# DEPLOYABLE TENSAIRITY STRUCTURES - Vrije Universiteit Brussel

DEPLOYABLE TENSAIRITY STRUCTURES - Vrije Universiteit Brussel

DEPLOYABLE TENSAIRITY STRUCTURES - Vrije Universiteit Brussel

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<strong>DEPLOYABLE</strong> <strong>TENSAIRITY</strong> <strong>STRUCTURES</strong><br />

development, design and analysis<br />

Lars De Laet

<strong>DEPLOYABLE</strong> <strong>TENSAIRITY</strong> <strong>STRUCTURES</strong><br />

Development, design and analysis

Print: Silhouet, Maldegem<br />

© 2011 Lars De Laet<br />

2011 Uitgeverij VUBPRESS <strong>Brussel</strong>s University Press<br />

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FACULTY OF ENGINEERING<br />

Department of Architectural Engineering Sciences<br />

<strong>DEPLOYABLE</strong> <strong>TENSAIRITY</strong> <strong>STRUCTURES</strong><br />

Development, design and analysis<br />

Thesis submitted in fulfilment of the requirements for the award of the degree of<br />

Doctor in Engineering (Doctor in de Ingenieurswetenschappen) by<br />

Lars De Laet<br />

March 2011<br />

Advisors: prof. dr. Marijke Mollaert<br />

dr. Rolf H. Luchsinger

ii<br />

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

Members of the jury<br />

prof. dr. ir. Marijke Mollaert (advisor)<br />

<strong>Vrije</strong> <strong>Universiteit</strong> <strong>Brussel</strong>, ARCH<br />

dr. Rolf H. Luchsinger (advisor)<br />

EMPA - Center for Synergetic Structures, Zurich, Switzerland<br />

prof. dr. ir. Dirk Lefeber (chairman)<br />

<strong>Vrije</strong> <strong>Universiteit</strong> <strong>Brussel</strong>, MECH<br />

prof. dr. ir. Rik Pintelon (vice-chairman)<br />

<strong>Vrije</strong> <strong>Universiteit</strong> <strong>Brussel</strong>, ELEC<br />

prof. dr. ir. arch. Ine Wouters (secretary)<br />

<strong>Vrije</strong> <strong>Universiteit</strong> <strong>Brussel</strong>, ARCH<br />

prof. dr. ir. René Motro<br />

Laboratoire de Mécanique et Génie Civil - Université de Montpellier II, France<br />

dr. ir. Frank Jensen<br />

Søren Jensen Consulting Engineering & Aarhus School of Architecture, Denmark<br />

iii

Acknowledgements<br />

There are many people I would like to thank for their contribution to this<br />

PhD research. Thanks to their support, encouragement, guidance, feedback,<br />

expertise, time, love, humor ... was this PhD research an interesting, enriching,<br />

unique and unforgettable project of which this dissertation is the final result.<br />

I would like to thank my advisor Prof. Marijke Mollaert for her scientific<br />

guidance, enthusiasm and continuous encouragement throughout the course<br />

of this research. Thank you, Marijke, for believing in me as a researcher and<br />

for all the opportunities you offered me. The freedom you gave me to explore<br />

and develop new ideas - at home and abroad - enriched me tremendously.<br />

Thank you for your kind and warm personality.<br />

I am very grateful to my co-advisor dr. Rolf Luchsinger for his invaluable<br />

scientific guidance throughout my many visits and stays at the EMPA-Center<br />

for Synergetic Structures. Thank you Rolf, for teaching me a great deal about<br />

Tensairity structures and showing me the importance of doing research in the<br />

most thorough and insightful way possible.<br />

I am honored that René Motro, Frank Jensen, Dirk Lefeber, Rik Pintelon and<br />

Ine Wouters accepted to be with Marijke and Rolf member of the jury of this<br />

work. Thank you for your scientific advice and for providing much valued<br />

comments and suggestions.<br />

I would like to thank the Research Foundation Flanders (FWO) for funding<br />

this research, as well my two-month stay at the EMPA-Center for Synergetic<br />

Structures in Zürich. I am also very grateful for the additional support of<br />

the department of Architectural Engineering Sciences of the <strong>Vrije</strong> <strong>Universiteit</strong><br />

<strong>Brussel</strong>.<br />

I would like to thank Daniel Debondt, Gabriël Van den Nest, Frans Boulpaep,<br />

v

Patrick Vanroose, Tinneke Van Thienen and Jan Roekens for their contribution<br />

to the design, manufacturing and assembling of the final prototype. Thank<br />

you Jan, for assisting me twice in conducting the experiments at EMPA-CSS<br />

and for your endless patience.<br />

During my many visits at the EMPA-CSS, I had the pleasure to work with an<br />

international group of young researchers. Thank you, Rolf, Markus, Cédric,<br />

René, Louis, Mircea, Roberto, Fanis, Joep and Antje for making me feel more<br />

than welcome in Zürich and considering me each time as part of the team. I<br />

enjoyed the many group meetings, interesting discussions and collaborations.<br />

Thank you for supporting me in conducting the experiments and numerical<br />

calculations. I am also grateful to the EMPA-CSS for allowing me to use their<br />

equipment to conduct experiments and materials to fabricate scale models. My<br />

trips to Zürich during the last year would never have been possible without the<br />

unrestrained hospitality of Philippe Block. Thank you Philippe, for allowing<br />

me to stay in style in Switzerland and offering me your amazing apartment.<br />

My most heartfelt sympathy goes out to my colleagues of the department of<br />

Architectural Engineering Sciences, whom I thank for providing a kind and<br />

stimulating environment, and for their friendship and support. Thank you<br />

all, for the relax discussions over lunch, for the (sometimes too) many tasteful<br />

beers after work in spring and summer and for being friends as much as<br />

colleagues.<br />

My friends and family deserve special thanks for always being there and<br />

encouraging me. Thank you Kenny and Lisa, for your support, humor and<br />

friendship. Thank you mama, papa, Jens, Katrien, Brigitte, Jos and Kelly, for<br />

your love and your unconditional support and encouragement.<br />

I wish to express my love and gratitude to Romy, my partner and friend.<br />

Thank you, for everything you did for me, especially those last couple of<br />

months. Thank you for your immense and unconditional support and above<br />

all, thank you for your love. Thank you Yano, for making worries fade away<br />

with your beautiful smile, for your love and showing what life is about.<br />

vi<br />

Lars De Laet<br />

<strong>Brussel</strong>s, Belgium<br />

February 2011

Abstract<br />

A Tensairity structure has most of the properties of a simple air-inflated beam,<br />

but can bear to hundred times more load. This makes Tensairity structures<br />

very suitable for temporary and mobile applications, where lightweight solutions<br />

and small transport volume are a requirement. However, the standard<br />

Tensairity structure, which is comprised of several interacting components<br />

such as an airbeam, cables and struts, can not be compacted to a small<br />

package without being disassembled. By replacing the standard compression<br />

and tension element with a mechanism, a deployable Tensairity structure is<br />

achieved that needs - besides changing the internal pressure of the airbeam -<br />

no additional handlings to compact or erect the structure.<br />

The development of such a deployable Tensairity structure is investigated in<br />

this research. Insight is gained in the structural and kinematic behaviour of<br />

this type of Tensairity structures by means of experimental and numerical<br />

investigations on small and large scale models.<br />

The first part of the dissertation focuses on the development of an appropriate<br />

mechanism for the deployable Tensairity structure. The exploration and<br />

analysis of ideas for deployable systems is presented by means of experiments<br />

on various scale models. By doing this, the boundary conditions and<br />

requirements that have to be taken into account in the design are observed and<br />

presented. These boundary conditions are imposed by the application where<br />

the structure is designed for and by the structural concept Tensairity. With<br />

this knowledge, a solution for a deployable Tensairity structure, the ‘foldable<br />

truss system’, is being improved with regard to its structural and kinematic<br />

behaviour. As a result, an easily foldable proposal for the deployable Tensairity<br />

structure is obtained.<br />

The second part investigates the structural behaviour of a Tensairity beam<br />

vii

y means of experiments on scale models and numerical investigations and<br />

identifies the influence of several design parameters. More precise, the contribution<br />

of hinges and cables to the structural behaviour is investigated, as<br />

well as the influence of the internal pressure, section of the strut and shape<br />

of the airbeam. Among other things is learned from the investigations that<br />

hinges decrease the structure’s stiffness and that a decent structural behaviour<br />

requires that all hinges are connected with a cable.<br />

A prototype of a deployable Tensairity beam is designed, fabricated, experimentally<br />

tested and evaluated in the third part. The detailing and manufacturing<br />

of the prototype is presented, as well how the experiments are conducted.<br />

The general behaviour of the deployable Tensairity beam, such as its stiffness<br />

and maximal load, is first discussed. Then, various configurations are studied<br />

experimentally and numerically to analyse the influence on the structural<br />

behaviour of the cables and hinges. Also here, the necessity of connecting<br />

hinges of the upper and lower strut with each other is shown. During the<br />

study, the experimental results are compared with the outcome of numerical<br />

calculations on the finite element model of the prototype.<br />

Finally, the deployable Tensairity beam is compared with other (non-deployable)<br />

Tensairity prototypes. The deployable structure shows to be a factor two less<br />

stiff than the same prototype with continuous strut. This difference can be<br />

attributed to the presence of hinges in the strut of the Tensairity structure.<br />

The proposal for the deployable Tensairity beam, containing thus the cable<br />

configuration that allows complete folding, is with regard to its structural<br />

behaviour insufficient. The main reason is that not all hinges are connected<br />

with a cable. As a consequence, the proposal for the deployable Tensairity<br />

structure should be adapted taken the formulated improvements into account.<br />

Nevertheless, new insights in the structural behaviour of Tensairity structures<br />

are created with this research. They form a solid base for further research on<br />

deployable Tensairity structures and bring us one step closer to the realisation<br />

of a deployable Tensairity beam.<br />

viii

Samenvatting<br />

Het constructief concept Tensairity bezit de meeste eigenschappen van een<br />

opblaasbare balk, maar kan tot honderd keer meer belasting dragen. Hierdoor<br />

zijn Tensairity constructies zeer geschikt voor tijdelijke en mobiele toepassingen,<br />

waar het lage gewicht en het compact volume bij transport een vereiste<br />

zijn. Echter, de standaard Tensairity constructie, die opgebouwd is uit verschillende<br />

samenwerkende delen zoals een opblaasbare balk, kabels en staven,<br />

kan niet opgeborgen worden tot een klein volume zonder de constructie te<br />

demonteren. Door middel van het vervangen van de standaard druk- en trek<br />

elementen door een mechanisme, wordt een opplooibare Tensairity constructie<br />

gerealiseerd die - behalve het aanpassen van de interne druk - geen bijkomende<br />

handelingen vereist om de constructie op te vouwen of te ontplooien.<br />

De ontwikkeling van een dergelijke opplooibare Tensairity constructie is het<br />

onderwerp van dit onderzoek. Inzichten in het constructief en kinematisch<br />

gedrag van dit type Tensairity constructies worden verworven door middel<br />

van experimenteel en numeriek onderzoek op kleine en grote schaalmodellen.<br />

Het eerste deel van dit onderzoek richt zich op de ontwikkeling van een<br />

geschikt mechanisme voor de opplooibare Tensairity constructie. De studie en<br />

analyse van ideeën voor opplooibare systemen is weergegeven door middel<br />

van experimenten op verscheidene schaalmodellen. De randvoorwaarden en<br />

vereisten die in rekening moeten gebracht worden bij het ontwerpen, worden<br />

op deze manier weergegeven. Deze randvoorwaarden worden opgelegd door<br />

de toepassing waarvoor de constructie ontworpen is en door het constructief<br />

concept Tensairity zelf. Een oplossing voor een opplooibare Tensairity constructie<br />

wordt met deze kennis verbeterd op het vlak van diens constructief<br />

en kinematisch gedrag. Het resultaat is een eenvoudig opplooibaar voorstel<br />

voor de opplooibare Tensairity constructie.<br />

ix

Het tweede deel onderzoekt het constructief gedrag van een opplooibare<br />

Tensairity constructie door middel van experimenten op schaalmodellen en<br />

numerieke simulaties, en identificeert de invloed van verscheidene ontwerpparameters.<br />

Meer precies wordt de bijdrage van scharnieren en kabels op<br />

het constructief gedrag bestudeerd, alsook de invloed van de interne druk,<br />

de sectie van het druk-en trekelement en de vorm van opblaasbare balk. Uit<br />

deze studie wordt onder andere geleerd dat de stijfheid van de constructie<br />

verminderd wordt door de aanwezige scharnieren en dat een degelijk constructief<br />

gedrag vereist dat alle scharnieren met kabels verbonden worden.<br />

Een prototype van een opplooibare Tensairity balk is ontworpen, vervaardigd,<br />

experimenteel onderzocht en geëvalueerd in het derde deel. De details en<br />

vervaardiging van dit prototype worden er weergegeven, alsook hoe de experimenten<br />

zijn uitgevoerd. Het algemeen constructief gedrag van de opplooibare<br />

Tensairity balk, zoals diens stijfheid en maximale belasting, worden<br />

eerst besproken. Daaropvolgend worden verschillende configuraties<br />

bestudeerd om de invloed van de kabels en scharnieren op het gedrag van de<br />

constructie te onderzoeken. Ook hier wordt de noodzaak duidelijk gemaakt<br />

om de scharnieren van de druk-en trekelementen door middel van een kabel<br />

te verbinden. De experimenteel verworven resultaten worden doorheen de<br />

hele studie vergeleken met de uitslag van numerieke berekeningen van een<br />

eindige elementen model van het prototype.<br />

De opplooibare Tensairity balk wordt vergeleken met andere (niet-opplooibare)<br />

Tensairity prototypes. De opplooibare configuratie is een factor twee minder<br />

stijf dan hetzelfde prototype met continue drukstaven. Dit verschil kan<br />

worden toegewezen aan de aanwezigheid van scharnieren. Het voorstel voor<br />

een opplooibare Tensairity constructie, bestaande uit de kabelconfiguratie die<br />

het volledig toevouwen van de structuur toelaat, is wat betreft het draagvermogen<br />

onvoldoende. De voornaamste reden is dat niet alle scharnieren<br />

verbonden zijn met een kabel. Dit heeft als gevolg dat het voorstel voor een<br />

opplooibare Tensairity constructie aangepast moet worden, rekening houdend<br />

met de geformuleerde voorstellen tot verbetering.<br />

Dit onderzoek heeft bijgedragen tot het creëeren van nieuwe inzichten met<br />

betrekking tot het constructief gedrag van opplooibare Tensairity structuren.<br />

Deze inzichten vormen een solide basis voor verder onderzoek naar opplooibare<br />

Tensairity structuren en brengen ons een stap dichter bij het realiseren<br />

van een opplooibare Tensairity structuur.<br />

x

Contents<br />

Introduction 1<br />

1 Introduction 3<br />

1.1 Lightweight structures . . . . . . . . . . . . . . . . . . . . . . . . 3<br />

1.2 Tensairity structures . . . . . . . . . . . . . . . . . . . . . . . . . 4<br />

1.3 Research goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6<br />

1.4 Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6<br />

2 Literature review & Basic principles 9<br />

2.1 Inflatable structures . . . . . . . . . . . . . . . . . . . . . . . . . . 9<br />

2.2 Tensairity structures . . . . . . . . . . . . . . . . . . . . . . . . . 18<br />

2.3 Deployable mechanisms for Tensairity structures . . . . . . . . . 26<br />

I Design of Deployable Tensairity Structures 33<br />

3 Mechanisms for deployable Tensairity structures 35<br />

3.1 Goal and boundary conditions . . . . . . . . . . . . . . . . . . . 35<br />

3.2 Folding segmented compression elements . . . . . . . . . . . . . 37<br />

3.3 Adapting the foldable truss system . . . . . . . . . . . . . . . . . 45<br />

3.4 Redesign of mechanism . . . . . . . . . . . . . . . . . . . . . . . 52<br />

xi

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60<br />

II Structural behaviour of the Deployable Tensairity Beam 63<br />

4 Experiments on scale models 65<br />

4.1 Experimental set-up . . . . . . . . . . . . . . . . . . . . . . . . . 65<br />

4.2 Observations and results . . . . . . . . . . . . . . . . . . . . . . . 68<br />

4.3 Physical and numerical model . . . . . . . . . . . . . . . . . . . . 72<br />

4.4 Cable configurations . . . . . . . . . . . . . . . . . . . . . . . . . 77<br />

4.5 Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86<br />

4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87<br />

5 Numerical Investigations 91<br />

5.1 Finite element analysis of Tensairity structures . . . . . . . . . . 91<br />

5.2 Modeling the Tensairity beams . . . . . . . . . . . . . . . . . . . 92<br />

5.3 Solving the models . . . . . . . . . . . . . . . . . . . . . . . . . . 99<br />

5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101<br />

6 Analysis of the structural behaviour 103<br />

6.1 Hull section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103<br />

6.2 Dimensioning the struts . . . . . . . . . . . . . . . . . . . . . . . 112<br />

6.3 Introducing hinges . . . . . . . . . . . . . . . . . . . . . . . . . . 123<br />

6.4 Cable configuration . . . . . . . . . . . . . . . . . . . . . . . . . . 135<br />

6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149<br />

6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150<br />

III Prototype of a Deployable Tensairity Beam 153<br />

7 Prototype and experimental set-up 155<br />

xii

7.1 Prototype of Tensairity beam . . . . . . . . . . . . . . . . . . . . 155<br />

7.2 The experimental set-up . . . . . . . . . . . . . . . . . . . . . . . 169<br />

7.3 Postprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173<br />

7.4 Goal of the experimental investigation . . . . . . . . . . . . . . . 174<br />

7.5 Observations and improvements . . . . . . . . . . . . . . . . . . 175<br />

7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176<br />

8 Experimental investigation of the prototype 179<br />

8.1 Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179<br />

8.2 General structural behaviour . . . . . . . . . . . . . . . . . . . . 179<br />

8.3 Influence of cables . . . . . . . . . . . . . . . . . . . . . . . . . . . 186<br />

8.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194<br />

9 Evaluation of the prototype 197<br />

9.1 Comparison with other Tensairity prototypes . . . . . . . . . . . 197<br />

9.2 Deployable versus non-deployable Tensairity beam . . . . . . . 199<br />

9.3 Concluding remarks and suggestions . . . . . . . . . . . . . . . 203<br />

Conclusions 205<br />

10 Conclusions 207<br />

10.1 Design of deployable Tensairity structures . . . . . . . . . . . . . 208<br />

10.2 Structural behaviour of the deployable Tensairity beam . . . . . 208<br />

10.3 Evaluation of the prototype . . . . . . . . . . . . . . . . . . . . . 211<br />

Bibliography 213<br />

List of publications 219<br />

xiii

xiv

Introduction<br />

1

Introduction<br />

1.1 LIGHTWEIGHT <strong>STRUCTURES</strong><br />

1<br />

Architectural and structural engineering have always been subject of the<br />

search for lighter, more efficient and more performing structural systems.<br />

Different types of structures have been developed with the goal of maximizing<br />

structural efficiency and thus reducing the weight of the structure and minimizing<br />

the use of resources. This maximization of the structural efficiency is a<br />

major goal of lightweight engineering.<br />

Various types of lightweight structures exist. They often thank their structural<br />

efficiency to the concept whereby the structure’s material is used as optimal<br />

as possible, ie. by means of loading the structural components only in tension<br />

or compression and this preferably until the material’s yield limit. This can<br />

be achieved by physically separating tension and compression within one<br />

structure, as in the case with tensegrity structures. Another option is using<br />

structures that are only loaded in compression, such as thin compression-only<br />

shells, or in tension, such as membrane structures (figure 1.1).<br />

Figure 1.1: Left: tensegrity structure, middle: compression-only shell,<br />

right: membrane structure<br />

Membrane structures are well known lightweight solutions for covering large<br />

3

4<br />

CHAPTER 1 INTRODUCTION<br />

areas or as component (cladding) for temporary constructions. Two main<br />

categories of tensile structures can be distinguished: the mechanically stressed<br />

and the pressurized membranes. In this latter category the membrane stress is<br />

caused by a pressure load (from a medium, like air or water). A well-known<br />

application of pressurized membranes in civil engineering are the inflatable<br />

structures, such as air-inflated tubes, also called airbeams, and air-inflated<br />

cushions.<br />

Inflatable structures have been used by engineers and architects for several<br />

decades. These structures offer lightweight solutions with a relatively high<br />

structural efficiency and provide several unique features, such as collapsibility,<br />

translucency, and a variety of shapes that can be produced. The assembling<br />

and dismantling of inflatable structures is fast by simple inflation and deflation,<br />

and the transport and storage volume are minimal.<br />

In spite of these exceptional properties, one of the major drawbacks of inflatable<br />

components is their limited load bearing capacity. After all, the external<br />

loading is only carried by the pretension in the membrane of the inflated<br />

component (caused by the internal overpressure). When the air-inflated beam,<br />

also called airbeam, is loaded distributed, compression will occur at one side<br />

of the beam and tension at the other, the same way as occurs in a beam<br />

made of rigid material. The compressive forces are then taken by a decrease<br />

in membrane stress, until this stress is zero. From this point on, additional<br />

compression causes wrinkling of the membrane, which has a large decrease of<br />

the structure’s load bearing capacity as result. Thus the initial prestress in the<br />

membrane of the inflated component is related with the maximum amount of<br />

compression the structure can bear.<br />

As a consequence, significant loads can only be carried with high pressures in<br />

the structure. This leads to very high fabric tensions that can only be supported<br />

by expensive fabrics. Moreover, with increasing pressure, the air tightness,<br />

pressure management and safety become issues. High pressure inflatable<br />

structural components are therefore generally not suitable and desirable for<br />

civil applications.<br />

1.2 <strong>TENSAIRITY</strong> <strong>STRUCTURES</strong><br />

The problem of the limited load bearing capacity of pneumatic structures can<br />

be overcome by avoiding the membrane to take up the external loading. This<br />

is achieved by combining the inflatable component, such as an airbeam, with<br />

traditional building elements such as cables and struts. The strut is applied as

1.2 <strong>TENSAIRITY</strong> <strong>STRUCTURES</strong><br />

compression element at one side of the airbeam, the cable is positioned at the<br />

opposite tension side 1 . Figure 1.2 shows this.<br />

The tension and compression element are thus separated by the air-inflated<br />

beam, which - when inflated to a low pressure 2 - pretensions the tension<br />

element and stabilizes the compression element against out-of-plane buckling.<br />

Indeed, because the compression element is connected continuously<br />

to the prestressed membrane, a movement out-of-plane in one direction of<br />

the element is counterbalanced by the tension of the membrane in the other<br />

direction.<br />

Thus, by combining the air-inflated element with cables and struts, a synergetic<br />

structure emerges whereby the components with different properties<br />

complement each other to a new entity by means of constructive interaction<br />

(Luchsinger and Crettol, 2007). This structural concept is called Tensairity, a<br />

combination of tension, air and integrity, that reflects the relationship with<br />

tensegrity (Luchsinger et al., 2004a).<br />

ession element<br />

compression element<br />

cable<br />

cable<br />

airbeam<br />

Figure 1.2: A basic Tensairity structure is constituted of a compres-<br />

sion and tension element, separated by a low pressurized airbeam<br />

(Luchsinger et al., 2004b).<br />

Due to this constructive interaction of the different elements it is constituted<br />

of, the Tensairity beam has a much higher load bearing capacity than an<br />

airbeam: the Tensairity beam is ten to one hundred times stronger than a<br />

simple airbeam with the same dimensions and pressure 3 . When comparing<br />

the Tensairity beam with a steel profile (HEB-profile) and truss, all designed<br />

for the same load, the Tensairity beam is a factor 6 lighter than the steel profile<br />

and a factor 2 lighter than the truss (Luchsinger et al., 2004a).<br />

Thus, a Tensairity structure has more or less the low weight and compact trans-<br />

1 In the case of a cylindrical airbeam is the cable winded around the airbeam in such a way that<br />

it is positioned at mid span at the tension side<br />

2 Order of magnitude between 100 and 500 mbar (10 - 50 kN<br />

m 2 )<br />

3 From equation 2.5 on page 18.<br />

5

6<br />

CHAPTER 1 INTRODUCTION<br />

port volume of an air-inflated beam, but a much higher load bearing capacity.<br />

These are properties interesting for adaptable or temporary applications, such<br />

as mobile shelters or retractable roofs.<br />

However, while the air beam is easily deployed by inflation, the basic Tensairity<br />

beam needs to be disassembled before it can be packed together. After all,<br />

the compression element of the Tensairity structure possesses some bending<br />

stiffness and must be tightly connected with the hull (in order to maintain its<br />

buckling-free behaviour). Thus, the basic Tensairity beam cannot be folded or<br />

rolled together when deflated.<br />

By replacing the standard compression and tension element with a mechanism,<br />

a deployable Tensairity structure is achieved that needs - besides changing the<br />

internal pressure of the airbeam - no additional handlings to compact or erect<br />

the structure. The development of such a deployable Tensairity structure is<br />

investigated in this research 4 .<br />

1.3 RESEARCH GOAL<br />

The main goal of this research is to gain insights in the structural and kinematic<br />

behaviour of deployable Tensairity structures and develop a first prototype of<br />

such a structure.<br />

Given the lack of general knowledge on deployable Tensairity structures, this<br />

investigation is pursued by means of experimental and numerical investigations<br />

on small and large scale models. Many issues regarding the design of a<br />

deployable system and its structural behaviour need to be investigated. These<br />

different issues are discussed in this dissertation in separate parts.<br />

1.4 OUTLINE OF THESIS<br />

The first part of the dissertation focuses on the development of a mechanism<br />

for the deployable Tensairity structure. The second part investigates the<br />

structural behaviour of a Tensairity beam by means of experiments on scale<br />

models and numerical investigations and identifies the influence of several<br />

design parameters. As a result, a prototype of a deployable Tensairity beam is<br />

designed, fabricated, experimentally tested and evaluated in the third part.<br />

4 Remark that in this research no distinction is made between the terms ‘foldable’ and<br />

‘deployable’: unfolding carries here the same meaning as deploying, just like folding and<br />

compacting. This is in contrast with other research whereby folding only occurs according a<br />

folding line (Jensen, 2004; De Temmerman, 2007)

1.4 OUTLINE OF THESIS<br />

In chapter 2, a literature review describes the context of this research. The<br />

basic principles of the inflatable and Tensairity technology are presented for a<br />

good understanding. A state-of-the-art with regard to deployable Tensairity<br />

structures is given as starting point for this research.<br />

part I - Design of deployable Tensairity structures<br />

Chapter 3 focuses on the development of an appropriate mechanism for the<br />

deployable Tensairity structure. The exploration and analysis of ideas for<br />

deployable systems is presented by means of experiments on various scale<br />

models. By improving the ‘foldable truss system’ with regard to its structural<br />

and kinematic behaviour, an easily foldable proposal for the deployable<br />

Tensairity structure is obtained and proposed.<br />

part II - Structural behaviour of the deployable Tensairity beam<br />

Chapter 4 investigates the structural behaviour of deployable Tensairity beams<br />

by means of experiments on scale models. The focus in the experiments<br />

is gaining qualitative insight in the load bearing behaviour of deployable<br />

Tensairity structures.<br />

In chapter 5, the most significant aspects regarding the finite element calculation<br />

of (deployable) Tensairity structures are discussed. The modelling of the<br />

deployable Tensairity beams, as well how they are solved, is presented.<br />

In chapter 6, the influence on the structural behaviour of several design parameters<br />

and configurations are investigated numerically. The main parameters<br />

whose influence is monitored are the struts’ and hull section, the hinges and<br />

the cable configuration.<br />

part III - Prototype of a deployable Tensairity beam<br />

Chapter 7 presents the prototype of the deployable Tensairity beam. The<br />

components it is constituted of are discussed in detail, as well the assembling<br />

and the way this prototype is investigated.<br />

Chapter 8 discusses the results of the experimental investigation of the prototype<br />

and compares it with the outcome of numerical calculations on the<br />

finite element model of the prototype. As well the general behaviour of the<br />

deployable Tensairity beam as the influence of several parameters, such as<br />

cables and hinges, is discussed.<br />

Chapter 9 evaluates the prototype by comparing experimental and numerical<br />

results with those of other (non-deployable) Tensairity structures. The<br />

7

8<br />

CHAPTER 1 INTRODUCTION<br />

discussion of the results leads to the proposal of improvements.<br />

Finally, chapter 10 compiles all those results and presents the conclusions. The<br />

main findings concerning the design and structural behaviour of deployable<br />

Tensairity structures and the general contributions to the field are discussed.

Literature review & Basic principles<br />

2<br />

First, the context of this research is described by means of a brief literature<br />

review. Then, the basic principles of the inflatable and Tensairity technology<br />

are presented for a good understanding. A state-of-the-art with regard to<br />

deployable Tensairity structures is finally given as starting point for this<br />

research.<br />

2.1 INFLATABLE <strong>STRUCTURES</strong><br />

2.1.1 HISTORICAL CONTEXT<br />

Inflatable structures are not new. The first technologically relevant realization<br />

of an inflatable device dates back to 1783, when the Montgolfier brothers were<br />

the first to venture the sky with their hot air balloon. The fascination of the sky<br />

in those days lead to further technological improvements and inventions such<br />

as hydrogen balloons and zeppelins (Topham, 2002). The idea of transposing<br />

the zeppelin technology to architecture tracks back to the English engineer F.<br />

W. Lanchester. His patent of a pneumatic system for campaign hospitals was<br />

approved in England in 1918, but was never constructed due to the lack of<br />

adequate membrane materials or appeal to possible clients (Herzog, 1976).<br />

On solid ground, pneumatic structures had a first breakthrough during the<br />

World War II. This mobile war demanded the invention of easily transportable<br />

and easy-to-store devices. The properties of lightness, durability and mobility<br />

which resulted in air balloons proved to be useful on the ground. As a result,<br />

inflatable structures were applied as decoys to distract the opponent and as<br />

emergency and radar shelters. In order to protect the radars from extreme<br />

weather conditions, thin non-metallic coverings were developed by Walter<br />

9

10<br />

CHAPTER 2 LITERATURE REVIEW & BASIC PRINCIPLES<br />

Bird. These spherical pneumatic domes, the so called radomes, were applied<br />

on many sites all over the world (figure 2.1).<br />

Figure 2.1: On solid ground, pneumatic structures had a first<br />

breakthrough during the World War II. Left: Walter Bird on top of a<br />

radome (1948), middle and right: inflatable decoys. (Topham, 2002)<br />

In the 1960’s, civil and commercial applications were established and empirical<br />

knowledge was gained with the development of pressurized airhouses as<br />

covering for warehouses, swimming pools and sport facilities. At the same<br />

time, the first academic investigations were undertaken by Frei Otto about the<br />

process of form finding. Through the IASS Pneumatic Colloquium (University<br />

of Stuttgart, 1967) and several publications and designs, Otto broadened<br />

the landscape, not only of pneumatics, but of tension structures in general.<br />

Pneumatics was also part of the repertoire of Richard Buckminster Fuller. His<br />

proposal of a pneumatic dome to cover New York (1962) is a famous example<br />

of Utopian pneumatic architecture (Herzog, 1976).<br />

Also during the 1960’s, a new generation of architects debuted which disagreed<br />

with the principles of Le Corbusier’s modernistic architecture. Radical<br />

architecture groups embraced inflatable forms as reaction to the traditional<br />

architecture (Topham, 2002). The Paris group Utopie formulated critics to the<br />

inertia of the post-war European society and its architecture, urbanism and<br />

daily life. As reaction, they reinterpreted the aesthetic of pneumatic structures<br />

to express ephemerality and mobility (Dessauce, 1999) (figure 2.2).<br />

Figure 2.2: Group Utopie: radical architecture groups embraced<br />

inflatable forms as reaction to the traditional architecture (Topham, 2002)

2.1 INFLATABLE <strong>STRUCTURES</strong><br />

The inherent portability of pneumatic structures soon inspired their use in<br />

temporary and itinerant exhibitions. Several pioneering pneumatic buildings<br />

using air-supported roofs and air-inflated elements were shown at the Expo ‘70<br />

in Osaka, which was the heyday of pneumatic architecture (figure 2.3). Inspired<br />

by the pavilions, the structural concept of the airhouse was applied in<br />

large sport arenas and coverings for tennis courts and swimming pools in the<br />

next decade.<br />

Figure 2.3: Expo ’70 in Osaka was the heyday of pneumatic architecture.<br />

Fuji Pavilion is shown. (Topham, 2002)<br />

However, since those days, no substantial progress in pneumatic architecture<br />

has been seen (Forster, 1994; Hamilton et al., 1994; Luchsinger et al., 2004c).<br />

A widespread exploitation of inflatable structures was setback due to (1) poor<br />

structural performance of the structure and the membrane material, (2) the<br />

complex and expensive pressure management, and (3) the absence of guide<br />

lines and design tools. The use of air-supported roofs decreased because of<br />

their high maintenance cost, vulnerability to strong wind- and snow loads<br />

and the inefficient use of volume by their spherical or cylindrical shape. For<br />

air-inflated elements, such as airbeams 1 , a major drawback was their limited<br />

load bearing capacity, especially for large structures in architecture and civil<br />

engineering. Substantial loads could only be carried with very high pressures<br />

in the airbeam. This leads to very high fabric tensions demanding for high<br />

strength and expensive fabrics.<br />

However today, an increased interest towards ‘inflatables’ can be identified.<br />

After all, in general they offer lightweight solutions with an optimal use of<br />

material. Because of their limited load bearing capacity with low pressure,<br />

pneumatics are nowadays applied as complementary elements to other stiff<br />

structural systems. Two categories can be distinguished here.<br />

One category deals with inflatables that are applied as secondary structural<br />

1 Airbeam is the short name for an air-inflated beam, like a cylindrical inflated tube.<br />

11

12<br />

CHAPTER 2 LITERATURE REVIEW & BASIC PRINCIPLES<br />

elements. An example of such pneumatic components are the air-inflated<br />

cushions. They are used as cladding and bracing of the primary structure and<br />

transfer wind and snow loads to this structure. In some recent large buildings,<br />

air-inflated cushions have shown a good performance as complementary<br />

elements to other stiff structural elements. The enormous domes built to house<br />

the Eden Project in Cornwall are an outstanding combination of rigid and<br />

pneumatic construction methods. Panels of inflated ETFE-foil - a lightweight,<br />

highly transparent foil - are held in place by a self-supporting network of<br />

tubular steel frames. Glass proved far too heavy to be supported by the<br />

lightweight skeleton, whereas an air-inflated cushion weighs less than one<br />

percent of a pane of glass of equivalent size (Liu et al., 2006). Other contemporary<br />

projects promoting the use of cushions as a cladding and covering system<br />

are the Allianz Arena in Munich (June 2005) and the National Aquatics Center<br />

in Beijing (January 2008) (figure 2.4).<br />

Figure 2.4: Application of air-inflated cushions as cladding. Left: Eden<br />

project, middle: Allianz Arena, right: National Aquatics Center.<br />

In the second category, the pneumatics are applied as a part of a structural<br />

element whereby different parts interact with each other. The structural<br />

concept Tensairity is such a combination of parts: cables, struts and an airbeam<br />

interact with each other. The result is a structure with more or less the low<br />

weight and compact transport volume of an air-inflated beam, but a much<br />

higher load bearing capacity. This structural system Tensairity is discussed in<br />

detail in section 2.2.<br />

2.1.2 INTRODUCTORY CONCEPTS<br />

A membrane that is tensioned by means of a pressure is called a pneumatic<br />

structure. The word ‘pneumatic’ comes from the Greek ‘pneuma’, meaning<br />

‘breath of air’ 2 . The pressure is a distributed load perpendicular to the surface<br />

of the membrane and positions the membrane to a form of equilibrium where<br />

2 However, the medium that tensions the membrane can be a liquid or gas.

2.1 INFLATABLE <strong>STRUCTURES</strong><br />

it is stable in both position and form. Due to the tension in the membrane, it<br />

is capable of resisting a certain amount of external loading.<br />

If a pneumatically stressed membrane does not form a closed cavity, it is called<br />

an open pneumatic structure (Herzog, 1976). Examples of such structures<br />

are sails, parachutes, kites. In the case of closed pneumatic structures, the<br />

membrane encloses a pressurized, airtight volume. In this research, only this<br />

latter group is taken into account when mentioning pneumatic structures.<br />

As building application, two different types of closed pneumatic structures can<br />

be distinguished, as illustrated in figure 2.5. In the first category is the space to<br />

be utilized and enclosed by the membrane in overpressure. This category<br />

is called ‘air-supported’ structures, because the single layer of membrane<br />

dividing the interior from the exterior is supported by the air. Often, these<br />

structures are called airhouses. Of course, the supporting medium must be air<br />

and the internal overpressure must be in a range acceptable and comfortable<br />

for the users. Typical overpressures here are in the range of 1 to 10 mbar 3 . In the<br />

second category, the area in overpressure is fully enclosed by a membrane and<br />

is not aimed to be utilized. These structures are called ‘air-inflated’. Another<br />

pressurizing medium than air can be applied here. Depending whether the<br />

horizontal components from the membrane stress are taken by a supporting<br />

structure, the distinction here can be made between ‘airbeams’ and ‘cushions’.<br />

After all, the ‘lens’-shape of a cushion can only be achieved when the horizontal<br />

reaction forces are supported by a primary structure, illustrated on the right<br />

of figure 2.5. The range of internal pressure in these structures depends on<br />

the geometry and the external loads that have be supported, but are typical at<br />

least a factor 10 larger than in the case of air-supported structures. The reason<br />

for this is explained next.<br />

a b<br />

Figure 2.5: Two different types of closed pneumatic structures can be<br />

distinguished. Left: air-supported; middle and right: air-inflated ((a)<br />

airbeam and (b) cushion).<br />

3 1 mbar = 100 N<br />

m 2 = ±1 × 10 −3 atm<br />

13

14<br />

CHAPTER 2 LITERATURE REVIEW & BASIC PRINCIPLES<br />

2.1.3 BASIC PRINCIPLES<br />

For a membrane in equilibrium, the following relation can be derived between<br />

the external loading and the membrane tensions:<br />

nxx<br />

Rx<br />

+ nyy<br />

Ry<br />

= p (2.1)<br />

Rx and Ry represent the radius of curvature [m] in two perpendicular directions<br />

in the plane of the membrane. Usually, the largest and smallest radii of<br />

curvature of the surface are chosen. nxx and nyy are the membrane tension<br />

in the corresponding directions [ N<br />

m ]. p is the load per area and represents in<br />

the case of pneumatic constructions (without additional external loading) the<br />

pressure, expressed in N<br />

m2 (or (m)bar).<br />

If above mentioned equation is applied for a sphere, whereby nxx = nyy and<br />

Rx = Ry, one obtains<br />

n = p.R<br />

2<br />

(2.2)<br />

The membrane tension is thus proportionally to the radius and internal pressure.<br />

As a result, to obtain the same membrane tension, a lower pressure can<br />

be applied in a pneumatic structure with a larger radius. This is the reason<br />

why the internal pressure of air-supported structures (with a larger radius) is<br />

relative small in comparison with airbeams.<br />

For cylindrical structures, the radii in longitudinal and radial direction differ.<br />

After all, the surface of the cylinder is singly curved and the radius in longitudinal<br />

direction can be regarded as infinite. With Rxx = ∞, the membrane stress<br />

in radial direction becomes nyy = p.Ry. When the cylinder is terminated with<br />

two half spheres as end caps, the membrane stresses can be calculated as:<br />

nradial = p.R (2.3)<br />

nlongitudinal = p.R<br />

2<br />

(2.4)<br />

The load-carrying mechanism of an air-inflated beam, also called airbeam<br />

in this research, is quite similar to that of a pretensioned beam (Schodek,<br />

2000). After all, the internal pressure introduces uniformly longitudinal tensile<br />

stresses, as illustrated on the left of figure 2.6. When loading, these stresses<br />

interact with the external loading. This loading causes compression stresses<br />

along the upper surface and tension stresses along the lower surface, the same

2.1 INFLATABLE <strong>STRUCTURES</strong><br />

way as occurs in a beam made of rigid material. As a consequence, tensile<br />

stresses originally present along the upper surface are reduced and those along<br />

the lower surface increased (figure 2.6). Clearly, the internal pressure must be<br />

such that no compressive stresses, which would manifest as wrinkles, are<br />

developed along the top surface. After all, once wrinkles are developed, the<br />

airbeam deflects considerably. The moment necessary to initiate wrinkling<br />

equals M = 1<br />

2 .π.p.R3 , the maximum bending moment the inflated beam can<br />

withstand is π.p.R 3 , with p the internal pressure and R the radius (Comer and<br />

Levy, 1963; Main et al., 1994).<br />

Figure 2.6: The load-carrying mechanism of an airbeam is quite similar<br />

to that of a prestressed beam (Schodek, 2000). Left: the inflated airbeam<br />

causes pretension in the membrane, right: under loading - the prestress<br />

decreases on the compression side and increases on the tension side.<br />

2.1.4 BUILT EXAMPLES OF AIR-INFLATED <strong>STRUCTURES</strong><br />

Most of the current research and development conducted on air-inflated structures<br />

can be traced to space, military, commercial, and recreational applications.<br />

Examples include air ships, weather balloons, inflatable antennas,<br />

temporary shelters, pneumatic muscles and actuators, inflatable boats, temporary<br />

bridging, and energy absorbers such as automotive air bags (Cavallaro<br />

and Sadegh, 2006). This brief overview focuses on the current application of<br />

airbeams in construction.<br />

A distinction can be made between the application of inflatable airbeams in<br />

temporary and permanent construction. As mentioned before, air-inflated<br />

components are in permanent large scale constructions mostly used as complementary<br />

elements to other stiff structural systems. Often is this in the form<br />

of cushions as cladding of the primary structure (Robinson-Gayle et al., 2001;<br />

Oñate and Kröplin, 2007). After all, they offer a lightweight alternative to glass<br />

with another aesthetical and prestigious appearance. Examples are illustrated<br />

in figure 2.4.<br />

With regard to airbeams for temporary constructions, the distinction can be<br />

made between employment on ground or in space. The inflatable technology<br />

15

16<br />

CHAPTER 2 LITERATURE REVIEW & BASIC PRINCIPLES<br />

proves to be useful for ephemeral and mobile constructions. After all, they<br />

provide a combination of features no other type of structure has, such as rapid<br />

and self-erecting deployment, enhanced mobility, large deployed-to-packaged<br />

volume ratios, fail-safe collapse, and possible rigidification.<br />

In the domain of aerospace, investigations are focusing on the development of<br />

large space-based inflatable structures for a variety of applications. Such applications<br />

include lunar habits, radar antennas, solar arrays, sunshields, telescope<br />

reflectors, etc. Reinforced air-supported structures serve as lunar habits, while<br />

airbeams are applied for the other applications. The inflated airbeams can<br />

remain pressurized to achieve stiffness for short missions; however, for longer<br />

missions they can be permanently rigidized to keep their configuration when<br />

space debris penetrates the structure and to avoid the influence of pressure<br />

decrease on structural accuracy (Natori et al., 1995; Cadogan et al., 1999; Salama<br />

et al., 2000; Darooka and Jensen, 2001; Benaroya et al., 2002; Hublitz et al., 2004;<br />

L’Garde, 2010).<br />

With regard to the application of airbeams on the ground, no substantial<br />

progress in the airbeam technology can be seen in scientific literature. Besides<br />

the new structural concept Tensairity, which will be discussed more in detail<br />

in section 2.2, a project worth mentioning is the ‘Airtecture Exhibition Hall’,<br />

developed by Festo (figure 2.7). The supporting structure of this hall is mainly<br />

constituted of inflated elements. The structure contains inflated columns,<br />

inflated wall components and airbeams as roof structure. The airbeams have a<br />

slenderness of approx. 8 and are pressurized till 500 mbar. The hall measures<br />

40 m x 20 m x 7,5 m in inflated state and can fit in a standardized container<br />

to be transported (Wagner, 1997; Schock, 1997). This design for temporary<br />

structures has not been developed further.<br />

Figure 2.7: ‘Airtecture Exhibition Hall’, developed by Festo (Tensinet<br />

Database, 2010).<br />

Most inflatable structures applied nowadays rely on the standard airbeam

2.1 INFLATABLE <strong>STRUCTURES</strong><br />

concept. They are developed for commercial purposes, such as coverings for<br />

events, or for military and emergency applications, such as rapidly deployable<br />

shelters (Inflate, 2010; Buildair, 2010) (figure 2.8). This latter one is usually<br />

composed of a frame of airbeams, covered with a membrane, as illustrated in<br />

figure 2.9. These tents typically have a span of 4-6 m, a height of ± 3 m and<br />

varying length. Their weight per area measures 3-5 kg<br />

m 2 . The internal pressure<br />

of the grid of airbeams varies from producer and ranges from 250 mbar -3 bar<br />

(Losberger, 2010; Eurovinil, 2010; HDT, 2010) 4 . For structures with a larger<br />

span and height, the necessary internal pressure quickly increases, leading to<br />

very high membrane forces and the need for expensive high tech fibres.<br />

Figure 2.8: Application of inflatables for events (Inflate, 2010).<br />

Figure 2.9: ‘Application of inflatables as structure for rapid deployable<br />

emergency and/or military shelters (Losberger, 2010).<br />

4 The Belgian Army applies the Losberger-rds inflatable tents from the typeTDR.<br />

17

18<br />

CHAPTER 2 LITERATURE REVIEW & BASIC PRINCIPLES<br />

2.2 <strong>TENSAIRITY</strong> <strong>STRUCTURES</strong><br />

2.2.1 PRINCIPLE<br />

As mentioned in section 2.1.3, a standard airbeam reaches its load limit for<br />

practical applications when wrinkling occurs at the compression side. After all,<br />

when the prestress becomes zero, the membrane cannot take any compressive<br />

forces. By connecting a stiff element on this compressed side, the load bearing<br />

behaviour is increased because the compressive forces are now partly taken<br />

by this stiff element. When this element is additionally connected with cables<br />

that are spiralled around the airbeam, the structural behaviour is even more<br />

optimised. The result is thus a structure composed of several parts interacting<br />

with each other. This combination of an airbeam and traditional building<br />

elements such as cables and struts is called a Tensairity structure, illustrated<br />

in figure 2.10.<br />

The tension and compression elements are thus separated by the air inflated<br />

beam, which - when inflated - pretensions the tension element and stabilizes<br />

the compression element against buckling. After all, the compression element<br />

is connected continuously to the prestressed membrane which has as result that<br />

a movement out-of-plane in one direction of the element is counterbalanced by<br />

the tension of the membrane in the other direction. The compression element<br />

is thus stabilized by the airbeam and not prone to buckle. As a result, as<br />

well the material of the cable and the compression element can be used to its<br />

yield limit and thus in its most efficient way. Because the airbeam only has<br />

a stabilizing function, the Tensairity beam can operate with low air pressure.<br />

Typical pressures are in order of 100 to 500 mbar.<br />

Because of the interaction of the different elements it is constituted of, the Tensairity<br />

beam has a higher load bearing capacity than an airbeam. Luchsinger<br />

et al. (2004a) calculated the ratio of the load bearing capacity of a Tensairity<br />

beam at a given pressure to a standard airbeam at the same pressure as being<br />

qa,Tensairity<br />

qa,airtube<br />

= 4<br />

.γ2<br />

π3 (2.5)<br />

with qa the load per area and γ the slenderness. Thus, depending on the<br />

slenderness (from 10 to 30 taken into account), the Tensairity beam is ten to<br />

one hundred times stronger than a simple airbeam with the same dimensions<br />

and pressure. When comparing the Tensairity beam with a steel profile (HEBprofile)<br />

and truss, all designed for the same load, the Tensairity beam is a factor

2.2 <strong>TENSAIRITY</strong> <strong>STRUCTURES</strong><br />

6 lighter than the steel profile and a factor 2 lighter than the truss (Luchsinger<br />

et al., 2004a). Thus, a Tensairity structure has more or less the low weight and<br />

compact transport volume of an air-inflated beam, but a much higher load<br />

bearing capacity.<br />

compression element<br />

cable<br />

ession element<br />

cable<br />

airbeam<br />

Figure 2.10: The basic cylindrical Tensairity beam (Luchsinger et al.,<br />

2004a).<br />

2.2.2 BASIC MODEL<br />

Luchsinger et al. (2004a) describes the interaction of the cables, struts and<br />

membrane by means of a simple analytical model for cylindrical Tensairity<br />

beams (figure 2.11). He derives in this model the maximal force T in the<br />

tension and compression element for a homogeneous distributed load q as<br />

T = q.L.γ<br />

8<br />

(2.6)<br />

with γ being the slenderness and L the length of the beam. The slenderness is<br />

determined as the ratio length over diameter (thus thickness): γ = L<br />

2.R .<br />

2 . R<br />

0<br />

p<br />

Figure 2.11: Clarification for the analytical model of the basic cylindrical<br />

Tensairity beam (Luchsinger et al., 2004a).<br />

The compression element is supported and stabilized by the air-inflated beam.<br />

As a result, the interaction between the compression element and the airbeam<br />

can be modeled as a beam on elastic foundation. Luchsinger et al. (2004a)<br />

calculated the spring constant k of this elastic foundation as being k = π.p,<br />

whereby p represents the internal pressure. The critical buckling load of this<br />

q<br />

T<br />

L<br />

19

20<br />

CHAPTER 2 LITERATURE REVIEW & BASIC PRINCIPLES<br />

compression element becomes then<br />

P = 2. π.p.E.I (2.7)<br />

with E the modulus of elasticity of the compression element and I its (area)<br />

moment of inertia.<br />

The buckling load of the compression element is a function of the square root<br />

of the internal pressure, the struts material and its geometry. Note that it is<br />

independent of the length of the beam. This implies that a proper choice of<br />

the bending stiffness of the compression element in relation with the applied<br />

internal pressure can lead to the compression element being loaded to its yield<br />

limit. This is called buckling free compression and an import feature of the<br />

Tensairity technology.<br />

By comparing the load-bearing behaviour of standard airbeams with Tensairity<br />

beams, the influence of the struts and cables becomes visible. The Tensairity<br />

beam is - depending on the slenderness - ten to hundred times stronger than<br />

a standard airbeam with the same dimensions and geometry Luchsinger et al.<br />

(2004a,b). Thus to withstand the same load, the pressure in an airbeam needs<br />

to be ten to hundred times larger than in a Tensairity beam. It is obvious that<br />

in relation to the Tensairity beam, the load bearing capacity of simple airbeams<br />

is very limited.<br />

2.2.3 SHAPE OF THE AIRBEAM<br />

The standard Tensairity beam has a cylindrical shape. Pedretti et al. (2004)<br />

investigated by means of finite element calculations various beam shapes,<br />

illustrated in figure 2.12. The cigar-shaped geometry (b)proves to be more<br />

efficient than the cylindrical form. The spindle shape (c-d), where the tube<br />

end converges to a point, is even stiffer. After all, the shape of the strut is an<br />

arch and transfers thus more optimal forces to the supports. As a result, it has<br />

lower deflections and contributes thus to the stiffness of the Tensairity beam.<br />

Another advantage of the spindle shape is that, compared to a cylindrical<br />

airbeam, the amount of expensive fabric (and thus weight) can be reduced<br />

by up to 33 percent. The volume of the compressed air can be reduced by<br />

up to 47 percent (Luchsinger and Crettol, 2006b). A third asset of the spindle<br />

configuration is that the cable becomes a straight line. As a consequence, a<br />

tension rod or compression element can be applied here. This is an interesting<br />

property for roofs whereby the load case can be negative and compression has<br />

to be taken at the lower side of the beam.

2.2 <strong>TENSAIRITY</strong> <strong>STRUCTURES</strong><br />

Given these advantages, many Tensairity applications are based on spindleshaped<br />

airbeams (Luchsinger et al., 2004b; Pedretti, 2005; Luchsinger and<br />

Crettol, 2006b).<br />

a<br />

b<br />

Figure 2.12: Various forms of Tensairity beams: (a) cylinder, (b)<br />

cigar-shaped, (c) symmetric spindleshaped and (d) asymmetric spindle-<br />

shaped (Luchsinger et al., 2004b).<br />

2.2.4 SPINDLE SHAPED <strong>TENSAIRITY</strong> <strong>STRUCTURES</strong><br />

The behaviour of a spindle shaped Tensairity beam under central point load is<br />

investigated by Luchsinger and Crettol (2006b). As well the load-displacement<br />

behaviour, the general deformation as the forces in the strut are noted and<br />

discussed in the paper.<br />

In these experiments can be seen that the behaviour of the membrane of<br />

the airbeam is reflected in the deformation behaviour of the spindle. The<br />

fabric adapts in a first load-cycle to the new loading, resulting in a residual<br />

displacement. The load-displacement response to subsequent loadings is from<br />

that point on identical. Thus, when investigating these fabric structures, the<br />

displacements should be noted in the second or third load cycle. Figure 2.13<br />

illustrates this.<br />

The results show that the displacement is a non-linear function of the load because<br />

the structure slightly softens with increasing load. The largest deflections<br />

are detected at the upper strut, where the load is applied. The displacement of<br />

the lower strut is smaller, approximately half of the order of the displacement<br />

of the upper strut. This reduction in thickness of the airbeam indicates the<br />

load transfer between upper and lower strut by means of the tensioned fabric.<br />

In addition, the upper strut deforms in a similar way to arches: when the load<br />

acts in the middle of the upper strut, the beam moves slightly upwards at the<br />

end regions. This is illustrated in figure 2.13.<br />

Luchsinger also determines the compression and tension forces in the struts<br />

as function of the applied load. He compares the analytical assumed values<br />

with numerically and experimentally derived values and good agreement is<br />

c<br />

d<br />

21

22<br />

CHAPTER 2 LITERATURE REVIEW & BASIC PRINCIPLES<br />

Load [kN]<br />

1.2<br />

1<br />

0.8<br />

0.6<br />

0.4<br />

0.2<br />

0<br />

−0.2<br />

0<br />

10<br />

p = 150 mbar<br />

20 30 40 50<br />

Displacement [mm]<br />

x [m]<br />

0.6<br />

0.4<br />

0.2<br />

0<br />

−0.2<br />

−0.4<br />

−0.6<br />

p = 150 mbar<br />

F = 0 kN<br />

F = 1 kN<br />

−2.5 −2 −1.5 −1 −0.5 0<br />

z [m]<br />

0.5 1 1.5 2 2.5<br />

Figure 2.13: Left: load-displacement response of the spindle, Right:<br />

experimental deformation of the spindle under central load of 1 kN.<br />

(Luchsinger and Crettol, 2006b).<br />

observed. If high loads at low pressures are not taken into account 5 , there<br />

can be stated that these forces in the struts are independent of the pressure.<br />

The equation whereby the struts’ forces are determined is an adaption of<br />

equation 2.6, taking the distributed load applied over a short distance l 6 into<br />

account: the compression force in the strut and tension force in cable equals<br />

T = Q.γ l<br />

.(2 − ) (2.8)<br />

8 L<br />

where Q is the total applied central load, γ the slenderness and L the length of<br />

the spindle.<br />

The role of the internal pressure on the structure’s behaviour is also determined<br />

in this research. From the experiments can be concluded that the displacements<br />

decrease with increasing pressure, as was also found in the numerical results.<br />

Finally, Luchsinger also presents the influence of the connection of the strut<br />

with the hull. By means of finite element calculations is shown that a connection<br />

whereby the strut can move relative to the membrane (in longitudinal<br />

and vertical direction) causes displacements almost twice as large as in the case<br />

of a fixed connection. An example of such a gliding connection is when the<br />

compression element is connected to the airbeam by means of a pocket (similar<br />

as it is done eg. with the poles in a camping tent). Thus, a tight connection of<br />

the compression element with the fabric can significantly stiffen the Tensairity<br />

structure.<br />

5 The analytical model does not take second order effects in account that occur for high loads<br />

at low pressures (large displacements).<br />

6 l=L for a distributed load and 0 for a point load.

2.2 <strong>TENSAIRITY</strong> <strong>STRUCTURES</strong><br />

Where Luchsinger and Crettol (2006b) investigated and described the spindle<br />

shaped Tensairity beam under point load, is this done for homogeneous load<br />

in Luchsinger and Teutsch (2009). By means of two coupled equations, the<br />

deflections of the upper and lower strut are derived analytically. The inflated<br />

body of the Tensairity structure is modeled as an elastic foundation with shear<br />

stiffness, while the bending stiffness of the struts is neglected. The deflection at<br />

mid span is found to be the sum of two terms. One term is due to the elasticity<br />

of the struts, i.e. the extensioning or shortening of the strut caused by the axial<br />

force. The other term is due to the deformation of the inflated body which<br />

depends on the air pressure. For smaller pressure values, the deformation<br />

of the upper strut is dominated by the deformation of the elastic foundation,<br />

while for higher pressure values, the elasticity of the strut is dominant.<br />

Another development in the domain of Tensairity beams is the application<br />

of a web inside the beam. A web is a connecting element between upper<br />

and lower strut that becomes prestressed due to the outwards movement of<br />

the struts, caused by inflation. This web is typically constituted of cables<br />

or a membrane. This web gives an additional support for the compression<br />

element increasing the value of the spring constant of the elastic foundation.<br />

This improves the stabilization of the compression element and can further<br />

reduce the necessary internal pressure. Breuer et al. (2007) investigated the<br />

application of a membrane web in an inflatable wing for kites. From the<br />

theoretical weight study in this research can be concluded that applying a web<br />

in the Tensairity structure increases the structure’s stiffness and thus efficiency.<br />

Wever et al. (2010) investigated the influence of internal fabric webs on the<br />

structural behaviour of a spindle shaped Tensairity column and showed an<br />

increase of both the axial stiffness and the buckling load.<br />

An airbeam, a Tensairity structure with and without web and a truss are<br />

compared in the theoretical weight study conducted by Breuer et al. (2007).<br />

This is done by means of calculating their weight for a given slenderness and<br />

load. All structures measure three meter, are optimised and are all constituted<br />

of the same materials. The slenderness is varied by varying the parameters<br />

specific for each structure. From this study can be concluded that the weight<br />

of a Tensairity structure is comparable to the weight of an optimized truss.<br />

The web Tensairity structure is lighter than the normal Tensairity structure<br />

due to a reduction of the air pressure, because less pressure is needed for the<br />

same stabilization of the struts. The airbeam is the heaviest configuration for<br />

slenderness values above 10. Figure 2.14 shows the four investigated cases<br />

and the results. More on this can be found in Breuer et al. (2007).<br />

23

24<br />

CHAPTER 2 LITERATURE REVIEW & BASIC PRINCIPLES<br />

weigth (kg)<br />

a b<br />

c<br />

0.09<br />

0.08<br />

0.07<br />

0.06<br />

0.05<br />

0.04<br />

Theoretical weight comparison ( L = 3 m q = 250 N/m)<br />

0.03<br />

Tensairity web spindle<br />

0.02<br />

Tensairity spindle<br />

0.01<br />

truss spindle<br />

air beam cylinder<br />

0<br />

0 10 20 30 40 50<br />

slenderness (-)<br />

60 70<br />

Figure 2.14: Upper: the four investigated cases (with an illustration of<br />

their section), lower: their weight for a given slenderness and under a<br />

given load (Breuer et al., 2007).<br />

As it is clear from this literature review, the structural concept Tensairity is very<br />

new (Luchsinger et al., 2004a; Pedretti et al., 2004). As a result, the research for<br />

this hybrid structure is still in its infancy (Luchsinger, 2009). All aspects of its<br />

structural behaviour are not yet fully understood and many issues are waiting<br />

to be tested, verified and modified.<br />

Among other things, the influence of the connection between upper and<br />

lower strut on the structural behaviour of a five meter Tensairity beam is<br />

currently being investigated. Various prototypes with different connection<br />

between upper and lower strut are tested experimentally. The research in this<br />

dissertation is part of these ongoing investigations.<br />

2.2.5 <strong>TENSAIRITY</strong> APPLICATIONS AND REALIZED PROJECTS<br />

Tensairity structures are not only applied as beams for roofs or bridges.<br />

Research is also being conducted on the application of this technology in<br />

columns, arches, kites and actuators (Plagianakos et al., 2009; Wever et al.,<br />

2010; Gauthier et al., 2009; Luchsinger, 2009; Breuer and Luchsinger, 2009;<br />

Luchsinger and Crettol, 2006a). It is beyond the scope of this research to<br />

present al these applications in detail.<br />

d

2.2 <strong>TENSAIRITY</strong> <strong>STRUCTURES</strong><br />

Figure 2.15: Patents for several Tensairity applications. Left: arch,<br />

middle: actuator, right: advertising column (Pedretti and Prospec-<br />

tiveConcepts, 2004a; Luchsinger and ProspectiveConcepts, 2005; Fuchs<br />

et al., 2005)<br />

When applying Tensairity structures in civil constructions like roofs or bridges,<br />

the safety is an important issue. Air leakages as a result of a damaged hull lead<br />

to a loss of air and a decrease in internal pressure. This can cause a collapse<br />

of the structure. However, because of the presence of the cable-strut structure<br />

in a Tensairity beam, such an unstable situation can be avoided. After all,<br />

the struts in a Tensairity structure have bending stiffness and can be designed<br />

such that the dead load of the structure is carried by the struts without the<br />

need of stabilization of the airbeam. The air pressure is needed for sustaining<br />

additional live loads (Luchsinger and Crettol, 2006a).<br />

In a lightweight Tensairity structure as the roof over the parking garage<br />

in Montreux, the live load is roughly ten times higher than the dead load<br />

(Luchsinger and Crettol, 2007). As a consequence, the structure is designed to<br />

take its dead load; the live load are taken by the interaction of the structure’s<br />

components. In structures like the roof in Montreux, high live load events are<br />

rare, as well the occurrence at the same time of a complete air loss and high<br />

live loads.<br />

Furthermore, all larger Tensairity structures are connected to an external fan,<br />

which is activated when the pressure needs to be increased. This fan can easily<br />

compensate undesired air loss caused eg. gunshot of vandals (Luchsinger and<br />

Crettol, 2007). In addition, the condition of Tensairity structures can be easily<br />

checked by a simple pressure sensor or even by eye.<br />

For reasons of illustration, two realized Tensairity projects are presented briefly.<br />

The first application, shown on the left of figure 2.16 is a roof over a parking<br />

garage in Montreux, Switzerland. Twelve spindle shaped inflated beams with<br />

span up to 28 m build up the structure. As a special feature, the Tensairity<br />

structures are illuminated during night. The roof was completed by the end<br />

25

26<br />

CHAPTER 2 LITERATURE REVIEW & BASIC PRINCIPLES<br />

of 2004 (Detail, 2005). The second construction, illustrated on the right of<br />

figure 2.16 is a skier bridge in the French Alps with 52 m span, supported by<br />

two asymmetric spindle shaped Tensairity girders. During the winter season,<br />

a ski slope runs over the bridge. The thick layer of snow covering the desk<br />

of the bridge during winter leads to high loads. The bridge was completed in<br />

2005 (Pedretti and Luscher, 2007).<br />

Figure 2.16: Left: roof over parking in Montreux, Switzerland - Luscher<br />

Architectes SA & Airlight Ltd, right: skier bridge in the French Alps<br />

- Charpente Concept SA, Barbayer Architect & Airlight Ltd (Airlight,<br />

2010).<br />

2.3 <strong>DEPLOYABLE</strong> MECHANISMS FOR <strong>TENSAIRITY</strong> <strong>STRUCTURES</strong><br />

To comply with the structural principles of Tensairity, the compression element<br />

should be attached for stabilizing reasons continuously to the airbeam.<br />

Otherwise, local buckling can occur. As a consequence, no relative movement<br />

between the struts and the membrane is allowed, which implies that the folding<br />

of the mechanism should be compatible with the inextensible membrane. This<br />

is discussed more in detail in section 3.2 on page 37.<br />

Because of this constraint, many investigated kinematic systems cannot be<br />

used as deployable alternative for the linear compression element. To give an<br />

example, solutions that erect in one dimension like e.g. telescopic compression<br />

elements require relative movement between strut and membrane and can thus<br />

not be applied. Also mechanisms whereby the membrane has to (over)stretch<br />

are not a solution, such as is the case with mechanisms comprised of scissor like<br />

elements. It is not in the scope of this literature review to name all investigated<br />

mechanisms that were rejected because of incompatibility with the folding of<br />

the membrane. Instead, a review is given from the mechanisms that finally<br />

led to a proposal for the deployable Tensairity structure.

2.3.1 EXPLORATIVE MODELS<br />

2.3 <strong>DEPLOYABLE</strong> MECHANISMS FOR <strong>TENSAIRITY</strong> <strong>STRUCTURES</strong><br />

The first developments regarding deployable Tensairity structures are conducted<br />

by Crettol and Luchsinger (2006). Three categories of deployable<br />

compression elements, based on the type of mechanism, are evaluated by<br />

means of experiments on small scale models. This section briefly presents<br />

these explorative experiments.<br />

The compression elements which allow folding of the Tensairity beam because<br />

of their low bending stiffness when the Tensairity beam is deflated are called<br />

flexible compression elements, of which two are discussed here briefly (the ‘hose’<br />

and ‘chain’) (figure 2.18). The insight gained with these flexible systems leads<br />

to the investigation of two discretized compression elements (the ‘segmented<br />

compression pipes’ and ‘separate wooden segments’) (figure 2.19). As a result<br />

of this exploration, the hinged compression element is developed by Crettol and<br />

Luchsinger (2006) (figure 2.20).<br />

These three categories of deployable compression elements are evaluated by<br />

means of experiments on small scale models. An air beam made from PVC<br />

coated polyester fabric of two meter length and 0,2m diameter is used for<br />

the experiments. The internal pressure of the beam is kept constant at 30 kN<br />

m 2<br />

(300 mbar). The investigated proposals are placed in a pocket attached at the<br />

upper side along the beams length (figure 2.17). Remark that the compression<br />

elements are for a first investigation only placed at the compression side and<br />

not as well at the lower side of the beam. A description of the different cases<br />

is given below, as well as a brief discussion of the test results.<br />

Figure 2.17: The two meter air beam with the various investigated<br />

compression elements (Crettol and Luchsinger, 2006).<br />

From the experiments can be concluded that the flexible compression elements,<br />

such as the ‘hose’ and ‘chain’, allow very well the folding of the hybrid structure<br />

27

28<br />

CHAPTER 2 LITERATURE REVIEW & BASIC PRINCIPLES<br />

(figure 2.18). The technology of pressurized compression elements has been<br />

patented (Pedretti and ProspectiveConcepts, 2004b). However, the maximal<br />

load these structures can bear and their stiffness is not sufficient for civil<br />

engineering applications. In addition, the complexity of the solution is too<br />

high. Therefore, solutions with a better load bearing behaviour and higher<br />

stiffness are investigated, like mechanisms constituted of stiff elements: the<br />

discretized systems.<br />

Figure 2.18: The pressurized hose and chain as flexible compression<br />

element (Crettol and Luchsinger, 2006).<br />

These discretized systems are - unlike the flexible ones - constituted of stiff elements<br />

(figure 2.19). Therefore, rolling or folding them together in deflated state<br />

is not so straightforward and a compact configuration is achieved less easy.<br />

When the compression element will be applied at the upper and lower side<br />

of the air beam, folding these discretized systems will be even more difficult.<br />

When looking at their structural behaviour however, a large improvement<br />

is achieved when comparing with the flexible compression elements. The<br />

results show a higher maximal load and stiffness. Applying stiff segments in<br />

the compression element is from a structural point of view a better choice.<br />

This technology of separate segments as compression element for a deployable<br />

Tensairity beam has been patented by Luchsinger et al. (2006).<br />

However, the structural behaviour of these discretized systems can still be<br />

improved by connecting the different segments. Because of the lack of this<br />

connection, the systems are poor in bearing the shear forces and high local<br />

loads. In addition, a good connection in inflated state can decrease the gaps<br />

between the segments and thus increase the stiffness of the structure.<br />

These suggested improvements lead to the design of the ‘composite compression<br />

element’. This compression element is constituted of wooden segments<br />

that are connected by a fabric/tape, attached at the lower side of the elements,<br />

illustrated in figure 2.20. Because of this fabric, the composite compression<br />

element is able to bear shear forces and the tension forces resulting from<br />

bending stresses at large deflections. Aluminum plates are placed on the

2.3 <strong>DEPLOYABLE</strong> MECHANISMS FOR <strong>TENSAIRITY</strong> <strong>STRUCTURES</strong><br />

Figure 2.19: The segmented compression pipes and separate wooden<br />

segments as compression element (Crettol and Luchsinger, 2006).<br />

upper side of the elements to bear the compressive forces. The compression<br />

element can thus be folded to one side and is bending stiff when loaded<br />

on the upper side. The stiffness of the structure is better than that of the<br />

Tensairity beam with separate wooden segments. After all, due to the presence<br />

of the tape, immediate contact is achieved between the segments in deployed<br />

configuration.<br />

Summary<br />

Figure 2.20: The composite compression element (Crettol and<br />

Luchsinger, 2006).<br />

Table 2.1 summarizes roughly and as indication the ‘experimental results’ of<br />

the investigated proposals mentioned before. The load at the deflection of 5 cm<br />

is given for those structures that did not fail yet at a smaller deflection. For<br />

those structures, the maximal load and reason of failure is given. The results<br />

from tests on the air beam without compression element and a Tensairity beam<br />

with wooden continuous compression element are included as reference.<br />

Summarizing the research, there can be stated that there is an evolution in the<br />

development of these foldable Tensairity structures. First, flexible solutions<br />

for a compression element are investigated. They show characteristics very<br />

appropriate for folding the Tensairity beam. However, their stiffness is too low<br />

and their complexity too high for applications in civil engineering. Secondly,<br />

29

30<br />

CHAPTER 2 LITERATURE REVIEW & BASIC PRINCIPLES<br />

Table 2.1: Indication of load bearing behaviour of the investigated<br />

proposals<br />

compression element maximal load deflection reason of limit<br />

air beam without compression element 18 kg - buckling<br />

hose 30 kg 5 cm -<br />

chain 30 kg - buckling<br />

segmented compression pipes 45 kg - buckling<br />

separate wooden segments 48 kg - buckling<br />

composite compression element 60 kg 5 cm -<br />

Tensairity with continuous compr. element 70 kg 3 cm material failure<br />

there is then observed that applying stiff segments in the foldable compression<br />

elements increases the stiffness and maximal load considerably. By connecting<br />

the various segments with a hinge, a mechanism with satisfying properties is<br />

developed.<br />

More details and discussions of the research by Crettol and Luchsinger can be<br />

found in De Laet et al. (2008).<br />

2.3.2 FOLDABLE TRUSS<br />

Santiago Calatrava developed in 1981 in his PhD-dissertation ‘Zur Faltbarkeit<br />

von Fachwerken’ (‘On the folding of trusses’) novel deployable structures<br />

by introducing hinges in trusses and by investigating their kinematics (Calatrava<br />

Valls, 1981). One of his deployable structures is a conventional truss<br />

where the horizontal tension and compression bars of each triangle are divided<br />

in two and reconnected with an intermediate hinge (figure 2.21). This way, the<br />

truss becomes a mechanism. After all, for a truss to be statically determinate,<br />

the number of equations must equal the number of unknowns, or: 2 × j = n + 3<br />

with j the number of joints and n the number of bars (members). As can be<br />

seen from figure 2.21, this is not the case here: 2 × 12 > 16 + 3 and the truss<br />

is as a consequence geometrical unstable. To stabilize the system in deployed<br />

configuration, Calatrava applied vertical bars and a locking mechanism at the<br />

intermediate hinges.<br />

A certain analogue can be detected between Tensairity structures and regular<br />

trusses: the airbeam fulfils the role of the diagonal struts. This reasoning<br />

brought Luchsinger to the idea of adapting Calatrava’s foldable truss system

Figure 2.21: Three steps in the unfolding sequence of the foldable plane<br />

truss (Calatrava Valls, 1981).<br />

for a deployable Tensairity structure (Luchsinger et al., 2007). He adjusted the<br />

system for applying it in a Tensairity structure by replacing the vertical bars<br />

with (pre-tensioned) cables, as illustrated in figure 2.22. The diagonals can<br />

be included or excluded. The linear compression and tension bars are in the<br />

deployable Tensairity structure continuously attached with the hull, and this<br />

way, the truss is stable when the air beam is fully inflated.<br />

Figure 2.22: Adaption of the foldable truss system for applying it in a<br />

Tensairity structure, by Luchsinger et al. (2007)<br />

This is a promising concept as alternative for a deployable Tensairity structure.<br />

The structure can be folded and unfolded without disassembling. In addition,<br />

the mechanism is present at tension and compression side, which allows the<br />

structure to withstand as well downwards loading as upwards loading (which<br />

is the case for wind loadings). An alternative for only downwards loading has<br />

also been developed. In this case, the lower strut is replaced by cables that are<br />

attached to the upper strut by means of a web (Luchsinger et al., 2009).<br />

31

32<br />

CHAPTER 2 LITERATURE REVIEW & BASIC PRINCIPLES

PART I<br />

Design of Deployable Tensairity Structures<br />

33

3<br />

Mechanisms for deployable Tensairity structures<br />

The standard Tensairity beam can not fold because of its continuous compression<br />

element. Therefore, the struts need to be replaced by a suitable<br />

deployment mechanism. This chapter presents the exploration and analysis<br />

of ideas for deployable systems by means of experiments on various scale<br />

models. By investigating these models, insight is gained to improve the<br />

existing foldable truss system. This system and the adaptations necessary<br />

to improve it are presented. But first, before exploring several solutions,<br />

the design requirements and boundary conditions that have to be taken into<br />

account are discussed.<br />

3.1 GOAL AND BOUNDARY CONDITIONS<br />

There is chosen in this research to design for civil engineering structures,<br />

like bridges or roofs. This choice imposes several restrictions on the design of<br />

deployable Tensairity structures. It is the challenge of this research to find a<br />

suitable solution that complies with these boundary conditions.<br />

Before addressing the various requirements, the definition of ‘deployable<br />

Tensairity structure’ in this research is clarified. After all, two categories of<br />

applying Tensairity elements in a deployable structure can be distinguished.<br />

The first one, on the left in figure 3.1, can be understood as a deployment<br />

of Tensairity structures. Here, the deployment does not occur at the level of<br />

the Tensairity elements, but at the connection of multiple Tensairity-elements.<br />

This kind of deployment can be seen as the replacement of bars or plates of<br />

conventional structures by Tensairity-beams or cushions. The second category,<br />

shown on the right of figure 3.1, is called in this research a ‘deployable<br />

Tensairity structure’. Here, the deployment of the continuously attached struts<br />

elements is the main focus. This latter category is investigated in this research.<br />

35

36<br />

CHAPTER 3 MECHANISMS FOR <strong>DEPLOYABLE</strong> <strong>TENSAIRITY</strong> <strong>STRUCTURES</strong><br />

Figure 3.1: Two categories of applying Tensairity elements in a<br />

deployable structure can be detected. Left: a deployment of Tensairity<br />

structures; right: a deployable Tensairity structure. The right category is<br />

the subject of this research.<br />

The unfolding of the deployable Tensairity beam has to be actuated by simple<br />

increase of the inner pressure of the air beam. Once this structure is inflated<br />

and thus completely unfolded, it should behave by the rules of the structural<br />

concept of Tensairity. The difference in structural behaviour between the<br />

deployable and standard Tensairity beam in inflated state should be as minimal<br />

as possible. On the other hand, once the deployable Tensairity structure is<br />

deflated, it should be able to be folded, rolled or packed together to a compact<br />

configuration without disassembling the different components it is constituted<br />

of. No additional operations (e.g. fixations with fasteners, mechanisms, ... )<br />

on the structure are allowed in this research.<br />

Since the deployable Tensairity structure will be applied in the context of civil<br />

engineering, requirements are also made on the type of solution. Specific for<br />

lightweight structures is that - due to wind suction - the load case can be<br />

upwards. Therefore, to be able to resist such an inversion of the load, the cable<br />

at the lower side of the beam is replaced with a bending stiff strut. When the<br />

structure is not loaded and operational, it should also withstand transportation<br />

and (un)folding. As a consequence, a durable and simple solution has to be<br />

proposed. In addition, the structure should fold and unfold for a large amount<br />

of times without significant damages to the membrane. A simple solution<br />

(with the less degree of freedom as possible) guarantees a straightforward, lowtech<br />

and thus robust solution. At the same time, the elegance of a Tensairity<br />

beam in general and the lightweight appearance of the struts in relation to the<br />

air beam should be preserved. Therefore, the visual presence of the mechanism<br />

has to be as minimal as possible.<br />

Above mentioned design constraints are all chosen to be imposed in this<br />

research. In addition to that, and in order to meet the requirements of the<br />

structural principle Tensairity, some other boundary conditions have to be<br />

taken into account. The most important is that the compression element for

3.2 FOLDING SEGMENTED COMPRESSION ELEMENTS<br />

stabilizing reasons should be attached along its length to the membrane as<br />

much and as tight as possible. This requirement has a large influence on the<br />

design of the mechanism, since no relative movement between the struts and<br />

the membrane is allowed. As a consequence, the folding of the membrane<br />

should be compatible with the proposed foldable compression element. After<br />

all, the fabric has to be regarded as an inextensible material and can therefore<br />

not stretch when e.g. it has to bend around a hinge.<br />

3.2 FOLDING SEGMENTED COMPRESSION ELEMENTS<br />

From the explorative models discussed in section 2.3.1 is concluded that stiff<br />

elements connected with hinges are - with regard to the structural behaviour<br />

- a good solution for the foldable compression element of the deployable<br />

Tensairity beam. This section investigates the folding of these systems by<br />

means of scale models.<br />

3.2.1 INTERACTION BETWEEN MECHANISM AND MEMBRANE<br />

To comply with the structural principles of Tensairity, the compression element<br />

should be attached for stabilizing reasons along its length to the membrane. As<br />

a consequence, no relative movement between the struts and the membrane is<br />

allowed, which implies that the folding of the mechanism should be compatible<br />

with the inextensible membrane. Because of this constraint, many kinematic<br />

systems can not be used as deployable compression element. Solutions that<br />

erect in one dimension like e.g. telescopic compression elements can thus not<br />

be applied. Also mechanisms whereby the membrane has to (over)stretch are<br />

not a solution.<br />

Overstretching occurs where the distance between two points of the membrane<br />

has to increase to comply with the kinematics of the mechanism. Consider<br />

two hinged struts which are coplanar with and attached to a membrane, as<br />

illustrated in figure 3.2. The axis of rotation of the hinge is perpendicular<br />

to the membrane, and it is obvious that - when rotating - the struts would<br />

move in such a way that the membrane has to stretch, which is not possible<br />

and/or allowable. Thus, the axis of rotation can only lie in the plane of the<br />

membrane 1 . However, fulfilling this condition is not sufficient. In addition,<br />

the membrane should be coplanar with the center line of the hinge. When this<br />

is not the case, the membrane will be stretched or wrinkled, as illustrated in<br />

1 When the membrane is curved, as is the case for a cylindrical airbeam, the axis of rotation<br />

should be tangential to the membrane at the position of the hinge.<br />

37

38<br />

CHAPTER 3 MECHANISMS FOR <strong>DEPLOYABLE</strong> <strong>TENSAIRITY</strong> <strong>STRUCTURES</strong><br />

figure 3.3. Stretching damages the membrane and wrinkling can obstruct the<br />

folding. Thus, a lot of attention has to be paid to the relative position of the<br />

center of rotation of the hinges with regard to the membrane.<br />

Figure 3.2: The folding of the mechanism should be compatible with<br />

the membrane.<br />

Figure 3.3: The membrane shoud be coplanar with the center line of the<br />

hinge (upper). When this is not the case, wrinkling (middle) or stretching<br />

(lower) of the membrane occurs.<br />

Another example, illustrating the consequences of the inextensible membrane

3.2 FOLDING SEGMENTED COMPRESSION ELEMENTS<br />

attached to the strut is illustrated in figure 3.4. Here, a cylindrical air beam<br />

is used as inflated component of the Tensairity beam. The upper and lower<br />

strut are positioned with a distance of 2 × R from each other. This distance<br />

can due to the membrane only be increased maximally to a value of π × R,<br />

thus approximately by fifty percent. As a consequence, there are limitations<br />

in applying mechanisms which move the upper and lower strut further from<br />

each other. Examples of such systems are concertina-type solutions (like an accordion)<br />

and (translational) scissor mechanisms in the airbeam for decreasing<br />

the degree of freedom of the folding process.<br />

2.R<br />

Figure 3.4: The folding of the mechanism should be compatible with<br />

the membrane. Left: inflated airbeam, right: most compact position due<br />

to stretching of membrane.<br />

Now the design consequences of the interaction between hinged stiff segments<br />

and membrane are understood, various systems of hinged folding can be<br />

developed and investigated.<br />

3.2.2 HINGED FOLDING<br />

The compression element, comprised of stiff elements connected with hinges,<br />

should be designed such that the Tensairity beam can fold undisturbed. In<br />

addition, as mentioned before, the kinematic system should be robust and<br />

simple. Therefore, in this research only hinges with just one axis of rotation<br />

are applied. With the x-axis being the longitudinal direction of the strut, the<br />

y- and z- axes are considered (figure 3.5). Three of the investigated models<br />

are discussed here briefly, two with hinges folding around the y-axis, called<br />

the ‘loop folding’ and ‘foldable truss’ and one around the z-axis, called ‘out of<br />

plane folding’.<br />

Out of plane folding<br />

Figure 3.6 shows the model whereby the hinges rotate around the z-axis (and<br />

thus allow movement ‘out-of-the-plane’). Since the external loading acts in<br />

the xz-plane, the hinged struts can take up bending moments around the yaxis<br />

and act as a continuous strut. Another advantage of this system is the<br />

straightforward and neat folding of the membrane when deflated, because the<br />

π.R<br />

39

40<br />

CHAPTER 3 MECHANISMS FOR <strong>DEPLOYABLE</strong> <strong>TENSAIRITY</strong> <strong>STRUCTURES</strong><br />

Figure 3.5: The coordinate system: the x-axis is the longitudinal direction<br />

of the strut, the loading acts in the xz-plane.<br />

beam can be folded like a flat membrane as illustrated in figure 3.6.<br />

However, this solution is not suitable from a structural point of view. After<br />

all, previous research on Tensairity structures with continuous compression<br />

elements showed that structural failure is mostly caused by global out-of-plane<br />

buckling. Introducing hinges that allow rotation out of the plane weaken the<br />

structure considerably. In addition, when bending moments occur in the<br />

upper strut under external loading, these hinges will also be bent. This is not<br />

favourable, since the smallest deformation of the kinematic system can cause<br />

the system to be jammed.<br />

Loop folding<br />

Figure 3.6: The out-of-plane folding.<br />

The second model is constituted of hinges that allow rotation around the yaxis<br />

(see fig. 3.5) and is very similar to the ‘composite compression element’<br />

discussed in section 2.3.1: the segments are connected at their lower side to<br />

each other by a fabric/tape. The compression element can thus be folded to<br />

one side and is bending stiff when loaded on the upper side (figure 3.7). Note

3.2 FOLDING SEGMENTED COMPRESSION ELEMENTS<br />

that when the separate segments are fully fixed to the air inflated membrane<br />

(glued, stitched, ..), the membrane takes up the role of the textile hinge and an<br />

additional connection is thus unnecessary.<br />

Figure 3.7: Tensairity beam with ‘loop-folding’ compression element<br />

(Crettol, EMPA).<br />

This system can be folded in different ways, as illustrated in figure 3.8. One<br />

way of folding is called the loop-folding (figure 3.8a). This implicates that all<br />

segments have different lengths, which is not desirable from a manufacturing<br />

point of view. Note the space between the different loops in figure 3.8a for<br />

‘accommodating’ the membrane. Another way of folding is called the closedloop<br />

folding (figure 3.8b). Here, the structure is folded until one loop is<br />

reached. The advantage of this proposal is that every segment has the same<br />

length. On the other hand, the radius of the packaging becomes very large in<br />

the case of long beams. A solution for folding the Tensairity beam where all<br />

segments can have the same length and where the folding is independent of<br />

the length of the Tensairity beam is called the ‘spiral folding mechanism’. This<br />

mechanism is a modified ‘closed-loop’ folding system: the hinge between the<br />

segments is inclined. This way, the mechanism will have the shape of a spiral<br />

in folded position (figure 3.8c). The stability of this packaging is guaranteed by<br />

using tapered segments and a fabric hinge at the starting point of the tapering<br />

(figure 3.9). More on the design of this ‘closed-loop’ folding system can also<br />

be found in De Laet et al. (2009).<br />

Because of the specificity of the hinges, this model folds only in one direction.<br />

The hinge is always positioned at the interface between the inflated beam<br />

and the compression element, because the membrane of the airbeam can<br />

not overstretch. The ‘segmented compression element’ can therefore only<br />

41

42<br />

Figure 3.8: Different folding possibilities: a. loop-folding, b. closed-loop<br />

folding, c. spiral folding mechanism.<br />

Figure 3.9: The stability of the ‘closed-loop’ folding system can be<br />

guaranteed by using tapered segments and a fabric hinge at the starting<br />

point of the tapering.<br />

be positioned at one side of the Tensairity beam. This concept is thus not<br />

suitable for applications where the direction of the loading can change (e.g.<br />

wind pressure and suction), because this implies that the Tensairity beam<br />

should have compression elements at both sides of the beam, which is not the<br />

case with this solution for a deployable Tensairity structure. Therefore, the<br />

folding concept is not developed further in this research.<br />

Foldable truss<br />

The third model is the foldable truss system, already introduced in chapter 2.<br />

Figure 3.10 shows the struts’ mechanism (Luchsinger et al., 2007). The foldable

3.2 FOLDING SEGMENTED COMPRESSION ELEMENTS<br />

truss can be seen as a conventional truss where the horizontal tension and<br />

compression bars are divided in two and reconnected with an intermediate<br />

hinge. This way, the truss becomes a mechanism. Figure 3.11 shows the<br />

deployment without airbeam of the foldable truss. Figure 3.12 shows the<br />

deployment sequence at inflation. The diagonals can be included or excluded.<br />

The compression and tension bars are in the deployable Tensairity structure<br />

continuously attached with the hull, and this way, the mechanism is stable<br />

when the air beam is fully inflated.<br />

Figure 3.10: The foldable truss system by Luchsinger et al. (2007).<br />

Figure 3.11: The deployment of the foldable truss system without air<br />

beam.<br />

As can be seen, this foldable solution allows the application of a compression<br />

element at the upper and lower side of the air beam, which is a requirement<br />

43

44<br />

Figure 3.12: The deployment sequence of the foldable truss (Luchsinger<br />

et al., 2007).<br />

for the design. Because the upper and lower strut move to each other when<br />

folding, the upper and lower strut can be directly connected with cables. No<br />

stretching of the circumferential membrane will occur because of this decrease<br />

in distance between upper and lower strut. Note that some local stretching at<br />

the hinges that rotate inwards occurs.<br />

The hinges in this solution fold down in the direction of the loading. There<br />

was expected that this would have a big influence on the structure’s stiffness.<br />

However, the explorative models show that the air beam counteracts - although<br />

not completely - the downwards movement of the hinges, which is beneficial<br />

for the structural behaviour. It is subject of further research (presented in the<br />

next section) to improve the design of the mechanism in general and the hinges<br />

in specific such that the stiffness is increased.

3.2.3 SUMMARY<br />

3.3 ADAPTING THE FOLDABLE TRUSS SYSTEM<br />

The deployable Tensairity beam, designed for being applied in the field of<br />

civil engineering, has to be robust and simple. Therefore, the compression<br />

element is chosen to be linear to accommodate and transfer the compressive<br />

forces the most optimal. This linear element is preferably constituted of<br />

stiff segments connected with a hinge and should have a straightforward<br />

deployment. Investigation of explorative models showed that the interaction<br />

between the membrane and the struts, which are fully connected together, is<br />

a major boundary condition that has to be taken into account in the design of<br />

the mechanism.<br />

Several things are learned from the explorative models. Folding out of the<br />

plane seems a solution that results in a structural sound beam because the<br />

hinges do not fold in the direction of the loading. However, out-of-plane<br />

buckling will then occur at smaller loads. In addition, the hinges are loaded<br />

in an unfavourable way, which should be avoided because deformation of the<br />

kinematic system will happen. Solutions whereby the hinges rotate only in the<br />

direction opposite to the loading have a good structural behaviour. However,<br />

mechanisms constituted only of these hinges just fold in one direction. Therefore,<br />

they are not fully applicable where upper and lower strut are required.<br />

The foldable truss mechanism shows characteristics, structural as well kinematic,<br />

that are all suitable for the deployable Tensairity beam. Off course,<br />

the system needs to be investigated further and adapted to be applied for a<br />

deployable Tensairity beam. This will be discussed in the next sections.<br />

3.3 ADAPTING THE FOLDABLE TRUSS SYSTEM<br />

The foldable truss system is identified as a suitable mechanism for the<br />

deployable Tensairity structure. Now, the configuration and detailing is<br />

adjusted to improve the deployment and structural behaviour.<br />

3.3.1 CONFIGURATION<br />

The prototype of the foldable truss, developed by Luchsinger et al. (2007),<br />

shows a straightforward deployment and a good load-bearing behaviour,<br />

especially when cables (or diagonal bars) are used to connect the upper and<br />

lower strut.<br />

However, there are also issues to be solved and optimized. The bars are<br />

positioned adjacent to each other with the type of hinge used (figure 3.13). As<br />

45

46<br />

a result, the hinges are loaded eccentric, which should be avoided. Another<br />

disadvantage of the present system is that it can not be fully folded, because<br />

the segments of upper and lower strut interfere with each other. In addition,<br />

the interaction between the mechanism and the membrane is such that when<br />

folding the strut, the membrane gets jammed between the struts. Cutting<br />

of the membrane can occur this way. Of course, this should be prevented.<br />

Figure 3.13 shows the issues to be solved with a new configuration.<br />

Figure 3.13: Issues to be solved in a new configuration: (a) eccentric<br />

loading of the hinges, (b) membrane is jammed between bars and (c) the<br />

system can not be fully folded because of interfering of the upper and<br />

lower strut.<br />

The hinge mechanism of the ‘foldable truss’-model is improved by adapting<br />

the initial configuration. The mechanism is changed in such a way that the<br />

membrane is positioned between the bars (instead of being cut by the bars)<br />

and that the bars lie in the same plane (instead of two parallel planes). With<br />

this configuration, the hinges are loaded centrically.<br />

The length of the bars with this configuration need to be derived from some<br />

basic equations in order to fold correctly. From figure 3.14 can be seen<br />

L1 = 2A + 3B + 4D (3.1)<br />

L2 = 2B + 5C + 6D (3.2)<br />

B = C + t (3.3)<br />

with t the thickness of the strut and C the distance (clearance) chosen by the<br />

designer to accommodate the membrane. In all cases has to be checked that<br />

A > D and preferably that A > D + C.<br />

The main issues of the previous model seem to be solved with this new<br />

configuration of the foldable truss. However, new problems are arisen. The

A A<br />

D<br />

C<br />

B<br />

B<br />

D D D<br />

C C C<br />

C<br />

D<br />

B<br />

C<br />

B B<br />

C<br />

3.3 ADAPTING THE FOLDABLE TRUSS SYSTEM<br />

L 1<br />

L 2<br />

D D membrane<br />

Figure 3.14: Left: the annotation of the bars of the new configuration,<br />

right: the membrane is accomodated between the bars.<br />

foldable truss is now constituted of almost twice as many hinges. As a result,<br />

more segments can rotate relative to each other which makes the packing of the<br />

structure to a compact configuration more complex. This way, some elements<br />

tend to deploy in other directions than they should, which hinders the compact<br />

folding.<br />

A solution for this issue is investigated in the next sections, first by decreasing<br />

the degree of freedom of the deployable Tensairity truss, then by designing<br />

the hinges such that a more straightforward deployment is reached.<br />

3.3.2 DEGREE OF FREEDOM<br />

The decrease in degree of freedom of the truss is investigated with the goal to<br />

improve and simplify the folding process of the deployable Tensairity beam.<br />

First, there is looked at the effect on the kinematic behaviour of introducing<br />

the diagonals (figure 3.15). The decrease in degree of freedom achieved by<br />

this is however not enough to have a major influence on the folding process.<br />

Mainly weight is added here.<br />

The diagonals of the deployable truss are replaced by a scissor mechanism,<br />

illustrated in figure 3.16. The resulting structure is a foldable truss with one<br />

degree of freedom, as intended. However, because the scissor mechanism lies<br />

in the same plane as the bars, the configuration can not completely fold to a<br />

47

48<br />

Figure 3.15: Diagonals are introduced to decrease the DOF.<br />

closed position. In addition, there is expected to have buckling in these slender<br />

diagonals when the Tensairity beam will be loaded.<br />

Figure 3.16: The diagonals of the deployable truss are replaced in this<br />

proposals by a scissor mechanism.<br />

More models are manufactured in the attempt to decrease the degree of<br />

freedom when folding. Sliders are applied to guide the intermediate hinges,<br />

cables transfer the rotations of the hinges, etc. The conclusion is always the<br />

same: to ease and simplify the packing of the deployable Tensairity structure,

3.3 ADAPTING THE FOLDABLE TRUSS SYSTEM<br />

more material, details, fragility and especially complexity needs to be added<br />

to the mechanism. This is in contradiction with the purpose of designing a<br />

lightweight, robust and simple deployable Tensairity beam. Therefore, a more<br />

straightforward packing of the structure is pursued by means of the geometry<br />

and configuration of the hinges.<br />

3.3.3 HINGES<br />

From the closed configuration of the mechanism, illustrated in 3.14, can be seen<br />

that there are two types of hinges, characterized by their range of deployment:<br />

some hinges allow the bars to rotate 180 degrees, some 90 degrees. To increase<br />

the control of the deployment and to create a better load-bearing behaviour,<br />

‘stops’ are introduced in the hinges. This way, the range whereby the bars can<br />

rotate around the hinges are limited. Several solutions for achieving the ‘90<br />

degree-stops’ are modeled and are shown in figure 3.17.<br />

In two proposals for these stops, extra elements are added around the rotating<br />

segments, such that the bars are hindered and can not rotate further than a<br />

predefined angle. In another solution, a stop is being made by the geometry of<br />

the contact surface of the two parts of the hinge. This stopping mechanism has<br />

already been applied for the ‘loop’-folding in section 3.2.2 and had satisfying<br />

effects.<br />

Figure 3.17: To increase the control of the deployment and to create a<br />

better load-bearing behaviour, ‘stops’ are introduced in the hinges.<br />

Up to now, the proposals of hinges have always been limited to being assem-<br />

49

50<br />

bled out of flat laser cut shapes. Because this is a limiting factor regarding the<br />

evaluation of the hinges, the new proposal is fabricated by means of computer<br />

numerical controlled (CNC) machine tools. Figure 3.18 shows the aluminum<br />

hinge where a square tube can be placed around. This proposal is redesigned<br />

to fit in a regular aluminum strut for a Tensairity structure, also illustrated in<br />

figure 3.18.<br />

Figure 3.18: Two proposals for hinges in aluminum fabricated by means<br />

of CNC machine tools.<br />

All solutions show an increase of control when packing the Tensairity beam to a<br />

compact configuration. After all, the hinges can now only rotate in a predefined<br />

interval. However, from the small scale laser cut models is seen that the least<br />

play in these ‘stops’ is sufficient for cancelling their effect. Attention should<br />

thus be paid to the materialization of the idea. The fabrication of the hinges in<br />

aluminum by means of CNC allows to better evaluate the solution.<br />

3.3.4 FROM STRAIGHT TO CURVED<br />

Up to now, the deployable Tensairity structure is composed of a cylindrical air<br />

beam and a truss with the shape of a trapezium. However, research showed<br />

that a Tensairity beam with a spindle shape and thus a curved upper and lower<br />

strut, is six times stiffer than a cylindrical Tensairity beam (Pedretti et al., 2004).<br />

From now on, this spindle shape will be applied for the deployable Tensairity<br />

structure. This does not have a great influence on the foldable configuration<br />

or the design of the hinges, as can be seen in figure 3.19.

3.3 ADAPTING THE FOLDABLE TRUSS SYSTEM<br />

Figure 3.19: A spindle shape will be applied for the deployable Tensairity<br />

structure for reasons of increase in stiffness.<br />

51

52<br />

3.4 REDESIGN OF MECHANISM<br />

The influence of hinges in the struts of a deployable Tensairity beam are<br />

investigated experimentally and numerically. These studies are presented<br />

further in this dissertation and show that the position of the hinge can have an<br />

influence on the load bearing behaviour of the structure: the maximum load<br />

the structure can bear decreases when the hinge is positioned too close to the<br />

ends of the beam. After all, the compressed strut will buckle outwards at this<br />

position. This is discussed more in detail in section 4.5 and section 6.3.<br />

3.4.1 COMPACTING THE CONFIGURATION<br />

Moving the hinges towards the middle improves thus the structural behaviour<br />

of the deployable Tensairity beam. For this reason it is chosen to redesign the<br />

configuration of the mechanism. The opportunity is taken at the same time to<br />

improve the kinematic behaviour of the mechanism. Indeed, after additional<br />

evaluations and explorative experiments, some issues open to improvement<br />

are detected. In the current configuration, the upper strut has to fold around<br />

the lower strut and vice versa. Even with the limitation of the rotation of the<br />

hinges to 90 degrees, this folding is in some cases still tough going. Another<br />

disadvantage of the current foldable truss is the large amount of small bars.<br />

There should be evaluated whether a new system is possible whereby there<br />

is less variation in bar length. At the same time is investigated if the folded<br />

configuration can be conceived more compact.<br />

In the new proposal, the segments of the upper strut are ‘moved upwards’ in<br />

their folded configuration, as illustrated in 3.20. This way, the upper strut does<br />

not has to fold anymore around the lower strut and vice versa. This means<br />

that only two layers of membrane have to be accommodated between the bars<br />

and hinges instead of a bunch of membrane. As a result, the small bars in<br />

the configuration are not necessary anymore and the hinges do not have to<br />

stop rotating after 90 degrees, but can close completely. This simplifies the<br />

design of the hinges and decreases their fragility. In addition, by shortening<br />

the upper middle struts and thus ‘moving’ the hinges in folded configuration<br />

upwards, the segments of the upper strut at the ends become longer, which<br />

means that the distance between the supports and the first hinge in the upper<br />

strut increases, which is aimed for as discussed before. Finally, another pluspoint<br />

of this solution is that the size of the folded configuration is decreased.<br />

Notice that asymmetrical upper and lower strut (with regard to the position<br />

of hinges) are required to have the supports at the same side of the folded

A<br />

C<br />

package.<br />

A<br />

B<br />

3.4 REDESIGN OF MECHANISM<br />

Figure 3.20: In the new proposal (annotated as C), the segments of the<br />

upper strut are ‘moved upwards’ in their folded configuration.<br />

As mentioned, a compact configuration is one of the objectives of this ‘redesign’.<br />

Therefore is chosen for symmetry between upper and lower strut with<br />

regard to the amount of segments folding inwards. This way, a configuration<br />

with parallel segments in folded configuration and segments with‘straightforward’<br />

length is achieved. With L being the length of a strut, the lower segments all<br />

measure L<br />

L<br />

6 , the middle upper segments have a length of 12 and the two side<br />

segments are L<br />

4 long.<br />

3.4.2 HINGES<br />

As already mentioned, all hinges are now allowed to rotate 180 degrees, which<br />

simplifies their design. However, they are not all the same. Two types of<br />

hinges exist, the hinges at the supports not taken into consideration. Their<br />

difference lies in the direction of rotation: one type rotates towards (or ‘in’ )<br />

the air beam, the other outwards (or ‘away’). As explained in section 3.2.1, the<br />

membrane should be coplanar with the center line of the hinge. Otherwise,<br />

the membrane will be stretched or wrinkled.<br />

C<br />

53

54<br />

In the case of the deployable Tensairity beam, the membrane lies on the same<br />

side of the struts for the hinges that rotate towards or away from the air beam.<br />

This means that for both cases, the axis of rotation has to be the same and<br />

collinear with the membrane. For the hinge that rotates outwards, no problem<br />

arises. However, for the hinge that rotates inwards, these conditions lead to the<br />

design whereby the hinge is constituted of two axes of rotation, as illustrated<br />

in figure 3.21.<br />

Figure 3.21: The two hinges, rotating 180 degrees. Left: the hinge<br />

rotating ‘away’ from the air beam. Right: the hinge rotating ‘inwards’.<br />

Because the axis of rotation has to be collinear with the membrane, this<br />

latter hinge is constituted of two axes of rotation.<br />

These hinges are first fabricated in wood as test case, later in aluminum. For<br />

reasons of manufacturing and cost, several proposals were made. A hinge<br />

out of one piece (solid) proved too complex and expensive to manufacture<br />

(figures 3.22). Therefore, the design is adapted to allow manufacturing out of<br />

flat pieces (subdivided), shown in figure 3.23.<br />

3.4.3 DIAGONAL CABLES<br />

The mechanism is constituted of hinges that move outwards and inwards.<br />

As a consequence, the distance between points of the upper and lower strut<br />

changes during deployment. As will be discussed further in this dissertation,<br />

cables connecting the upper and lower strut may increase the stiffness of the<br />

deployable Tensairity beam under loading. Off course, when points move<br />

away from each other, cables can not be applied here. Therefore, the distance<br />

between hinges before, during and after folding is calculated.<br />

Because the mechanism has many degrees of freedom, it is unpredictable what<br />

the exact intermediate configurations are during the folding and unfolding.<br />

Therefore, there is assumed that angles between the struts rotate with the<br />

same aspect ratio from open till closed position, thus in relative values from

Figure 3.22: The designs of the aluminum solid hinges<br />

Figure 3.23: The subdivided hinges.<br />

3.4 REDESIGN OF MECHANISM<br />

55

56<br />

0 percent to 100 percent closed. Because this is an assumption, the distances<br />

during deployment may be incorrect. After all, it is possible that one hinge is<br />

already completely folded, while the other has not rotated yet. This does not<br />

pose a problem however, since it are especially the distances in the fully open<br />

and closed position that are relevant for this research.<br />

The position of each point is first calculated during deployment. Then the<br />

distances between the hinges are plotted. Figure 3.24 presents the annotation<br />

of the bars and angles. The coordinates of the points are given as:<br />

x0 = 0 y0 = 0 (3.4)<br />

x1 = L1. cos(α) y1 = L1. sin(α) (3.5)<br />

x2 = x1 + L2. cos(α + γ − π) y2 = y1 − L2. sin(α + γ) (3.6)<br />

x3 = x2 + L2. cos(α + γ − ε) y3 = y2 + L2. sin(α + γ − ε) (3.7)<br />

x4 = x3 + L2. cos(π − α − γ + ε) y4 = y3 − L2. sin(π − α − γ + ε) (3.8)<br />

xa = L3 ya = 0 (3.9)<br />

xb = xa + L4. cos(π − β) yb = ya − L4. sin(π − β) (3.10)<br />

xc = xb + L5. cos(β − δ) yc = yb + L5. sin(β − δ) (3.11)<br />

For clarity, the angles mentioned above are the angles at a certain deployment.<br />

Thus when the structure is folded by 50%, α = α50% = αopen<br />

2 . By means of above<br />

coordinates, the distances between the various points on opposite struts are<br />

calculated and plotted, see figure 3.25.<br />

y<br />

x<br />

α<br />

L 3<br />

a<br />

β<br />

L 1<br />

Figure 3.24: The annotation of the bars and angles for calculating the<br />

distances between the hinges during deployment.<br />

From the graph can be seen that the distance between the upper and lower<br />

hinges varies. All links from the upper strut to hinge b increase in length<br />

during folding. Also the length of 1 − c and 3 − c increases. However, these<br />

links shorten again after a certain amount of folding. It can be that when the<br />

folding is not uniform, these links do not increase in length. The links whereby<br />

1<br />

γ<br />

L 4<br />

L 2<br />

δ<br />

b<br />

ε<br />

2<br />

L 2<br />

3<br />

φ<br />

L 5<br />

L 2<br />

4<br />

c

Distance between hinge (1,2,3,4) and (a,b,c) [mm]<br />

70<br />

60<br />

50<br />

40<br />

30<br />

20<br />

10<br />

a<br />

1<br />

4c<br />

0<br />

1 2 3 4 5 6 7 8 9 10<br />

γ − folding : from open ( value 1) to closed position (value 10) [/]<br />

2<br />

b<br />

3<br />

1b<br />

3b<br />

4b<br />

2b<br />

1c<br />

3a<br />

3c<br />

4a<br />

2c<br />

2a<br />

4<br />

1a<br />

3.4 REDESIGN OF MECHANISM<br />

Figure 3.25: Plot of the distances between the various points on opposite<br />

struts for a 2 m spindle with slenderness 10.<br />

the distance increases during folding can thus not be connected with a cable.<br />

Figure 3.26 shows all cable configuration allowing (upper) and obstructing<br />

(lower) the folding.<br />

3.4.4 FOLDING THE HULL<br />

Some points of the upper strut move away from the lower strut when folding,<br />

as shown above. This implies that the outer hull has to be designed such<br />

that it allows for this increase in length. As can be seen in figure 3.25, the<br />

links whose length increases most are 1 − b and 3 − b. The increase of both<br />

lengths is identical. It is thus sufficient to design the membrane such that it can<br />

c<br />

57

58<br />

allow folding<br />

obstruct folding<br />

Figure 3.26: Cables connecting the upper and lower strut that can be<br />

used in this configuration, because their length decreases when folding.<br />

accommodate the increase in length between 1−b. After all, due to the spindle<br />

shape, the amount of outer hull between 1 − b is smaller than 3 − b. Thus, if<br />

the membrane will accommodate the outwards movement between 1 − b, no<br />

problem will arise between 3 − b. In figure 3.27 can be seen that stretching of<br />

the membrane occurs between 1 − b.<br />

Figure 3.27: Stretching of the membrane occurs between 1 − b.<br />

To avoid this stretching of the membrane and thus obstruction of the folding,<br />

the outer hull should be dimensioned such that enough membrane is available<br />

between 1 − b. In the case of a standard spindle where the section is circular,<br />

the hull is determined by the shape of the struts and can thus not be changed.<br />

As a consequence, stretching of the membrane will occur and the standard<br />

spindle can not be folded completely with this configuration. In the case of a<br />

web spindle, the radii of the outer hull along the length of the spindle beam<br />

can be chosen. After all, the struts are kept in position by means of cables<br />

between the upper and lower strut.<br />

To calculate the distance along the hull between point 1 and b, the segment<br />

of the spindle is considered as a cone. The error made by this approximation<br />

is low due to the small curvature of the spindle in the longitudinal direction.<br />

Consider radii R1 and Rb as given, as well the height of the spindle h1 and<br />

hb 2 . Points 1 and b are positioned a distance l from each other, this is thus the<br />

height of the cone (figure 3.28).<br />

2 Remark that the height of the spindle, thus the distance between the upper and lower strut

1<br />

h 1<br />

R 1<br />

l<br />

b<br />

h b<br />

3.4 REDESIGN OF MECHANISM<br />

Figure 3.28: Stretching of the membrane occurs between 1 − b.<br />

2πRb<br />

α 1<br />

2πR1 1<br />

s<br />

Figure 3.29: The cone can be flattened and from this, the distance s can<br />

be calculated.<br />

The cone can be flattened and the geodesic line between points 1 and b becomes<br />

a straight line, as illustrated in figure 3.29. From this figure, the distance s can<br />

be calculated as:<br />

s =<br />

α<br />

β<br />

R b<br />

αb<br />

l’<br />

b<br />

l<br />

L<br />

<br />

l ′ 2 + L 2 − 2.L.l ′ cos β (3.12)<br />

When the distance s does not match the length necessary for a complete folding,<br />

the radii need to be changed. As will be discussed in section 6.1, the radii of<br />

every section along the spindle’s length are dependent: once a radius at a<br />

certain position along the length is chosen (R1 for example), all other radii are<br />

along the spindle’s length, is always determined and known by design.<br />

h<br />

R<br />

l<br />

ϕ<br />

59

60<br />

determined. Thus by means of iterative calculation, the appropriate radii can<br />

be determined for achieving a distance s between the points 1 and b. This<br />

means that the cutting pattern of the hull of the air beam can not be chosen,<br />

but is determined by the geometry and kinematics of the foldable compression<br />

element.<br />

The cutting pattern is redesigned, taking the above mentioned design rules<br />

into account. It is applied in a two meter long Tensairity beam by way of<br />

evaluation. The Tensairity beam can completely be folded, which indicates<br />

that the cutting pattern of the airbeam allows the outwards movement of the<br />

points 1-b. However, as already observed before, the folding of the structure<br />

and thus the packing of the membrane to a dense bunch does not go very<br />

smoothly. This is because the membrane contains a certain amount of bending<br />

stiffness. In addition, the outer hull of the membrane tends to move between<br />

the struts, which can obstruct folding. With this new cutting pattern for the<br />

model, this difficult folding is more obvious than before, since a membrane<br />

with larger bending stiffness (coating on both sides) is applied and because<br />

more membrane is used in the airbeam with the new cutting pattern.<br />

Therefore, research is conducted on the linear folding of the cylindrical and<br />

spindle shaped membrane according a predefined folding pattern (figure 3.30).<br />

After all, folding a membrane along lines goes much easier than folding it to<br />

a random bunch. An additional advantage is that the folding occurs in a<br />

controlled way, i.e. the folding pattern dictates how the membane moves.<br />

However, all solutions investigated imply mechanisms with many hinges<br />

or acquire a complex mechanism which would result in complicated and<br />

fragile hinges. Although the folding of the adapted foldable truss does not<br />

go smoothly, it is still experienced to be the most appropriate solution so far.<br />

Therefore is chosen to develop and investigate this system further. Figure 3.31<br />

illustrates de model for the deployable Tensairity beam.<br />

3.5 CONCLUSIONS<br />

Several models were designed and investigated in this research to gain insight<br />

in the development of an appropriate mechanism for the deployable Tensairity<br />

structure. These explorations resulted in the ‘redesign’ of the foldable truss<br />

system with regard to the structural and kinematical behaviour.<br />

The explorations show the necessity of having in mind the application where<br />

the structure is designed for. After all, many different solutions exist for a

3.5 CONCLUSIONS<br />

Figure 3.30: Research is conducted on the linear folding of the cylindrical<br />

and spindle shaped membrane according a predefined folding pattern.<br />

Figure 3.31: Folding sequence of the proposal for a deployable Tensairity<br />

beam.<br />

deployable Tensairity structure, however, they are not all suitable for the same<br />

use. In this research is chosen to design for an application in the field of<br />

civil engineering (like a roof or bridge structure), and therefore is aimed for<br />

a solution that is as strong, robust, low-tech and straightforward as possible.<br />

Besides the boundary conditions dictated by the application purpose and/or<br />

designer, there are also some requirements and consequences inherent to the<br />

structural concept of Tensairity. The study cases show that the continuous<br />

connection of the membrane with the compression element is the most crucial<br />

and imperative requirement. The folding of the membrane should thus be<br />

compatible with the proposed foldable compression element.<br />

The investigation of the structural behaviour of the various models indicates<br />

the ‘hinged stiff segments’ as being an appropriate mechanism as compression<br />

61

62<br />

element. The folding of this system occurs preferably in the same plane as the<br />

loading, because otherwise deformation of the kinematic system is risked. As<br />

a result, the hinges fold towards or away from the air beam. When designing<br />

the hinges, the inextensibility of the membrane has to be taken into account.<br />

The centre line of the hinge has to lie in the same plane as the membrane,<br />

otherwise stretching or wrinkling will occur.<br />

From various studied models with hinged elements, the ‘foldable truss system’<br />

proves to be the most appropriate. Evaluations of the system lead to the<br />

improvement with regard to the structural and kinematic behaviour. The<br />

system’s load bearing capacities are ameliorated by changing the longitudinal<br />

shape from cylindrical to spindle, by decreasing the amounts of hinges and<br />

segments and by positioning hinges on compression side towards the middle.<br />

The kinematic behaviour is improved by adapting the configuration in such<br />

a way that the deployment occurs more straightforward. This is achieved<br />

with a redesign of the foldable truss which has as result that the segments of<br />

upper and lower strut do not have to ‘fold’ into each other. In addition, less<br />

hinges and less complicated joints that are compatible with the folding of the<br />

membrane are necessary.<br />

The result of this chapter is a proposal for a compression and tension element<br />

(both capable of taking compression) for a deployable Tensairity structure,<br />

achieved by means of investigations on scale models. However, before developing<br />

further the deployable Tensairity structure and building a prototype,<br />

the influence of several design parameters (such as the configuration of cables<br />

or the amount and position of hinges) on its structural behaviour needs to be<br />

understood. This is investigated in the next part.

PART II<br />

Structural behaviour of the<br />

Deployable Tensairity Beam<br />

63

Experiments on scale models<br />

4<br />

Before simulating or developing a large scale model for a deployable Tensairity<br />

beam, it is necessary to gain some first insights in the structural behaviour of<br />

this complex synergetic structure. This chapter investigates the structural<br />

behaviour of these beams by means of experiments and numerical investigations<br />

on small scale models. The focus in this chapter is gaining qualitative<br />

insight in the load bearing behaviour of Tensairity structures in general and<br />

in the influence of the different parameters on the load-bearing behaviour of<br />

the deployable Tensairity structure in specific. After all, no experiments have<br />

been conducted yet on deployable Tensairity beams.<br />

Experiments are conducted on a two meter deployable cylindrical and spindle<br />

Tensairity beam. The experimental set-up and the models are first described in<br />

more detail. Then the influence on the structural behaviour of the parameters<br />

such as the amount of hinges, the presence of internal cables connecting the<br />

upper and lower strut of the Tensairity beam, the configuration of these cables,<br />

the shape of the beam etc. are discussed. Finally, some conclusions are made<br />

that will provide the basis for a proposal for a deployable Tensairity structure.<br />

4.1 EXPERIMENTAL SET-UP<br />

4.1.1 SMALL SCALE MODELS<br />

Two deployable Tensairity beams, a cylindrical and spindle shaped are investigated<br />

experimentally. The studied models are two meter long and the<br />

maximum height in the middle of the beams is 0,25m. The air beams are<br />

composed of two layers. An outer PU coated polyester fabric 1 carries the<br />

1 ‘Riverjacket 200’ from Rivertex<br />

65

66<br />

CHAPTER 4 EXPERIMENTS ON SCALE MODELS<br />

tension forces due to the air pressure and surrounds two internal bags made<br />

of thin PU foil that keep the structure airtight. These internal bags are<br />

positioned adjacent to each other in the longitudinal direction. This way,<br />

elements connecting the compression and tension side of the beam can be<br />

placed between the two bags. This is illustrated in figure 4.1. These two bags<br />

are also called air chambers.<br />

Figure 4.1: Left: Longitudinal view of the investigated cylindrical and<br />

spindle shaped deployable Tensairity beams (isostatically supported),<br />

right: section of the beams<br />

The internal pressure of these air chambers is kept constant at 100 mbar<br />

during the experiments by means of a compressed air regulator. The internal<br />

overpressure is monitored continuously by means of a water column. Cables<br />

and bars used are made of steel and the bars have a rectangular cross section<br />

(10 mm height × 6 mm wide). The cables connecting the upper and lower strut<br />

have a diameter of 2 mm. The compression and tension elements are placed<br />

in a pocket. Holes are made in these pockets to connect upper and lower side<br />

of the beam by cables or struts.<br />

Various configurations of the cylindrical Tensairity beam are tested in order to<br />

reveal the influence of the different parameters on the load bearing behaviour,<br />

such as the amount of hinges and the presence of pretensioned cables that<br />

connect the upper and lower hinges. The investigated configurations are<br />

illustrated in figure 4.2. Only the struts and the connection between them are<br />

illustrated. The hull is not shown but is the same for all configurations. Full<br />

lines are used to indicate struts, the dotted line represents cable elements and<br />

black dots illustrate the hinges.<br />

Configurations 1, 15 and 16 are used to investigate the influence of the amount<br />

of hinges on the load bearing behaviour. The results of configurations 2 to 14<br />

are used to study the influence of vertical and diagonal cables and diagonal<br />

bars on the load-bearing behaviour of the deployable Tensairity beam.<br />

h = 0.25 m<br />

outer hull<br />

pocket<br />

strut<br />

inner layer<br />

cable

1<br />

2<br />

3<br />

4<br />

4.1.2 SET-UP<br />

5<br />

6<br />

7<br />

8<br />

9<br />

10<br />

11<br />

12<br />

4.1 EXPERIMENTAL SET-UP<br />

Figure 4.2: Investigated configurations: they differ in amount of hinges<br />

and/or cable configuration.<br />

The two meter long statically determined deployable Tensairity beam is supported<br />

in its endpoints by a stiff steel frame. The loads are applied manually in<br />

the upper hinges. Depending on the load case, 50 N, 30 N or 10 N is applied in<br />

each load step for respectively distributed, asymmetrical and point load. The<br />

deflections of the beams are measured in ten points at each load step by means<br />

of analogue clock gauges, distributed evenly at the upper and lower side of<br />

the beam. Figure 4.3 illustrates the test set-up. Figure 4.4 gives an illustration<br />

of the data gained for one configuration and one load case. The displacement<br />

of one case is plotted throughout loading.<br />

Figure 4.3: Set-up for load-deflection experiments on scale model.<br />

There must be remarked that the results presented in this chapter are quantita-<br />

13<br />

14<br />

15<br />

16<br />

67

68<br />

CHAPTER 4 EXPERIMENTS ON SCALE MODELS<br />

displacement [mm]<br />

0<br />

5<br />

10<br />

15<br />

20<br />

25<br />

30<br />

-1 0<br />

position along x axis [m]<br />

1<br />

Figure 4.4: Illustration of the data gained for one configuration and one<br />

load case. The displacement is noted after each load step of 10 N per<br />

node.<br />

tively not exact as it would be for large scale models. After all, the scale models<br />

can contain more easily manufacture imperfections and approximations due<br />

to their size, such as using a pocket for the connection with the strut. In<br />

addition, it is difficult to quantify these influences on the test results. Also<br />

the procedure for conducting the experiments is very basic; think about<br />

applying loads in discrete steps and reading the deflections from analogue<br />

clock gauges. However, it is not the purpose of this chapter to retrieve very<br />

accurate quantitative data, but to gain insight in the structural behaviour by<br />

comparing the results qualitatively.<br />

4.2 OBSERVATIONS AND RESULTS<br />

This section presents the observations and conclusions drawn from the experimental<br />

investigations. The influence of various parameters on the load<br />

bearing behaviour is discussed.<br />

4.2.1 APPLYING THE LOAD<br />

The load-displacement response of the deployable Tensairity beam is monitored<br />

when loading and unloading the structure several times. Figure 4.5<br />

(left) shows the applied load as function of the time. The load is applied and<br />

released in three cycles. The response of the deployable Tensairity structure,<br />

illustrated on the right of figure 4.5, is very similar to the behaviour of a spindle<br />

shaped Tensairity beam with continuous compression element under a point<br />

load, investigated experimentally by Luchsinger and Crettol (2006c). The xaxis<br />

of the load-displacement response represents the average displacement of<br />

the five points on the upper strut, the y-axis shows the applied load.

Load [N]<br />

350<br />

300<br />

250<br />

200<br />

150<br />

100<br />

50<br />

0<br />

0 50 100<br />

time [min]<br />

150 200<br />

Load [N]<br />

350<br />

300<br />

250<br />

200<br />

150<br />

100<br />

50<br />

4.2 OBSERVATIONS AND RESULTS<br />

0<br />

0 10 20 30 40 50<br />

Average deflection [mm]<br />

Figure 4.5: Left: The applied load as function of the time. Right: The<br />

load-displacement response of the spindle.<br />

The graph shows that the Tensairity structure adapts to the new load condition<br />

during the first load cycle. This can be seen from the residual displacement<br />

of the unloaded structure after the first cycle. The displacements under the<br />

second and third load cycle are almost identical. The deflections of the second<br />

or third cycle are noted and used in the next sections. The graph also shows that<br />

the structure has another load-displacement path during loading than during<br />

unloading. This phenomenon, called hysteresis, indicates energy dissipation.<br />

It has to be attributed to the air beam, since the steel chords of the Tensairity<br />

girder have an elastic behaviour. It can result from the fabric. The details<br />

behind the hysteresis are not in the scope of this research. It is in any case an<br />

interesting aspect regarding the dynamic properties and damping behaviour<br />

of Tensairity girders. The residual displacements as well the hysteresis are<br />

known behaviours of fabrics and can also be seen in the biaxial testing of<br />

fabrics.<br />

4.2.2 INFLUENCE OF POINT(S) OF APPLICATION AND LOAD CASE<br />

It is evident that the displacement of the beam along its length is dependent<br />

of the load case. In addition, in the case of a Tensairity beam is the deflection<br />

of the upper strut also dependent whether the load is applied on the upper or<br />

lower strut of the beam. Figure 4.6 shows the deformed shape of a deployable<br />

Tensairity beam loaded at the upper strut and at the lower strut. A larger<br />

displacement arises at the side where the load is applied, which indicates that<br />

load is being taken up by the stressed hull.<br />

To be able to compare the various configurations from figure 4.2, the average<br />

displacement of the upper strut is taken into account (unless mentioned<br />

69

70<br />

CHAPTER 4 EXPERIMENTS ON SCALE MODELS<br />

F<br />

Figure 4.6: Different deflection when the load is applied on the upper or<br />

lower side of the beam.<br />

differently). This is the average value of the displacement of several evenly<br />

distributed points on the upper strut. In the case of these explorative models<br />

are the values taken into account of the five analogue clock gauges at the upper<br />

strut.<br />

In the experiments, the load will always be applied at the upper strut of<br />

the beam. The displacements under three different load cases, illustrated in<br />

figure 4.7, were registered to verify whether the conclusions derived from the<br />

comparison of the various configurations are dependent of the applied load<br />

cases. The results are incorporated in the following sections.<br />

4.2.3 INTERNAL PRESSURE<br />

Figure 4.7: The three applied load cases.<br />

The internal pressure of the beam is varied during the experiments. The deflections<br />

of various configurations were measured with an internal pressure of 75,<br />

100 and 125 mbars. The load-deflection behaviour of case 1 under the different<br />

pressures is illustrated in figure 4.8 (left). From the results can be concluded<br />

that the stiffness of the structure is increased with increasing pressure, such as it<br />

is the case for Tensairity beams with continuous compression element (Breuer<br />

et al., 2007; Luchsinger and Crettol, 2006a,b). After all, a higher pressure and<br />

thus a more pretensioned membrane leads to a more constant spacing between<br />

tension and compression element. Moreover, the friction between the pocket<br />

and the compression element increases with higher pressure values, which<br />

results in a stiffer structure.<br />

F

load [N]<br />

7<br />

6<br />

5<br />

4<br />

3<br />

2<br />

1<br />

075 mbar<br />

100 mbar<br />

125 mbar<br />

0<br />

0 10 20 30 40 50<br />

deflection [mm]<br />

deflection [mm]<br />

0<br />

10<br />

20<br />

30<br />

40<br />

4.2 OBSERVATIONS AND RESULTS<br />

50<br />

075 mbar<br />

100 mbar<br />

125 mbar<br />

60<br />

-1 0<br />

position along beam [m]<br />

1<br />

Figure 4.8: Influence of the internal pressure on the stiffness of the<br />

deployable Tensairity beam (case 1). Left: load-displacement curves;<br />

right: the displaced shape of the upper strut at 300 N.<br />

4.2.4 INFLUENCE OF NUMBER OF HINGES<br />

To study the influence of the number of hinges on the load bearing behaviour<br />

of the structure, three cases with different amount of hinges in the upper<br />

and lower strut are investigated. Their load-displacement response under<br />

a distributed load and with an internal pressure of 100 mbar is illustrated<br />

in figure 4.9 (left). The displacement of the compression chord at mid span<br />

is given on the x-axis, the y-axis represents the applied load. The (scaled)<br />

displacement of the upper chord at 300 N (60 N per node) is given on the<br />

right of figure 4.9. From the results can be seen that the cases 1 and 15 have<br />

similar deflections, despite the different number of hinges. The presence and<br />

influence of the middle hinge on the stiffness is thus much greater than that of<br />

other hinges. To conclude more on the influence of the amount of hinges on<br />

the structural behaviour of the Tensairity beam, more investigations need to<br />

be done. This will be done by means of simulations in chapter 6.<br />

4.2.5 INFLUENCE OF CABLES<br />

The compression and tension element of a Tensairity beam can be connected<br />

by means of a fabric web or cables. This way, the section of the airbeam is no<br />

circle anymore, but seems constituted of two circle segments. Figure 4.1(right)<br />

shows this. The connecting element, a fabric web or cable, becomes stressed<br />

when the airbeam is inflated. Previous research showed that using a fabric web<br />

improves the structural behaviour of the Tensairity beam (Breuer et al., 2007;<br />

71

72<br />

CHAPTER 4 EXPERIMENTS ON SCALE MODELS<br />

load [N]<br />

7<br />

6<br />

5<br />

4<br />

3<br />

2<br />

1<br />

case 1<br />

0<br />

0 10 20 30 40<br />

displacement [mm]<br />

case 15<br />

case 1 case 15 case 16 case 1 case 15 case 16<br />

displacement [mm]<br />

0<br />

10<br />

20<br />

30<br />

40<br />

case 16<br />

-1 0<br />

position along beam [m]<br />

1<br />

Figure 4.9: Influence of the amount of hinges on the stiffness of the<br />

structure. The internal pressure measures 100 mbar. Left: load-<br />

displacement graph, right: deformed shape of the upper strut under<br />

300 N (60 N per node).<br />

Wever et al., 2010) (see p. 23). Since using a continuous web would obstruct<br />

the folding of the Tensairity beam, cables are used to connect the upper and<br />

lower hinges.<br />

The influence of this connection between upper and lower strut on the load<br />

bearing behaviour of a deployable Tensairity structure is investigated. Before<br />

discussing the experimental results of the various cable configurations, the<br />

influence of cables on the load-bearing behaviour of a deployable Tensairity<br />

beam is first investigated and understood by means of a physical and numerical<br />

model.<br />

4.3 PHYSICAL AND NUMERICAL MODEL<br />

Since the structural behaviour of deployable Tensairity beams has never been<br />

studied before, the experimental investigations from the previous section help<br />

to gain general insight in these structures. However, it is difficult from these<br />

experimental results to derive how the structure really ‘works’. Therefore, a<br />

physical model (with its simplifications and approximations) is developed in<br />

the next section for a basic understanding of the effect of interactions between

4.3 PHYSICAL AND NUMERICAL MODEL<br />

load, pressure, membrane, compression element and cables. Conclusions<br />

derived from this simplified model are verified by means of a numerical<br />

model 2 .<br />

Many work on the basic understanding of regular Tensairity beams has already<br />

been conducted by Luchsinger et al. (2004a). This section focuses on the loaddisplacement<br />

behaviour of the deployable Tensairity beam: its stiffness and<br />

maximal load. More investigations with regard to the hinges, the effect of the<br />

section of the struts and geometry of the section of the hull are presented in<br />

chapter 6.<br />

As seen before in this chapter, the investigated deployable Tensairity beam is<br />

constituted of an airbeam, an upper and lower strut and cables connecting the<br />

hinges of upper and lower strut. The shape of the section of the airbeam is<br />

constituted of two circle segments. Figure 4.10 illustrates a longitudinal and<br />

sectional view of the deployable Tensairity beam (case 10).<br />

1 a 2 b 3 c 4 d 5<br />

Figure 4.10: A longitudinal and sectional view of the deployable<br />

Tensairity beam.<br />

4.3.1 INFLATION<br />

When inflating the airbeam, the overpressure pushes the upper and lower<br />

struts outwards and the hull tends to become a circle. This action will<br />

be counterbalanced by the cables that connect the compression and tension<br />

element. As a result, the cables become tensioned and experience thus a<br />

tensile force. The value of this force can easily be calculated.<br />

In chapter 2 (p. 14) is seen that the radial membrane tension nradial [ N<br />

m ] is the<br />

product of the radius of the hull and the internal overpressure: nradial = p × R.<br />

As a result, the cable force F in one cable equals<br />

F = 2 × p × R × l × sinα (4.1)<br />

with α being the angle between the membrane (tangential) and the horizontal<br />

(indicated in figure 4.10) and l the distance between two adjacent cables. This<br />

2 Details on the development of finite element models in this research can be found in chapter 5.<br />

h<br />

R<br />

α<br />

73

74<br />

CHAPTER 4 EXPERIMENTS ON SCALE MODELS<br />

normal force F [N] can also be called the pretension in the cable due to inflation<br />

(Fpre).<br />

The deployable Tensairity beam with vertical and diagonal cables connecting<br />

upper and lower strut (as shown in figure 4.10) is investigated by means of<br />

finite element calculations. The finite element model, illustrated in figure 4.11,<br />

is inflated with an internal overpressure of 100 mbar. From the figure can be<br />

seen that the membrane is not fully closed at the ends and only modeled until<br />

the first cable. Otherwise, convergence was an issue. This approximation has<br />

as result that the cables closest to the ends (cable 1 and 5) experience half of the<br />

calculated and expected pretension ( 1<br />

2 Fpre). The cable pretension derived from<br />

the numerical calculations is 149 N for the middle cables and 74,5 N for cables<br />

1 and 5. All forces introduced by inflation are taken by the vertical cables.<br />

The diagonals are not pretensioned under inflation because of their angulated<br />

position.<br />

The pretension in the vertical cables is also calculated with equation 4.1. With<br />

using the same parameters as in the finite element model 3 , one obtains the<br />

value of 147 N, which is a good approximation of the numerical value.<br />

Figure 4.11: The deployable Tensairity beam is also investigated<br />

numerically. The airbeam is only modeled until the first and last cable<br />

for reasons of convergence.<br />

4.3.2 LOADING<br />

The cable pretension is decreased by loading the deployable Tensairity beam<br />

(downwards). When the amount of external load taken by one cable is equal<br />

to the pretension in this cable (Fpre), it becomes slack. This means that the<br />

cable has from that point on zero stiffness and cannot support any additional<br />

loading. As a consequence, the cable does not contribute anymore to the<br />

3 An angle α between the hull and the horizontal of 10 ◦ , a height h between upper and lower<br />

strut of 0.25 m, a radius R of 0.127 m, length l between adjacent cables of 0.333 m, and internal<br />

pressure p of 10 kN<br />

m 2 .

4.3 PHYSICAL AND NUMERICAL MODEL<br />

structural behaviour and one can expect the stiffness of the Tensairity beam to<br />

change at the value whereby the cables become slack.<br />

The deployable Tensairity beam from figure 4.10 is loaded with a point load<br />

in each upper hinge. If Fext is the total amount applied load, then in each<br />

hinge a load of 1<br />

5 Fext is applied. From standard trusses, one knows that the<br />

first verticals (cable 1 and 5) experience a compression force of 1<br />

2 Fext. This<br />

means that these cables become slack when Fpre = 1<br />

2 Fext or Fext = 2Fpre. Since<br />

the cables 1 and 5 are tensioned with a normal force of 74,5 N, the maximal<br />

load this deployable Tensairity beam can bear before changing stiffness is thus<br />

149 N.<br />

This is also investigated by means of finite element calculations on the model<br />

presented in figure 4.11. The displacement of all upper hinges is noted and<br />

the average value in relation to the applied load is illustrated in figure 4.12.<br />

From the curve can clearly be seen that the stiffness changes at a total load of<br />

approximately 150 N, which corresponds with the analytical derived value.<br />

Figure 4.13, plotting the tension in the cables throughout loading, shows that<br />

cables 1 and 5 indeed reach zero tension at this value. The graph also shows<br />

that the diagonal cables are tensioned under loading, as is also the case for a<br />

truss with the same configuration of diagonals. This holds true until cable 1<br />

and 5 become slack. From that point on, the diagonals are compressed.<br />

load [N]<br />

250<br />

200<br />

150<br />

100<br />

50<br />

0<br />

truss<br />

case 10<br />

airbeam<br />

0 0.2 0.4 0.6 0.8 1<br />

x 10 −5<br />

average displacement [m]<br />

Figure 4.12: Average displacement of the upper strut of case 10 in relation<br />

to the applied load. (pressure is 100 mbar, five point loads in upper<br />

hinges). (Numerical results).<br />

Case 10 has also been investigated numerically under various pressures.<br />

Figure 4.14 shows the load-displacement graph of the case under 50, 100<br />

75

76<br />

CHAPTER 4 EXPERIMENTS ON SCALE MODELS<br />

cable tension [N]<br />

160<br />

140<br />

120<br />

100<br />

80<br />

60<br />

40<br />

20<br />

cable 3<br />

cable 2 & 4<br />

cable 1 & 5<br />

0<br />

0 50 100 150 200 250 300 350<br />

load [N]<br />

cable tension [N]<br />

160<br />

140<br />

120<br />

100<br />

80<br />

60<br />

40<br />

20<br />

cable a & d<br />

cable b & c<br />

0<br />

0 50 100 150 200 250 300 350<br />

load [N]<br />

Figure 4.13: The tension in the cables of case 10 in relation to the applied<br />

load. (Numerical results).<br />

load [N]<br />

350<br />

300<br />

250<br />

200<br />

150<br />

100<br />

50<br />

200 mbar<br />

100 mbar<br />

050 mbar<br />

0 0.5 1 1.5 2 x 10 −5<br />

0<br />

average displacement [m]<br />

Figure 4.14: The load-displacement graph of the case 11 under 50, 100<br />

and 200 mbar (loaded with five point loads in the hinges) (Numerical<br />

results).<br />

and 200 mbar. As long as all cables are pretensioned, all curves have the<br />

same stiffness. Because the pretension in the cable is dependent of the internal<br />

pressure (equation 4.1), the case with the lowest internal pressure experiences<br />

as first a slack cable and thus another stiffness.<br />

When all cables are pretensioned and thus able to take compressive forces,<br />

the deployable Tensairity beam has the same stiffness as a truss (with the<br />

same configuration of diagonals and with the same sections). This can be seen<br />

in figure 4.12. Once a cable does not contribute anymore to the structural

4.4 CABLE CONFIGURATIONS<br />

behaviour, the bar structure becomes a ‘mechanism’. This is illustrated in<br />

figure 4.15. However, the deployable Tensairity beam does not collapse<br />

immediately since it is still supported by the airbeam. This is why the stiffness<br />

of the beam is similar to the stiffness of an airbeam after the first cables are<br />

slack (figure 4.12).<br />

Figure 4.15: Once a cable does not contribute anymore to the structural<br />

behaviour, the bar structure becomes a ‘mechanism’.<br />

Thus, a relation between the structure’s load-displacement behaviour and the<br />

contribution of pretensioned cables to this structural behaviour is shown. The<br />

cables of the deployable Tensairity beam are pretensioned when the airbeam is<br />

inflated. When both diagonal and vertical cables are present, only the vertical<br />

cables become tensioned. These tensioned cables are able to take compressive<br />

forces, by the same amount as their initial pretension. This has as result<br />

that these cables avoid the hinges to deflect under compression. Or in other<br />

words, the pretensioned cables ‘block’ the hinges. Once the external load has<br />

reached the value whereby the value of the pretension becomes zero in at least<br />

one cable, the hinge is not blocked or supported anymore by this cable. The<br />

hinge will experience larger displacements and the stiffness of the deployable<br />

Tensairity beam decreases.<br />

Now the contribution of cables to the structural behaviour is understood,<br />

the influence of the various cable configurations on the structure’s loaddisplacement<br />

is studied.<br />

4.4 CABLE CONFIGURATIONS<br />

The deployable Tensairity beams with various cable configurations are investigated<br />

numerically and experimentally. Note that the experimental and<br />

numerical results are not compared quantitative, but qualitative. After all, the<br />

77

78<br />

CHAPTER 4 EXPERIMENTS ON SCALE MODELS<br />

values do not correspond due to imperfections in the two meter prototype<br />

(movement of the struts in the pockets), approximations in the finite element<br />

model (perfect connection between membrane, struts and cables), and a slight<br />

different geometry of the section of the airbeam in the FEA (a higher value for<br />

α is chosen to achieve higher cable tensions for reasons of clarity). The next<br />

sections explain the effect of diagonal and vertical cable configurations on the<br />

structural behaviour.<br />

4.4.1 VERTICAL CABLES<br />

The load-displacement behaviour of case 6, containing only vertical cables,<br />

is investigated numerically. The value of the pretension of the cables after<br />

inflation is identical as in case 10. This is because in case 10 only the vertical<br />

cables were pretensioned during inflation.<br />

However, the load-displacement behaviour of these two cases is different.<br />

After all, no diagonals are present in case 6 to contribute to the behaviour under<br />

loading. This has as result that the structure experiences a larger deformation.<br />

Figure 4.16 illustrates the deformation of the structure under point loads on<br />

the five upper hinges. The total load is F, thus each point load has a value of<br />

F<br />

5 . Because a part of the external loading is taken by the deformation of the<br />

airbeam, cables 1 and 5 typically become slack when4 Fpre = 1<br />

3Fext or Fext = 3Fpre.<br />

The load at which the cable - pretensioned by 74,5 N - becomes slack is 223,5 N,<br />

as can be seen in figure 4.17. This figure shows the load-displacement of case<br />

6 and the cable tensions during loading.<br />

4.4.2 DIAGONAL CABLES<br />

Cases 2, 3, 4 and 5 are investigated experimentally and numerically to determine<br />

the effect of the configuration of the diagonal cables on the loadbearing<br />

behaviour. Figure 4.18 shows the load-displacement of the cases<br />

under distributed load, derived from the finite element analysis. It is clear<br />

that the configuration of the diagonal cables has an influence on the stiffness<br />

of the structure. Cases 4 and 5 have similar load-displacement behaviour and<br />

are less stiff than cases 2 and 3.<br />

Before explaining the behaviour of the different cases, it is useful to discuss<br />

the action of the external loading on the cables. After all, depending on the<br />

configuration of the cable, it will experience compression or tension when<br />

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<br />

load on cable 1, the second term from the point load on cable 2 and the third on cable 3. The<br />

numerical results (verified at different pressures) confirm this assumption.

load [N]<br />

350<br />

300<br />

250<br />

200<br />

150<br />

100<br />

50<br />

4.4 CABLE CONFIGURATIONS<br />

Figure 4.16: Deformation of case 6 under loading, caused by the absence<br />

of diagonal cables.<br />

case 6 − p = 100 mbar<br />

0<br />

0 0.01 0.02 0.03 0.04 0.05<br />

average displacement [m]<br />

cable tension [N]<br />

160<br />

140<br />

120<br />

100<br />

80<br />

60<br />

40<br />

20<br />

cable c<br />

cable b<br />

cable a<br />

0<br />

0 100 200<br />

load [N]<br />

300 400<br />

Figure 4.17: The load-displacement of case 6 and the cable tensions<br />

during loading. (Numerical results).<br />

79

80<br />

CHAPTER 4 EXPERIMENTS ON SCALE MODELS<br />

case 2<br />

case 3<br />

case 4<br />

case 5<br />

load [N]<br />

350<br />

300<br />

250<br />

200<br />

150<br />

100<br />

case 2<br />

case 3<br />

50<br />

case 4<br />

case 5<br />

0<br />

0 0.02 0.04 0.06 0.08<br />

average displacement [m]<br />

Figure 4.18: The load-displacement of cases 2, 3, 4 and 5 under five point<br />

loads and with an internal pressure of 100 mbar. (Numerical results).<br />

loading and this has an influence on the load-bearing behaviour, as mentioned<br />

in section 4.3. Whether the pretensioned cable is tensioned or compressed under<br />

external loading is determined by means of the finite element calculations<br />

and by observing the tension in the cables throughout the experiments: a loose<br />

cable indicates that the cable is ‘compressed’. Both methods have the same<br />

result and correspond to the signs of the normal forces in a conventional truss<br />

with similar diagonal configurations under distributed load. The sign of the<br />

forces are illustrated in figure 4.19. A ‘+’ means that the pretensioned cable<br />

experiences tension during loading, a ‘–’ indicates that this cable is compressed.<br />

- + + - + - - +<br />

case 2<br />

- - - -<br />

case 3<br />

+<br />

tension<br />

-<br />

compression<br />

+<br />

case 4<br />

+<br />

case 5<br />

Figure 4.19: The tension (+) and compression (–) forces from the cases<br />

with diagonal cables indicated. (Isostatically supported)<br />

The lower stiffness of cases 4 and 5 is not caused by the action of the external<br />

loading on the cables, but has to be attributed to the cable configuration, and<br />

more precise to the deformation of the first and last ‘mesh’ of the truss. After<br />

all, this has the shape of a parallelogram and deforms much easier than the<br />

triangles in cases 2 and 3. Figure 4.20 illustrates this. This can also be seen<br />

+<br />

+

4.4 CABLE CONFIGURATIONS<br />

in figure 4.21 (left), where the displacement of the first hinge of each case is<br />

plotted throughout loading: the hinge of cases 4 and 5 experience a larger<br />

displacement. Thus in cases 4 and 5, the stiffness of the structure is not really<br />

defined by a prestressed cable that becomes slack at a certain load, but by a<br />

geometrical deformation that is made possible by the cable configuration. In<br />

fact, the cables of case 5 did not become slack during the loading.<br />

This behaviour can also be seen in the experimental results. Figure 4.21 (right)<br />

plots the displacement of the first hinge of the four cases. Also here, the hinge<br />

of cases 4 and 5 experience a larger displacement than cases 2 and 3.<br />

load [N]<br />

350<br />

300<br />

250<br />

200<br />

150<br />

Figure 4.20: The unstable shape of the first mesh of cases 4 and 5 causes<br />

a deformation of the truss mechanism. Case 5 is shown.<br />

numerical − p = 100 mbar<br />

100<br />

case 2<br />

case 3<br />

50<br />

case 4<br />

case 5<br />

0<br />

0 0.02 0.04 0.06 0.08<br />

displacement [m]<br />

load [N]<br />

350<br />

300<br />

250<br />

200<br />

150<br />

experimental − p = 100 mbar<br />

100<br />

case 2<br />

case 3<br />

50<br />

case 4<br />

case 5<br />

0<br />

0 0.005 0.01 0.015 0.02<br />

displacement [m]<br />

Figure 4.21: The first hinge of cases 4 and 5 deflects more than cases 2<br />

and 3 because of the unstable shape of the first mesh. Left: numerical<br />

results, right: experimental results.<br />

When having a closer look at the load-displacement behaviour of cases 2<br />

81

82<br />

CHAPTER 4 EXPERIMENTS ON SCALE MODELS<br />

and 3 (figure 4.18), one can see that case 2 is initial stiffer than case 3, but<br />

changes stiffness at a lower load. The higher stiffness of case 2 can also here be<br />

attributed to the configuration of the cables: they are positioned such that all<br />

meshes in the ‘truss’ are triangles. In case 3, the second and fourth meshes are<br />

parallelograms and deform easier, hence the lower stiffness. Because case 2<br />

possess cables that are tensioned when the structure is loaded, the pretension<br />

in the compressed cables reaches zero value at a lower load. This can be seen<br />

in the graph by the change in stiffness of case 2 at a lower load (100 N).<br />

To summarize, the investigation of the diagonal cables revealed that the<br />

displacement behaviour of the structure is strongly influenced by the shape of<br />

the first mesh of the truss. The shape of a parallelogram should be avoided,<br />

since large displacements will occur at the level of the first hinge. The diagonal<br />

cables are pretensioned under inflation, when no vertical cables are present.<br />

Diagonal cables versus bars<br />

case 11<br />

case 12<br />

case 13<br />

case 14<br />

load [N]<br />

350<br />

300<br />

250<br />

200<br />

150<br />

100<br />

case 11<br />

case 12<br />

50<br />

case 13<br />

case 14<br />

0<br />

0 5 10 15 20 25<br />

average displacement [mm]<br />

Figure 4.22: The average displacement in relation with the applied load<br />

of the cases with diagonal bars. The cases are uniformly distributed<br />

loaded, each load step represents 50 N. (Experimental results).<br />

Experiments are also conducted on cases with diagonal stiff bars instead of<br />

cables. Cases 11, 12, 13 and 14 are the analogue of respectively cases 2, 3, 4<br />

and 5. Note that only case 11 is deployable. Figure 4.22 shows the average<br />

displacement in relation to the applied load. From the graph can clearly be<br />

seen that two cases (13 and 14) have much larger displacement than the other<br />

two configurations. As already mentioned in the previous section, this is<br />

because of the shape of the first mesh of the strut, formed by the outer struts

4.4 CABLE CONFIGURATIONS<br />

and the first diagonal (illustrated on the right of figure 4.21). After all, this<br />

parallelogram cannot retain its shape as in the case of a triangle and thus<br />

experiences larger deformation.<br />

When comparing the configurations with diagonal bars and cables, interesting<br />

similarities can be detected. Figure 4.23 shows the deflected configurations<br />

under the same distributed load. In the case of configurations 10 and 14, all<br />

connecting elements are under tension. As a result and as can be expected,<br />

the deflected shape is identical. The figure also shows clearly the larger<br />

deformation at the location of the first hinge of cases 13 and 14 due to the<br />

presence of the unstable parallelogram. For cases 2 and 3, it is obvious that<br />

their counterpart with diagonals, able to take compression, is much stiffer.<br />

displacement [mm]<br />

displacement [mm]<br />

0<br />

10<br />

20<br />

30<br />

40<br />

0 1 2 3 4 5 6<br />

x position along axis [mm]<br />

0<br />

10<br />

20<br />

30<br />

case 2 case 11 case 3 case 12<br />

40<br />

0 1 2 3 4 5 6<br />

x position along axis [mm]<br />

displacement [mm]<br />

0<br />

10<br />

20<br />

30<br />

40<br />

0 1 2 3 4 5 6<br />

x position along axis [mm]<br />

case 4 case 13 case 5 case 14<br />

displacement [mm]<br />

0<br />

10<br />

20<br />

30<br />

40<br />

0 1 2 3 4 5 6<br />

x position along axis [mm]<br />

Figure 4.23: Comparison of configurations with diagonal cables and<br />

struts. (Experimental results).<br />

83

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CHAPTER 4 EXPERIMENTS ON SCALE MODELS<br />

4.4.3 DIAGONAL AND VERTICAL CABLES<br />

Cases 7, 8, 9 and 10 contain vertical cables in every hinge combined with the<br />

diagonal cables of resp. cases 2, 3, 4 and 5. Their load-displacement behaviour<br />

is investigated numerically and plotted in figure 4.24.<br />

case 7<br />

case 8<br />

case 9<br />

case 10<br />

load [N]<br />

350<br />

300<br />

250<br />

200<br />

150<br />

p = 100 mbar<br />

100<br />

case 7<br />

case 8<br />

50<br />

case 9<br />

case 10<br />

0<br />

0 0.005 0.01 0.015 0.02<br />

average displacement [m]<br />

Figure 4.24: Load-displacement behaviour of cases 7, 8, 9 and 10 at<br />

pressure of 100 mbar and loaded distributed (point load in each upper<br />

hinge). (Numerical results).<br />

From the graphs can be concluded that cases 9 and 10 are stiffer than cases<br />

7 and 8. This is remarkable, because this is the opposite of their analogue<br />

cases without vertical cables (figure 4.18). The explanation of this different<br />

behaviour can be found in the pretension of the cables. After all, as seen in<br />

section 4.3, the diagonal cables are not pretensioned due to inflation when<br />

vertical cables are also connecting the hinges. This means that under loading,<br />

only the tensioned diagonal cables contribute to the stiffness. The diagonals<br />

under compression can be considered as non-existing. Figure 4.25 illustrates<br />

the cases without the compressed diagonals and their deformed shape.<br />

From this figure becomes clear that the presence of a mesh with the shape of a<br />

quadrangle (here mainly a rectangle or parallelogram) weakens the structure.<br />

In the cases 7 and 8 is this mesh near the end supports, which results in a large<br />

decrease of stiffness. In fact, case 8 has identical load-bearing behaviour as<br />

case 6. Cases 9 and 10 are stiffer, because they have triangular meshes near<br />

the end supports. Case 10 is the stiffest case, since it contains only triangular<br />

meshes under loading. This results in a stiffness identical to that of a truss<br />

with the same configuration, as illustrated in figure 4.12.

7<br />

8<br />

9<br />

10<br />

4.4 CABLE CONFIGURATIONS<br />

Figure 4.25: The diagonals that are compressed during loading can be<br />

considered as non-existing. The cases without the compressed diagonals<br />

and their deformed shape.<br />

This contribution of the vertical and diagonal cables to the stiffness of the<br />

deployable Tensairity beam is also investigated experimentally. Figure 4.26<br />

shows the stiffness and the deflection of cases 10 (vertical + diagonal cables),<br />

case 6 (only vertical cables) and case 5 (only diagonals). As can be expected,<br />

the case with both diagonal and vertical cables has a larger stiffness than the<br />

other two cases.<br />

load step<br />

7<br />

6<br />

5<br />

4<br />

3<br />

case 5 case 10 case 11<br />

2<br />

1<br />

case 5<br />

case 10<br />

case 11<br />

0<br />

0 5 10 15 20 25<br />

average displacement [mm]<br />

displacement [mm]<br />

0<br />

5<br />

10<br />

15<br />

20<br />

25<br />

30<br />

0 1 2 3 4 5 6<br />

postition along x axis [mm]<br />

Figure 4.26: The stiffness and deflection of the cases 5, 6 and 10.<br />

(Experimental results).<br />

4.4.4 SUMMARY<br />

The numerical and experimental investigations show clearly the effect of the<br />

configuration of cables on the load-displacement behaviour. In brief, there can<br />

be concluded that the shape of the ‘mesh’ - formed by the pretensioned cables<br />

85

86<br />

CHAPTER 4 EXPERIMENTS ON SCALE MODELS<br />

and the cables that will become tensioned under loading - has a large influence<br />

on the structure’s stiffness. If this mesh is a rectangle or parallelogram, it can<br />

easily be deformed under external loading and causes larger displacements of<br />

the structure. The more this quadrangle mesh is located at the end supports (as<br />

in the case of 4 and 5) the higher the impact of this mesh on the deformation.<br />

To give an overview and to compare, the load-displacement of cases 2 to 10 is<br />

plotted in figure 4.27.<br />

4.5 SHAPE<br />

load [N]<br />

350<br />

300<br />

250<br />

200<br />

150<br />

distributed load − p = 100 mbar<br />

100<br />

case 2<br />

case 3<br />

case 4<br />

case 5<br />

case 6<br />

50<br />

case 7<br />

case 8<br />

case 9<br />

case 10<br />

0<br />

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07<br />

average displacement [m]<br />

Figure 4.27: Load-displacement of cases 2 to 10. (Numerical results).<br />

Until now, the experiments are all conducted on the foldable truss system<br />

with cylindrical airbeam. However, as already mentioned in the previous part,<br />

research showed that the spindle shaped Tensairity beam has a better loadbearing<br />

behaviour than the cylindrical one (Pedretti et al., 2004). Therefore, a<br />

two meter deployable spindle is investigated experimentally. The upper and<br />

lower strut of the spindle, having a section of 6 mm width by 10 mm height,<br />

contain five evenly distributed hinges. No connecting cables between upper<br />

and lower hinges are applied here. Figure 4.28 shows a picture of the investigated<br />

spindle, as well its load-displacement behaviour and deflection. For<br />

reasons of comparison, the stiffness and deflection of a cylindrical deployable<br />

Tensairity beam are included. From the graphs can clearly be seen that the<br />

spindle shaped beam has a larger stiffness than the cylindrical.

load step<br />

7<br />

6<br />

5<br />

4<br />

3<br />

2<br />

1<br />

0<br />

cylinder<br />

spindle<br />

0 10 20 30 40<br />

average displacement [mm]<br />

displacement [mm]<br />

0<br />

10<br />

20<br />

30<br />

40<br />

4.6 CONCLUSIONS<br />

0 1 2 3 4 5 6<br />

postition along x axis [mm]<br />

Figure 4.28: The investigated spindle and its load-displacement be-<br />

haviour and deflection under distributed load. Each load step represents<br />

50 N.<br />

During the experiments on the spindle shaped structure, some problems were<br />

encountered. Since the upper strut is now loaded as an arch under distributed<br />

loading, the bending moment forces the struts near the supports to bend<br />

outwards. Because the investigated model possesses a hinge in this zone and<br />

because this hinge is not well supported by the membrane, the struts ‘buckled’<br />

outwards. After all, due to the smaller radius of the hull at the ends of the<br />

spindle beam, the membrane stress is at this place lower. This implies that the<br />

strut and hinge, positioned in a pocket, are less supported by the membrane<br />

and can thus more easily buckle outwards. This ‘buckling’ of the strut can be<br />

seen on the deformed shape of the strut in figure 4.28 (right), indicated with an<br />

arrow. Another observation during the experiments is that under distributed<br />

loading, the spindle had the tendency to rotate. After all, the centre of gravity<br />

lies on the same line as the supports, which has as consequence that the smallest<br />

eccentricity out of the plane of the loading is sufficient to rotate the structure.<br />

From a structural point of view, the spindle shaped deployable Tensairity beam<br />

proves to be more efficient than the cylindrical. However, the specificities of<br />

the shape have to be taken into account when developing further.<br />

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CHAPTER 4 EXPERIMENTS ON SCALE MODELS<br />

4.6 CONCLUSIONS<br />

Before simulating or developing a large scale model for a deployable Tensairity<br />

beam, it is necessary to gain some first insights in the structural behaviour<br />

of this complex synergetic structure by means of small scale experimental and<br />

numerical investigations. Therefore, this chapter presents the experiments on<br />

two meter long cylindrical and spindle shaped deployable Tensairity beams<br />

with various configurations.<br />

The experiments show us that the location of applying the load has an influence<br />

on the outcome of the experiment. Because of the compressibility of the<br />

airbeam, the results are different when the load is applied at the top or bottom<br />

of the beam. This is unlike conventional structural elements such as wooden,<br />

concrete or steel beams. Also different from these conventional elements is<br />

that the displacements of the loaded structure should be noted in the second<br />

or third load cycle. After all, the structure first adapts to the new load condition,<br />

which results in a residual displacement of the unloaded structure after the<br />

first cycle.<br />

To detect the influence of several parameters - such as the internal pressure,<br />

the amount of hinges in the struts and the configuration of cables - on the<br />

structural behaviour of the beam, various configurations were investigated.<br />

The experimental results show that the internal pressure influences the stiffness<br />

of the structure, as is also the case for Tensairity beams with continuous<br />

compression element. A higher internal pressure results in a stiffer structure.<br />

Regarding the influence of the hinges in the strut, no conclusions can yet be<br />

derived. This is because the amount of hinges was not varied enough in the<br />

experiments.<br />

Numerical simulations and analytical approximations revealed the contribution<br />

of pretensioned cables to the structure’s load-displacement behaviour. As<br />

long as the cables connecting upper and lower hinges are pretensioned due to<br />

inflation, they can take compressive forces. This means that these cables can<br />

support (or ‘block’) the hinges. Once the pretension in a cable is zero, the hinge<br />

is not supported anymore and the structure experiences larger displacement.<br />

This has as result that the stiffness of the deployable Tensairity beam changes<br />

when one cable becomes slack.<br />

Several cable configurations connecting the upper and lower strut of the<br />

airbeam are investigated experimentally and numerically. The study showed<br />

that the shape of the ‘mesh’ - formed by the pretensioned cables and the

4.6 CONCLUSIONS<br />

cables that will become tensioned under loading - has a large influence on the<br />

structure’s stiffness. If this mesh is a quadrangle, it can easily be deformed<br />

under external loading and cause large displacements of the structure. When<br />

these deformations occur, the deployable Tensairity beam is supported by<br />

the airbeam and has stiffness similar to that one of an airbeam. When all<br />

cables are prestressed and thus able to take compressive forces, the deployable<br />

Tensairity beam has the same stiffness as a truss (with the same configuration of<br />

diagonals and with the same sections). The stiffest configuration is constituted<br />

of diagonal cables that are tensioned during loading and vertical cables.<br />

This configuration contains, as long as the verticals are pretensioned, only<br />

triangular stable meshes.<br />

There can be concluded that the configuration influences the structure’s stiffness.<br />

Vertical cables have more influence on the stiffness than diagonals, also<br />

for asymmetrical load cases. In the case of the cylindrical foldable truss, it is<br />

beneficial for the stiffness to have the first mesh of the truss being a triangle<br />

and not a parallelogram. In the latter case, the first hinge will experience large<br />

displacements. Applying struts as connecting element increases the stiffness<br />

when they are positioned such that they will be loaded in compression. If not,<br />

the stiffness will be identical to the configuration with cables and only weight<br />

will be added.<br />

In addition to the cylindrical beam, a spindle shaped deployable Tensairity<br />

structure is investigated. The results show an increase in stiffness in comparison<br />

with the cylindrical beam (with similar distance between upper and<br />

lower strut in its middle). The observations during the experiments reveal<br />

also some consequences inherent to this more optimal shape. Because of the<br />

bending moment of the curved strut under distributed load, the position of the<br />

hinges has an influence of the structure’s behaviour and should be investigated<br />

thoroughly.<br />

Now the first qualitative insight of the structural behaviour of a deployable<br />

Tensairity beam is gained with these scale models, it is time to investigate more<br />

configurations on a larger structure by means of simulations in finite element<br />

analysis. This is done in the next two chapters: the finite element modeling is<br />

considered in chapter 5, the results are presented and discussed in chapter 6.<br />

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CHAPTER 4 EXPERIMENTS ON SCALE MODELS

Numerical Investigations<br />

5<br />

Conducting experiments on Tensairity structures is a very instructive way of<br />

gaining insight in this new structural concept. However, these experiments<br />

are time consuming and thus not always feasible when many different configurations<br />

have to be investigated. Besides, there are also often manufacturing<br />

imperfections involved with these experiments which can cause difficulties in<br />

comparing experimental results from different models. This can be avoided<br />

by investigating the structure numerically by means of finite element models<br />

which simulate the structural behaviour. It is thus essential to ‘build’ the<br />

finite element model as accurate as possible in order to have reliable results.<br />

Therefore, it is important to know how the structure is modeled, which<br />

materials are used, which assumptions are made, etc. This chapter discusses<br />

all these aspects regarding the finite element analysis of (deployable) Tensairity<br />

structures.<br />

5.1 FINITE ELEMENT ANALYSIS OF <strong>TENSAIRITY</strong> <strong>STRUCTURES</strong><br />

The finite element analysis (FEA) method is a computational technique whereby<br />

approximate solutions of a variety of “real-world” engineering problems can<br />

be calculated. This is also the case for the structural behaviour of Tensairity<br />

structures.<br />

As far as the finite element analysis of Tensairity structures is concerned,<br />

some numerical research is published. Pedretti et al. (2004) introduces the<br />

simulation of cylindrical Tensairity structures by means of the commercial<br />

finite element code ‘Ansys’. He discusses the modeling of the compression<br />

element, the cables and the membrane, as well as the interaction and contact<br />

between these materials. Pedretti also addresses convergence issues as a<br />

result of the membrane-element (able to resist only tension) and how to avoid<br />

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CHAPTER 5 NUMERICAL INVESTIGATIONS<br />

this. Luchsinger and Crettol (2006b) compare the numerical, experimental<br />

and analytical results of a spindle shaped Tensairity beam. The used finite<br />

element model is an improvement of the one Pedretti presented, more precise<br />

by using a more specific shell-element 1 instead of a membrane-element for the<br />

membrane. Although Luchsinger models the membrane as linear isotropic<br />

material, the experimental non-linear load displacement behaviour of the<br />

structure is well reproduced by the numerical results. Plagianakos et al. (2009)<br />

and Wever et al. (2010) used in the finite element model of a spindle-shaped<br />

Tensairity column the same element types as Luchsinger, but modeled the<br />

membrane with linear orthotropic properties. Also here, good agreement<br />

between the numerical and experimental results is observed.<br />

The finite element models in this research use the same material types and<br />

parameters as in the research of Plagianakos. The material properties of the<br />

membrane however are derived from mean values of biaxial testing measurements<br />

conducted on the same fabric as the one that will be used for<br />

the prototype in Part III. Besides, these models will primarily be used for a<br />

qualitative comparison. This means that the value derived from a simulation<br />

on itself is not the main focus (quantitative), but that comparing different<br />

models should give us insight in the structural behaviour of the Tensairity<br />

structure.<br />

There are three main steps in a typical finite element analysis: modeling<br />

the structure (also known as preprocessing), solving the finite model and<br />

reviewing the results (postprocessing). Reviewing the results is subject of<br />

the next chapter. modeling and solving the Tensairity structures in FEA is<br />

discussed in the next sections.<br />

5.2 MODELING THE <strong>TENSAIRITY</strong> BEAMS<br />

The purpose of a finite element analysis is to recreate numerically the behavior<br />

of an actual engineering system. In other words, the analysis must be an<br />

accurate numerical model of a physical prototype. However, small details that<br />

are unimportant to the analysis should not be included in the finite element<br />

model, since they will only make the model more complicated than necessary<br />

and increase the calculation time.<br />

1 The applied shell-element (with membrane option activated) is a 4-node element with<br />

translational degrees of freedom. The element is well-suited for linear and large strain nonlinear<br />

applications. It accounts for load stiffness effects of distributed pressures.

5.2 MODELING THE <strong>TENSAIRITY</strong> BEAMS<br />

To gain insight in the structural behaviour of deployable Tensairity structures,<br />

various cases are modeled and compared. These cases however differ often<br />

only in one parameter from each other, for example the amount of hinges, the<br />

section of the strut, etc. This means that with exception of these parameters, the<br />

modeling explained here holds true for all cases. modeling Tensairity beams<br />

comprises creating and specifying the geometry, all the nodes, elements, material<br />

properties, real constants (thickness and cross-sectional area), boundary<br />

conditions, and other features that are used to represent the physical system.<br />

5.2.1 CREATING THE MODEL GEOMETRY<br />

The spindle shaped Tensairity beam has a length of five meter and a maximal<br />

diameter in the middle of 50 cm, representing a slenderness of ten. Figure 5.1<br />

shows the Tensairity beam.<br />

Figure 5.1: The Tensairity beam in the finite element software Ansys.<br />

The model is built up from the bottom, beginning by defining the lowest-order<br />

model entities, keypoints. By connecting these keypoints, lines, areas and<br />

volumes can then be defined. First, the keypoints are positioned to construct<br />

the spindle shape and the struts. Their coordinates are calculated by means of<br />

an arc of a circle with radius of 12,625 m and subtending angle of 0,3984 rad.<br />

Figure 5.2 shows this. The keypoints are then connected with lines. The upper<br />

line will be used to model the upper strut, the lower line for the lower strut.<br />

In the case of a spindle with one circular air chamber (and thus no cables<br />

connecting upper and lower strut), the upper line is rotated around the axis<br />

connecting the ends of the arc. As a result, an area is constructed representing<br />

the hull of the membrane, see figure 5.2. In the case of an airbeam with<br />

cables connecting upper and lower strut, the section of the airbeam is no circle<br />

anymore. Therefore, arcs are applied which determine the shape of the hull.<br />

The spindle with the two hull shapes is shown in figure 5.3.<br />

From the figure 5.2 can be seen that the hull of the airbeam ends in a point<br />

at the supports. This should be avoided, since converging issues occur due<br />

to local phenomenon. Besides, end pieces like this are hard to manufacture<br />

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CHAPTER 5 NUMERICAL INVESTIGATIONS<br />

Figure 5.2: Some snapshots during the modeling of the Tensairity beam<br />

with circular cross section.<br />

Figure 5.3: The spindle with the two hull shapes: a circular section and<br />

a section composed of two circle segments.<br />

and will normally not be used in practice. The ends are usually finished by<br />

means of round end caps, as illustrated in figure 5.4. To reduce the modeling<br />

complexity and ‘gain’ calculation time, there is chosen not to model the ends<br />

of the membrane hull, but to apply an equivalent tangential line loading along<br />

the ends. Figure 5.5 represents this. Numerical investigation of both cases<br />

shows that their structural behaviour is identical and the approximation with<br />

an equivalent line load justified is.<br />

Many of the investigated finite element models should also contain hinges.<br />

These hinges are modeled by positioning two keypoints at the same location.<br />

The line (strut) at one side of the coincident points will contain a different<br />

one than the strut at the other side. The nodes have coupled degree of freedom,<br />

which means that these nodes are forced to take the same translational

5.2 MODELING THE <strong>TENSAIRITY</strong> BEAMS<br />

Figure 5.4: The ends are usually finished by means of round end caps.<br />

Figure 5.5: The ends of the membrane hull are replaced by the action of<br />

an equivalent tangential line loading.<br />

displacement. Of course, they have different rotational degree of freedom<br />

(around the three main axes) in order to ‘create’ the hinge.<br />

The Tensairity structure is in all investigated models statically determined:<br />

both endpoints allow rotation around the z-axis (axis perpendicular to the<br />

plane), whereby one endpoint also has a translational degree of freedom in the<br />

x-direction (longitudinal direction).<br />

5.2.2 MESHING THE MODEL<br />

Analytical solutions give at any point within a system the calculated value<br />

(like a displacement). In contrast, numerical calculations approximate the<br />

exact solutions only at discrete points, called nodes. Therefore, the model<br />

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CHAPTER 5 NUMERICAL INVESTIGATIONS<br />

needs to be discretized, also known as ‘meshed’. This process divides the<br />

model into a number of nodes and small subregions, called elements.<br />

Analytical solutions show the exact behaviour of a system at any point within<br />

that system. In contrast, numerical calculations approximate the exact solutions<br />

only at discrete points, called nodes. Therefore, the model needs to be<br />

discretized, also known as ‘meshed’. This process divides the model into a<br />

number of nodes and small subregions, called elements.<br />

The size of these subregions and thus the mesh density is very important in<br />

finite element calculations. If the mesh is too coarse, the approximate solution<br />

can contain errors. A mesh which is too fine on the other hand, results in a<br />

excessively long calculation time. To avoid this, an appropriate mesh density<br />

should be used.<br />

The question that rises is of course what an appropriate mesh density is. To<br />

find this out, a first analysis is performed with what seems a ‘reasonable’<br />

mesh. Then, the analysis is reanalyzed using twice as many elements and the<br />

results of these two analysis are compared. The mesh is most likely adequate<br />

if the two meshes give nearly the same results. If the results are substantially<br />

different, further mesh refinement is required until nearly identical results for<br />

succeeding meshes are obtained. It is of course also useful and wise to compare<br />

the numerical result with a known accurate analytical result, if possible and<br />

existing.<br />

In order to study the effect of mesh refinement for a Tensairity structure, four<br />

different mesh densities were applied on the fabric hull: 40 × 6, 60 × 12, 120<br />

× 24 and 240 × 48 elements along the axial and radial direction respectively.<br />

Figure 5.6 illustrates the cases. The central membrane hoop stress after inflation<br />

and the central displacement under distributed load (200 N<br />

m ) were noted and are<br />

given in table 5.1. As can be seen from the table, all mesh densities produce<br />

nearly identical results regarding the middle displacement. The analytical<br />

value, derived from the analytical model in Luchsinger and Teutsch (2009),<br />

corresponds with the calculated values. The hoop stress on the other hand<br />

is affected by the mesh density. A larger element size produces a coarser<br />

approximation. This is because the hoop stress in the centre of the beam is the<br />

maximal hoop stress of the hull. The larger the element size, more areas with<br />

a lower stress are comprised in this element, which results in a lower mean<br />

value. When comparing the results with the analytical value of the hoop stress<br />

for a cylindrical airbeam2 , it is clear that a denser mesh size approximates<br />

2 The analytical value for the hoop stress of a cylindrical airbeam is nhoop = p.r, with n hoop being

5.2 MODELING THE <strong>TENSAIRITY</strong> BEAMS<br />

Figure 5.6: Four different mesh densities were applied on the fabric hull<br />

to study the effect of mesh refinement for a Tensairity structure.<br />

better the theoretical value.<br />

There can be concluded that the simulation of the structural behaviour is<br />

rather insensitive to mesh refinement of the hull, in contrast with the membrane<br />

stress. Because in this research, the focus will mainly be on the loaddisplacement<br />

behaviour of the structure, a ‘coarser’ mesh size is sufficient to<br />

obtain reliable results. The mesh with 60 × 12 elements is implemented in all<br />

investigated finite element models. Regarding the struts, a non-uniform mesh<br />

was applied being finer near the ends of the beam to avoid approximation<br />

errors at the ends due to the coarseness of the mesh.<br />

Table 5.1: Results reveal very little influence of the investigated mesh<br />

densities on the value of the middle displacement of the beam.<br />

40 × 6 60 × 12 120 × 24 240 × 48 analytical<br />

middle displacement [mm] 2,92 2,96 2,98 2,98 2,9<br />

hoop stress [ N ] 3202 3570 3688 3700 3750<br />

m<br />

After creating nodes and elements by means of meshing, the nodes of struts<br />

and hull are merged together. This implies a perfect connection between these<br />

components. Luchsinger and Crettol (2006b) investigated by means of finite<br />

element analysis for a spindle shaped Tensairity beam under central point load<br />

the influence on the structural behaviour of an imperfect connection between<br />

the membrane stress [ N m ], p the air pressure and r the radius.<br />

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CHAPTER 5 NUMERICAL INVESTIGATIONS<br />

struts and hull. Results show that the displacement at the compression side of<br />

the model with imperfect connection is almost a factor of two larger compared<br />

to the model with perfect connection. On the tension side is this effect<br />

even more pronounced. A tight connection of the struts and hull can thus<br />

significantly stiffen the Tensairity spindle. Physical models will therefore be<br />

manufactured with a good connection between the elements. Regarding the<br />

numerical model, a tight connection between hull and struts is thus assumed<br />

in all cases, modeled by means of merging the coincident nodes of membrane<br />

and strut.<br />

Meshing does not only discretizes the structural components, but also assigns<br />

the element type, element real constants and material properties to the elements.<br />

5.2.3 ASSIGNING ELEMENTS AND MATERIALS<br />

The appropriate element attributes need to be defined and assigned to the respective<br />

elements. The element attributes comprises the element type, the real<br />

constants and the material properties. The element type determines, among<br />

other things, the degree of freedom set and whether the element lies in 2D or<br />

3D space. The element’s geometric properties, such as thickness and crosssectional<br />

area, are categorized under the real constant set. Material properties<br />

include for instance the Young’s and shear modulus, poisson coefficient and<br />

the density.<br />

The finite element model of a Tensairity structure in ansys consists of shell<br />

elements for the membrane, beam elements for the struts, and link elements<br />

for cables when needed.<br />

The fabric hull is modeled by four-node shell elements (shell 181) whose<br />

bending stiffness is not taken into account in the solution by means of a feature<br />

provided in the element properties. The 0.85 millimeter thick membrane,<br />

which is a coated woven fabric, is modeled with linear elastic orthotropic<br />

properties. In fact, the membrane should be modeled as non-linear, but this<br />

would increase the complexity of the finite element model considerably. The<br />

approximation with a linear model however shows fairly good agreement with<br />

experimental results ((Luchsinger and Crettol, 2006b; Plagianakos et al., 2009)).<br />

The struts are represented by linear two-node Timoshenko beam elements<br />

(beam 188), where shear deformation effects are included. The sections have<br />

a full rectangular shape. Some investigated models have cables connecting<br />

the upper and lower strut . These cables are modeled by two-node three

5.3 SOLVING THE MODELS<br />

dimensional elements (link 10) with the tension-only option activated, which<br />

means that stiffness is removed if the element goes into compression. The<br />

material properties for fabric (PVC-coated polyester fabric), struts (Aluminum)<br />

and cables (Steel) are listed in table 5.2.<br />

Table 5.2: In-plane properties of materials used in the finite element<br />

models.<br />

element type E11 [GPa] E22 [Gpa] G12 [GPa] ν12 size [mm]<br />

membrane shell 181 1,06 0,53 2 × 10 −2 0,313 t = 0,85<br />

strut beam 188 69 69 26,2 0,330 b × h = 30 × 10<br />

cable link 10 100 - - 0,300 φ = 5<br />

5.2.4 SCRIPTING THE GENERATION OF THE FINITE ELEMENT MODEL<br />

When building the finite element model in ansys, there can be communicated<br />

with the software by means of the graphical user interface or by means of<br />

commands. These commands can be written in a text file, which is then called<br />

a ‘script’, that can be loaded entirely in the software. The successive commands<br />

are then executed automatically.<br />

The advantage of producing and using these scripts, is that the model does<br />

not has to be generated all over again manually when performing identical<br />

or comparable analysis. Loading the file is sufficient to run the prescribed<br />

list of commands. If the model is slightly different, the script can be altered<br />

quickly where necessary. Script files are also used to parameterize the variables<br />

involved in the construction and analysis of each case so that different designs<br />

can easily be obtained by changing these parameters. Examples of such a<br />

parameters in this research are the load, the pressure and the amount of hinges.<br />

Off course, writing such a parameterized script is time consuming, but permits<br />

to gain a lot of effort and time once it is ‘built’.<br />

All finite element models in this research are built by means of scripted files.<br />

5.3 SOLVING THE MODELS<br />

Now the finite element model is entirely generated, it is almost ready to<br />

be solved. But the loads have to be applied first and the analysis type and<br />

options have to be specified. Then the finite element solution can be initiated.<br />

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CHAPTER 5 NUMERICAL INVESTIGATIONS<br />

5.3.1 APPLYING LOADS<br />

The Tensairity structure is loaded in two successive steps. The internal<br />

overpressure is first applied in the airbeam, then the upper strut is loaded<br />

with a distributed line load. The overpressure is modeled by means of an<br />

outwards surface load on the membrane and is applied in twenty substeps.<br />

Substeps are used in non-linear analysis to apply the loads gradually so that an<br />

accurate solution can be obtained and for convergence purposes. The pressure<br />

forces acts as so called ‘following’ forces, which means that they remain normal<br />

to the surface during displacements under inflation. As already mentioned,<br />

the end caps were replaced by an equivalent tangential line loading along the<br />

ends, with respect to the value of the internal pressure.<br />

After inflation, the distributed line load is applied. This load consists of<br />

vertical point loads on every node of the compression strut. In the case of<br />

an asymmetric load, only the nodes at one side of the strut are loaded. The<br />

point loads on every node are calculated such that the load per length remains<br />

constant. Especially at the ends of the struts has this to be taken into account,<br />

where the nodes are closer to each other because of a finer meshing. Also<br />

the distributed load is applied in discrete substeps, up to the point where the<br />

geometrical non-linear solution diverges.<br />

5.3.2 OBTAINING THE SOLUTION<br />

The combination of compression elements, membranes and cables in Tensairity<br />

make the finite element model by no means trivial. Converging to a solution<br />

and obtaining reliable results is a critical issue in such a nonlinear model.<br />

Therefore it is essential to specify the principles that have to be taken into<br />

account in the solving. In all finite element models in this research, the<br />

structural static analysis is specified to take geometric nonlinearities and the<br />

effects of stress stiffening into account. Geometric nonlinearities refer to the<br />

large deflection effects which have to be considered during the solving. Stress<br />

stiffening is the out-of-plane stiffening of a structure due to its in-plane-stress<br />

(e.g. the out-of-plane stiffness of a cable or membrane is dependent on its inplane-stress).<br />

This coupling is called stress stiffening and is taken into account<br />

in all analyses.<br />

The solving can start once the options are specified. Due to the tension-only<br />

behavior of the membrane and cables, the problem is nonlinear and Newton-<br />

Raphson 3 iterative method is utilized to solve the nonlinear FE equations.<br />

3 The Newton-Raphson method works by iterating the equation [Kt].{u} = {Fa} − {Fnr}, where

5.4 DISCUSSION<br />

As already mentioned before, converging to a solution is a critical issue in such<br />

a nonlinear model. Once the system matrix has a singularity, the solving is<br />

terminated and no convergence can occur. Singularities in the solution process<br />

may be caused by cables, membranes, etc. When for example wrinkling in a<br />

membrane occurs, the structure of the matrix experiences a singularity. In<br />

reality however, the structure still has approximately its full load bearing<br />

capacity when the first wrinkles appear. The load at the last converging<br />

step is thus not the real maximal load. It is therefore with this analysis very<br />

hard to find a reliable value for the maximal load the structure can bear. As<br />

a consequence, this research does not focus on this value, but investigates<br />

the load-displacement behaviour of the various models. If one would like<br />

to calculate the maximal load the structure can bear – mostly this load will<br />

initiate global buckling – finite element investigations should be conducted<br />

with ‘explicit’ finite element analysis tools, like eg. ‘Abaqus/Explicit’. The<br />

software used in this research, Ansys, calculates according to the rules of<br />

implicit analysis methods 4 .<br />

5.4 DISCUSSION<br />

Conducting experiments is very time consuming, especially when the influence<br />

of many parameters on the structural behaviour needs to be investigated.<br />

Therefore, the behaviour of structures can be simulated by means of finite<br />

element calculations. However, before conducting these investigations, it is<br />

important to determine how the structure is investigated and which assumptions<br />

and approximations are made.<br />

In this chapter, the modeling and solving of deployable Tensairity structures<br />

in the finite element software ‘Ansys’ are briefly described.<br />

The development of the spindle shaped Tensairity beam is described step by<br />

step. To reduce the modeling complexity and gain calculation time, there is<br />

chosen not to model the ends of the membrane hull, but to apply an equivalent<br />

tangential line loading along the ends. Also, investigations revealed that an<br />

appropriate mesh density is chosen, which makes the results more reliable. The<br />

mesh density is preferably chosen to be not too dense, since this increases the<br />

[Kt] is the tangential stiffness matrix, {Fa} is the applied load vector and {Fnr} is the internal load<br />

vector, until the residual, {Fa} − {Fnr}, falls within a certain convergence criterion. The Newton-<br />

Raphson method increments the load a finite amount at each substep and keeps that load fixed<br />

throughout the equilibrium iterations (Moaveni, 2008).<br />

4 Implicit: u t+∆t = f (u t+∆t , u t ) - iterative, convergence not guaranteed, ∆t is not limited;<br />

Explicit: u t+∆t = f (u t ) - solved directly, no need to converge, ∆t is limited.<br />

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CHAPTER 5 NUMERICAL INVESTIGATIONS<br />

calculation time. The hinges are modeled by means of two coincident points<br />

that share identical translational degrees of freedom, but different rotational<br />

degree of freedom. Furthermore, this chapter describes which materials are<br />

applied in the research.<br />

The Tensairity structure is loaded in two successive steps. The internal<br />

overpressure is first applied in the airbeam, then the upper strut is loaded<br />

with a distributed line load. By applying the load in discrete substeps, large<br />

deformations between successive steps are prevented to avoid divergence of<br />

the calculations. Divergence occurs when a singularity in the model occurs,<br />

for example wrinkling of the membrane. As a consequence, with these finite<br />

element calculations, it is not possible to calculate the realistic maximal load.<br />

Many simulations are conducted in this research to gain insight in the general<br />

behaviour of Tensairity structures and to monitor the influence of specific<br />

design parameters on the structural behaviour of the deployable beam. These<br />

results are presented in the next chapter.

Analysis of the structural behaviour<br />

6<br />

The design of the deployable Tensairity beam does not only rely on the<br />

foldability constraint. Decisions should also be made from a structural point<br />

of view. It is therefore important to investigate and understand the influence<br />

of the various design parameters on the structural behaviour of the deployable<br />

Tensairity beam. A first study on this is already conducted in chapter 4 by<br />

means of experiments on scale models. The influence of cables and the shape<br />

of the airbeam on the structural behaviour of the deployable Tensairity beam<br />

were discussed.<br />

In this chapter, more parameters and configurations with regard to the deployable<br />

Tensairity beam are studied to understand their influence on its structural<br />

behaviour. The main parameters whose influence will be monitored by means<br />

of numerical investigations are the hinges, the cable configuration, the struts’<br />

and hull section. Figure 6.1 illustrates the different investigated elements in<br />

this chapter. These elements are all interacting with each other in the Tensairity<br />

beam: the hull section has an influence on the cable tension, the cables and<br />

hinges have an influence on the section of the strut, etc. Therefore is chosen for<br />

reasons of clarity to discuss each element separately in a section. The insights<br />

gained will then be brought together at the end of this chapter, discussing the<br />

effect of the cables, hinges, hull and strut section on each other. These insights<br />

will lead to the design of the prototype of the deployable Tensairity beam,<br />

presented in the next chapters.<br />

6.1 HULL SECTION<br />

The section of an inflatable volume always tends to become a circle. This<br />

is because the pressure acts perpendicular to the surface of the outer airtight<br />

hull. This is also the case for a Tensairity structure with no connection between<br />

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CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR<br />

cables<br />

hinges<br />

strut section<br />

hull section<br />

Figure 6.1: When two parts of the hull are connected by means of a<br />

tension element (e.g. cable, membrane) shorter than the diameter, the<br />

section will not be a circle anymore, but a combination of two circles.<br />

the upper and lower strut: the standard spindle shaped beam is an object of<br />

revolution and has thus a circular section (with the section plane perpendicular<br />

on the longitudinal axis of the structure). This standard spindle is also called<br />

‘O-spindle’ in this research. When two parts of the hull are connected by<br />

means of a tension element (e.g. cable, membrane) shorter than the diameter,<br />

the section will not be a circle anymore, but a combination of two circles.<br />

Figure 6.2 illustrates this.<br />

By connecting the upper and lower strut of a Tensairity beam with such a tension<br />

element, this element becomes stressed when the airbeam is inflated and<br />

contributes to the structural behaviour of the beam (Breuer et al., 2007; Wever<br />

et al., 2010). After all, this connecting element gives an additional support for<br />

the compression element, increasing the value of the spring constant of the<br />

elastic foundation. This improves the stabilization of the compression element<br />

and the structure’s stiffness. This section investigates what the radius of the<br />

airbeam along the spindle length should be to pretension the connecting web<br />

in such a way that it contributes as much as possible to the stiffness of the<br />

Tensairity beam.<br />

The elements connecting the upper and lower strut of the Tensairity beam can<br />

be cables or membrane and is generally called ‘web’ in this section, no matter<br />

how many elements, the material or the configuration. This web is stressed

inflating<br />

6.1 HULL SECTION<br />

Figure 6.2: When two parts of the hull are connected by means of a<br />

tension element (e.g. cable, membrane) shorter than the diameter, the<br />

section will not be a circle anymore, but a combination of two circles.<br />

due to the action of the inflated hull on this web: by inflating the airbeam,<br />

the hull tends to give the section a circular shape and thus moving the struts<br />

outwards. This way, the web becomes stressed and counteracts this outwards<br />

movement of the struts.<br />

The stress in the hull and the angle between the web and the hull determine<br />

the stress in the web. The stress in the hull depends on its radius and internal<br />

overpressure (see equation 2.3 on page 14). The angle between the web and<br />

the hull is - for a fixed height h - determined by the radius of the hull: the larger<br />

the radius, the larger the angle. Figure 6.3 shows the action from the stressed<br />

hull on the web. The horizontal components of this action, represented by nh,<br />

counterbalance each other. The vertical components, represented by nv, are<br />

superposed and are thus the tension force that act on the web.<br />

Two cases are investigated and compared. In the first model, the ratio h<br />

R is kept<br />

constant along the spindle’s length, whereby height h represents the distance<br />

between the upper and lower strut and R the radius of the hull at one side<br />

of the struts. The second case is modeled in such a way that the vertical stress<br />

in the web (membrane or cables) due to the inflation is constant throughout<br />

the length of the beam. The geometry of the middle section of the beam<br />

of the two cases are chosen identical for comparison reasons. This means<br />

that the stress in the web of the second case is identical to the middle web<br />

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CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR<br />

stress of the first model. Figure 6.4 illustrates the outer hull of the two cases,<br />

figures 6.5 and 6.6 show sections of each case. The shape of the struts - which<br />

defines the ‘longitudinal shape’ of the structure - will not be modified here.<br />

The struts cover a span of 5 m and are in the middle of the beam 0,5 m apart<br />

from each other.<br />

6.1.1 CONSTANT H R -RATIO<br />

The shape of the outer hull is determined by choosing the h<br />

R-ratio. After all, at<br />

each position along the length of the beam is the height h between the upper<br />

and lower strut defined. The radius can then be easily derived from the above<br />

mentioned relation. Figure 6.5 shows the hull of the Tensairity beam at two<br />

positions along the beams length. Because h<br />

R is constant, the angle is the same<br />

for both sections. However, since the stress of the hull is larger in the case of<br />

a larger radius, the tension force F on the web is larger in the case of a larger<br />

radius. This means that in the case of a spindle shaped Tensairity beam with<br />

constant h<br />

R-ratio, the web stress varies with the height (and thus the length) of<br />

the spindle and thus from large in the middle towards smaller at the ends.<br />

6.1.2 CONSTANT STRESS<br />

The spindle shaped Tensairity beam is modeled in such a way that the vertical<br />

stress in the web – membrane or cables – due to the inflation is constant<br />

throughout the length of the beam. Since the pressure is the same in the<br />

whole spindle airbeam, the ratio h<br />

R has to decrease towards the ends. Once the<br />

wanted stress in the web is decided (or once the radius at a certain point of the<br />

spindle (and thus height of the web) is chosen), the radii at other web heights<br />

can be calculated easily. Figure 6.6 shows sections of the Tensairity beam at<br />

two different locations along the beams length. The height and radius are thus<br />

chosen in such a way that the stress in the web is equal in both sections.<br />

Suppose that the radius Ra at a certain height of web ha is given. Then, in order<br />

to have everywhere the same web stress, the radius Rb at spindle height hb can<br />

be calculated by the following expression:<br />

<br />

Rb =<br />

R 2 a − h2 a<br />

4<br />

+ h2<br />

b<br />

4<br />

(6.1)<br />

To calculate the normal force F in the web at a certain internal pressure p, the<br />

following expression can be used:<br />

<br />

F = 2p.<br />

(R2 − h2<br />

) (6.2)<br />

4

hull<br />

n<br />

α<br />

h R<br />

web<br />

F<br />

n v<br />

n h<br />

n<br />

6.1 HULL SECTION<br />

Figure 6.3: The action of the hull (stress n) on the web. This results in<br />

the tension force F on the web.<br />

Figure 6.4: The outer hull (and a zoom of the ends) of the two<br />

investigated cases. Upper: constant stress, lower: constant h<br />

R -ratio.<br />

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CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR<br />

6.1.3 NORMAL FORCE IN WEB<br />

Membrane as web<br />

The configurations with constant h<br />

R-ratio and constant stress are both numerically<br />

investigated in a spindle shaped beam without hinges in the struts. The<br />

y-component of the stress in the membrane connecting the upper and lower<br />

strut is illustrated in figure 6.7 and visualizes very well the above mentioned<br />

difference between the cases. After inflation and loading, the web is equally<br />

stressed along the spindle length in the ‘constant stress’-case, while this is<br />

not true for the ‘constant h<br />

R-ratio’-case where the web has smaller membrane<br />

stresses at smaller spindle heights.<br />

Cables as web<br />

It is obvious that the Tensairity beam constituted of continuous struts and a<br />

membrane as web is not deployable. Therefore, the two hull configurations<br />

are also investigated with a Tensairity beam containing hinges in the struts<br />

and cables connecting the upper and lower strut. The beam is illustrated<br />

in figure 6.8. The forces acting in the cables throughout the loading are<br />

represented in figure 6.9.<br />

The cases are being inflated to 150 mbar in the first loadstep of the finite element<br />

model. This is shown on the left of the dotted line in the graphs of figure 6.9 and<br />

represented by an increasing cable tension. The cable numbers are indicated<br />

and correspond with the cable numbers in figure 6.8. The cable forces under<br />

distributed loading represent very well the influence of the geometry of the<br />

two cases. The forces in the constant ‘ h<br />

R-ratio case’ are higher for the cables<br />

positioned at a larger height of the spindle. After all, as illustrated in figure 6.5<br />

is the vertical tensile component then larger. The cable forces of the ‘constant<br />

stress’-case are all nearly identical, as expected. Notice that the force in cable 3<br />

is identical for both cases, which was expected since both configurations have<br />

the same hull section in the middle of the beam because they were designed<br />

for an identical web stress at that position.<br />

6.1.4 DISPLACEMENTS AND STIFFNESS<br />

The deformation of the upper strut of the Tensairity beam from figure 6.8 is<br />

monitored during loading. This is done by measuring the displacement of<br />

88 points, distributed evenly over the strut. The average displacement of the<br />

upper strut of the two hull configurations is plotted on the left of figure 6.10.

n a<br />

h a<br />

F a<br />

n a<br />

R a<br />

n b<br />

h b<br />

F b<br />

6.1 HULL SECTION<br />

Figure 6.5: The hull of the spindle shaped beam with constant h<br />

R -ratio.<br />

n a<br />

h a<br />

Fa Fa = Fb Fb n a n b n b<br />

R a<br />

Figure 6.6: The hull of the spindle shaped Tensairity beam is modeled<br />

in such a way that the stress in the web is constant along the length of<br />

the beam.<br />

h b<br />

R b<br />

n b<br />

R b<br />

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CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR<br />

cable tension [N]<br />

1500<br />

1000<br />

500<br />

Figure 6.7: The difference between the two cases can be seen in the<br />

y-component of the stress of the web. Upper: after inflation, lower: after<br />

loading.<br />

1 2 3<br />

Figure 6.8: The Tensairity beam with hinges and cables connecting<br />

upper strut is used in the cases of constant stress and constant h<br />

R -ratio.<br />

The outer hull is not shown.<br />

constant stress<br />

2<br />

1<br />

3<br />

0<br />

0 0.5 1<br />

loadstep<br />

1.5 2<br />

cable tension [N]<br />

1500<br />

1000<br />

500<br />

constant<br />

h<br />

R -ratio.<br />

3<br />

2<br />

0<br />

0 0.5 1<br />

loadstep<br />

1.5 2<br />

Figure 6.9: The cable forces of the two configurations along the loading<br />

show very well the influence of their difference in geometry. The cases<br />

are inflated to 150 mbar and loaded with a distributed load of 5000 N .<br />

1

6.1 HULL SECTION<br />

The average stiffness of both cases is identical until a certain load. At higher<br />

loads is the case with constant web stress more stiff than the case with constant<br />

h<br />

R -ratio.<br />

As discussed in chapter 4 can this behaviour be attributed to the influence of the<br />

pretensioned cables. When all cables in both cases are still pretensioned, their<br />

load-displacement behaviour is identical because the cables support all hinges.<br />

From a certain load on, the less pretensioned cables from the h<br />

R -configuration<br />

(which are the cables at the sides) will become slack and allow the hinge to<br />

displace more, hence the lower stiffness. Figure 6.10 (right) illustrates the<br />

displaced shape of the two cases.<br />

load [N]<br />

7000<br />

6000<br />

5000<br />

4000<br />

3000<br />

2000<br />

1000<br />

constant stress−case<br />

h/R − case<br />

0<br />

0 0.002 0.004 0.006 0.008 0.01<br />

average displacement [m]<br />

constant h<br />

R -ratio<br />

constant stress<br />

Figure 6.10: Left: the average displacement of the upper strut of the<br />

two hull configurations, right: the displaced shape of the two cases at<br />

5000 N. (Displacements are scaled with factor 10. Also displacements<br />

due to inflation, hence the outwards movement of the strut. The outer<br />

hull is not shown).<br />

This investigation showed the influence of the shape of the section of the<br />

airbeam on the forces in the elements connecting the upper and lower strut of<br />

the Tensairity beam. The ‘constant stress’ configuration proves to be the most<br />

optimal, since all cables are tensioned by the same amount. After all, in the<br />

case with a constant h<br />

R-ratio are the cables towards the ends less pretensioned<br />

under inflation. As a result, these cables will become slack under a smaller<br />

load and large displacements will occur. Therefore is chosen to apply the<br />

‘constant stress’-configuration for further numerical models and the physical<br />

prototype.<br />

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CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR<br />

6.2 DIMENSIONING THE STRUTS<br />

In this section, the influence of the dimensions of the struts on the structural<br />

behaviour of a deployable Tensairity beam is investigated. After all, the<br />

deployable structure differs from a standard Tensairity beam by containing<br />

hinges and cables that connect the upper and lower strut. No investigations<br />

are conducted yet on the influence of these additional elements on the strut’s<br />

behaviour.<br />

The upper and lower strut of a uniformly vertical loaded standard Tensairity<br />

beam are respectively in compression and tension. The strength, stability and<br />

stiffness have to be considered when dimensioning the strut in compression<br />

while only the strength requirement has to be taken into account for the tension<br />

element. In this research, upper and lower struts are chosen to be identical.<br />

This way, the structure can also bear negative load cases, such as wind suction.<br />

Therefore, only the dimensioning of the compression element is discussed here.<br />

First, the strength and stability requirement are considered. Then the influence<br />

of its dimensions on the global stiffness of the structure is investigated.<br />

6.2.1 STRENGTH AND STABILITY<br />

The strut has to be dimensioned to withstand the compressive forces. They<br />

can be determined by means of finite element models or by analytical approximation.<br />

The compressive force for a cylindrical Tensairity beam can be<br />

approximated by<br />

N = q.L.γ<br />

, (6.3)<br />

8<br />

with q the applied distributed load [ N<br />

m ], L the length of the Tensairity beam [m]<br />

and γ the slenderness [/], defined as the beams length divided by its height<br />

(Luchsinger et al., 2004a). This value approximates also well the axial forces in<br />

the middle of the struts of a spindle shaped Tensairity beam (Luchsinger and<br />

Crettol, 2006b). The compressive forces in the middle of the strut are thus a<br />

linear function of the applied load, but independent of the pressure.<br />

A compressed structural element is prone to buckling. The compression<br />

element of a Tensairity beam is tightly connected to the inflated hull along<br />

its length. This stressed membrane acts as a continuous elastic support for the<br />

strut. The buckling load for such a beam on elastic foundation is P = 2 √ k.E.I<br />

with k the spring constant of the elastic foundation [ N<br />

m 2 ], E the modulus of

6.2 DIMENSIONING THE STRUTS<br />

elasticity [ N<br />

m 2 ] and I the moment of inertia 1 [m 4 ]. In Tensairity structures,<br />

the spring constant depends on the overpressure p of the airbeam and is<br />

determined by Luchsinger et al. (2004a) as p×π. Hence, the maximum buckling<br />

load for a Tensairity beam is<br />

Pbuckling = 2 p.π.E.I (6.4)<br />

This buckling load increases with increasing square root of the pressure and<br />

the moment of inertia of the beam. Notice that it is independent of the length<br />

of the beam. When dimensioning the compression strut, the moment of inertia<br />

for a given material and overpressure can be chosen such that the buckling<br />

load is higher than the yield load. This way, the compression element can be<br />

loaded to its yield limit and the yield load becomes then the limiting factor of<br />

the compression element. This is called buckling free compression (Luchsinger<br />

et al., 2004a). The compressive force is thus<br />

P = σyield.A (6.5)<br />

with σyield the yield stress and A the cross sectional area of the compression<br />

element.<br />

6.2.2 INFLUENCE OF CABLES<br />

Because of the constructive separation of tension and compression, no bending<br />

occurs in the struts of a standard spindle shaped Tensairity beam under<br />

homogeneous load. As a result, the bending stiffness of the struts is in these<br />

cases of little importance and can be neglected when investigating the load<br />

displacement behaviour. The deflection at mid span of such a Tensairity beam<br />

is proposed analytically by Luchsinger and Teutsch (2009). They find that the<br />

deflection is the sum of two terms, one due to the elasticity of the struts and<br />

one due to the deformation of the inflated body which depends on the air<br />

pressure. The bending stiffness of the struts is neglected in this model of a<br />

standard spindle shaped Tensairity beam and comparisons with finite element<br />

results show that this is an allowable approximation.<br />

However, when cables connect the upper and lower strut of the Tensairity<br />

beam, additional forces are introduced. These point loads introduce large<br />

bending moments in the struts and the bending stiffness can not be neglected<br />

anymore. Figure 6.11 illustrates the moment diagrams of four cases. Two<br />

cases have a membrane connecting the upper and lower strut, the other two<br />

have cables as web. Each web type has a case without hinges and one with<br />

1 With moment of inertia is meant in this research the ‘area moment of inertia.’<br />

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CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR<br />

five hinges. From the figure can be seen that the cases with a membrane as<br />

connection have very little bending moments in the struts. After all, the forces<br />

of the web are introduced homogeneous over the struts. This is in contrast<br />

with the cases with cables, because point loads occur which introduce bending<br />

moments in the struts. Since hinges do not transfer bending moments, the cases<br />

with and without hinges have different bending moments.<br />

cables<br />

membrane<br />

with hinges<br />

without hinges<br />

Figure 6.11: The bending moment of four Tensairity beams under<br />

homogeneous load. Two cases have a membrane connecting the upper<br />

and lower strut, the other two have cables as connection. Each connection<br />

type has a case without hinges and one with five hinges.<br />

The bending moments in the struts, introduced by the cables, have an influence<br />

on the load-displacement behaviour of the deployable Tensairity beam. It is<br />

clear that the struts should be dimensioned taking the bending moments into<br />

account. If not, the struts will deform considerable and the Tensairity beam<br />

will experience larger displacements than the same configuration without<br />

cables. This is in contradiction with the purpose of adding cables, namely<br />

increasing the stiffness of the deployable Tensairity beam by blocking the<br />

hinges as discussed in chapter 4and as will be presented in section 6.3.<br />

6.2.3 BENDING STIFFNESS<br />

The influence of the bending stiffness of the struts on the load displacement<br />

behaviour of the deployable Tensairity structure is studied numerically. A<br />

statically determined five meter Tensairity beam is inflated by an overpressure<br />

of 150 mbar (15 kN<br />

m 2 ) and loaded with a uniform distributed load. The shape of<br />

the hull is chosen in such a way that the pretension in the cables throughout<br />

the length is constant. The upper and lower struts are connected by means<br />

of vertical cables which are attached to the strut at the location of the five<br />

upper and lower hinges, see figure 6.12. The length of the cables is equal to<br />

the distance of the hinges that are connected by that cable which means that<br />

there is no prestress present in the cables before inflation. Inflating the airbeam

introduces a tensile force in the cables.<br />

6.2 DIMENSIONING THE STRUTS<br />

The structural response of several Tensairity beams is simulated and the<br />

average stiffness of each case is noted. The cases differ only in one thing,<br />

the cross-section of the struts. All struts have identical cross-sectional area<br />

(of 600 mm 2 ), but different bending stiffness. Since the Young’s modulus is<br />

constant for all cases (all struts are made of aluminum), only the moment of<br />

inertia is varied. The dimensions of the cross sections of the cases can be<br />

found in table 6.1. It is important to note that out-of-plane buckling was not<br />

taken into account in these simulations (by blocking the struts to move outof-plane).<br />

After all, the focus in this study is on the influence of the struts’<br />

bending stiffness on the load-displacement behaviour of the beam. The results<br />

also hold true for strut sections with the same bending stiffness, but other<br />

profile geometry (e.g. hollow rectangular sections).<br />

Figure 6.12: The configuration of the investigated Tensairity beam. The<br />

dots represent hinges, the vertical lines are cables.<br />

Table 6.1: The investigated struts’ cross sections.<br />

case 1 = A case 2 case 3 case 4 case 5 case 6<br />

B [mm] 50,00 45,00 40,00 35,00 32,00 30,00<br />

H [mm] 12,00 13,33 15,00 17,14 18,75 20,00<br />

Iz × 10 3 [mm 4 ] 7,20 8,88 11,25 14,69 17,58 20,00<br />

case 7 case 8 case 9 case 10 case 11 case 12<br />

B [mm] 28,00 26,00 24,00 22,00 20,00 18,00<br />

H [mm] 21,43 23,08 25,00 27,27 30,00 33,33<br />

Iz × 10 3 [mm 4 ] 22,96 26,63 31,25 37,19 45,00 55,55<br />

case 13 case 14 case 15 case 16 case 17<br />

B [mm] 16,00 14,00 12,00 11,00 10,00<br />

H [mm] 37,50 42,86 50,0 54,55 60,0<br />

Iz × 10 3 [mm 4 ] 70,31 91,84 125,00 148,76 180,00<br />

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CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR<br />

The average stiffness versus the moment of inertia of all the cases is plotted in<br />

figure 6.13. The curve, which results from connecting the average stiffness of<br />

the cases, can be divided in three parts. In the first part, a small rise of Iz stiffens<br />

the structure considerably. Increasing the moment of inertia in the third part<br />

of the graph, even by a large amount, has very little influence on the stiffness<br />

of the structure. The reason for this can be found in the structural behaviour<br />

of the segments of the struts of the Tensairity beam. This is discussed here<br />

more in detail by looking closer at their deflection under loading (inflation and<br />

external loading).<br />

Inflation<br />

Average stiffness [N/m]<br />

x 10<br />

3<br />

6<br />

2.5<br />

2<br />

1.5<br />

1<br />

0.5<br />

0<br />

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8<br />

I y [m 4 ]<br />

150 mbar<br />

x 10 −7<br />

Figure 6.13: The average stiffness of the various cases is plotted in<br />

relation with the area moment of inertia of the strut.<br />

As already mentioned in section 6.1, the hull of the Tensairity beam moves<br />

outwards when inflating and acts as a distributed load on the struts in the<br />

outwards direction. At the places where the upper and lower strut are<br />

connected by means of a cable, this outwards movement is counterbalanced<br />

and taken up by the cable. In the other parts, the struts will deflect under the<br />

loading. The size of this deflection depends - besides the pressure - on the<br />

bending stiffness of the strut: the larger the bending stiffness, the lower the<br />

deflection of the strut.<br />

The deflection of the struts of two cases, case A and case B, are taken as<br />

illustration. Case A has a strut with dimensions (H × B) 12 mm × 50 mm<br />

(Iz = 7, 2 × 10 3 mm 4 ) and case B is 25 times stiffer with a hollow rectangle cross<br />

section of 50 mm × 35 mm and t=2 mm (Iz = 180 × 10 3 mm 4 ). The deflection

6.2 DIMENSIONING THE STRUTS<br />

of case A (21,63 mm) is measured 16 times larger than case B (1,31 mm) in the<br />

simulations. Figure 6.14 shows the deflection of a segment of the upper strut<br />

of both cases. For clearness, the displacement magnified by a factor 10 is also<br />

shown.<br />

strut<br />

section<br />

1 x<br />

10 x<br />

A<br />

Figure 6.14: The deflection of a segment of the upper strut of case A and<br />

case B after inflation.<br />

This deflection can also be approximated analytical. Because the cables on<br />

both sides of the segment are tensioned with the same force and elongate thus<br />

by the same amount (which is by the way very little compared with the central<br />

deflection), the real situation can be approximated by the model of a simply<br />

supported beam.<br />

The deflection of such a beam under distributed line load is known as:<br />

δ = 5ql4<br />

384EIz<br />

B<br />

(6.6)<br />

whereby q represents the distributed load [ N<br />

m ], l the length of the segment<br />

[m], E the Young’s modulus [ N<br />

m2 ] and I the area moment of inertia [m4 ]. For<br />

this case, the action of the inflated membrane on the strut can be calculated<br />

as 2600 N<br />

m for a pressure of 150 mbar. The length of the beam equals 0,84 m<br />

(length of segment of strut between two hinges), the Young’s modulus of Alu<br />

9 N<br />

is 69 × 10 m2 . Case B has a moment of inertia of 180 ×103 mm4 , which results<br />

in an analytical displacement δcase B = 1,35 mm, which is close to the numerical<br />

117

118<br />

CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR<br />

value of 1,31 mm. For case A, the analytical value (δcase A= 33,74 mm) differs<br />

more from the numerical of 21,63 mm. This is due to non-linear effects of<br />

the large deformation. Figure 6.15 shows the analytical and numerical central<br />

displacement for all investigated cases. The two curves match reasonably well,<br />

especially for larger moments of inertia. For smaller values is the deflection<br />

overestimated by the analytical approximation.<br />

deflection [m]<br />

0.035<br />

0.03<br />

0.025<br />

0.02<br />

0.015<br />

0.01<br />

0.005<br />

numerical displacement<br />

analytical displacement<br />

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 x 10 −7<br />

0<br />

area moment of inertia [m 4 ]<br />

Figure 6.15: The numerical and analytical value of the central<br />

displacement of the strut’s segment after inflation.<br />

External loading<br />

It is this deflected shape, illustrated in figure 6.14, that is loaded vertical and<br />

distributed. The upper strut takes the compressive forces, the lower strut is<br />

tensioned. In case B, the deflection due to inflation is so little that the upper<br />

strut takes up compression more or less centric. Therefore, it shows the same<br />

behaviour and has similar stiffness as in the case of a regular spindle shaped<br />

Tensairity beam with membrane web and continuous strut. A small change<br />

in bending stiffness of the strut has in this case very little influence on the<br />

stiffness of the Tensairity beam. This is reflected in the third part of the graph<br />

illustrated in figure 6.13. In case A, the compressive force acts eccentric on<br />

the strut because of the large deformation due to inflation. The eccentric<br />

compressive loading results in an increase of bending and deflection of the<br />

strut. Figure 6.16 illustrates this. As a result, the Tensairity beam as a whole<br />

experiences this deformation and has a lower average stiffness than case B.<br />

Increasing the bending stiffness of the strut in case A decreases the deflection

6.2 DIMENSIONING THE STRUTS<br />

caused by inflation and lowers thus the eccentricity of the compressive loading.<br />

As can be seen in the first part of the graph in figure 6.13, a small increase<br />

of bending stiffness increases the average stiffness of the Tensairity beam<br />

considerable. Part two of the curve is the transition whereby the eccentricity<br />

of the compressive force decreases and thus also the effect of this eccentricity<br />

on the deflection.<br />

after inflation after loading<br />

Figure 6.16: A schematic overview explaining the second order effects<br />

in case A.<br />

This behaviour is also reflected in the occurring bending moments of the cases,<br />

as illustrated in figure 6.17. The bending moment in the struts of case B, where<br />

the strut is loaded more or less centric, decreased after distributed loading. This<br />

is because the external loading is in opposite direction of the inflation load.<br />

The ‘centric’ compressive force has little influence on the bending moment.<br />

In case A, where the compressive force acts eccentric on the segments due<br />

to large deflections, is the bending moment doubled at the compression side.<br />

Despite the opposite (downwards) direction of the external loading, the struts<br />

are more bended. This is due to the eccentric compressive force, as illustrated<br />

and explained in figure 6.16.<br />

inflation<br />

loading<br />

case A<br />

case B<br />

Figure 6.17: The bending moments of case A and B after inflation and<br />

loading. The bending moments after loading reflect well the influence<br />

of the eccentricity on the strut’s behaviour.<br />

119

120<br />

CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR<br />

The same cases from table 6.1 are also investigated under various internal<br />

overpressures. Figure 6.18 shows the average stiffness of the cases in relation<br />

to the moment of inertia of the struts. From the curves can be concluded that<br />

for a Tensairity beam with hinges and cables, a higher pressure means a lower<br />

stiffness. This is not at all what would be expected intuitive, since this is the<br />

opposite of the behaviour of regular Tensairity structures and airbeams, where<br />

an increase of internal overpressure stiffens the structure. The explanation for<br />

this can be found in the previous discussion: a higher internal pressure deflects<br />

the struts more and increases the eccentricity of the compressive forces, which<br />

results in a less stiff structure. At high values of Iz, the influence of the deflection<br />

due to inflation is minimized and all cases have similar average stiffness.<br />

However, as will be seen in chapter 8 where the experimental investigations<br />

are discussed, is this lowering of the stiffness at higher pressures not occurring<br />

in reality. From the experiments (where struts with an Iz = 11,25 ×10 3 mm 4<br />

are used) is observed that the deflection of the struts after inflation is not as<br />

large as the finite element models predict and has as a consequence no (or no<br />

visible) influence on the structure’s stiffness.<br />

average stiffness [N/m]<br />

x 106<br />

2.5<br />

2<br />

1.5<br />

1<br />

050 mbar<br />

100 mbar<br />

150 mbar<br />

200 mbar<br />

250 mbar<br />

0.5<br />

300 mbar<br />

400 mbar<br />

500 mbar<br />

0<br />

0 0.5 1 1.5 2<br />

I y [m 4 ]<br />

x 10 −7<br />

Figure 6.18: The average stiffness of the various cases is plotted in<br />

relation with the area moment of inertia of the strut. Various pressure<br />

levels are shown.

Influence of configuration<br />

6.2 DIMENSIONING THE STRUTS<br />

The previous discussion showed that using cables instead of a membrane<br />

as web type changes the structural behaviour and thus the average stiffness<br />

of the Tensairity beam considerable. As illustration, the average stiffness is<br />

calculated of the four Tensairity cases whose bending moments are discussed<br />

in section 6.2.2 and illustrated in figure 6.11. This comparison, plotted in<br />

figure 6.19, permits to illustrate the influence of using cables or a membrane as<br />

connection type and the influence of hinges in the struts. The average stiffness<br />

of the four cases is plotted in relation to the moment of inertia. The curves<br />

clearly show that the cases with a web (and thus very low bending moment<br />

in the strut) are more or less independent of the area moment of inertia of the<br />

strut and thus its bending stiffness. This is clearly not the case for the Tensairity<br />

beam with cables, as explained. For high values of the moment of inertia are<br />

the cases with cables stiffer than with membrane. This can be attributed to<br />

the higher E-modulus of the steel cable than the one of the membrane. From<br />

the graph can also be seen that the cases with hinges are less stiff than the<br />

cases with continuous strut. The influence on the structural behaviour of<br />

introducing hinges will be discussed in the next section.<br />

average stiffness [N/m]<br />

x 106<br />

2.5<br />

2<br />

1.5<br />

1<br />

0.5<br />

cable hinge<br />

cable no hinge<br />

web hinge<br />

web no hinge<br />

0<br />

0 0.5 1 1.5 2<br />

I y [m 4 ]<br />

x 10 −7<br />

Figure 6.19: The average stiffness of the various cases is plotted in<br />

relation with the area moment of inertia of the strut. The cases differ in<br />

web type (cable and membrane) and the presence of hinges.<br />

121

122<br />

CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR<br />

6.2.4 CONCLUSION<br />

By investigating the influence of the dimensions of the struts on the structural<br />

behaviour of a deployable Tensairity beam, it became clear that its structural<br />

behaviour is very different from that of a regular Tensairity beam. Introducing<br />

cables as connecting element between upper and lower strut changes the forces<br />

and moments in the struts drastically.<br />

In the case of a Tensairity beam whereby the upper and lower strut are<br />

connected by means of a membrane, the outwards movement of the inflated<br />

membrane (and thus force) is taken up by this membrane. The struts of the<br />

beam experience no bending and are only loaded in tension and compression,<br />

just like a regular Tensairity beam with circular cross section of the hull. In<br />

the case of a Tensairity beam whereby cables are used as connecting element,<br />

the segments of the struts have to take up the outwards force of the inflating<br />

membrane and are at the ends supported by the vertical cables. This situation<br />

makes that the segments of a deployable Tensairity beam, connected to each<br />

other with hinges, behave like a simply supported beam (supported by the<br />

cables), loaded under a distributed line load (the action of the membrane).<br />

The size of this deflection is determined by the load acting on the strut and<br />

the bending stiffness of the strut’s segment. The bending stiffness is defined<br />

by the product of the strut’s Young modulus and moment of inertia. The size<br />

of the deflection caused by inflation has a great influence on the behaviour<br />

of the strut under external loading. The larger the deflection, the larger the<br />

eccentricity of the compressive forces in the strut and thus the influence of<br />

second order effects on the bending.<br />

If the deflection is small due to a high bending stiffness of the strut (compared<br />

to the load), the eccentricity of the compressive force is negligible and has no<br />

influence on the structural behaviour under external loading. The average<br />

stiffness of such cases is very comparable with that of a Tensairity beam with<br />

membrane as web. If the bending stiffness is on the other hand quite small, a<br />

large deflection will occur during inflation and the compressive load during<br />

external loading will be eccentric. Second order effects will cause large bending<br />

moments and displacements in the upper strut of the Tensairity beam.

6.3 INTRODUCING HINGES<br />

6.3 INTRODUCING HINGES<br />

When developing a deployable Tensairity beam, the influence of introducing a<br />

mechanism as compression and tension element on the load-bearing behaviour<br />

of the Tensairity structure should be known and understood. Therefore,<br />

Tensairity beams with varying amount of hinges in the upper and lower struts<br />

and a varying location of the hinges are investigated. First, the influence of<br />

introducing hinges on the structural behaviour is studied.<br />

6.3.1 INFLUENCE OF HINGES<br />

Two different Tensairity beams are investigated. One beam has a circular crosssection<br />

of the hull, also called the ‘O-spindle’ in this research, and the second<br />

has cables connecting the upper and lower strut, the ‘discretized web spindle’<br />

or ‘D-spindle’. Figure 6.20 illustrates both cases.<br />

O-spindle<br />

O-spindle<br />

D-spindle<br />

Figure 6.20: The influence of hinges on the structural behaviour of two<br />

cases is investigated: the ‘O-spindle’ and ‘D-spindle’.<br />

To investigate the influence of hinges on the structural behaviour of the Ospindle<br />

Tensairity beam, two configurations are simulated. They differ in<br />

the presence of hinges in the upper and lower strut: one case contains five<br />

hinges in upper and lower strut, the other none. Both cases are inflated to an<br />

overpressure of 150 mbar and loaded by a distributed line load of 240 N<br />

m . The<br />

struts have a width of 30 mm and a height of 20 mm. The moment diagrams of<br />

the struts after distributed loading are noted, as well the average displacement<br />

of the structures throughout loading.<br />

Figure 6.21 shows the moment diagrams of the two cases after loading. As<br />

can be expected, the bending moment of the continuous strut (thus without<br />

hinges) is similar to the one of an arch under distributed loading. However,<br />

123

124<br />

CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR<br />

due do the specificity of a Tensairity beam 2 , the value of the maximum bending<br />

moment is very small (7 N.m in this case). In the case of a segmented strut, the<br />

hinges can not transfer bending moments and the maximum bending moment<br />

is thus smaller than in the case of a continuous strut (5 N.m).<br />

Figure 6.21: The bending moment of a standard spindle with and<br />

without hinges after distributed loading (240 N<br />

m ).<br />

Figure 6.22 plots the average displacement of both cases throughout the<br />

loading. The difference between both cases is very small, because there are<br />

mainly normal forces and almost no bending moments in the struts. However<br />

small, the two configurations have different average displacement and, as<br />

expected, the Tensairity beam with continuous strut is stiffer. Obvious, this is<br />

because the hinges do not transfer bending moments and experience a larger<br />

displacement, as can be seen in figure 6.23. This larger displacement takes<br />

especially place near the ends of the beam, where the radius is smaller. The<br />

consequence of this smaller radius is a smaller membrane stress and thus a<br />

lower ‘support’ of the hinge by the membrane.<br />

D-spindle<br />

As already mentioned in section 6.2.2, the bending moment diagrams of a<br />

web Tensairity 3 are different than in the case of a standard Tensairity structure<br />

(O-spindle). After all, the cables are responsible for additional forces on the<br />

struts. The values of the bending moments in the struts are in the case of a<br />

D-spindle 200 times larger than in the case of a standard Tensairity beam.<br />

Figure 6.25 illustrates the bending moments of a D-spindle with and without<br />

hinges. Also here, the strut’s section measures 30 mm width by 20 mm height.<br />

2 The bending is taken up by the complete Tensairity beam and the upper strut takes up the<br />

compressive forces, the lower one the tensile forces. However, due to the deflection of the hull<br />

under external loading, a minor bending moment occurs in the upper strut.<br />

3 This is a Tensairity beam whereby the upper and lower strut are connected by means of a<br />

membrane or cable. In this research, except mentioned differently, cables are always used as<br />

connecting element. This is then called the ‘D-spindle’, from discretized web.

load [N]<br />

1400<br />

1200<br />

1000<br />

800<br />

600<br />

400<br />

O−spindle − pressure 150 mbar<br />

0 1 2 3 4 x 10 −3<br />

200<br />

0 hinge<br />

5 hinge<br />

0<br />

average displacement [m]<br />

6.3 INTRODUCING HINGES<br />

Figure 6.22: The average displacement of the O-spindle with and<br />

without hinges.<br />

Figure 6.23: The displaced struts after loading. Upper: the case with<br />

hinges, lower: without hinges. The displacement is for reasons of<br />

clearness magnified by a factor 20.<br />

A large difference in moment diagram between the two cases can be seen.<br />

For the D-spindle with hinges, the segments of the struts behave as simply<br />

supported beam. From the bending moments of the case without hinges can<br />

be stated that their behaviour is analogue with hyperstatic beams. Figure 6.24<br />

illustrates this.<br />

This can be verified by means of comparing the analytical maximum bending<br />

moment of a simply supported beam and a hyperstatic beam with the numerical<br />

bending moments after inflation. The analytical maximum bending<br />

moments can be calculated as follow:<br />

125

126<br />

CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR<br />

.<br />

strut without hinges strut with hinges<br />

hyperstatic isostatic<br />

Figure 6.24: The deformation and bending of the segments of the struts<br />

can be modeled as simply supported or hyperstatic beams, depending<br />

whether the segments are connected by hinges.<br />

Miso = q.l2<br />

8<br />

Mhyper = q.l2<br />

12<br />

= 229, 32 N.m (6.7)<br />

= 152, 88 N.m (6.8)<br />

with q the action of the inflated membrane on the strut (2600 N<br />

m for a pressure of<br />

150 mbar) and l the distance between two vertical cables ( 0, 84 m). The values<br />

of the respective maximum bending moments, derived by means of numerical<br />

simulation, are Miso = 223, 81N.m and Mhyper = 153, 92N.m, which is reasonably<br />

close to the analytical values. Thus above comparison of the behaviour of the<br />

segments with isostatical and hyperstatical beams, holds true.<br />

One can expect to have larger displacements in the case with hinges than the<br />

case without, because of a larger bending moment in the struts. Figure 6.26<br />

(left) plots the average displacement of the two cases during the loading. It is<br />

clear that the case with hinges is less stiff than the case with continuous strut.<br />

Figure 6.27 shows the displaced struts of the two cases after loading. The<br />

displacement is scaled by a factor 20. Because the case with hinges deflects<br />

much more under inflation than the case without, it will experience a larger<br />

global deformation under loading. As a consequence, if stiffer struts are used<br />

and the deformation after inflation is smaller, the difference between the case<br />

with and without hinges decreases. This is illustrated on the right of figure 6.26,

6.3 INTRODUCING HINGES<br />

Figure 6.25: The bending moments of a D-spindle with and without<br />

hinges after inflation (150 mbar). The influence of the hinges can be<br />

clearly seen.<br />

where the average displacement of the two cases is plotted with a much stiffer<br />

section of 10 mm width × 60 mm height.<br />

load [N]<br />

6000<br />

5000<br />

4000<br />

3000<br />

2000<br />

section 30mm width × 20mm height<br />

load [N]<br />

0 1 2 3 4 5 x 10 −3<br />

1000<br />

0<br />

0 1 2 3 4 5 x 10<br />

average displacement [m]<br />

−3<br />

0<br />

0 hinge 1000<br />

0 hinge<br />

5 hinges<br />

5 hinges<br />

average displacement [m]<br />

6000<br />

5000<br />

4000<br />

3000<br />

2000<br />

section 10mm width × 60mm height<br />

Figure 6.26: The average displacement of the upper strut with and<br />

without hinges during loading. Left: the cases with section 30 mm ×<br />

20 mm; right: 10 mm × 60 mm.<br />

There can thus be concluded that hinges do have an influence on the structural<br />

behaviour of the Tensairity beam. After all, hinges do not transfer bending<br />

moments. Generally speaking, the Tensairity beams with hinges are less<br />

stiff than their equivalent with continuous strut. In the case of a standard<br />

spindle beam under loading, (almost) no bending moments occur in the struts.<br />

Therefore, the influence of the hinges on the stiffness is rather limited. In the<br />

case of a web-Tensairity, the difference between the case with and without<br />

hinges is dependent on the deflection under loading of the struts, and thus of<br />

the bending stiffness of the struts.<br />

Now the effect of hinges in the struts of a Tensairity beam is known and un-<br />

127

128<br />

CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR<br />

hinges<br />

no hinges<br />

Figure 6.27: The displaced upper and lower struts after loading of<br />

the case with and without hinges (with section 30 mm × 20 mm). The<br />

displacement is for reasons of clearness magnified by a factor 20.<br />

derstood, there is investigated what the influence on the structural behaviour<br />

is of the amount of hinges in the struts.<br />

6.3.2 AMOUNT OF HINGES<br />

The amount of hinges in the upper and lower strut of a spindle shaped<br />

Tensairity beam are varied to monitor the influence of the presence of these<br />

hinges on the load-bearing behaviour of the structure. Two configurations,<br />

illustrated in figure 6.28 are investigated numerically: the O- and D-spindle.<br />

This latter has cables connecting the upper and lower strut at regular distances,<br />

the other configuration does not. Within these two configurations, fifteen<br />

cases are simulated under a distributed load: the amount of hinges taken<br />

into account during the numerical investigations varies between 0 and 14, as<br />

illustrated in figure 6.29.<br />

O-spindle<br />

D-spindle<br />

Figure 6.28: The influence of the amount of hinges is investigated with<br />

the O- and D-spindle.<br />

The average stiffness of the cases is plotted in figure 6.30. The x-axis plots the<br />

amount of hinges, the average stiffness is shown on the y-axis. For both cases,<br />

a small decrease in stiffness for increasing amount of hinges can be detected.<br />

Thus, the amount of hinges has very little influence on the stiffness of the<br />

Tensairity beam. Notice that the case without connecting cables has a much<br />

lower average stiffness than the other case. This will be discussed more in

0<br />

1<br />

2<br />

3<br />

5<br />

7<br />

9<br />

11<br />

14<br />

Figure 6.29: The amount of hinges vary between 0 and 14.<br />

6.3 INTRODUCING HINGES<br />

129

130<br />

CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR<br />

average stiffness [N/m]<br />

2.5<br />

2<br />

1.5<br />

1<br />

0.5<br />

x 106<br />

3<br />

p = 150 mbar<br />

D−spindle<br />

O−spindle<br />

0<br />

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14<br />

amount of hinges<br />

Figure 6.30: The influence of the amount of hinges on the average<br />

stiffness of the cases is little.<br />

detail in section 6.4.3.<br />

maximum load [N]<br />

3000<br />

2500<br />

2000<br />

1500<br />

1000<br />

500<br />

O−spindle − pressure 150 mbar<br />

0<br />

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15<br />

amount of hinges<br />

Figure 6.31: The maximal load for the O-spindle for the various amount<br />

of hinges.<br />

The load at which the finite element calculation stops (and thus no convergence<br />

is achieved anymore) is presented here as being the maximal load.<br />

As discussed in chapter 5, this is the load whereby a large deformation or<br />

singularity occurs, like eg. large displacement of a hinge or wrinkling of the<br />

membrane. This value is different for the various configurations of figure 6.29.<br />

The maximal load for the O-spindle for the various amount of hinges is plotted<br />

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.<br />

6.3 INTRODUCING HINGES<br />

The results show that there is a relation between the amount of hinges in the<br />

struts and the maximum load that the structure can bear before ‘collapse’: the<br />

more the hinges, the lower the value of maximal load. However, more in<br />

depth investigation, presented in section 6.3.3, shows that it is the position of<br />

the hinge closest to the endpoints that influences the maximum load and not<br />

the amount of hinges. The reason why it seems that the amount of hinges<br />

determines the maximum load the structure can bear, is that (as can be seen<br />

in figure 6.29) in the case of evenly distributed hinges, the position of the last<br />

hinge and the amount of hinges in the strut are related: the more hinges the<br />

structure has, the smaller the distance between the last hinge and the ends of<br />

the Tensairity beam. Figure 6.32 plots the maximum loads in relation to the<br />

position of the last hinge along the length of the Tensairity beam. The zero<br />

represents the middle of the beam, the x-coordinate of 2,5 m represents the<br />

endpoint of the beam.<br />

maximum load [N]<br />

3000<br />

2500<br />

2000<br />

1500<br />

1000<br />

500<br />

2 hinges<br />

0<br />

0 0.5 1 1.5 2 2.5<br />

position along length of beam [m]<br />

3 hinges<br />

9 hinges<br />

O−spindle − pressure 150 mbar<br />

Figure 6.32: The maximum load in relation to the position of the last<br />

hinge along the length of the Tensairity beam. The configuration with 3<br />

3<br />

5<br />

7<br />

9<br />

11<br />

14<br />

and 9 hinges are illustrating the position of the last hinge.<br />

131

132<br />

CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR<br />

6.3.3 POSITION OF HINGES<br />

The influence of the location of the hinge on the maximal load of the deployable<br />

Tensairity beam is investigated. This is done by means of eleven cases which<br />

differ in location of hinge in upper and lower strut. All cases contain two<br />

hinges in their strut, but the position of these hinges changes throughout the<br />

cases. Note that the hinges are symmetric in relation to the middle of the beam.<br />

Three of these cases are illustrated in figure 6.33.<br />

2<br />

6<br />

11<br />

Figure 6.33: Three of the eleven cases, whereby the position of two<br />

hinges (symmetrical) on upper and lower strut is varied. The black dots<br />

represent hinges. Cases 2, 6 and 11 are shown.<br />

The maximal load of the O-spindle (fig. 6.28) for the eleven cases is plotted<br />

in figure 6.34 (left). The maximal load is plotted at the position of the hinge<br />

of each case along the Tensairity beam. From the figure can be seen that the<br />

maximal load decreases when the hinge is more positioned to the ends.<br />

A similar decrease of maximal load was detected in the study on the influence<br />

of the amount of hinges (figure 6.32). There has been discussed that not<br />

the amount of hinges directly determines the maximum load, but mainly the<br />

position of the last hinge. This is shown in the right graph in figure 6.34 where<br />

the curves of both cases are illustrated.<br />

The curve that represents the maximal load of the structure with only two hinges<br />

- but with variable location (A) - and the curve that represents the maximal load of<br />

the structure with different amount of hinges (B) are very similar. They illustrate<br />

a relation between the position of the last hinge and the maximum load of the

maximal load [N]<br />

3000<br />

2500<br />

2000<br />

1500<br />

1000<br />

500<br />

0<br />

0 0.5 1 1.5 2 2.5<br />

x−coordinate of hinge [m]<br />

3000<br />

2500<br />

2000<br />

1500<br />

1000<br />

6.3 INTRODUCING HINGES<br />

500<br />

amount of hinges<br />

location of hinges<br />

0<br />

0 0.5 1 1.5 2 2.5<br />

x−coordinate of hinge [m]<br />

Figure 6.34: Left: The maximum load of the eleven cases with different<br />

position of the two hinges. Right: Maximum load of the cases with<br />

different position of the two hinges (A) and with different amount of<br />

hinges (B) (figure 6.32. The maximum load is plotted at the value of the<br />

x-coordinate of the (last) hinge.<br />

structure. The value of curve A is somewhat higher than that of B, because the<br />

influence of the other hinges probably decreases the maximum load at which<br />

convergence is achieved.<br />

An explanation for the relation between the position of the (last) hinge and<br />

the so-called maximal load has not been proven yet by means of experimental<br />

investigations. A hypothesis put forward here is that the maximal load -<br />

which is the highest load at which the finite element calculations converged -<br />

is reached when the membrane starts to wrinkle due to deflections. Since the<br />

hull has a lower membrane stress at lower radii of the airbeam, the hinges are<br />

less supported towards the ends. Thus, the hinges near the ends experience<br />

larger deflections.<br />

6.3.4 CONCLUSION<br />

Introducing hinges in the strut of a Tensairity beam decreases the structure’s<br />

stiffness. In the case of an O-spindle, thus without connection between upper<br />

and lower strut, are the struts mainly loaded in compression and tension, and<br />

experience very little bending. As a result, the influence of hinges is relative<br />

small here. When cables connect the upper and lower strut, large bending<br />

moments are introduced in the struts and the influence of the hinges is more<br />

noticeable.<br />

maximal load [N]<br />

B<br />

A<br />

133

134<br />

CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR<br />

In this study, the amount of hinges were varied, as well the location of two<br />

hinges along the beam’s length. Regarding the stiffness of the structure, there<br />

can be concluded that the amount of hinges has very little influence on the<br />

stiffness. The investigations also revealed that the maximum load whereby<br />

the structure is stable under a distributed line load is related to the position<br />

of the hinge that is located nearest the endpoint: the closer the hinge to this<br />

endpoint, the lower the load at which the structure will reach its maximum<br />

load.<br />

It is with this knowledge in mind that the foldable mechanism for the deployable<br />

Tensairity beam has been redesigned in chapter 3 (section 3.4, p. 52). The<br />

influence of the position of the hinges and the very low impact of the amount of<br />

hinges on the structural behaviour of the structure lead to a redesign whereby<br />

the hinges in the compressed strut were moved towards the middle, i.e. a<br />

larger distance between the last hinge and the end support was achieved. To<br />

allow the mechanism with this new arrangement of hinges to be folded to a<br />

compact configuration, two more hinges than the previous design in the upper<br />

strut are required. However, as seen in this section, this increase in amount<br />

has negligible influence on the structural behaviour. Figure 6.35 illustrates the<br />

proposed configuration for a foldable mechanism of the deployable Tensairity<br />

beam.<br />

Figure 6.35: The influence of the position of the hinges and the very<br />

low impact of the amount of hinges on the structural behaviour of the<br />

structure lead to this proposed configuration for a foldable mechanism<br />

of the deployable Tensairity beam.

6.4 CABLE CONFIGURATION<br />

6.4 CABLE CONFIGURATION<br />

The influence of cables that connect the upper and lower strut on the structural<br />

behaviour of the deployable Tensairity beam is already addressed in the<br />

previous sections. When discussing the shape of the hull, cables are mentioned<br />

as appropriate elements for connecting the upper and lower strut and thus<br />

preventing the airbeam to become a circle. With the investigations on the<br />

struts’ cross section is shown that the cables introduce point loads on the<br />

struts, resulting in bending moments and deformation of the strut. In the<br />

research on the influence of the amount of hinges on the structural behaviour<br />

were two configurations applied, they differed in the presence of cables. The<br />

comparison of the stiffness of these two configurations shows very clearly the<br />

influence of cables: the stiffness of the configuration with cables was a factor<br />

five larger than the case without cables. Thus the cables have an influence on<br />

the structural behaviour of the Tensairity beam.<br />

This was also shown by means of experimental and numerical results in<br />

chapter 4. The cable tension throughout inflation and loading is discussed<br />

there, as well the contribution of pretensioned cables to the structure’s loaddisplacement<br />

behaviour. The research, conducted on a cylindrical deployable<br />

Tensairity beam 4 , showed that as long as the cables connecting upper and<br />

lower hinges are pretensioned due to inflation, they can take compressive<br />

forces. This means that these cables can support or ‘block’ the hinges. Once<br />

the pretension in a cable is zero, the hinge is not supported anymore and the<br />

structure experiences larger displacement. This has as result that the stiffness<br />

of the cylindrical deployable Tensairity beam changes when one cable becomes<br />

slack.<br />

This section verifies first whether these conclusions also hold true for the<br />

spindle shaped Tensairity beam by means of applying the same cable configurations<br />

in the spindle as in chapter 4. Then, the influence of (other)<br />

cable configurations in the redesigned foldable mechanism are investigated<br />

(figure 6.35). But first, a closer look is taken on the behaviour of one cable in<br />

the spindle throughout inflation and loading, and its influence on the hinges<br />

it connects.<br />

4 Containing segments with a high bending sitffness in relation to the scale of the beam.<br />

135

136<br />

CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR<br />

6.4.1 CABLE TENSION<br />

As discussed several times in this dissertation, some cables become pretensioned<br />

under inflation and release tension when being compressed under<br />

external loading. The value of the tension in the cables is monitored during<br />

inflation and loading of the deployable Tensairity beam by means of finite<br />

element analysis.<br />

Figure 6.36 plots the tension in the vertical and diagonal cables of case 2<br />

during inflation (150 mbar) and distributed loading. The investigated cables<br />

are indicated in the figure. Note that all diagonal cables are connected with a<br />

hinge that is also linked with a vertical cable. The bending stiffness of the strut<br />

is in the finite element model chosen to be high in order to avoid influences of<br />

the bending of the strut on the cable tension.<br />

From the figure can be seen that all forces developed by inflation are taken<br />

by the vertical cables, which corresponds with the findings of chapter 4. The<br />

x-axis of the figure represents the time steps of the loading: the structure is<br />

being inflated between time 0 − 1 and loaded between time 1 − 2. The y-axis<br />

represents the tension in a cable.<br />

case 2<br />

cable tension [N]<br />

vertical cables<br />

1000<br />

p=150 mbar<br />

800<br />

600<br />

400<br />

200<br />

inflation<br />

1<br />

cable 1 & 5<br />

cable 2 & 4<br />

0<br />

0<br />

cable 3<br />

0.5 1 1.5 2<br />

load step (0−1 inflation, 1−2 loading)<br />

a<br />

2<br />

b<br />

3<br />

c<br />

4<br />

d<br />

5<br />

1000<br />

loading 800 inflation loading<br />

cable tension [N]<br />

600<br />

400<br />

200<br />

diagonal cables<br />

0<br />

0 0.5 1 1.5 2<br />

load step (0−1 inflation, 1−2 loading)<br />

Figure 6.36: The tension in the vertical (left) and diagonal (right) cables<br />

during inflation (0-1) and loading (1-2).

6.4 CABLE CONFIGURATION<br />

While some diagonals in the cylindrical models of chapter 4 contributed 5 to<br />

the structural behaviour under distributed loading , remain the diagonals in<br />

the case of spindle shaped deployable Tensairity beam slack throughout this<br />

distributed loading. This is remarkable, because there could be expected to<br />

have the two middle cables being loaded in tension.<br />

This can be attributed to the spindle shape of the structure: the arched<br />

compression strut moves outwards when being loaded, which results in an<br />

upwards movement of the tension element, despite the downwards loading<br />

transferred by the airbeam to the lower strut.(The struts compress thus inwards<br />

in the airbeam). Figure 6.37 illustrates this.<br />

Figure 6.37: The diagonals of the spindle shaped Tensairity beam are not<br />

loaded in tension under distributed load due to the upwards movement<br />

of the lower strut.<br />

The value of the pretension of the vertical cable after inflation can be calculated<br />

analytically with equation 4.1 from chapter 4:<br />

F = 2 × p × R × l × sinα (6.9)<br />

With p=150 mbar (15 kN<br />

m 2 ), R=0,252 m, l=0,838 m and α = 7, 5 ◦ , one obtains the<br />

analytical value of Fpre=827 N. The numerical value of the pretension in the<br />

vertical cable measures 845 N, which is close to the analytical approximation.<br />

When the deployable Tensairity beam is loaded asymmetric, the lower strut<br />

does not move upwards along the whole length of the beam as in the case of<br />

distributed loading, which results in diagonals being tensioned under loading.<br />

This can be seen in figure 6.38, plotting the cable tension of the vertical (left)<br />

and diagonal (right) cables. During inflation (loadstep from 0 till 1 on the<br />

x-axis), the cables are not tensioned, during loading (from 1 till 2 on the y-axis)<br />

become some cables tensioned.<br />

5 When loaded in tension. This depends on their position/configuration in the structure.<br />

137

138<br />

CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR<br />

cable tension [N]<br />

1000<br />

800<br />

600<br />

400<br />

200<br />

cable 1<br />

cable 2<br />

cable 3<br />

cable 4<br />

cable 5<br />

vertical cables<br />

0<br />

0 0.5 1 1.5 2<br />

load step (0−1 inflation, 1−2 loading)<br />

cable tension [N]<br />

1000<br />

800<br />

600<br />

400<br />

200<br />

diagonal cables<br />

cable a<br />

cable b<br />

cable c<br />

cable d<br />

0<br />

0 0.5 1 1.5 2<br />

load step (0−1 inflation, 1−2 loading)<br />

Figure 6.38: The cable tensions of case 2 (left) en case 7 (right) under<br />

inflation and asymmetrical load.<br />

cable tension [N]<br />

1000<br />

800<br />

600<br />

400<br />

200<br />

p = 150 mbar<br />

cable 2<br />

cable a<br />

0<br />

0 0.5 1 1.5 2<br />

load step (0−1 inflation, 1−2 loading)<br />

Figure 6.39: The interaction between vertical and diagonal cables under<br />

asymmetrical load: once the vertical cables becomes slack, the tensioned<br />

diagonal is being compressed.(load step from 1 till 2 : 0.1 step represents<br />

500 N external loading)<br />

The interesting interplay between the vertical and diagonal cables can be seen<br />

in this configuration under asymmetrical loading. Figure 6.39 illustrates this<br />

by plotting the cable tension of cable 2 and A in relation to the loading (load step<br />

from 1 till 2 : 0.1 step represents 500 N external loading). From the figure can be<br />

seen that the pretension of the vertical cable decreases under external loading.<br />

At the same time is the diagonal being tensioned, as one can expect under the<br />

asymmetrical loading (it prevents the mesh to become parallelogram) . Once<br />

the vertical cable has become slack, the forces in the diagonal changes and it<br />

becomes compressed. Figure 6.40 shows the deformed shape of the structure<br />

under asymmetrical load with the cables 2 and A indicated.

a<br />

2<br />

6.4 CABLE CONFIGURATION<br />

Figure 6.40: The deformed shape of the structure under asymmetrical<br />

load with the cables 2 and A indicated. (The deformation is scaled by a<br />

factor 2.)<br />

As discussed in chapter 4 is the pretension of the cables related with the internal<br />

pressure: the higher the internal pressure of the airbeam, the more the cables<br />

are pretensioned. As a result, the cables can support more compressive forces.<br />

Figure 6.41 shows the tension in the central vertical cable of configuration 2 at<br />

50, 150 and 250 mbar. Notice that the reduction in prestress is proportional to<br />

the applied external load and independent of the cable’s prestress.<br />

cable tension [N]<br />

1500<br />

1000<br />

500<br />

50 mbar 150 mbar 250 mbar<br />

inflation loading<br />

0<br />

0 0.5 1 1.5 2<br />

load step (0−1 inflation, 1−2 loading)<br />

Figure 6.41: The tension in cable 3 of case 2 , at 50, 150 and 250 mbar.<br />

The higher the internal pressure, the higher the prestress. The reduction<br />

in prestress is proportional to the applied external load and independent<br />

of the prestress.<br />

6.4.2 STIFFNESS<br />

In chapter 4 is shown that the stiffness of the structure is related with the<br />

behaviour of the cables in the structure. Once a cable becomes slack, it can<br />

no longer support or ‘block’ the hinge it connects. As a consequence, larger<br />

displacements occur and the stiffness of the structure changes.<br />

139

140<br />

CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR<br />

This is also the case for a spindle shaped deployable Tensairity structure.<br />

Figure 6.42 shows on the left the load-displacement behaviour of the asymmetrical<br />

loaded case 2. On the right is the cable tension plotted of cable 2 in<br />

relation with the applied load. Note that for this graph, the cable tension is<br />

represented on the x-axis and the applied load on the y-axis. It shows that<br />

the cable tension decreases with increasing load (as discussed before), until it<br />

reaches zero. On the left graph can be seen that this is exactly the same load<br />

whereby the stiffness of the structure changes.<br />

load [N]<br />

3500<br />

3000<br />

2500<br />

2000<br />

1500<br />

1000<br />

500<br />

load−displacement<br />

0<br />

0 0.01 0.02 0.03 0.04<br />

average displacement [m]<br />

load [N]<br />

3500<br />

3000<br />

2500<br />

2000<br />

1500<br />

1000<br />

500<br />

cable tension<br />

0<br />

0 200 400 600 800 1000<br />

cable tension [N]<br />

Figure 6.42: The influence of the cable tension on the structure’s stiffness.<br />

Left: the load-displacement behaviour of the asymmetrical loaded case<br />

2 (pressure 150 mbar). Right: the cable tension of cable 2 throughtout<br />

loading.<br />

To investigate the influence of the pressure on the stiffness of the structure<br />

is the average displacement noted of case 2 under 50, 150 and 250 mbar.<br />

Figure 6.43 illustrates this. From this figure can be seen that the conclusions<br />

from chapter 4 also hold true for the spindle shaped deployable Tensairity<br />

beam. As long as all vertical cables are pretensioned, is the stiffness of the<br />

structure independent of the pressure (but influenced by the stiffness of the<br />

cables). Because the cables in the case with a higher internal pressure are more<br />

pretensioned, they experience the zero pretension in a cable at a higher load.<br />

Therefore, the change in stiffness occurs at a higher load in the case of a higher<br />

internal pressure, as can be seen in figure 6.43.<br />

6.4.3 CABLE CONFIGURATION<br />

The influence of various cable configurations on the structural behaviour<br />

of the spindle shaped deployable Tensairity beam are investigated. Fig-

load [N]<br />

5000<br />

4000<br />

3000<br />

2000<br />

1000<br />

50 mbar<br />

150 mbar<br />

250 mbar<br />

0<br />

0 0.02 0.04 0.06 0.08<br />

average displacement [m]<br />

6.4 CABLE CONFIGURATION<br />

Figure 6.43: As long as all cables are pretensioned, is the stiffness<br />

independent of the pressure. The load at which the stiffness changes<br />

is related with the internal pressure.<br />

ure 6.44illustrates the cases. First, the cases without vertical cables are discussed<br />

(cases 2–5), then the cases with verticals ( cases 6–10).<br />

2<br />

3<br />

4<br />

5<br />

Cases 2 – 5<br />

6<br />

Figure 6.44: Various cable configurations are investigated numerically.<br />

The tension in the cables of cases 2, 3, 4 and 5 is investigated numerically,<br />

as well the load-bearing behaviour of the cases. All the diagonal cables<br />

become pretensioned under inflation, just as the cylindrical cases 2 till 5 in<br />

chapter 4. However, when loading the spindle shaped beam, all the diagonals<br />

become compressed, regardless their configuration. This is different from the<br />

cylindrical models and can be attributed to the spindle shape, as illustrated<br />

already in figure 6.37.<br />

7<br />

8<br />

9<br />

10<br />

141

142<br />

CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR<br />

Figure 6.45 illustrates the load-displacement behaviour of the four cases. An<br />

explanation for the difference in stiffness between the cases does not lie so<br />

explicit in the shape of the ‘mesh’ or in cables becoming more tensioned or<br />

compressed, as was the case with the cylindrical models. The structure’s<br />

stiffness is influenced here by the deformation of the cases under inflation.<br />

Indeed, because the hull of the airbeam is designed such to have an outwards<br />

action on the cables (see section 6.1), the hinges that are not connected with<br />

a cable move outwards and deform the spindle shape. Figure 6.46 illustrates<br />

the four cases after inflation.<br />

load [N]<br />

6000<br />

5000<br />

4000<br />

3000<br />

distributed load − pressure 150 mbar<br />

2000<br />

case 2<br />

1000<br />

case 3<br />

case 4<br />

case 5<br />

0<br />

0 0.005 0.01 0.015 0.02<br />

average displacement [m]<br />

Figure 6.45: The load-displacement behaviour of cases 2, 3, 4 and 5.<br />

2<br />

3<br />

Figure 6.46: The hinges not connected with a cable will move outwards<br />

when inflating the structure. The cable configuration determines the<br />

deformed shape after inflation. The displacements are scaled with a<br />

factor 5.<br />

When loading the deformed spindle beams uniformly, their upper strut is<br />

compressed and deforms the shape even more. After all, the hinges moving<br />

outwards are then pushed further outwards and vice versa. Thus, the cable<br />

configuration determines the deformed shape after inflation and as conse-<br />

4<br />

5

6.4 CABLE CONFIGURATION<br />

quence the structure’s stiffness. As can be seen in figure 6.46 is case 3 the<br />

most optimal, since the three successive middle hinges are connected with<br />

a cable, which prevents a zig-zag shape of the upper strut as in the case of<br />

configurations 2 and 4.<br />

Cases 6 – 10<br />

The stiffness of cases 6, 7, 8, 9 and 10, inflated with an internal pressure of<br />

150 mbar and loaded distributed, is plotted in figure 6.47. The influence of<br />

the vertical pretensioned cables can clearly be seen: all cases have exactly the<br />

same stiffness and load whereby the first (vertical) cable becomes slack (the<br />

‘maximum load’). Note that the diagonals are not contributing to the structural<br />

behaviour: they are not pretensioned (due to the presence of the vertical cables<br />

in the same hinges) and because they are all loaded in compression when the<br />

external load is applied.<br />

load [N]<br />

12000<br />

10000<br />

8000<br />

6000<br />

distributed load − pressure 150 mbar<br />

0 0.5 1 1.5 2 2.5 3<br />

x 10 −3<br />

4000<br />

case 6<br />

case 7<br />

case 8<br />

2000<br />

case 9<br />

case 10<br />

0<br />

average displacement [m]<br />

Figure 6.47: The load-displacement behaviour of cases 6, 7, 8, 9 and 10.<br />

The influence of the vertical pretensioned cables can clearly be seen: all<br />

cases have exactly the same stiffness.<br />

In the case of asymmetrical loading are some diagonals contributing to the<br />

structural behaviour of the deployable Tensairity beam. Numerical investigation<br />

showed that the cables one can expect to work in tension during loading,<br />

do contribute to the stiffness of the structure. Figure 6.48 illustrates the cables<br />

in tension and compression(and thus slack) during asymmetrical loading.<br />

Figure 6.49 shows the load-displacement behaviour of the cases under asymmetrical<br />

loading. It shows that case 6 has the lowest stiffness, which can<br />

be expected since this case has no diagonals to contribute to the structural<br />

143

144<br />

CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR<br />

7<br />

8<br />

+<br />

tension<br />

+ +<br />

+ +<br />

o o<br />

o slack<br />

o<br />

9<br />

o 10 o o<br />

o<br />

+ o +<br />

Figure 6.48: The cables in tension and compression(and thus slack)<br />

during asymmetrical loading.<br />

behaviour. Case 10 has the largest stiffness, followed by case 8. Both cases<br />

have two successive ‘meshes’ of the truss with the shape of a triangle. At least<br />

two successive hinges are stabilized here by a tensioned cable, which prevents<br />

‘zig-zag’ deformation of the upper strut. Figure 6.50 illustrates the deformed<br />

spindles under asymmetrical loading.<br />

load [N]<br />

4000<br />

3500<br />

3000<br />

2500<br />

2000<br />

asymmetric load − pressure 150 mbar<br />

1500<br />

case 6<br />

1000<br />

case 7<br />

case 8<br />

500<br />

case 9<br />

case 10<br />

0<br />

0 0.01 0.02 0.03 0.04<br />

average displacement [m]<br />

Figure 6.49: The load-displacement behaviour of the cases 6 to 10 under<br />

asymmetrical loading. (internal pressure measures 150 mbar).<br />

6.4.4 PROTOTYPE CONFIGURATIONS<br />

The prototype for a deployable Tensairity beam has a different hinge configuration<br />

and thus folding mechanism than the cases discussed so far in this<br />

section. In chapter 3, where this configuration for a mechanism is discussed,<br />

is shown that not all cable connections between upper and lower hinges allow<br />

the structure to be folded. Figure 6.51 illustrates the cable configurations that<br />

allow the structure to be folded (left) and the cables that would obstruct this<br />

folding (right).<br />

+<br />

+

6<br />

7<br />

8<br />

9<br />

10<br />

6.4 CABLE CONFIGURATION<br />

Figure 6.50: The deformed spindles under asymmetrical loading. (The<br />

displacements are scaled.)<br />

In this section, three different cable configurations are studied numerically.<br />

These cases will also be investigated experimentally in chapter 8. Although<br />

they don’t allow folding of the structure, two of the three cases contain vertical<br />

cables connecting upper and lower strut. The third case contains only diagonal<br />

cables. Figure 6.52 illustrates the three cases: A, B and D. By comparing case<br />

A and B, the influence on the diagonals will be discussed. Comparing B and<br />

D will allow us to evaluate the effect of the vertical cables.<br />

Diagonal cables<br />

Figure 6.53 shows the numerical load-displacement behaviour of cases A and<br />

B under distributed (left) and asymmetrical load (right). From the figure can<br />

be seen that case B is stiffer in both load cases. The diagonals thus contribute<br />

to the structure’s stiffness.<br />

In section 6.4.3 is seen that - in the configurations with diagonal and vertical<br />

cables - the diagonals are not pretensioned under inflation and thus not<br />

145

146<br />

CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR<br />

links allowing folding<br />

links obstructing folding<br />

Figure 6.51: The cable configurations that allow the structure to be folded<br />

(left) and the cables that would obstruct this folding (right).<br />

A<br />

B<br />

D<br />

Figure 6.52: Three cable configurations of the prototype: A, B and D.<br />

contribute to the structural behaviour under distributed loading. This is<br />

because the outwards movement by the inflation is in those cases taken entirely<br />

by the vertical cables, which are present in every hinge. However, in the case<br />

of configuration B are not all hinges connected with a vertical cable. (These<br />

are the hinges on the lower strut closest to the supports). This has as a result<br />

that the diagonals connecting these hinges - cables a, d, e and f (indicated<br />

in figure 6.52) - are pretensioned. As a consequence, these cables can take a<br />

certain amount of compressive forces when the structure is loaded. Because<br />

they form ‘triangular meshes’ in the truss and support an inwards rotating<br />

hinge, they contribute to the stiffness.<br />

As one can expected are these four diagonal cables also contributing to the<br />

stiffness of configuration B under asymmetrical loading.<br />

Vertical cables<br />

Cases B and D differ in the presence of vertical cables. As seen before in<br />

section 6.4.3 and illustrated in figure 6.46 are the cases 2–5 deformed under

load [N]<br />

4500<br />

4000<br />

3500<br />

3000<br />

2500<br />

2000<br />

1500<br />

distributed loading − pressure 150 mbar<br />

1000<br />

500<br />

case A<br />

case B<br />

0<br />

0 0.01 0.02 0.03 0.04<br />

average displacement [m]<br />

load [N]<br />

4000<br />

3500<br />

3000<br />

2500<br />

2000<br />

1500<br />

6.4 CABLE CONFIGURATION<br />

asymmetric loading − pressure 150 mbar<br />

4500<br />

1000<br />

500<br />

case A<br />

case B<br />

0<br />

0 0.01 0.02 0.03 0.04<br />

average displacement [m]<br />

Figure 6.53: Load-displacement behaviour of cases A and B under<br />

distributed (left) and asymmetrical load (right) (pressure 150 mbar).<br />

inflation, due to a lack of vertical cables preventing the free hinges to move<br />

outwards. This deformation under inflation can also be seen in the case of<br />

configuration D, shown in figure 6.54.<br />

Figure 6.54: Due to a lack of vertical cables preventing the free hinges<br />

to move outwards is case D deformed after inflation. (No scaling of<br />

displacements. Pressure 150 mbar).<br />

By analogy with cases 2–5, one can expect case D to have a low stiffness.<br />

After all, when the upper strut is compressed under loading, the hinges will<br />

rotate further and deform the upper strut. The load-displacement behaviour<br />

of case D is plotted in relation with cases A and B under distributed load in<br />

figure 6.55. The lack of cables preventing all hinges to go outwards decreases<br />

thus the structure’s stiffness.<br />

6.4.5 CONCLUSION<br />

This section investigated numerically the influence of cables and their configuration<br />

on the structural behaviour of the deployable Tensairity beam.<br />

The study revealed that the pretensioned cables contribute to the structure’s<br />

stiffness by supporting the hinges (until their pretension becomes zero) and<br />

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CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR<br />

load [N]<br />

4500<br />

4000<br />

3500<br />

3000<br />

2500<br />

2000<br />

1500<br />

distributed load − pressure 150 mbar<br />

1000<br />

case A<br />

500<br />

case B<br />

case D<br />

0<br />

0 0.01 0.02 0.03 0.04<br />

average displacement [m]<br />

Figure 6.55: The load-displacement behaviour of case D in relation with<br />

cases A and B under distributed load and a pressure of 150 mbar.<br />

by preventing the structure to deform under inflation.<br />

When not every hinge is ‘blocked’ by a cable from moving outwards, the strut<br />

experiences a deformation which results in a lower stiffness of the structure<br />

than in the case whereby every hinge is connected with a cable.<br />

In the case where all hinges are supported by a cable, is the stiffness of<br />

the structure independent of the internal pressure, as long as all cables are<br />

pretensioned. Once a cable becomes slack, the stiffness changes. The value<br />

whereby this stiffness changes, is related with the pretension of the cable and<br />

thus the internal pressure.<br />

The investigation of the cable configurations and the cable tension showed that<br />

diagonal cables are not pretensioned under inflation when they connect hinges<br />

that are already linked with vertical cables. This has as a consequence that<br />

only the vertical cables are in that case contributing to the structure’s stiffness<br />

under distributed load. Under asymmetrical loading on the other hand, are<br />

the diagonal cables (loaded in tension) contributing. Diagonal cables do have<br />

a positive influence on the structure’s stiffness under distributed loading when<br />

they are not already linked with vertical cables and thus pretensioned.

6.5 CONCLUSIONS<br />

6.5 CONCLUSIONS<br />

As the introduction of a deployment mechanism changes the structural<br />

behaviour of a traditional Tensairity structure, it is one of the objectives of<br />

this research to explore the general behaviour of the deployable Tensairity<br />

beam. This is done in this chapter by means of numerical investigations. More<br />

specifically, the influence of various design parameters on the beams structural<br />

behaviour are investigated.<br />

The hull section of the Tensairity beam defines the tension forces present in the<br />

connection between upper and lower strut. Numerical investigations revealed<br />

that the shape of the hull of the Tensairity beam whereby the web experiences<br />

a constant stress along the beam’s length, uses the material more optimal and<br />

has the largest average stiffness. This can be clarified by the fact that a higher<br />

stress in the web near the ends supports the struts and hinges up to a higher<br />

load.<br />

The influence of the struts section was shown to be related to the occurrence<br />

or absence of bending moments in the struts. The strut’s cross section proved<br />

to have no influence in the cases without bending moment in the struts. When<br />

bending moments are introduced (mostly by cables), however, the bending<br />

stiffness, and thus the cross section, plays an import role on the stiffness of<br />

the deployable Tensairity beam. For cases with large deflections of the strut,<br />

second order effects occurred and a large decrease in stiffness was detected. As<br />

a result, the struts should be dimensioned to avoid these second order effects.<br />

Finite element calculations showed that introducing hinges in the Tensairity<br />

technology decreases the structure’s stiffness. The influence of the hinges is<br />

relatively small when no bending moment occurs in the struts. However,<br />

when large bending moments are introduced in the strut (eg. by means of<br />

cables), the influence of the hinges on the deflection is more noticeable. A<br />

variation in the amount of hinges showed little influence on the structure’s<br />

stiffness. Furthermore, the numerical experiments revealed that the location<br />

of the hinge along the beam determines the maximal load of the structure: the<br />

closer the hinge is to the endpoint of the beam, the lower the load will be at<br />

which no convergence of the finite element calculation is reached. Hinges at<br />

the ends of the beam should thus be avoided as they are less stabilized by the<br />

membrane.<br />

The numerical investigations revealed the influence of cables and their configuration<br />

on the structural behaviour of the deployable Tensairity beam. The<br />

149

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CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR<br />

pretensioned cables contribute to the structure’s stiffness (until their pretension<br />

becomes zero) by supporting the struts and hinges. As long as all cables<br />

are pretensioned is the stiffness of the structure independent of the internal<br />

pressure. Once a cable becomes slack, the stiffness changes. The maximum<br />

load of the structure is thus related with the internal pressure.<br />

Furthermore, the numerical investigations showed the importance of connecting<br />

all the hinges of the upper strut with the lower strut by means of cables.<br />

This keeps the hinges in position when inflating the airbeam and makes the<br />

struts maintain their spindle shape. A deformation of the strut under inflation<br />

decreases the structure’s stiffness considerably. The investigation of the cable<br />

configurations and the cable tension also showed that diagonal cables are<br />

not pretensioned under inflation when they connect hinges that are already<br />

linked with vertical cables. This has as a consequence that only the vertical<br />

cables are in that case contributing to the structure’s stiffness under distributed<br />

load. Under asymmetrical loading on the other hand, are the diagonal cables<br />

(loaded in tension) contributing. Diagonal cables do have a positive influence<br />

on the structure’s stiffness under distributed loading when they are not already<br />

linked with vertical cables and thus being pretensioned.<br />

6.6 DISCUSSION<br />

Although discussed separately in this chapter, the various design parameters<br />

are related and influenced by each other. The shape of the hull section<br />

determines together with the internal pressure the load by which the struts<br />

are pushed outwards and thus the pretension in the cables that prevent this<br />

outwards movement of the upper and lower strut. Because these pretensioned<br />

cables can take compressive forces, they are able to block the hinges. The larger<br />

the pretension in the cables, the higher the external load at which the hinge<br />

will cause large displacements. As a consequence, one can conclude that an<br />

increase in cable tension, caused by the outwards movement of the hull under<br />

inflation, increases the structural performance.<br />

However, the segments of the struts are also pushed outwards. This introduces<br />

bending moments in these strut, which deform the segments of the strut. Their<br />

deformation is thus related with the internal pressure. In the investigation<br />

is shown that the stiffness of the Tensairity beam decreases with increasing<br />

deformation of the strut. Thus, one can conclude from this point of view that<br />

the increase of internal pressure has a negative influence on the structural<br />

behaviour.

6.6 DISCUSSION<br />

Two opposite influences of the internal pressure can thus be concluded: one<br />

has a stabilizing function, i.e. the cable pretension, while the other destabilizes<br />

or weakens the structure, i.e. the deflection of the strut. From the numerical<br />

investigations seems that the destabilizing effect has the upper hand, while<br />

experimental results (see chapter 8 prove that the pressure has a positive<br />

influence on the structural behaviour. Because numerical simulations reflect<br />

the ideal situation (no manufacturing imperfections) and the experimental<br />

investigations the reality, more reliability is given here to the stabilizing effect<br />

of the internal pressure and the cables. It is possible that this destabilizing<br />

effect will become more pronounces for experiments under higher pressures.<br />

These stabilizing and destabilizing effects need to be investigated more in<br />

detail in further studies to be understood better and to be able to calculate<br />

and predict which effect dominates in which situation (pressure, shape of hull<br />

section, bending stiffness, ... ).<br />

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CHAPTER 6 ANALYSIS OF THE STRUCTURAL BEHAVIOUR

PART III<br />

Prototype of a Deployable Tensairity Beam<br />

153

Prototype and experimental set-up<br />

7<br />

In the previous two parts, deployable Tensairity structures were developed<br />

and investigated by means of small scale models or numerical investigations.<br />

This was in each case an approximation of how the structure would behave<br />

in reality. In this part, a prototype for the deployable Tensairity structure is<br />

developed and investigated on a larger scale. This way, insight will be gained<br />

in the behaviour of real deployable Tensairity structures, manufactured with<br />

the same materials and in the same way as real structures, and loaded with<br />

representative loads.<br />

The results of the experimental investigations are presented in the next chapter.<br />

This chapter presents the prototype and the components it is constituted of in<br />

detail, as well the way this prototype is investigated.<br />

7.1 PROTOTYPE OF <strong>TENSAIRITY</strong> BEAM<br />

The prototype of the five meter spindle shaped deployable Tensairity beam<br />

is illustrated in figure 7.1. It has a slenderness of 10 and is constituted of<br />

an airbeam with two airtight chambers, struts, hinges and cables. Because<br />

it is one of the objectives of the experimental investigations to compare the<br />

structural behaviour of the deployable Tensairity beam with the behaviour of<br />

other already existing non-deployable Tensairity prototypes, the design of the<br />

prototype and thus the various components will take the specifications of these<br />

other prototypes into account. Examples are the shape of the hull, the position<br />

of the cables and the strut’s section. This way, the prototypes are comparable<br />

and allow to investigate the influence of one parameter, such as the effect of<br />

hinges on the structure. This is discussed more in detail in chapters 8 and 9.<br />

The various components are discussed in detail in the next sections.<br />

155

156<br />

CHAPTER 7 PROTOTYPE AND EXPERIMENTAL SET-UP<br />

Figure 7.1: The five meter spindle prototype for a deployable Tensairity<br />

beam.<br />

7.1.1 MEMBRANE<br />

The airbeam is in this prototype made of PVC coated polyester fabric (Valmex<br />

7318) 1 . This technical membrane is widely used in the field of inflatable<br />

structures and very likely to be applied for Tensairity structures. This is<br />

because among other types of fabric, the PVC coated polyester fabric is relative<br />

inexpensive and easily weldable. In addition, the polyester fibres are not brittle<br />

(which is the case with PTFE coated glass fibres), which results in a membrane<br />

that can be folded repeatedly without damage.<br />

The spindle shaped airbeam is constituted of two separate airtight chambers,<br />

adjacent to each other along the spindle’s length. This way, the upper and<br />

lower strut can be connected by cables without risk for air leakages. Moreover,<br />

holes can be made to accommodate the hinges in such a way that the membrane<br />

lies in the same plane as the axis of rotation of the hinges 2 . This is illustrated<br />

in figure 7.2.<br />

The shape of the hull is chosen such that the normal force in the connecting<br />

1 kN Producer: Mehler Texnologies, warp and fill tensile strength: 60 m , thickness: 0.85 mm,<br />

Ewarp = 904 kN m , E f ill = 452 kN m , ν = 0.386<br />

2In section 3.2 was seen that this is a requirement to guarantee a correct folding of the membrane<br />

without stretching or wrinkling. Figure 3.3 on page 38 illustrates this.

1<br />

2<br />

3<br />

4<br />

7.1 PROTOTYPE OF <strong>TENSAIRITY</strong> BEAM<br />

1 keder<br />

2 strut<br />

3 web for airtight<br />

chambers - no stress<br />

4 outer hull<br />

4<br />

50mm weld<br />

Figure 7.2: The section of the airbeam: two separate airtight airbeams<br />

are positioned adjacent to each other along the spindle’s length.<br />

element between upper and lower strut is identical along the spindle’s length.<br />

As explained in section 6.1, this implies that the radius is dependent of the<br />

normal force desired in the connecting element at a certain pressure and the<br />

height between the upper and lower strut.<br />

In addition, to fold the Tensairity structure completely, the hull of the membrane<br />

should allow an increase in length between upper and lower strut<br />

when folding. Remember, as seen in section 3.4.4 (p. 57), some points of<br />

the upper strut move away from the lower strut when folding the foldable<br />

truss mechanism.<br />

However, this latter requirement was not taken into account for the cutting<br />

pattern for the prototype. There is chosen consciously to apply the same shape<br />

of hull of other ongoing experiments on five meter Tensairity spindles. This<br />

way, the experimental results of this prototype for a deployable Tensairity<br />

beam are comparable with other investigations. Unfortunately, this implies<br />

that the prototype will not be completely foldable.<br />

The cutting pattern is developed by calculating at several positions along the<br />

spindle’s length the circumference of one quarter of the hull. This is done by<br />

means of equation 6.1 (p. 106). This length is then plotted along the spindle’s<br />

curvature forming the cutting pattern, as illustrated in figure 7.3. The pattern<br />

157

158<br />

CHAPTER 7 PROTOTYPE AND EXPERIMENTAL SET-UP<br />

of the two identical internal membranes closing the air chamber, called webs,<br />

is more simple. They should be wider than the height between the upper<br />

and lower struts. After all, these webs are used to make the two chambers<br />

airtight and will thus not bear any tensile forces. Once the cutting patterns are<br />

developed, the welding lines and markers are added.<br />

496<br />

15 mm welding area for connecting two halves and keder<br />

4726<br />

50 mm weld area for connecting two quarters<br />

Figure 7.3: Development of the cutting pattern of a quarter of the spindle.<br />

To obtain an airtight airbeam, capable of taking the stresses introduced by<br />

the internal pressure, attention has to be paid to the welding procedure and<br />

sequence. From a manufacturing point of view, the welding should occur in<br />

a straightforward and logic way, whereby the welding is not obstructed by<br />

other parts of the airbeam. The sequence of welding the cutting patterns is<br />

illustrated in figure 7.4 and described here briefly. First, the quarters from one<br />

chamber are welded together with an overlap of 5 cm (A). Then, the membrane<br />

closing the air chamber (web) is welded on the inside of the hull (B). Finally,<br />

the two halves are welded to a keder and each other (C). Notice that this<br />

connection contains a weld that is perpendicular to the direction of membrane<br />

stress. As a consequence, peeling of the membrane is risked here. However,<br />

this weld and the keder will be clammed between the two struts. This way,<br />

the tensile forces from the outer hull are taken by the struts and not the weld.<br />

The outer hull of the airbeam contains two valves each chamber. This way,<br />

the pressure can be monitored when inflating the airbeam. The hull is also<br />

printed with a speckle pattern for measuring purposes, as will be mentioned<br />

further in this chapter.<br />

7.1.2 HINGES<br />

As already discussed in detail in section 3.4.2 (p. 53), two types of hinges<br />

exist, the end supports not taken into account. One type rotates towards the

A<br />

C<br />

Figure 7.4: Sequence of welding the cutting patterns.<br />

7.1 PROTOTYPE OF <strong>TENSAIRITY</strong> BEAM<br />

air beam, the other outwards. Because the membrane has to lie in the same<br />

plane as the center of rotation of the hinge, this has as consequence that each<br />

type requires another design. Otherwise, the membrane will be stretched or<br />

wrinkled. Figure 7.5 and 7.6 shows both hinges in combination with the<br />

membrane.<br />

Figure 7.5: Geometry of the two hinges in combination with the<br />

membrane.<br />

The hinges are first modeled in such a way that they can be fixed around the<br />

end of a segment. This way, no tooling of the segments are necessary and<br />

an easy connection can be achieved. This design is illustrated in figure 3.22<br />

(p. 55). However, this design is very complex and is as consequence difficult<br />

to manufacture and thus relative expensive. Therefore, some design iterations<br />

B<br />

159

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CHAPTER 7 PROTOTYPE AND EXPERIMENTAL SET-UP<br />

Figure 7.6: The folding sequence of the two hinges.<br />

were conducted. The kinematic of the hinge is identical to the first version, only<br />

the materialization differs. The final design is not a solid three dimensional<br />

hinge, but composed of several flat plates. The main advantage of this design<br />

is a simplified, faster and thus less expensive manufacturing 3 . The new design<br />

requires the struts to be manufactured such that the plates can be fixed to it.<br />

Figure 7.7 gives a more detailed view of the hinges. As can be seen, the hinges<br />

are composed of flat plates, manufactured by CNC in aluminum, and steel<br />

pins, that form the axes of rotation. The middle plates in hinge 02 keep the two<br />

axes at a constant distance and limit the rotation of the hinge to 180 degrees.<br />

This latter is achieved at hinge 01 by means of the geometry of the flat plates<br />

themselves.<br />

In the figure can also be seen that holes are made in the steel pins to accommodate<br />

the cables that connect upper and lower strut. This way, the cables can<br />

rotate and their normal forces are transferred directly to the hinges. Note that<br />

the holes in the three pins of hinge 02 are shifted from each other such that the<br />

ends of the cables are adjacent and do not obstruct each other. Because of the<br />

cable ends are positioned next to each other, enough space has to be provided.<br />

As a consequence, the outer plates of hinge 02 are fixed to the sides of the strut.<br />

This way, the forces need to be taken up by the connection between the hinge<br />

and the strut, which is not optimal.<br />

7.1.3 STRUTS<br />

Segments<br />

The upper and lower strut of the Tensairity beam consist of several aluminum<br />

segments, connected with hinges. The cross section of the segments is chosen<br />

3 The hinge constituted of plates costs 4,5 times less than the solid hinge.

1<br />

2<br />

1 2<br />

Figure 7.7: Detailed view of the two hinges.<br />

7.1 PROTOTYPE OF <strong>TENSAIRITY</strong> BEAM<br />

161

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CHAPTER 7 PROTOTYPE AND EXPERIMENTAL SET-UP<br />

to be identical to ongoing experimental investigations on Tensairity structures<br />

with continuous struts, obvious for reasons of comparison. Their section<br />

measures 40 mm wide and 15 mm high.<br />

As mentioned in the introduction on Tensairity structures, it is important<br />

for the stabilization of the compression element to have a good connection<br />

between this compression element and the hull of the airbeam. From previous<br />

experiments is learned that connecting the compression element to the hull by<br />

means of a pocket (as with tent poles of an iglo-tent) is not an optimal solution<br />

because play between the elements still exist and decreases the structure’s<br />

stiffness considerable. Therefore is chosen to clamp the membrane and a keder<br />

between the struts. These struts are bolted together at regular distances. This<br />

is illustrated in figure 7.8. This way, a tight connection between the membrane<br />

and the struts is realized. The membrane stress is taken up by the struts, the<br />

keder prevents the membrane from slipping from the clamped struts.<br />

1<br />

2<br />

3<br />

1 keder<br />

2 strut<br />

3 web for airtight<br />

chambers - no stress<br />

4 outer hull<br />

4<br />

Figure 7.8: Detail of the connection between strut and membrane.<br />

The length of the segments is dictated from the foldable truss mechanism,<br />

illustrated in figure 7.9 and discussed in chapter 3. The mechanism was<br />

initially constituted of segments with three different sizes. The segments of<br />

the lower strut were all 1<br />

6th of the struts length (L), the middle upper segments<br />

measured each L<br />

12<br />

and the two side segments were L<br />

4<br />

long. However, because<br />

of reasons of comparison with another Tensairity prototype, there is chosen<br />

to align the second lower hinge with the second upper hinge, indicated in<br />

figure 7.9. Therefore, the mechanism is adapted, which has as consequence<br />

that a larger variation of length is needed.<br />

The segments are manufactured such to accommodate the hinges, that are<br />

composed of plates. In the case of hinge 01, material of the segment is milled

D<br />

A B C B B C B<br />

E<br />

F<br />

F<br />

7.1 PROTOTYPE OF <strong>TENSAIRITY</strong> BEAM<br />

Figure 7.9: The geometry of the segments is dictated by the foldable<br />

truss mechanism.<br />

away to fit the plates. This is not necessary in the case of hinge 02, since the<br />

plates are attached on the sides of the segments. Off course, holes are provided<br />

to connect both elements together. Figure 7.10 gives a detailed view of two<br />

segments.<br />

From straight to curved<br />

Figure 7.10: A detailed view of two segments.<br />

In the case of a regular Tensairity beam, a straight and continuous aluminum<br />

strut is used as compression element. The section of the strut is such that the<br />

bending stiffness allows it to be curved along the spindle when being attached<br />

to the airbeam and the supports. With the struts of the deployable Tensairity<br />

beam, this bending will not occur due to the presence of the hinges. Indeed,<br />

these hinges can not transfer bending moments. But because these hinges<br />

can rotate the struts along the curve of the spindle and because the maximum<br />

angle between the rotated struts would be less than 4 degrees, there is chosen to<br />

design as well the upper and lower strut of the prototype as straight elements.<br />

The struts are illustrated in figure 7.11.<br />

However, when inflating the prototype for the deployable Tensairity beam,<br />

the straight segments cause problems. After all, because the straight segments<br />

E<br />

A<br />

D<br />

163

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CHAPTER 7 PROTOTYPE AND EXPERIMENTAL SET-UP<br />

Figure 7.11: The upper and lower strut of the prototype are conceived<br />

as straight struts, as is done with regular Tensairity beams. Half of the<br />

upper and lower strut is shown.<br />

are positioned along a curve, they are not perfectly in line with each other<br />

but already angulated. Note that the direction of rotation is fixed, because the<br />

hinges are only allowed to rotate in one direction.<br />

When loading this Tensairity beam with a uniform distributed load, the upper<br />

strut becomes compressed. As a result, the segments rotate further. This means<br />

that the external loading is not fully transferred to an increase in compression<br />

in the struts themselves. Instead, the loading is also partially taken by a<br />

change in geometry of the segments and by the airbeam where some hinges<br />

are pressed into.<br />

The prototype with these straight and thus kinked segments is investigated<br />

experimentally. Figure 7.12 shows the prototype with straight segments under<br />

inflation. Notice the kinks in the struts. Because of the initial compression in<br />

the struts after inflation, it is clear that - even without external loading - the<br />

hinges are already loaded considerable. As a result, the connection between a<br />

hinge and strut failed after some loadings.<br />

Figure 7.12: The prototype with straight segments under inflation. The<br />

kinks are very visible.<br />

Off course, for decent experimental investigations and reliable results that<br />

represent well the structural behaviour of the deployable Tensairity beam, the<br />

prototype needs to be adapted. The aluminum struts are bended according<br />

the spindle’s curve by means of a three-point bending: a deflection is applied<br />

between two points of the strut and this several times along this strut. The<br />

deflection is such that the segment is plastically deformed and that residual<br />

deformation, also called sag, corresponds with a predefined value. When

7.1 PROTOTYPE OF <strong>TENSAIRITY</strong> BEAM<br />

the strut is bended with this value at regular distances, the curvature is well<br />

approximated. To give an idea of the curvature: with a distance of 30 cm<br />

between the supports, the sag has to measure 0,8911 cm.<br />

The bended segments are repositioned along the membrane and keder. Where<br />

necessary, new holes in the membrane are made one or two millimeters next<br />

to the old ones. Through these holes, the bolts are placed that clamp the two<br />

halves of the strut together with the membrane and keder. After inflation, the<br />

result is a spindle shaped Tensairity beam with unstressed, compatible, curved<br />

struts , as can be seen in figure 7.13<br />

7.1.4 CABLES<br />

Figure 7.13: After bending and repositioning the struts is a spindle shape<br />

achieved with unstressed, compatible, curved struts.<br />

A connection between upper and lower strut is necessary in this prototype for<br />

multiple reasons. Without connection, the struts are moved outwards under<br />

inflation and the section of the inflatable volume tends to become a circle. This<br />

implies that, as discussed in section 7.1.1, not enough membrane would be<br />

present to allow the hinges of the mechanism to move away from each. As a<br />

result, the membrane can not fold completely. Another reason is that such a<br />

connection between upper and lower strut increases the structure’s stiffness,<br />

as shown in section 6.1. Because a membrane as connecting element between<br />

the struts would obstruct the folding, cables are applied for this prototype.<br />

The cables are stainless steel ropes with diameter of 5 mm, containing (M6)<br />

threaded ends. They fit through the holes of the pins of the hinges, as described<br />

before (figure 7.14). This way, they can rotate free and are connected directly<br />

with the hinges. By rotating the nut, the length of the cable can be adjusted.<br />

Figure 7.15 shows all cables applied in the prototype. Various combinations are<br />

tested, as will be shown in the next chapter. As can be seen from the drawing,<br />

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Figure 7.14: The stainless steel ropes fit through the holes of the pins of<br />

the hinges. They contain threaded ends to adjust their length.

7.1 PROTOTYPE OF <strong>TENSAIRITY</strong> BEAM<br />

as well vertical as diagonal cables are used. The vertical cables are applied<br />

in the prototype for reasons of comparison with other Tensairity prototypes.<br />

The diagonal cables are (together with the middle vertical cable) those that<br />

allow complete folding of the Tensairity structure, as illustrated in figure 3.26<br />

on page 58.<br />

1 2 3 4<br />

D C<br />

B A<br />

Figure 7.15: The stainless steel ropes fit through the holes of the pins of<br />

the hinges. They contain threaded ends to adjust their length.<br />

7.1.5 END PIECES<br />

The upper and lower strut are at their both ends connected with each other<br />

and with the supports by means of a steel end piece, illustrated in figure 7.16.<br />

This is conceived such that it directs the struts along the curve of the spindle.<br />

In addition, its geometry makes sure that the center lines of the struts intersect<br />

each other in the axis of rotation of the supports. Otherwise, additional<br />

bending moments at the ends will occur, as shown in Teutsch (2009).<br />

Figure 7.16: The upper and lower strut are at their both ends connected<br />

with each other and with the supports by means of a steel end piece.<br />

Notice that the end piece does not allow rotation of the segments that are<br />

connected with it. This means that with these end pieces, the Tensairity<br />

beam can not fold completely. However, for reasons of comparison with<br />

other Tensairity prototypes, this end piece is applied for the experimental<br />

investigations. Another ending that allows rotation of the segments is designed<br />

and manufactured, but as mentioned not applied in the experimental<br />

investigations. After all, finite element investigations shows no big influence<br />

on the structural behaviour of allowing the segments to rotate in relation with<br />

the end pieces or not.<br />

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7.1.6 WEIGHT OF PROTOTYPE<br />

The previous sections show that the prototype for a deployable Tensairity<br />

structure is constituted of various elements, all necessary for an optimal<br />

kinematic and structural behaviour. Table 7.1 gives the weight of the various<br />

elements the prototype is constituted of and their contribution to the total<br />

weight of this light weight structure.<br />

Table 7.1: Weight of the various elements the Tensairity prototype is<br />

constituted of and their contribution to the total weight.<br />

element weight [kg] percentage of total weight [%]<br />

membrane 13,0 41,1<br />

segments 14,1 44,4<br />

hinges 2,3 7,3<br />

cables 1,2 3,8<br />

bolts & nuts 1,1 3,4<br />

total 31,7 100

7.2 THE EXPERIMENTAL SET-UP<br />

7.2 THE EXPERIMENTAL SET-UP<br />

The prototype for a deployable Tensairity beam, explained in detail in the<br />

previous section, is investigated experimentally. This section describes the<br />

experimental set-up. More precise, the test-rig wherein the structure is placed,<br />

the way loads are applied and how the displacements are measured, is discussed<br />

here.<br />

7.2.1 TEST RIG<br />

The Tensairity beam is investigated experimentally in the test rig of the ‘Empa-<br />

Center for Synergetic Structures’ in Zurich (Switzerland), illustrated in figure<br />

7.17. The test rig is a stiff steel frame that contains the supports for the<br />

Tensairity beam. The five meter long structure is isostatically supported by<br />

means of the end pieces that are attached to the roller and hinged support.<br />

In addition, this frame holds the motors and the ‘balance system’ that control<br />

the loading of the beam. Loading the structure is done upside-down in this<br />

experimental set-up: the beam is positioned upside-down in the test rig and<br />

the loads are thus applied upwards at the compression strut (placed at the<br />

bottom of the beam). Two motors are used - one for the left and one for the<br />

right part of the beam - because this way asymmetric loads can be applied.<br />

The upwards pulling of the motors is transferred to the compression strut by<br />

means of the balance system. This system is like a balance in equilibrium,<br />

which makes sure that the same load is applied at all points along the curved<br />

strut. A more detailed view of the balance system can be found in figure 7.18.<br />

7.2.2 LOADING<br />

The loading of the structure is displacement controlled: the motors pull the<br />

balance system in a certain time over a certain distance (and thus with a<br />

certain speed). The user can control this time and distance by means of input<br />

parameters. The motors monitor then which forces are applied to reach the<br />

predefined displacement of the balance system (and thus the deformation of<br />

the structure).<br />

The value of this chosen displacement varies during the experiments, because<br />

it depends on the configuration (cable configuration and internal pressure )<br />

and the purpose of the experiment (measuring stiffness and/or the maximum<br />

load). The time for one load cycle is for all experiments identical and measures<br />

180 seconds. Several load cycles are applied on one structure, to allow the<br />

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Figure 7.17: The test rig contains the supports, motors and balance<br />

system.

Figure 7.18: Detail of the balance system.<br />

7.2 THE EXPERIMENTAL SET-UP<br />

structure to ‘accommodate’ to the loading, as explained in section 4.2.1 on<br />

page 68. Figure 7.19 gives an example of the applied load as function of<br />

the time (left) and the load in function of the displacement (right). The small<br />

circles represent the displacement of the motor in each picture taken with the<br />

optical measurement instrument. This is explained in section 7.3.<br />

7.2.3 MEASURING<br />

The structural behaviour of the investigated Tensairity beam can be derived<br />

from the data obtained from the motors: the forces and displacements. However,<br />

these displacements are the average displacement of one half of the beam.<br />

No information regarding the displacements of specific points can be derived<br />

from this data.<br />

Therefore, displacements of the structure are at the same time measured by<br />

means of the optical ‘digital image correlation’ method (DIC), that employs<br />

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Displacement (mm)<br />

25<br />

20<br />

15<br />

10<br />

5<br />

0<br />

Displacement of the motors<br />

−5<br />

−200 −100 0 100<br />

Time (s)<br />

200 300 400<br />

load [N]<br />

5000<br />

4000<br />

3000<br />

2000<br />

1000<br />

0<br />

0 5 10 15 20 25<br />

displacement [mm]<br />

Figure 7.19: Left: an example of the applied load as function of the time,<br />

right: the applied load in function of the displacement.<br />

tracking and image registration techniques for accurate 2D and 3D measurements.<br />

Images from the structure during the load cycle are captured every<br />

ten seconds. The pictures are taken with two camera’s, because this way,<br />

measurements in 3D can be done.<br />

As can be seen in figure 7.20, the displacements of specific points are tracked<br />

by means of markers attached to the struts. The hull of the membrane contains<br />

also a random speckle pattern, for measuring displacements of the membrane.<br />

The image captured from the left and right camera are shown in figure 7.20<br />

Figure 7.20: Markers and a speckle pattern are applied on the prototype<br />

for measuring the displacements by means of DIC. The image captured<br />

from the left and right camera are shown.<br />

load [N]

7.3 POSTPROCESSING<br />

7.3 POSTPROCESSING<br />

The data obtained during the measurements, can now be processed to become<br />

useful results. The forces and displacements of the motors are acquired<br />

immediately and do not need many additional handling. The data from the<br />

optical measurements need to be postprocessed before the displacements can<br />

be visualized.<br />

The markers are selected and numbered in a reference picture in the DICsoftware<br />

4 . Then, the software will track the markers in the other images<br />

captured during that load cycle. This is possible, because these markers have<br />

a specific pattern and contrast that allows the software more easily to recognize<br />

them. However, this is sometimes not the case. The markers need to be selected<br />

manually. There must be noticed that in this latter case, the measurements are<br />

less accurate than in the case of automatically tracking by the software.<br />

The speckle pattern of the hull of the airbeam can also be processed in the<br />

DIC-software to investigate the displacements of the hull. However, since the<br />

focus of this research is mainly on the structural behaviour of the structure<br />

and thus the displacement of the struts, the behaviour of the membrane is not<br />

processed for every configuration.<br />

When all markers have been tracked in all pictures, their position is exported.<br />

This position is in relation to a global fixed coordinate system that was defined<br />

during calibration. This calibration is also necessary to allow the software to<br />

calculate the distance between markers in reality. Now the position of the<br />

markers and thus the displacements is known (in relation to the time), the<br />

data from the motors needs to be synchronized with the data from the optical<br />

measurement instrument to determine the load at which each picture is taken.<br />

This is done by means of a routine scripted in the mathematical software<br />

‘Matlab’ that compares the displacement of the motor with the displacement<br />

of the motor in the images.<br />

The graph on the left in figure 7.19 visualises the outcome of this synchronization.<br />

The displacement of the motor obtained from the optical measurements<br />

is indicated with a small circle, the continuous line represents the actual force<br />

displacement of the motor. When synchronized, the displacements of the two<br />

measurement instruments (motor and DIC) are identical. This way, the load<br />

at which each picture is taken can be determined.<br />

4 VIC-3D 2009 from Correlated Solutions is used.<br />

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The position of each marker can be tracked in every picture, and thus also<br />

the displacement of the markers. By doing so, graphs can be generated to<br />

investigate eg. the displacement of marked points on the upper strut during<br />

loading or to compare several configurations at a certain load. An example is<br />

given as illustration in figure 7.21. These results are analyzed and discussed<br />

in the next chapter.<br />

Vertical displacement (mm)<br />

Vertical displacement (mm)<br />

0<br />

10<br />

20<br />

30<br />

B dist150<br />

40<br />

−2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 2500<br />

Position x (mm)<br />

0<br />

10<br />

20<br />

30<br />

40<br />

50<br />

60<br />

70<br />

A dist150<br />

80<br />

−2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 2500<br />

Position x (mm)<br />

Figure 7.21: Example of graphs that can be generated with the<br />

experimental data to investigate the structural behaviour of several<br />

cases. Left: displacement of the strut of one case during loading, right:<br />

displaced strut of several cases at one load.<br />

7.4 GOAL OF THE EXPERIMENTAL INVESTIGATION<br />

Before conducting experiments and analyzing results, it is important to know<br />

what is aimed for to learn from these experiments.<br />

The main focus of this part of the research is to gain insight in the general<br />

structural behaviour of the deployable Tensairity beam. After all, such a<br />

structure has not been investigated yet. Therefore, the maximal load and

7.5 OBSERVATIONS AND IMPROVEMENTS<br />

stiffness of some representative configurations under various pressures are<br />

noted during the experiments. These data are then compared with other nondeployable<br />

Tensairity structures, manufactured with the same materials and<br />

with comparable geometry. This way, the deployable prototype is evaluated.<br />

In addition, it is also the goal of the experiments to study the influence of the<br />

configuration of the cables and hinges on the structural behaviour. Therefore,<br />

the displacements of several points along the struts from some configurations<br />

are measured by means of the markers and the optical measurement instruments.<br />

This way, insight is gained in the contribution of vertical and diagonal<br />

cables and the influence of the position of the hinges.<br />

These data are then analyzed in the next chapter to evaluate the current<br />

proposal for a deployable Tensairity beam and to formulate improvements<br />

for future designs.<br />

7.5 OBSERVATIONS AND IMPROVEMENTS<br />

The prototype is being observed during the experiments. In general, the<br />

design of the deployable Tensairity structure is satisfying, but some issues<br />

- open to improvement - are detected. This can be taken into account in<br />

a next prototype for a (deployable) Tensairity structure. Two main issues<br />

originate from the friction between the webs and the hull under pressurized<br />

air chambers.<br />

When the airbeam is not fully pressurized, the webs are - due to gravity -<br />

hanging from the struts downwards, as illustrated in figure 7.22. As a result, a<br />

large part of the web is lying at the bottom of the airbeam. When inflating the<br />

air chambers, there can be expected that the pressure distributes the webs along<br />

the center, as shown in the middle of figure 7.22. This also happens in reality,<br />

but not completely. At the moment the chambers become pressurized and<br />

contact between the webs occur, they do not move anymore. As illustrated on<br />

the right of figure 7.22, the weld of the web becomes stressed as consequence.<br />

The welding in this prototype is not designed to take up tensile forces, since<br />

peeling can occur. Therefore, in a new design, there has to be taken into<br />

account that the web also wil carry some tension and the welding to the outer<br />

hull has to be conceived as such.<br />

Another issue caused by the friction between the webs, is the obstruction of the<br />

cables. When no overpressure is present in the airbeam, the segments of the<br />

struts are not yet pushed outwards. This means that the cables are slack and<br />

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CHAPTER 7 PROTOTYPE AND EXPERIMENTAL SET-UP<br />

web<br />

before inflation as it should be after inflation<br />

as it is after inflation<br />

Figure 7.22: Due to friction of the web, the upper welding between outer<br />

hull and web may become stressed.<br />

position themselves ‘randomly curved’ between the web. When the airbeam<br />

is being inflated, the struts are pushed outwards and the webs are pressed<br />

together. There can be expected that the cables straighten themselves when<br />

being tensioned. However, observations show that the friction between the<br />

cable and the webs is such that the cable is jammed between the webs. As a<br />

result, the cables are not straighten and the struts are held in a wrong position<br />

by the cables, because not enough cable length is available. This phenomenon<br />

is avoided in these experiments by manually straightening the cables when<br />

the airbeam is being inflated. In a new design, this can be avoided by placing<br />

the cables in tubes that stay in their correct position. This way, the cables are<br />

not hindered anymore by the webs.<br />

7.6 SUMMARY<br />

When investigating a prototype experimentally, attention should not only<br />

be paid to the final results and conclusions of these experiments, but also to<br />

the prototype itself and how it is investigated. After all, not only are the results<br />

influenced by the manufacturing and testing, the development of a prototype<br />

is also very instructive and a research on its own. Therefore, the prototype and<br />

the components it is constituted of were discussed in detail in this chapter, as<br />

well the way of investigating this prototype.<br />

The prototype contains an airbeam made from pvc coated polyester fabric, a<br />

common and flexible material suitable for inflatable structures. The airbeam<br />

showed to have a well thought-out cutting pattern and welding procedure,<br />

which is a requirement for an airtight component. However, because of friction<br />

between the webs, the welding procedure should take into account that the<br />

web can become stressed. The two separate chambers proved to be efficient<br />

for allowing holes in the hull. This way, connecting the struts with cables can<br />

be done easily and some segments of the hinges can be accommodated.

7.6 SUMMARY<br />

The two types of hinges were materialized by flat aluminum plates. After<br />

all, a solid three-dimensional design for the hinges proved to be too complex<br />

for manufacturing. Therefore, a redesign constituted of flat elements was<br />

proposed and manufactured. The hinges demonstrated to have the kinematics<br />

as they were conceived. Also the compatibility with the airbeam and thus the<br />

correct folding of the membrane is demonstrated with the hinges.<br />

Because the strut of a Tensairity beam needs to be stabilized as much as possible<br />

by the membrane, attention was paid to the connection of those two. The<br />

prototype proves that the clamping of the membrane with a keder between<br />

two halves of the strut is a decent solution. However, this tight connection<br />

can cause problems when the segments of the struts are straight. After all, the<br />

incompatibility between straight segments and a curved airbeam results in a<br />

deformed prototype with an incorrect, kinked shape and additional, unwanted<br />

stresses in the components. From this prototype is learned that the hinged<br />

segments should be curved according the spindle’s shape and that attention<br />

should be paid to a careful positioning of the struts along the airbeam.<br />

The connection between the upper and lower strut proved to be well designed.<br />

The fixation of the cables to the hinges by means of a threaded end and a nut<br />

was satisfying. However, future designs should take the friction between the<br />

webs and the cables in inflated state into account. Otherwise, the cables risk to<br />

be jammed between the webs in a ‘curved’ position, which has consequences<br />

for the final shape and structural behaviour of the prototype.<br />

The total weight of the prototype was calculated, as well the contribution of<br />

each element separately to this total weight. As can be expected, the struts<br />

are the heaviest components with a contribution of 44,4% to the total weight<br />

of 31,7 kg. Somehow surprising is the large contribution of the membrane<br />

(41,1%) to this total weight.<br />

In addition to the prototype, the experimental set-up was discussed. The testrig,<br />

constituted of a stiff steel frame, the supports, the balance system and the<br />

motors responsible for the loading, was presented in detail. The loads are<br />

applied upside-down to the Tensairity beam by means of a balance system<br />

to make sure the same force is applied everywhere. This balance system is<br />

controlled by the displacement-driven motors.<br />

The applied load to reach a certain displacement are recorded by the motors.<br />

An optical measurement instrument, constituted of two camera’s, records the<br />

displacement of predefined points of the strut. The data from this optical<br />

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measurement instrument needs first to be postprocessed before it can be used<br />

for analysis.<br />

Now is understood how the prototype is manufactured and how it is investigated<br />

experimentally, the results from these experiments can be discussed.<br />

Various configurations are tested in order to reveal the influence on the load<br />

bearing behaviour of the deployable Tensairity beam. This is presented in the<br />

next chapter.

Experimental investigation of the prototype<br />

8<br />

The five meter long prototype of the deployable Tensairity beam is investigated<br />

experimentally. This chapter presents the results of these experiments. The<br />

general behaviour of the deployable Tensairity beam, such as its stiffness and<br />

maximal load, is first discussed. Then, various configurations are studied to<br />

analyse the influence on the structural behaviour of the cables and hinges.<br />

The experimental results are throughout the discussions compared with the<br />

outcome of numerical calculations on the finite element model of the prototype.<br />

8.1 CONFIGURATIONS<br />

To study the influence on the structural behaviour of different parameters,<br />

various cases are investigated experimentally. Figure 8.1 gives an overview.<br />

Ten configurations, indicated with a letter from A - J, differ in cable configuration<br />

or location of hinge, but are all constituted of the same airbeam and<br />

struts. Different pressures and motor displacements are applied with these<br />

configurations. The combination of configuration, pressure and displacement,<br />

results in more than sixty investigated cases.<br />

All combinations are investigated experimentally to gain insight in the structural<br />

behaviour. However, not all data is presented in this chapter. A synthesis<br />

of the findings is given, together with plots of several cases as illustration.<br />

8.2 GENERAL STRUCTURAL BEHAVIOUR<br />

In this section, the general structural behaviour, such as the characteristics<br />

of the load-deflection behaviour, the influence of the pressure on the stiffness<br />

and the maximal load the structure can bear, is discussed.<br />

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CHAPTER 8 EXPERIMENTAL INVESTIGATION OF THE PROTOTYPE<br />

A<br />

B<br />

C<br />

D<br />

E<br />

F<br />

G<br />

H<br />

I<br />

J<br />

Figure 8.1: Overview of all investigated combinations. The configura-<br />

tions are illustrated on the left and indicated with a letter, the pressures<br />

and displacements applied in the experiment are given on the right.

8.2.1 LOAD-DEFLECTION BEHAVIOUR<br />

8.2 GENERAL STRUCTURAL BEHAVIOUR<br />

As already mentioned in the previous chapter, the deployable Tensairity beam<br />

is loaded and reloaded several times. This is because the prototype, just like<br />

the small scale models in chapter 4, has to adapt first to the new load condition<br />

during the first load cycle, creating a residual displacement. This can be<br />

seen in figure 8.2 (left), showing the experimentally obtained load-deflection<br />

behaviour of a configuration during the three load-cycles . As a consequence,<br />

only the load-deflection curve of the third load-cycle is used for analysis of the<br />

results in this research.<br />

load [N]<br />

5500<br />

5000<br />

4500<br />

4000<br />

3500<br />

3000<br />

2500<br />

2000<br />

1500<br />

1000<br />

500<br />

0<br />

0 5 10 15 20 25<br />

average displacement [mm]<br />

load [N]<br />

5500<br />

5000 initial bending failure<br />

4500<br />

4000<br />

3500<br />

3000<br />

2500<br />

2000<br />

1500<br />

1000<br />

500<br />

0<br />

0 5 10 15 20 25<br />

average displacement [mm]<br />

Figure 8.2: The experimentally obtained load-deflection behaviour of<br />

one configuration. Left: the different curves during the three load-cycles.<br />

Right: three parts can be distinguished in the load-deflection curve.<br />

This curve is illustrated on the right of figure 8.2. Three main parts can<br />

be distinguished in this curve: the initial, the bending and the failure part.<br />

The data used for the calculation of the average stiffness are taken in the<br />

bending part, where the structure is loaded with representative loads. The<br />

structure’s response to the load is linear here. The initial part shows generally<br />

a different average stiffness than in the bending part when the structure is<br />

loaded distributed. An explanation for this other behaviour at low loads could<br />

not be verified with these experiments and will thus not be covered in this<br />

research 1 . The failure part contains the ‘collapse’ of the structure. The stiffness<br />

of the structure decreases drastically here due to failure or deformation. This<br />

1 Possible influences can be the arched struts or the pretensioned cables.<br />

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CHAPTER 8 EXPERIMENTAL INVESTIGATION OF THE PROTOTYPE<br />

‘collapse’ and the bending part will be discussed more in detail further in this<br />

chapter.<br />

8.2.2 MAXIMUM LOAD<br />

The ‘maximum load’ is in this research considered as the load whereby the<br />

stiffness of the structure decreases drastically, due to failure or deformation.<br />

This load can be seen in the average load-deflection behaviour by a ‘flattening’<br />

of the curve, as illustrated in figure 8.3. The experimental maximum<br />

distributed load of configuration A is plotted in this figure for three pressures.<br />

The average deflection is determined by the average value of the displacement<br />

of nine measurement points on the loaded strut.<br />

Total load (N)<br />

8000<br />

7000<br />

6000<br />

5000<br />

4000<br />

3000<br />

2000<br />

1000<br />

050 mbar<br />

150 mbar<br />

250 mbar<br />

0<br />

0 10 20 30 40<br />

Vertical displacement (mm)<br />

Figure 8.3: The maximum load is reached when the stiffness decreases<br />

drastically. The load-deflection of configuration A is plotted for three<br />

pressures under distributed load. (Experimental results).<br />

In chapter 4 (section 4.3) and chapter 6 (section 6.4.3) is shown that a change<br />

in stiffness of the deployable Tensairity beam is caused by the slackening of<br />

a pretensioned cable under loading. This has as result that the hinge is not<br />

supported anymore and experiences larger displacements under the same load<br />

increment. From the experiments is observed that the decrease in stiffness of<br />

the structure is indeed caused by the inwards rotation of one hinge (hinge<br />

2). Figure 8.4 illustrates this by plotting the experimental displacement of<br />

the hinges (and other measurement points) of the loaded strut at several load<br />

values. The hinges are indicated. The larger deflection of hinge 2 from a<br />

certain load on is noticeable.<br />

This configuration A of the deployable Tensairity spindle is also investigated

A<br />

Vertical displacement (mm)<br />

0<br />

5<br />

10<br />

15<br />

20<br />

25<br />

30<br />

35<br />

1<br />

1<br />

hinge 1 2<br />

3<br />

4<br />

40<br />

−2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 2500<br />

Position x (mm)<br />

5<br />

6<br />

7<br />

8.2 GENERAL STRUCTURAL BEHAVIOUR<br />

2 3 4 5 6 7 hinge moving<br />

outwards of beam<br />

2 3 4 5 6 7<br />

hinge moving<br />

inwards of beam<br />

Force = 0N<br />

Force = 7N<br />

Force = 417N<br />

Force = 845N<br />

Force = 1228N<br />

Force = 1601N<br />

Force = 2013N<br />

Force = 2484N<br />

Force = 2984N<br />

Force = 3485N<br />

Force = 3985N<br />

Force = 4489N<br />

Force = 4999N<br />

Force = 5191N<br />

Force = 5069N<br />

Figure 8.4: The experimental displacement of the hinges (and other<br />

measurement points) of the loaded upper strut at several load values of<br />

case A under 150 mbar. The hinges are indicated. The larger deflection<br />

of hinge 2 from a certain load on is noticeable.<br />

with a finite element model 2 . The calculations do not converge anymore when<br />

cables 2 and 6 become slack and hinges 2 and 6 as consequence experience too<br />

large displacements. Thus, the same hinge as in the experiments causes in the<br />

finite element model the large displacements. Figure 8.5 illustrates the shape<br />

of the spindle after loading, derived from the finite element analysis.<br />

Figure 8.6 plots the numerical derived load-displacement of the case A under<br />

50, 150 and 250 mbar.<br />

Figure 8.7 compares the experimental and numerical load-displacement of<br />

case A under 150 mbar. The stiffness of the numerical model is higher than<br />

the experimental, as can be expected. After all, the numerical value contains<br />

no (manufacturing) imperfections. The experimental maximum load is higher<br />

than the numerical. This can also be attributed to the imperfections of the<br />

experimental model. After all, it is possible due to these imperfections that not<br />

2 The hinges 1, 3, 5 and 7 of the upper strut are blocked to move inwards, like it in reality.<br />

183

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CHAPTER 8 EXPERIMENTAL INVESTIGATION OF THE PROTOTYPE<br />

2 6<br />

Figure 8.5: The same hinge as in the experiments causes in the finite<br />

element model the large displacements. This shape after loading is<br />

derived from the finite element analysis.<br />

load [N]<br />

8000<br />

7000<br />

6000<br />

5000<br />

4000<br />

3000<br />

numerical results − distributed load<br />

2000<br />

050 mbar<br />

1000<br />

150 mbar<br />

250 mbar<br />

0<br />

0 0.01 0.02 0.03 0.04<br />

average displacement [m]<br />

Figure 8.6: The numerical derived load-displacement of the case A<br />

under 50, 150 and 250 mbar.<br />

8000<br />

7000<br />

6000<br />

5000<br />

4000<br />

3000<br />

2000<br />

1000<br />

numerical<br />

experimental<br />

0<br />

0 0.01 0.02 0.03 0.04<br />

Figure 8.7: The experimental and numerical load-displacement of case<br />

A under 150 mbar.

8.2 GENERAL STRUCTURAL BEHAVIOUR<br />

all forces are transferred completely to the decrease in cable tension. However,<br />

this is an assumption that can not be verified because the cable tension of the<br />

experimental model could not be measured.<br />

8.2.3 STIFFNESS<br />

In chapter 4 and 6 is discussed that the stiffness of the structure under distributed<br />

load is independent of the internal pressure, as long as all vertical<br />

cables are pretensioned and thus supporting the struts and hinges. This is<br />

also the case with the experimental investigated prototype: the stiffness (in the<br />

linear part of the load-displacement curve) of the three pressure cases are very<br />

similar, as shown in figure 8.3. This confirms the influence of the prestressed<br />

cables.<br />

In the case of asymmetrical loading, more influence on the stiffness from the<br />

internal pressure can be detected. After all, the ‘meshes’ of the struts are<br />

deformed in a parallelogram and the support of the airbeam has an influence<br />

on this. The higher the internal pressure, the less easy these ‘meshes’ will<br />

deform. Figure 8.8 plots the experimental derived stiffness of the asymmetrical<br />

loaded configuration A under 50 and 150 mbar.<br />

Total load (N)<br />

3500<br />

3000<br />

2500<br />

2000<br />

1500<br />

1000<br />

500<br />

experimental results − asymmetric loading<br />

case A − 150<br />

case A − 050<br />

0<br />

0 5 10 15 20<br />

Vertical displacement (mm)<br />

Figure 8.8: The experimental derived stiffness of the asymmetrical<br />

loaded configuration A under 50 and 150 mbar.<br />

185

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CHAPTER 8 EXPERIMENTAL INVESTIGATION OF THE PROTOTYPE<br />

8.3 INFLUENCE OF CABLES<br />

In the previous section, the general load-bearing behaviour of the deployable<br />

Tensairity beam (case A) has been related with the influence of pretensioned<br />

cables connecting upper and lower strut. In this section, various experimental<br />

investigated configurations are compared to illustrate and confirm the effect<br />

of the configuration of cables on the structure’s behaviour. First, the vertical<br />

cables are discussed, then the diagonals.<br />

8.3.1 VERTICAL CABLES<br />

The load-deflection curves of cases B, C and D, pressurized till 150 mbar and<br />

loaded to a displacement of 10 mm, are compared. The configurations have<br />

the same diagonal cables, but differ in amount of vertical cables: case B has<br />

7 vertical cables, case C one and case D zero. From the curves, illustrated<br />

in figure 8.9 is clear that, for both load cases, there is a large difference in<br />

average stiffness between the case with 7 vertical cables (B) and with one (C)<br />

or zero (D). Comparison of cases C and D shows that one extra vertical middle<br />

cable has little influence on the structures stiffness, as well for distributed as<br />

asymmetrical loading. The comparison of cases E and F confirms this.<br />

Configuration B is approximately a factor 4 stiffer than configurations C and<br />

D. This can be attributed to the presence of the vertical cables, due to a<br />

combination of reasons.<br />

load [N]<br />

4500<br />

4000<br />

3500<br />

3000<br />

2500<br />

2000<br />

1500<br />

1000<br />

500<br />

0<br />

0 2 4 6 8 10<br />

average displacement [mm]<br />

B<br />

D<br />

C<br />

load [N]<br />

4500<br />

4000<br />

3500<br />

3000<br />

2500<br />

2000<br />

1500<br />

1000<br />

500<br />

0<br />

0 2 4 6 8 10<br />

average displacement [mm]<br />

Figure 8.9: The load-deflection curves of configurations B, C and D,<br />

pressurized till 150 mbar and loaded to a displacement of 10 mm. Left:<br />

distributed load, right: asymmetrical load. (Experimental results).<br />

B<br />

D<br />

C

8.3 INFLUENCE OF CABLES<br />

All hinges in case B are supported by pretensioned (mainly vertical) cables<br />

that can take up compressive forces. As a result, the loaded strut is better<br />

supported. Because not every hinge in cases C and D is connected with a<br />

cable, the struts do not keep their spindle shape after inflation. The hinges are<br />

pushed outwards by the internal pressure and the struts deform noticeable in a<br />

zigzag-shape, as already discussed in section 7.5 and illustrated in figure 8.10<br />

(upper). This has a negative influence on the structural behaviour. After all,<br />

when external load is applied on the strut, this loading is not fully transferred<br />

to compressive forces in the strut (as is the case for configuration B), but also<br />

to a change in geometry of the strut. Figure 8.10 (lower) shows the displaced<br />

upper struts of cases B, C and D at the same distributed load (1200 N).<br />

y postion [mm]<br />

400<br />

300<br />

200<br />

100<br />

0<br />

−100<br />

−200<br />

−300<br />

Spindle shape of cases after inflation<br />

−400<br />

−3000 −2000 −1000 0 1000 2000 3000<br />

x postion [mm]<br />

Vertical displacement (mm)<br />

0<br />

2<br />

4<br />

6<br />

8<br />

10<br />

12<br />

14<br />

−2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 2500<br />

Position x (mm)<br />

Figure 8.10: The displaced struts of cases B, C and D. Upper: after<br />

inflation, Lower: at 1200N. (Experimental results).<br />

B<br />

D<br />

BD150<br />

CD150<br />

DD150<br />

This deformation of the upper strut under inflation when not all hinges<br />

are connected with a cable is also studied numerically and concluded in<br />

section 6.4.3. The simulations confirm a decrease in stiffness due to the<br />

deformation of the strut, as mentioned above.<br />

187

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CHAPTER 8 EXPERIMENTAL INVESTIGATION OF THE PROTOTYPE<br />

Configurations G and H differ in the presence of three vertical cables. However,<br />

because these vertical cables connect the hinges that are already linked with a<br />

diagonal cable and because there are still hinges not connected with a cable,<br />

the same issues as described above are present. This has as result that the three<br />

extra vertical cables of case H do not contribute to an increase in stiffness and<br />

cases G and H have as result very similar load-deflection behaviour.<br />

8.3.2 DIAGONAL CABLES<br />

Configurations A and B differ in the presence of diagonal cables and are thus<br />

ideal to verify the influence of these diagonals. The load-displacement of these<br />

configurations pressurized till 150 mbar is presented in figure 8.11.<br />

From the graphs can be seen that for an asymmetrical load case, the configuration<br />

with diagonals (B), has a higher stiffness at higher loads. However,<br />

this difference in stiffness between A and B is very little. In the case of a<br />

symmetrical loading is the configuration without diagonals stiffer.<br />

Total load (N)<br />

6000<br />

5000<br />

4000<br />

3000<br />

2000<br />

1000<br />

distributed load asymmetric load<br />

6000<br />

A dist150<br />

0<br />

0 10 20<br />

B dist150<br />

30 40<br />

Vertical displacement (mm)<br />

Total load (N)<br />

5000<br />

4000<br />

3000<br />

2000<br />

1000<br />

A asymm 150<br />

B asymm 150<br />

0<br />

0 10 20 30 40<br />

Vertical displacement (mm)<br />

Figure 8.11: The load-displacement curves of configurations A and B.<br />

Left: distributed load, right: asymmetrical load. (Experimental results).<br />

The displacement of the loaded (upper) strut of configurations A and B under<br />

asymmetric load (2950 N and 150 mbar) is shown in figure 8.12. From the<br />

displaced struts can be seen that the displacements are lower in case B, as<br />

well on the loaded side (right half), as the unloaded. This is due to the<br />

positive influence of the diagonal cables. This is confirmed by the numerical<br />

investigations in section 6.4.3, where configuration B indeed has a higher<br />

stiffness that configuration A due to the presence of diagonal cables.<br />

Figure 8.13 shows the displacement of the loaded (upper) strut of configurations<br />

A and B under a distributed load of 3950 N. From this figure and the

Vertical displacement (mm)<br />

0<br />

10<br />

20<br />

30<br />

40<br />

50<br />

A asymm 150<br />

B asymm 150<br />

8.3 INFLUENCE OF CABLES<br />

60<br />

−2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 2500<br />

Position x (mm)<br />

Figure 8.12: The displaced struts of configurations A and B under<br />

asymmetrical load of 2950 N and internal pressure of 150 mbar.<br />

(Experimental results).<br />

load-displacement graph in figure 8.11 (left) can be seen that configuration B<br />

experiences a larger displacement under distributed load than configuration<br />

A. This larger displacement in the case of configuration B is located mainly<br />

at the hinge that is allowed to move downwards and that is connected with<br />

diagonal cables. There can be assumed that the diagonal cables cause this<br />

larger displacement.<br />

However, from the numerical investigations of configuration A and B conducted<br />

in chapter 6, is concluded that configuration B is stiffer under distributed<br />

loading than A. This is because some diagonal cables are there tensioned<br />

under inflation (because they are connected with a hinge not connected<br />

with a vertical cable), which has as result that they can take some loading and<br />

thus support the hinges.<br />

Vertical displacement (mm)<br />

0<br />

10<br />

20<br />

30<br />

A dist150<br />

B dist150<br />

40<br />

−2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 2500<br />

Position x (mm)<br />

Figure 8.13: The displaced struts of configurations A and B under<br />

symmetrical load of 3950 N and internal pressure of 150 mbar.<br />

189

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CHAPTER 8 EXPERIMENTAL INVESTIGATION OF THE PROTOTYPE<br />

Configurations I and J are respectively the configurations B and A turned<br />

upside-down. These two cases can thus also be used to discuss and illustrate<br />

the influence of diagonal cables on the structural behaviour of the deployable<br />

Tensairity beam. Figure 8.14 plots the experimental load-displacement curve<br />

of both cases under distributed load. From the graph can be seen that case I is<br />

stiffer than case J. It is obvious that this is also for asymmetric loading. Thus<br />

also here is confirmed that the diagonals contribute to the structural behaviour.<br />

Load (N)<br />

6000<br />

5000<br />

4000<br />

3000<br />

2000<br />

Distributed load − pressure 150 mbar<br />

1000<br />

case I<br />

case J<br />

0<br />

0 10 20 30 40<br />

Vertical displacement (mm)<br />

Figure 8.14: The experimental load-displacement curve of case I and J<br />

under distributed load. The internal pressure measures 150 mbar.<br />

The support of the diagonal cables can also be seen in figure 8.15 where the<br />

displaced upper and lower strut of both cases are illustrated. Take for example<br />

the middle hinge on the upper strut. In the case of configuration I is this hinge<br />

supported by two diagonal and one vertical cable. In the case of J is the hinge<br />

only supported by the vertical cable, hence the larger displacement of this<br />

point.<br />

Configurations D, F and G - which differ in a diagonal cable - are also<br />

investigated experimentally. Their stiffness is plotted in figure 8.16. Not<br />

all hinges of these cases are connected with a cable and one can thus expect<br />

these cases to have a deformed strut under inflation. Their stiffness is thus<br />

mainly dominated by this deformation of the strut under loading, and not by<br />

the structural contribution of the cables.<br />

8.3.3 POSITION OF HINGES<br />

Because only the position of the hinges in the lower and upper strut of cases A<br />

and J differ from each other, these two configurations can be used to investigate

Vertical displacement (mm)<br />

Vertical displacement (mm)<br />

0<br />

10<br />

20<br />

30<br />

40<br />

50<br />

60<br />

0<br />

5<br />

10<br />

15<br />

20<br />

25<br />

30<br />

Displacement of upper (loaded) strut<br />

8.3 INFLUENCE OF CABLES<br />

70<br />

−2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 2500<br />

Position x (mm)<br />

Displacement of lower (unloaded) strut<br />

case I<br />

case J<br />

case I<br />

case J<br />

35<br />

−2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 2500<br />

Position x (mm)<br />

Figure 8.15: The displaced upper and lower strut of cases I and J under an<br />

external distributed load of 4500 N and an internal pressure of 150 mbar<br />

the influence on the structural behaviour of the position of the hinges.<br />

Figure 8.17 shows the load-deflection behaviour of the two configurations<br />

under distributed and asymmetric load. The stiffness of both configurations<br />

is under distributed load identical till a certain load. From that load on,<br />

configuration A is stiffer than J. In addition, the maximum load of configuration<br />

A is higher than J. In the case of an asymmetrical load, the two configurations<br />

show very similar stiffness. Here, the maximal load of J is higher than A.<br />

Figure 8.18 plots the displaced upper strut of both configurations during<br />

loading. It is clear from the figure that in case J the middle hinge (3) causes<br />

large displacements, while in case A it is the second hinge (the first hinge from<br />

the support that can rotate inwards). There was expected that the first hinge<br />

of case J would experience large displacements, since it is not supported by a<br />

191

192<br />

CHAPTER 8 EXPERIMENTAL INVESTIGATION OF THE PROTOTYPE<br />

load [N]<br />

load [N]<br />

1500<br />

1000<br />

500<br />

0<br />

distributed load − pressure 150 mbar<br />

0 2 4 6 8 10<br />

displacement [mm]<br />

D<br />

G<br />

F<br />

load [N]<br />

1500<br />

1000<br />

500<br />

asymmetric load − pressure 150 mbar<br />

0<br />

0 2 4 6 8 10<br />

displacement [mm]<br />

Figure 8.16: The load-displacement curves of configurations D, F and G<br />

under distributed (left) and asymmetrical load (right).<br />

6000<br />

5000<br />

4000<br />

3000<br />

2000<br />

1000<br />

distributed load − pressure 150 mbar<br />

A<br />

0<br />

0 10 20 30 40<br />

displacement [mm]<br />

J<br />

load [N]<br />

6000<br />

5000<br />

4000<br />

3000<br />

2000<br />

1000<br />

asymmetric load − pressure 150 mbar<br />

0<br />

0 10 20 30 40<br />

displacement [mm]<br />

Figure 8.17: Load-deflection behaviour of configurations A and J under<br />

distributed and asymmetric load. Pressure is 150 mbar.<br />

A<br />

G<br />

F<br />

J<br />

D

8.3 INFLUENCE OF CABLES<br />

cable. Explanations for this behaviour could not be found experimentally and<br />

numerically and are an interest topic for further in-depth investigations 3 . An<br />

explanation for the similar stiffness of cases A and J under asymmetrical load<br />

is that the airbeam plays an important role in preventing the ‘meshes’ to be<br />

deformed to parallelograms.<br />

Vertical displacement (mm)<br />

Vertical displacement (mm)<br />

upper strut J<br />

0<br />

10<br />

20<br />

30<br />

40<br />

50<br />

60<br />

70<br />

80<br />

−2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 2500<br />

Position x (mm)<br />

upper strut A<br />

0<br />

10<br />

20<br />

30<br />

40<br />

50<br />

60<br />

70<br />

1<br />

2 3 4 5<br />

1 2 3 4 5 6 7<br />

80<br />

−2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 2500<br />

Position x (mm)<br />

hinge moving outwards (up) hinge moving inwards (down)<br />

Force = 0N<br />

Force = 4550N<br />

Force = 0N<br />

Force = 5069N<br />

Figure 8.18: The displaced upper strut of configurations A and J during<br />

external distributed loading and with internal pressure of 150 mbar.<br />

3 An assumption why the first hinge of case J does not causes large displacements, is that it is<br />

pushed outwards by the inflation since it is not connected with a cable.<br />

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CHAPTER 8 EXPERIMENTAL INVESTIGATION OF THE PROTOTYPE<br />

8.4 CONCLUSIONS<br />

Experimental investigations were conducted on the prototype to evaluate<br />

the proposed mechanism for a deployable Tensairity structure and to gain<br />

insight in its structural behaviour. The influence of various parameters, such<br />

as the internal pressure, the cable configuration, and the position of the hinges,<br />

is analyzed by means of various configurations. The results were compared<br />

with conclusions from previous chapters and with finite element models of<br />

the prototype.<br />

First, the influence of the cable tension on the general load-bearing behaviour<br />

was studied. The experimental results confirm the outcome of the investigations<br />

from chapter4 and chapter 6: the cables support the hinges and are<br />

responsible for the structure’s stiffness, as long as they are pretensioned. Once<br />

the load is sufficient high to decrease a cable’s tension to zero, the stiffness of<br />

the structure changes. The maximum load of the structure is thus related with<br />

the internal pressure of the beam, while the stiffness of the structure (in the<br />

linear part) independent is of the pressure. As was expected, the maximal load<br />

of the deployable Tensairity structure is reached when one hinge starts to rotate<br />

inwards. This ‘collapse’ of the segments occurs obviously at the hinge that<br />

was designed for rotating inwards and at higher loads for higher pressures.<br />

In chapter 6 is shown by means of numerical investigations that the bending<br />

stiffness of the strut has an influence on the structure’s stiffness. More precise,<br />

for higher pressures will the strut deflect more and cause a decrease of<br />

stiffness in the structure. This behaviour is not observed during the experiments.<br />

Again, in the experimental investigations is found that the stiffness<br />

is independent of the pressure as long as the cables are pretensioned. As<br />

already mentioned in chapter 6, more reliability is given to the experimental<br />

investigations than the numerical outcome (which reflects the ideal situation<br />

without imperfection). It is possible that this negative influence of the pressure<br />

becomes more visible for experiments under higher pressures (where a larger<br />

deflection of the strut will occur).<br />

Various cable configurations were investigated. The experiments showed the<br />

importance of connecting all the hinges of the upper strut with the lower strut<br />

by means of cables. This way, the hinges are kept in position after being inflated<br />

and the struts maintain their spindle shape. After all, during the experiments<br />

is observed that the hinges without cables are pushed outwards by the inflated<br />

airbeam. As a consequence, the segments of the strut rotate relative to each

8.4 CONCLUSIONS<br />

other and the strut is deformed. When loading the structure, the loads are<br />

not fully transferred to compression and tension in the struts, but also to a<br />

further deformation of the strut. Obvious, this has a negative influence on the<br />

structural behaviour, which can be seen in the comparison of configurations B<br />

and D.<br />

From this comparison was expected to detect the contribution of vertical cables<br />

to the structure’s stiffness. After all, the two configurations (B and D) differ<br />

only in the presence of vertical cables. However, because configuration D<br />

has not in all hinges a cable and thus an incorrect initial shape after inflation,<br />

no conclusion can be drawn. To investigate this, a comparison should be<br />

made between a configuration that contains in all hinges a diagonal and a<br />

configuration with as well diagonals in all hinges as verticals.<br />

The influence of diagonal cables on the behaviour of the deployable Tensairity<br />

structure is investigated from the comparison of cases A and B, and cases I<br />

and J. From cases I and J can be concluded that the diagonals contribute to the<br />

structural behaviour, as well under distributed as asymmetrical loading. After<br />

all, because the diagonals are connected with a hinge that has no link with a<br />

vertical cable, it is pretensioned and thus able to support some compressive<br />

forces.<br />

With regard to the influence of the position of the hinges, not much could<br />

be concluded. It is obvious that only the hinges that are allowed to rotate<br />

inwards are causing the large deflections. In the case of A and B is this the first<br />

inwards rotating hinge from the supports on. In cases I and J is this the middle<br />

hinge. This latter is not expected, since numerical investigations revealed<br />

that the hinge closest to the ends will cause the ‘collapse’ of the structure. An<br />

explanation for this behaviour can be that the first hinge is pushed outwards by<br />

the air pressure, since no cable is connected to this hinge. Future experiments<br />

need to investigate the influence of the position of hinges more in-depth,<br />

especially since it has a large impact on the design of the deployable Tensairity<br />

beam.<br />

With these experiments, a five meter prototype for a deployable Tensairity<br />

structure is investigated for the first time. This structure differs on some crucial<br />

points from the regular, better known Tensairity structures, namely on the<br />

presence of the hinges and the curved segments of the strut. Therefore, it was<br />

not obvious what to expect from the experiments. With the observations made<br />

during the experiments and the analysis of the experimental data, insight is<br />

gained in the structural behaviour and the prototype is evaluated. However, at<br />

195

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CHAPTER 8 EXPERIMENTAL INVESTIGATION OF THE PROTOTYPE<br />

the same time, more questions are raised with regard to the specific behaviour<br />

of the deployable structure and the influence of various phenomenon that<br />

apparently occur during the loading of the structure. These experiments<br />

pointed out what has to be investigated more in detail in future experiments<br />

on the deployable Tensairity structure.

Evaluation of the prototype<br />

9<br />

The prototype for a deployable Tensairity beam, investigated experimentally<br />

and numerically in this dissertation, is evaluated in this chapter. This is done<br />

by comparing the structural behaviour of the deployable Tensairity beam with<br />

experimental and numerical results of its non-deployable counterpart, the ‘D’spindle.<br />

The experimental results will by way of illustration also be compared<br />

with those of other Tensairity prototypes. Finally, this evaluation will lead to<br />

some concluding remarks and suggestions for the further development of a<br />

deployable Tensairity beam.<br />

9.1 COMPARISON WITH OTHER <strong>TENSAIRITY</strong> PROTOTYPES<br />

The experimental load-deflection behaviour of configuration A of the deployable<br />

Tensairity structure is compared with three other five meter spindle<br />

shaped Tensairity prototypes. The three cases differ in connection between<br />

upper and lower strut: the ‘O-spindle’ has no connection between the struts,<br />

the ‘C-spindle’ has a continuous membrane (web) as connecting element and<br />

the ‘D-spindle’ contains a discrete connection between upper and lower strut,<br />

namely a certain amount of cables at regular distances (Crisbasanu, 2010).<br />

The configuration of the D-spindle that will be used for the comparisons<br />

has the same amount of cables and at the same position as configuration<br />

A of the deployable Tensairity structure, called here ‘F-spindle’ (for Foldable).<br />

Figure 9.2 illustrates the four cases.<br />

Figure 9.1 shows the load-deflection curves of the four Tensairity types under<br />

distributed and asymmetrical load for three pressures. Note that the value of<br />

the axes are different in each graph for reasons of clarity.<br />

The curves show clearly that the stiffness of the deployable Tensairity structure<br />

is under distributed load in the same range as the O-spindle, however the F-<br />

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CHAPTER 9 EVALUATION OF THE PROTOTYPE<br />

load [N]<br />

load [N]<br />

load [N]<br />

8000<br />

7000<br />

6000<br />

5000<br />

4000<br />

3000<br />

2000<br />

1000<br />

0<br />

0 5 10 15 20 25 30<br />

displacement [mm]<br />

16000<br />

14000<br />

12000<br />

10000<br />

8000<br />

6000<br />

4000<br />

2000<br />

20000<br />

18000<br />

16000<br />

14000<br />

12000<br />

10000<br />

8000<br />

6000<br />

4000<br />

2000<br />

D<br />

50 mbar − distributed<br />

150 mbar − distributed<br />

D<br />

F<br />

C<br />

C<br />

F<br />

0<br />

0 5 10 15 20 25 30<br />

displacement [mm]<br />

250 mbar − distributed<br />

D<br />

0<br />

0 5 10 15 20 25 30<br />

displacement [mm]<br />

C<br />

O<br />

O<br />

O<br />

F<br />

load [N]<br />

load [N]<br />

load [N]<br />

3000<br />

2500<br />

2000<br />

1500<br />

1000<br />

500<br />

50 mbar − asymmetric<br />

0<br />

0 10 20 30 40 50 60<br />

displacement [mm]<br />

6000<br />

5000<br />

4000<br />

3000<br />

2000<br />

1000<br />

7000<br />

6000<br />

5000<br />

4000<br />

3000<br />

2000<br />

1000<br />

150 mbar − asymmetric<br />

0<br />

0 10 20 30 40 50<br />

displacement [mm]<br />

F<br />

F<br />

250 mbar − asymmetric<br />

0<br />

0 10 20 30 40<br />

displacement [mm]<br />

Figure 9.1: Load-deflection curves of the four Tensairity types under<br />

distributed and asymmetrical load for three pressures.<br />

D<br />

D<br />

D<br />

C<br />

F<br />

O<br />

C<br />

C<br />

O<br />

O

9.2 <strong>DEPLOYABLE</strong> VERSUS NON-<strong>DEPLOYABLE</strong> <strong>TENSAIRITY</strong> BEAM<br />

O-spindle C-spindle D-spindle F-spindle<br />

Figure 9.2: The four Tensairity cases which structural behaviour is<br />

compared.<br />

spindle is stiffer under low pressures. The D- and C-spindle have a much<br />

higher stiffness. The O-spindle has under asymmetrical load case the lowest<br />

stiffness in all pressure cases, while the F-spindle has stiffness comparable<br />

with the D- and C-spindle. Thus, although the structural behaviour of the<br />

F-spindle is less optimal than the D and C spindle, it provides a foldable<br />

Tensairity structure with at least the same structural performance of a standard<br />

O-spindle.<br />

The D- and F-spindle contain the same cable configuration, but differ in the<br />

presence of hinges. To evaluate the deployable Tensairity beam, the D- and Fspindle<br />

are compared numerically and experimentally.<br />

9.2 <strong>DEPLOYABLE</strong> VERSUS NON-<strong>DEPLOYABLE</strong> <strong>TENSAIRITY</strong> BEAM<br />

The numerical model for the prototype of the deployable Tensairity beam<br />

has already been applied in the numerical study of its structural behaviour<br />

(chapter 6) and the comparison with the experimental results (chapter 8). This<br />

latter showed that the numerical model shows good qualitative agreement<br />

with the experimental results.<br />

Load-displacement behaviour<br />

Here, the behaviour of configuration A (F-spindle) is compared with the Dspindle<br />

by means of finite element models. Figure 9.3 plots the stiffness of the<br />

D- and F-spindle under distributed and asymmetric loading with an internal<br />

pressure of 150 mbar. Figure 9.4 compares the experimental and numerical<br />

derived load-displacement behaviour of both cases. The figure shows that the<br />

finite element results agree very well with the experimental outcome in the<br />

case of distributed load. In the case of asymmetrical loading is the difference<br />

between the numerical and experimental curves larger, approximately a factor<br />

3.<br />

199

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CHAPTER 9 EVALUATION OF THE PROTOTYPE<br />

load [N]<br />

load [N]<br />

12000<br />

10000<br />

8000<br />

6000<br />

4000<br />

2000<br />

12000<br />

10000<br />

distributed load − pressure 150 mbar<br />

0<br />

0 0.01 0.02 0.03 0.04 0.05<br />

average displacement [m]<br />

8000<br />

6000<br />

4000<br />

2000<br />

load [N]<br />

12000<br />

10000<br />

8000<br />

6000<br />

4000<br />

2000<br />

asymmetric load − pressure 150 mbar<br />

0<br />

0 0.01 0.02 0.03 0.04 0.05<br />

average displacement [m]<br />

Figure 9.3: Numerical load-deflection curves of the D- and F-spindle<br />

under distributed and asymmetrical load. The internal pressure<br />

measures 150 mbar.<br />

distributed load − pressure 150 mbar<br />

D-num<br />

D-exp<br />

F-exp<br />

F-num<br />

0<br />

0 0.01 0.02 0.03 0.04 0.05<br />

average displacement [m]<br />

load [N]<br />

12000<br />

10000<br />

8000<br />

6000<br />

4000<br />

2000<br />

asymmetric load − pressure 150 mbar<br />

D-exp<br />

D-num<br />

F-exp<br />

F-num<br />

0<br />

0 0.01 0.02 0.03 0.04 0.05<br />

average displacement [m]<br />

Figure 9.4: Comparison between the numerical and experimental<br />

derived load-displacement curves. The finite element results agree very<br />

well with the experimental in the case of distributed load (left). In the<br />

case of asymmetrical loading differs the stiffness of the cases by a factor<br />

2 (right).<br />

Figure 9.3 shows that the D-spindle has a larger stiffness and a higher maximum<br />

load for both load cases. Thus, the hinges make the structure ‘softer’,<br />

produce larger displacements and cause the structure to ‘collapse’ at a lower<br />

load. This collapse is different for the two cases, as observed experimentally<br />

and numerically. As seen in the previous chapters will, in the case of the Fspindle,<br />

one hinge become unstable (not supported anymore by a pretensioned

9.2 <strong>DEPLOYABLE</strong> VERSUS NON-<strong>DEPLOYABLE</strong> <strong>TENSAIRITY</strong> BEAM<br />

cable) and rotate inwards. In the case of the D-spindle will the compressed<br />

upper strut buckle out-of-plane.<br />

This has as result that in the case of the D-spindle, the maximum load is difficult<br />

to predict visually. In addition, once the structure has collapsed, the struts are<br />

deformed irreversible. In the case of the F-spindle, the inwards rotation of<br />

one hinge can be observed visually. When the load is released, the F-spindle<br />

returns to its initial shape.<br />

Influence of hinges<br />

To illustrate the influence of the pretensioned cables on the structural behaviour<br />

of both cases and to show the effect of hinges on the cable pretension<br />

is the tension in cables 2 and 3 of the D- and F-spindle plotted throughout<br />

loading (figure 9.5).<br />

cable tension [N]<br />

600<br />

500<br />

400<br />

300<br />

200<br />

100<br />

2 3<br />

cable tension throughout distributed loading − pressure 150 mbar<br />

cable 2 − D−spindle<br />

cable 3 − D−spindle<br />

cable 2 − F−spindle<br />

cable 3 − F−spindle<br />

0<br />

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000<br />

total applied load [N]<br />

Figure 9.5: The tension of cables 2 and 3 of the D- and F-spindle<br />

throughout distributed loading. (Internal pressure measures 150 mbar.)<br />

It shows that the decrease of pretension in cable 3 is initially similar for both<br />

cases. However, once a cable becomes slack (cable 2), the hinge at cable 2 is not<br />

supported anymore by the cable and larger deformation occurs. This results<br />

in a large decrease of cable tension, as shown for cable 3 in the figure. The<br />

slackening of one cable occurs faster in the case of the F-spindle. In the case of<br />

the D-spindle has this slackening of the cable little influence on the decrease<br />

in pretension of the other cables, as can be seen for cable 3.<br />

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CHAPTER 9 EVALUATION OF THE PROTOTYPE<br />

As already discussed in chapter 6, is the influence of hinges also reflected in<br />

the bending moments in the struts. Figure 9.6 illustrates the moments of both<br />

cases under a distributed load of 4100 N and internal pressure of 150 mbar.<br />

Note that the upper hinges, allowed to rotate outwards, are blocked to move<br />

inwards in this finite element model. Hence the moments at these ‘hinges’.<br />

Figure 9.6: The influence of hinges in the upper and lower strut is<br />

also reflected in the bending moments of the struts. (Under distributed<br />

loading of 4100 N.)<br />

Finally, the difference in structural behaviour between the D- and F-spindle<br />

is illustrated in figure 9.7 where their deformed shape under a distributed<br />

load of 4100 N is plotted. As discussed, the difference in defection is caused<br />

by the hinges because of their effect on the bending moments and the cable<br />

pretension.<br />

D-spindle<br />

F-spindle<br />

Summary<br />

Figure 9.7: The displaced struts of the D- and F-spindle, magnified by a<br />

factor 5. The influence of the hinges is obvious.<br />

It is obvious that the spindle without hinges in upper and lower strut, the<br />

D-spindle, is stiffer and can bear more loads than its deployable variant, the<br />

F-spindle. This section showed that the D-spindle is under distributed loading<br />

a factor three stiffer than the F-spindle.<br />

This difference in structural behaviour is attributed to the influence of hinges.<br />

After all, these are the only parameters in which these cases differ. The hinges

9.3 CONCLUDING REMARKS AND SUGGESTIONS<br />

have an influence on the bending moments in the struts and thus in the<br />

pretension (and thus support) of the cables throughout loading. Once a hinge<br />

becomes unstable, the F-spindle experiences with little increase of load larger<br />

displacements.<br />

9.3 CONCLUDING REMARKS AND SUGGESTIONS<br />

The proposal for a deployable Tensairity structure has been evaluated throughout<br />

this research. This is not only done by comparing the experimental results<br />

with other Tensairity prototypes, but also by investigating numerically and<br />

experimentally various configurations of the prototype.<br />

The research clearly showed the impact on the structural behaviour of cable<br />

configurations. All hinges need to be connected with a cable in order to avoid<br />

deformation of the struts during inflation, which causes a drop in stiffness.<br />

Especially vertical pretensioned cables are contributing largely to the structural<br />

behaviour.<br />

However, the cable configuration that allows the deployable Tensairity beam<br />

to fold completely contains only diagonal cables. In addition, not all hinges<br />

are connected with a cable. As a consequence, this configuration has a<br />

relative low stiffness, approximately one fourth of the value of the stiffness<br />

of the configuration with vertical cables in every hinge. The configuration<br />

of the cables in the mechanism was, however, mainly determined by the<br />

kinematics of the foldable system. Thus, the mechanism for the deployable<br />

Tensairity beam needs to be redesigned, taking into account the requirement<br />

of connecting all hinges with vertical cables.<br />

This is possible with the original folding mechanism of the foldable truss<br />

(cylindrical and spindle shape), where upper and lower strut ‘fold’ into each<br />

other. After all, upper and lower strut (and by extension all points of the<br />

structure) move closer to each other when folding, which allows every possible<br />

cable configuration and shape of the hull. The issue here however is that<br />

this configuration needs to possess more hinges, located closer to the ends.<br />

Numerical research showed that this has a negative influence on the maximal<br />

load of the structure. Thus to conclude, the original foldable truss mechanism<br />

is a good suggestion for the foldable Tensairity beam, taking into account that<br />

hinges near the end supports are to be avoided.<br />

This new mechanism can have as well a cylindrical as a spindle shape. The<br />

spindle shape has a more optimal structural behaviour due to its curved<br />

203

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CHAPTER 9 EVALUATION OF THE PROTOTYPE<br />

shaped, but can be unwanted for some practical applications, like the deck<br />

of a bridge. In addition, because of the upwards movement of the lower<br />

strut of a spindle shaped structure under distributed loading, are all cables,<br />

regardless of their configuration, loaded in compression. This is not the case<br />

for a cylindrical deployable Tensairity beam. Depending on the load case<br />

and the application can a cylindrical deployable Tensairity beam thus be more<br />

appropriate.<br />

When developing a new deployable Tensairity beam, this research showed<br />

that it is crucial to first fully understand the influence of every parameter on<br />

the structural and kinematic behaviour. Therefore, it is advisable to start with<br />

very simple models, e.g. containing one or two hinges and or cables. Then,<br />

by systematically increasing the complexity of the structure, one can achieve<br />

a design for an optimal deployable Tensairity beam.

Conclusions<br />

205

Conclusions<br />

10<br />

A Tensairity structure has most of the properties of a simple air-inflated beam,<br />

but can bear to hundred times more load. This makes Tensairity structures very<br />

suitable for temporary and mobile applications, where lightweight solutions<br />

and small transport volume are desired. However, the standard Tensairity<br />

structure, which is comprised of several interacting components such as an<br />

airbeam, cables and struts, can not be compacted to a small volume without<br />

being disassembled. By replacing the standard compression and tension<br />

element with a mechanism, a deployable Tensairity structure is achieved that<br />

needs no additional handlings to compact or erect the structure.<br />

The main goal of this research was to gain insights in the structural and<br />

kinematic behaviour of deployable Tensairity structures and develop a first<br />

prototype of such a structure. Given the lack of general knowledge on<br />

deployable Tensairity structures, this investigation was pursued by means<br />

of experimental and numerical investigations on small and large scale models.<br />

The first part of the dissertation focused on the development of an appropriate<br />

mechanism for the deployable Tensairity structure. The second part investigated<br />

the structural behaviour of a Tensairity beam by means of experiments<br />

on scale models and numerical investigations and identified the influence of<br />

several design parameters. As a result, a prototype of a deployable Tensairity<br />

beam was designed, fabricated, experimentally tested and evaluated in the<br />

third part. The main findings concerning the design and structural behavior<br />

of deployable Tensairity structures and the general contributions to the field<br />

are discussed below.<br />

207

208<br />

CHAPTER 10 CONCLUSIONS<br />

10.1 DESIGN OF <strong>DEPLOYABLE</strong> <strong>TENSAIRITY</strong> <strong>STRUCTURES</strong><br />

In order to create a Tensairity structure that is deployable, the typical continuous<br />

compression element needs to be replaced by a suitable deployment<br />

mechanism. The first research goal was to gain insight in the development<br />

of an appropriate mechanism for the deployable Tensairity structure. Several<br />

deployment mechanism were designed and investigated in relation to the<br />

boundary conditions and requirements imposed by the structural concept<br />

Tensairity and the structural applications. The investigated models show that<br />

the continuous connection of the membrane with the compression element is<br />

the most crucial and imperative requirement. The folding of the membrane<br />

should thus be compatible with the proposed foldable compression element<br />

and vice versa. As a result, the inextensibility of the membrane has to be taken<br />

into account to avoid damaging the membrane or obstruction of the folding.<br />

More concretely, the hinges centre line has to lie in the same plane as the<br />

membrane to prevent stretching or wrinkling.<br />

A mechanism constituted of stiff segments, connected to each other by hinges,<br />

proves to be an appropriate solution for a foldable compression element. The<br />

’foldable truss system’ is being improved with regard to its structural and<br />

kinematic behaviour. The system’s load bearing capacities are ameliorated by<br />

changing the longitudinal shape from cylindrical to spindle, by decreasing the<br />

amount of hinges and segments and by positioning hinges on compression<br />

side towards the middle. By means of a redesign of the configuration of the<br />

foldable truss, the kinematic behaviour is improved. In this new proposal,<br />

the segments of upper and lower strut do not have to ‘fold’ into each other<br />

anymore. In addition, less hinges and less complicated joints are necessary.<br />

As a result, an easily foldable proposal for the deployable Tensairity structure<br />

was obtained.<br />

10.2 STRUCTURAL BEHAVIOUR OF THE <strong>DEPLOYABLE</strong> <strong>TENSAIRITY</strong> BEAM<br />

As the introduction of a deployment mechanism changes the structural<br />

behaviour of a traditional Tensairity structure, the second goal of this research<br />

was to explore the general behaviour of the deployable Tensairity beam. More<br />

specifically, the influence of various design parameters on the beams structural<br />

behaviour was investigated by means of numerical and experimental<br />

investigations on small scale models as well as a large scale prototype. These<br />

investigations provided specific insights in the effects of the hinges, the cables,

10.2 STRUCTURAL BEHAVIOUR OF THE <strong>DEPLOYABLE</strong> <strong>TENSAIRITY</strong> BEAM<br />

the internal pressure and the sections of the hull and struts.<br />

Hinges<br />

The introduction of hinges in the upper and lower strut (to create a deployable<br />

Tensairity beam) has several consequences on the structural behaviour of the<br />

beam. Finite element calculations showed that these hinges decrease the<br />

structure’s stiffness. The influence of the hinges is relatively small when no<br />

bending moment occurs in the struts. However, when large bending moments<br />

are introduced in the strut (eg. by means of cables), the influence of the hinges<br />

on the deflection is more noticeable. A variation in the amount of hinges<br />

showed little influence on the structure’s stiffness. Furthermore, the numerical<br />

experiments revealed that the location of the hinge along the beam determines<br />

the maximal load of the structure: the closer the hinge is to the endpoint of<br />

the beam, the lower the load will be at which no convergence of the finite<br />

element calculation is reached (and thus a certain collapse occurs). Hinges at<br />

the ends of the beam should thus be avoided as they are less stabilized by the<br />

membrane.<br />

A comparison of the experimental results of the (deployable) prototype with<br />

those of non-deployable Tensairity beams confirmed this significant influence<br />

of the hinges on the structural behaviour. The stiffness of the prototype with<br />

hinges was less than half of the stiffness of the same configuration without<br />

hinges.<br />

Cables<br />

The numerical and experimental investigations on large and small scale models<br />

revealed the influence of cables and their configuration on the structural behaviour<br />

of the deployable Tensairity beam. The pretensioned cables contribute<br />

to the structure’s stiffness (until their pretension becomes zero) by increasing<br />

the spring constant of the elastic foundation and thus supporting the struts and<br />

hinges. As long as all cables are pretensioned is the stiffness of the structure<br />

independent of the internal pressure. Once a cable becomes slack, the stiffness<br />

changes. The maximum load of the structure is thus related with the internal<br />

pressure.<br />

The experiments showed the importance of connecting all the hinges of the<br />

upper strut with the lower strut by means of cables. This keeps the hinges<br />

in position when inflating the airbeam and makes the struts maintain their<br />

spindle shape. When this is not the case, the hinges without cable connection<br />

will move outwards and deform the struts. This has as result that the upper<br />

209

210<br />

CHAPTER 10 CONCLUSIONS<br />

strut will deform further under external loading, hence the lower stiffness of<br />

these configurations.<br />

The investigation of the cable configurations and the cable tension showed a<br />

difference in behaviour between the cylindrical and spindle shaped deployable<br />

Tensairity beam. In the case of a cylindrical shaped beam can the diagonals<br />

become tensioned under distributed load, depending on their configuration.<br />

In the case of a spindle shaped Tensairity beam become all cables compressed<br />

under distributed loading, regardless their configuration. This is due to the<br />

upwards movement of the lower strut, caused by the horizontal displacement<br />

of the upper strut.<br />

Because diagonal cables are not pretensioned under inflation when they connect<br />

hinges that are already linked with vertical cables, only the vertical cables<br />

are in the case of a spindle shaped structure contributing to the structure’s<br />

stiffness under distributed load. Under asymmetrical loading on the other<br />

hand, are the diagonal cables (loaded in tension) contributing. Diagonal cables<br />

do have a positive influence on the structure’s stiffness under distributed<br />

loading when they are not already linked with vertical cables and thus being<br />

pretensioned.<br />

The investigation of cylindrical Tensairity beams showed that the shape of the<br />

‘mesh’ - formed by the pretensioned cables and the cables that will become<br />

tensioned under loading - has a large influence on the structure’s stiffness. If<br />

this mesh is a quadrangle, it can easily be deformed under external loading and<br />

cause large displacements of the structure. When all cables are pretensioned<br />

and thus able to take compressive forces, the deployable Tensairity beam has<br />

the same stiffness as a truss (with the same configuration of diagonals and<br />

with the same strut sections).<br />

Hull and strut section<br />

Also the sections of the hull and the struts proved to have an influence on<br />

the structural behavior. Numerical investigations revealed that the shape of<br />

the hull of the Tensairity beam whereby the web experiences a constant stress<br />

along the beam’s length, uses the material more optimal and has the largest<br />

average stiffness. This can be clarified by the fact that a higher stress in the<br />

web near the ends supports the struts and hinges up to a higher load.<br />

The influence of the struts section was shown to be related to the occurrence<br />

or absence of bending moments in the struts. The strut’s cross section proved<br />

to have no influence in the cases without bending moment in the struts. When

10.3 EVALUATION OF THE PROTOTYPE<br />

bending moments are introduced (mostly by cables), however, the bending<br />

stiffness, and thus the cross section, plays an import role on the stiffness of<br />

the deployable Tensairity beam. For cases with large deflections of the strut,<br />

second order effects occurred and a large decrease in stiffness was detected.<br />

As a result, the struts should be dimensioned to avoid these second order<br />

effects. This bending stiffness of the strut has not been noticed to be that<br />

influential in the experiments. While numerical calculations predict a decrease<br />

in stiffness under higher pressure, due to the bending of the strut, show the<br />

experimental investigations a higher stiffness when increasing the pressure.<br />

This contradiction can be attributed to the difference between a perfect ideal<br />

numerical model and a prototype with its imperfections. This subject is an<br />

interesting aspect for further research.<br />

The experimental investigation on the prototype pointed out some other<br />

specific details with regard to the behaviour of the deployable Tensairity beam<br />

that need to be studied in more detail. The influence of the pretension of<br />

cables on the structure’s stiffness is one aspect that deserves a closer look<br />

by continuously measuring the tension in the cables during inflation and<br />

loading, and relating this with the load-displacement behaviour. In addition,<br />

the influence of the position of the hinges on the structure’s stiffness should be<br />

investigated experimentally more in detail.<br />

10.3 EVALUATION OF THE PROTOTYPE<br />

As a result of the different findings from the numerical studies and experimental<br />

investigations on scale models, a five meter long prototype has been<br />

developed during the final phase of this research. This prototype is the result<br />

of an iterative process, taking into consideration general design issues, as well<br />

as aiming for a structurally sound structure. The result is a working prototype,<br />

which has been tested and is ready for further improvements.<br />

The design and the assembling of the various components the structure is<br />

constituted of are evaluated positive. The connection of the airbeam with<br />

the curved struts proved to be satisfying. The experiments revealed the<br />

necessity of applying curved segments along the spindle beam and of carefully<br />

positioning these segments along the beam. The hinges demonstrated to have<br />

the kinematics as they were conceived. Also the compatibility with the airbeam<br />

and thus the correct folding of the membrane at the position of the hinges is<br />

demonstrated. In addition, they proved to withstand considerable loadings.<br />

Overall, the prototype weighs 31,7 kg.<br />

211

212<br />

CHAPTER 10 CONCLUSIONS<br />

With regard to its structural behavior, the proposal for the deployable Tensairity<br />

beam is insufficient. The cable configuration that allows the mechanism to<br />

fold completely contains only diagonal cables and has hinges not connected<br />

with any cable. As a consequence, this configuration has a relative low stiffness,<br />

approximately one fourth of the value of the stiffness of the configuration<br />

with vertical cables. The configuration of the cables was , however, mainly<br />

determined by the kinematics of the foldable system. This influence was not<br />

expected to be this large and decisive. The influence of the position of the<br />

hinges on the maximal load was taken into account in the development of the<br />

prototype, hence the redesign of the mechanism and the more central location<br />

of the hinges.<br />

The folding of the structure and thus the packing of the membrane to a dense<br />

bunch was not as smooth as intended due to the bending stiffness of the<br />

PVC coated polyester (although small). Guiding the membrane during the<br />

compacting of the structure was necessary at some points. Thus, a neat,<br />

repeatable and controlled folding of the membrane according to a predefined<br />

folding pattern is advisable.<br />

It has been shown that the proposal for the deployable Tensairity structure can<br />

be improved regarding its structural and kinematical aspects. After all, this<br />

is the first investigated prototype for such a deployable structure and some<br />

design iterations need to be done. With regard to the kinematics, a closer look<br />

should be taken to the folding of the airbeam according a predefined folding<br />

pattern to ease the folding of the technical membrane. At the same time,<br />

the proposal should take the requirements for a structurally sound Tensairity<br />

beam into account, like the connection of the hinges with vertical cables.<br />

With this research, the first step is taken towards a functional large scale<br />

deployable Tensairity beam. However, many aspects in this research, such as<br />

the detailing and the gained insight, can also be applied in the development<br />

and research of other structures, such as Tensairity arches, cushions and<br />

grids. The application of the deployable technology on other scales or in<br />

other domains than civil engineering will bring forward new questions and<br />

knowledge and is worth investigating.

Bibliography<br />

Airlight (2010). Airlight sa - lightweight and large span: Tensairity.<br />

http://www.airlight.biz.<br />

Benaroya, H., Bernold, L., and Chua, K. M. (2002). Engineering, design and<br />

construction of lunar bases. Journal of Aerospace Engineering, 15(2):33–45.<br />

Breuer, J., Ockels, W., and Luchsinger, R. (2007). An inflatable wing using<br />

the principle of tensairity. In 48th AIAA/ASME/ASCE/AHS/ASC Structures,<br />

Structural Dynamics, and Materials Conference, Honolulu, Hawaii.<br />

Breuer, J. C. M. and Luchsinger, R. H. (2009). Inflatable kites with tensairity.<br />

In Kröplin, B. and Oñate, E., editors, International Conference on Textile<br />

Composites and Inflatable Structures IV - Structural Membranes 2009, Stuttgart.<br />

Cimne.<br />

Buildair (2010). Inflatable buildings for events - military and emergency<br />

applications. http://www.buildair.com/.<br />

Cadogan, D., Stein, J., and Grahne, M. (1999). Inflatable composite habitat<br />

structures for lunar and mars exploration. Acta Astronautica - Pacific RIM: A<br />

Rapidly Expanding Space Market, 44(7-12):399–406.<br />

Calatrava Valls, S. (1981). Zur Faltbarkeit von Fachwerken. Phd, Dissertation<br />

number 6870, ETH-Zürich, Switzerland.<br />

Cavallaro, P. V. and Sadegh, A. M. (2006). Air-inflated fabric structures. NUWC-<br />

NPT Report 11774.<br />

Comer, R. L. and Levy, S. (1963). Deflections of an inflated circular cylindrical<br />

cantilever beam. AIAA Journal, 1(5):1652–1655.<br />

213

214<br />

BIBLIOGRAPHY<br />

Crettol, R. and Luchsinger, R. (2006). Flexible druckelemente - unpublished.<br />

Technical report, ProspectiveConcepts AG.<br />

Crisbasanu, M. (2010). The effect of discrete reinforcement on the load-bearing<br />

behavior of a spindle-shapedtensairity beam. Master’s thesis, T.U. Delft.<br />

Darooka, D. K. and Jensen, D. W. (2001). Advanced space structure<br />

concepts and their development. In AIAA/ASME/ASCE/AHS/ASC Structures,<br />

Structural Dynamics, and Materials Conference and Exhibit, 42nd, Seattle.<br />

De Laet, L., Luchsinger, R., Crettol, R., Mollaert, M., and De Temmerman, N.<br />

(2008). Deployable tensairity structures. Journal of the International Association<br />

for Shell and Spatial Structures, 50(2):121–128.<br />

De Laet, L., Mollaert, M., Luchsinger, R. H., De Temmerman, N., and<br />

Guldentops, L. (2009). Mechanisms for deployable tensairity structures.<br />

In Domingo, A. and Lazaro, C., editors, International Association for Shell and<br />

Spatial Structures (IASS) Symposium 2009 - Evolution and Trends in Design,<br />

Analysis and Construction of Shell and Spatial Structures, Valencia.<br />

De Temmerman, N. (2007). Design and Analysis of Deployable Bar Structures for<br />

Mobile Architectural Applications. Phd, <strong>Vrije</strong> <strong>Universiteit</strong> <strong>Brussel</strong>.<br />

Dessauce, M. (1999). The Inflatable Moment: Pneumatics and Protest in ’68.<br />

Princeton Architectural Press.<br />

Detail (2005). Roof over parking area in montreux. Detail, 7+8:756–759.<br />

Eurovinil (2010). Inflatable tents. http://www.eurovinil.it/.<br />

Forster, B. (1994). Cable and membrane roofs - a historical survey. Structural<br />

Engineering Review, 6(3/4):145174.<br />

Fuchs, F., Ruttiman, R., Mauro, P., Luchsinger, R. H., and ProspectiveConcepts<br />

(2005). Advertising medium - patent wo 2005 031682 a1.<br />

Gauthier, L. P., Wever, T. E., and Luchsinger, R. H. (2009). Study of the buckling<br />

behaviour of compression elements in tensairity. In Kröplin, B. and Oñate, E.,<br />

editors, International Conference on Textile Composites and Inflatable Structures<br />

IV - Structural Membranes 2009, Stuttgart. Cimne. (paper on cd).<br />

Hamilton, K. P., Champbell, D. M., and Gossen, P. A. (1994). Current state<br />

of development and future trends in employment of air-supported roofs<br />

in long-span applications. In IASS International Symposium, pages 612–621,<br />

Atlanta, US.

BIBLIOGRAPHY<br />

HDT (2010). Hdt airbeam shelters. http://www.hdtglobal.com/hdt-airbeam-shelters.<br />

Herzog, T. (1976). Pneumatic Structures: A Handbook of Inflatable Architecture.<br />

Oxford University Press.<br />

Hublitz, I., Henninger, D., Drake, B., and Eckart, P. (2004). Engineering<br />

concepts for inflatable mars surface greenhouses. Advances in Space Research<br />

- Space Life Sciences: Life Support Systems and Biological Systems under Influence<br />

of Physical Factors, 34(7):1546–1551.<br />

Inflate (2010). Design and manufacture of inflatable structures and architecture.<br />

http://www.inflate.co.uk/structures.htm.<br />

Jensen, F. V. (2004). Concepts for Retractable Roof Structures. PhD thesis,<br />

University of Cambridge.<br />

L’Garde (2010). Inflatable space structural systems. http://www.lgarde.com/.<br />

Liu, J., Wu, M., and Zhang, Q. (2006). Structural analysis for double-layer etfe<br />

cushion. In International Symposium on New Perspectives for Shell and Spatial<br />

Structures: International Association for Shell and Spatial Structures, Beijing,<br />

China.<br />

Losberger (2010). Losberger rapid deployment systems: Shelters, inflatable<br />

tents, fast erecting shelters and hangars. http://www.losberger-rds.com/.<br />

Luchsinger, R. H. (2009). Recent progress in the r&d of tensairity. In Kröplin,<br />

B. and Oñate, E., editors, International Conference on Textile Composites and<br />

Inflatable Structures IV - Structural Membranes 2009, Stuttgart. Cimne. (paper<br />

on cd).<br />

Luchsinger, R. H. and Crettol, R. (2006a). Adaptable tensairity. In Scheublin,<br />

F., Pronk, A., Borgard, A., and Houtman, R., editors, Adaptables 2006<br />

- International Conference On Adaptable Building Structures, pages 3–7,<br />

Eindhoven, The Netherlands.<br />

Luchsinger, R. H. and Crettol, R. (2006b). Experimental and numerical study<br />

of spindle shaped tensairity girders. International Journal of Space Structures,<br />

21(3):119–130.<br />

Luchsinger, R. H. and Crettol, R. (2006c). The role of fabrics in tensairity. In<br />

International Symposium on New Perspectives for Shell and Spatial Structures:<br />

International Association for Shell and Spatial Structures, Beijing, China.<br />

215

216<br />

BIBLIOGRAPHY<br />

Luchsinger, R. H. and Crettol, R. (2007). Synergetic structures and tensairity.<br />

In TensiNet Symposium 2007 - Ephemeral Architecture: Time and Textiles, pages<br />

207–216, Milano, Italy.<br />

Luchsinger, R. H., Crettol, R., and ProspectiveConcepts (2006). Collapsible<br />

pneumatically stabilised supports - patent wo 2006 099764 a1.<br />

Luchsinger, R. H., Crettol, R., and ProspectiveConcepts (2007). Pneumatic<br />

support structure - patent wo 2007/147270 a1.<br />

Luchsinger, R. H., Crettol, R., and ProspectiveConcepts (2009). Foldable<br />

pneumatic support - patent wo 2009 065238 a2.<br />

Luchsinger, R. H., Pedretti, A., Pedretti, M., and Steingruber, P. (2004a).<br />

The new structural concept tensairity: Basic principles. In Zingoni, A.,<br />

editor, Second International Conference on Structural Engineering, Mechanics and<br />

Computation, pages 323–328, Lisse, The Netherlands. A. A. Balkema/Swetz<br />

Zeitlinger.<br />

Luchsinger, R. H., Pedretti, A., Steingruber, P., and Pedretti, M. (2004b).<br />

Lightweight structures with tensairity. In Motro, R., editor, Proceedings of the<br />

International Association for Shell and Spatial Structures (IASS) : Shell and Spatial<br />

Structures: From Models to Realization, pages 80–81, Montpellier, France.<br />

Luchsinger, R. H., Pedretti, M., and Reinhard, A. (2004c). Pressure induced<br />

stability: From pneumatic structures to tensairity. Journal of Bionics<br />

Engineering, 1:141–148.<br />

Luchsinger, R. H. and ProspectiveConcepts (2005). Pneumatic actuator - patent<br />

wo 2005 031173 a1.<br />

Luchsinger, R. H. and Teutsch, U. (2009). An analytical model for tensairity<br />

girders. In Lazaro, A. D. and Carlos, editors, Proceedings of the International<br />

Association for Shell and Spatial Structures (IASS) : Evolution and Trends in<br />

Design, Analysis and Construction of Shell and Spatial Structures, Valencia.<br />

Main, J., Peterson, S., and Strauss, A. (1994). Load deflection behavior of spacebased<br />

inflatable fabric beams. Journal of Aerospace Engineering, 7(2):225–238.<br />

Moaveni, S. (2008). Finite Element Analysis Theory and Application with ANSYS.<br />

Pearson - Prentice Hall, third edition.

BIBLIOGRAPHY<br />

Natori, M. C., Higuchi, K., Sekine, K., and Okazaki, K. (1995). Adaptivity<br />

demonstration of inflatable rigidized integrated structures (iris). Acta<br />

Astronautica - 45th International Astronautical Federation Congress, 37:59–67.<br />

Oñate, E. and Kröplin, B., editors (2007). Structural Membranes 2007 - Textile<br />

Composites and Inflatable Structures, Barcelona. CIMNE.<br />

Pedretti, A., Steingruber, P., Pedretti, M., and Luchsinger, R. H. (2004). The new<br />

structural concept tensairity: Fe-modeling and applications. In Zingoni, A.,<br />

editor, Second International Conference on Structural Engineering, Mechanics and<br />

Computation, pages 329–333, Lisse, The Netherlands. A. A. Balkema/Swetz<br />

Zeitlinger.<br />

Pedretti, M. (2005). The basic physical principles of tensairity. In E. Oñate, B. K.,<br />

editor, International Conference on Textile Composites and Inflatable Structures,<br />

Structural Membranes 2005, pages 25–28, Stuttgart. CIMNE, Barcelona.<br />

Pedretti, M. and Luscher, R. (2007). Tensairity-patent - eine pneumatische<br />

tenso-struktur. Stahlbau, 76(5):314–319.<br />

Pedretti, M. and ProspectiveConcepts (2004a). Bow-like support consisting of<br />

a pneumatic structural element - patent wo 2004 083569 a1.<br />

Pedretti, M. and ProspectiveConcepts (2004b). Flexible compression member<br />

for a flexible pneumatic structural element and means for erecting pneumatic<br />

element structures - patent wo 2004 083568 a1.<br />

Plagianakos, T. S., Teutsch, U., Crettol, R., and Luchsinger, R. H. (2009).<br />

Static response of a spindle-shaped tensairity column to axial compression.<br />

Engineering Structures, 31(8):1822–1831.<br />

Robinson-Gayle, S., Kolokotroni, M., Cripps, A., and Tanno, S. (2001). Etfe foil<br />

cushions in roofs and atria. Construction and Building Materials, 15(7):323–327.<br />

Salama, M., Lou, M. C., and Fang, H. (2000). Deployment of inflatable<br />

space structures - a review of recent developments. In<br />

AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials<br />

Conference and Exhibit, 41st, Atlanta.<br />

Schock, H.-J. (1997). Soft Shells: Design and Technology of Tensile Architecture.<br />

Birkhäuser.<br />

Schodek, D. L. (2000). Structures. Prentice Hall.<br />

217

218<br />

BIBLIOGRAPHY<br />

Teutsch, U. (2009). Tragverhalten von Tensairity Trägern. Phd, Dissertation<br />

number 18679, ETH-Zürich, Switzerland.<br />

Topham, S. (2002). Blow-Up: Inflatable Art, Architecture, and Design. Prestel<br />

Publishing.<br />

Wagner, R. (1997). Future developments. In Textile Roofs 1997.<br />

Wever, T. E., Plagianakos, T. S., Luchsinger, R. H., and Marti, P. (2010). Effect<br />

of fabric webs on the static response of spindle-shaped tensairity columns.<br />

Journal of Structural Engineering ASCE, 136(4):410–418.

List of publications<br />

De Laet, L., Luchsinger, R. H., Crettol, R., Mollaert, M., and De Temmerman,<br />

N. (2009a). Deployable tensairity structures. Journal of the International<br />

Association for Shell and Spatial Structures, 50:121–128.<br />

De Laet, L., Mollaert, M., and De Temmerman, N. (2008b). Do it yourself!<br />

tensile structures. Fabric Architecture, 20 (July/August - 4).<br />

De Laet, L., Mollaert, M., Luchsinger, R. H., De Temmerman, N., Van Mele, T.,<br />

and Guldentops, L. (2009d). Folding mechanisms for deployable tensairity<br />

structures. In Oñate, E. and Kröplin, B., editors, Proceeding of the International<br />

Conference on Textile Composites and Inflatable Structures, pages 244–247.<br />

published by CIMNE.<br />

De Laet, L., Mollaert, M., Luchsinger, R. H., De Temmerman, N., and<br />

Guldentops, L. (2009b). Mechanisms for deployable tensairity structures.<br />

In Domingo, A. and Lzaro, C., editors, Proceedings of the 50th Anniversary<br />

Symposium of the IASS - ”Evolution and trends in Design, Analysis and<br />

Construction of Shells and Spatial Structures”, Valencia, Spain. (paper on cd).<br />

De Laet, L., Mollaert, M., Luchsinger, R. H., De Temmerman, N., and<br />

Guldentops, L. (2009c). Mechanisms for deployable tensairity structures. In<br />

Proceedings of the 8th National Congress on Theoretical and Applied Mechanics,<br />

pages 601–605, <strong>Brussel</strong>s, Belgium.<br />

De Laet, L., Luchsinger, R., Mollaert, M., and De Temmerman, N. (2008a).<br />

Deployable tensairity structures. In Salinas, J. G. O., editor, Proceedings of the<br />

IASS-SLTE Symposium 2008 - ”New materials and technologies, new designs and<br />

innovations”, Acapulco, Mexico. (paper on cd).<br />

De Laet, L., Mollaert, M., De Temmerman, N., and Van Mele, T. (2007). Analysis<br />

of the adaptability of air-inflated components. In Oñate, E. and Kröplin, B.,<br />

219

220<br />

LIST OF PUBLICATIONS<br />

editors, Proceeding of the III International Conference on Textile Composites and<br />

Inflatable Structures, pages 244–247. CIMNE, Barcelona.<br />

De Laet, L. (2009). Student’s research on tensairity structures. In Jaarboek<br />

Vakgroep ARCH 2008-2009, pages 46–47, <strong>Vrije</strong> <strong>Universiteit</strong> <strong>Brussel</strong>. Sintjoris.<br />

De Laet, L. and Mollaert, M. (2008). Tensairtent. In Jaarboek Vakgroep ARCH<br />

2007-2008, pages 44–45, <strong>Vrije</strong> <strong>Universiteit</strong> <strong>Brussel</strong>. Sintjoris, Gent.<br />

De Laet, L. (2008). Deployable tensairity structures. In Jaarboek Vakgroep ARCH<br />

2007 - 2008, pages 46–47, <strong>Vrije</strong> <strong>Universiteit</strong> <strong>Brussel</strong>. Sintjoris, Gent.<br />

De Laet, L. and Mollaert, M. (2007). Education in lightweight structures. In<br />

Jaarboek Vakgroep ARCH 2006-2007, pages 55–56, <strong>Vrije</strong> <strong>Universiteit</strong> <strong>Brussel</strong>.<br />

Sintjoris, Gent.<br />

De Laet, L. (2007). blow up - inflatable structures. In Jaarboek Vakgroep ARCH<br />

2006-2007, pages 44–45, <strong>Vrije</strong> <strong>Universiteit</strong> <strong>Brussel</strong>. Sintjoris, Gent.<br />

De Temmerman, N., Alegria Mira, L., Mollaert, M., De Laet, L., and Van Mele,<br />

T. (2010). A state-of-the-art of deployable scissor structures for architectural<br />

applications. In Proceedings of the International Association for Shell and Spatial<br />

Structures (IASS) Symposium 2010, Shanghai ’Spatial Structures - Permanent<br />

and Temporary’, Shangai, Chine. (paper on cd).<br />

De Temmerman, N., Mollaert, M., De Laet, L., Henrotay, C., Paduart, A.,<br />

Guldentops, L., Van Mele, T., and De Backer, W. (2009a). A deployable mast<br />

for kinetimatic architecture. In Proceedings of the 8th National Congress on<br />

Theoretical and Applied Mechanics, pages 585–591, <strong>Brussel</strong>s, Belgium.<br />

De Temmerman, N., Mollaert, M., De Laet, L., and Van Mele, T. (2007a).<br />

Development of a novel deployable bar structure with foldable articulated<br />

joints. In Proceeding of the IASS 2007 International Symposium on Structural<br />

Architecture - Towards the future looking to the past, Venice, Italy. (paper on cd).<br />

De Temmerman, N., Mollaert, M., De Laet, L., Van Mele, T., Guldentops, L.,<br />

Henrotay, C., De Backer, W., Paduart, A., Hendrickx, H., and De Wilde, W. P.<br />

(2009b). A deployable mast for adaptable textile architecture. In Domingo,<br />

A. and Lzaro, C., editors, Proceedings of the 50th Anniversary Symposium of the<br />

IASS - ”Evolution and trends in Design, Analysis and Construction of Shells and<br />

Spatial Structures”, Valencia, Spain. (paper on cd).

LIST OF PUBLICATIONS<br />

De Temmerman, N., Mollaert, M., Van Mele, T., and De Laet, L. (2006a). A<br />

concept for a foldable mobile shelter system. In Proceedings of the IASS-APCS<br />

2006 Int. Symposium on the New Olympics and New Shell and Spatial Structures,<br />

Bejing, China. IASS-APCS. (paper on cd).<br />

De Temmerman, N., Mollaert, M., Van Mele, T., and De Laet, L. (2006b).<br />

Development of a foldable mobile shelter system. In Adaptables 2006:<br />

Proceedings of the International Conference on Adaptability in Design and<br />

Construction, volume 2, pages 13–17, Eindhoven University of Technology,<br />

The Netherlands.<br />

De Temmerman, N., Mollaert, M., Van Mele, T., and De Laet, L. (2007b). Design<br />

and analysis of a foldable mobile shelter system. International Journal of Space<br />

Structures,Special Issue: Adaptable Structures, 22:161–168.<br />

Guldentops, L., M., M., and De Laet, L. (2009a). Fabric-formed concrete shell<br />

structures. In Proceedings of the 14th International Workshop on the Design and<br />

Practical Realisation of Architectural Membrane Structures, Textile Roofs 2008.<br />

Technical University Berlin, Germany.<br />

Guldentops, L., Mollaert, M., Adriaenssens, S., De Laet, L., and De Temmerman,<br />

N. (2009b). Textile formwork for concrete shells. In Domingo, A. and<br />

Lzaro, C., editors, Proceedings of the 50th Anniversary Symposium of the IASS -<br />

”Evolution and trends in Design, Analysis and Construction of Shells and Spatial<br />

Structures”, Valencia, Spain. (paper on cd).<br />

Mollaert, M., De Temmerman, N., De Laet, L., Guldentops, L., and Vanthienen,<br />

T. (2010). Analysis of the deployment of a frame and a foldable membrane:<br />

the integrated model of a contex-t demonstrator. In Proceedings of the<br />

International Association for Shell and Spatial Structures (IASS) Symposium 2010,<br />

Shanghai ’Spatial Structures - Permanent and Temporary, Shangai, Chine. (paper<br />

on cd).<br />

Van Mele, T., De Temmerman, N., De Laet, L., and Mollaert, M. (2007).<br />

Retractable roofs: Scissor-hinged membrane structures. In Proceedings of the<br />

TensiNet Symposium 2007, ”Ephemeral Architecture, Time and Textiles”, pages<br />

283–291. Libreria, Milan.<br />

Van Mele, T., De Temmerman, N., De Laet, L., and Mollaert, M. (2010). Scissorhinged<br />

retractable membrane structures. International Journal of Structural<br />

Engineering - Special Issue on ”New Structures and New Analysis Methods”,<br />

1(3/4).<br />

221

222<br />

LIST OF PUBLICATIONS<br />

Van Mele, T., Mollaert, M., De Temmerman, N., and De Laet, L. (2006). Design<br />

of scissor structures for retractable roofs. In Adaptables 2006: Proceedings of the<br />

International Conference on Adaptability in Design and Construction, volume 2,<br />

pages 13–17, Eindhoven University of Technology, The Netherlands.

<strong>DEPLOYABLE</strong> <strong>TENSAIRITY</strong> <strong>STRUCTURES</strong><br />

development, design and analysis<br />

Inflatable structures have been used by engineers and architects for several decades. These structures offer<br />

lightweight solutions and provide several unique features, such as collapsibility, translucency and a minimal transport<br />

and storage volume. In spite of these exceptional properties, one of the major drawbacks of inflatable structures is<br />

their limited load bearing capacity. This is overcome by combining the inflatable structure with cables and struts,<br />

which results in the structural principle called Tensairity.<br />

A Tensairity structure has most of the properties of a simple air-inflated beam, but can bear to hundred times more<br />

load. This makes Tensairity structures very suitable for temporary and mobile applications, where lightweight<br />

solutions that can be compacted to a small volume are a requirement. However, the standard Tensairity structure<br />

cannot be compacted without being disassembled. By replacing the standard compression and tension element<br />

with a mechanism, a deployable Tensairity structure is achieved that needs - besides changing the internal pressure<br />

of the airbeam - no additional handlings to compact or erect the structure.<br />

The development of such a deployable Tensairity structure is investigated in this research. Insight is gained in the<br />

structural and kinematic behavior of this type of Tensairity structures by means of experimental and numerical<br />

investigations on small and large scale models. The first part of this dissertation focuses on the development of an<br />

appropriate mechanism for the deployable Tensairity structure. The second part investigates by means of<br />

experiments on scale models and numerical investigations the structural behavior of a deployable Tensairity beam<br />

and the effect of several design parameters on it. The insight gained in the first two parts results finally in the development<br />

of a full scale prototype of a deployable Tensairity beam, which is evaluated experimentally in the third part of<br />

this research.<br />

New insights in the structural and kinematic behavior of Tensairity structures are created with this research. They<br />

form a solid base for further research on deployable Tensairity structures and bring us one step closer to the<br />

realization of an optimal deployable Tensairity beam.<br />

March 2011<br />

Thesis submitted in fullment of the requirements for the awar d of the degree of Doctor in Engineering<br />

(Doctor in de Ingenieurswetenschappen)<br />

Advisors: prof. dr. Marijke Mollaert and dr. Rolf Luchsinger<br />

<strong>Vrije</strong> <strong>Universiteit</strong> <strong>Brussel</strong><br />

Faculty of Engineering<br />

Department of Architectural Engineering Sciences<br />

ISBN 978 90 5487 879 7<br />

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