiffness
of the structure. The reason for this can be found in the structural behaviour
of the segments of the struts of the Tensairity beam. This is discussed here
more in detail by looking closer at their deflection under loading (inflation and
external loading).
Inflation
Average stiffness [N/m]
x 10
3
6
2.5
2
1.5
1
0.5
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
I y [m 4 ]
150 mbar
x 10 −7
Figure 6.13: The average stiffness of the various cases is plotted in
relation with the area moment of inertia of the strut.
As already mentioned in section 6.1, the hull of the Tensairity beam moves
outwards when inflating and acts as a distributed load on the struts in the
outwards direction. At the places where the upper and lower strut are
connected by means of a cable, this outwards movement is counterbalanced
and taken up by the cable. In the other parts, the struts will deflect under the
loading. The size of this deflection depends - besides the pressure - on the
bending stiffness of the strut: the larger the bending stiffness, the lower the
deflection of the strut.
The deflection of the struts of two cases, case A and case B, are taken as
illustration. Case A has a strut with dimensions (H × B) 12 mm × 50 mm
(Iz = 7, 2 × 10 3 mm 4 ) and case B is 25 times stiffer with a hollow rectangle cross
section of 50 mm × 35 mm and t=2 mm (Iz = 180 × 10 3 mm 4 ). The deflection

6.2 DIMENSIONING THE STRUTS
of case A (21,63 mm) is measured 16 times larger than case B (1,31 mm) in the
simulations. Figure 6.14 shows the deflection of a segment of the upper strut
of both cases. For clearness, the displacement magnified by a factor 10 is also
shown.
strut
section
1 x
10 x
A
Figure 6.14: The deflection of a segment of the upper strut of case A and
case B after inflation.
This deflection can also be approximated analytical. Because the cables on
both sides of the segment are tensioned with the same force and elongate thus
by the same amount (which is by the way very little compared with the central
deflection), the real situation can be approximated by the model of a simply
supported beam.
The deflection of such a beam under distributed line load is known as:
δ = 5ql4
384EIz
B
(6.6)
whereby q represents the distributed load [ N
m ], l the length of the segment
[m], E the Young’s modulus [ N
m2 ] and I the area moment of inertia [m4 ]. For
this case, the action of the inflated membrane on the strut can be calculated
as 2600 N
m for a pressure of 150 mbar. The length of the beam equals 0,84 m
(length of segment of strut between two hinges), the Young’s modulus of Alu
9 N
is 69 × 10 m2 . Case B has a moment of inertia of 180 ×103 mm4 , which results
in an analytical displacement δcase B = 1,35 mm, which is close to the numerical
117

118
CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR
value of 1,31 mm. For case A, the analytical value (δcase A= 33,74 mm) differs
more from the numerical of 21,63 mm. This is due to non-linear effects of
the large deformation. Figure 6.15 shows the analytical and numerical central
displacement for all investigated cases. The two curves match reasonably well,
especially for larger moments of inertia. For smaller values is the deflection
overestimated by the analytical approximation.
deflection [m]
0.035
0.03
0.025
0.02
0.015
0.01
0.005
numerical displacement
analytical displacement
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 x 10 −7
0
area moment of inertia [m 4 ]
Figure 6.15: The numerical and analytical value of the central
displacement of the strut’s segment after inflation.
External loading
It is this deflected shape, illustrated in figure 6.14, that is loaded vertical and
distributed. The upper strut takes the compressive forces, the lower strut is
tensioned. In case B, the deflection due to inflation is so little that the upper
strut takes up compression more or less centric. Therefore, it shows the same
behaviour and has similar stiffness as in the case of a regular spindle shaped
Tensairity beam with membrane web and continuous strut. A small change
in bending stiffness of the strut has in this case very little influence on the
stiffness of the Tensairity beam. This is reflected in the third part of the graph
illustrated in figure 6.13. In case A, the compressive force acts eccentric on
the strut because of the large deformation due to inflation. The eccentric
compressive loading results in an increase of bending and deflection of the
strut. Figure 6.16 illustrates this. As a result, the Tensairity beam as a whole
experiences this deformation and has a lower average stiffness than case B.
Increasing the bending stiffness of the strut in case A decreases the deflection

6.2 DIMENSIONING THE STRUTS
caused by inflation and lowers thus the eccentricity of the compressive loading.
As can be seen in the first part of the graph in figure 6.13, a small increase
of bending stiffness increases the average stiffness of the Tensairity beam
considerable. Part two of the curve is the transition whereby the eccentricity
of the compressive force decreases and thus also the effect of this eccentricity
on the deflection.
after inflation after loading
Figure 6.16: A schematic overview explaining the second order effects
in case A.
This behaviour is also reflected in the occurring bending moments of the cases,
as illustrated in figure 6.17. The bending moment in the struts of case B, where
the strut is loaded more or less centric, decreased after distributed loading. This
is because the external loading is in opposite direction of the inflation load.
The ‘centric’ compressive force has little influence on the bending moment.
In case A, where the compressive force acts eccentric on the segments due
to large deflections, is the bending moment doubled at the compression side.
Despite the opposite (downwards) direction of the external loading, the struts
are more bended. This is due to the eccentric compressive force, as illustrated
and explained in figure 6.16.
inflation
loading
case A
case B
Figure 6.17: The bending moments of case A and B after inflation and
loading. The bending moments after loading reflect well the influence
of the eccentricity on the strut’s behaviour.
119

120
CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR
The same cases from table 6.1 are also investigated under various internal
overpressures. Figure 6.18 shows the average stiffness of the cases in relation
to the moment of inertia of the struts. From the curves can be concluded that
for a Tensairity beam with hinges and cables, a higher pressure means a lower
stiffness. This is not at all what would be expected intuitive, since this is the
opposite of the behaviour of regular Tensairity structures and airbeams, where
an increase of internal overpressure stiffens the structure. The explanation for
this can be found in the previous discussion: a higher internal pressure deflects
the struts more and increases the eccentricity of the compressive forces, which
results in a less stiff structure. At high values of Iz, the influence of the deflection
due to inflation is minimized and all cases have similar average stiffness.
However, as will be seen in chapter 8 where the experimental investigations
are discussed, is this lowering of the stiffness at higher pressures not occurring
in reality. From the experiments (where struts with an Iz = 11,25 ×10 3 mm 4
are used) is observed that the deflection of the struts after inflation is not as
large as the finite element models predict and has as a consequence no (or no
visible) influence on the structure’s stiffness.
average stiffness [N/m]
x 106
2.5
2
1.5
1
050 mbar
100 mbar
150 mbar
200 mbar
250 mbar
0.5
300 mbar
400 mbar
500 mbar
0
0 0.5 1 1.5 2
I y [m 4 ]
x 10 −7
Figure 6.18: The average stiffness of the various cases is plotted in
relation with the area moment of inertia of the strut. Various pressure
levels are shown.

Influence of configuration
6.2 DIMENSIONING THE STRUTS
The previous discussion showed that using cables instead of a membrane
as web type changes the structural behaviour and thus the average stiffness
of the Tensairity beam considerable. As illustration, the average stiffness is
calculated of the four Tensairity cases whose bending moments are discussed
in section 6.2.2 and illustrated in figure 6.11. This comparison, plotted in
figure 6.19, permits to illustrate the influence of using cables or a membrane as
connection type and the influence of hinges in the struts. The average stiffness
of the four cases is plotted in relation to the moment of inertia. The curves
clearly show that the cases with a web (and thus very low bending moment
in the strut) are more or less independent of the area moment of inertia of the
strut and thus its bending stiffness. This is clearly not the case for the Tensairity
beam with cables, as explained. For high values of the moment of inertia are
the cases with cables stiffer than with membrane. This can be attributed to
the higher E-modulus of the steel cable than the one of the membrane. From
the graph can also be seen that the cases with hinges are less stiff than the
cases with continuous strut. The influence on the structural behaviour of
introducing hinges will be discussed in the next section.
average stiffness [N/m]
x 106
2.5
2
1.5
1
0.5
cable hinge
cable no hinge
web hinge
web no hinge
0
0 0.5 1 1.5 2
I y [m 4 ]
x 10 −7
Figure 6.19: The average stiffness of the various cases is plotted in
relation with the area moment of inertia of the strut. The cases differ in
web type (cable and membrane) and the presence of hinges.
121

122
CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR
6.2.4 CONCLUSION
By investigating the influence of the dimensions of the struts on the structural
behaviour of a deployable Tensairity beam, it became clear that its structural
behaviour is very different from that of a regular Tensairity beam. Introducing
cables as connecting element between upper and lower strut changes the forces
and moments in the struts drastically.
In the case of a Tensairity beam whereby the upper and lower strut are
connected by means of a membrane, the outwards movement of the inflated
membrane (and thus force) is taken up by this membrane. The struts of the
beam experience no bending and are only loaded in tension and compression,
just like a regular Tensairity beam with circular cross section of the hull. In
the case of a Tensairity beam whereby cables are used as connecting element,
the segments of the struts have to take up the outwards force of the inflating
membrane and are at the ends supported by the vertical cables. This situation
makes that the segments of a deployable Tensairity beam, connected to each
other with hinges, behave like a simply supported beam (supported by the
cables), loaded under a distributed line load (the action of the membrane).
The size of this deflection is determined by the load acting on the strut and
the bending stiffness of the strut’s segment. The bending stiffness is defined
by the product of the strut’s Young modulus and moment of inertia. The size
of the deflection caused by inflation has a great influence on the behaviour
of the strut under external loading. The larger the deflection, the larger the
eccentricity of the compressive forces in the strut and thus the influence of
second order effects on the bending.
If the deflection is small due to a high bending stiffness of the strut (compared
to the load), the eccentricity of the compressive force is negligible and has no
influence on the structural behaviour under external loading. The average
stiffness of such cases is very comparable with that of a Tensairity beam with
membrane as web. If the bending stiffness is on the other hand quite small, a
large deflection will occur during inflation and the compressive load during
external loading will be eccentric. Second order effects will cause large bending
moments and displacements in the upper strut of the Tensairity beam.

6.3 INTRODUCING HINGES
6.3 INTRODUCING HINGES
When developing a deployable Tensairity beam, the influence of introducing a
mechanism as compression and tension element on the load-bearing behaviour
of the Tensairity structure should be known and understood. Therefore,
Tensairity beams with varying amount of hinges in the upper and lower struts
and a varying location of the hinges are investigated. First, the influence of
introducing hinges on the structural behaviour is studied.
6.3.1 INFLUENCE OF HINGES
Two different Tensairity beams are investigated. One beam has a circular crosssection
of the hull, also called the ‘O-spindle’ in this research, and the second
has cables connecting the upper and lower strut, the ‘discretized web spindle’
or ‘D-spindle’. Figure 6.20 illustrates both cases.
O-spindle
O-spindle
D-spindle
Figure 6.20: The influence of hinges on the structural behaviour of two
cases is investigated: the ‘O-spindle’ and ‘D-spindle’.
To investigate the influence of hinges on the structural behaviour of the Ospindle
Tensairity beam, two configurations are simulated. They differ in
the presence of hinges in the upper and lower strut: one case contains five
hinges in upper and lower strut, the other none. Both cases are inflated to an
overpressure of 150 mbar and loaded by a distributed line load of 240 N
m . The
struts have a width of 30 mm and a height of 20 mm. The moment diagrams of
the struts after distributed loading are noted, as well the average displacement
of the structures throughout loading.
Figure 6.21 shows the moment diagrams of the two cases after loading. As
can be expected, the bending moment of the continuous strut (thus without
hinges) is similar to the one of an arch under distributed loading. However,
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CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR
due do the specificity of a Tensairity beam 2 , the value of the maximum bending
moment is very small (7 N.m in this case). In the case of a segmented strut, the
hinges can not transfer bending moments and the maximum bending moment
is thus smaller than in the case of a continuous strut (5 N.m).
Figure 6.21: The bending moment of a standard spindle with and
without hinges after distributed loading (240 N
m ).
Figure 6.22 plots the average displacement of both cases throughout the
loading. The difference between both cases is very small, because there are
mainly normal forces and almost no bending moments in the struts. However
small, the two configurations have different average displacement and, as
expected, the Tensairity beam with continuous strut is stiffer. Obvious, this is
because the hinges do not transfer bending moments and experience a larger
displacement, as can be seen in figure 6.23. This larger displacement takes
especially place near the ends of the beam, where the radius is smaller. The
consequence of this smaller radius is a smaller membrane stress and thus a
lower ‘support’ of the hinge by the membrane.
D-spindle
As already mentioned in section 6.2.2, the bending moment diagrams of a
web Tensairity 3 are different than in the case of a standard Tensairity structure
(O-spindle). After all, the cables are responsible for additional forces on the
struts. The values of the bending moments in the struts are in the case of a
D-spindle 200 times larger than in the case of a standard Tensairity beam.
Figure 6.25 illustrates the bending moments of a D-spindle with and without
hinges. Also here, the strut’s section measures 30 mm width by 20 mm height.
2 The bending is taken up by the complete Tensairity beam and the upper strut takes up the
compressive forces, the lower one the tensile forces. However, due to the deflection of the hull
under external loading, a minor bending moment occurs in the upper strut.
3 This is a Tensairity beam whereby the upper and lower strut are connected by means of a
membrane or cable. In this research, except mentioned differently, cables are always used as
connecting element. This is then called the ‘D-spindle’, from discretized web.

load [N]
1400
1200
1000
800
600
400
O−spindle − pressure 150 mbar
0 1 2 3 4 x 10 −3
200
0 hinge
5 hinge
0
average displacement [m]
6.3 INTRODUCING HINGES
Figure 6.22: The average displacement of the O-spindle with and
without hinges.
Figure 6.23: The displaced struts after loading. Upper: the case with
hinges, lower: without hinges. The displacement is for reasons of
clearness magnified by a factor 20.
A large difference in moment diagram between the two cases can be seen.
For the D-spindle with hinges, the segments of the struts behave as simply
supported beam. From the bending moments of the case without hinges can
be stated that their behaviour is analogue with hyperstatic beams. Figure 6.24
illustrates this.
This can be verified by means of comparing the analytical maximum bending
moment of a simply supported beam and a hyperstatic beam with the numerical
bending moments after inflation. The analytical maximum bending
moments can be calculated as follow:
125

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CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR
.
strut without hinges strut with hinges
hyperstatic isostatic
Figure 6.24: The deformation and bending of the segments of the struts
can be modeled as simply supported or hyperstatic beams, depending
whether the segments are connected by hinges.
Miso = q.l2
8
Mhyper = q.l2
12
= 229, 32 N.m (6.7)
= 152, 88 N.m (6.8)
with q the action of the inflated membrane on the strut (2600 N
m for a pressure of
150 mbar) and l the distance between two vertical cables ( 0, 84 m). The values
of the respective maximum bending moments, derived by means of numerical
simulation, are Miso = 223, 81N.m and Mhyper = 153, 92N.m, which is reasonably
close to the analytical values. Thus above comparison of the behaviour of the
segments with isostatical and hyperstatical beams, holds true.
One can expect to have larger displacements in the case with hinges than the
case without, because of a larger bending moment in the struts. Figure 6.26
(left) plots the average displacement of the two cases during the loading. It is
clear that the case with hinges is less stiff than the case with continuous strut.
Figure 6.27 shows the displaced struts of the two cases after loading. The
displacement is scaled by a factor 20. Because the case with hinges deflects
much more under inflation than the case without, it will experience a larger
global deformation under loading. As a consequence, if stiffer struts are used
and the deformation after inflation is smaller, the difference between the case
with and without hinges decreases. This is illustrated on the right of figure 6.26,

6.3 INTRODUCING HINGES
Figure 6.25: The bending moments of a D-spindle with and without
hinges after inflation (150 mbar). The influence of the hinges can be
clearly seen.
where the average displacement of the two cases is plotted with a much stiffer
section of 10 mm width × 60 mm height.
load [N]
6000
5000
4000
3000
2000
section 30mm width × 20mm height
load [N]
0 1 2 3 4 5 x 10 −3
1000
0
0 1 2 3 4 5 x 10
average displacement [m]
−3
0
0 hinge 1000
0 hinge
5 hinges
5 hinges
average displacement [m]
6000
5000
4000
3000
2000
section 10mm width × 60mm height
Figure 6.26: The average displacement of the upper strut with and
without hinges during loading. Left: the cases with section 30 mm ×
20 mm; right: 10 mm × 60 mm.
There can thus be concluded that hinges do have an influence on the structural
behaviour of the Tensairity beam. After all, hinges do not transfer bending
moments. Generally speaking, the Tensairity beams with hinges are less
stiff than their equivalent with continuous strut. In the case of a standard
spindle beam under loading, (almost) no bending moments occur in the struts.
Therefore, the influence of the hinges on the stiffness is rather limited. In the
case of a web-Tensairity, the difference between the case with and without
hinges is dependent on the deflection under loading of the struts, and thus of
the bending stiffness of the struts.
Now the effect of hinges in the struts of a Tensairity beam is known and un-
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CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR
hinges
no hinges
Figure 6.27: The displaced upper and lower struts after loading of
the case with and without hinges (with section 30 mm × 20 mm). The
displacement is for reasons of clearness magnified by a factor 20.
derstood, there is investigated what the influence on the structural behaviour
is of the amount of hinges in the struts.
6.3.2 AMOUNT OF HINGES
The amount of hinges in the upper and lower strut of a spindle shaped
Tensairity beam are varied to monitor the influence of the presence of these
hinges on the load-bearing behaviour of the structure. Two configurations,
illustrated in figure 6.28 are investigated numerically: the O- and D-spindle.
This latter has cables connecting the upper and lower strut at regular distances,
the other configuration does not. Within these two configurations, fifteen
cases are simulated under a distributed load: the amount of hinges taken
into account during the numerical investigations varies between 0 and 14, as
illustrated in figure 6.29.
O-spindle
D-spindle
Figure 6.28: The influence of the amount of hinges is investigated with
the O- and D-spindle.
The average stiffness of the cases is plotted in figure 6.30. The x-axis plots the
amount of hinges, the average stiffness is shown on the y-axis. For both cases,
a small decrease in stiffness for increasing amount of hinges can be detected.
Thus, the amount of hinges has very little influence on the stiffness of the
Tensairity beam. Notice that the case without connecting cables has a much
lower average stiffness than the other case. This will be discussed more in

0
1
2
3
5
7
9
11
14
Figure 6.29: The amount of hinges vary between 0 and 14.
6.3 INTRODUCING HINGES
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CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR
average stiffness [N/m]
2.5
2
1.5
1
0.5
x 106
3
p = 150 mbar
D−spindle
O−spindle
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
amount of hinges
Figure 6.30: The influence of the amount of hinges on the average
stiffness of the cases is little.
detail in section 6.4.3.
maximum load [N]
3000
2500
2000
1500
1000
500
O−spindle − pressure 150 mbar
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
amount of hinges
Figure 6.31: The maximal load for the O-spindle for the various amount
of hinges.
The load at which the finite element calculation stops (and thus no convergence
is achieved anymore) is presented here as being the maximal load.
As discussed in chapter 5, this is the load whereby a large deformation or
singularity occurs, like eg. large displacement of a hinge or wrinkling of the
membrane. This value is different for the various configurations of figure 6.29.
The maximal load for the O-spindle for the various amount of hinges is plotted
in figure 6.31. The total load is plotted on the y-axis, the x-axis represents the

amount of hinges in the upper strut of the Tensairity beam.
6.3 INTRODUCING HINGES
The results show that there is a relation between the amount of hinges in the
struts and the maximum load that the structure can bear before ‘collapse’: the
more the hinges, the lower the value of maximal load. However, more in
depth investigation, presented in section 6.3.3, shows that it is the position of
the hinge closest to the endpoints that influences the maximum load and not
the amount of hinges. The reason why it seems that the amount of hinges
determines the maximum load the structure can bear, is that (as can be seen
in figure 6.29) in the case of evenly distributed hinges, the position of the last
hinge and the amount of hinges in the strut are related: the more hinges the
structure has, the smaller the distance between the last hinge and the ends of
the Tensairity beam. Figure 6.32 plots the maximum loads in relation to the
position of the last hinge along the length of the Tensairity beam. The zero
represents the middle of the beam, the x-coordinate of 2,5 m represents the
endpoint of the beam.
maximum load [N]
3000
2500
2000
1500
1000
500
2 hinges
0
0 0.5 1 1.5 2 2.5
position along length of beam [m]
3 hinges
9 hinges
O−spindle − pressure 150 mbar
Figure 6.32: The maximum load in relation to the position of the last
hinge along the length of the Tensairity beam. The configuration with 3
3
5
7
9
11
14
and 9 hinges are illustrating the position of the last hinge.
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132
CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR
6.3.3 POSITION OF HINGES
The influence of the location of the hinge on the maximal load of the deployable
Tensairity beam is investigated. This is done by means of eleven cases which
differ in location of hinge in upper and lower strut. All cases contain two
hinges in their strut, but the position of these hinges changes throughout the
cases. Note that the hinges are symmetric in relation to the middle of the beam.
Three of these cases are illustrated in figure 6.33.
2
6
11
Figure 6.33: Three of the eleven cases, whereby the position of two
hinges (symmetrical) on upper and lower strut is varied. The black dots
represent hinges. Cases 2, 6 and 11 are shown.
The maximal load of the O-spindle (fig. 6.28) for the eleven cases is plotted
in figure 6.34 (left). The maximal load is plotted at the position of the hinge
of each case along the Tensairity beam. From the figure can be seen that the
maximal load decreases when the hinge is more positioned to the ends.
A similar decrease of maximal load was detected in the study on the influence
of the amount of hinges (figure 6.32). There has been discussed that not
the amount of hinges directly determines the maximum load, but mainly the
position of the last hinge. This is shown in the right graph in figure 6.34 where
the curves of both cases are illustrated.
The curve that represents the maximal load of the structure with only two hinges
- but with variable location (A) - and the curve that represents the maximal load of
the structure with different amount of hinges (B) are very similar. They illustrate
a relation between the position of the last hinge and the maximum load of the

maximal load [N]
3000
2500
2000
1500
1000
500
0
0 0.5 1 1.5 2 2.5
x−coordinate of hinge [m]
3000
2500
2000
1500
1000
6.3 INTRODUCING HINGES
500
amount of hinges
location of hinges
0
0 0.5 1 1.5 2 2.5
x−coordinate of hinge [m]
Figure 6.34: Left: The maximum load of the eleven cases with different
position of the two hinges. Right: Maximum load of the cases with
different position of the two hinges (A) and with different amount of
hinges (B) (figure 6.32. The maximum load is plotted at the value of the
x-coordinate of the (last) hinge.
structure. The value of curve A is somewhat higher than that of B, because the
influence of the other hinges probably decreases the maximum load at which
convergence is achieved.
An explanation for the relation between the position of the (last) hinge and
the so-called maximal load has not been proven yet by means of experimental
investigations. A hypothesis put forward here is that the maximal load -
which is the highest load at which the finite element calculations converged -
is reached when the membrane starts to wrinkle due to deflections. Since the
hull has a lower membrane stress at lower radii of the airbeam, the hinges are
less supported towards the ends. Thus, the hinges near the ends experience
larger deflections.
6.3.4 CONCLUSION
Introducing hinges in the strut of a Tensairity beam decreases the structure’s
stiffness. In the case of an O-spindle, thus without connection between upper
and lower strut, are the struts mainly loaded in compression and tension, and
experience very little bending. As a result, the influence of hinges is relative
small here. When cables connect the upper and lower strut, large bending
moments are introduced in the struts and the influence of the hinges is more
noticeable.
maximal load [N]
B
A
133

134
CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR
In this study, the amount of hinges were varied, as well the location of two
hinges along the beam’s length. Regarding the stiffness of the structure, there
can be concluded that the amount of hinges has very little influence on the
stiffness. The investigations also revealed that the maximum load whereby
the structure is stable under a distributed line load is related to the position
of the hinge that is located nearest the endpoint: the closer the hinge to this
endpoint, the lower the load at which the structure will reach its maximum
load.
It is with this knowledge in mind that the foldable mechanism for the deployable
Tensairity beam has been redesigned in chapter 3 (section 3.4, p. 52). The
influence of the position of the hinges and the very low impact of the amount of
hinges on the structural behaviour of the structure lead to a redesign whereby
the hinges in the compressed strut were moved towards the middle, i.e. a
larger distance between the last hinge and the end support was achieved. To
allow the mechanism with this new arrangement of hinges to be folded to a
compact configuration, two more hinges than the previous design in the upper
strut are required. However, as seen in this section, this increase in amount
has negligible influence on the structural behaviour. Figure 6.35 illustrates the
proposed configuration for a foldable mechanism of the deployable Tensairity
beam.
Figure 6.35: The influence of the position of the hinges and the very
low impact of the amount of hinges on the structural behaviour of the
structure lead to this proposed configuration for a foldable mechanism
of the deployable Tensairity beam.

6.4 CABLE CONFIGURATION
6.4 CABLE CONFIGURATION
The influence of cables that connect the upper and lower strut on the structural
behaviour of the deployable Tensairity beam is already addressed in the
previous sections. When discussing the shape of the hull, cables are mentioned
as appropriate elements for connecting the upper and lower strut and thus
preventing the airbeam to become a circle. With the investigations on the
struts’ cross section is shown that the cables introduce point loads on the
struts, resulting in bending moments and deformation of the strut. In the
research on the influence of the amount of hinges on the structural behaviour
were two configurations applied, they differed in the presence of cables. The
comparison of the stiffness of these two configurations shows very clearly the
influence of cables: the stiffness of the configuration with cables was a factor
five larger than the case without cables. Thus the cables have an influence on
the structural behaviour of the Tensairity beam.
This was also shown by means of experimental and numerical results in
chapter 4. The cable tension throughout inflation and loading is discussed
there, as well the contribution of pretensioned cables to the structure’s loaddisplacement
behaviour. The research, conducted on a cylindrical deployable
Tensairity beam 4 , showed that as long as the cables connecting upper and
lower hinges are pretensioned due to inflation, they can take compressive
forces. This means that these cables can support or ‘block’ the hinges. Once
the pretension in a cable is zero, the hinge is not supported anymore and the
structure experiences larger displacement. This has as result that the stiffness
of the cylindrical deployable Tensairity beam changes when one cable becomes
slack.
This section verifies first whether these conclusions also hold true for the
spindle shaped Tensairity beam by means of applying the same cable configurations
in the spindle as in chapter 4. Then, the influence of (other)
cable configurations in the redesigned foldable mechanism are investigated
(figure 6.35). But first, a closer look is taken on the behaviour of one cable in
the spindle throughout inflation and loading, and its influence on the hinges
it connects.
4 Containing segments with a high bending sitffness in relation to the scale of the beam.
135

136
CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR
6.4.1 CABLE TENSION
As discussed several times in this dissertation, some cables become pretensioned
under inflation and release tension when being compressed under
external loading. The value of the tension in the cables is monitored during
inflation and loading of the deployable Tensairity beam by means of finite
element analysis.
Figure 6.36 plots the tension in the vertical and diagonal cables of case 2
during inflation (150 mbar) and distributed loading. The investigated cables
are indicated in the figure. Note that all diagonal cables are connected with a
hinge that is also linked with a vertical cable. The bending stiffness of the strut
is in the finite element model chosen to be high in order to avoid influences of
the bending of the strut on the cable tension.
From the figure can be seen that all forces developed by inflation are taken
by the vertical cables, which corresponds with the findings of chapter 4. The
x-axis of the figure represents the time steps of the loading: the structure is
being inflated between time 0 − 1 and loaded between time 1 − 2. The y-axis
represents the tension in a cable.
case 2
cable tension [N]
vertical cables
1000
p=150 mbar
800
600
400
200
inflation
1
cable 1 & 5
cable 2 & 4
0
0
cable 3
0.5 1 1.5 2
load step (0−1 inflation, 1−2 loading)
a
2
b
3
c
4
d
5
1000
loading 800 inflation loading
cable tension [N]
600
400
200
diagonal cables
0
0 0.5 1 1.5 2
load step (0−1 inflation, 1−2 loading)
Figure 6.36: The tension in the vertical (left) and diagonal (right) cables
during inflation (0-1) and loading (1-2).

6.4 CABLE CONFIGURATION
While some diagonals in the cylindrical models of chapter 4 contributed 5 to
the structural behaviour under distributed loading , remain the diagonals in
the case of spindle shaped deployable Tensairity beam slack throughout this
distributed loading. This is remarkable, because there could be expected to
have the two middle cables being loaded in tension.
This can be attributed to the spindle shape of the structure: the arched
compression strut moves outwards when being loaded, which results in an
upwards movement of the tension element, despite the downwards loading
transferred by the airbeam to the lower strut.(The struts compress thus inwards
in the airbeam). Figure 6.37 illustrates this.
Figure 6.37: The diagonals of the spindle shaped Tensairity beam are not
loaded in tension under distributed load due to the upwards movement
of the lower strut.
The value of the pretension of the vertical cable after inflation can be calculated
analytically with equation 4.1 from chapter 4:
F = 2 × p × R × l × sinα (6.9)
With p=150 mbar (15 kN
m 2 ), R=0,252 m, l=0,838 m and α = 7, 5 ◦ , one obtains the
analytical value of Fpre=827 N. The numerical value of the pretension in the
vertical cable measures 845 N, which is close to the analytical approximation.
When the deployable Tensairity beam is loaded asymmetric, the lower strut
does not move upwards along the whole length of the beam as in the case of
distributed loading, which results in diagonals being tensioned under loading.
This can be seen in figure 6.38, plotting the cable tension of the vertical (left)
and diagonal (right) cables. During inflation (loadstep from 0 till 1 on the
x-axis), the cables are not tensioned, during loading (from 1 till 2 on the y-axis)
become some cables tensioned.
5 When loaded in tension. This depends on their position/configuration in the structure.
137

138
CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR
cable tension [N]
1000
800
600
400
200
cable 1
cable 2
cable 3
cable 4
cable 5
vertical cables
0
0 0.5 1 1.5 2
load step (0−1 inflation, 1−2 loading)
cable tension [N]
1000
800
600
400
200
diagonal cables
cable a
cable b
cable c
cable d
0
0 0.5 1 1.5 2
load step (0−1 inflation, 1−2 loading)
Figure 6.38: The cable tensions of case 2 (left) en case 7 (right) under
inflation and asymmetrical load.
cable tension [N]
1000
800
600
400
200
p = 150 mbar
cable 2
cable a
0
0 0.5 1 1.5 2
load step (0−1 inflation, 1−2 loading)
Figure 6.39: The interaction between vertical and diagonal cables under
asymmetrical load: once the vertical cables becomes slack, the tensioned
diagonal is being compressed.(load step from 1 till 2 : 0.1 step represents
500 N external loading)
The interesting interplay between the vertical and diagonal cables can be seen
in this configuration under asymmetrical loading. Figure 6.39 illustrates this
by plotting the cable tension of cable 2 and A in relation to the loading (load step
from 1 till 2 : 0.1 step represents 500 N external loading). From the figure can be
seen that the pretension of the vertical cable decreases under external loading.
At the same time is the diagonal being tensioned, as one can expect under the
asymmetrical loading (it prevents the mesh to become parallelogram) . Once
the vertical cable has become slack, the forces in the diagonal changes and it
becomes compressed. Figure 6.40 shows the deformed shape of the structure
under asymmetrical load with the cables 2 and A indicated.

a
2
6.4 CABLE CONFIGURATION
Figure 6.40: The deformed shape of the structure under asymmetrical
load with the cables 2 and A indicated. (The deformation is scaled by a
factor 2.)
As discussed in chapter 4 is the pretension of the cables related with the internal
pressure: the higher the internal pressure of the airbeam, the more the cables
are pretensioned. As a result, the cables can support more compressive forces.
Figure 6.41 shows the tension in the central vertical cable of configuration 2 at
50, 150 and 250 mbar. Notice that the reduction in prestress is proportional to
the applied external load and independent of the cable’s prestress.
cable tension [N]
1500
1000
500
50 mbar 150 mbar 250 mbar
inflation loading
0
0 0.5 1 1.5 2
load step (0−1 inflation, 1−2 loading)
Figure 6.41: The tension in cable 3 of case 2 , at 50, 150 and 250 mbar.
The higher the internal pressure, the higher the prestress. The reduction
in prestress is proportional to the applied external load and independent
of the prestress.
6.4.2 STIFFNESS
In chapter 4 is shown that the stiffness of the structure is related with the
behaviour of the cables in the structure. Once a cable becomes slack, it can
no longer support or ‘block’ the hinge it connects. As a consequence, larger
displacements occur and the stiffness of the structure changes.
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CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR
This is also the case for a spindle shaped deployable Tensairity structure.
Figure 6.42 shows on the left the load-displacement behaviour of the asymmetrical
loaded case 2. On the right is the cable tension plotted of cable 2 in
relation with the applied load. Note that for this graph, the cable tension is
represented on the x-axis and the applied load on the y-axis. It shows that
the cable tension decreases with increasing load (as discussed before), until it
reaches zero. On the left graph can be seen that this is exactly the same load
whereby the stiffness of the structure changes.
load [N]
3500
3000
2500
2000
1500
1000
500
load−displacement
0
0 0.01 0.02 0.03 0.04
average displacement [m]
load [N]
3500
3000
2500
2000
1500
1000
500
cable tension
0
0 200 400 600 800 1000
cable tension [N]
Figure 6.42: The influence of the cable tension on the structure’s stiffness.
Left: the load-displacement behaviour of the asymmetrical loaded case
2 (pressure 150 mbar). Right: the cable tension of cable 2 throughtout
loading.
To investigate the influence of the pressure on the stiffness of the structure
is the average displacement noted of case 2 under 50, 150 and 250 mbar.
Figure 6.43 illustrates this. From this figure can be seen that the conclusions
from chapter 4 also hold true for the spindle shaped deployable Tensairity
beam. As long as all vertical cables are pretensioned, is the stiffness of the
structure independent of the pressure (but influenced by the stiffness of the
cables). Because the cables in the case with a higher internal pressure are more
pretensioned, they experience the zero pretension in a cable at a higher load.
Therefore, the change in stiffness occurs at a higher load in the case of a higher
internal pressure, as can be seen in figure 6.43.
6.4.3 CABLE CONFIGURATION
The influence of various cable configurations on the structural behaviour
of the spindle shaped deployable Tensairity beam are investigated. Fig-

load [N]
5000
4000
3000
2000
1000
50 mbar
150 mbar
250 mbar
0
0 0.02 0.04 0.06 0.08
average displacement [m]
6.4 CABLE CONFIGURATION
Figure 6.43: As long as all cables are pretensioned, is the stiffness
independent of the pressure. The load at which the stiffness changes
is related with the internal pressure.
ure 6.44illustrates the cases. First, the cases without vertical cables are discussed
(cases 2–5), then the cases with verticals ( cases 6–10).
2
3
4
5
Cases 2 – 5
6
Figure 6.44: Various cable configurations are investigated numerically.
The tension in the cables of cases 2, 3, 4 and 5 is investigated numerically,
as well the load-bearing behaviour of the cases. All the diagonal cables
become pretensioned under inflation, just as the cylindrical cases 2 till 5 in
chapter 4. However, when loading the spindle shaped beam, all the diagonals
become compressed, regardless their configuration. This is different from the
cylindrical models and can be attributed to the spindle shape, as illustrated
already in figure 6.37.
7
8
9
10
141

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CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR
Figure 6.45 illustrates the load-displacement behaviour of the four cases. An
explanation for the difference in stiffness between the cases does not lie so
explicit in the shape of the ‘mesh’ or in cables becoming more tensioned or
compressed, as was the case with the cylindrical models. The structure’s
stiffness is influenced here by the deformation of the cases under inflation.
Indeed, because the hull of the airbeam is designed such to have an outwards
action on the cables (see section 6.1), the hinges that are not connected with
a cable move outwards and deform the spindle shape. Figure 6.46 illustrates
the four cases after inflation.
load [N]
6000
5000
4000
3000
distributed load − pressure 150 mbar
2000
case 2
1000
case 3
case 4
case 5
0
0 0.005 0.01 0.015 0.02
average displacement [m]
Figure 6.45: The load-displacement behaviour of cases 2, 3, 4 and 5.
2
3
Figure 6.46: The hinges not connected with a cable will move outwards
when inflating the structure. The cable configuration determines the
deformed shape after inflation. The displacements are scaled with a
factor 5.
When loading the deformed spindle beams uniformly, their upper strut is
compressed and deforms the shape even more. After all, the hinges moving
outwards are then pushed further outwards and vice versa. Thus, the cable
configuration determines the deformed shape after inflation and as conse-
4
5

6.4 CABLE CONFIGURATION
quence the structure’s stiffness. As can be seen in figure 6.46 is case 3 the
most optimal, since the three successive middle hinges are connected with
a cable, which prevents a zig-zag shape of the upper strut as in the case of
configurations 2 and 4.
Cases 6 – 10
The stiffness of cases 6, 7, 8, 9 and 10, inflated with an internal pressure of
150 mbar and loaded distributed, is plotted in figure 6.47. The influence of
the vertical pretensioned cables can clearly be seen: all cases have exactly the
same stiffness and load whereby the first (vertical) cable becomes slack (the
‘maximum load’). Note that the diagonals are not contributing to the structural
behaviour: they are not pretensioned (due to the presence of the vertical cables
in the same hinges) and because they are all loaded in compression when the
external load is applied.
load [N]
12000
10000
8000
6000
distributed load − pressure 150 mbar
0 0.5 1 1.5 2 2.5 3
x 10 −3
4000
case 6
case 7
case 8
2000
case 9
case 10
0
average displacement [m]
Figure 6.47: The load-displacement behaviour of cases 6, 7, 8, 9 and 10.
The influence of the vertical pretensioned cables can clearly be seen: all
cases have exactly the same stiffness.
In the case of asymmetrical loading are some diagonals contributing to the
structural behaviour of the deployable Tensairity beam. Numerical investigation
showed that the cables one can expect to work in tension during loading,
do contribute to the stiffness of the structure. Figure 6.48 illustrates the cables
in tension and compression(and thus slack) during asymmetrical loading.
Figure 6.49 shows the load-displacement behaviour of the cases under asymmetrical
loading. It shows that case 6 has the lowest stiffness, which can
be expected since this case has no diagonals to contribute to the structural
143

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CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR
7
8
+
tension
+ +
+ +
o o
o slack
o
9
o 10 o o
o
+ o +
Figure 6.48: The cables in tension and compression(and thus slack)
during asymmetrical loading.
behaviour. Case 10 has the largest stiffness, followed by case 8. Both cases
have two successive ‘meshes’ of the truss with the shape of a triangle. At least
two successive hinges are stabilized here by a tensioned cable, which prevents
‘zig-zag’ deformation of the upper strut. Figure 6.50 illustrates the deformed
spindles under asymmetrical loading.
load [N]
4000
3500
3000
2500
2000
asymmetric load − pressure 150 mbar
1500
case 6
1000
case 7
case 8
500
case 9
case 10
0
0 0.01 0.02 0.03 0.04
average displacement [m]
Figure 6.49: The load-displacement behaviour of the cases 6 to 10 under
asymmetrical loading. (internal pressure measures 150 mbar).
6.4.4 PROTOTYPE CONFIGURATIONS
The prototype for a deployable Tensairity beam has a different hinge configuration
and thus folding mechanism than the cases discussed so far in this
section. In chapter 3, where this configuration for a mechanism is discussed,
is shown that not all cable connections between upper and lower hinges allow
the structure to be folded. Figure 6.51 illustrates the cable configurations that
allow the structure to be folded (left) and the cables that would obstruct this
folding (right).
+
+

6
7
8
9
10
6.4 CABLE CONFIGURATION
Figure 6.50: The deformed spindles under asymmetrical loading. (The
displacements are scaled.)
In this section, three different cable configurations are studied numerically.
These cases will also be investigated experimentally in chapter 8. Although
they don’t allow folding of the structure, two of the three cases contain vertical
cables connecting upper and lower strut. The third case contains only diagonal
cables. Figure 6.52 illustrates the three cases: A, B and D. By comparing case
A and B, the influence on the diagonals will be discussed. Comparing B and
D will allow us to evaluate the effect of the vertical cables.
Diagonal cables
Figure 6.53 shows the numerical load-displacement behaviour of cases A and
B under distributed (left) and asymmetrical load (right). From the figure can
be seen that case B is stiffer in both load cases. The diagonals thus contribute
to the structure’s stiffness.
In section 6.4.3 is seen that - in the configurations with diagonal and vertical
cables - the diagonals are not pretensioned under inflation and thus not
145

146
CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR
links allowing folding
links obstructing folding
Figure 6.51: The cable configurations that allow the structure to be folded
(left) and the cables that would obstruct this folding (right).
A
B
D
Figure 6.52: Three cable configurations of the prototype: A, B and D.
contribute to the structural behaviour under distributed loading. This is
because the outwards movement by the inflation is in those cases taken entirely
by the vertical cables, which are present in every hinge. However, in the case
of configuration B are not all hinges connected with a vertical cable. (These
are the hinges on the lower strut closest to the supports). This has as a result
that the diagonals connecting these hinges - cables a, d, e and f (indicated
in figure 6.52) - are pretensioned. As a consequence, these cables can take a
certain amount of compressive forces when the structure is loaded. Because
they form ‘triangular meshes’ in the truss and support an inwards rotating
hinge, they contribute to the stiffness.
As one can expected are these four diagonal cables also contributing to the
stiffness of configuration B under asymmetrical loading.
Vertical cables
Cases B and D differ in the presence of vertical cables. As seen before in
section 6.4.3 and illustrated in figure 6.46 are the cases 2–5 deformed under

load [N]
4500
4000
3500
3000
2500
2000
1500
distributed loading − pressure 150 mbar
1000
500
case A
case B
0
0 0.01 0.02 0.03 0.04
average displacement [m]
load [N]
4000
3500
3000
2500
2000
1500
6.4 CABLE CONFIGURATION
asymmetric loading − pressure 150 mbar
4500
1000
500
case A
case B
0
0 0.01 0.02 0.03 0.04
average displacement [m]
Figure 6.53: Load-displacement behaviour of cases A and B under
distributed (left) and asymmetrical load (right) (pressure 150 mbar).
inflation, due to a lack of vertical cables preventing the free hinges to move
outwards. This deformation under inflation can also be seen in the case of
configuration D, shown in figure 6.54.
Figure 6.54: Due to a lack of vertical cables preventing the free hinges
to move outwards is case D deformed after inflation. (No scaling of
displacements. Pressure 150 mbar).
By analogy with cases 2–5, one can expect case D to have a low stiffness.
After all, when the upper strut is compressed under loading, the hinges will
rotate further and deform the upper strut. The load-displacement behaviour
of case D is plotted in relation with cases A and B under distributed load in
figure 6.55. The lack of cables preventing all hinges to go outwards decreases
thus the structure’s stiffness.
6.4.5 CONCLUSION
This section investigated numerically the influence of cables and their configuration
on the structural behaviour of the deployable Tensairity beam.
The study revealed that the pretensioned cables contribute to the structure’s
stiffness by supporting the hinges (until their pretension becomes zero) and
147

148
CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR
load [N]
4500
4000
3500
3000
2500
2000
1500
distributed load − pressure 150 mbar
1000
case A
500
case B
case D
0
0 0.01 0.02 0.03 0.04
average displacement [m]
Figure 6.55: The load-displacement behaviour of case D in relation with
cases A and B under distributed load and a pressure of 150 mbar.
by preventing the structure to deform under inflation.
When not every hinge is ‘blocked’ by a cable from moving outwards, the strut
experiences a deformation which results in a lower stiffness of the structure
than in the case whereby every hinge is connected with a cable.
In the case where all hinges are supported by a cable, is the stiffness of
the structure independent of the internal pressure, as long as all cables are
pretensioned. Once a cable becomes slack, the stiffness changes. The value
whereby this stiffness changes, is related with the pretension of the cable and
thus the internal pressure.
The investigation of the cable configurations and the cable tension showed that
diagonal cables are not pretensioned under inflation when they connect hinges
that are already linked with vertical cables. This has as a consequence that
only the vertical cables are in that case contributing to the structure’s stiffness
under distributed load. Under asymmetrical loading on the other hand, are
the diagonal cables (loaded in tension) contributing. Diagonal cables do have
a positive influence on the structure’s stiffness under distributed loading when
they are not already linked with vertical cables and thus pretensioned.

6.5 CONCLUSIONS
6.5 CONCLUSIONS
As the introduction of a deployment mechanism changes the structural
behaviour of a traditional Tensairity structure, it is one of the objectives of
this research to explore the general behaviour of the deployable Tensairity
beam. This is done in this chapter by means of numerical investigations. More
specifically, the influence of various design parameters on the beams structural
behaviour are investigated.
The hull section of the Tensairity beam defines the tension forces present in the
connection between upper and lower strut. Numerical investigations revealed
that the shape of the hull of the Tensairity beam whereby the web experiences
a constant stress along the beam’s length, uses the material more optimal and
has the largest average stiffness. This can be clarified by the fact that a higher
stress in the web near the ends supports the struts and hinges up to a higher
load.
The influence of the struts section was shown to be related to the occurrence
or absence of bending moments in the struts. The strut’s cross section proved
to have no influence in the cases without bending moment in the struts. When
bending moments are introduced (mostly by cables), however, the bending
stiffness, and thus the cross section, plays an import role on the stiffness of
the deployable Tensairity beam. For cases with large deflections of the strut,
second order effects occurred and a large decrease in stiffness was detected. As
a result, the struts should be dimensioned to avoid these second order effects.
Finite element calculations showed that introducing hinges in the Tensairity
technology decreases the structure’s stiffness. The influence of the hinges is
relatively small when no bending moment occurs in the struts. However,
when large bending moments are introduced in the strut (eg. by means of
cables), the influence of the hinges on the deflection is more noticeable. A
variation in the amount of hinges showed little influence on the structure’s
stiffness. Furthermore, the numerical experiments revealed that the location
of the hinge along the beam determines the maximal load of the structure: the
closer the hinge is to the endpoint of the beam, the lower the load will be at
which no convergence of the finite element calculation is reached. Hinges at
the ends of the beam should thus be avoided as they are less stabilized by the
membrane.
The numerical investigations revealed the influence of cables and their configuration
on the structural behaviour of the deployable Tensairity beam. The
149

150
CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR
pretensioned cables contribute to the structure’s stiffness (until their pretension
becomes zero) by supporting the struts and hinges. As long as all cables
are pretensioned is the stiffness of the structure independent of the internal
pressure. Once a cable becomes slack, the stiffness changes. The maximum
load of the structure is thus related with the internal pressure.
Furthermore, the numerical investigations showed the importance of connecting
all the hinges of the upper strut with the lower strut by means of cables.
This keeps the hinges in position when inflating the airbeam and makes the
struts maintain their spindle shape. A deformation of the strut under inflation
decreases the structure’s stiffness considerably. The investigation of the cable
configurations and the cable tension also showed that diagonal cables are
not pretensioned under inflation when they connect hinges that are already
linked with vertical cables. This has as a consequence that only the vertical
cables are in that case contributing to the structure’s stiffness under distributed
load. Under asymmetrical loading on the other hand, are the diagonal cables
(loaded in tension) contributing. Diagonal cables do have a positive influence
on the structure’s stiffness under distributed loading when they are not already
linked with vertical cables and thus being pretensioned.
6.6 DISCUSSION
Although discussed separately in this chapter, the various design parameters
are related and influenced by each other. The shape of the hull section
determines together with the internal pressure the load by which the struts
are pushed outwards and thus the pretension in the cables that prevent this
outwards movement of the upper and lower strut. Because these pretensioned
cables can take compressive forces, they are able to block the hinges. The larger
the pretension in the cables, the higher the external load at which the hinge
will cause large displacements. As a consequence, one can conclude that an
increase in cable tension, caused by the outwards movement of the hull under
inflation, increases the structural performance.
However, the segments of the struts are also pushed outwards. This introduces
bending moments in these strut, which deform the segments of the strut. Their
deformation is thus related with the internal pressure. In the investigation
is shown that the stiffness of the Tensairity beam decreases with increasing
deformation of the strut. Thus, one can conclude from this point of view that
the increase of internal pressure has a negative influence on the structural
behaviour.

6.6 DISCUSSION
Two opposite influences of the internal pressure can thus be concluded: one
has a stabilizing function, i.e. the cable pretension, while the other destabilizes
or weakens the structure, i.e. the deflection of the strut. From the numerical
investigations seems that the destabilizing effect has the upper hand, while
experimental results (see chapter 8 prove that the pressure has a positive
influence on the structural behaviour. Because numerical simulations reflect
the ideal situation (no manufacturing imperfections) and the experimental
investigations the reality, more reliability is given here to the stabilizing effect
of the internal pressure and the cables. It is possible that this destabilizing
effect will become more pronounces for experiments under higher pressures.
These stabilizing and destabilizing effects need to be investigated more in
detail in further studies to be understood better and to be able to calculate
and predict which effect dominates in which situation (pressure, shape of hull
section, bending stiffness, ... ).
151

152
CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR

PART III
Prototype of a Deployable Tensairity Beam
153

Prototype and experimental set-up
7
In the previous two parts, deployable Tensairity structures were developed
and investigated by means of small scale models or numerical investigations.
This was in each case an approximation of how the structure would behave
in reality. In this part, a prototype for the deployable Tensairity structure is
developed and investigated on a larger scale. This way, insight will be gained
in the behaviour of real deployable Tensairity structures, manufactured with
the same materials and in the same way as real structures, and loaded with
representative loads.
The results of the experimental investigations are presented in the next chapter.
This chapter presents the prototype and the components it is constituted of in
detail, as well the way this prototype is investigated.
7.1 PROTOTYPE OF TENSAIRITY BEAM
The prototype of the five meter spindle shaped deployable Tensairity beam
is illustrated in figure 7.1. It has a slenderness of 10 and is constituted of
an airbeam with two airtight chambers, struts, hinges and cables. Because
it is one of the objectives of the experimental investigations to compare the
structural behaviour of the deployable Tensairity beam with the behaviour of
other already existing non-deployable Tensairity prototypes, the design of the
prototype and thus the various components will take the specifications of these
other prototypes into account. Examples are the shape of the hull, the position
of the cables and the strut’s section. This way, the prototypes are comparable
and allow to investigate the influence of one parameter, such as the effect of
hinges on the structure. This is discussed more in detail in chapters 8 and 9.
The various components are discussed in detail in the next sections.
155

156
CHAPTER 7 PROTOTYPE AND EXPERIMENTAL SET-UP
Figure 7.1: The five meter spindle prototype for a deployable Tensairity
beam.
7.1.1 MEMBRANE
The airbeam is in this prototype made of PVC coated polyester fabric (Valmex
7318) 1 . This technical membrane is widely used in the field of inflatable
structures and very likely to be applied for Tensairity structures. This is
because among other types of fabric, the PVC coated polyester fabric is relative
inexpensive and easily weldable. In addition, the polyester fibres are not brittle
(which is the case with PTFE coated glass fibres), which results in a membrane
that can be folded repeatedly without damage.
The spindle shaped airbeam is constituted of two separate airtight chambers,
adjacent to each other along the spindle’s length. This way, the upper and
lower strut can be connected by cables without risk for air leakages. Moreover,
holes can be made to accommodate the hinges in such a way that the membrane
lies in the same plane as the axis of rotation of the hinges 2 . This is illustrated
in figure 7.2.
The shape of the hull is chosen such that the normal force in the connecting
1 kN Producer: Mehler Texnologies, warp and fill tensile strength: 60 m , thickness: 0.85 mm,
Ewarp = 904 kN m , E f ill = 452 kN m , ν = 0.386
2In section 3.2 was seen that this is a requirement to guarantee a correct folding of the membrane
without stretching or wrinkling. Figure 3.3 on page 38 illustrates this.

1
2
3
4
7.1 PROTOTYPE OF TENSAIRITY BEAM
1 keder
2 strut
3 web for airtight
chambers - no stress
4 outer hull
4
50mm weld
Figure 7.2: The section of the airbeam: two separate airtight airbeams
are positioned adjacent to each other along the spindle’s length.
element between upper and lower strut is identical along the spindle’s length.
As explained in section 6.1, this implies that the radius is dependent of the
normal force desired in the connecting element at a certain pressure and the
height between the upper and lower strut.
In addition, to fold the Tensairity structure completely, the hull of the membrane
should allow an increase in length between upper and lower strut
when folding. Remember, as seen in section 3.4.4 (p. 57), some points of
the upper strut move away from the lower strut when folding the foldable
truss mechanism.
However, this latter requirement was not taken into account for the cutting
pattern for the prototype. There is chosen consciously to apply the same shape
of hull of other ongoing experiments on five meter Tensairity spindles. This
way, the experimental results of this prototype for a deployable Tensairity
beam are comparable with other investigations. Unfortunately, this implies
that the prototype will not be completely foldable.
The cutting pattern is developed by calculating at several positions along the
spindle’s length the circumference of one quarter of the hull. This is done by
means of equation 6.1 (p. 106). This length is then plotted along the spindle’s
curvature forming the cutting pattern, as illustrated in figure 7.3. The pattern
157

158
CHAPTER 7 PROTOTYPE AND EXPERIMENTAL SET-UP
of the two identical internal membranes closing the air chamber, called webs,
is more simple. They should be wider than the height between the upper
and lower struts. After all, these webs are used to make the two chambers
airtight and will thus not bear any tensile forces. Once the cutting patterns are
developed, the welding lines and markers are added.
496
15 mm welding area for connecting two halves and keder
4726
50 mm weld area for connecting two quarters
Figure 7.3: Development of the cutting pattern of a quarter of the spindle.
To obtain an airtight airbeam, capable of taking the stresses introduced by
the internal pressure, attention has to be paid to the welding procedure and
sequence. From a manufacturing point of view, the welding should occur in
a straightforward and logic way, whereby the welding is not obstructed by
other parts of the airbeam. The sequence of welding the cutting patterns is
illustrated in figure 7.4 and described here briefly. First, the quarters from one
chamber are welded together with an overlap of 5 cm (A). Then, the membrane
closing the air chamber (web) is welded on the inside of the hull (B). Finally,
the two halves are welded to a keder and each other (C). Notice that this
connection contains a weld that is perpendicular to the direction of membrane
stress. As a consequence, peeling of the membrane is risked here. However,
this weld and the keder will be clammed between the two struts. This way,
the tensile forces from the outer hull are taken by the struts and not the weld.
The outer hull of the airbeam contains two valves each chamber. This way,
the pressure can be monitored when inflating the airbeam. The hull is also
printed with a speckle pattern for measuring purposes, as will be mentioned
further in this chapter.
7.1.2 HINGES
As already discussed in detail in section 3.4.2 (p. 53), two types of hinges
exist, the end supports not taken into account. One type rotates towards the

A
C
Figure 7.4: Sequence of welding the cutting patterns.
7.1 PROTOTYPE OF TENSAIRITY BEAM
air beam, the other outwards. Because the membrane has to lie in the same
plane as the center of rotation of the hinge, this has as consequence that each
type requires another design. Otherwise, the membrane will be stretched or
wrinkled. Figure 7.5 and 7.6 shows both hinges in combination with the
membrane.
Figure 7.5: Geometry of the two hinges in combination with the
membrane.
The hinges are first modeled in such a way that they can be fixed around the
end of a segment. This way, no tooling of the segments are necessary and
an easy connection can be achieved. This design is illustrated in figure 3.22
(p. 55). However, this design is very complex and is as consequence difficult
to manufacture and thus relative expensive. Therefore, some design iterations
B
159

160
CHAPTER 7 PROTOTYPE AND EXPERIMENTAL SET-UP
Figure 7.6: The folding sequence of the two hinges.
were conducted. The kinematic of the hinge is identical to the first version, only
the materialization differs. The final design is not a solid three dimensional
hinge, but composed of several flat plates. The main advantage of this design
is a simplified, faster and thus less expensive manufacturing 3 . The new design
requires the struts to be manufactured such that the plates can be fixed to it.
Figure 7.7 gives a more detailed view of the hinges. As can be seen, the hinges
are composed of flat plates, manufactured by CNC in aluminum, and steel
pins, that form the axes of rotation. The middle plates in hinge 02 keep the two
axes at a constant distance and limit the rotation of the hinge to 180 degrees.
This latter is achieved at hinge 01 by means of the geometry of the flat plates
themselves.
In the figure can also be seen that holes are made in the steel pins to accommodate
the cables that connect upper and lower strut. This way, the cables can
rotate and their normal forces are transferred directly to the hinges. Note that
the holes in the three pins of hinge 02 are shifted from each other such that the
ends of the cables are adjacent and do not obstruct each other. Because of the
cable ends are positioned next to each other, enough space has to be provided.
As a consequence, the outer plates of hinge 02 are fixed to the sides of the strut.
This way, the forces need to be taken up by the connection between the hinge
and the strut, which is not optimal.
7.1.3 STRUTS
Segments
The upper and lower strut of the Tensairity beam consist of several aluminum
segments, connected with hinges. The cross section of the segments is chosen
3 The hinge constituted of plates costs 4,5 times less than the solid hinge.

1
2
1 2
Figure 7.7: Detailed view of the two hinges.
7.1 PROTOTYPE OF TENSAIRITY BEAM
161

162
CHAPTER 7 PROTOTYPE AND EXPERIMENTAL SET-UP
to be identical to ongoing experimental investigations on Tensairity structures
with continuous struts, obvious for reasons of comparison. Their section
measures 40 mm wide and 15 mm high.
As mentioned in the introduction on Tensairity structures, it is important
for the stabilization of the compression element to have a good connection
between this compression element and the hull of the airbeam. From previous
experiments is learned that connecting the compression element to the hull by
means of a pocket (as with tent poles of an iglo-tent) is not an optimal solution
because play between the elements still exist and decreases the structure’s
stiffness considerable. Therefore is chosen to clamp the membrane and a keder
between the struts. These struts are bolted together at regular distances. This
is illustrated in figure 7.8. This way, a tight connection between the membrane
and the struts is realized. The membrane stress is taken up by the struts, the
keder prevents the membrane from slipping from the clamped struts.
1
2
3
1 keder
2 strut
3 web for airtight
chambers - no stress
4 outer hull
4
Figure 7.8: Detail of the connection between strut and membrane.
The length of the segments is dictated from the foldable truss mechanism,
illustrated in figure 7.9 and discussed in chapter 3. The mechanism was
initially constituted of segments with three different sizes. The segments of
the lower strut were all 1
6th of the struts length (L), the middle upper segments
measured each L
12
and the two side segments were L
4
long. However, because
of reasons of comparison with another Tensairity prototype, there is chosen
to align the second lower hinge with the second upper hinge, indicated in
figure 7.9. Therefore, the mechanism is adapted, which has as consequence
that a larger variation of length is needed.
The segments are manufactured such to accommodate the hinges, that are
composed of plates. In the case of hinge 01, material of the segment is milled

D
A B C B B C B
E
F
F
7.1 PROTOTYPE OF TENSAIRITY BEAM
Figure 7.9: The geometry of the segments is dictated by the foldable
truss mechanism.
away to fit the plates. This is not necessary in the case of hinge 02, since the
plates are attached on the sides of the segments. Off course, holes are provided
to connect both elements together. Figure 7.10 gives a detailed view of two
segments.
From straight to curved
Figure 7.10: A detailed view of two segments.
In the case of a regular Tensairity beam, a straight and continuous aluminum
strut is used as compression element. The section of the strut is such that the
bending stiffness allows it to be curved along the spindle when being attached
to the airbeam and the supports. With the struts of the deployable Tensairity
beam, this bending will not occur due to the presence of the hinges. Indeed,
these hinges can not transfer bending moments. But because these hinges
can rotate the struts along the curve of the spindle and because the maximum
angle between the rotated struts would be less than 4 degrees, there is chosen to
design as well the upper and lower strut of the prototype as straight elements.
The struts are illustrated in figure 7.11.
However, when inflating the prototype for the deployable Tensairity beam,
the straight segments cause problems. After all, because the straight segments
E
A
D
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Figure 7.11: The upper and lower strut of the prototype are conceived
as straight struts, as is done with regular Tensairity beams. Half of the
upper and lower strut is shown.
are positioned along a curve, they are not perfectly in line with each other
but already angulated. Note that the direction of rotation is fixed, because the
hinges are only allowed to rotate in one direction.
When loading this Tensairity beam with a uniform distributed load, the upper
strut becomes compressed. As a result, the segments rotate further. This means
that the external loading is not fully transferred to an increase in compression
in the struts themselves. Instead, the loading is also partially taken by a
change in geometry of the segments and by the airbeam where some hinges
are pressed into.
The prototype with these straight and thus kinked segments is investigated
experimentally. Figure 7.12 shows the prototype with straight segments under
inflation. Notice the kinks in the struts. Because of the initial compression in
the struts after inflation, it is clear that - even without external loading - the
hinges are already loaded considerable. As a result, the connection between a
hinge and strut failed after some loadings.
Figure 7.12: The prototype with straight segments under inflation. The
kinks are very visible.
Off course, for decent experimental investigations and reliable results that
represent well the structural behaviour of the deployable Tensairity beam, the
prototype needs to be adapted. The aluminum struts are bended according
the spindle’s curve by means of a three-point bending: a deflection is applied
between two points of the strut and this several times along this strut. The
deflection is such that the segment is plastically deformed and that residual
deformation, also called sag, corresponds with a predefined value. When

7.1 PROTOTYPE OF TENSAIRITY BEAM
the strut is bended with this value at regular distances, the curvature is well
approximated. To give an idea of the curvature: with a distance of 30 cm
between the supports, the sag has to measure 0,8911 cm.
The bended segments are repositioned along the membrane and keder. Where
necessary, new holes in the membrane are made one or two millimeters next
to the old ones. Through these holes, the bolts are placed that clamp the two
halves of the strut together with the membrane and keder. After inflation, the
result is a spindle shaped Tensairity beam with unstressed, compatible, curved
struts , as can be seen in figure 7.13
7.1.4 CABLES
Figure 7.13: After bending and repositioning the struts is a spindle shape
achieved with unstressed, compatible, curved struts.
A connection between upper and lower strut is necessary in this prototype for
multiple reasons. Without connection, the struts are moved outwards under
inflation and the section of the inflatable volume tends to become a circle. This
implies that, as discussed in section 7.1.1, not enough membrane would be
present to allow the hinges of the mechanism to move away from each. As a
result, the membrane can not fold completely. Another reason is that such a
connection between upper and lower strut increases the structure’s stiffness,
as shown in section 6.1. Because a membrane as connecting element between
the struts would obstruct the folding, cables are applied for this prototype.
The cables are stainless steel ropes with diameter of 5 mm, containing (M6)
threaded ends. They fit through the holes of the pins of the hinges, as described
before (figure 7.14). This way, they can rotate free and are connected directly
with the hinges. By rotating the nut, the length of the cable can be adjusted.
Figure 7.15 shows all cables applied in the prototype. Various combinations are
tested, as will be shown in the next chapter. As can be seen from the drawing,
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Figure 7.14: The stainless steel ropes fit through the holes of the pins of
the hinges. They contain threaded ends to adjust their length.

7.1 PROTOTYPE OF TENSAIRITY BEAM
as well vertical as diagonal cables are used. The vertical cables are applied
in the prototype for reasons of comparison with other Tensairity prototypes.
The diagonal cables are (together with the middle vertical cable) those that
allow complete folding of the Tensairity structure, as illustrated in figure 3.26
on page 58.
1 2 3 4
D C
B A
Figure 7.15: The stainless steel ropes fit through the holes of the pins of
the hinges. They contain threaded ends to adjust their length.
7.1.5 END PIECES
The upper and lower strut are at their both ends connected with each other
and with the supports by means of a steel end piece, illustrated in figure 7.16.
This is conceived such that it directs the struts along the curve of the spindle.
In addition, its geometry makes sure that the center lines of the struts intersect
each other in the axis of rotation of the supports. Otherwise, additional
bending moments at the ends will occur, as shown in Teutsch (2009).
Figure 7.16: The upper and lower strut are at their both ends connected
with each other and with the supports by means of a steel end piece.
Notice that the end piece does not allow rotation of the segments that are
connected with it. This means that with these end pieces, the Tensairity
beam can not fold completely. However, for reasons of comparison with
other Tensairity prototypes, this end piece is applied for the experimental
investigations. Another ending that allows rotation of the segments is designed
and manufactured, but as mentioned not applied in the experimental
investigations. After all, finite element investigations shows no big influence
on the structural behaviour of allowing the segments to rotate in relation with
the end pieces or not.
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7.1.6 WEIGHT OF PROTOTYPE
The previous sections show that the prototype for a deployable Tensairity
structure is constituted of various elements, all necessary for an optimal
kinematic and structural behaviour. Table 7.1 gives the weight of the various
elements the prototype is constituted of and their contribution to the total
weight of this light weight structure.
Table 7.1: Weight of the various elements the Tensairity prototype is
constituted of and their contribution to the total weight.
element weight [kg] percentage of total weight [%]
membrane 13,0 41,1
segments 14,1 44,4
hinges 2,3 7,3
cables 1,2 3,8
bolts & nuts 1,1 3,4
total 31,7 100

7.2 THE EXPERIMENTAL SET-UP
7.2 THE EXPERIMENTAL SET-UP
The prototype for a deployable Tensairity beam, explained in detail in the
previous section, is investigated experimentally. This section describes the
experimental set-up. More precise, the test-rig wherein the structure is placed,
the way loads are applied and how the displacements are measured, is discussed
here.
7.2.1 TEST RIG
The Tensairity beam is investigated experimentally in the test rig of the ‘Empa-
Center for Synergetic Structures’ in Zurich (Switzerland), illustrated in figure
7.17. The test rig is a stiff steel frame that contains the supports for the
Tensairity beam. The five meter long structure is isostatically supported by
means of the end pieces that are attached to the roller and hinged support.
In addition, this frame holds the motors and the ‘balance system’ that control
the loading of the beam. Loading the structure is done upside-down in this
experimental set-up: the beam is positioned upside-down in the test rig and
the loads are thus applied upwards at the compression strut (placed at the
bottom of the beam). Two motors are used - one for the left and one for the
right part of the beam - because this way asymmetric loads can be applied.
The upwards pulling of the motors is transferred to the compression strut by
means of the balance system. This system is like a balance in equilibrium,
which makes sure that the same load is applied at all points along the curved
strut. A more detailed view of the balance system can be found in figure 7.18.
7.2.2 LOADING
The loading of the structure is displacement controlled: the motors pull the
balance system in a certain time over a certain distance (and thus with a
certain speed). The user can control this time and distance by means of input
parameters. The motors monitor then which forces are applied to reach the
predefined displacement of the balance system (and thus the deformation of
the structure).
The value of this chosen displacement varies during the experiments, because
it depends on the configuration (cable configuration and internal pressure )
and the purpose of the experiment (measuring stiffness and/or the maximum
load). The time for one load cycle is for all experiments identical and measures
180 seconds. Several load cycles are applied on one structure, to allow the
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Figure 7.17: The test rig contains the supports, motors and balance
system.

Figure 7.18: Detail of the balance system.
7.2 THE EXPERIMENTAL SET-UP
structure to ‘accommodate’ to the loading, as explained in section 4.2.1 on
page 68. Figure 7.19 gives an example of the applied load as function of
the time (left) and the load in function of the displacement (right). The small
circles represent the displacement of the motor in each picture taken with the
optical measurement instrument. This is explained in section 7.3.
7.2.3 MEASURING
The structural behaviour of the investigated Tensairity beam can be derived
from the data obtained from the motors: the forces and displacements. However,
these displacements are the average displacement of one half of the beam.
No information regarding the displacements of specific points can be derived
from this data.
Therefore, displacements of the structure are at the same time measured by
means of the optical ‘digital image correlation’ method (DIC), that employs
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Displacement (mm)
25
20
15
10
5
0
Displacement of the motors
−5
−200 −100 0 100
Time (s)
200 300 400
load [N]
5000
4000
3000
2000
1000
0
0 5 10 15 20 25
displacement [mm]
Figure 7.19: Left: an example of the applied load as function of the time,
right: the applied load in function of the displacement.
tracking and image registration techniques for accurate 2D and 3D measurements.
Images from the structure during the load cycle are captured every
ten seconds. The pictures are taken with two camera’s, because this way,
measurements in 3D can be done.
As can be seen in figure 7.20, the displacements of specific points are tracked
by means of markers attached to the struts. The hull of the membrane contains
also a random speckle pattern, for measuring displacements of the membrane.
The image captured from the left and right camera are shown in figure 7.20
Figure 7.20: Markers and a speckle pattern are applied on the prototype
for measuring the displacements by means of DIC. The image captured
from the left and right camera are shown.
load [N]

7.3 POSTPROCESSING
7.3 POSTPROCESSING
The data obtained during the measurements, can now be processed to become
useful results. The forces and displacements of the motors are acquired
immediately and do not need many additional handling. The data from the
optical measurements need to be postprocessed before the displacements can
be visualized.
The markers are selected and numbered in a reference picture in the DICsoftware
4 . Then, the software will track the markers in the other images
captured during that load cycle. This is possible, because these markers have
a specific pattern and contrast that allows the software more easily to recognize
them. However, this is sometimes not the case. The markers need to be selected
manually. There must be noticed that in this latter case, the measurements are
less accurate than in the case of automatically tracking by the software.
The speckle pattern of the hull of the airbeam can also be processed in the
DIC-software to investigate the displacements of the hull. However, since the
focus of this research is mainly on the structural behaviour of the structure
and thus the displacement of the struts, the behaviour of the membrane is not
processed for every configuration.
When all markers have been tracked in all pictures, their position is exported.
This position is in relation to a global fixed coordinate system that was defined
during calibration. This calibration is also necessary to allow the software to
calculate the distance between markers in reality. Now the position of the
markers and thus the displacements is known (in relation to the time), the
data from the motors needs to be synchronized with the data from the optical
measurement instrument to determine the load at which each picture is taken.
This is done by means of a routine scripted in the mathematical software
‘Matlab’ that compares the displacement of the motor with the displacement
of the motor in the images.
The graph on the left in figure 7.19 visualises the outcome of this synchronization.
The displacement of the motor obtained from the optical measurements
is indicated with a small circle, the continuous line represents the actual force
displacement of the motor. When synchronized, the displacements of the two
measurement instruments (motor and DIC) are identical. This way, the load
at which each picture is taken can be determined.
4 VIC-3D 2009 from Correlated Solutions is used.
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The position of each marker can be tracked in every picture, and thus also
the displacement of the markers. By doing so, graphs can be generated to
investigate eg. the displacement of marked points on the upper strut during
loading or to compare several configurations at a certain load. An example is
given as illustration in figure 7.21. These results are analyzed and discussed
in the next chapter.
Vertical displacement (mm)
Vertical displacement (mm)
0
10
20
30
B dist150
40
−2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 2500
Position x (mm)
0
10
20
30
40
50
60
70
A dist150
80
−2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 2500
Position x (mm)
Figure 7.21: Example of graphs that can be generated with the
experimental data to investigate the structural behaviour of several
cases. Left: displacement of the strut of one case during loading, right:
displaced strut of several cases at one load.
7.4 GOAL OF THE EXPERIMENTAL INVESTIGATION
Before conducting experiments and analyzing results, it is important to know
what is aimed for to learn from these experiments.
The main focus of this part of the research is to gain insight in the general
structural behaviour of the deployable Tensairity beam. After all, such a
structure has not been investigated yet. Therefore, the maximal load and

7.5 OBSERVATIONS AND IMPROVEMENTS
stiffness of some representative configurations under various pressures are
noted during the experiments. These data are then compared with other nondeployable
Tensairity structures, manufactured with the same materials and
with comparable geometry. This way, the deployable prototype is evaluated.
In addition, it is also the goal of the experiments to study the influence of the
configuration of the cables and hinges on the structural behaviour. Therefore,
the displacements of several points along the struts from some configurations
are measured by means of the markers and the optical measurement instruments.
This way, insight is gained in the contribution of vertical and diagonal
cables and the influence of the position of the hinges.
These data are then analyzed in the next chapter to evaluate the current
proposal for a deployable Tensairity beam and to formulate improvements
for future designs.
7.5 OBSERVATIONS AND IMPROVEMENTS
The prototype is being observed during the experiments. In general, the
design of the deployable Tensairity structure is satisfying, but some issues
- open to improvement - are detected. This can be taken into account in
a next prototype for a (deployable) Tensairity structure. Two main issues
originate from the friction between the webs and the hull under pressurized
air chambers.
When the airbeam is not fully pressurized, the webs are - due to gravity -
hanging from the struts downwards, as illustrated in figure 7.22. As a result, a
large part of the web is lying at the bottom of the airbeam. When inflating the
air chambers, there can be expected that the pressure distributes the webs along
the center, as shown in the middle of figure 7.22. This also happens in reality,
but not completely. At the moment the chambers become pressurized and
contact between the webs occur, they do not move anymore. As illustrated on
the right of figure 7.22, the weld of the web becomes stressed as consequence.
The welding in this prototype is not designed to take up tensile forces, since
peeling can occur. Therefore, in a new design, there has to be taken into
account that the web also wil carry some tension and the welding to the outer
hull has to be conceived as such.
Another issue caused by the friction between the webs, is the obstruction of the
cables. When no overpressure is present in the airbeam, the segments of the
struts are not yet pushed outwards. This means that the cables are slack and
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web
before inflation as it should be after inflation
as it is after inflation
Figure 7.22: Due to friction of the web, the upper welding between outer
hull and web may become stressed.
position themselves ‘randomly curved’ between the web. When the airbeam
is being inflated, the struts are pushed outwards and the webs are pressed
together. There can be expected that the cables straighten themselves when
being tensioned. However, observations show that the friction between the
cable and the webs is such that the cable is jammed between the webs. As a
result, the cables are not straighten and the struts are held in a wrong position
by the cables, because not enough cable length is available. This phenomenon
is avoided in these experiments by manually straightening the cables when
the airbeam is being inflated. In a new design, this can be avoided by placing
the cables in tubes that stay in their correct position. This way, the cables are
not hindered anymore by the webs.
7.6 SUMMARY
When investigating a prototype experimentally, attention should not only
be paid to the final results and conclusions of these experiments, but also to
the prototype itself and how it is investigated. After all, not only are the results
influenced by the manufacturing and testing, the development of a prototype
is also very instructive and a research on its own. Therefore, the prototype and
the components it is constituted of were discussed in detail in this chapter, as
well the way of investigating this prototype.
The prototype contains an airbeam made from pvc coated polyester fabric, a
common and flexible material suitable for inflatable structures. The airbeam
showed to have a well thought-out cutting pattern and welding procedure,
which is a requirement for an airtight component. However, because of friction
between the webs, the welding procedure should take into account that the
web can become stressed. The two separate chambers proved to be efficient
for allowing holes in the hull. This way, connecting the struts with cables can
be done easily and some segments of the hinges can be accommodated.

7.6 SUMMARY
The two types of hinges were materialized by flat aluminum plates. After
all, a solid three-dimensional design for the hinges proved to be too complex
for manufacturing. Therefore, a redesign constituted of flat elements was
proposed and manufactured. The hinges demonstrated to have the kinematics
as they were conceived. Also the compatibility with the airbeam and thus the
correct folding of the membrane is demonstrated with the hinges.
Because the strut of a Tensairity beam needs to be stabilized as much as possible
by the membrane, attention was paid to the connection of those two. The
prototype proves that the clamping of the membrane with a keder between
two halves of the strut is a decent solution. However, this tight connection
can cause problems when the segments of the struts are straight. After all, the
incompatibility between straight segments and a curved airbeam results in a
deformed prototype with an incorrect, kinked shape and additional, unwanted
stresses in the components. From this prototype is learned that the hinged
segments should be curved according the spindle’s shape and that attention
should be paid to a careful positioning of the struts along the airbeam.
The connection between the upper and lower strut proved to be well designed.
The fixation of the cables to the hinges by means of a threaded end and a nut
was satisfying. However, future designs should take the friction between the
webs and the cables in inflated state into account. Otherwise, the cables risk to
be jammed between the webs in a ‘curved’ position, which has consequences
for the final shape and structural behaviour of the prototype.
The total weight of the prototype was calculated, as well the contribution of
each element separately to this total weight. As can be expected, the struts
are the heaviest components with a contribution of 44,4% to the total weight
of 31,7 kg. Somehow surprising is the large contribution of the membrane
(41,1%) to this total weight.
In addition to the prototype, the experimental set-up was discussed. The testrig,
constituted of a stiff steel frame, the supports, the balance system and the
motors responsible for the loading, was presented in detail. The loads are
applied upside-down to the Tensairity beam by means of a balance system
to make sure the same force is applied everywhere. This balance system is
controlled by the displacement-driven motors.
The applied load to reach a certain displacement are recorded by the motors.
An optical measurement instrument, constituted of two camera’s, records the
displacement of predefined points of the strut. The data from this optical
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measurement instrument needs first to be postprocessed before it can be used
for analysis.
Now is understood how the prototype is manufactured and how it is investigated
experimentally, the results from these experiments can be discussed.
Various configurations are tested in order to reveal the influence on the load
bearing behaviour of the deployable Tensairity beam. This is presented in the
next chapter.

Experimental investigation of the prototype
8
The five meter long prototype of the deployable Tensairity beam is investigated
experimentally. This chapter presents the results of these experiments. The
general behaviour of the deployable Tensairity beam, such as its stiffness and
maximal load, is first discussed. Then, various configurations are studied to
analyse the influence on the structural behaviour of the cables and hinges.
The experimental results are throughout the discussions compared with the
outcome of numerical calculations on the finite element model of the prototype.
8.1 CONFIGURATIONS
To study the influence on the structural behaviour of different parameters,
various cases are investigated experimentally. Figure 8.1 gives an overview.
Ten configurations, indicated with a letter from A - J, differ in cable configuration
or location of hinge, but are all constituted of the same airbeam and
struts. Different pressures and motor displacements are applied with these
configurations. The combination of configuration, pressure and displacement,
results in more than sixty investigated cases.
All combinations are investigated experimentally to gain insight in the structural
behaviour. However, not all data is presented in this chapter. A synthesis
of the findings is given, together with plots of several cases as illustration.
8.2 GENERAL STRUCTURAL BEHAVIOUR
In this section, the general structural behaviour, such as the characteristics
of the load-deflection behaviour, the influence of the pressure on the stiffness
and the maximal load the structure can bear, is discussed.
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A
B
C
D
E
F
G
H
I
J
Figure 8.1: Overview of all investigated combinations. The configura-
tions are illustrated on the left and indicated with a letter, the pressures
and displacements applied in the experiment are given on the right.

8.2.1 LOAD-DEFLECTION BEHAVIOUR
8.2 GENERAL STRUCTURAL BEHAVIOUR
As already mentioned in the previous chapter, the deployable Tensairity beam
is loaded and reloaded several times. This is because the prototype, just like
the small scale models in chapter 4, has to adapt first to the new load condition
during the first load cycle, creating a residual displacement. This can be
seen in figure 8.2 (left), showing the experimentally obtained load-deflection
behaviour of a configuration during the three load-cycles . As a consequence,
only the load-deflection curve of the third load-cycle is used for analysis of the
results in this research.
load [N]
5500
5000
4500
4000
3500
3000
2500
2000
1500
1000
500
0
0 5 10 15 20 25
average displacement [mm]
load [N]
5500
5000 initial bending failure
4500
4000
3500
3000
2500
2000
1500
1000
500
0
0 5 10 15 20 25
average displacement [mm]
Figure 8.2: The experimentally obtained load-deflection behaviour of
one configuration. Left: the different curves during the three load-cycles.
Right: three parts can be distinguished in the load-deflection curve.
This curve is illustrated on the right of figure 8.2. Three main parts can
be distinguished in this curve: the initial, the bending and the failure part.
The data used for the calculation of the average stiffness are taken in the
bending part, where the structure is loaded with representative loads. The
structure’s response to the load is linear here. The initial part shows generally
a different average stiffness than in the bending part when the structure is
loaded distributed. An explanation for this other behaviour at low loads could
not be verified with these experiments and will thus not be covered in this
research 1 . The failure part contains the ‘collapse’ of the structure. The stiffness
of the structure decreases drastically here due to failure or deformation. This
1 Possible influences can be the arched struts or the pretensioned cables.
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‘collapse’ and the bending part will be discussed more in detail further in this
chapter.
8.2.2 MAXIMUM LOAD
The ‘maximum load’ is in this research considered as the load whereby the
stiffness of the structure decreases drastically, due to failure or deformation.
This load can be seen in the average load-deflection behaviour by a ‘flattening’
of the curve, as illustrated in figure 8.3. The experimental maximum
distributed load of configuration A is plotted in this figure for three pressures.
The average deflection is determined by the average value of the displacement
of nine measurement points on the loaded strut.
Total load (N)
8000
7000
6000
5000
4000
3000
2000
1000
050 mbar
150 mbar
250 mbar
0
0 10 20 30 40
Vertical displacement (mm)
Figure 8.3: The maximum load is reached when the stiffness decreases
drastically. The load-deflection of configuration A is plotted for three
pressures under distributed load. (Experimental results).
In chapter 4 (section 4.3) and chapter 6 (section 6.4.3) is shown that a change
in stiffness of the deployable Tensairity beam is caused by the slackening of
a pretensioned cable under loading. This has as result that the hinge is not
supported anymore and experiences larger displacements under the same load
increment. From the experiments is observed that the decrease in stiffness of
the structure is indeed caused by the inwards rotation of one hinge (hinge
2). Figure 8.4 illustrates this by plotting the experimental displacement of
the hinges (and other measurement points) of the loaded strut at several load
values. The hinges are indicated. The larger deflection of hinge 2 from a
certain load on is noticeable.
This configuration A of the deployable Tensairity spindle is also investigated

A
Vertical displacement (mm)
0
5
10
15
20
25
30
35
1
1
hinge 1 2
3
4
40
−2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 2500
Position x (mm)
5
6
7
8.2 GENERAL STRUCTURAL BEHAVIOUR
2 3 4 5 6 7 hinge moving
outwards of beam
2 3 4 5 6 7
hinge moving
inwards of beam
Force = 0N
Force = 7N
Force = 417N
Force = 845N
Force = 1228N
Force = 1601N
Force = 2013N
Force = 2484N
Force = 2984N
Force = 3485N
Force = 3985N
Force = 4489N
Force = 4999N
Force = 5191N
Force = 5069N
Figure 8.4: The experimental displacement of the hinges (and other
measurement points) of the loaded upper strut at several load values of
case A under 150 mbar. The hinges are indicated. The larger deflection
of hinge 2 from a certain load on is noticeable.
with a finite element model 2 . The calculations do not converge anymore when
cables 2 and 6 become slack and hinges 2 and 6 as consequence experience too
large displacements. Thus, the same hinge as in the experiments causes in the
finite element model the large displacements. Figure 8.5 illustrates the shape
of the spindle after loading, derived from the finite element analysis.
Figure 8.6 plots the numerical derived load-displacement of the case A under
50, 150 and 250 mbar.
Figure 8.7 compares the experimental and numerical load-displacement of
case A under 150 mbar. The stiffness of the numerical model is higher than
the experimental, as can be expected. After all, the numerical value contains
no (manufacturing) imperfections. The experimental maximum load is higher
than the numerical. This can also be attributed to the imperfections of the
experimental model. After all, it is possible due to these imperfections that not
2 The hinges 1, 3, 5 and 7 of the upper strut are blocked to move inwards, like it in reality.
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2 6
Figure 8.5: The same hinge as in the experiments causes in the finite
element model the large displacements. This shape after loading is
derived from the finite element analysis.
load [N]
8000
7000
6000
5000
4000
3000
numerical results − distributed load
2000
050 mbar
1000
150 mbar
250 mbar
0
0 0.01 0.02 0.03 0.04
average displacement [m]
Figure 8.6: The numerical derived load-displacement of the case A
under 50, 150 and 250 mbar.
8000
7000
6000
5000
4000
3000
2000
1000
numerical
experimental
0
0 0.01 0.02 0.03 0.04
Figure 8.7: The experimental and numerical load-displacement of case
A under 150 mbar.

8.2 GENERAL STRUCTURAL BEHAVIOUR
all forces are transferred completely to the decrease in cable tension. However,
this is an assumption that can not be verified because the cable tension of the
experimental model could not be measured.
8.2.3 STIFFNESS
In chapter 4 and 6 is discussed that the stiffness of the structure under distributed
load is independent of the internal pressure, as long as all vertical
cables are pretensioned and thus supporting the struts and hinges. This is
also the case with the experimental investigated prototype: the stiffness (in the
linear part of the load-displacement curve) of the three pressure cases are very
similar, as shown in figure 8.3. This confirms the influence of the prestressed
cables.
In the case of asymmetrical loading, more influence on the stiffness from the
internal pressure can be detected. After all, the ‘meshes’ of the struts are
deformed in a parallelogram and the support of the airbeam has an influence
on this. The higher the internal pressure, the less easy these ‘meshes’ will
deform. Figure 8.8 plots the experimental derived stiffness of the asymmetrical
loaded configuration A under 50 and 150 mbar.
Total load (N)
3500
3000
2500
2000
1500
1000
500
experimental results − asymmetric loading
case A − 150
case A − 050
0
0 5 10 15 20
Vertical displacement (mm)
Figure 8.8: The experimental derived stiffness of the asymmetrical
loaded configuration A under 50 and 150 mbar.
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8.3 INFLUENCE OF CABLES
In the previous section, the general load-bearing behaviour of the deployable
Tensairity beam (case A) has been related with the influence of pretensioned
cables connecting upper and lower strut. In this section, various experimental
investigated configurations are compared to illustrate and confirm the effect
of the configuration of cables on the structure’s behaviour. First, the vertical
cables are discussed, then the diagonals.
8.3.1 VERTICAL CABLES
The load-deflection curves of cases B, C and D, pressurized till 150 mbar and
loaded to a displacement of 10 mm, are compared. The configurations have
the same diagonal cables, but differ in amount of vertical cables: case B has
7 vertical cables, case C one and case D zero. From the curves, illustrated
in figure 8.9 is clear that, for both load cases, there is a large difference in
average stiffness between the case with 7 vertical cables (B) and with one (C)
or zero (D). Comparison of cases C and D shows that one extra vertical middle
cable has little influence on the structures stiffness, as well for distributed as
asymmetrical loading. The comparison of cases E and F confirms this.
Configuration B is approximately a factor 4 stiffer than configurations C and
D. This can be attributed to the presence of the vertical cables, due to a
combination of reasons.
load [N]
4500
4000
3500
3000
2500
2000
1500
1000
500
0
0 2 4 6 8 10
average displacement [mm]
B
D
C
load [N]
4500
4000
3500
3000
2500
2000
1500
1000
500
0
0 2 4 6 8 10
average displacement [mm]
Figure 8.9: The load-deflection curves of configurations B, C and D,
pressurized till 150 mbar and loaded to a displacement of 10 mm. Left:
distributed load, right: asymmetrical load. (Experimental results).
B
D
C

8.3 INFLUENCE OF CABLES
All hinges in case B are supported by pretensioned (mainly vertical) cables
that can take up compressive forces. As a result, the loaded strut is better
supported. Because not every hinge in cases C and D is connected with a
cable, the struts do not keep their spindle shape after inflation. The hinges are
pushed outwards by the internal pressure and the struts deform noticeable in a
zigzag-shape, as already discussed in section 7.5 and illustrated in figure 8.10
(upper). This has a negative influence on the structural behaviour. After all,
when external load is applied on the strut, this loading is not fully transferred
to compressive forces in the strut (as is the case for configuration B), but also
to a change in geometry of the strut. Figure 8.10 (lower) shows the displaced
upper struts of cases B, C and D at the same distributed load (1200 N).
y postion [mm]
400
300
200
100
0
−100
−200
−300
Spindle shape of cases after inflation
−400
−3000 −2000 −1000 0 1000 2000 3000
x postion [mm]
Vertical displacement (mm)
0
2
4
6
8
10
12
14
−2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 2500
Position x (mm)
Figure 8.10: The displaced struts of cases B, C and D. Upper: after
inflation, Lower: at 1200N. (Experimental results).
B
D
BD150
CD150
DD150
This deformation of the upper strut under inflation when not all hinges
are connected with a cable is also studied numerically and concluded in
section 6.4.3. The simulations confirm a decrease in stiffness due to the
deformation of the strut, as mentioned above.
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CHAPTER 8 EXPERIMENTAL INVESTIGATION OF THE PROTOTYPE
Configurations G and H differ in the presence of three vertical cables. However,
because these vertical cables connect the hinges that are already linked with a
diagonal cable and because there are still hinges not connected with a cable,
the same issues as described above are present. This has as result that the three
extra vertical cables of case H do not contribute to an increase in stiffness and
cases G and H have as result very similar load-deflection behaviour.
8.3.2 DIAGONAL CABLES
Configurations A and B differ in the presence of diagonal cables and are thus
ideal to verify the influence of these diagonals. The load-displacement of these
configurations pressurized till 150 mbar is presented in figure 8.11.
From the graphs can be seen that for an asymmetrical load case, the configuration
with diagonals (B), has a higher stiffness at higher loads. However,
this difference in stiffness between A and B is very little. In the case of a
symmetrical loading is the configuration without diagonals stiffer.
Total load (N)
6000
5000
4000
3000
2000
1000
distributed load asymmetric load
6000
A dist150
0
0 10 20
B dist150
30 40
Vertical displacement (mm)
Total load (N)
5000
4000
3000
2000
1000
A asymm 150
B asymm 150
0
0 10 20 30 40
Vertical displacement (mm)
Figure 8.11: The load-displacement curves of configurations A and B.
Left: distributed load, right: asymmetrical load. (Experimental results).
The displacement of the loaded (upper) strut of configurations A and B under
asymmetric load (2950 N and 150 mbar) is shown in figure 8.12. From the
displaced struts can be seen that the displacements are lower in case B, as
well on the loaded side (right half), as the unloaded. This is due to the
positive influence of the diagonal cables. This is confirmed by the numerical
investigations in section 6.4.3, where configuration B indeed has a higher
stiffness that configuration A due to the presence of diagonal cables.
Figure 8.13 shows the displacement of the loaded (upper) strut of configurations
A and B under a distributed load of 3950 N. From this figure and the

Vertical displacement (mm)
0
10
20
30
40
50
A asymm 150
B asymm 150
8.3 INFLUENCE OF CABLES
60
−2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 2500
Position x (mm)
Figure 8.12: The displaced struts of configurations A and B under
asymmetrical load of 2950 N and internal pressure of 150 mbar.
(Experimental results).
load-displacement graph in figure 8.11 (left) can be seen that configuration B
experiences a larger displacement under distributed load than configuration
A. This larger displacement in the case of configuration B is located mainly
at the hinge that is allowed to move downwards and that is connected with
diagonal cables. There can be assumed that the diagonal cables cause this
larger displacement.
However, from the numerical investigations of configuration A and B conducted
in chapter 6, is concluded that configuration B is stiffer under distributed
loading than A. This is because some diagonal cables are there tensioned
under inflation (because they are connected with a hinge not connected
with a vertical cable), which has as result that they can take some loading and
thus support the hinges.
Vertical displacement (mm)
0
10
20
30
A dist150
B dist150
40
−2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 2500
Position x (mm)
Figure 8.13: The displaced struts of configurations A and B under
symmetrical load of 3950 N and internal pressure of 150 mbar.
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CHAPTER 8 EXPERIMENTAL INVESTIGATION OF THE PROTOTYPE
Configurations I and J are respectively the configurations B and A turned
upside-down. These two cases can thus also be used to discuss and illustrate
the influence of diagonal cables on the structural behaviour of the deployable
Tensairity beam. Figure 8.14 plots the experimental load-displacement curve
of both cases under distributed load. From the graph can be seen that case I is
stiffer than case J. It is obvious that this is also for asymmetric loading. Thus
also here is confirmed that the diagonals contribute to the structural behaviour.
Load (N)
6000
5000
4000
3000
2000
Distributed load − pressure 150 mbar
1000
case I
case J
0
0 10 20 30 40
Vertical displacement (mm)
Figure 8.14: The experimental load-displacement curve of case I and J
under distributed load. The internal pressure measures 150 mbar.
The support of the diagonal cables can also be seen in figure 8.15 where the
displaced upper and lower strut of both cases are illustrated. Take for example
the middle hinge on the upper strut. In the case of configuration I is this hinge
supported by two diagonal and one vertical cable. In the case of J is the hinge
only supported by the vertical cable, hence the larger displacement of this
point.
Configurations D, F and G - which differ in a diagonal cable - are also
investigated experimentally. Their stiffness is plotted in figure 8.16. Not
all hinges of these cases are connected with a cable and one can thus expect
these cases to have a deformed strut under inflation. Their stiffness is thus
mainly dominated by this deformation of the strut under loading, and not by
the structural contribution of the cables.
8.3.3 POSITION OF HINGES
Because only the position of the hinges in the lower and upper strut of cases A
and J differ from each other, these two configurations can be used to investigate

Vertical displacement (mm)
Vertical displacement (mm)
0
10
20
30
40
50
60
0
5
10
15
20
25
30
Displacement of upper (loaded) strut
8.3 INFLUENCE OF CABLES
70
−2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 2500
Position x (mm)
Displacement of lower (unloaded) strut
case I
case J
case I
case J
35
−2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 2500
Position x (mm)
Figure 8.15: The displaced upper and lower strut of cases I and J under an
external distributed load of 4500 N and an internal pressure of 150 mbar
the influence on the structural behaviour of the position of the hinges.
Figure 8.17 shows the load-deflection behaviour of the two configurations
under distributed and asymmetric load. The stiffness of both configurations
is under distributed load identical till a certain load. From that load on,
configuration A is stiffer than J. In addition, the maximum load of configuration
A is higher than J. In the case of an asymmetrical load, the two configurations
show very similar stiffness. Here, the maximal load of J is higher than A.
Figure 8.18 plots the displaced upper strut of both configurations during
loading. It is clear from the figure that in case J the middle hinge (3) causes
large displacements, while in case A it is the second hinge (the first hinge from
the support that can rotate inwards). There was expected that the first hinge
of case J would experience large displacements, since it is not supported by a
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CHAPTER 8 EXPERIMENTAL INVESTIGATION OF THE PROTOTYPE
load [N]
load [N]
1500
1000
500
0
distributed load − pressure 150 mbar
0 2 4 6 8 10
displacement [mm]
D
G
F
load [N]
1500
1000
500
asymmetric load − pressure 150 mbar
0
0 2 4 6 8 10
displacement [mm]
Figure 8.16: The load-displacement curves of configurations D, F and G
under distributed (left) and asymmetrical load (right).
6000
5000
4000
3000
2000
1000
distributed load − pressure 150 mbar
A
0
0 10 20 30 40
displacement [mm]
J
load [N]
6000
5000
4000
3000
2000
1000
asymmetric load − pressure 150 mbar
0
0 10 20 30 40
displacement [mm]
Figure 8.17: Load-deflection behaviour of configurations A and J under
distributed and asymmetric load. Pressure is 150 mbar.
A
G
F
J
D

8.3 INFLUENCE OF CABLES
cable. Explanations for this behaviour could not be found experimentally and
numerically and are an interest topic for further in-depth investigations 3 . An
explanation for the similar stiffness of cases A and J under asymmetrical load
is that the airbeam plays an important role in preventing the ‘meshes’ to be
deformed to parallelograms.
Vertical displacement (mm)
Vertical displacement (mm)
upper strut J
0
10
20
30
40
50
60
70
80
−2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 2500
Position x (mm)
upper strut A
0
10
20
30
40
50
60
70
1
2 3 4 5
1 2 3 4 5 6 7
80
−2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 2500
Position x (mm)
hinge moving outwards (up) hinge moving inwards (down)
Force = 0N
Force = 4550N
Force = 0N
Force = 5069N
Figure 8.18: The displaced upper strut of configurations A and J during
external distributed loading and with internal pressure of 150 mbar.
3 An assumption why the first hinge of case J does not causes large displacements, is that it is
pushed outwards by the inflation since it is not connected with a cable.
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CHAPTER 8 EXPERIMENTAL INVESTIGATION OF THE PROTOTYPE
8.4 CONCLUSIONS
Experimental investigations were conducted on the prototype to evaluate
the proposed mechanism for a deployable Tensairity structure and to gain
insight in its structural behaviour. The influence of various parameters, such
as the internal pressure, the cable configuration, and the position of the hinges,
is analyzed by means of various configurations. The results were compared
with conclusions from previous chapters and with finite element models of
the prototype.
First, the influence of the cable tension on the general load-bearing behaviour
was studied. The experimental results confirm the outcome of the investigations
from chapter4 and chapter 6: the cables support the hinges and are
responsible for the structure’s stiffness, as long as they are pretensioned. Once
the load is sufficient high to decrease a cable’s tension to zero, the stiffness of
the structure changes. The maximum load of the structure is thus related with
the internal pressure of the beam, while the stiffness of the structure (in the
linear part) independent is of the pressure. As was expected, the maximal load
of the deployable Tensairity structure is reached when one hinge starts to rotate
inwards. This ‘collapse’ of the segments occurs obviously at the hinge that
was designed for rotating inwards and at higher loads for higher pressures.
In chapter 6 is shown by means of numerical investigations that the bending
stiffness of the strut has an influence on the structure’s stiffness. More precise,
for higher pressures will the strut deflect more and cause a decrease of
stiffness in the structure. This behaviour is not observed during the experiments.
Again, in the experimental investigations is found that the stiffness
is independent of the pressure as long as the cables are pretensioned. As
already mentioned in chapter 6, more reliability is given to the experimental
investigations than the numerical outcome (which reflects the ideal situation
without imperfection). It is possible that this negative influence of the pressure
becomes more visible for experiments under higher pressures (where a larger
deflection of the strut will occur).
Various cable configurations were investigated. The experiments showed the
importance of connecting all the hinges of the upper strut with the lower strut
by means of cables. This way, the hinges are kept in position after being inflated
and the struts maintain their spindle shape. After all, during the experiments
is observed that the hinges without cables are pushed outwards by the inflated
airbeam. As a consequence, the segments of the strut rotate relative to each

8.4 CONCLUSIONS
other and the strut is deformed. When loading the structure, the loads are
not fully transferred to compression and tension in the struts, but also to a
further deformation of the strut. Obvious, this has a negative influence on the
structural behaviour, which can be seen in the comparison of configurations B
and D.
From this comparison was expected to detect the contribution of vertical cables
to the structure’s stiffness. After all, the two configurations (B and D) differ
only in the presence of vertical cables. However, because configuration D
has not in all hinges a cable and thus an incorrect initial shape after inflation,
no conclusion can be drawn. To investigate this, a comparison should be
made between a configuration that contains in all hinges a diagonal and a
configuration with as well diagonals in all hinges as verticals.
The influence of diagonal cables on the behaviour of the deployable Tensairity
structure is investigated from the comparison of cases A and B, and cases I
and J. From cases I and J can be concluded that the diagonals contribute to the
structural behaviour, as well under distributed as asymmetrical loading. After
all, because the diagonals are connected with a hinge that has no link with a
vertical cable, it is pretensioned and thus able to support some compressive
forces.
With regard to the influence of the position of the hinges, not much could
be concluded. It is obvious that only the hinges that are allowed to rotate
inwards are causing the large deflections. In the case of A and B is this the first
inwards rotating hinge from the supports on. In cases I and J is this the middle
hinge. This latter is not expected, since numerical investigations revealed
that the hinge closest to the ends will cause the ‘collapse’ of the structure. An
explanation for this behaviour can be that the first hinge is pushed outwards by
the air pressure, since no cable is connected to this hinge. Future experiments
need to investigate the influence of the position of hinges more in-depth,
especially since it has a large impact on the design of the deployable Tensairity
beam.
With these experiments, a five meter prototype for a deployable Tensairity
structure is investigated for the first time. This structure differs on some crucial
points from the regular, better known Tensairity structures, namely on the
presence of the hinges and the curved segments of the strut. Therefore, it was
not obvious what to expect from the experiments. With the observations made
during the experiments and the analysis of the experimental data, insight is
gained in the structural behaviour and the prototype is evaluated. However, at
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CHAPTER 8 EXPERIMENTAL INVESTIGATION OF THE PROTOTYPE
the same time, more questions are raised with regard to the specific behaviour
of the deployable structure and the influence of various phenomenon that
apparently occur during the loading of the structure. These experiments
pointed out what has to be investigated more in detail in future experiments
on the deployable Tensairity structure.

Evaluation of the prototype
9
The prototype for a deployable Tensairity beam, investigated experimentally
and numerically in this dissertation, is evaluated in this chapter. This is done
by comparing the structural behaviour of the deployable Tensairity beam with
experimental and numerical results of its non-deployable counterpart, the ‘D’spindle.
The experimental results will by way of illustration also be compared
with those of other Tensairity prototypes. Finally, this evaluation will lead to
some concluding remarks and suggestions for the further development of a
deployable Tensairity beam.
9.1 COMPARISON WITH OTHER TENSAIRITY PROTOTYPES
The experimental load-deflection behaviour of configuration A of the deployable
Tensairity structure is compared with three other five meter spindle
shaped Tensairity prototypes. The three cases differ in connection between
upper and lower strut: the ‘O-spindle’ has no connection between the struts,
the ‘C-spindle’ has a continuous membrane (web) as connecting element and
the ‘D-spindle’ contains a discrete connection between upper and lower strut,
namely a certain amount of cables at regular distances (Crisbasanu, 2010).
The configuration of the D-spindle that will be used for the comparisons
has the same amount of cables and at the same position as configuration
A of the deployable Tensairity structure, called here ‘F-spindle’ (for Foldable).
Figure 9.2 illustrates the four cases.
Figure 9.1 shows the load-deflection curves of the four Tensairity types under
distributed and asymmetrical load for three pressures. Note that the value of
the axes are different in each graph for reasons of clarity.
The curves show clearly that the stiffness of the deployable Tensairity structure
is under distributed load in the same range as the O-spindle, however the F-
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CHAPTER 9 EVALUATION OF THE PROTOTYPE
load [N]
load [N]
load [N]
8000
7000
6000
5000
4000
3000
2000
1000
0
0 5 10 15 20 25 30
displacement [mm]
16000
14000
12000
10000
8000
6000
4000
2000
20000
18000
16000
14000
12000
10000
8000
6000
4000
2000
D
50 mbar − distributed
150 mbar − distributed
D
F
C
C
F
0
0 5 10 15 20 25 30
displacement [mm]
250 mbar − distributed
D
0
0 5 10 15 20 25 30
displacement [mm]
C
O
O
O
F
load [N]
load [N]
load [N]
3000
2500
2000
1500
1000
500
50 mbar − asymmetric
0
0 10 20 30 40 50 60
displacement [mm]
6000
5000
4000
3000
2000
1000
7000
6000
5000
4000
3000
2000
1000
150 mbar − asymmetric
0
0 10 20 30 40 50
displacement [mm]
F
F
250 mbar − asymmetric
0
0 10 20 30 40
displacement [mm]
Figure 9.1: Load-deflection curves of the four Tensairity types under
distributed and asymmetrical load for three pressures.
D
D
D
C
F
O
C
C
O
O

9.2 DEPLOYABLE VERSUS NON-DEPLOYABLE TENSAIRITY BEAM
O-spindle C-spindle D-spindle F-spindle
Figure 9.2: The four Tensairity cases which structural behaviour is
compared.
spindle is stiffer under low pressures. The D- and C-spindle have a much
higher stiffness. The O-spindle has under asymmetrical load case the lowest
stiffness in all pressure cases, while the F-spindle has stiffness comparable
with the D- and C-spindle. Thus, although the structural behaviour of the
F-spindle is less optimal than the D and C spindle, it provides a foldable
Tensairity structure with at least the same structural performance of a standard
O-spindle.
The D- and F-spindle contain the same cable configuration, but differ in the
presence of hinges. To evaluate the deployable Tensairity beam, the D- and Fspindle
are compared numerically and experimentally.
9.2 DEPLOYABLE VERSUS NON-DEPLOYABLE TENSAIRITY BEAM
The numerical model for the prototype of the deployable Tensairity beam
has already been applied in the numerical study of its structural behaviour
(chapter 6) and the comparison with the experimental results (chapter 8). This
latter showed that the numerical model shows good qualitative agreement
with the experimental results.
Load-displacement behaviour
Here, the behaviour of configuration A (F-spindle) is compared with the Dspindle
by means of finite element models. Figure 9.3 plots the stiffness of the
D- and F-spindle under distributed and asymmetric loading with an internal
pressure of 150 mbar. Figure 9.4 compares the experimental and numerical
derived load-displacement behaviour of both cases. The figure shows that the
finite element results agree very well with the experimental outcome in the
case of distributed load. In the case of asymmetrical loading is the difference
between the numerical and experimental curves larger, approximately a factor
3.
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CHAPTER 9 EVALUATION OF THE PROTOTYPE
load [N]
load [N]
12000
10000
8000
6000
4000
2000
12000
10000
distributed load − pressure 150 mbar
0
0 0.01 0.02 0.03 0.04 0.05
average displacement [m]
8000
6000
4000
2000
load [N]
12000
10000
8000
6000
4000
2000
asymmetric load − pressure 150 mbar
0
0 0.01 0.02 0.03 0.04 0.05
average displacement [m]
Figure 9.3: Numerical load-deflection curves of the D- and F-spindle
under distributed and asymmetrical load. The internal pressure
measures 150 mbar.
distributed load − pressure 150 mbar
D-num
D-exp
F-exp
F-num
0
0 0.01 0.02 0.03 0.04 0.05
average displacement [m]
load [N]
12000
10000
8000
6000
4000
2000
asymmetric load − pressure 150 mbar
D-exp
D-num
F-exp
F-num
0
0 0.01 0.02 0.03 0.04 0.05
average displacement [m]
Figure 9.4: Comparison between the numerical and experimental
derived load-displacement curves. The finite element results agree very
well with the experimental in the case of distributed load (left). In the
case of asymmetrical loading differs the stiffness of the cases by a factor
2 (right).
Figure 9.3 shows that the D-spindle has a larger stiffness and a higher maximum
load for both load cases. Thus, the hinges make the structure ‘softer’,
produce larger displacements and cause the structure to ‘collapse’ at a lower
load. This collapse is different for the two cases, as observed experimentally
and numerically. As seen in the previous chapters will, in the case of the Fspindle,
one hinge become unstable (not supported anymore by a pretensioned

9.2 DEPLOYABLE VERSUS NON-DEPLOYABLE TENSAIRITY BEAM
cable) and rotate inwards. In the case of the D-spindle will the compressed
upper strut buckle out-of-plane.
This has as result that in the case of the D-spindle, the maximum load is difficult
to predict visually. In addition, once the structure has collapsed, the struts are
deformed irreversible. In the case of the F-spindle, the inwards rotation of
one hinge can be observed visually. When the load is released, the F-spindle
returns to its initial shape.
Influence of hinges
To illustrate the influence of the pretensioned cables on the structural behaviour
of both cases and to show the effect of hinges on the cable pretension
is the tension in cables 2 and 3 of the D- and F-spindle plotted throughout
loading (figure 9.5).
cable tension [N]
600
500
400
300
200
100
2 3
cable tension throughout distributed loading − pressure 150 mbar
cable 2 − D−spindle
cable 3 − D−spindle
cable 2 − F−spindle
cable 3 − F−spindle
0
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
total applied load [N]
Figure 9.5: The tension of cables 2 and 3 of the D- and F-spindle
throughout distributed loading. (Internal pressure measures 150 mbar.)
It shows that the decrease of pretension in cable 3 is initially similar for both
cases. However, once a cable becomes slack (cable 2), the hinge at cable 2 is not
supported anymore by the cable and larger deformation occurs. This results
in a large decrease of cable tension, as shown for cable 3 in the figure. The
slackening of one cable occurs faster in the case of the F-spindle. In the case of
the D-spindle has this slackening of the cable little influence on the decrease
in pretension of the other cables, as can be seen for cable 3.
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As already discussed in chapter 6, is the influence of hinges also reflected in
the bending moments in the struts. Figure 9.6 illustrates the moments of both
cases under a distributed load of 4100 N and internal pressure of 150 mbar.
Note that the upper hinges, allowed to rotate outwards, are blocked to move
inwards in this finite element model. Hence the moments at these ‘hinges’.
Figure 9.6: The influence of hinges in the upper and lower strut is
also reflected in the bending moments of the struts. (Under distributed
loading of 4100 N.)
Finally, the difference in structural behaviour between the D- and F-spindle
is illustrated in figure 9.7 where their deformed shape under a distributed
load of 4100 N is plotted. As discussed, the difference in defection is caused
by the hinges because of their effect on the bending moments and the cable
pretension.
D-spindle
F-spindle
Summary
Figure 9.7: The displaced struts of the D- and F-spindle, magnified by a
factor 5. The influence of the hinges is obvious.
It is obvious that the spindle without hinges in upper and lower strut, the
D-spindle, is stiffer and can bear more loads than its deployable variant, the
F-spindle. This section showed that the D-spindle is under distributed loading
a factor three stiffer than the F-spindle.
This difference in structural behaviour is attributed to the influence of hinges.
After all, these are the only parameters in which these cases differ. The hinges

9.3 CONCLUDING REMARKS AND SUGGESTIONS
have an influence on the bending moments in the struts and thus in the
pretension (and thus support) of the cables throughout loading. Once a hinge
becomes unstable, the F-spindle experiences with little increase of load larger
displacements.
9.3 CONCLUDING REMARKS AND SUGGESTIONS
The proposal for a deployable Tensairity structure has been evaluated throughout
this research. This is not only done by comparing the experimental results
with other Tensairity prototypes, but also by investigating numerically and
experimentally various configurations of the prototype.
The research clearly showed the impact on the structural behaviour of cable
configurations. All hinges need to be connected with a cable in order to avoid
deformation of the struts during inflation, which causes a drop in stiffness.
Especially vertical pretensioned cables are contributing largely to the structural
behaviour.
However, the cable configuration that allows the deployable Tensairity beam
to fold completely contains only diagonal cables. In addition, not all hinges
are connected with a cable. As a consequence, this configuration has a
relative low stiffness, approximately one fourth of the value of the stiffness
of the configuration with vertical cables in every hinge. The configuration
of the cables in the mechanism was, however, mainly determined by the
kinematics of the foldable system. Thus, the mechanism for the deployable
Tensairity beam needs to be redesigned, taking into account the requirement
of connecting all hinges with vertical cables.
This is possible with the original folding mechanism of the foldable truss
(cylindrical and spindle shape), where upper and lower strut ‘fold’ into each
other. After all, upper and lower strut (and by extension all points of the
structure) move closer to each other when folding, which allows every possible
cable configuration and shape of the hull. The issue here however is that
this configuration needs to possess more hinges, located closer to the ends.
Numerical research showed that this has a negative influence on the maximal
load of the structure. Thus to conclude, the original foldable truss mechanism
is a good suggestion for the foldable Tensairity beam, taking into account that
hinges near the end supports are to be avoided.
This new mechanism can have as well a cylindrical as a spindle shape. The
spindle shape has a more optimal structural behaviour due to its curved
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shaped, but can be unwanted for some practical applications, like the deck
of a bridge. In addition, because of the upwards movement of the lower
strut of a spindle shaped structure under distributed loading, are all cables,
regardless of their configuration, loaded in compression. This is not the case
for a cylindrical deployable Tensairity beam. Depending on the load case
and the application can a cylindrical deployable Tensairity beam thus be more
appropriate.
When developing a new deployable Tensairity beam, this research showed
that it is crucial to first fully understand the influence of every parameter on
the structural and kinematic behaviour. Therefore, it is advisable to start with
very simple models, e.g. containing one or two hinges and or cables. Then,
by systematically increasing the complexity of the structure, one can achieve
a design for an optimal deployable Tensairity beam.

Conclusions
205

Conclusions
10
A Tensairity structure has most of the properties of a simple air-inflated beam,
but can bear to hundred times more load. This makes Tensairity structures very
suitable for temporary and mobile applications, where lightweight solutions
and small transport volume are desired. However, the standard Tensairity
structure, which is comprised of several interacting components such as an
airbeam, cables and struts, can not be compacted to a small volume without
being disassembled. By replacing the standard compression and tension
element with a mechanism, a deployable Tensairity structure is achieved that
needs no additional handlings to compact or erect the structure.
The main goal of this research was to gain insights in the structural and
kinematic behaviour of deployable Tensairity structures and develop a first
prototype of such a structure. Given the lack of general knowledge on
deployable Tensairity structures, this investigation was pursued by means
of experimental and numerical investigations on small and large scale models.
The first part of the dissertation focused on the development of an appropriate
mechanism for the deployable Tensairity structure. The second part investigated
the structural behaviour of a Tensairity beam by means of experiments
on scale models and numerical investigations and identified the influence of
several design parameters. As a result, a prototype of a deployable Tensairity
beam was designed, fabricated, experimentally tested and evaluated in the
third part. The main findings concerning the design and structural behavior
of deployable Tensairity structures and the general contributions to the field
are discussed below.
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CHAPTER 10 CONCLUSIONS
10.1 DESIGN OF DEPLOYABLE TENSAIRITY STRUCTURES
In order to create a Tensairity structure that is deployable, the typical continuous
compression element needs to be replaced by a suitable deployment
mechanism. The first research goal was to gain insight in the development
of an appropriate mechanism for the deployable Tensairity structure. Several
deployment mechanism were designed and investigated in relation to the
boundary conditions and requirements imposed by the structural concept
Tensairity and the structural applications. The investigated models show that
the continuous connection of the membrane with the compression element is
the most crucial and imperative requirement. The folding of the membrane
should thus be compatible with the proposed foldable compression element
and vice versa. As a result, the inextensibility of the membrane has to be taken
into account to avoid damaging the membrane or obstruction of the folding.
More concretely, the hinges centre line has to lie in the same plane as the
membrane to prevent stretching or wrinkling.
A mechanism constituted of stiff segments, connected to each other by hinges,
proves to be an appropriate solution for a foldable compression element. The
’foldable truss system’ is being improved with regard to its structural and
kinematic behaviour. The system’s load bearing capacities are ameliorated by
changing the longitudinal shape from cylindrical to spindle, by decreasing the
amount of hinges and segments and by positioning hinges on compression
side towards the middle. By means of a redesign of the configuration of the
foldable truss, the kinematic behaviour is improved. In this new proposal,
the segments of upper and lower strut do not have to ‘fold’ into each other
anymore. In addition, less hinges and less complicated joints are necessary.
As a result, an easily foldable proposal for the deployable Tensairity structure
was obtained.
10.2 STRUCTURAL BEHAVIOUR OF THE DEPLOYABLE TENSAIRITY BEAM
As the introduction of a deployment mechanism changes the structural
behaviour of a traditional Tensairity structure, the second goal of this research
was to explore the general behaviour of the deployable Tensairity beam. More
specifically, the influence of various design parameters on the beams structural
behaviour was investigated by means of numerical and experimental
investigations on small scale models as well as a large scale prototype. These
investigations provided specific insights in the effects of the hinges, the cables,

10.2 STRUCTURAL BEHAVIOUR OF THE DEPLOYABLE TENSAIRITY BEAM
the internal pressure and the sections of the hull and struts.
Hinges
The introduction of hinges in the upper and lower strut (to create a deployable
Tensairity beam) has several consequences on the structural behaviour of the
beam. Finite element calculations showed that these hinges decrease the
structure’s stiffness. The influence of the hinges is relatively small when no
bending moment occurs in the struts. However, when large bending moments
are introduced in the strut (eg. by means of cables), the influence of the hinges
on the deflection is more noticeable. A variation in the amount of hinges
showed little influence on the structure’s stiffness. Furthermore, the numerical
experiments revealed that the location of the hinge along the beam determines
the maximal load of the structure: the closer the hinge is to the endpoint of
the beam, the lower the load will be at which no convergence of the finite
element calculation is reached (and thus a certain collapse occurs). Hinges at
the ends of the beam should thus be avoided as they are less stabilized by the
membrane.
A comparison of the experimental results of the (deployable) prototype with
those of non-deployable Tensairity beams confirmed this significant influence
of the hinges on the structural behaviour. The stiffness of the prototype with
hinges was less than half of the stiffness of the same configuration without
hinges.
Cables
The numerical and experimental investigations on large and small scale models
revealed the influence of cables and their configuration on the structural behaviour
of the deployable Tensairity beam. The pretensioned cables contribute
to the structure’s stiffness (until their pretension becomes zero) by increasing
the spring constant of the elastic foundation and thus supporting the struts and
hinges. As long as all cables are pretensioned is the stiffness of the structure
independent of the internal pressure. Once a cable becomes slack, the stiffness
changes. The maximum load of the structure is thus related with the internal
pressure.
The experiments showed the importance of connecting all the hinges of the
upper strut with the lower strut by means of cables. This keeps the hinges
in position when inflating the airbeam and makes the struts maintain their
spindle shape. When this is not the case, the hinges without cable connection
will move outwards and deform the struts. This has as result that the upper
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CHAPTER 10 CONCLUSIONS
strut will deform further under external loading, hence the lower stiffness of
these configurations.
The investigation of the cable configurations and the cable tension showed a
difference in behaviour between the cylindrical and spindle shaped deployable
Tensairity beam. In the case of a cylindrical shaped beam can the diagonals
become tensioned under distributed load, depending on their configuration.
In the case of a spindle shaped Tensairity beam become all cables compressed
under distributed loading, regardless their configuration. This is due to the
upwards movement of the lower strut, caused by the horizontal displacement
of the upper strut.
Because diagonal cables are not pretensioned under inflation when they connect
hinges that are already linked with vertical cables, only the vertical cables
are in the case of a spindle shaped structure contributing to the structure’s
stiffness under distributed load. Under asymmetrical loading on the other
hand, are the diagonal cables (loaded in tension) contributing. Diagonal cables
do have a positive influence on the structure’s stiffness under distributed
loading when they are not already linked with vertical cables and thus being
pretensioned.
The investigation of cylindrical Tensairity beams showed that the shape of the
‘mesh’ - formed by the pretensioned cables and the cables that will become
tensioned under loading - has a large influence on the structure’s stiffness. If
this mesh is a quadrangle, it can easily be deformed under external loading and
cause large displacements of the structure. When all cables are pretensioned
and thus able to take compressive forces, the deployable Tensairity beam has
the same stiffness as a truss (with the same configuration of diagonals and
with the same strut sections).
Hull and strut section
Also the sections of the hull and the struts proved to have an influence on
the structural behavior. Numerical investigations revealed that the shape of
the hull of the Tensairity beam whereby the web experiences a constant stress
along the beam’s length, uses the material more optimal and has the largest
average stiffness. This can be clarified by the fact that a higher stress in the
web near the ends supports the struts and hinges up to a higher load.
The influence of the struts section was shown to be related to the occurrence
or absence of bending moments in the struts. The strut’s cross section proved
to have no influence in the cases without bending moment in the struts. When

10.3 EVALUATION OF THE PROTOTYPE
bending moments are introduced (mostly by cables), however, the bending
stiffness, and thus the cross section, plays an import role on the stiffness of
the deployable Tensairity beam. For cases with large deflections of the strut,
second order effects occurred and a large decrease in stiffness was detected.
As a result, the struts should be dimensioned to avoid these second order
effects. This bending stiffness of the strut has not been noticed to be that
influential in the experiments. While numerical calculations predict a decrease
in stiffness under higher pressure, due to the bending of the strut, show the
experimental investigations a higher stiffness when increasing the pressure.
This contradiction can be attributed to the difference between a perfect ideal
numerical model and a prototype with its imperfections. This subject is an
interesting aspect for further research.
The experimental investigation on the prototype pointed out some other
specific details with regard to the behaviour of the deployable Tensairity beam
that need to be studied in more detail. The influence of the pretension of
cables on the structure’s stiffness is one aspect that deserves a closer look
by continuously measuring the tension in the cables during inflation and
loading, and relating this with the load-displacement behaviour. In addition,
the influence of the position of the hinges on the structure’s stiffness should be
investigated experimentally more in detail.
10.3 EVALUATION OF THE PROTOTYPE
As a result of the different findings from the numerical studies and experimental
investigations on scale models, a five meter long prototype has been
developed during the final phase of this research. This prototype is the result
of an iterative process, taking into consideration general design issues, as well
as aiming for a structurally sound structure. The result is a working prototype,
which has been tested and is ready for further improvements.
The design and the assembling of the various components the structure is
constituted of are evaluated positive. The connection of the airbeam with
the curved struts proved to be satisfying. The experiments revealed the
necessity of applying curved segments along the spindle beam and of carefully
positioning these segments along the beam. The hinges demonstrated to have
the kinematics as they were conceived. Also the compatibility with the airbeam
and thus the correct folding of the membrane at the position of the hinges is
demonstrated. In addition, they proved to withstand considerable loadings.
Overall, the prototype weighs 31,7 kg.
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CHAPTER 10 CONCLUSIONS
With regard to its structural behavior, the proposal for the deployable Tensairity
beam is insufficient. The cable configuration that allows the mechanism to
fold completely contains only diagonal cables and has hinges not connected
with any cable. As a consequence, this configuration has a relative low stiffness,
approximately one fourth of the value of the stiffness of the configuration
with vertical cables. The configuration of the cables was , however, mainly
determined by the kinematics of the foldable system. This influence was not
expected to be this large and decisive. The influence of the position of the
hinges on the maximal load was taken into account in the development of the
prototype, hence the redesign of the mechanism and the more central location
of the hinges.
The folding of the structure and thus the packing of the membrane to a dense
bunch was not as smooth as intended due to the bending stiffness of the
PVC coated polyester (although small). Guiding the membrane during the
compacting of the structure was necessary at some points. Thus, a neat,
repeatable and controlled folding of the membrane according to a predefined
folding pattern is advisable.
It has been shown that the proposal for the deployable Tensairity structure can
be improved regarding its structural and kinematical aspects. After all, this
is the first investigated prototype for such a deployable structure and some
design iterations need to be done. With regard to the kinematics, a closer look
should be taken to the folding of the airbeam according a predefined folding
pattern to ease the folding of the technical membrane. At the same time,
the proposal should take the requirements for a structurally sound Tensairity
beam into account, like the connection of the hinges with vertical cables.
With this research, the first step is taken towards a functional large scale
deployable Tensairity beam. However, many aspects in this research, such as
the detailing and the gained insight, can also be applied in the development
and research of other structures, such as Tensairity arches, cushions and
grids. The application of the deployable technology on other scales or in
other domains than civil engineering will bring forward new questions and
knowledge and is worth investigating.

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List of publications
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T. (2010). A state-of-the-art of deployable scissor structures for architectural
applications. In Proceedings of the International Association for Shell and Spatial
Structures (IASS) Symposium 2010, Shanghai ’Spatial Structures - Permanent
and Temporary’, Shangai, Chine. (paper on cd).
De Temmerman, N., Mollaert, M., De Laet, L., Henrotay, C., Paduart, A.,
Guldentops, L., Van Mele, T., and De Backer, W. (2009a). A deployable mast
for kinetimatic architecture. In Proceedings of the 8th National Congress on
Theoretical and Applied Mechanics, pages 585–591, Brussels, Belgium.
De Temmerman, N., Mollaert, M., De Laet, L., and Van Mele, T. (2007a).
Development of a novel deployable bar structure with foldable articulated
joints. In Proceeding of the IASS 2007 International Symposium on Structural
Architecture - Towards the future looking to the past, Venice, Italy. (paper on cd).
De Temmerman, N., Mollaert, M., De Laet, L., Van Mele, T., Guldentops, L.,
Henrotay, C., De Backer, W., Paduart, A., Hendrickx, H., and De Wilde, W. P.
(2009b). A deployable mast for adaptable textile architecture. In Domingo,
A. and Lzaro, C., editors, Proceedings of the 50th Anniversary Symposium of the
IASS - ”Evolution and trends in Design, Analysis and Construction of Shells and
Spatial Structures”, Valencia, Spain. (paper on cd).

LIST OF PUBLICATIONS
De Temmerman, N., Mollaert, M., Van Mele, T., and De Laet, L. (2006a). A
concept for a foldable mobile shelter system. In Proceedings of the IASS-APCS
2006 Int. Symposium on the New Olympics and New Shell and Spatial Structures,
Bejing, China. IASS-APCS. (paper on cd).
De Temmerman, N., Mollaert, M., Van Mele, T., and De Laet, L. (2006b).
Development of a foldable mobile shelter system. In Adaptables 2006:
Proceedings of the International Conference on Adaptability in Design and
Construction, volume 2, pages 13–17, Eindhoven University of Technology,
The Netherlands.
De Temmerman, N., Mollaert, M., Van Mele, T., and De Laet, L. (2007b). Design
and analysis of a foldable mobile shelter system. International Journal of Space
Structures,Special Issue: Adaptable Structures, 22:161–168.
Guldentops, L., M., M., and De Laet, L. (2009a). Fabric-formed concrete shell
structures. In Proceedings of the 14th International Workshop on the Design and
Practical Realisation of Architectural Membrane Structures, Textile Roofs 2008.
Technical University Berlin, Germany.
Guldentops, L., Mollaert, M., Adriaenssens, S., De Laet, L., and De Temmerman,
N. (2009b). Textile formwork for concrete shells. In Domingo, A. and
Lzaro, C., editors, Proceedings of the 50th Anniversary Symposium of the IASS -
”Evolution and trends in Design, Analysis and Construction of Shells and Spatial
Structures”, Valencia, Spain. (paper on cd).
Mollaert, M., De Temmerman, N., De Laet, L., Guldentops, L., and Vanthienen,
T. (2010). Analysis of the deployment of a frame and a foldable membrane:
the integrated model of a contex-t demonstrator. In Proceedings of the
International Association for Shell and Spatial Structures (IASS) Symposium 2010,
Shanghai ’Spatial Structures - Permanent and Temporary, Shangai, Chine. (paper
on cd).
Van Mele, T., De Temmerman, N., De Laet, L., and Mollaert, M. (2007).
Retractable roofs: Scissor-hinged membrane structures. In Proceedings of the
TensiNet Symposium 2007, ”Ephemeral Architecture, Time and Textiles”, pages
283–291. Libreria, Milan.
Van Mele, T., De Temmerman, N., De Laet, L., and Mollaert, M. (2010). Scissorhinged
retractable membrane structures. International Journal of Structural
Engineering - Special Issue on ”New Structures and New Analysis Methods”,
1(3/4).
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LIST OF PUBLICATIONS
Van Mele, T., Mollaert, M., De Temmerman, N., and De Laet, L. (2006). Design
of scissor structures for retractable roofs. In Adaptables 2006: Proceedings of the
International Conference on Adaptability in Design and Construction, volume 2,
pages 13–17, Eindhoven University of Technology, The Netherlands.

DEPLOYABLE TENSAIRITY STRUCTURES
development, design and analysis
Inflatable structures have been used by engineers and architects for several decades. These structures offer
lightweight solutions and provide several unique features, such as collapsibility, translucency and a minimal transport
and storage volume. In spite of these exceptional properties, one of the major drawbacks of inflatable structures is
their limited load bearing capacity. This is overcome by combining the inflatable structure with cables and struts,
which results in the structural principle called Tensairity.
A Tensairity structure has most of the properties of a simple air-inflated beam, but can bear to hundred times more
load. This makes Tensairity structures very suitable for temporary and mobile applications, where lightweight
solutions that can be compacted to a small volume are a requirement. However, the standard Tensairity structure
cannot be compacted without being disassembled. By replacing the standard compression and tension element
with a mechanism, a deployable Tensairity structure is achieved that needs - besides changing the internal pressure
of the airbeam - no additional handlings to compact or erect the structure.
The development of such a deployable Tensairity structure is investigated in this research. Insight is gained in the
structural and kinematic behavior of this type of Tensairity structures by means of experimental and numerical
investigations on small and large scale models. The first part of this dissertation focuses on the development of an
appropriate mechanism for the deployable Tensairity structure. The second part investigates by means of
experiments on scale models and numerical investigations the structural behavior of a deployable Tensairity beam
and the effect of several design parameters on it. The insight gained in the first two parts results finally in the development
of a full scale prototype of a deployable Tensairity beam, which is evaluated experimentally in the third part of
this research.
New insights in the structural and kinematic behavior of Tensairity structures are created with this research. They
form a solid base for further research on deployable Tensairity structures and bring us one step closer to the
realization of an optimal deployable Tensairity beam.
March 2011
Thesis submitted in fullment of the requirements for the awar d of the degree of Doctor in Engineering
(Doctor in de Ingenieurswetenschappen)
Advisors: prof. dr. Marijke Mollaert and dr. Rolf Luchsinger
Vrije Universiteit Brussel
Faculty of Engineering
Department of Architectural Engineering Sciences
ISBN 978 90 5487 879 7
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