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Design and Analysis Methodologies for

Inflated Beams

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Design and Analysis Methodologies for Inflated Beams

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector magnificus prof. dr. ir. J.T. Fokkema, voorzitter van het College voor Promoties

in het openbaar te verdedigen op vrijdag 17 juni 2005 om 13:00 uur door

Sebastiaan Leopold VELDMAN ingenieur luchtvaart en ruimtevaart

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Dit proefschrift is goedgekeurd door de promotor: Prof. ir. A. Beukers

Samenstelling promotiecommissie:

Rector magnificus Voorzitter

Prof. ir. A. Beukers Technische Universiteit Delft, promotor

Prof. dr. –ing. K. Drechsler Technische Universität Stuttgart

Prof. dr. J. Arbocz Technische Universiteit Delft

Prof. dr. Z. Gürdal Technische Universiteit Delft

Prof. dr. ir. R. Marissen Technische Universiteit Delft

Prof. dr. J. Loughlan Loughborough University

Dr. ir. O.K. Bergsma Technische Universiteit Delft

Dr. ir. O.K. Bergsma heeft als begeleider in belangrijke mate aan de totstandkoming van het proefschrift bijgedragen.

Published and distributed by DUP Science DUP Science is an imprint of

Delft University Press P.O. Box 98 2600 GM Delft The Netherlands Telephone: +31 152785678 Telefax: +31 152785706 E-mail: DUP@Library.TUDelft.NL ISBN: 90-407-2586-1

Keywords: Inflated beams, Braids, Cones Copyright © 2005 by S.L. Veldman

All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilised in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the publisher: Delft University Press. If any questions arise about the figures used in the thesis please contact S.L. Veldman, email: SLVeldman@yahoo.com

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Acknowledgements

After finishing my masters degree at TU Delft I took a holiday in Australia. In January 2001 I remember sitting on a beach and considering my goals for the future. I wanted to continue developing my skills as an engineer, preferably by focussing on one field, and to have the opportunity to work with and perhaps lead a team and to further develop my presentation skills. Not long after that I was offered the opportunity to undertake this Ph.D. at TU Delft. Though challenging at times during the four years of research I am pleased to say that I have fulfilled my goals as well as learned many other skills. I could not have achieved all of this so successfully without the help of the following people.

I am very grateful to my promotor Prof. Ir. Adriaan Beukers for the financial support and the privilege to explore the scientific world involved in inflatable structures at conferences all around the world. His talent for out-of-the-box thinking has helped me a lot. I am most appreciative of my direct supervisor Dr. ir. Otto Bergsma and of Dr. ir. Coen Vermeeren who fulfilled this role during the first two years of my research. The interesting discussions and their help have aided to the completion of this work.

I would like to thank Hans Weerheim who has been most helpful in establishing the test set-up. Furthermore I would like to thank Hans van Schie of TNO and FESTO for providing me the pneumatic equipment. Also the help of Peter de Regt of the Adhesion Institute and the donation of the bonding agents by DELO B.V. are most appreciated. After I performed the biaxial tube tests I was given the opportunity of comparing the test method with a cruciform biaxial test method at the Faculty of Civil Engineering. The help of ir. R. Houtman, ir. P. de Vries, ir. Ch. Nederpelt of Tentech B.V. and the Faculty of Civil Engineering has been most appreciated. The braided beams were created at August Herzog Maschinenfabrik GmbH & Co.KG with the help of V. Witzel and R. Kehrle. It was all made possible due to intervention of Prof. Dr. -ing. K. Drechsler. It has been most kind of A. Voskamp and S. Voskamp of Eurocarbon B.V. to show me their braiding company and sending me some sample braids.

As said in the begin paragraph one of my goals was to develop my skills as a team leader. This was made possible through supervising master students. So I am very pleased to have supervised Hessel van Kleffens, Béate Heru Utomo, Jeroen Breukels, Bas de Groot, Jaap Postma, and Brent Vermeulen. Their master thesis work resulted in various designs incorporating inflatable technology and working with them has been very motivating. Furthermore I would like to thank two interns: Kai Heinlein of the University of Stuttgart and David Sabolish of the University of Michigan for their assistance with the test set-up and experiments. I am most appreciative for the advice and support of Prof. Dr. J. Arbocz, ir. Jan Hol, ir. Paolo Tiso, and Dr. ir. Eelco Jansen. Their expertise has aided a lot in the

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theoretical and finite element analysis of cylindrical beams. Furthermore I would like to thank Dr. ir. Sotiris Koussios for his advice. Also the various papers dealing with inflatable beams send by Dr. J. A. Main are most appreciated.

I would like to offer my warm thanks to my parents-in-law, Tony and June Tentori for their care and support. I want to thank my family for their love and support and in particular my parents Piet and Marian Veldman for their guidance throughout the various stages of my life and for being there when I needed them most.

Lastly I would like to express my gratitude to my wife Renée for her love and unlimited support.

Sebastiaan L. Veldman June 2005

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Summary

Design and Analysis Methodologies for Inflated Beams By Sebastiaan L. Veldman

Inflated beams are part of the inflatable structures family. These structures are characterised by being made from a relatively thin skin which, unless pre-stressed, has very little capacity to bear compressive loads. In order to provide some more general knowledge of this family of structures, an overview is given of the various applications and related technologies. Three distinctive fields of applications have seen some renewed interest over the past decade. They are: civil, aeronautical and space applications. In civil engineering the pre-stressing of the skin is achieved by either a pressure differential over the skin or by loads applied to the edges. The latter method has wide-spread application in tent structures and therefore a majority of the research is focused on patterning and the effect of loads such as wind on the skin. In aeronautical engineering inflatable structures can be found in applications such as airships and inflatable wings. Equipping unmanned small aircraft with inflatable wings may reduce the weight and may be beneficial over rigid systems in term of energy consumption during wing deployment. A number of research projects have been undertaken since the mid nineteen-eighties in relation to using inflatable technology in space. Some of these projects have resulted in actual flight experiments. In 1996, a 14m diameter inflatable antenna was successfully deployed in space and in 2000 an inflatable re-entry vehicle was tested. Typical research areas are: wrinkling behaviour; static and dynamic mechanical behaviour; surface accuracy; and deployment modelling and control. The analysis of inflated beams as is the topic of this thesis focuses on static mechanical behaviour and optimisation for minimum deflection using the same volume of material.

Chapter Three is devoted to the bending behaviour of a straight inflated beam, which is the most primary shape of the family of inflated beams treated in this thesis. There is already a theory that predicts the bending behaviour of inflated beams. This theory is however limited to pure membrane materials. A modification was made to incorporate shell-like materials that may exhibit orthotropic material behaviour. Besides the way the material is treated as a membrane or as a shell there are also two versions on how to define the wrinkling threshold. One version assumes that the wrinkling threshold is related to the lowest principal stress becoming zero and the other assumes that this threshold is reached when the lowest principal strain becomes zero. It was found that the stress criterion should be used because the strain criterion does not predict a realistic behaviour in the post-wrinkled situation. Experiments have been conducted to verify the modified theory. Three different materials: PC; PPS; and PEI -have been used to make three 1m long inflatable beams. All beams were tested at various pressure levels on a displacement controlled testing device. The

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results were in agreement with the proposed theory.

Instead of making a beam out of a homogeneous material, it can also be made of fibres. As known from pressure vessels, the use of fibres can create new opportunities to optimise a structure. Chapter Four studies the behaviour of braided beams when bent and optimises a beam for minimum deflection using a constant amount of material. Two braided inflatable beams have been made that differ in the number of fibres that are placed along the length, and in the angle at which the bias fibres are placed. Each beam consisted of a silicone rubber bladder, two end caps and a dry carbon fibre braid placed over the silicone bladder. Experimental and theoretical analysis of the beams revealed that due to bending, the beams initially deflect in a linear manner like the Euler-Bernoulli beam model predicts. Once the stress in the axial fibres becomes zero, wrinkling occurs resulting in a significant loss of bending stiffness. The two beams that were tested were optimised for minimum deflection at a constant volume of fibres. The stiffest design has the maximum possible volume of fibres placed parallel to the beam.

Tailoring the material is one way to improve the performance of an inflated beam. Modifying the geometry is another way and this method is covered in the fifth Chapter -in which the optimum conical beam is derived. The load deflection model proposed in Chapter Three has been modified for isotropic truncated conical beams. The collapse load of a conical beam in bending is derived in a similar way as the collapse load of an axially compressed truncated cone. The collapse load is equal to the collapse load of a cylindrical beam having the same local radius and thickness. Optimising for minimum deflection using the same amount of PPS material resulted in a beam with a taper ratio of 0.5. The taper ratio is the ratio between minimum and maximum radius. Instead of verifying the result with an experiment the result was compared to a finite element model made in ABAQUS. A straight cylinder made of PPS foil having an identical geometry as the beam used in Chapter Three is analysed and subsequently compared to experimental results. Both results correlate reasonably well. Next, a finite element model for the optimum conical beam is analysed. The load deflection curve obtained using the finite element model is in agreement with the analytical model. A series of inflated beams may be used to form a wing structure. Chapter Six analyses the behaviour of such a series, with the goal of deriving a theory that can be used in the design of an inflatable wing. The beams are placed parallel to each other and are connected only at the ends. It is shown that the analytical theory correlates well with the finite element results for a structure consisting of three beams each having a different radius. Furthermore, the influence of taper on the mechanical behaviour is studied. The system of beams has been optimised for minimum deflection using the same volume of foil. The parameters that have been allowed to vary are the taper ratio and the scaling factors for the maximum radius and skin thickness. It is shown that the optimum taper ratio varies for different combinations of transverse shear force and torque moment.

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Nomenclature

A [m2] area B [-] cover factor b [m] width C [-] t r E v p f ax g

= ; Empirical correction factor

C [-] constant, Cbp [-] 2 rt E Mcr p

= ; Bending collapse coefficient for

pressurised cylinders

d [m] distance from the neutral axis of a circular beam

E [N/m2] modulus of elasticity; Youngs modulus

F [N] force G [N/m2] shear modulus L [m] length M [Nm] moment N [N/m] membrane force n [-] number of fibres p [N/m2] pressure

p* [-] = p/E(r/t)2 pressure parameter

Q [N] shear force

r [m] radius

R [-] ratio

s [m] coordinate in meridional direction

T [J/m] potential energy of the pressure difference

t [m] thickness

TR [-] taper ratio

U [J/m] strain energy per unit length

u,v,w [m] displacement functions in x,y,z direction

V [J/m] potential energy per unit length

V [m3] volume

v [-] volume fraction

W [J/m] applied moment per unit length

W [kg] weight

x,y,z [-] cartesian coordinates

Greek symbols

α [rad] semi vertex angle

β [rad] bias angle

γ [-] correlation factor

γ [-] shear strain

ε [-] strain

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system)

θ [rad] circumferential angle (cylindrical coordinate

system)

κ [m-1] curvature parameter

λ [-] ‘artificial’ Poisson ratio

λ [-] lagrange multiplier

σ [N/m2] stress

τ [N/m2] shear stress

υ [-] poisson ratio

φ [rad] circumferential angle (spherical coordinate system)

Suffixes a applied a axial b bias b bending c centre coll collapse cr critical cyl cylinder f fibre

fa fibre in axial direction

fb fibre in bias direction

g.cap gravity of end cap

i number of a fibre max maximum MD machine direction min minimum ref reference t torque TD transverse direction tot total w wrinkle

x,y,z cartesian coordinates

Abbreviations

AID Attached Inflated Decelerator

BC Boundary Condition

exp Experimental

FEM Finite Element Method

IRDT Inflatable Re-entry and Descent Technology

IRV Inflatable Recovery Vehicle

ISIS Inflatable Sunshield In Space

LVTD Linear Variable Displacement Transducor

MPC Multiple Point Constraint

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PC PolyCarbonate

PEI PolyEtherImide

PPS PolyPhenyleneSulphide

Turgor The force of the cell fluids of a plant pressing against the cell wall

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Contents

1. INTRODUCTION ... 1

1.1 GLOBAL OVERVIEW OF THE RESEARCH... 1

1.2 STRUCTURE OF THE THESIS... 2

1.3 LOAD DEFLECTION THEORY FOR STRAIGHT CYLINDRICAL BEAMS... 3

1.4 INFLUENCE OF MATERIAL... 3

1.5 INFLUENCE OF GEOMETRY... 4

1.6 BEHAVIOUR OF A SERIES OF BEAMS PLACED PARALLEL TO EACH OTHER... 5

2. OVERVIEW OF APPLICATIONS OF INFLATABLE STRUCTURES AND ASSOCIATED TECHNOLOGIES... 7

2.1 INTRODUCTION... 7

2.2 TENSILE STRUCTURES IN CIVIL ENGINEERING... 8

2.2.1 Air-supported structures... 8

2.2.2 Air-inflated structures ... 9

2.2.3 Tent structures... 11

2.3 INFLATABLE STRUCTURES IN AERONAUTICS... 12

2.3.1 Airships ... 12

2.3.2 Inflatable wings ... 14

2.4 GOSSAMER SPACECRAFT... 15

2.4.1 The Echo balloons... 15

2.4.1 Inflatable antenna experiment... 16

2.4.2 Inflatable sunshield in space ... 17

2.4.3 Re-entry vehicles ... 19

2.4.4 Inflatable habitats ... 21

3. BENDING BEHAVIOUR OF ANISOTROPIC INFLATED CYLINDRICAL BEAMS... 23

3.1 INTRODUCTION... 23

3.2 THEORY... 24

3.2.1 Equilibrium of forces and moments... 24

3.2.2 Collapse bending moment ... 27

3.2.3 Load deflection model ... 30

3.3 EXPERIMENTS... 33

3.3.1 Bi-axial testing of PC, PPS and PEI ... 33

3.3.2 Static bending test set-up... 36

3.4DISCUSSION... 38

3.5 CONCLUSION... 42

4. BENDING AND OPTIMISATION OF AN INFLATED BRAIDED BEAM ... 43

4.1 INTRODUCTION... 43

4.2 MANUFACTURING OF A BRAIDED INFLATABLE BEAM... 44

4.3 BENDING THEORY FOR BRAIDED INFLATED BEAMS... 45

4.4 BENDING EXPERIMENTS ON BRAIDED INFLATED BEAMS... 51

4.5 DESIGN FOR MINIMUM DEFLECTION... 54

4.5.1 Type 1: constant bias angle, no geometrical constraints to size of axial fibres ... 56

4.5.2 Type 2: constant bias angle, geometrical constraints on size of axial fibres ... 59

4.5.3 Comparison of the two types ... 59

4.6 CONCLUSION... 60

5. ANALYSIS OF INFLATED CONICAL CANTILEVER BEAMS IN BENDING... 61

5.1 INTRODUCTION... 61

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5.3 GEOMETRICAL OPTIMISATION FOR MINIMUM DEFLECTION... 66

5.4 COMPARISON OF A FINITE ELEMENT ANALYSIS OF A STRAIGHT AND A CONICAL BEAM... 69

5.4.1 Finite Element Analysis of a straight cylindrical beam... 69

5.4.2 Finite element analysis of a conical beam... 72

5.5 CONCLUSION... 74

6 STRUCTURAL ANALYSIS OF A SERIES OF INFLATED BEAMS PLACED PARALLEL TO EACH OTHER ... 75

6.1 INTRODUCTION... 75

6.2 ANALYTICAL THEORY OF A SERIES OF BEAMS LOADED IN BENDING... 76

6.2.1 Linear deflection theory ... 76

6.2.2 Wrinkling threshold... 80

6.2.3 Design for minimum deflection ... 82

6.3 LOAD DEFLECTION BEHAVIOUR OF A SYSTEM OF TAPERED BEAMS... 84

6.3.2 Wrinkling load prediction ... 86

6.3.3 Design for minimum deflection by altering the taper ratio ... 87

6.4 CONCLUSION... 88

7. CONCLUSION AND RECOMMENDATIONS ... 91

7.1 CONCLUSIONS... 91

7.2 RECOMMENDATIONS... 94

APPENDIX A: ENVELOPE DESIGN CRITERIA FOR AIRSHIPS... 97

APPENDIX B: BENDING OF INFLATED BEAMS USING A STRAIN CRITERION... 99

APPENDIX C: BRAZIER EFFECT ON ORTHOTROPIC PRESSURISED BEAMS ... 107

APPENDIX D: DERIVATION OF THE MEMBRANE PRE-BUCKLING STATE OF AXI-SYMMETRICAL BEAMS LOADED BY INTERNAL PRESSURE AND A TRANSVERSE SHEAR LOAD ... 113

D.1 FORCE EQUILIBRIUM EQUATIONS ON A SHELL ELEMENT... 113

D.2 INTRODUCTION OF EDGE LOADS... 117

D.3 TRANSFORMATION FROM POLAR TO CYLINDRICAL COORDINATES... 121

APPENDIX E: COMBINED BENDING AND TORSION ... 123

E.1 THE CASE OF PURE BENDING AND TORSION ON A BEAM MADE OF A MEMBRANE MATERIAL123 E.2 WRINKLING LOAD PREDICTION OF A CYLINDRICAL PRESSURISED SHELL LOADED BY PURE TORSION AND A SHEAR LOAD. ... 129

E.3 INFLUENCE OF TORSION ON A CONICAL BEAM SUBJECTED TO A SHEAR LOAD... 134

APPENDIX F: GRAPHICAL OUTPUT OF THE MODAL ANALYSIS OF A SYSTEM OF BEAMS SUBJECTED TO A SHEAR LOAD. ... 141

REFERENCES... 147

SAMENVATTING ... 153

ABOUT THE AUTHOR ... 155

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1. Introduction

Inflatable structures are part of a characteristic group called tensile structures. A tensile structure is a membrane-like structure that requires tensile pre-stresses in order to bear externally applied compressive loads. An inflatable structure creates these pre-stresses by means of a pressure differential over the skin. Inflatable structures offer advantages such as being lightweight, being easy to erect and having a low storage volume. Furthermore, the production of these structures relatively cheap so for some applications, like for instance in space, the use of inflatable technology may offer significant cost savings. Inflatable technology can be used for applications in the areas of civil, nautical, aeronautical and space engineering.

1.1 Global overview of the research

Out of all main application areas, space is likely to gain the most benefit from the typical advantages that inflatable technology offers. The conditions for structures in space in general are challenging. It is therefore not surprising that the most research undertaken is in this field. Examples of research are modelling the wrinkling behaviour of thin films (Wong et.al. 2003), and modelling deployment of thin film structures (Lienard 2005). Large thin film structures in space are often supported using inflatable beams. In aeronautical and civil applications, inflatable beams have a wide spread use as well. All these application areas may therefore benefit from research of inflatable beams.

Figure 1.1 Research topics associated with inflatable beams

Figure 1.1 provides a schematic overview of several research topics related to inflatable beams. This thesis focuses on the static behaviour of inflated beams. A different research topic is for instance research to rigidizing the material. Rigidizing in this context means that the material transforms from a flexible state

Material -rigidizing Inflatable beams Static behaviour Dynamic behaviour Deployment Theory: Experiments: Finite elements:

Straight, conical, and braided beams Straight and braided beams

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to a rigid state. Using this technology there will be no need for inflation pressure once a structure is inflated so it can increase the mission duration substantially. The dynamic analysis focuses on the response of inflated structures to vibrations. Deployment on the other hand deals with modelling the deployment of a structure and also researches ways to control this deployment.

An inflatable beam relies on an overpressure to deal with externally applied compressive loads therefore the structural behaviour is more influenced by the kind of material, geometry and applied loads than rigid structures. Figure 1.2 shows the trinity between material, geometry and applied loads / turgor combination. Turgor is a term that originates from biology and represents the force of the cell fluids of a plant pressing against the cell wall. A plant derives a lot of its solidity from this turgor effect and it is analogous to the function of internal overpressure of an inflatable beam.

Figure 1.2 Trinity between geometry, material, and loads / turgor

The title of this thesis is: ”Design and Analysis Methodologies for inflated beams”. The design part of the thesis focuses on the influence of material and geometry on the mechanical behaviour of an inflatable beam structure and the analysis part covers theoretical, finite element and to a lesser extend experimental analysis.

1.2 Structure of the thesis

The structure of this thesis is as follows: Firstly an overview is given of the various applications of inflatable structures in civil, aeronautical and space engineering in Chapter Two. It is intended to provide a more broad view on inflatable structures. Chapter Three discusses the load deflection behaviour of an inflated beam. For this purpose an existing theory based on membranes is modified to meet shell behaviour of the material. Chapter Four investigates how the skin material can be tailored to minimise the deflection. It provides a theoretical model that is used to predict the deflection behaviour of a tri-axial

Loads / Turgor Material

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braided inflated beam. Chapter Five examines the role of geometry on the deflection behaviour. It deals with truncated conical beams and besides the theoretical analysis, it also deals with finite element analysis. Chapter Six involves the interaction between a series of parallel placed inflated beams. Such a series could be used to form an inflated wing. The last Chapter (Chapter Seven) provides the conclusions and recommendations. The next sections provide a more detailed introduction to the various topics addressed in this thesis.

1.3 Load deflection theory for straight cylindrical beams

There is a number of analytical theories that one may use to analyse the bending behaviour of pressurised thin-walled beams. The best known theory is probably the membrane theory first derived by Stein et.al (1961) and later extended to a load deflection theory by Comer et.al. (1963). This theory treats the material as a membrane and assumes that wrinkles are formed as soon as the lowest principal stress becomes zero. Main et.al (1995) used the membrane load deflection theory but instead of using the stress wrinkling criterion they used a criterion based on the lowest principal strain becoming zero. The peculiar thing about the latter theory is that it provides a wrinkling moment that is dependent on the Poisson ratio, and as soon as this ratio becomes 0.5 the wrinkling moment becomes zero. This is highly unlikely so a stress criterion is selected. The membrane theory has a drawback that it predicts collapse loads that are solely dependent on the applied pressure and not on the material used. Various literature (Axelrad 1980, Baruch et.al. 1992, Brazier 1927) has provided several different expressions for the collapse load for circular cylindrical beams that are either pressurised, or not, or consist of isotropic or anisotropic material. Using some experimental results a new semi-empirical expression for the collapse load of an orthotropic pressurised beam is presented in this thesis. This expression is used to modify the membrane theory by Stein et.al. (1961). Another novel aspect of the load deflection of straight cylindrical beams addressed here is the testing method. The experimental results on the bending of inflated beams published by Main et.al. (1995) and Webber (1982) were obtained using a force controlled test bench. The experimental results presented in this theses however are obtained using a displacement controlled test bench. This type of test bench is advantageous over the force controlled one because it allows for retrieving test data in the region of the load deflection curve where a very small increment in load results in a relatively large increment in displacement.

1.4 Influence of material

Like the trinity figure depicts, the material used plays an important role in the mechanical behaviour of an inflated beam. This is, of course, also the case with rigid unpressurised structures however inflatable structures rely on pre-stresses whose magnitude determines whether wrinkles occur or not. Textile materials can be tailored to direct the pre-stresses to the location where they are most effective. Braiding is a textile production process that is suitable for producing tubular

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structures. It allows fibres to be placed along the axis of the beam and at a particular angle with respect to the axis of the beam. This angle is called the bias angle. Relatively little has been published on the mechanical behaviour of a braided inflated beam. Brown et.al provide a limited theoretical analysis and some experimental results. They stated that increasing the bias angle of the braided beams might increase the moment of incipient wrinkling. To uncover more about the mechanical behaviour of braided inflated beams, an analytical theory has been derived which has been verified to experimental results. In addition some optimisation has been performed to provide understanding of the various parameters.

1.5 Influence of geometry

The geometry is of more influence to the mechanical behaviour of inflated beams than to rigid beams because the pre-stresses depend on radii of curvature. So altering the radii of curvature allows a designer to tailor the pre-stress distribution. This thesis is limited to single curved beams such as truncated cones. Only a small number of publications have been found on the bending of pressurised truncated cones. On the other hand, quite some research has been published on axial compressed unpressured truncated cones. Weingarten et.al. (1968) provide general design criteria for buckling of thin-walled truncated cones. They provide a conservative expression for the collapse moment of a pressurised conical shell under pure bending. A different expression for the collapse of a conical shell in bending is provided by Seide (1962). He states that the critical stress for a conical shell under axial compression should be equal to the critical stress of a cylinder having the same local radius of curvature. This is a similar approach for relating the buckling load of axially compressed cylinders to that of truncated cones (Esslinger et.al. 1980, Hausrath et.al. 1962, Seide (1962), Spagnoli 1999, Zhang 1989, 1993). Arbocz (1968) confirmed that such an approach is valid for cones with a semi-vertex angle smaller than 45˚. This approach is adopted in this thesis as well. A load deflection theory is derived using the newly derived theory for straight cylindrical beams. No experiments have been conducted so to provide some support to the theory, finite element computations are made. Firstly, finite element results of a straight cylindrical beam are compared to experimental and theoretical results. All three analysis methods give a comparable load deflection curve. Secondly, a finite element model is made of a tapered beam and compared to theoretical results. Again reasonable correlation has been found. The taper ratio of the tapered beam has been determined by an optimisation process. Having a straight beam as a reference geometry, the taper ratio, maximum radius and skin thickness have been varied such that the total weight of the beam remains constant, the maximum stress of the straight beam is not exceeded and no wrinkling takes place. As a result a taper ratio of 0.5 has been determined to be the optimum one.

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1.6 Behaviour of a series of beams placed parallel to each

other

As outlined in section 1.1, inflatable beams can be regarded as building blocks for larger structures. A typical example of such a structure is an inflatable wing, where a series of beams, are placed parallel to each other form the wing. This principle has already been applied in a few designs (Brown et.al. 2001, Murray et.al. 2002). Inflatable wings can be used successfully for specific aircraft types that require their wing to be deployable. There are only a limited number of publications on the structural behaviour of inflated wings. Brown et.al. (2001) and Murray et.al. (2002) describe the concept of an inflatable wing used for a small unmanned aircraft. This wing consists of 5 inflated beams that support a fabric which, in combination with open cell foam, creates an airfoil. Although some results of the bending tests of the inflated wing are published, the publications do not provide an in depth theoretical analysis. Crimi (2000) provides a technical note on the divergence of an inflated wing. Although his publication provides some insights in the interaction of the beams more research is necessary to provide a complete analytical model that can describe the load deflection behaviour. This thesis presents such a model for a series of straight and tapered beams. The load deflection curve obtained with the analytical theory correlates well to the one obtained with a finite element model. In order to improve the mechanical characteristics of a series of beams some optimisation has been performed. The series of straight beams have been optimised for minimum deflection thereby keeping the volume of skin material and applied load and pressure constant. Only the thicknesses of the beams were used as optimisation variables. The structure consisting of cones was optimised under the same criterion. The optimisation variables were the taper ratio, the factor with which the maximum radii change and the factor with which the thicknesses of the beams changed.

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2. Overview of applications of inflatable structures

and associated technologies

This Chapter provides an overview of applications of inflatable structures and the relevant technologies associated with these applications. Three distinctive fields of applications will be discussed: civil, aeronautical and space applications. For civil applications two types of inflatable structures are used air-supported structures and air-inflated structures. In aeronautical engineering inflatable structures are found in applications such as airships and wings. Lots of research projects have been started from the mid nineteen eighties to use inflatable technology in space.

2.1 Introduction

In general, inflatable structures offer advantages such as being lightweight and simple to deploy and additionally have a high packaging efficiency. These advantages are incorporated in many different applications of inflatable technology. One can think of a commonly known product such as life jackets and rafts. For these applications the three main advantages are fully utilised which makes the use of inflatable technology very successful.

Figure 2.1: Characterisation of membrane structures

As said before the overview given in this Chapter focuses on three distinctive fields of applications; civil engineering, aeronautical, and space. From the mid 1990s there has been a significant increase in the research in these three areas. On each specific field, organisations were formed to accommodate workshops and conferences for knowledge sharing. To provide a global overview of the entire field of engineering figure 2.1 was created. Figure 2.1 characterises the different types of membrane structures. There are basically three prime types of membrane structures. The first two pre-stress the membrane using a pressure differential over

Air-supported structures Space antennae/reflectors

Tents, Solar sails Dual wall structures

Inflatable beams Tensioning due to pressure Membrane structures 3. Tensioning due to edge loads

2. Large radii, low pressure differential 1. Small radius, high

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the membrane and the third provides the membrane with pre-stresses by loading the edges. The two pressure stabilised types are further subdivided. The first type has a small radius and a high pressure differential like inflatable beams. The second one is characterised by relatively large radii and a low pressure differential such as for example an air-supported roof structure for a tennis hall.

The following sections will provide examples of typical structures in all three distinctive engineering fields. Besides looking at the examples attention will also be paid to specific research area’s involved.

2.2 Tensile structures in civil engineering

Inflatable structures in civil engineering can be divided into two categories, air-supported structures and air-inflated structures.

2.2.1 Air-supported structures

Air-supported structures are a form of pressure stabilised membrane structures that rely on a pressure difference between the interior and the exterior of the building. Probably the earliest attempt to use air-supported structures in architecture was a proposal for an air-supported hall by Lanchester in 1917 (Price et.al. 1971). His design however could not be realised partly due to lack of availability of suitable materials and manufacturing processes but mainly due to lack of acceptance of this type of construction by the public. In the mid nineteen forties, Walter Bird realised an air-supported radome (Dent 1971). The elegant way of dealing with forces of an air-supported structure has drawn the attention of architects ever since as it offers them a way to create temporary structures, portable buildings and to span large areas.

An air-supported structure consists of a single membrane held in place by a relatively low-pressure differential. The air-supported structure differs from more conventional structural forms by the membrane material that does not directly resist the externally applied loads. The membrane is used to enclose the internal volume of air, which in return supports the applied loads. In theory, if the external loads where uniform and equal to the internal pressure, the membrane would just be a separating medium. For this case the membrane would be free of any tensile stress and therefore the span of an air-supported structure would theoretically have no limit. In practice however, the loading is never uniform and the pressure must be maintained at a higher level to be able to bear e.g. wind or snow loads on the structure.

A typical example of an air-supported structure is the roof system of a sports stadium. Figure 2.2 shows the interior of an air-supported tennis hall in Delft. The roof of an air-supported structure consists of a membrane that is held in place by an internal overpressure. Special entrances to the building such as airlocks are required to maintain a specific internal overpressure. The anchorage cables used are an essential part of the structure as a whole. Variation of the anchorage system

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layout changes the radius of curvature of the membrane, which in the end allows a designer to tailor the behaviour or the structure with respect to the loads.

Figure 2.2: Interior of an air-inflated tennis hall at the TUDelft Sports centre

2.2.2 Air-inflated structures

An air-supported structure differs from an air-inflated one by the continuous replenishment of air. This is because with an air-supported structure the inhabitants are in a higher air pressure environment than the atmospheric pressure and by movements to inside and outside loss of air pressure will occur. Air-inflated structures are closer related to more conventional structures. Closed membrane sections are inflated to form structural elements like beams, columns, arches and walls. The load bearing capacity of these structural elements is dependent on:

· The internal pressure level

· The membrane material properties · The shape of the element

Air-inflated structures can be further divided into the sub-category of tubular frame structures or dual wall cushion structures, see figure 2.3. Tubular structures are made up of a framework of inflated hoses under high pressure, often supporting a fabric membrane. The supported membrane can add considerably to the stability of the structure. Inflated dual wall structures consist of two membrane

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walls, often connected by drop threads or diaphragms, under internal pressure. With this type of construction more complex shapes are possible.

Figure 2.3: Cross sections of three types of air-inflated structures

Figure 2.4: An inflatable stage cover

The structural elements in air-inflated construction are self-contained. In case of air-inflated architectural applications this means that provision of special entrances as in air-supported structures are not required. Because of the absence of an uplift force an anchorage is only needed to withstand external loads like wind. Furthermore, because of the self-containment of the structural elements, a constant air-supply would not be necessary in theory. It is however impossible to construct a completely airtight structure and a periodic replenishment of air (every 3 to 6 months) is usually needed. In case of larger contained volumes of air, a constant air supply might be required. Temperature variations will cause a change in pressure. Air-inflated structures at a relative low pressure might therefore require a continuous pressure control, for higher pressures this effect will be less pronounced.

Tubular frame Dual wall

diaphragms

Dual wall drop threads

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An inflatable stage, see figure 2.4 is a typical example of an air-inflated structure for temporary use. The stage cover is built up from a series of inflatable beams. This example shows the full utilisation of the three advantages of inflatable structures namely being lightweight and easy to deploy or erect and having a low storage volume. The structure needs less people to erect it and because of the abundance of a rigid elements it weighs less than a conventional stage. The inflatable stage however does require a compressor.

2.2.3 Tent structures

Mankind has been using fabric structures in civil engineering for a long period of time. Nomads used fabrics to create tents (see figure 2.5). The tent poles pre-stresses the membrane in order to overcome the loads acting on the fabric such as dead weight, wind, rain and snow. The structure is an example of a division between tension and compression members. The tent poles are in compression whereas the fabric is in tension.

Figure 2.5: The black tent

The tent poles stabilise the fabric of a tent. With an inflatable structure, this stabilisation proceeds through air-pressure. Most of the research in tensile structures for civil engineering applications is done in the field of tent structures. Tensinet, an EU funded institution was established in 2001 to connect architects, civil engineers, universities and companies (Tensinet). On of the aims was to identify gaps in the current knowledge of tensile structures. Examples of specific research topics are: form finding and patterning, structural analysis, creep behaviour of fabrics, and acoustics.

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Figure 2.6 shows an example of a modern tent, an open air theatre in Soest, the Netherlands. This tent is a combination of a high point, and an arch type tent (Tentech). The arch is a hollow double tapered and curved beam. It is manufactured by creating an inflatable beam with the desired shape, next drape it with fibres and then apply vacuum infusion to create a composite beam. The same principle could also be used to manufacture blob buildings (Pronk et.al. 2002).

Figure 2.6: Open air theatre in Soest (left); composite arch (middle); vacuum injection of the composite arch (right) (Tentech)

2.3 Inflatable structures in aeronautics

Inflatable structures have a long history in aeronautical applications such as airships and balloons. From the mid 1990s, research to airships has seen some revival. When the German company Cargolifter experienced some financial problems at the end of 2002, the initial rise in research activities seems to be in decline. Another application of inflatable structures in aeronautics is an inflatable wing. Due to the relative ease of deployment and low storage volumes these wings might successfully be used on Uninhabited Aeronautical Vehicles (UAVs).

2.3.1 Airships

One of the best-known applications of an inflatable structure in aeronautical engineering is the airship. The research on airships has seen some revival since the mid 1990s as well. Since then almost every year a conference was held entirely devoted to airships. In the Netherlands the platform of airships was established to promote the use of airships in this country (platform of airships). There are several application areas for airships. One of the best known is for advertisement. With the increasing need for security, surveillance from the air has become another role for the airship. Research has been carried out to use an airship for transporting heavy loads and to use it as a high altitude platform for communication purposes (Veldman et.al. 2001a). The latter application involves an unmanned airship that has to maintain a specific position at 20 km altitude for a period of 5 years. Because of the renewed interest in airships in Germany, the airworthiness authorities developed airworthiness requirements for airships. Especially the requirements posed to the envelope materials might be very useful for the design of inflatable structures (Miller et.al. 2000). Appendix A states some

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design requirements for airship envelopes. Despite the enthusiasm seen with regard to airships, lack of funding remains an issue. It therefore remains uncertain where all efforts will lead to.

Figure 2.7: Unmanned Airship at Duxford airfield

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2.3.2 Inflatable wings

Several attempts have been undertaken to apply inflatable technology to aircraft wings in the late fifties and early sixties. The ‘Inflatoplane’ program of Goodyear (see figure 2.8) started in the 1950s and lasted until 1973. In that period of time a total of twelve ‘Inflatoplanes’ were developed, tested and evaluated. The ‘Inflatoplane’ is probably one of the most successfully designed inflatable aircraft (Brown et.al. 2001). The ‘Inflatoplane’ could be dropped in a container to rescue downed pilots. The wing was constructed out of the 3D fabric called ‘Airmat’ (Taylor 1960) and was inflated with a differential pressure of about 48 kPa. ‘Airmat’ consisted of two fabric panels joined together by thousands of dropped threads. The length of these threads could be varied to obtain a curved surface. A recent project of an inflatable wing for a manned flight is ‘Pneuwing’, which is an ultra light aeroplane designed by the Swiss company prospective concepts (Prospective concepts) and German based Festo. The plane made its first flight in 1998 and its purpose is to gain more knowledge on inflatable wings. It has a wingspan of 8.2 m and the pressure differential inside the wing is 0.7 bar.

Figure 2.9: Inflatable wing on the I-2000 craft (left) inflatable wing in manufacturing jig (right) (Brown et.al. 2000)

Inflatable technology is also applied for unmanned aeronautical vehicles as well. Vertigo Inc. designed and constructed inflatable deployable wings for a US Navy program (Murray et.al. 2002). This program utilised the wings to add glide capacity to gun-launched munitions. The wings of the so-called Gun Launched Observation Vehicle were to be inflated near the target area to guide and slow down the aeroplane. The deployment took place in less than a second. The construction of the inflatable wing is depicted on the right side of figure 2.9. The wing was built up from braided inflatable tubes covered by a fabric to provide a smooth surface. The gaps between the tubes and the fabric were filled with open cell foam. The wings developed for the US Navy program were later used in the

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I-2000 aircraft also depicted in figure 2.9 (Murray et.al. 2002). The I-I-2000 test program has provided more understanding of the structural, aerodynamic and operational aspects of an inflatable winged aircraft. Only a small number of publications were found on the mechanical behaviour of an inflated wing. Chapter Six is therefore devoted to uncover more of this behaviour for a set of parallel placed inflated beams.

2.4 Gossamer spacecraft

For space applications, inflatable structures must be seen as part of a much wider research into thin film structures. These structures are also known as Gossamer structures. Since 2000, each year a conference called AIAA Gossamer Spacecraft Forum is held that deals specifically with thin film structures for space applications. Currently, they are trying to set up an AIAA Technical Committee specialised for Gossamer spacecraft structures. In Europe, the European Space Agency (ESA) started to organise biannually workshops on inflatable structures since 2002. Because the four key advantages of inflatable structures may fully be exploited for space applications, much more research is done to space applications than to the other two applications. This section will describe several applications for membrane structures in space. The following research area’s can be observed: · Analysis of wrinkling of thin films

· Modelling deployment of inflatable structures · Rigidizing technologies

· Membrane optics

· Static and dynamic analysis

This section will only briefly discuss the engineering field of space applications. Jenkins (2001) provides a complete work on the developments on gossamer spacecraft.

2.4.1 The Echo balloons

In the early days of space flight, rockets had limited power and the payload had to be small and lightweight. Industry initiated the idea to use inflatable structures to meet these limitations. The Echo I and II missions launched the first successful inflatable structures in space (Veldman et.al. 2001b, 2002). The Echo I was a passive communication satellite in space. The spherical shape of the Echo I is made using 12µm-thick Mylar gores, coated with vapour-deposited aluminium. The technological developments in this project incorporate handling, processing, manufacturing and high-precision assembly of an ultra thin membrane structure. The sphere had a diameter of 30 m, weighed 63 kg and was stowed in a 66 cm spherical container for launch. To inflate the balloon, sublimating powder was used (Jenkins 2001). The echo I balloon clearly demonstrated the significant advantages that inflatable structures have over rigid structures.

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Figure 2.10: Echo I balloon

2.4.1 Inflatable antenna experiment

The IN-STEP Inflatable Antenna Experiment (Freeland et.al. 1996, 1998b) is an example of an inflatable structure in space that involves many of the associated research areas. On May 20, 1996, the STS 77 mission successfully deployed an inflatable antenna (fig 2.11). The experiment was intended to demonstrate the maturity of the inflatable technology. The antenna consists of an inflatable torus, which supports a 14-meter diameter inflatable parabolic membrane reflector structure. One half of the inflatable reflector facing the payload bus is transparent whereas the other half is given a reflecting coating on the side that faced the payload bus. The inflatable torus was connected to the spacecraft with three 28-meter long inflatable struts. The canopy and reflector were respectively fabricated from 7-microns thick Mylar and aluminised Mylar. The large curved surfaces were constructed from segments (gores) that we joined with Mylar tape. The torus and struts were made of Neoprene rubber coated Kevlar fabric with a thickness of 0.3 mm. The cost of the whole experiment was in the order of $10 million (Freeland et.al. 1996, 1998b). The inflatable antenna experiment proved to be very valuable for inflatable technology to become mature to be used in space engineering. The surface accuracy for instance plays an important role in the success of an inflatable antenna. It is dependent on the construction method, the material properties and the boundary conditions at the support of the membrane (Freeland 1996). Some important lessons were learned from the deployment of the inflatable antenna as it deployed prematurely. The magnitude of the residual gas and strain energy of the structures was not anticipated correctly. This caused uncontrolled deployment of the antenna, which could have led to failure of the

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entire mission. The hazard of uncontrolled deployment is that the rigid payload bus might tear the fragile membrane.

Figure 2.11: Inflatable Antennae deploying (Jenkins, 2001).

2.4.2 Inflatable sunshield in space

NASAs Goddard Space Flight Centre has been studying an Inflatable Sunshield In Space (ISIS) as it may be used to passively cool the Next Generation Space Telescope (NGST) (Pacini et.al. 1999). The NGST is planned to be launched in 2007. Figure 6.12 depicts a model of the ISIS. The sunshield consists of four diamond-shaped membrane layers mounted to a central rigid container. The membranes are made of aluminised polyimide film. Fully deployed the system measures 15 x 35 x 1 meters.

Figure 2.12: Deploying the Inflatable Sunshield In Space (Cadogan et.al. 1999).

Four inflatable circular cylindrical beams support the sunshield. Together they form a crucifix joined in the centre by a rigid container. The design of the NGST sunshield is governed by issues such as mission life, weight and systems control. The analysis of wrinkling formation is important for the NGST sunshield design since wrinkling and creasing of the membranes may degrade the surface coatings.

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They should therefore be minimised. (Pacini et.al. 1999) Two other key research areas that are incorporated in the design of the inflatable sunshield are deployment control and rigidization techniques.

Figure 2.12 shows the deployment of the sunshield. Current deployment techniques are usually focused on members such as booms and struts (Freeland 1998a). These members are used to position other elements such as the sunshield or an antenna. A deployment method must keep the structure in a predictable envelope during inflation and provide a slow deployment rate to prevent significant load on the spacecraft. There are several techniques available to deploy an inflatable boom (Cadogan et.al. 1999) and only a few of them are discussed here.

· Roll out method · Mandrel method · Compartmentalisation

The roll out method is similar to the well-known party favour. The inflatable beam is rolled up in its storage phase and rolled out in deployment phase. Rolling out can be controlled by using a brake mechanism at the end of the beam. This type of deployment mechanism is used on the beams of the ISIS depicted in figure 2.12. The mandrel method involves folding of the beam beneath a mandrel. During deployment the beam is pulled over the mandrel by the pressure on the other side of the mandrel (Freeland et.al. 1998a, Cadogan et.al. 1999) see figure 2.13. Figure 2.14 shows the method of compartmentalisation. Compartmentalisation is the method where the beam is divided into compartments. Deployment proceeds when the burst patches or pressure relieve valves that divide the sections allow each compartment to be pressurised.

Figure 2.13: Mandrel method. Figure 2.14: Internal Compartmentalisation.

1). Clamp 2). Outer sleeve 3). Deformable seals 4). Collapsed tube 5). Mandrel

1

2

3

5 4

1). Clamp 2). Collapsed tube 3). Fabric bulkhead 4). Burst patches 4). Break cords

1

2

5

4

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Rigidisation is a technology developed to increase the lifespan of an inflatable structure in space. Because inflatable structures derive their stiffness from a pressure differential over the membrane they are vulnerable to impact from for instance micro-meteorites. Cadogan et.al. (2001) define rigidizable materials as materials that are initially flexible to facilitate inflation or deployment, and become rigid when exposed to an external influence.

There are several rigidization techniques available and they can largely be divided into the next four groups.

· Thermoset curing

· Thermoplastic rigidisation · Foam inflation

· Aluminium laminate

Both the thermoset and thermoplastic rigidization systems are based on forming a composite in space. Curing of thermoset systems can proceed by exposure to light or heat. The beams of the ISIS use a thermoset rigidization system that uses heat. Each boom is heated for 45 minutes at 120 ºC to cure the resin (Pacini et.al. 1999). A thermoplastic rigidization system requires a heating device to heat up the resin above it glass transition temperature where it is flexible enough to allow the beam to be deployed. The foam inflation technique involves inflation a beam with foam instead of gas. The aluminium laminate technique uses a laminate of aluminium and a plastic foil. During inflation the aluminium is stressed above its yielding point after which the pressure is relieved. The plastically deformed aluminium is responsible for the stiffness after deflation.

2.4.3 Re-entry vehicles

The benefits of inflatable deceleration systems were already recognised in the nineteen-sixties. NASA studied a re-entry vehicle using an Attached Inflated Decelerator (AID) (Deiveikis et.al. 1970, Bohon et.al. 1970, Gillis 1968). The studies concentrated on a ram-air inflated decelerator system for entry of the Mars atmosphere. An AID (figure 2.15) is a low-mass inflatable canopy directly attached to a payload. The AID is designed as an efficient deceleration system at high supersonic speeds in low-density planetary atmospheres. Its canopy is made of Viton coated Nomex fabric (Mikulas et.al. 1969).

In the late nineteen eighties Aerospace Recovery Systems, Inc. developed an Inflatable Recovery Vehicle (IRV) (Kendall et.al. 1990, 1991). A truncated cone is created with a small torus linked by inflatable cylindrical elements to a larger torus (figure 2.15). The canopy can be made of Nextel or Kevlar fabric that is coated with a suitable insulative coating such as silicone. The payload is located at the top of the truncated cone to ensure that the IRV will stabilise in flight like a badminton shuttlecock.

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Figure 2.15: Attached Inflatable Decelerator (left); Inflatable Recovery Vehicle (middle and right) (Kendall et.al. 1990).

In 1996 the Russian Mars-96 project carried two small penetrators designed to land on the surface of Mars. The penetrators were designed with an inflatable re-entry shield. Unfortunately, the spacecraft failed to reach its nominal orbit and the inflatable re-entry system could not be tested. The experience of the Mars-96 project was used in the IRDT (Inflatable Re-entry and Descent Technology) demonstrator. (Marraffa et.al. 2000). The IRDT (figure 2.16) project is developed by a co-operation between Astrium-I, NPO Lavochkin and ESA.

Figure 2.16: Inflatable Re-entry and Descent Technology demonstrator (Marraffa et.al. 2000).

In February 2000 a Soyuz Fregat launcher took the IRDT demonstrator into orbit. After re-entry of the demonstrator it was found that the envelope was partly damaged. Despite this, the flight data and hardware were successfully recovered. The construction of the IRDT consists of a small ablative rigid nose shield and a conical shaped inflatable envelope, which is inflated in two stages. The first stage is an entry shield consisting of an internal network of rubber hoses pressurised with nitrogen. The inflation of the first stage increases the diameter of the vehicle from 0.8m to 2.3m. The first stage is covered with a silica-based fabric impregnated with an ablative material. The second stage replaces a parachute system. When it is inflated the diameter of the vehicle increases to 3.8m.

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2.4.4 Inflatable habitats

Any habitat in space will be a pressure vessel by virtue of having an internal pressure higher than the external atmospheric pressure. The use of a flexible membrane instead of a metallic membrane may offer a reduction of mass and storage volume compared to metallic counter systems. The Centre for Engineering Infrastructure and Sciences in Space has made lots of effort in developing of an inflatable prismoidal module concept (Criswell, et.al.). The structure consists of a spherical roof, sub-floor and prismoidal side-wall members connected by a pressurised frame comprising inflated columns and arches (figure 2.17). The structure offers a lightweight and small stowage volume, modularity for base expansion, a large usable floor area relative to total interior volume, easy application of a layer of regolith for shielding and minimum deployment, operation and maintenance.

Figure 2.17: Proposal for prismoidal inflatable Lunar/Martian space habitat (left)(Criswell 1993); Transhab (middle and right (Jenkins 2001)).

Space habitat proposals involve utilisation in three different environments: a habitat for the International Space Station (ISS), a lunar habitat and a Martian habitat. NASA Johnson Space Centre has researched an inflatable habitat called TransHab which is short for Transit Habitation Vehicle (figure 2.17). TransHab can be used as a transfer vehicle to Mars, or as a habitation module on the ISS (Cadogan et.al. 1998). TransHab is build around a central core to decrease its volume for launch. When inflated, TransHab is approximately 9.1 m in length and 7.6 m in diameter. The structural layer of the membrane consists of a series of Kevlar webbings that are interwoven and indexed to one another to form a shell. The TransHab offers three times more living space than its one-room aluminium competitor and will feel much more like a real home.

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3. Bending behaviour of orthotropic inflated

cylindrical beams

The bending behaviour of inflated beams has been studied for several decades. Several models have been developed to predict the load deflection behaviour of these beams. Each models treats the problem differently, particularly in the way the wrinkling and collapse moment are defined. This Chapter relates the wrinkling threshold to a stress criterion rather than a strain criterion. An important issue with respect to the collapse load is whether to regard the material as a true membrane or as a very thin shell. It will be shown that the three thin films used in this research PC, PPS, and PEI should be regarded as thin shells. A new model that predicts the collapse moment for these kind of materials has been developed, which incorporates orthotropic material properties. These material properties were obtained via biaxial testing by means of axial loading of a pressurised tube. Experiments were conducted to obtain the load deflection curves at five different pressure levels. The results were shown to correlate reasonably well with the proposed theory.

3.1 Introduction

Various papers have been published on the load deflection behaviour of circular cylindrical inflated beams (Comer et.al. 1963, Main et.al. 1995, McComb et.al. 1962, Stein et.al. 1961, Webber 1982, Wielgosz, et.al. 2002). When a beam is bent, the material can be in two distinctive states: the taut state and the wrinkled state. In the taut state the load deflection curve of the beam shows a linear trend whereas in the wrinkled state it shows non-linear behaviour. The wrinkling threshold of a membrane may be based on a strain criterion (Main et.al. 1995) or on a stress criterion (Comer et.al. 1963, Stein et.al. 1961, Webber 1982). The strain criterion (i.e. principal strains become zero) applied by Main et.al. (1995) showed good correlation with the test results they obtained. However, they found a value for the Poisson ratio of 0.5 for the fabric used. This Poisson ratio resulted in a wrinkling moment equal to zero. This is considered to be highly unlikely and therefore a criterion based on stresses is used. Appendix B discusses this in more detail. The collapse moment has been modelled in many different ways (Axelrad 1980, Baruch et.al. 1992, Brazier 1927, Comer et.al. 1963 Donnell 1933, Jamal, 1998, Main et. al. 1995, Mcomb et.al. 1962, Seide et.al. 1968, Stein et.al. 1961, Suer et.al. 1958, Webber 1982, Wielgosz et.al. 2002, Wood 1958, Zender 1962). When the material is considered as a membrane the collapse moment becomes a linear function of the pressure. The collapse moment will become zero as soon as the pressure becomes zero. Wielsgosz et.al. reduced the collapse moment with a

factor p/2 because their experiments with highly pressured fabric beams showed

that the wrinkled region was different than the theory predicted. The problem with these membrane formulations of the collapse moment is that they are independent of the material used. When the material is modelled as a shell instead of a

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membrane, a beam may have bending stiffness even when the pressure is zero. Also a distinction between various materials can be made. Zender (1962) and McComb et.al. (1962) used a membrane approach to model the collapse moment of an isotropic pressurised shell. This implied that the collapse moment of a pressurised membrane was added to the collapse moment of an unpressurised shell.

This Chapter is intended to provide a reasonably accurate load deflection model for inflated circular cylindrical beams, that takes material properties into account. The structure of the Chapter is as follows. Firstly the theoretical bending model will be discussed. In the subsequent section, the methods used for the biaxial tests and the static bending tests will be explained followed by a discussion and a conclusion.

3.2 Theory

3.2.1 Equilibrium of forces and moments

This sub-section provides expressions of equilibrium of forces and moments in the taut region and in the wrinkled region. The theory presented here is based on the theory described by Stein et.al. (1961) with some minor modifications.

Figure 3.1: Sign conventions.

Figure 3.1 shows the sign conventions used for the derivation. Equilibrium of forces and moments gives:

òs q = p 2p 0 x 2 t rd r p (3.1a) òs q q -= 2p 0 x 2t cos d r M (3.1b)

In an unwrinkled situation the stress distribution due to pressurisation and a transverse tip load “Q” will be as follows:

x, u y z θ, v Q M θw w r θ, v

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(

) ( )

q p -= s cos t r x L Q t 2 pr 2 x (3.2a) t pr = sq (3.2b) rt sin Q x p q -= t q (3.2c)

The stress strain relations are:

x x x x x E Eq q s u -s = e (3.3a) q q q q q = s -u s e E E x x (3.3b) q q q = t g x x x G (3.3c)

In this case the Poisson’s ratios are defined according to Maxwell’s law, see Eq. (3.36). The linear strain displacement relations are defined as:

x u x ¶ ¶ = e (3.4a) r w v r 1 + q ¶ ¶ = eq (3.4b) x v u r 1 x ¶ ¶ + q ¶ ¶ = g q (3.4c)

As shown by Donnell (1933) the term “w/r” in Eq. (3.4b) is generated by the change in circumferential dimensions. The compatibility equation becomes:

2 2 x 2 2 2 x 2 2 x w r 1 x r 1 x r 1 ¶ ¶ + q ¶ ¶ g ¶ = ¶ e ¶ + q ¶ e ¶ q q (3.5)

Eq. (3.2c) shows that the beam is loaded with a constant transverse tip load hence

“τxθ” is only dependent on “θ” and independent on “x”. Eq. (3.3c) shows that “γxθ”

is only dependent on “θ”. The stress “σx” is a linear function of “x” implying “εθ”

is also a linear function of “x”. Therefore the compatibility equation can be reduced to:

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2 2 2 x 2 x w r 1 ¶ ¶ = q ¶ e ¶ (3.6)

The displacement functions in the taut region can be found by combining Eqs. (3.2a-c), Eqs.(3.3a-c) and Eqs.(3.4a-c). After solving the differential equations the displacement functions are found to be in the following form:

( )

q 1 2 0 2! u x Lx xu u ÷÷ø ö ççè æ + -+ = (3.7a)

( )

q 2

( )

q 2 3 1 ! 2 ! 3 v L x x xv v ÷÷ø ö ççè æ -+ = (3.7b)

( ) (

q

) ( )

q 3

( )

q 3 2 2 1 0 2! 3! w x L x w x L xw w w ÷÷ø ö ççè æ -+ -+ + = (3.7c)

Where the second derivative of “w” to “x” is related to a bending term.

( )

(

) ( )

2 2 3 cos cos x w t r E x L Q ¶ ¶ = -= q p q k (3.8)

Substitution of Eqs. (3.4a, 3.7a, 3.7c, 3.8) in Eq. (3.6) and integration yields:

( )

q =-k

( )

q + 1 + 2q

1 rcos C C

u (3.9)

Because of the condition that “u1” has to be symmetrical with respect to “θ=0” it

follows that the second integration constant “C2” equals zero. Rewriting Eq. (3.3a)

gives: q qs u + e = sx Ex x x (3.10)

Substitution of Eqs. (3.9, 3.2b and 3.4a) into Eq. (3.10) yields:

( )

(

rcos C

)

prt

Ex 1 x

x = -k q + +u q

s (3.11)

Stein et.al. (1961) suggested that in the wrinkled region the Poisson ratio will be

replaced by a parameter “λxθ” so that the stress “σx” would remain zero. This is for

the case that the material is assumed to be a true membrane, i.e. having no out-of-plane bending stiffness. For the membrane case wrinkles are assumed to initiate when the lowest in-plane principal stress becomes zero. The artificial Poisson

ratio “λxθ” for membranes proposed by Stein et.al. (1961) follows from Eq. (3.11)

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( )

(

rcos C1

)

pr t Ex x = k q -l q (3.12)

At the wrinkling threshold “θ=θw” and “λxθ=υxθ” hence the constant “C1”

becomes:

( )

Eprt r C x x w u q q k -= cos 1 (3.13)

Substitution in Eq. (3.11) gives:

( )

( )

(

q q

)

k

sx =Ex r cos w -cos qw £q£2p-qw (3.14)

Eq. (3.14) is valid for the taut region. Substitution in Eqs. (3.1a,b) gives:

( ) (

) ( )

[

w w w

]

xtr

E r

pp 2 =2 2k sinq + p -q cosq (3.15a)

( ) ( )

[

w w w

]

3 xtr cos sin E M= kp-q + q q (3.15b)

Where Eq. (3.15a) and (3.15b) describe the balance of forces and moments respectively.

3.2.2 Collapse bending moment

The bending moment at which collapse takes place is regarded as the moment at which an increase in deflection does not result in an increase in moment. Several different expressions for the collapse moment can be found in the literature. The expressions differ in the manner they regard the material as a shell or as a membrane, or whether they concern isotropic or anisotropic material. Other differences are whether they deal with pressurised beams or not, whether it is an analytical or empirical expression, or whether the beam is of finite length or not. This section is devoted to highlighting various expressions for the collapse bending moment and to derive a new expression for the collapse bending moment. In 1927, a paper by Brazier (1927) was published in which he derived a collapse (or critical) bending moment by minimising the strain energy per unit length of a shell. This strain energy is dependent on the axial curvature. In this way Brazier (1927) described the flattening of infinite long cylinders subjected to bending. For an unpressurised isotropic shell he found the collapse moment to be:

2 2 Brazier 1 Ert 9 2 2 M n -p = (3.16)

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3 2 2 Wood t r E p 4 1 1 Ert 9 2 2 M ÷ ø ö ç è æ + n -p = (3.17)

Baruch et.al. (1992) modified Brazier’s (1927) expression for orthotropic materials. x x x 2 Baruch 1 E E rt 9 2 2 M q q q u u -p = (3.18)

In a similar way, the expression of Wood (1958) Eq. (3.17) is modified (see appendix C) for orthotropic materials to give:

3 3 x x x 2 x cr t E pr 4 ) 1 ( 1 E E rt E 2 9 2 M q q q q + u u -p = (3.19)

Eqs. (3.16 – 3.19) model the flattening of a long beam subjected to bending. This collapse behaviour is dependent on circumferential deformations of a shell. A second deformation / collapse mode takes place in longitudinal direction. It is known as short-wave axial buckling (Stephens et.al. 1975). For this kind of collapse Stein et.al. (1961) derived a moment for a pressurised beam made of a true membrane.

3

pr

MStein =p (3.20)

Zender (1962) derived a semi-empirical expression for the collapse moment of a pressurised cylindrical shell based on a membrane approach. The second term in this expression, Eq. (3.21) is related to the axial compressive buckling collapse of cylindrical shells. This is a semi-empirical expression which is a factor four lower than the theoretical axial compressive buckling load.

(

2

)

2 3 1 3 2 -u p + p = pr Ert MZender (3.21)

Seide et.al. (1968) derived the following collapse moment for pressurised circular cylindrical shells in bending:

(

)

(

)

MSeide pr rE tx e x x = + - -- + æ è ç ç ö ø ÷ ÷ -0 8 1 0 731 1 3 1 3 2 . p p . u u g f q q D (3.22) Where

(45)

t r 8 . 29 12 4 = f (3.23a)

and “∆γ” is a function of the pressure parameter which has to be obtained from figure 3.2. ÷ ÷ ø ö ç ç è æ ÷ ø ö ç è æ = g 2 t r E p f D (3.23b)

The factor 0.8 in Eq. (3.22) in front of the ‘classical’ collapse load for pressurised membranes is a design parameter.

Figure 3.2: Increase in axial-compressive buckling-stress coefficient of cylinders due to internal overpressure (Seide 1968)

Wielsgosz et.al. (2002) found that the ‘classical’ collapse moment for pressurised membranes is too large and in order to correlate to their experiments it should be reduced to:

3

4 pr

MWielsgosz =pp (3.24)

They noticed that collapse of their highly pressured fabric beams occurred when wrinkles progressed half way around the circumference. For comparison their factor of 0.25π = 0.785 is close to the factor of 0.8 which Seide et.al. (1968) suggested. 10-2 10-1 1 10 10-2 10-1 1 10 ∆γ 2 ÷ ø ö ç è æ t r E p

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