# On the design and feasibility of a pneumatically supportedactively guided space tower

код для вставкиСкачатьON THE DESIGN AND FEASIBILITY OF A PNEUMATICALLY SUPPORTED ACTIVELY GUIDED SPACE TOWER RAJ KUMAR SETH A DISSERTATION SUBMITTED TO THE FACULITY OF GRADUATE STUDIES IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY GRADUATE PROGRAMME IN PHYSICS AND ASTRONOMY YORK UNIVERSITY, TORONTO, ONTARIO AUGUST 2010 1*1 Library and Archives Canada Bibliotheque et Archives Canada Published Heritage Branch Direction du Patrimoine de I'edition 395 Wellington Street Ottawa ON K1A 0N4 Canada 395, rue Wellington Ottawa ON K1A0N4 Canada Your file Votre r6f6mnce ISBN: 978-0-494-68582-2 Our file Notre r6f6rence ISBN: 978-0-494-68582-2 NOTICE: AVIS: The author has granted a nonexclusive license allowing Library and Archives Canada to reproduce, publish, archive, preserve, conserve, communicate to the public by telecommunication or on the Internet, loan, distribute and sell theses worldwide, for commercial or noncommercial purposes, in microform, paper, electronic and/or any other formats. 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Bien que ces formulaires aient inclus dans la pagination, il n'y aura aucun contenu manquant. 1+1 Canada On the Design and Feasibility of a Pneumatically Supported Actively Guided Space Tower By Raj Kumar Seth a dissertation submitted to the Faculty of Graduate Studies of York University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY © 2010 Permission has been granted to: a) YORK UNIVERSITY LIBRARIES to lend or sell copies of this dissertation in paper, microform or electronic formats, and b) LIBRARY AND ARCHIVES CANADA to reproduce, lend, distribute, or sell copies of this dissertation anywhere in the world in microform, paper or electronic formats and to authorize or procure the reproduction, loan, distribution or sale of copies of this dissertation anywhere in the world in microform, paper or electronic formats. The author reserves other publication rights, and neither the dissertation nor extensive extracts from it may be printed or otherwise reproduced without the author's written permission. ABSTRACT Space tethers have been investigated widely as a means to provide easy access to space. However, the design and construction of such a device presents significant unsolved technological challenges. An alternative approach is proposed to the construction of a space elevator that utilises a free-standing core structure to provide access to near space regions and to reduce the cost of space launch. The theoretical and experimental investigation of the bending of inflatable cylindrical cantilevered beams made of modern fabric materials provides the basis for the design of an inflatable space tower. Experimental model structures were deployed and tested in order to determine design guidelines for the core structure. The feasibility of the construction of a thin walled inflatable space tower of 20 km vertical extent comprised of pneumatically inflated sections that are actively controlled and stabilised to balance external disturbances and support the structure is discussed. The response of the structure under wind loads is analyzed and taken into account for determining design guidelines. Such an approach avoids problems associated with a space tether including material strength constraints, the need for in-space construction, the fabrication of a cable at least 50,000 km in length, and the ageing and meteorite damage effects associated with a thin tether or cable in Low Earth Orbit. A suborbital tower of 20 km height would provide an ideal mounting point where a geostationary orbital space tether could be attached without experiencing atmospheric turbulence and weathering in the lower atmosphere. The tower can be utilized as a platform for various scientific and space missions or as an elevator to carry payloads and tourists. In addition, space towers can significantly be utilized to generate electrical power by harvesting high altitude renewable energy sources. iv Keywords: Space Elevator, Inflatable Space Tower, Inflatable Structure, Inflatable Beam, Inflatable Multiple-beam Structure, Cantilevered Beam, Pneumatic Structures. ACKNOWLEDGMENTS First and foremost I would like to express my deepest gratitude to my supervisor Dr. Brendan M. Quine for his continuous valuable guidance, motivation and support towards my research. The faith Dr. Quine had shown in me and my work for the past four years empowered me to work hard and achieve my academic goals. I would also like to thank my research committee members, Dr. Zheng Hong Zhu and Dr. JinJun Shan for their assistance, advice, and support by reviewing my work and for their valuable suggestions. I am indebted to my collaborator Dr. Zhu for his engagement and excitement in my research. His ubiquitous support throughout this process has been invaluable. I would like to acknowledge inhabitants of PSE 012, Rajinder Jagpal, Sanjar Abrarov and Farid Ahmed for their insight and generous support. A voluntarily assistance in developing electronics for active control mechanism for the pneumatic structure provided by space engineering students Michael Ligori and Christy Fernando is gratefully acknowledged. I appreciate their timely help in deploying the experimental structures also. I would like to thank Ator Sarkisoff for his help in machine shop and Nick Balaskas where availability and readiness to assist voluntarily is unsurpasssed. I would also like to thank Marlene Caplan for her timely assistance in various administrative matters by providing all the relevant answers. I am very much thankful to Dr. Caroline Roberts, CEO Thoth Technology Inc., who came forward to collaborate with our research team for materializing the 20 km space tower in to reality. I appreciate her invaluable support and encouragement. vi I would like to dedicate this dissertation to my parents for their continuous support and encouragement over these long years. I would especially like to thank my wife Manita Seth for her emotional support and all our family members and well wishers in India for their support and creative attitude towards my research. Finally, I would like to extend my sincere gratitude to everyone who made this research work possible. It is a matter of immense pleasure for the research team and York community for the worldwide media coverage of this innovative research in renowned magazines including New Scientist and Popular Science, and also in TV channels including CNN. I am very much thankful to the media reporters for their contribution in promoting the research. vn TABLE OF CONTENTS ABSTRACT iv ACKNOWLEDGEMENTS vi LIST OF FIGURES xi LIST OF TABLES xv NOMENCLATURE xvi CHAPTER 1. INTRODUCTION 1 1.1 Introduction 1 1.2 Motivation 4 1.3 Dissertation overview 5 CHAPTER 2. HISTORICAL BACKGROUND 7 2.1 Introduction 7 2.2 Historical Background of Tower Construction 7 2.3 Historical background of space elevator tower 10 CHAPTER 3. LITERATURE REVIEW OF INFLATABLE STRUCTURES 19 3.1 Introduction 19 3.2 Developments in Inflatable Technology 20 3.3 Inflatable Terrestrial Structures 22 3.4 Inflatable structures for aerospace applications 24 3.5 Inflatable Space-based Structures 30 3.6 Inflatable Beam Technology 38 3.7 Challenges in Inflatable Structure Design 40 3.8 Future Evolution of Space Inflatable Structures 41 3.9 Conclusion 43 CHAPTER 4. INFLATED BEAM ANALYSIS 45 4.1 Introduction 45 4.2 Critical Bending Moments under Lateral Loads 46 4.2.1 Wrinkle and Collapse Moments of Inflatable Cylindrical Beam 46 4.2.2 Experimental Set Up 48 4.2.3 Experimental Results 50 vm 4.2.4 4.3 Summary of Critical Bending Moments Euler's Critical Load Analysis 56 57 4.3.1 Theoretical view of critical buckling load 57 4.3.2 Effective ElasticModulus 59 4.3.3 Experimental Set up for Determining Effective Elastic Modulus 60 4.3.4 Experimental results of critical buckling load 61 4.3.5 Summary of Euler's Buckling Load 66 4.4 Conclusion 67 CHAPTER 5. CONSTRUCTION OF INFLATED MULTIPLE-BEAM STRUCTURES 68 5.1 Introduction 5.2 68 Theoretical formulation 69 5.2.7 Inflated multiple-beam structure 72 5.2.2 Equivalency of multiple-beam structure to a single beam 75 5.2.3 Comparison of bending moments of single inflated beam and multiple-beam structure 76 5.3 Experimental setup 76 5.4 Experimental Results 80 5.5 Free standing 7 m inflatable multiple-beam structure 85 5.6 Conclusion 86 CHAPTER 6. ACTIVE CONTROL MECHANISM METHODOLOGIES 88 6.1 Introduction 88 6.2 Theoretical background 89 6.2.1 89 6.2.1.1 Mathematical model of a system of gyros 93 6.2.1.2 Multi-gyro system design 94 6.2.1.3 Rotational frequency of gyros for vertical stabilization 95 6.2.2 6.3 Gyroscopic control mechanism Pressure control mechansim 98 6.2.2.1 Generating restoring moments 103 6.2.2.2 Pressure coefficients 103 Experimental Setup 6.3.1 106 Experimental set up for gyroscopic control mechanism IX 106 6.3.2 Experimental set up for pressure control mechansim 6.3.2.1 6.4 Electronic circuit for performing control operation Experimental results 109 Ill 113 6.4.1 Results and analysis of gyroscopic control mechanism 113 6.4.2 Demonstration and analysis of pressure control mechanism 119 6.5 Thermal efffects of sunlight and UV radiation 122 6.6 Conclusion 122 CHAPTER 7. FEASIBILITY OF FREE-STANDING INFLATABLE SPACE TOWER 123 7.1 Introduction 123 7.2 Material Selection 124 7.3 Practical Structural Concept 126 7.4 Theory of Inflatable Structure in the atmosphere 127 7.5 Critical Bending Moment 134 7.5.7 Bending Moments due to Wind Loads 135 7.5.2 Bending moments due to dead load contribution of the tower 145 7.6 Inflated Multiple-Beam Structure 146 7.7 Cost of space elevator structure 151 7.8 Conclusion 151 CHAPTER 8. SUMMARY AND FUTURE WORK 153 8.1 Summary 153 8.2 Future Work 155 8.3 Future Innovation 156 REFERENCES 160 x LIST OF FIGURES Figure 1.1 - 7.0 m demonstration device installed in stairwell (Quine et al., 2009) 2 Figure 2.1 - The Tower of Babel by Pieter Brueghel the Elder (1563) 8 Figure 2.2 - The angels climb Jacob's Ladder on the west front of Bath Abbey 8 Figure 2.3 - Toronto's CN Tower 9 Figure 2.4 - Burj Dubai, 818 m high, on 20 March 2009 9 Figure 2.5 - Artist Pat Rawling's concept of a space elevator viewed from the geostationary transfer station looking down along the length of the elevator toward Earth ..11 Figure 2.6 - Equatorial base sites are essential for space elevators 12 Figure 2.7 - "Bridge to Space" concept for a LEO Space Elevator by LMCO (Smitherman,2004) 16 Figure 3.1 - 'Millennium Arches' in Stockholm, Sweden: entirely self-supporting inflated structure (Lindstrand, 2000) 23 Figure 3.2 - Aviation Inflatable Maintenance Shelter (AIMS) (Vertigo Inc, 2009) 23 Figure 3.3 - World's Strongest AirBeam as claimed by Vertigo (Vertigo Inc, 2009) 24 2 Figure 3.4 - Jack Cust Baseball Facility - Flemington, NJ - 12040 m (Air Structures American Technologies, Inc., 2009) 24 Figure 3.5 - Inflatable Search Radar (Jenkins, 2001) 25 Figure 3.6 - Goodyear Inflatable Pyramidal Horn (Jenkins, 2001) 25 Figure 3.7 - Goodyear Radar Calibration Sphere (Jenkins, 2001) 26 Figure 3.8 - Lenticular Inflatable Parabolic Reflector (Jenkins, 2001) 26 Figure 3.9 - Airborne Communication Extender (ACE) airship (LTA Projects, 2009) 27 Figure 3.10(a) - COMET, COMmunity Emergency Tower (LTA Projects, 2009) 27 Figure 3.10(b) - RAFT (Rapid Air Inflated Tower) shown in various size configurations (Rapid Air Inflated Tower, 2009) 28 Figure 3.11(a) - The inflated airplane Stingray (Luchsinger et al., 2004) 29 Figure 3.11(b) - The inflated wing Pneuwing (Luchsinger et al., 2004) 29 Figure 3.12 - Tensairity demonstration bridge with 8 m span and 3.5 tons maximal load (Luchsinger et al., 2004) 30 Figure 3.13 - Tensairity Advertisement Pillaeight (Tensairity, 2009) 30 Figure 3.14 - Echo 1 Balloon (Freeland et al. 1998) 31 XI Figure 3.15 - Inflatable Very Long Baseline Interferometry Antenna 32 Figure 3.16 - Telescope Sun Shade Support Structure (Freeland et al., 1998) 33 Figure 3.17 - Inflatable Land Mobile Communications Reflector Antenna 33 Figure 3.18 - Thin-film inflated torus used as a support structure for optical reflector 33 Figure 3.19 - Inflatable Antenna Experiment on Orbit (Dornheim, 1999) 34 Figure 3.20 - Examples of ILC's pneumatic structures (ILC Dover Inc., 2010) 35 Figure 3.21 - Inflated and deployed Genesis II spacecraft launched on June 28,2007 36 Figure 3.22 - Mars Lander Airbag Deployment System (Thoth Technology Inc., 2009) 37 Figure 3.23 - A section of 7.0 m strcuture installed in stairwell 43 Figure 4.1 - Schematic of experimental setup 48 Figure 4.2 - Wrinkled fabric beam 53 Figure 4.3 - Experimental load - deflection curves of inflated beam Ll=1.484 m 54 Figure 4.4 - Experimental load - deflection curves of inflated beam L2=0.983 m 55 Figure 4.5 - Dimensionless load - deflection curves for the inflated beams 55 Figure 4.6 - Experimental arrangement to determine effective elastic modulus of the inflated beam 60 Figure 4.7 - Dependence of effective elastic modulus of inflated beam on inflation pressure 63 Figure 4.8 - Critical buckling theoretical load predictions of inflated beams of different lengths 64 Figure 4.9 - Demonstrating stability of inflated column with axial load at the top 65 Figure 5.1 - Multiple - beam structure cross section consisting of three inflated beams 72 Figure 5.2 - Experimental set up of inflated multiple-beam structure 77 Figure 5.3 - Bracket design to maintain the structure symmetry 78 Figure 5.4 - Three pressure gauges are mounted at the top of each inflated beam 78 Figure 5.5 - Structure base bolted and clamped rigidly to the heavy wooden base 79 Figure 5.6 - Lateral load-deflection data of inflated multiple-beam structure of length 4 m. 81 Figure 5.7 - Lateral load-deflection data of inflated multiple-beam structure of length 5 m. 81 Figure 5.8 - Dimensionless load-deflection curves of inflated multiple beam structure 82 Figure 5.9 - A 7 m structure pressurized using compressor that stands on a heavy base 85 Figure 6.1 - Gyroscope spin axis along with torque and precession axis 90 xii Figure 6.2 - Bending of the spin axis (vertical axis) and angular momentum of the spinning wheel 91 Figure 6.3 - System of gyros mounted on a vertical axis 94 Figure 6.4 - Top view of the multiple-beam structure comprising inflated beams 99 Figure 6.5 - Inflated multiple-beam structure force diagram 101 Figure 6.6 - Experimental set up using flywheel assembly 107 Figure 6.7 - Spinning wheel mounted at the top on the inflated beam of length 1 m 107 Figure 6.8(a) - Experimental set up of pressure control mechanism of an inflated 2 m structure 110 Figure 6.8(b) - Three inlet channels (Right) and three outlet channels (Left) through the solenoids mounted at the base 110 Figure 6.8(c) - Three pressure sensors mounted at top of the structure along with string and pulley arrangement 110 Figure 6.8(d) - Computer screen display along with electronics box and power supplies.. 110 Figure 6.9 - Electronics circuit designed to control solenoids inlet and outlet valves 111 Figure 7.1 - Core structure configurations (A,B and C) 127 Figure 7.2 - Maximum limiting gas pressure at various R/t ratios using different materials 129 Figure 7.3 - Variation of gas density with R/t ratio using Kevlar49(p = 1440Ag/m 3 130 Figure 7.4 - Maximum load capacity per unit cross-section area for particular tower height and gas at different R/t ratios using Kevlar 49 (p = \440kg/m3) 133 Figure 7.5 - Variation of critical wrinkling bending moment with radius at various pressures corresponding to different R/t ratios using Kevlar 49 Figure 7.6 - Cylindrical section of the inflated beam exposed to wind 135 137 Figure 7.7 - Inflatable 15 km high space tower at an altitude of 5 km accessing 20 km altitude under different wind loads in various regions Figure 7.8 - Wind load variation with radius of the tower for different heights 139 141 Figure 7.9 - Variation of lateral tip displacement of the tower of different heights and thickness 0.1m, with radius of the tower xm 143 Figure 7.10 - Variation of angle of inclination of the tower of different heights and thickness 0.1m, with radius of tower 144 Figure 7.11 - A structure configuration of 15 km high space tower comprising inflated gas cells each of radii 0.5 m 147 Figure 7.12 - The vertical space through the centre of inflated multi-beam structure xiv 150 LIST OF TABLES Table 4.1 - Tensile test results of beam fabric 50 Table 4.2 - Length and radius of inflated beams against inflation pressure 51 Table 4.3 - Experimental measurements of deflection of inflated beam L2 = 0.983 52 Table 4.4 - Experimental measurements of deflection of inflated beam LI = 1.484 52 Table 4.5 - Boundary conditions to determine the effective length constant of the column. 59 Table 4.6 - Load-depression data for the inflated beam of length 1.5 m 62 Table 4.7 - Load-depression data for the inflated beam of length 2.5 m 62 Table 4.8 - Average values of effective elastic modulus at different pressures 62 Table 5.1 - Numerical values of parameters involved in developing a free-standing 7.0 m prototype inflated structure 86 Table 6.1 - Numerical values of parameters involved in gyroscopic control of the inflated beam 108 Table 6.2 - Load - deflection observations using flywheel assembly 116 Table 6.3 - Theoretical result for a single gyro system, radius = 100 m, Total mass of the system = 200,000 kg 110 Table 6.4 - Number of gyros = 3, radius = 100 m, mass of lower most gyro = 100,000 kg, Total mass of the system = 272,220 kg 117 Table 7.1 - Mechanical properties of Kevlar 125 Table 7.2 - Average wind velocity and atmospheric density 140 xv NOMENCLATURE The following is the list of symbols used in thesis. S.I. units are used throughout unless otherwise indicated. A = area of cross section Cd = drag coefficient d = lateral tip deflection dLy = change in angular momentum along y-axis E = Young's modulus of elasticity Ex = Axial Young's modulus Eg = Circumferential Young's Modulus of elasticity (ER) E' = Effective elastic modulus EI = flexural rigidity of the inflated multiple-beam structure Er = total energy stored in inflated multiple-beam / = frequency of revolution F = external lateral tip load applied Fx = force acting perpendicular to spin axis Fc = critical force for an inflated structure Fd = wind load drag force g = acceleration due to gravity h = length of cylindrical surface hg = height of flywheel from ground H = height of tower I = area moment of inertia Ix = area moment of inertia of multiple -beam structure about x-axis xvi ly = area moment of inertia of multiple - b e a m structure about y-axis /' beam = area moment of inertia of equivalent single beam corresponding to multiplestructure Imx = mass moment of inertia about x-axis Imz = mass moment of inertia about z-axis K = structure parameter factor Ke = effective length constant L = length of inflated beam L = angular momentum Lcm = height of centre of mass of the system with respect to ground Le = effective length of the inflated beam Lg = height of gyro from ground Lz = angular momentum of gyro about z-axis m = dimensionless load for single inflated beam m = dimensionless load for multiple-beam structure MactiVe = bending moment generated in active pressure control mechanism Mc = stress based collapse moment of inflated beam MC_NASA= collapse moment recommended by NASA M„„ = strain based collapse moment mg = mass of gyro Mgyros = total mass of gyros Ms = mass of structure excluding mass of gyros Mloi = total mass of system comprising mass of structure and mass of gyros Mw = stress based wrinkle moment of inflated beam MK Imdn = critical bending moment for multiple-beam structure xvii Mw_st„ = strain based wrinkle moment Mwind = n = number of gyros N = number of inflated beams p = pressure at planet's surface wind load bending moment p = inflation pressure in the fabric beam p = internal gas pressure corresponding given values of tensile strength and R/t ratio pa = pressure corresponding to axial stress Ph = pressure corresponding to hoop stress Pmax = Ppcr = critical buckling load Prmax = maximum pressure energy stored in the inflated structure maximum limiting value of internal gas pressure Pi, P2,P3= inflation gas pressure in three inflated beams comprising multiple-beam structure r = radius of inflated structure rg = radius of gyro R = radius of cylindrical beam Re = radius equivalent to single beam corresponding to the multiple-beam structure Rg = gas constant S = surface area projected to air flow (body frontal area) T = temperature in Kelvin V = tension in the tower wall t = thickness of cylindrical beam v = air flow velocity xviii V = volume of the inflated beam Vu V2, Vi= volume of the three beams comprising multiple-beam structure W' = payload capacity of the inflated tower Wext = external work done Wioad = weight of payload Wi = weight applied in Newton y = vertical distance from ground yd = vertical depression AE = increase in pressure energy AF = differential coupled force Ap = change in pressure T = torque required to bend vertical axis comprising spinning flywheel T = r = torque about y-axis co = angular velocity of rotation flywheel (gyro) 00' = angular rate of bending of spin axis a = tensile strength cra = axial stress ,es resistive torque og, ah = hoop stress 8 = dimensionless tip deflection 89 = angle of deviation 8p = increase in pressure in the inflated beam ST = increase in tension in the inflated beam XIX v ev = P ° i s s o n ra tio of the fabric beam p = density of material pf = air density pg = density of gas fi = molecular mass 6 = angle of inclination aL = pressure coefficient of length aR = pressure coefficient of radius XX CHAPTER 1 INTRODUCTION 1.1 Introduction This thesis focuses on the design guidelines for the construction and operation of an actively guided inflatable space tower on Earth that would provide access to an altitude of 20 km. A space tower is a physical connection between the surface of Earth and a point in the near-space region. The proposed design comprises a self-supporting pneumatically pressurised core structure that consists of horizontal and vertical inflated sections. The pneumatically inflated compartments utilize tubes made from strong material such as Kevlar pressurized with air, helium and/or hydrogen. The structure would be self-guided by an active control mechanism utilizing pressure balancing in the inflated columns and gyroscopic stabilization, to maintain the centre of gravity and negate external forces on the structure. A 7.0 m prototype of the proposed structure developed by the author is shown in Figure 1.1 (Quine et al, 2009). This 1:2000 free standing scale model consists of three columns made of laminated polyethylene pressurised with air and braced at suitable intervals to avoid buckling. Along with traditional applications of the towers including communication networks and tourism, the space tower could potentially be used for launching spacecraft and satellites using a 20 km high launching pad. Launching from such an altitude could save up to 30% of energy, which is otherwise wastage in overcoming atmospheric drag. Fixed high-altitude space observatories can be installed at different heights along the length of the tower. Further, technology could also be used for harnessing environmental friendly renewable energy sources such as wind energy. The utilization of high altitude wind turbines for harnessing wind energy for electrical power generation is proposed by Bolonkin (2004). Turbines can be installed in the region 10-15 km above the surface using these space towers, in order to access air stream where wind power is available 100% of the time. The proposed towers would likely save a rocket stage and would benefit by reducing space junk due to spent rocket stages. While the space tower, because of its significant load capacity can be utilized for multiple applications, space tether concepts are limited typically to space transportation. Figure 1. 1 - 7.0 m demonstration device installed in stairwell (Quine et al., 2009) Current skyscrapers, Burj Dubai (828 m high) located in Dubai and CN tower (553 m high) in Toronto, are constructed using conventional building technologies that employ a massive steel-reinforced concrete structure. The tower can be built to any height utilizing current materials. However, the base of the tower should be sufficiently 2 large and that makes it very expensive costing tens of billions of dollars (Bolonkin, 2003). These towers are constructed to improve communication networks, tourism and to supply high-end commercial and residential accommodations. The traditional construction technology being cumbersome and expensive is not appropriate for the construction of space towers as it is prohibitively heavy (Lou and Feria, 1998). Compared to conventional structures, the inflatable structures have several distinct advantages, such as light-weight, lower life cycle costs, and simpler design with fewer parts (Lou and Feria, 1998). A 20 km tall space tower provides a practical alternative to the more popular space tether concept that would span the distance between an equatorial surface point and geostationary orbit. Using a rocket propulsion system, it costs about $400 million to launch a satellite into geo synchronous orbit or approximately $100,000 per kg (Edwards, 2000). The utilization of space tether could reduce this launch costs to as low as $10 per kg of material transported into space (Smitherman, 2000). The vision has inspired scientists around the world and government organizations including NASA. Japan Space Elevator Association, recently in 2008, announced to develop and design a space elevator (The Times, 2008). The idea of a space tower is similar to the concept of a space tether which provides a physical connection from a point on Earth to a point beyond geostationary orbit (GEO) in space and an elevator that provides a means of transportation up and down the structure. A space tether however has only a single anchor point; the other end is supported by tension induced by the centrifugal force induced by Earth rotation. Space tethers must extend significantly past geostationary orbit 3 (33,000 km above Earth) in order to induce the required tension, and consequently have a significant length exceeding 50,000 km in most practical designs (Edwards, 2000). Carbon nanotubes (CNT's) were found to be the most suitable candidate material for the construction of a space tether. Currently, the construction of a geostationary space tether utilizing CNT's is not feasible because carbon-nanotubes are not commercially available. Moreover, Pugno (2006) argues that the presence of microscale defects alone will prevent the fabrication of a cable and concludes that a space tether, if built as designed today will certainly break. The impacts of high speed meteors and hardly detectable micro meteors could break the cable within weeks. Lightening strike would be another obstacle in developing the space ribbon. Likely to be unavoidable, a single lightening strike could be responsible for failure of the entire geosynchronous tether. Atomic oxygen will cause a severe damage by removing epoxy/nanotube material at a rate of approximately lum/month (Edwards, 2000). The technological and engineering constraints prevent the construction of space elevator tether. Compared to the construction of a geostationary space tether, which is not feasible yet, the construction of a 20 km high inflatable space tower is quite feasible using current strong fibers. 1.2 Motivation Research ideas covering wide range of applications of space instruments including small and large space based devices such as match box size spectrometers and space elevator construction technology had been discussed at the very first time when author met with Prof. Brendan Quine in his office to join his research team. Revolutionary and out of box, the fascinating idea of constructing a pneumatically supported space tower, as 4 explained by Prof. Quine, inspired author to conduct research on an innovative methodology of building a space tower. Construction methodologies to build a space tower involve a good combination of physics and engineering including material science technology. The experimental investigations of inflatable structures to build space tower in particular is not seen in literature so far. At first, building a space elevator tower would seem to be very risky because of lack of experiences in space construction techniques. The numerous cost effective applications of space tower along with technological challenges motivate the author to design and develop a prototype scale model of the space tower structure, which could be used as a fundamental technology basis for the construction of 20 km high space towers. 1.3 Dissertation Overview The objective of the dissertation is to provide design guidelines for the construction and feasibility of an inflatable space tower on Earth. First, developments in the construction of such a tower are reviewed and described in Chapter 2. Inflatable structure technology is found to be one of the most promising and emerging technologies and is recommended for the construction of a space tower. The literature review of inflatable structure technology is presented in Chapter 3. For providing practical guidelines to the structural concept and construction methodologies as first proposed by Quine, Seth and Zhu (2009), experimentations on inflatable beams are performed. The theoretical and experimental analysis of mechanics of inflated beams of different sizes is presented in Chapter 4. This work is drawn from Zhu, Seth and Quine (2008). The experimentations, on inflated beams of different sizes have been conducted by the author for the analysis 5 and validation of the results obtained theoretically. The results obtained based on single beam analysis as investigated by Zhu, Seth and Quine (2008) are further carried out to analyse the behaviour of inflated multiple-beam structure. The theoretical and experimental analysis of inflated multiple-beam structures for its utilization as a core structure in the proposed space tower is presented in Chapter 5. It has been found that multiple-inflated structures can be efficiently controlled using active control mechanism. Active control mechanism utilizing gyro control stabilization and pressure balancing techniques for structural control are analysed and subsequently theoretical and experimental investigations are provided in Chapter 6. After successful performance of experimentation of the designed inflated space tower prototype structure and active control mechanisms, the feasibility of the free-standing inflated space tower is analysed and validated by Seth, Quine and Zhu (2009) and presented in Chapter 7. Finally, a summary and description of future work presented in Chapter 8. Supplementary supporting information can be found in References. 6 CHAPTER 2 HISTORICAL BACKGROUND 2.1 Introduction Towers are tall "artificial"structures built to take advantage of their height and can stand alone or as a part of a larger structure. Towers have been used by mankind since ancient times. The historical background of tower technology is categorized into two parts. First developments in building towers since ancient times are discussed. Second, the background of modern concept of space elevator tower is described. 2.2 Historical Background of Tower Construction The idea of building a tower high above the Earth into the heavens is very old (Smitherman, 2000). The Book of Genesis refers to an idea of "stairway to heavens" and includes The Tower of Babel and Jacob's Ladder. The Tower of Babel was an enormous tower built in approximately 2100 BC at the city of Babel and was depicted by Pieter Brueghel, in the year 1563 as shown in Figure 2.1 (The Yorck Project, 2002 and Harris, 1985). The Tower of Babel was not built for the worship and praise of God, but was dedicated to the glory of man, with a motive of making a 'name' for the builders: "Then they said, 'Come, let us build ourselves a city, and a tower with its top in the heavens". The Book of Genesis is the first book of the Hebrew Bible or Christian Old Testament, and the first of five books of the Jewish Torah or Pentateuch. In Genesis, it has also been described that around 1900 BC, Jacob had a dream about a staircase or ladder built to heaven. This construction was called Jacob's Ladder and is shown in Figure 2.2. "Jacob 7 left Beersheba, and went towards Haran. He came to the place and stayed there that night because the Sun had set. Taking one of the stones of the place, he put it under his head and lay down in that place to sleep. And he dreamed that there was a ladder set up on the earth, and the top of it reached to heaven" (Book of Genesis (28:11-19)). K -,L . V . . . ; • in i .• •* ...: . T '.. iSE^V*:*"- Figure 2.1 - The Tower of Babel depicted by artist Pieter Brueghel the Elder in 1563 (The Yorck Project, 2002). syg£| Figure 2.2 - The angels climb Jacob's Ladder on the west front of Bath Abbey (Haukurth en:User). The CN Tower in Toronto, Ontario, Canada as shown in Figure 2.3 was the world's tallest free-standing structure from 1975 until 2007, when it was surpassed in height by Burj Khalifa. It is 553 m in height, and has the world's highest observation deck at 447 m. The tower structure is concrete up to the observation deck level. Above is a steel structure supporting radio, television, and communication antennas. The total weight of the tower is 3,000,000 metric tons. Burj Dubai, shown in Figure 2.4 is a supertall skyscraper under construction in Dubai, United Arab Emirates, and is the tallest man-made structure ever built, at 828 m high. The building officially opened on Jan 2, 2010 (Burj Khalifa, 2010). The tower has communication antennas and will also be used for habitation. Figure 2.3 - Toronto's CN Tower, 553 m high. Figure 2.4 - Burj Dubai, 828 m high, The world's other well known towers built mostly for the purpose to improve communication network are described here. The Ostankin Tower, constructed in 1967 in Moscow is 540 m in height and has an observation deck at 370 m. The world's tallest office building is the Petronas Towers in Kuala Lumpur, Malasia. The twin towers, constructed in 1998 are 452 m in height. They are 10 m taller than The Sears Tower in Chicago, Illinois, USA, constructed in 1974. The Skyscrapers Taipei 101, constructed in 2004 in Taiwan has a height of 509 m. Eiffel Tower in Paris, has a height of 300 m and its construction was completed in 1889. Another skyscraper, the Empire State Building in 9 New York, USA (1930-1931) has a height of 381 m including a TV mast of 61 m. All these towers are mostly used for communication purpose and for tourist attraction as well. These towers utilize advanced methodologies for the construction of skyscrapers using current materials such as steel reinforced concrete. 2.3 Historical Background of Space Elevator Tower In 1895, the concept of space elevator tower was proposed by the Russian physicist K.E. Tsiolkovski in his manuscript "Speculation about Earth and Sky and on Vesta," (Tsiolkovski, 1999). At that time, Tsiolkovski was a school teacher in St. Petersburg, Russia, who provided a "thought experiment" on a tower into space. Tsiolkovski imagined tall towers and cosmic railways and discovered what we now call the geostationary altitude for a spacecraft, at which point the gravitational and centrifugal forces on a body in a one-day orbit are in balance. Tsiolkovski calculated the synchronous altitudes for the five visible planets, and the sun, but concluded that building such a real tower into orbit was impossible. The concept was revisited by the American engineer, Pearson (1975). Pearson conceived the idea of the space elevator in 1969 at the NASA Ames Research Center, and refined the concept in the early 1970s, when he was at the Air Force Research Laboratory in Ohio. He published his ideas in Acta Astronautica that first brought the idea to the attention of the entire world of spaceflight researchers. Pearson provided a physical basis for the construction of such a device. He reasoned that the tower must be constructed from Geostationary orbit outwards in both directions in order to keep the structure in 10 gravitational balance. Using force derivatives he calculated that the tower or cable would need to be at least 144,000 km long in order to be in balance with a net weight of zero and characteristic of the Earth's radius, surface gravity, and period of rotation. He proposed an area taper exponent for the structure of 0.116r0lh where rg is the radius of the Earth and h is the specific strength or characteristic height to which a constant diameter tower could be built to in a lg field. Suitable materials can be classified from the relation h = ajpg0 where a is the maximum allowable material stress and g0 is the surface gravity. Pearson proposed graphite crystals with a material stress of 46.5 GNm" and density 2200 kgm"3 yield a solution with h = 2150 km and a taper ratio of 10 without safety factors applied. Such a structure would certainly have significant utility; payloads ascending to the top of the structure could even be injected into escape orbits without the use of rocketry. However, the construction of the device, requiring innovations in materials and in space manufacture as well as at least 24,000 flights of a modified space shuttle with geostationary capability to raise the construction material place severe constraints regarding practicality. Figure 2.5 - Artist Pat Rawling's concept of a space elevator viewed from the geostationary transfer station looking down along the length of the elevator toward Earth. 11 Novelist Sir Arthur Clarke, consulted with Pearson when writing his novel, "The Fountains of Paradise" (Clarke, 1978). The novel publication brought Pearson's idea of the space elevator to wide population audience. Clarke proposed that a space elevator could be constructed using a cable and counter-balanced mass system. Built on the mythical island of Taprobane, closely based on Sri Lanka, a real island near the southern tip of India, Clarke made one important change to the geography of Sri Lanka/Taprobane: he moved the island 800 km south so that it straddles the equator. At the moment, Sri Lanka lies between 6 and 10 degrees north. Other suitable locations include equatorial mountain sites in South America and Africa. The elevator could be built at 5 km altitude in one of four regions on the equator (Quine et al., 2009). For Earth's gravity and spin rate, such a solution requires a cable of at least 35,000 km in length and a counter balance mass similar to a small asteroid. Such a system could be constructed by launching the cable into space or manufacturing it in situ and lowering it into contact with Earth. However, the technological obstacles that must be overcome, including the construction of a cable with suitable strength characteristics or the in-space construction of the apparatus, have not been realized since this concept was popularized by Clarke. Known materials are simply not strong enough to enable the construction of a cable of that length that would even be capable of supporting its own weight. Figure 2.6 - Equatorial base sites are essential for space elevators 12 Since the original concept was popularized, several alternative elevator concepts have been proposed. In the 1960s, Leningrad engineer Yuri Artsutanov (Artsutanov, 1960) theorised how to build a practical space elevator structure and his ideas published in Pravda. John Isaacs (Issacs et al, 1966) discovered that proposed thin wire tether would not be realizable because it would not sustain for a long time due to micro-meteorites impacts. A concept of lowering a stationary satellite utilizing a long cable was also proposed by Sutton and Diederich (1967) and Collar and Flower (1969), but idea of utilizing such a long cable was not extended to a space elevator (Star Technology and Research, Inc.). Pearson later extended space elevator idea to the moon (Pearson, 1978) utilizing Lagrangian points as balance points. The idea of low cost "lunar anchored satellite" as proposed by Pearson could be utilized to transport lunar material into Earth orbit (Pearson, 1979). The concept of lunar space elevator was discovered independently by Artsutanov (1979) who also published during the same time period. The concept of a rotating tube to retrieve asteroidal material was proposed by Pearson (1980) and Pignolet (1979). The proposed idea utilizes thrust by making use of centripetal force of the rotating tube. Birch (1982) proposed a system to support a lowaltitude space elevator. The proposed system comprises a hollow ring around Earth encompassing a high speed wire that generates a centrifugal force to support massive space elevator structure. The payloads could be carried to the ring utilizing space elevator to launch into orbit with required orbital speed. During the same period of time another 13 dynamic space elevator system utilizing electromagnetic disks was proposed by Hyde et al. (1982). The system comprises a pair of electromagnetic propulsion mechanisms that would maintain vertical stability of the tower. Penzo (1984) extended the idea of space elevator tethers to Phobos. A launch loop comprising a 2000 km magnetic levitation cable transport structure system for launching humans for space tourism, space exploration and to develop space colonization is describe by Lofstrom (1985). Anchored at both ends and at intervals along its length that extends 80 km vertically, the structure is maintained by energy and momentum exchange with a electromagnetically moving iron rotor of mass 15,600 tonnes which is accelerated to 14 kms" inside the cable. Quine, Seth and Zhu (2009) estimated the mass of the structure as 1.5xl07 kg from the data provided assuming a single ribbon system. The failure of the guiding system will release 1.5xl015 J enough to boil 400,000 m3 of water and will be equivalent to an explosion of 350 kilotons of TNT. The 2000 km launch loop cannot be constructed near habitation area due the expected occasional catastrophic failure of accelerating guiding system. A realizable launch loop technology can be utilized as a method for energy storage. The fundamental problem in building a space elevator is the availability of appropriate material that would be capable to support a massive cable over 36,000 km in height. After the discovery of carbon nanotubes (CNTs) in last decade of 20' century, scientists realized that carbon nanotube structures would make it possible to build a space 14 elevator to access physically geo-stationary orbits due to their enormous strength (Yakobson and Smalley, 1997). American scientist Bradley C. Edwards proposed a 100,000 km long ribbon-like structure comprising 1.5um ribbons made of a carbon nanotube composite material (Edwards, 2000). Edwards suggested tensile strength and density of comprising carbonnanotubes as 130 GPa (including a safety factor of 2) and 1300 kgm"3 respectively, however tests indicate that the strongest multi-walled carbon nanotube have a measured tensile strength of 63 GPa (Yu et al, 2000). However, Edwards suggests that meteorites, atomic oxygen and lightening strike would cause severe damage to the proposed tether within weeks. In addition, other technological and engineering challenges for deploying a tether up to beyond geostationary level would seem to place severe constraints regarding the practicality of such a device. Edwards and Westling (2003) received NASA research funding to study their proposed practical scheme for the construction of a space elevator. Along with these concepts, Boyd and Thomas, in 2000 proposed a design for a space elevator that moves payloads between locations located at substantially fixed orbital distances from the Earth (Boyd and Thomas, 2002). The device incorporates a cable system capable of transporting payloads between orbital locations leveraging the energy efficiency of an elevator device that climbs the tether. As the device has no part attached to the Earth's surface, a secondary means must be utilized to attach initially the payload to the elevator. 15 2mG m ...... ^ •> • -•- i p ^ : o ' r T * ' ' T f ; j ' ''-rf"*>- ; - •• ^ - ^ ' J?-;-- * ,. • . ' a - f t > - * / • »' .• • * - ' * W ? ' •'*"" J *•:• •<£... . . -' - .? -< • - * !•*« r., • asr S ' , . . £ ^ " ; *• •- » ' « ».-• •*' / . \ T * . •. • • . - * • . . , • •• »•-' . . . jr- fel •• : cM'•'•'••3::± ;,i;.;- , : --.: :* • ^*'.--ssr. "^rrara Figure 2.7 - "Bridge to Space" concept for a LEO Space Elevator by LMCO (Smitherman, 2004). 16 A system for developing a space elevator consisting of a flexible tension structure deployed above and below geosynchronous altitude has also been proposed by Dempsey (Dempsey, 2006) A mechanism of a transport tether configuration forms a harmonic oscillator using a combination of gravitational and centripetal forces. The transport of payloads along the tether utilizes centripetal force energy and does not require any additional power; however substantially longer cable in comparison to other tether concepts are subject to an even greater engineering and technological challenges. A "Bridge to Space" concept envisioned by the Lockheed Martin Company (LMCO) is shown in Figure 2.7. This concept is proposed to capture payloads from a future suborbital space plane at 100 km altitude, and then lift the payload to the upper end of the elevator at 4000 km altitude, where when released it would enter a GEO transfer orbit. The LEO Deployment Demonstration would help test the feasibility of this concept (Smitherman, 2004). Due to non-availability of carbon nanotubes (CNTs) on mass scale, the existing materials could be used to construct large space structures, such as space towers many kilometers in height using inflatable technology. The theoretical basis for the construction of an inexpensive suborbital space tower with 3 to 100 km height has also been proposed by Bolonkin (2003). Pearson's lunar space elevator by contrast appears feasible using current materials and could be used to develop lunar resources and for lunar far side communication. It does not require the super-strength of carbon nanotubes (CNT). The use of rotating tethers for asteroid retrieval is also possible now and could even be used to 17 remove asteroids in danger of collision with Earth, such as Apophis which will pass within 6 Earth radii in 2029 (Near Earth Object Program, JPL, NASA). Recently, there have been attempts to turn the idea of space elevator into a reality by New Jersey's Liftport Group and the Japan Space Elevator Association (The Times, 2008). More recently, a guided stabilised tower approach is proposed by Quine (2009) for the construction of suborbital inflatable towers to access altitudes above 20 km that is realisable utilising current material technologies. The proposed structure comprised of pneumatically inflated sections that are actively controlled and stabilised by using gyrocontrol machinery to balance external disturbances and support the structure. This concept is described in Quine, Seth and Zhu (2009); A free standing space elevator structure: A practical alternative to the space tether. The proposed structure provides a fixed link between ground and near-space locations enabling the transportation of the equipment, personnel and other objects or people to the platforms or pods above the surface of the Earth for the purpose of scientific research, communication and tourism. 18 CHAPTER 3 LITERATURE REVIEW OF INFLATABLE STRUCTURES 3.1 Introduction This chapter presents the developments in inflatable structure technology that may be utilized as the basis for the construction of a space tower. Conventional construction technology is not appropriate for the construction of a space tower as traditional steelreinforced concrete structure technology is prohibitively heavy, cumbersome and expensive (Thomas, 1992). By contrast, these new emerging technologies, based on inflatable structure systems can potentially solve design, control and load carrying capacity problems. Successive developments in inflatable land structures and inflatable structures for space applications are reviewed and their applicability for the construction of a space tower is assessed. Current skyscrapers on Earth are constructed using conventional building technologies that employ massive steel-reinforced columns. Most of the current analytical tools, specialized building designs, and the methods of construction cannot be easily accepted to build space structures (Thomas, 1992). To build large space structures such as a space tower, an inflatable structure technology must employ specially designed control mechanisms. These new emerging inflatable technologies have the advantages over other approaches including extremely low weight, suitability for on-orbit deployable and low volume during launch in addition to having reduced construction cost (Lou and Feria, 1998). A number of different technology developments have taken place over the years focused on inflatable deployable space structures. The success of various space missions 19 based on inflatable technology has led NASA and other space organizations to begin to use space inflatable structure technology in order to achieve the advanced goals of future space missions (Freeland et al. 1998). The chapter consists of nine sections. Following the introduction in Section 3.1, the Section 3.2 presents the developments in inflatable structure technology. Inflatable terrestrial structures are presented in Section 3.3. Aerospace applications of inflatable structures are presented in Section 3.4. Problems related to designing complex shape structures and load bearing capacity are also reviewed in this section. Inflatable spacebased structures are presented in Section 3.5. A description of the efforts in development of inflatable beam technology is presented in Section 3.6. Problems and challenges related to rigidization of the inflatable structures are also reviewed and subsequently, discussed in Section 3.7. After a discussion of future evolution of space inflatable structures as an application for the construction of space tower in Section 3.8, the chapter is concluded in Section 3.9. 3.2 Developments in Inflatable Technology In 1783, Mongolfier brothers ventured successfully into sky with their hot air balloon (Gillispie, 1983). Later on, the first hydrogen filled balloon capable to carry people was demonstrated by Charles (CIA Balloon and Airship Hall of Fame, 2000) and subsequently after a period of two years, Blanchard, a French balloonist, crossed the English Channel using hydrogen filled balloon technology (Encyclopaedia Britannica, 2010). In 1784, the first successful non-rigid airship was envisioned by Jean Baptiste Marie Meusnier, a French aeronautical theorist and military general (US Centennial of 20 Flight commission). After almost one hundred years, gas filled airships started to be used in air transport for carrying passengers and cargo. The golden age of airship started in July, 1900 with the launch of the Luftschiff Zeppelin LZ1 from Lake Constance at Manzell in Southern Germany (Lueger, 1920). Airships for sky transportation were popular until the first quarter of 20* century, when the fast upcoming aircraft industry began to dominate the air transportation business. The tragedy of the Hindenburg, an airship 245 m long, filled with over 200,000 m3 of hydro gen in Lakehurst occurred on May 6, 1937 and marked the abrupt end of the airship boom (Luchsinger et al, 2004). The primary cause of the accident was its highly combustible skin coated with extremely flammable cellulose nitrate or cellulose acetate, which caught fire from an electric spark releasing hydrogen that became the fuel for the already existing fire (Bain and Vorst, 1999). As material science technology has advanced, there were many developments in inflatable technology in space programs. Among them, antennas developed by Goodyear, Echo Balloons, Contraves Antennas, Sunshades, the NASA Inflatable Antenna Experiment and the successful deployment of Genesis I and Genesis II spacecrafts are significant achievements. Developments in inflatable structure technologies can be categorized into terrestrial, aerospace and space corresponding to their applications in respective fields. 21 3.3 Inflatable Terrestrial Structures Inflatable ground-based structures are typically completely self-supported, requiring no solid structural members, few mechanical parts and are supported only by internal air pressure. These structures are currently in use in military and architectural designs for tents, hangars, roofs, and small buildings. In the 1960s, German architect and engineers (Otto and Trostel, 1962) explored the potential of inflatable structures. An excellent overview of the pneumatic structure and architecture up to 1970s is given by Herzog et al. (1976). One of the prime times for the development of inflatable land structures and pneumatic architecture was the 1970 Expo in Osaka (Luchsinger et al., 2004). These groundbreaking pneumatic buildings demonstrated innovative architectural ideas. In sports, for the construction of tennis courts and large inflatable domes for the use in various types of indoor games, the inflatable structures have played a significant role. Within the past few decades, new inflatable applications for land structures are being explored and have gained acceptance because of their low weight, smaller stowed volume and ease of assembly. Today, it is possible to find inflatable housing, concert venues, exhibition halls, museums, and space-age science displays, without solid supporting members. Figure 3.1 shows an entirely self-supporting inflatable structure called Millennium Arches (Lindstrand, 2000). The whole building is 100 m long, 18 m wide and 17 m high. The main arches are 11.5 m high, and 18 m wide while the end arches are 12.5 m long and the center arch is 50 m long. 22 Figure 3.1 - 'Millennium Arches' in Stockholm, Sweden: entirely self-supporting inflated structure (Lindstrand, 2000). The design and development of air beam and pneumatic shelters by Vertigo Inc. uses high pressure air beam (AirBeam™) Systems that use fiber reinforced elastomeric composites, capable of containing high gas pressure and resisting bending (Vertigo Inc., 2009). These shelters are rapidly gaining acceptance by military, aerospace and commercial operators. The Aviation Inflatable Maintenance Shelter (AIMS), shown in Figure 3.2, is an air-transportable, rapid-deployment military shelter system, designed to house maintenance operations for helicopters and aircrafts. The inflatable beam as designed and developed by Vertigo is claimed to be one of the strongest and is shown in Figure 3.3 (Vertigo Inc., 2009). A sports dome built by Air Structures American Technologies, Inc. (2009) is a fraction of the cost of conventional buildings of similar size and is shown in Figure 3.4. Figure 3.2 - Aviation Inflatable Maintenance Shelter (AIMS) (Vertigo Inc., 2009). 23 Figure 3.3 - World's Strongest AirBeam as claimed by Vertigo (Vertigo Inc., 2009). Figure 3.4 - Jack Cust Baseball Facility - Flemington, NJ - 12040 m2 (Air Structure American Technologies, Inc., 2009). 3.4 Inflatable Structures for Aerospace Applications In the late 1950s, Goodyear started investigating light weight terrestrial inflatable structures for observation and analysis of space activities. Goodyear developed inflatable structural concepts for their Search Radar Antenna, Pyramidal Horn, Radar Calibration Sphere, Lenticular Inflatable Parabolic Reflector. The inflatable Search Radar Antenna has a parabolic profile and an aperture of 10 m in length and 3 m in width. This truss type deployable rigidizable structure is also foldable in stowed configuration after its use and is shown in Figure 3.5 (Freeland et al, 1998; Jenkins, 2001). An inflated Pyramidal Horn designed and developed by Goodyear is also shown in Figure 3.6. 24 The Radar Calibration Sphere design utilizes a large number of flat membrane panels. The panels are joined at their perimeters to form a larger structure that, when inflated, would take the shape of sphere. The Radar Calibration Sphere is approximately 6 m in diameter and is shown in Figure 3.7 (Freeland et al., 1998; Jenkins, 2001). The Lenticular Inflatable Parabolic Reflector is a structure of size 12m diameter comprises a lenticular reflector having diameter 10 m. The reflector is supported by a circumferentially bonded toroidal structure using different rigidization techniques. The fully inflated structure is shown in Figure 3.8 (Freeland et al., 1998; Jenkins, 2001). The images shown in Figures 3.5 - 3.8 are reproduced from a thesis (New ultra-light stiff panels for space apertures (Black, J.T, 2006)). Figure 3.6 - Inflatable Pyramidal Horn (Jenkins, 2001). 25 Figure 3.7 - Radar Calibration Sphere (Jenkins, 2001). Figure 3.8 - Lenticular Inflatable Parabolic Reflector (Jenkins, 2001). LTA Projects' partnership with Bosch Aerospace has produced airships and aerostats for a variety of applications. The 38.1 m blimp can operate at an altitude of 1.6 km, move at speeds of up to 80 kph, and stay aloft for more than 24 hours without pilot and is shown in Figure 3.9 (LTA Projects, 2009). 26 Figure 3.9 - Airborne Communication Extender (ACE) airship (LTA Projects, 2009). COMET (COMmunity Emergency Tower), one of the product developed by LTA, shown in Figure 3.10(a), provides 360°, 28x zoom low light viewing and control system using the internet to connect to a remote command center. Its height and visibility make it ideal for ambulance/medical services and security centers. It is easy to use and portable, can be mounted in the bed of a truck, on a trailer, or on the ground (LTA Projects, 2009). Rapid Air Inflated Tower (RAFT) is designed and manufactured by ISL, a Division of Bosch Aerospace, for crowd control, local surveillance and communications extensions is shown in Figure 3.10(b). These air inflated towers have range in height from 11 m to 30.5 m and lift payloads from 14 kg to 100 kg. Logistics free and highly mobile, these towers can be inflated in minutes (Rapid Air Inflated Tower, 2009). Figure 3.10(a) - COMET, COMmunity Emergency Tower (LTA Projects, 2009). 27 Vii' * ' swsto m vyl ' j§£|§ 0 ^< : > ^ ^^I^JIF Figure 3.10(b) - RAFT (Rapid Air Inflated Tower) shown in various size configurations (Rapid Air Inflated Tower, 2009). To build a prescribed complex shaped inflated structure, the inflated volume may be divided into a number of chambers by webs. The desired shape of the inflated structure can be precisely obtained by appropriate design of the number and form of the webs. Prospective Concepts has advanced this technology in designing the manned pneumatic aircrafts with the successful launch of their pneumatic aircraft, Stingray and Pneuwing in 1998 as shown in Figure 3.11 (a) and Figure 3.11 (b) respectively (Luchsinger et ai, 2004). The wings of the aircraft are inflated using air as filling gas with a mechanism to maintain pressure. The use of Helium gas makes the aircraft lighter and adds the static buoyancy and dynamic lift. The flying wing having an area of 70 m~, enclosing gas volume of 68 m3, can take off at a speed of 47 km/h and reaches a maximum speed of 130 km/h. Stingray is a two persons vehicle and can be accelerated in the range of up to + 4.5 g / -3.0 g (Luchsinger et al, 2004). 28 Figure 3.11(a) - The inflated airplane Stingray (Luchsinger et al, 2004). , jgE. ' i ^ Figure 3.11(b) - The inflated wing Pneuwing (Luchsinger et al., 2004). Tensairity, a trade marked word combination of tension, air and integrity, serves as a hybrid technology between the air-beam and conventional beam structures. The load bearing capacity of the air-beam can be improved by up to two orders of magnitude (Luchsinger et al., 2004). The air-beam is modified in such a way that it has the same load bearing capacity as a steel beam, with an additional advantage of substantial weight reduction. The swiss company Airlight has developed a new pneumatic beam which enables novel applications especially in structural engineering. The synergetic combination of air filled flexible tube, cables and rods with low pressure gives a very light but strong structure element. The load bearing capacity of Tensairity is so high that with a pressure of 20 kPa (0.2 bar), it is possible to build temporary bridges suitable for vehicles. Compared with traditional air-beams, Tensairity claims that beams need just 1% of air pressure. With such a low pressure, air losses are easily compensated. This technology is ideally suited 29 for wide span structures and for deployable applications such as temporary bridges, scaffolds or large tents and opens up many new technical opportunities for pressure induced stability (Luchsinger et al., 2004). A pneumatic demonstration bridge with 8 m span and 3500 kg load is shown in Figure 3.12. The new concept of Tensairity can also be used to significantly improve the load bearing capacity of inflatable wings (Breuer et al., 2007). The load bearing capacity in this middle pressure range is high enough for many applications including towers as shown in Figure 3.13 (Tensairity, 2009). Figure 3.12 - Tensairity demonstration bridge with 8 m span and 3.5 tons maximum load (Luchsinger et al., 2004). Figure 3.13- Tensairity Advertisement Pillaeight (Tensairity, 2009). 3.5 Inflatable Space-based Structures Work to establish space habitats using inflatable technology along with the other space applications such as sunshield, antennas, solar sails and structural booms is ongoing. Space applications take advantage of several features of inflatable structures. 30 The large space structures such as 300 m antennas or solar sails may not be feasible with mechanical deployment, but may be realizable with inflatable technology (Dornheim, 1999). Inflatable structure technology has favourable characteristics for the construction of large space structures that can be employed to design and develop a space tower, the largest man-made structure proposed so far. In 1960 NASA launched into space successfully Echo I, the first balloon, of a series with an objective to provide passive, space based communications reflectors. The compact volume of the balloon packed in a spherical container was only 0.66 m. The small container consisting of balloon was launched on a Delta rocket for its deployment in LEO. Upon achieving orbital altitude, the balloon was inflated to its maximum diameter of 30.5 m and separated from the launch vehicle. The 61.7 kg balloon was constructed using 12 micrometer thick mylar gores bonded together to form a sphere and coated with a very thin layer (200 nm) of aluminum (Freeland et ai, 1998; Jenkins, 2001; Leonard, 1961; Wilson, 1981 and Hansen, 1994) Echo I is shown in Figure 3.14. Figure 3.14 - Echo 1 Balloon (Freeland et ah, 1998). 31 The European Space Agency's (ESA) Contraves Space Division developed inflatable reflector antenna and sun shade structural concepts during the 1980s. Initially, a reflector antenna of 6 m diameter sized for Very Long Baseline Interferometry (VLBI), was developed and is shown in Figure 3.15. The Telescope Sun Shade Support Structure, consisting of 2 m diameter flexible thermal panels and inflation systems is shown in Figure 3.16. During the same period, a 10 by 12 m offset reflector antenna for land mobile communications was built and evaluated for surface precision and other mechanical characteristics. The construction of these antennas was based on two parabolic membranes, supported circumferentially by a toroidal structure as shown in Figure 3.17. The load carrying fibers in the gores were of Kevlar and matrix material was so designed that the structure became rigid on orbit during test (Freeland et al., 1998 and Bemasconi et al., 1987). The images shown in Figures 3.14 - 3.17 are reproduced from a thesis (New ultra-light stiff panels for space apertures (Black, J.T, 2006)). Figure 3.15 - Inflatable Very Long Baseline Interferometry Antenna (Freeland et al, 1998) 32 \ IV J Figure 3.16 - Telescope Sun Shade Support Structure (Freeland et al., 1998). '«e A 0 X -, 1 / Figure 3.17- Inflatable Land Mobile Communications Reflector Antenna (Freeland et al, 1998). Another thin film antenna to be used as optical reflector was designed and developed by Smalley et al. (2001) and is shown in Figure 3.18. The 5 m optical reflector is supported by inflated torus concentrator. Figure 3.18 - Thin-film inflated torus used as a support structure for optical reflector (Smalley et al., 2001). A joint STS-77, shuttle mission of NASA, L'Garde and Jet Propulsion Laboratory (JPL) successfully launched the Inflatable Antenna Experiment (IAE) in May, 1996. In 33 the experimental set up, three 28 m long inflatable struts are designed and arranged to support 14 m diameter inflatable reflector during its deployment as shown in Figure 3.19 (Dornheim, 1999). The experiment was intended to demonstrate the maturity of the inflatable technology. The torus and struts were made of 0.3 mm thick Kevlar with Neoprene coating. The cost of the whole experiment was in the order of $10 million (Freeland et al., 1997; Freeland and Veal, 1989). The experiment of the inflatable antenna proved to be very valuable for inflatable technology utilized in space engineering. The high tensile strength of Kevlar resisted punctures caused by high velocity space debris. The interwoven design of Kevlar resists ripping, tearing, and cracking (Tanner et al., 1989). Figure 3.19- Inflatable Antenna Experiment on Orbit (Dornheim, 1999). Pneumatic technology has also been utilized in high-altitude ballooning since the 1960s. A typical example is MANTRA (The Middle Atmosphere Nitrogen TRend Assessment), a series of Canadian high-altitude balloon flights undertaken to investigate odd-nitrogen budget of the stratosphere. The first MANTRA mission took place in August 1998, with the balloon flight on August 24, 1998 and was the first Canadian launch of a high-altitude balloon in about 15 years (Stong et al., 2005). The mission carried a new balloon-borne pointing system, scanning the Earth's limb, which pointed a platform of optical instruments at an inertial target with accuracy better than 0.1 in 34 elevation and 3 in azimuth. (Quine et al., 2002). High-altitude balloon missions provide a cost-effective means of making space observations. For 50 years, ILC Dover has been manufacturing gossamer materials having diverse range of aerospace and defence experience. ILC has been manufacturing a number of products for NASA for different kinds of space missions. Their wide range of products envelops almost all gossamer technologies, including structural and laminated materials, space and other protective suits, planetary entry and landing systems, and unmanned aerial vehicle wings (ILC Dover Inc., 2009). Recent examples of ILC's pneumatic structures are shown in Figure 3.20. Hi (t) Figure 3.20 - Examples of ILC's pneumatic structures; (a) Space suit garments, (b) Lighter than air; airships and blimps, (c) Impact absorbing systems, (d) Package of Mars airbag landing system. In 2006, Bigelow Aerospace, launched Genesis I (Berardelli, 2006), a one-third scale prototype of an inflatable habitat and later on launched another inflatable spacecraft Genesis II on June 28, 2007 for testing and validation of the technologies to set the stage for future manned orbital complexes. Both the inflatable spacecrafts represent the use of expandable space habitat technology demonstrated on orbit. These inflatable spacecrafts 35 look alike and have dimensions of 4.4 m in length and 2.6 m in diameter. The internal habitable space volume was 11.5m , when inflated and deployed. (K- •w • I" > f A Figure 3.21 - Inflated and deployed Genesis II spacecraft launched on June 28, 2007. These spacecrafts are comprised of Kevlar and Vectran and are intended to have a five year orbital life time. The objective of these low cost commercial missions is to test the procedures for maintaining proper internal temperature and pressure and also to test their validity as a full scale inflatable habitation. The use of Kevlar and Vectran for its manufacture makes the spacecraft resistant to micrometeorites and other space debris. An image of the deployed structure is shown in Figure 3.21. Thoth Technology Inc. develops Robotic Landers for remote sensing applications. Thoth's EDL (Entry, Descent, and Landing) system employs parachutes and airbags along control mechanisms, that eliminates the bouncing traditionally associated with airbag landings (Thoth Technology Inc., 2009). Mars Lander Airbag Deployment System as developed by Thoth is shown in Figure 3.22. 36 Figure 3.22 - Mars Lander Airbag Deployment System (Thoth Technology Inc., 2009). NASA plans inflatable technology developments as the basis for future radiometer technology with operational frequencies 1.4-300 GHz, may be compared from a 25-50 m array made of mesh or membrane with an areal density less than 2 kg/m". Space optics for the visible range of light are expected to be 6-10 m in diameter and have an areal density of 5 kg/m2. DARPA (Defence Advanced Research Projects Agency) is developing an ultralarge, ultralight weight phased-radar antenna to sense incoming missiles at ranges 600 km and deployed ground troop movements at 300 km range. In addition, as a part of a multidisciplinary approach to understand the many facets of inflatable structures, new efforts are integrating active smart materials into the design process of future inflatable large spacecraft systems and optics (Ruggiero and Inman, 2006). The Canadian Space Agency (CSA) is developing foldable large size membrane SAR (Synthetic Aperture Radar) antennas in space, which have the advantages of achieving large gain and good resolution with ultra-light mass. The motion of deployed membranes can be controlled by mechanisms in longitudinal as well as lateral directions, for the control of large membrane deployable space structures (Shen et ai, 2007). 37 3.6 Inflatable Beam Technology Inflatable beams have been widely used as load-carrying members in space and aerospace applications (Fang et al., 2004). These structures are usually made of modern synthetic fabric materials and the inflation air provides structural capacity by pretensioning the fabric. Compared with the conventional beams, the inflatable beams offer benefits such as easy packing, deployment, and low weight and low costs. However, the inflatable beam structures are easy to deform when subjected to external bending loads and even collapse by local buckling (wrinkle) of the fabric wall, especially for the inflatable cantilevered beams. Therefore, accurate and efficient prediction of the bending moments of wrinkle and collapse of the inflatable beams is critical to the wider application of inflatable beam structures (Zhu et al., 2008). Many efforts have been devoted to the development of mechanics of the inflatable cylindrical beams. There are two types of approaches found in the literature: the beamtype and the membrane/shell-type. Leonard et al. (1960), and Comer and Levy (1963) studied the inflatable cylindrical cantilevered beams by the Euler beam theory. Main et al. (1994-95) further studied the inflatable cylindrical cantilevered beams with the consideration of the biaxial stress state in the beam fabric due to the combination of pressurization and structural loads. Wielgosz and Thomas (2004) modeled the inflatable cylindrical beams by considering the pressure as a follower force and using Timoshenko's beam theory to account for the shear deformation of the fabric. Consequently, an inflatable beam element was developed and applied for the analysis of simply supported inflatable cylindrical beam with central load. Recently, Davids and co38 workers (2007) developed an inflatable beam element by considering the internal pressure through the volume change and the local fabric wrinkle using the Stein and Hedgepeth (1961) taut and wrinkled criterion. Fichter (1966) modeled the inflated cylindrical beams using membrane theory and considered the effect of inflation pressure using the variational approach. Later, Veldman (2000) studied the inflatable beam using thin shell theory and his results agreed with the experimental data of inflatable thin films reasonably. More recently, Yoo et al. (2007) implemented Stein and Hedgepeth (1961) taut and wrinkled criterion into a membrane element with commercial finite element codes and modeled the wrinkle of inflatable cantilevered beams made of thin film. The numerical predictions agree with the experiments. However, their results predicted a higher wrinkling moment than the beam theory although the predicted collapse moment does not exceed the collapse moment derived by the beam theory. These existing efforts represent substantial and novel contributions to the field of inflatable beams. However, differences in the theoretical predictions of the wrinkle and collapse moments between different approaches and discrepancies between the theoretical and experimental wrinkle and collapse moments of the inflated beams result in a great uncertainty in the design of the inflatable structures. Further, the experimental data of the inflatable cylindrical cantilevered beams in the literature are in raw form and are not easy to compare to each other and to be used as guidelines for design work. 39 Parametric experimental investigations with different internal pressure, lateral loads, and beam lengths have also been conducted by Zhu, Seth and Quine (2008). In this work, a dimensionless form of the experimental load-deflection data is introduced to characterize and generalize the load-deflection relationship in a unified way to make the experimental data easy for design application. The study provides new design guidelines for the construction of inflatable structures. 3.7 Challenges in Inflatable Structure Design One important problem with inflatable structures is gas leakage in the pressurized structure. The leakage of gas may be due to cracks or punctures in the membrane when, for example, debris hits the structure. There may also be loss of gas due to mechanical problems in inlets, outlets and seams. Due to these losses, the shape of the inflated structure may not be maintained as designed. The mechanical problems with inlets and outlets can be mitigated by regular maintenance, but for repairing cracks and punctures, special rigidization techniques are required. Immediate attention and repair may be required to fix leakage problems. Ground-based structures are easier to repair using resinbased composites. However, in space, repair is difficult because of temperature and pressure conditions. Advanced techniques are therefore required for repair in orbit for the future space missions to provide further heritage to the inflatable structure technology (Cadogan et ai, 2001). Another problem that causes hindrances in the inflation process is the bend in the fold or a twist in the flow passage made by a flexible material. Kinks in flexible pipes 40 may cause problems during the inflation process and there may not be streamlined flow of gas in the structure to be inflated. Folding and unfolding methods for flexible structure can eliminate problem involving in inflation process. To assure structural accuracy, the introduction of active control methods, for example, the differential change of gas pressure in various structural elements, the combination of cable network and membrane or complementing shape-reproducibility using shape-memory alloys or resins and other rigidization techniques may be applicable. In addition, it is advantageous that the distribution of the mass in the structure is symmetric for balance purposes. 3.8 Future Evolution of Space Inflatable Structures The success of past space missions based on inflatable technology and the current trend in new inflatable structure applications are the initial basis to study the preliminary engineering design analysis of the future space tower structure. A free standing space elevator structure: A practical alternative to the space tether has also been proposed most recently by Quine, Seth and Zhu (2009). The proposed device provides access to the near-space and space environment utilising a self-supporting pneumatically pressurized core structure. In space-elevator configuration, the core structure will be arranged along a linear axis such that the sum of centripetal, gravity and external forces are minimized in the horizontal axes. The mass of the structure and other vertical forces are counteracted by the pneumatic pressure in the cells of the core. The self-guiding approach comprises of 41 active control machinery including gyro stabilisation is desirable to stabilize the structure against buckling or falling and to couple disturbance torques into other axes. The proposed structure provides a fixed link between ground and near-space locations enabling the transportation of the equipment, personnel and other objects or people to the platforms or pods above the surface of the Earth for the purpose of scientific research, communication and tourism. A 7.0 m scale model of a structure is shown in Figure 3.23 and was developed by the author. This 1:2000 scale model comprises three 0.082 m diameter cores braced at intervals of 1.0 m (first four intervals) and 1.5m (last two intervals). The overall diameter of the structure is 0.34 m. Constructed of laminated polyethylene (Young's Modulus of 280 MPa) with an average wall thickness of 0.0013 m the structure has a total mass of 17 kg excluding the base support and is freestanding when pressurized with air above 48,000 Pa (7 PSI). Although many investigations are focussed on the development of a space tether 36000 km in height using carbon nanotubes, the feasibility of a space tower of 20 km in height in Earth's atmosphere may be significantly more practical as assessed by Seth, Quine and Zhu (2009). Such a structure can also provide an ideal surface mounting point where a geostationary orbital space tether could be attached without experiencing the atmospheric turbulence, lightening and weathering. The success of various past inflatable space structures may be utilized in the construction of such a space tower. 42 Figure 3.23 - A section of 7.0 m structure installed in stairwell (Quine et al, 2009). 3.9 Conclusion The pneumatic space structure technology has the potential to replace existing construction methodologies and to extend the limitations of contemporary structural designs. With the continuous developments in materials science, the inflatable structure technology is directly applicable to the construction of large space structures such as space towers. Complex shape inflatable structures with sufficient load carrying capacity can be designed easily using the latest technology involving various design strategies. Inflatable technology being light weight, cost effective and compatible with the space environment is most suitable for large space structural systems. Major improvements in pneumatic structures along with the advancement in inflatable beam technology and the 43 new inventions in material science, and control mechanism, make the dream of Space Elevator Tower realizable in the near future. 44 CHAPTER 4 INFLATED BEAM ANALYSIS 4.1 Introduction This chapter investigates experimentally the bending of inflatable cylindrical cantilevered beams and Euler's buckling as the basis to develop design guidelines to construct an inflatable tower and is derived from work developed collaboratively by the author, Dr. B. M Quine and Dr. Z. H Zhu. A detailed inflated beam analysis is presented in this chapter in collaboration with Dr. Zhu. A dimensionless form of load vs deflection as developed by Zhu, Seth and Quine (2008) generalize the bending behaviour of the inflatable cylindrical cantilevered beams of different sizes, materials, and inflation pressures in a unified way for easy application. The author performed experimentation for validation of the results obtained theoretically. The experimental results demonstrate that the inflatable beams, highly or lightly inflated, can be modeled by the Euler beam theory accurately before wrinkle occurs. The initial wrinkle is hardly noticeable in the experiments and the transition from non-wrinkle to wrinkle is mainly defined by the slope change of load-deflection curve. The actual collapse moment is hard to measure in experiments because the inflatable beam becomes unstable in the collapsed stage. Inflated beam satisfying Euler beam conditions with respect to its length to diameter ratio and critical bending moments can be analysed as a load carrying member based on its load carrying capability. Euler's buckling load is a function of elasticity of material of the column along with its geometrical parameters. Although elasticity is the inherent characteristic of the material, the elasticity of inflated fabric beams depends upon internal 45 gas pressure and therefore can be termed as an effective elastic modulus of the inflated beam. The variation in effective elastic modulus of the inflated beam has been investigated experimentally at different levels of inflation air pressure and results obtained are further carried out to analyse Euler's buckling (critical) load. This chapter is divided into two parts. First part of the chapter presents detailed theoretical and experimental analysis of the inflated beams in terms of its critical bending moments under lateral loads in Section 4.2. Euler's critical axial load analysis is presented in the second part of the chapter presented in section 4.3. The chapter is concluded in Section 4.4. 4.2 Critical Bending Moments under Lateral Loads A brief description of critical bending moments in terms of wrinkle and collapse moments of inflated beam is provided as follows 4.2.1 Wrinkle and Collapse Moments of Inflatable Cylindrical Beam Consider an inflated cylindrical cantilevered beam subject to a lateral tip load. Assume the cross-section of the inflated cylindrical beam unchanged during the deformation. As the applied load increases beyond a certain critical value, the wrinkling region of the inflated cylindrical beam expands in the axial and circumferential directions. When the wrinkled region extends completely around the cross-section area, the inflated cylindrical beam collapses and the corresponding collapse moment can be given directly as (Comer and Levy, 1963) Mc=FL = npR\ 46 (4.1) In addition, several semi-empirical expressions of collapse moment can also be found in literature. For instance, the NASA (1968) recommended a design formula for the collapse moment and is given as McNASA=0.SnpR\ (4.2) In order to generalize the information from experimental results, the dimensionless load m and tip deflections Save introduced as (Zhu. Seth and Quine, 2008) FL Etd where E, L, R, t are the Young's modulus, length, radius, thickness of beam, p is the inflation pressure, F is the transverse tip load and d is the transverse tip deflection of beam, respectively. The critical (wrinkling) bending moment is given as M=mnpR\ (4.4) The value of critical wrinkle moment as derived by Comer and levy (1963) is given as MM=Q.5npR\ (4.5) The tip load - deflection relationship predicted by Euler beam theory is given by FL1 d =- ^ - . (4.6) Equation (4.6) can be expressed in dimensionless form by setting the values from Equation (4.3) as follows 47 5 =m (4.7) This equation is applicable to determine the critical bending moment of the inflated beam that corresponds to its critical wrinkle stage. Consequently, this analysis provides a limit for the application of Euler beam theory and limiting critical bending moment. 4.2.2 Experimental Set Up The experimental set up is shown in Figure 4.1. The experiment involves an inflatable cylindrical cantilevered beam made of fiber reinforced polyethylene hose, an air compressor with regulated air supply, two air pressure gauges and a loading fixture. The inflatable beam was placed vertically with the lower end of the beam being firmly clamped to an aluminum plug that is rigidly bolted to a heavy base with the air supply. The hose was firstly glued to the aluminum plug with epoxy and then clamped tightly by two hose clamps. >1 5L Pressure Gauge String Pulley 2R13 "to Pressure Gauge Load^J CP Air Compressor 31 ////////////////// T7T, ////////// 77" T, Figure 4.1 - Schematic of experimental set up. 48 TO O The upper end of the beam is closed by another aluminum plug using the same procedure. The mass of the upper plug is m = 0.685 kg. An air pressure gauge is mounted on the upper plug to observe the inflation pressure of the cell directly rather than utilizing the pressure reading at the air compressor. The beam is inflated with air up to a certain level using the regulated air supply. At the upper free end of the beam, the lateral load is applied using a string (greater than 1.5 times the beam length) and pulley arrangement. The pulley is positioned at the same height relative to the free end to ensure that the applied load will be perpendicular to the beam. A pointer is attached to the mass hanger on the other end of the string to record the deflection on a vertical scale. The transverse deflection of the upper free end is measured at each applied load immediately after the load is applied in order to minimize the effect of creeping. The load is then removed and the beam restores to a position slightly offset from its undeformed position due to fabric hysteresis. The residual deflection of the inflatable fabric beam is about 5% of the total deflection under load. The beam is manually restored to its undeformed position and then next load is applied again. The experiments were conducted at the following inflation pressures: 69 kPa (10 psi), 103 kPa (15 psi), 138 kPa (20 psi), 172 kPa (25 psi), 207 kPa (30 psi) and 241 kPa (35 psi). The accuracy of length and deflection measurement is estimated as 0.5 mm, while the accuracy for the pressure and load measure is estimated as 7 kPa (1 psi) and 0.01 N, respectively. 49 4.2.3 Experimental Results The mechanical and geometric parameters of inflatable beams are obtained experimentally. The thickness of the beam is uneven in the length and circumferential directions. The variation ranges from 1.0 mm to 1.9 mm. An average thickness, t = 1.22 mm, is used in the data post-processing. For consistency the averaged inflated radius of the beam is R = 0.041 m. The Young's modulus and Poisson's ratio of the beam fabric were measured by tensile tests as per ASTM D638-03 (2003). The measurements are shown in Table 4.1. Five samples were die cut into Type-C samples as per ASTM D638-03 from the beam in the axial and circumferential directions respectively and tested at a pull speed of 5 millimeters per minute. Table 4.1 - Tensile test results of beam fabric. Circumferential Direction Axial Direction Test No. 1 2 3 4 5 Average Young's Modulus Poisson's Ratio Young's Modulus Poisson's Ratio Ex (MPa) E0 (MTIa) V0^ 229.5 0.15 0.16 214.8 0.13 0.11 289.3 211.8 343.7 0.17 0.16 250.5 0.12 313.3 0.20 267.4 0.14 0.14 209.5 243.1 0.14 277.1 0.16 242.9 Three beam samples of different lengths were used to predict lateral load-deflection of the inflated beams. The nominal lengths of three samples are: 1.5m, 1.0m, and 0.5m. Table 4.2 lists the measured length and radius of the inflated beams as a function of inflation pressure. 50 Table 4.2 - Length and radius of inflated beams against inflation pressure. Inflation Pressure Length of Beam (m) Radius (m) (Psi) (kPa) LI L2 L3 R 5 34 1.470 0.973 0.452 0.0402 10 69 1.473 0.977 0.454 0.0404 15 103 1.478 0.978 0.456 0.0407 20 138 1.484 0.980 0.457 0.041 25 172 1.490 0.985 0.458 0.0418 30 207 1.493 0.990 0.460 0.0419 35 241 1.497 0.995 0.463 0.0420 The bending experiments of the cantilevered beam were conducted using two beam samples of different lengths, namely, LI = 1.484m and L2 = 0.983m. The ratios of length to diameter of the beams are L1/(2R) « 18 and L2/(2R) = 12 and therefore satisfy the ratio requirement of Euler beam theory. An inflated beam of length 0.5m does not satisfy the condition and the beam is not taken in further analysis. The transverse tip deflections of the inflated beams of different lengths were measured at the different loads. The results are shown in Tables 4.3 - 4.4. Figures 4.3 - 4.4 show the plots of the external load against the transverse tip deflection curves at different values of inflation pressure. The load capacity of the inflated beam is proportional to the inflation pressure as expected. The strength of the inflated beam increases with increase in internal gas pressure with appropriate geometrical parameters. 51 Table 4.3 - Experimental measurements of deflection of inflated beamZ2 = 0.983 m. P = 69 kPa (lOpsi) P = 103 kPa (15 psi) P=138kPa (20 psi) P = 172kPa (25 psi) P = 207 kPa (30 psi) P = 241kPa (35 psi) F(N) d(m) F(N) d(m) F(N) d(m) F(N) d(m) F(N) d(m) F(N) d(m) 0 0 0 0 0 0 0 0 0 0 0 0 0.49 0.003 0.49 0.003 0.49 0.002 0.49 0.002 0.49 0.001 0.49 0.001 2.45 0.011 2.45 0.011 2.45 0.009 2.45 0.009 2.45 0.008 2.45 0.008 4.41 0.021 4.41 0.021 4.41 0.018 4.41 0.018 4.41 0.016 4.41 0.016 6.37 0.033 6.37 0.029 6.37 0.027 6.37 0.025 6.37 0.024 6.37 0.024 8.33 0.046 8.33 0.039 8.33 0.035 8.33 0.033 8.33 0.032 8.33 0.032 10.29 0.068 10.29 0.052 10.29 0.043 10.29 0.042 10.29 0.04 10.29 0.04 12.25 0.11 12.25 0.066 12.25 0.055 12.25 0.052 12.25 0.05 12.25 0.049 12.66 0.137 15.56 0.129 15.56 0.09 15.56 0.083 19.57 0.102 19.57 0.102 14.62 0.253 19.57 19.57 0.144 19.57 0.133 28.58 0.201 28.58 0.166 0.281 28.58 0.244 43.18 0.362 43.18 0.312 0.23 21.53 0.265 26.4 Table 4.4 - Experimental measurements of deflection of inflated beam LI = 1.484 m. P = 69 kPa P = 103 kPa P = 138kPa (10 psi) (15 psi) (20 psi) P = 172kPa (25 psi) P = 207 kPa (30 psi) P = 241 kPa (35 psi) F(N) d(m) F(N) d(m) F(N) d(m) F(N) d(m) F(N) d(m) F(N) d(m) 0 2.45 4.41 6.37 8.33 9.31 0 0.043 0.083 0.135 0.24 0.32 0 0 0 0 0 0 2.45 0.042 2.45 0.042 2.45 0.04 4.41 0.082 4.41 0.082 4.41 0.065 6.37 0.13 6.37 0.12 6.37 0.1 8.33 0.2 8.33 0.17 8.33 0.155 9.31 0.27 10.29 0.255 10.29 0.235 12.25 0.335 12.25 0.314 14.21 0.53 14.21 0.44 52 0 2.45 4.41 6.37 8.33 10.29 12.25 14.21 16.71 0 0.037 0.069 0.11 0.145 0.195 0.245 0.3 0.475 0 2.45 4.41 6.37 8.33 10.29 12.25 14.21 16.71 0 0.035 0.065 0.1 0.14 0.185 0.225 0.27 0.34 The load - deflection relationships were linear when the load was low. It is interesting to note that all the linear parts of load - deflection curves of different inflation pressure have similar slopes. This suggests that the unwrinkled inflatable beams could be modeled with the Euler beam theory. As the load increased, the load - deflection relationships became nonlinear due to the wrinkle of the fabric and eventually the beam approached to its collapse state. The collapse loads were not obtainable in the experiments because the beam started to pivot at the cantilevered point before reaching the stress-based theoretical collapse load. It was observed in the experiments that the maximum loads obtained before the beam became unstable corresponding to the situation where approximately half of the hose wrinkled as shown in Fig. 4.2. mmam i?:M Figure 4.2 - Wrinkled fabric beam. The raw experimental data of two beams shown in Fig. 4.3 - 4.4 are not very useful in terms of characterizing and generalizing the bending of the inflatable fabric beams. With further processing into a dimensionless load vs deflection form, they are more informative regarding generalised properties and are shown in Fig. 4.5. For comparison, the load - deflection relationship of the Euler beam theory in Equation (4.7) is also 53 shown in Fig. 4.5. The dimensionless experimental data in Fig. 4.5 clearly demonstrate that the dimensionless m - 5 relationship may be approximately fit into a single curve. Compared with the Euler theory, the deflection 8 is linearly dependant on the external load m and agrees very well with the Euler theory up to m «0.4. Beyond that value, the m - 5 relationship gradually becomes nonlinear due to the wrinkle of beam fabric until the external load m approaches the theoretical collapse moment of beam theory mc = 1. In Fig. 4.5, there is a divergence after the critical dimension load of 0.4. At loads exceeding this limit, the inflated beam starts wrinkling under the load losing strength due to loss in effective elastic modulus of the inflated beam. Due to highly non-wrinkle nature of wrinkle effects, the load-deflection curves are seen to diverge in behaviour after the onset of wrinkling. I P = 69 kPa 1 — P = 103 kPa e — p = 138 kPa *—P= 172 kPa B — p = 207 kPa A — p = 241 kPa 0.3 0.4 Deflection in meter 0.7 Figure 4.3 - Experimental load - deflection curves of inflated beam LI = 1.484 m. 54 0.15 0.2 0.25 Deflection in meter 4.4 - Experimental load - deflection curves of inflated beam L2 = 0.983 m. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Etd/pL2 ure 4.5 - Dimensionless load - deflection curves for the inflated beams. 55 4.2.4 Summary of Critical Bending Moments Experimental investigation of inflatable cylindrical cantilevered fabric beams has been conducted in order to obtain design guidelines for the inflated cylindrical beam structures. A dimensionless form of load - deflection relationship has been used to characterize and generalize the behaviour of inflatable cylindrical cantilevered fabric beams of different sizes and different inflation pressures in a unified way for easy application. The experimental results indicate that the wrinkle of the inflated fabric beam occurs because the fabric cannot resist compression. The current and previous (Yoo et al., 2007) experimental results show that the inflatable beam, either highly or lightly inflated, can be modeled by the simple Euler beam theory accurately before the wrinkle occurs. The experimental investigations demonstrate that the bending of inflatable cylindrical beam can be modeled simply using the Euler beam theory before the beam starts to wrinkle. It has been found that the value of dimensionless load (m=0.4) gives the critical point beyond which inflated beam potentially loses its strength under compressive lateral loads. In a practical space tower of height 20 km as proposed by Quine, Seth and Zhu (2009), wind loads would be taken as lateral loads for analysing the behaviour of inflated space tower. The dimensionless form of load-deflection data provides a good tool to reveal some common characteristics of inflatable beams made of different materials with different sizes and working in different conditions. The results obtained here would be very useful in designing the inflated structure for the space tower and further guidelines. 56 4.3 Euler's Critical Load Analysis Buckling causes catastrophic failure of structures and is particularly dangerous due to large buckling deflections. It depends significantly upon geometrical parameters of the structures as discussed further in Chapter 5 involving inflated structures comprising inflated beams. The structural design must satisfy both strength and buckling safety constraints. In inflated structures, these conditions can be achieved by selecting appropriate values of the inflation gas pressure and geometrical parameters along with strong fabric material of the beam. Although buckling is a complicated phenomenon, and any individual inflated column free from defects in the fabric wall material, may be influenced by misalignment in loading, variations in straightness of inflated member, presence of initial unknown stresses in the inflated column, yet it can be controlled by differentially changing the gas pressure inside the inflated columns comprising the structure. 4.3.1 Theoretical View of Critical Buckling Load In 1757, Leonard Euler developed a relationship for the critical column load to analyse catastrophic failure of the columns due to buckling. The critical load is usually regarded as the maximum sustainable load before causing buckling. As applied critical load causes the column to buckle, the column cannot restore its original shape and maintains its deflected shape after the application of critical load. Theoretically, any buckling mode is possible, but the column will ordinarily deflect into its first mode. In pneumatically inflated structures, the critical buckling load depends upon gas pressure inside the columns responsible for strength and stiffness of the inflated structure. The 57 elasticity of the inflated column varies with the level of pressure inside the columns. Therefore, by varying internal gas pressure, the value of critical bucking load can be appropriately maintained to avoid sudden failure of inflated structures. The critical buckling load can be calculated using Euler's formula (Quine, Seth and Zhu, 2009): where £"be the effective elastic modulus of the inflated column depending upon internal gas pressure. Critical buckling load depends upon the second moment of inertia / and Le, effective length of the column. The effective length depends upon the boundary conditions of the column and is given by (Brush and Almroth, 1975) Le = KeL, (4.9) where Ke is a constant called effective length constant which depends upon the boundary conditions (column end conditions) of the column of natural length L. An inflated cantilever beam deployed vertically can be regarded as an inflated column satisfying Euler beam conditions for length to diameter ratio of(>10). To determine the effective length of vertically deployed inflated column, boundary conditions are given in Table 4.5. The theoretical value of effective length constant Ke is usually less than that of AISC (American Institute of Steel and Construction) and a conservative estimate of these parameters are significant in determining critical buckling load. 58 Table 4.5 - Boundary conditions to determine the effective length constant of the column End Conditions Theoretical Value Ke AISC Value Conservative value 4.3.2 Pinned-Pinned 1 1 1 Fixed-Pinned 0.707 0.8 1 Fixed-Fixed 0.5 0.65 1 Fixed-Free 2 2.1 2.4 Effective Elastic Modulus Effective elastic moduls of the inflated beam depends directly upon the level of gas pressure inside the beam. The internal gas pressure maintains the strength and stiffness of inflated beam. The effective elastic modulus E' (measured in Nm" ) of the inflated beam can be determined experimentally using the double cantilevered beam method. The effective elastic modulus at different pressure levels of the inflated beam can be measured using the following relationship (Shanley, 1967): E'= WL Tl (4.10) 48^7 where £"be the effective elastic modulus of the inflated beam of length L and area moment of inertia I (I = 7rR3t). Depression yd at the centre of the inflated beam is caused by applying a load force WL at the centre of the double cantilevered beam clamped rigidly at both ends. R is the radius of the inflated beam. Effective elastic modulus of the inflated beam of circular cross section of radius R and thickness t is given by E' WLU 4SnydR3t (4.11) 59 4.3.3 Experimental Set up for Determining Effective Elastic Modulus The experimental set up for determining the effective elastic modulus of inflated fabric reinforced polyethylene beam is shown in Figure 4.6. The two ends of hose pipe are sealed using rigidly clamped aluminum plugs. An air supply is provided by air compressor equipped with Figure 4.6 - Experimental arrangement to determine effective elastic modulus of the inflated beam. pressure gauge. The horizontally deployed inflated beam is supported by two heavy stands by rigidly clamping two sealed ends of the beam. A mass hanger along with meter scale is provided to measure load applied at the centre of horizontally deployed beam and subsequent measurement of depression at the centre of inflated beam. A pointer is attached to the hanger to record the depression on a vertical meter scale. The depression is measured at each applied load immediately after the load is applied in order to minimize the creeping effect. The load was then removed and the beam returned to a position slightly off its undeformed horizontal position. The beam was restored to its undeformed horizontal position and then next load was applied again to measure corresponding depression in the inflated beam. 60 4.3.4 Experimental Results of Critical Buckling Load At first effective elastic modulus of the inflated beam is determined experimentally. The experiment was conducted by taking two samples of beams of lengths 1.5 m and 2.5 m at the following inflation pressures: 152 kPa (22 psi), 221 kPa (32 psi) and 290 kPa (42 psi). The load-depression data at different levels of inflation pressure is shown in Table 4.6 and Table 4.7. The value of effective elastic modulus of two inflated beams at various levels of inflation air pressure is calculated using equation (4.11). The undeformed symmetric shape of the inflated beam is maintained after the removal of each applied load. It is very hard to predict effective elastic modulus precisely. Its value decreases as compressive applied lateral loads increases. Also increasing length results in decrease in effective elastic modulus of inflated beam because depression in double cantilevered inflated beam increases under loads. Inflated beams cannot regain their original undeformed state after a certain critical lateral load corresponding to its wrinkle stage. It has been observed that effective elastic modulus of inflated beam increases with increase in internal air pressure. As pressure inside the inflated tube increases, its resistance to applied load deformation also increases. At an appropriate internal pressure, the effective elastic modulus of the inflated beam becomes equal to elasticity of the fabric material and increases with increase in pressure. The average values of effective elastic modulus at different pressures are presented in Table 4.8. 61 Table 4.6 - Load-depression data for the inflated beam of length 1.5 m. Pressure 152 kPa (22 psi) Pressure 221 kPa (32 psi) Pressure 290 kPa (42 psi) Load(kg) Load Depression(m) Load(m) Depression(m) Depression (m) (kg) 2.450 0.022 2.450 0.018 2.450 0.014 3.450 0.032 3.450 0.025 3.450 0.02 Table 4.7 - Load-depression data for the inflated beam of length 2.5 m. Pressure 22 psi (152 kPa) Load(kg) Depression (m) Pressure 32 psi (221 kPa) Pressure 42 psi (290 kPa) Load Depression (m) Load(m) Depression (m) (kg) 2.450 0.10 2.450 0.09 2.450 0.07 3.450 0.16 3.450 0.11 3.450 0.09 Table 4.8 - Average values of effective elastic modulus at different pressures. Pressure psi (kPa) Effective elastic modulus of the inflated beam (Nm~2) 22(152) 2.8xl0 8 32(221) 3.5xl0 8 42(290) 4.45 xlO 8 62 Figure 4.7 shows that effective elastic modulus of the inflated beams has approximately linear relationship with internal air pressure. It is interesting to note that inflated beam at high pressures has more effective elastic modulus. It is due to the fact that inflated beam at high pressures have more ability to restore its configuration after the removal of deforming force. The increase in effective elastic modulus of inflated beams shows that beams could resist more to bending loads. However, elastic behavior of inflated beams is limited by inflation pressure constraints depending upon tensile strength of the material. In addition, the critical bending moment parameter predicts wrinkle stage on the skin of inflated beams which depends upon elasticity of the material along with geometrical parameters. 4.6 x 10 A a. * Lenylh - 1.5m 4.4 4.2 B 7y _ —7- 38 7^ ^ ~~ <u 3.4 | 3.2 ^ LU 2.8 ^ T 2.6 1.5 _ / ::: 2 2.5 Pressure in Inflated Tube (Pa) x 10 Figure 4.7 - Dependence of effective elastic modulus of inflated beam on inflation pressure. 63 •^— Fixed-Free * — Fixed-Guided • Experimental \ralue 1.5 2 Original Length of Inflated Column (m) Figure 4.8 - Critical buckling theoretical load predictions of inflated beams and experimentally derived value (71N) for 1.5 m tower shown in black. Euler's equation is used to determine axial critical load. In order to determine applicability of the relationship to determine critical load, the value of length to diameter ratio should be more than 10. The length to diameter ratio of inflated columns of lengths 1.5 m and 2 m are 18 and 30 and found to be always greater than the recommended value of 10, which justifies the applicability of Euler's equation (4.8) to predict critical buckling load. Considering a 15 km high tower, diameter 230 m, anchored at an altitude of 5 km as proposed by (Quine et al. 2009), length to diameter ratio is 65 indicating that Euler's critical load is applicable to the proposed tower. Critical buckling load of inflated beams of various lengths can be predicted by selecting different values of effective elastic modulus at different pressures. Using conservative value of effective length constant 2.4 for fixed-free end conditions 64 (Table 4.5), critical buckling loads of inflated beams of various lengths at 221 kPa (32 psi) pressure has been predicted and results are shown in Figure 4.8. For fixed-pinned (guided) end conditions, the conservative value of effective length constant is 1 (Table 4.5) and therefore, effective length would be equivalent to original length of the inflated beam. The critical load capacity of inflated beam at a particular pressure would also increase. Fixed-guided end conditions can be achieved by mounting an arrangement of gyro control machinery at the upper free end (Quine et al. 2009). For experimental verification of results predicted for the critical axial load, an inflated beam of length 1.5 m, inflated at a pressure of 221 kPa (32 psi) is deployed vertically as shown in Figure 4.9. The lower end of inflated column is clamped rigidly to a heavy support. At upper free end, a wooden platform is mounted for axial loading on the inflated column. A mass of 7.7 kg is placed over the platform to test the strength of inflated beam. It has been observed that inflated beam with fixed-free end conditions did not buckle and was quite stable, and is consistent with the result as predicted as 71 N (7.2 kg). Figure 4.9 - Demonstrating stability of inflated column with axial load at the top. 65 At a high pressure of 290 kPa (42 psi), load capacity should be much larger because stiffness and strength of the inflated beam increases at high pressures. Axial force exerted by the internal air pressure of inflated beam of radius 0.041 m with internal pressure 221 kPa is found to be 1167N, approximately 17 times the Euler's buckling load. Considering a 15 km high tower at an altitude of 5 km, made of boron (Elasticity 450 GPa) with a cylindrical configuration as proposed by Quine, Seth and Zhu, (2009), the critical buckling load is 4.1xl0 9 N (fixed-guided), however critical buckling load changes with change in material properties (Young's modulus of elasticity) and geometrical parameters and also changes with end conditions determining effective length of the tower. Experimental analysis shows that axial load capacity of the inflated beam should be sufficiently large in order to maintain stability under applied axial loads. 4.3.5 Summary ofEuler 's Buckling Load Buckling in particular is a critical phenomenon and is difficult to predict specifically in inflatable structures. Although critical buckling load depends upon geometrical parameters of the columns, yet inflation gas pressure in the inflated columns is significantly important for maintaining effective elastic modulus to avoid buckling. The strength and stiffness of inflated beam depend upon the level of pressure inside the beam. It is important to note that inflated beam maintains its geometrical parameters when inflated appropriately above a certain threshold pressure which maintains its shape. Critical buckling load decreases with increase in length of the inflated beam. Load capacity of inflated beam depends upon level of internal gas pressure and area of cross 66 section of the beam, and is always greater than Euler's buckling load being dependent upon length, elasticity and other geometrical parameters. 4.4 Conclusion The theoretical and experimental investigations show that inflated beams can be modelled as Euler beams. The conditions for an inflated beam to act as an Euler beam depend upon the mode of loadings in addition to length to diameter ratio. The two main modes of loading are lateral and axial loading. Experimental investigations show that inflated beam can be regarded as Euler beam before wrinkling starts under lateral loading. As expected, the critical lateral wrinkling load and critical axial buckling load depend significantly upon internal gas pressure. Geometrical parameters of the inflated beam and properties of the wall material are also significant. 67 CHAPTER 5 CONSTRUCTION OF INFLATED MULTIPLE- BEAM STRUTURES 5.1 Introduction This chapter describes experimental investigations of inflatable multiple-beam structure geometries to be used as a core structure for the construction of a 20 km high space elevator tower. The result of investigations of a single inflatable beam based on nonlinear moment-curvature relationship is further applied to design inflatable multiplebeam structures. To design a stable inflatable multiple-beam structure, intermittent bracing system is needed to stabilize each inflatable beam and their spacing is optimized for maintaining shape and buckling strength of the inflated structure. The bending of inflatable towers of different sizes comprising inflated beams are investigated experimentally. Experimental design guidelines are applied to develop a 7 m high prototype structure, demonstrating the capability of inflatable multiple-beam structure to construct a core structure for the design of a pneumatically supported 20 km high space tower. The existing literature shows the theoretical and experimental investigations of the single inflated beams (Chapter 3; Section 3.6). The experimental analysis of the inflated multiple-beam structures is rarely seen in the literature so far and multiple-beam behavior under the applied loads needs to be investigated. The inflatable multiple-beam structures are suitable for the construction of a core structure for building large inflatable structural systems. 68 In parametric experimental investigations of single inflated beams, Zhu, Seth and Quine (2008) introduced a dimensionless form of the experimental load-deflection data to characterize and generalize the load-deflection relationship in a unified way and to make the experimental data easy for design applications. The proposed design guidelines are used further for experimental investigations of inflatable multiple-beam structures for their significant use in constructing a pneumatically supported space tower to access 20 km altitude. This chapter consists of six sections. Following this brief introduction in Section 5.1, Section 5.2 presents the theoretical view of the inflatable multiple-beam structures. The dimensionless load-deflection mechanics to be applied for its utilization in constructing inflated multiple-beam structures is presented in this section. Section 5.3 describes the experimental set up. The experimental results are presented in Section 5.4 and subsequent analysis of the results is also presented in this Section. The lateral deflections of inflated structures of different lengths were measured with different lateral tip loads at different inflation pressures. Consequently, the result of the analysis has been applied to design and develop a free standing 7.0 m inflated multiple-beam structure. The details of the free standing 7.0 m inflatable structure comprising three inflated beams is described in Section 5.5. Finally, the chapter is concluded in Section 5.6. 5.2 Theoretical Formulation The load capacity and geometrical stiffness of inflatable structures depend upon internal gas pressure of the comprised inflated beams. The maximum value of applied 69 internal gas pressure depends upon tensile strength of the material and geometrical parameters like radius and thickness of the structure wall. When the cylindrical fabric beam is inflated, the material comprising the beam is subjected to pressure loading, and hence stressed in all directions. The stresses resulting from this pressure are typically parameterized as functions of the radius and thickness of the beam element under consideration, the shape of the inflated structure (currently cylindrical), and the applied pressure. The most common method to analyse the inflated cylindrical beam is based on a simple mechanics approach which is applicable to "thin wall" pressure vessels that by definition have a ratio of inner radius, R, to wall thickness, t, of RJt > 10 (Young et al, 2001). When an internal gas pressure exists in the inflated cylindrical beam, two types of stresses are generated: axial stress cra and hoop stress ah, such as (Young et al., 2001): pR pR Let a be the tensile strength of the material. Then, the corresponding maximum gas pressures in the inflatable beam can be obtained as: lot Pa=~, at P„=-- (5-2) The value of pressure corresponding to hoop stress ph is less than that corresponding to axial stress pa. Therefore, the pressure value corresponding to hoop stress determines the maximum safe limit of the internal gas pressure with an additional applicable safety 70 factor. Therefore, ph gives the maximum limiting value of the internal gas pressure corresponding to a given RJt ratio as P ^ ~ - (5-3) It is assumed here that the internal gas pressure is kept constant throughout the entire inflated column of a given height. Experimental investigation of single inflatable cylindrical cantilevered fabric beams provides the design guidelines for the construction of inflated multiple-beam structures. The bending of inflatable cylindrical beam can be modeled simply using the Euler beam theory before the wall of beam starts to wrinkle. The dimensionless form of loaddeflection data provides a good tool to reveal some common characteristics of inflatable beams made of different materials with different sizes and working in different conditions. The dimensionless load m and tip deflections 8 are introduced as (Zhu et ai, 2008) FL Etd s m= r ; 8 =—-. (5.4) pL2 TtpR' where E, L, R, t are the Young's modulus, length, radius, thickness of beam, p is the inflation pressure, F is the transverse tip load and d is the transverse tip deflection of beam, respectively. In comparison to the Euler beam theory, the deflection 8 is linearly dependant on the external load m and agrees very well with the Euler theory up to m a 0.4. Beyond that value, the m - 8 relationship gradually becomes nonlinear due to 71 the wrinkle of beam fabric until the external load m approaches the theoretical collapse moment (mc=l). 5.2.1 Inflated Multiple-Beam Structure The expression for critical (wrinkling) bending moment of the inflated multiplebeam can be derived in terms of inflation pressure and geometric parameters of the structure comprising the inflated beams. The wrinkling bending moment of the structure depends upon the applied pressure p, the number of inflated beams N, radius of each inflated beam R and the radius r of the structure. The rigidity of the inflated beam structure is maintained by internal gas pressure. The second moment of inertia of the structure is significantly involved in the Euler beam equation and therefore, initially, expression for second moment of inertia of the structure consisting of a number of beams positioned circumferentially symmetric as shown in Figure 5.1 is derived. • X Figure 5.1 - Multiple-beam structure cross section under lateral load F such as Fi, F?, F3. in any direction consisting of three inflated beams. 72 The area moment of inertia of the cross section of the multiple-beam structure about any arbitrary axis passing through the centre O (e.g. the axes OX, OY and OX') is found to be same. Three beams A, B and C, each having radii R are equidistant from each other and centered at the circumference of the circle of radius r, such that AB = BC = CA as shown in Figure 5.1. The inertia moment of the cross section area of the multi-beam structure about OX-axis is given by rrV /.. = nB}t + InRtr1 + 2 nR3t + 2nRt\ v^y = 3nRh + 3nRtr2 = 3nRt(R2 +r2). (5.5) The inertia moment about OY-axis is also given as Iy =7rR3t + 2 7TRit + 27iRt(rcos30°)2\ ( n; V I =nRit + 2, nRU + 2nRt\ S V " = 37rRt(R2+r2). J (5.6) Equations (5.5) and (5.6) show that the inertia moment of the cross section area is independent of the axes and the result is significant in the sense that structure would give same lateral deflection irrespective of the chosen axes. The equations (5.5) and (5.6) can be generalized to any number of beams forming the structure. Further, the multiple-beam structure consisting of 4, 5, 6...TV inflated beams of radius R, centered symmetrically at the circumference of the circle of radius r, the area moment of inertia of the configuration 73 about any axis passing through the center of the circle in its plane of cross-section is directly proportional to N, the number of inflated beams in the multi-beam structure. Therefore, the generalized form of inertia moment of cross section area for N number of beams is given as I = N[nRt(R2 +r2)\ (5.7) The flexural rigidity EI (Nm2) of the inflated multi-beam structure is given by {EI)muln =NEnRt(R2+r2). (5.8) The tip load-deflection relationship as predicted by the Euler beam theory can be expressed in the dimensionless form in the form of composite beam equation for the inflated multiple-beam structure with the introduction of pressure p as follows (Zhu et al, 2008) FL NnpR{R2+r2) 3Etd pL2 (5.9) Following equation (5.9), the critical dimensionless load m for the multiple-beam structure consisting of TV inflated beams can be introduced as FI m' = £^ _ NnpR(R2 +r2) (5.10) The critical bending moment for the inflated multiple-beam structure is given by Mu. 2 im„„=m'N7ipR(R 74 +r2). (5.11) The critical bending moment of the inflated structure therefore depends upon inflation pressure, geometry of the structure and also the number of inflated beams used in the structure. 5.2.2 Equivalency of Multiple-Beam Structure to a Single Beam The multi-beam structure consisting of any number of inflated beams can be equivalent to a single beam by equating the area moment of inertia as follows. Let R = radius of the single beam t = thickness of the beam The area moment of inertia of the single beam is given by /' = > < / . (5.12) By equating equations (5.7) and (5.12), we get, nR]qt = NnRt(r2 +R2) Req=[N{R(r2+R2)}j. (5.13) Equation (5.13) is valid when the thickness of the material using in both the single and multi-beam structure is same, otherwise thickness can also be introduced for determining the equivalency. 75 5.2.3 Comparison of Bending Moments of Single Inflated Beam and MulitpleBeam Structure The inflated multiple-beam structure consisting of TV single beams inflated with pressure p, can be reduced to an equivalent single inflated beam by taking into account of the respective critical bending moments by using Eqs (5.4) and (5.10). _m'NR(R2+r2) mRl Mw_mM M,„ (5.14) The above equation can be written as M i 2 A f —N 1 mum= m M„. m v r H R2J Mwmulli=KMw, 2 A ( m K = —N l + ^2 m v R J (5.15) where K is a factor, which depends upon the structure parameters. Therefore, the critical value of the bending moment for the inflated multiple-beam structure is K times the critical bending moment of the single inflated beam. The inflated multiple-beam structure can be designed as required with respect to its bending moments, by choosing suitable values of the parameters involved in Eq. (5.15). 5.3 Experimental Set Up The experimental set up is shown in Figure 5.2. The inflated multiple-beam structure is comprised of three equal size fiber reinforced polyethylene tubes. The brackets to hold the structure are designed and developed as shown in Figure 5.3 in such a way that all the 76 three inflated beams could be positioned symmetrically in the structure. The center of each inflated beam is located at the corners of an equilateral triangle. Figure 5.2 - Experimental set up of inflated multiple-beam structure. Three hoses are put through the brackets placed equidistant from each other on ground. The ends of all the three hoses are sealed with aluminum plugs using epoxy and clamped using three hose clamps at each end. The pressure gauges are mounted at the top to observe the pressure in the inflated beams as shown in Figure 5.4. 77 rf*"' X"^N Figure 5.3 - Bracket to maintain structural symmetry. MHJiiiWMFffl Figure 5.4 - Three pressure gauges mounted at the top of each inflated beam. At first stage, the structure consisting of three hoses is inflated by deploying it horizontally on the ground. The positions of the brackets to hold the three beams are optimized approximately at a distance of 1 m. A dedicated air compressor is used to inflate the beams. The output of air compressor is divided into three identical channels to provide separate air supply for each of the three columns. The air supply in each line can 78 be controlled manually by using the mechanical valves provided in each of the three channels. During the inflation process, the bracket-hose arrangement provides a strong grip between the brackets and the inflated beams due to increase in air pressure in each of the beams. After proper horizontal alignment of the inflated structure, the structure consisting of three inflated beams is then deployed vertically with its base bolted and clamped rigidly to the heavy wooden base with an air supply as shown in Figure 5.5. Figure 5.5 - Structure base bolted and clamped rigidly to the heavy wooden base. The structures of lengths 4 m and 5 m are then deployed vertically in a stairwell to analyze the lateral tip load- deflection behavior at different pressures. The lateral tip loaddeflection behaviour is analyzed using string and pulley arrangement as shown in Figure 5.2. The pulley is installed at the same height relative to the free end to ensure that the applied load will be perpendicular to the structure. A pointer is attached to the weight hanger on the other end of the string to record the deflection on a vertical scale. The transverse deflection of the upper free end is measured at each applied load immediately after the load is applied in order to minimize the creeping effect. The load was then removed and the structure returned to a position slightly off its undeformed position due 79 to transverse component of the dead load of the structure and the hysteresis of the fabric of the inflated beams. The residual deflection of the inflated structure was about 10 % of the total deflection under load. The inflated structure was manually restored to its undeformed position and then the next load was applied again. The experiments were conducted at the following inflation pressures: 69 kPa (10 psi), 103 kPa (15 psi), 138 kPa (20 psi), 172 kPa (25 psi), 207 kPa (30 psi) and 241 kPa (35 psi). The accuracy of length and deflection measurement is 0.5 mm, while the accuracy for the pressure and load measurement is 7 kPa (1 psi) and 0.01 N, respectively. 5.4 Experimental Results The bending experiments of the inflated structure consisting of three inflated beams were conducted using structure samples of lengths 4 m and 5 m respectively. The total mass of the structure includes mass of brackets, material of the beams, aluminum plugs and mass of pressure gauges mounted at the top. The comprehensive mass of the structures of lengths 4 m and 5 m is measured as 11.5 kg and 13.8 kg respectively. The transverse tip deflections of the inflated beam structure of different lengths were measured at the different loads and inflation pressures. The transverse component of weight of the structure (mgsinO ~ mgd/L, angle of deviation 6 being small), though small also makes its comparatively small contribution in lateral deflection and thus is taken into account in the measurement. The total lateral tip load F on the structure is equal to the sum of applied load and transverse component of the weight of the structure during bending. The results are shown in Figure 5.6 and Figure 5.7. 80 45 40 35 'f 30 25 d ^ ^ 20 „ 69 kPa 103 kPa 138 kPa * P - 172 kPa — B — P - 207 kPa 15 10 5 P - 241 kPa o 0.2 0.1 0.3 0.4 Deflection (m) 0.5 0.6 0.7 Figure 5.6 - Lateral load-deflection data of inflated multiple-beam structure of length 4 m. 30 25 15 10 -P= -P = -P = -P = 5 69 kPa 103 kPa 138 kPa 172 kPa - P = 207 kPa 0 0.05 0.1 0.15 0.2 0.25 0.3 Deflection (m) 0.35 0.4 0.45 0.5 Figure 5.7 - Lateral load-deflection data of inflated multiple-beam structure of length 5 m. 81 The raw data as shown in Figure 5.6 and Figure 5.7 as such does not provide much information for structure design guidelines [Zhu et al. 2008]. Dimensionless loaddeflection curves using raw data are drawn to obtain further design guidelines for building large space structures. Figure 5.8 shows the plots of the dimensionless load and deflection at different values of inflation pressure. The load capacity of the inflated structure is proportional to the inflation pressure as expected. The transverse tip load is applied at the free end of the structure along different lateral directions. It is found that lateral tip deflection in all the different directions is symmetrical as predicted by theory. P = 69 kPa P = 103 kPa \—P= 138 kPa a—p= 172 kPa e — p = 207 kPa — A — p = 241 kPa 0.12 Etd/pl/ Figure 5.8 - Dimensionless load-deflection curves of inflated multiple beam structures. The dimensionless experimental data in Figure 5.8 demonstrates that the dimensionless m -8 relationship of the inflated structure approximately linear up to a certain critical stage when wrinkle starts to develop in the fabric beam. Compared with 82 the Euler beam theory again, the deflection 8 is linearly dependant on the external load m and agrees very well with the Euler beam theory up to nearly m = 0.2. Beyond that value, the m - 8 relationship gradually becomes nonlinear due to the development of the wrinkles in the inflated beams forming the structure. The value of dimensionless load m is considerably less (nearly 50%) than the value found in the experimental investigations of the single inflated beam (m « 0.4). The dimensionless load depends upon geometrical parameters of the structure and inflation pressure. By substituting r = 0.125 m, R = 0.041 m and N = 3 in equation (5.13), the value of radius of single beam {Req) equivalent to the multiple-beam structure found to be 0.13 m. Length to Diameter ratio for the structures of lengths 4 m and 5 m is calculated as 15 and 19 respectively, which satisfy Euler beam condition (Length/Diameter>10). The results are more satisfactory with an inflated structure of length 5 m having with more length to diameter ratio. Even though these sample structures satisfy the Euler beam condition of length to diameter ratio, for the structure to behave as an Euler beam under lateral loads, the inflatable multiple-beam structures are not as consistent and uniform as compared to an inflated single beam. The isotropic and homogeneous conditions of the structure material are not satisfied by the designed inflated multiple-beam structure being constructed using fabric reinforced polyethylene as a beam material and overlapped intermittently by brackets required to hold the inflated beams. Intermittently positioned brackets are beneficial to maintain cross section of the structures, but part of the composite structure in between two successive brackets does not maintain cross section precisely. 83 Consequently, the inflated multiple-beam structure does not maintain its axis of symmetry in the plane of bending under lateral loads beyond a dimensionless load ofm' « 0.2 . In addition, cross section of the composite inflated structure cannot remain plane during bending due to bending of different inflated beams comprising the structure beyond a dimensionless load of m « 0.2 as compared to single inflated beamm ~ 0.4. Experimental investigation of hybrid inflated multiple-beam structure is particularly important due to its characteristic of maintaining its stability and attitude by differential variation of pressure in the comprising beams as explained in Chapter 6. The geometrical parameters, though varies due to change in internal pressure and applied loads, found to have significant role in determining the behaviour of inflatable structures more precisely. This investigation directed that Euler beam behaviour in inflated multiple-beam structures should be expected for stresses of only 20% of the critical collapse load of inflated beam (see Equation (4.1) in Chapter 4). A 15 km tall tower located at an altitude of 5 km as proposed by Quine, Seth and Zhu (2009) comprises about 700 inflated beams each of radius 0.5 m and wall thickness 0.012 m. Considering a cylindrical configuration, these inflated beams are arranged symmetrically on a circumference of radius 114.5m. Maximum limiting value of pressure using Equation (5.3) in each beam is 83300 kPa (833 bar) (Boron as a wall material, tensile strength = 3.5 GPa) and 85700 kPa (857 bar) (Kevlar as a wall material, tensile strength = 3.6 GPa). The equivalent radius of the structure using Equation (5.13) is found to be 166.2 m. The limiting value of critical bending moment using Equation (5.11) is found to be 7.91xlO l3 Nm (for tower made of Boron) and 8.0xl0 13 Nm (for 84 tower made of Kevlar). The wind load bending moments are considerably less than limiting (critical) bending moments and their feasibility along with lateral deflections due to wind loads is presented in Chapter 7. 5.5 Free Standing 7 m Inflatable Multiple-Beam Structure A free standing 7.0 m high inflatable multiple-beam structure comprising of three inflated beams is designed and installed in the stairwell for demonstration as shown in Figure 1.1 (Chapter 1). The inflated structure is pressurized using regulated air compressor and stands on a heavy wooden base as shown in Figure 5.9. A video of the Figure 5.9 - 7 m structure pressurized using compressor that stands on a heavy base. 7 m structure is also made in order to record the behaviour of structure. The inflated structure clearly demonstrates its capability to restore its position after immediate removal of deflecting forces. The detail of the parameters involved in developing prototype structure is shown in Table 5.1. 85 Table 5.1 -Numerical values of physical quantities involved in developing a free- standing 7.0 m prototype inflated structure. Name of physical quantity Numerical value Height of inflated multiple-beam structure 7.0 m Outer radius of structure comprising of three inflated beams 0.17m Total mass of the structure 17.06 kg Minimum pressure required in each of the inflated columns 48 kPa(7psi) for free standing structure Mass of each bracket 1.093 kg Size (length) of each bracket and thickness 0.125 m, 0.002 m Average radius of each inflated beam 0.041 m Average thickness of the inflated beam 0.00122 m Young's Modulus of the beam fabric material 277 MPa 5.6 Conclusion The load-deflection behaviour of inflated structures of different lengths has been studied. Geometrical parameters of inflated structures have significant contribution in determining dimensionless load-deflection relationships, when pressurized appropriately. Flexural rigidity of the structure can be maintained by changing geometrical parameters in addition to the inflation pressure. The inflated structure behaves as an Euler beam structure before wrinkle (critical stage) occurs in the fabric of comprised inflated beams. The critical stage of bending moment of the inflated multiple-beam structure is 86 represented by dimensionless load and provides design guidelines to build pneumatically supported space towers. The low-precision inflatable beam structures have certain limitations in predicting bending moments; nevertheless, a 7 m high prototype structure successfully demonstrates behaviour. As the walls are not fully populated with inflated tubes, this 2000:1 scale model is quite similar to the proposed structure in aspect ratio. Experimental investigation of inflated multiple-beam structures in terms of its bending moments is an important subject to study for designing large inflatable structural systems and requires further research to predict their behaviour under loads with more accuracy. Advantageously, inflated multiple-beam structures can be actively controlled by differential change of pressure in the individual inflated beams for its vertical stabilization. 87 CHAPTER 6 ACTIVE CONTROL MECHANISM METHODOLOGIES 6.1 Introduction Active control mechanisms can significantly aid in controlling the attitude of inflatable structures. Control options include gyroscopic inertial stabilization utilizing a series of gyros rotating at high speed and pressure balancing stabilization induced by differential change of pressure in inflated columns of the structure to maintain its centre of gravity at desired position. The attitude control of unstructured objects in navigation, aeronautical and space engineering utilizes gyroscopes. The active control mechanism utilizing gyros can also be used to control wind induced vibrations in the free-standing tall structures (Higashiyama et ai, 1998; Moon et al., 2005). The pressure control mechanism can potentially be used to control attitude of the inflated space structures (Cadogan et ai, 1999). Theoretical results have been developed in order to analyse the behaviour of inflated structures controlled using gyroscopic and pressure balancing mechanism under the application of lateral loads. The results so obtained are applied on a system consisting of rotating flywheel mounted at the top of an inflated beam. To enhance gyroscopic effect, the utilization of a multiple-gyro approach has been studied for its application in proposed space tower. Space tower comprised of inflated multiple-beam core structure maintains its strength utilizing gas pressure at an appropriate level. Complex wind loads can cause lateral deflections at various regions along the length of the inflated structure. To mitigate the effects of wind induced lateral deflections and vibrations in the inflated structure, 88 bending moments can be generated in the tower structure by differentially varying the pressure in the inflated columns comprising the inflated structure. An experimental set up is designed for stabilizing the structure under the influence of lateral load by utilizing the methodology of differential change of pressure inside the inflated multiple-beam structure for restoring vertical orientation. Following this brief introduction, Section 6.2 presents theoretical background of gyroscopic control mechanism and pressure control mechanism. Mathematical model of a system of gyros for active control mechanism of a space tower is presented in this Section. Experimental set up of active control mechanisms involving rotating flywheel assembly mounted on an inflated beam and pressure balancing mechanism are presented in Section 6.3. Following the experimental set ups, the experimental results obtained and their subsequent analysis are described in Section 6.4. Thermal effects are briefly discussed in Section 6.5. Finally, the chapter is concluded in Section 6.6. 6.2 Theoretical Background 6.2.1. Gyroscopic Control Mechanism Consider a flywheel rotating about a vertical axis with an angular momentum depending upon its mass moment of inertia and rotation speed as shown in Fig. 6.1, where LT represents angular momentum of the flywheel about z-axis. r and ImT represent the torque about y-axis and mass moment of inertia of the wheel about z-axis respectively. The system comprising a rotating flywheel about a vertical axis tends to maintain its vertical orientation unless an external torque is applied to change its attitude. Assume an external force Fv is applied perpendicular to the spin axis of the flywheel. 89 This force causes a torque about an axis perpendicular to the spin axis leading to gyroscopic precession. The effect may be used to control the orientation of a structure. +X Figure 6.1 - Gyroscope spin axis along with torque and precession axis. With the application of a force Fx acting perpendicular to the spin axis at an arbitrary point P, a torque about y-axis (perpendicular to z-axis) r is developed as dLv (6.1) dt where dL is the change in angular momentum along the j-direction in time interval dt. The final angular momentum due to the effect of torque developed can be written as L = L. +dl, . 90 Subject to the force Fx, the vertical axis bends with an angle dd as shown in Fig. 6.2, such that, dLyv d6- (6.2) Z_ Pr dL U Lz + dLy O (a) Lz O O (b) (c) Figure 6.2 - Bending of the spin axis (vertical axis) and angular momentum of the spinning wheel. The scalar form of Equation (6.1) and Equation (6.2) may be combined as d0_ = ^ dt L_ ' (6.3) Equation (6.3) gives the rate of angular displacement. This equation shows that rotational motion about x-axis can be obtained by the application of torque about y-axis (or applying force along x-axis) and leads to precession. The torque required to deflect the vertical axis through a certain angle can be determined by considering angular momentum of the spinning wheel, such as 91 4 = ' > , (6-4) where co is the angular velocity of the rotating wheel in rad/sec. Since the magnitude i.e., \L7 = Imzco = \LT+dL of the angular momentum remains the as shown in Fig. 6.2(b) and the lengths OP = OP'-L, same, are the same because of the same magnitude of angular momentum, see Fig. 6.2(c), the angular displacement of the spin axis becomes de = — , (6.5) K where hg denotes the height of gyro from ground and dS the lateral displacement of the loading point P, respectively. Equating equations (6.2) and (6.5) gives dLy _ dS L. ~ h' (6.6) Dividing the above equation by dt and substituting Eq. (6.1) gives the torque required to bend the vertical axis, such that L X = T=— ; dLy dt L_(dS\ =^ h\dt — L_h0d9 =—— h„ dt d6 T , = L- — = L_0) • ' dt df) where co' = — is the angular rate of bending of the spin axis measured in rad/sec. dt 92 l£ n. (6.7) Setting Ly = Im7co = 1njlm_ in Eq. (6.7), the torque required for bending of spin axis which is numerically equal to resistive torque (r res ) is given as r,es=2nfImzco', (6.8) w h e r e / i s the frequency of revolution of the flywheel in hertz (rev/sec). Considering the torque, such as r = Fxhg, (6.9) Then, the force required for a particular lateral displacement of the vertical axis can be calculated by substituting value of r from Eq. (6.7) into Eq. (6.9), such that Equations (6.8) and (6.10) can be applied for experimental investigations of the system comprising rotating flywheel mounted at the top of the inflated beam. 6.2.1.1. Mathematical Model of a System of Gyros Gyroscopes are known for their gravity stabilization characteristic utilizing angular momentum. A system comprising of an appropriate number of high frequency spinning wheels may be advantageous in maintaining vertical stabilization of the space tower. Angular momentum of rotating flywheel mounted on a vertical stabilized system cannot be changed unless external torque is applied. A mathematical model is developed by keeping in view of centre of gravity of the tower in relation to its height and mass. The number of gyros to be used depends upon the height of the tower to be achieved. 93 6.2.1.2. Multi-Gyro System Design A typical design comprises a number of gyros (flywheels) mounted on a vertical axis as shown in Fig. 6.3. The configuration of the whole system depends upon the mass mg, radius rg and height Lg (from ground) of the lower-most gyro (first gyro). VJ J 3L g ^=> Vz J * 2Lg mg Lg Figure 6.3 - System of gyros mounted on a vertical axis. Let n be the number of gyros. The number of gyros is taken to be odd because the mathematical series developed simplifies well for odd number of gyros; however, any number of gyros can be used depending upon user's design strategies. The masses of the 94 2 nd , 3 rd , 4lh, 5th ... gyros are taken to be mg/4, mg/9, mg/16, mg/25... (mass divided by square of each successive number of gyro). In addition to mass of gyros, the mass of the structure is also taken in to account to demonstrate the practicality of the design strategy. The height of 2 nd , 3 rd , 4th, 5th ... gyros is Lg+2Lg, Lg+2Lg+3Lg, Lg+2Lg+3 Lg+4Lg, Lg+2Lg+3Lg+4Lg+5Lg ... respectively. Radius rg of each gyro is taken to be same for cylindrical symmetry. The frequency required to keep the system vertically stable in the presence of gravitational force can be calculated using the following equation (Quine et al. 2009): \4M,o,gLcJm, ( 6 H ) where Mmg is weight of the core structure comprising gyros, Lcm is the height of centre of mass of structure, Imx is mass moment of inertia about x-axis and Imz is mass moment of inertia about z-axis of the core structure respectively. The mass moment of inertia of the whole system about the horizontal axis and vertical axis is adjusted to obtain vertical stabilization of the structure. For stability of the structure, the centre of mass of the system is lowered by decreasing the mass of each gyro in a series with increase in height of the tower. 6.2.1.3. Rotational Frequency of Gyros for Vertical Stabilization Denote Llol as the total vertical achievable length of the tower and Mayros as total mass of the gyros, such that, 95 L,o, = Lg +2Lg +3Lg +4Lg +5Lg+ + nLg ^1- (6.12) v=l ( M„ m m ma m m0s +—— + —- + —- + —'- + . 4 9 16 25 n 1 (6.13) M gyros=mg\YJ-Y .v=l •* J Let Ms be the mass of structure excluding the mass of gyros. Mass Ms can be taken as a factor of the total mass of gyros, such as Af (6.14) =iMr where / is positive real number value. The total mass of the system consists of mass of the structure and mass of gyros, such as (6.15) Mm=M^Mwns Substituting the values from equations (6.13) and (6.14) in Eq. (15), the total mass is given by Mm=mg{i + 1){\±^ A= l A (6.16) J The moment of inertia of the system about z - axis (vertical axis) is given by ma m0 ma m0 m + —- + —- + —- + —- +. * 4 9 16 25 96 f n I =mgrS na z- 2 X\ (6.17) V -v=l •* J Also, the moment of inertia of the system about x - axis (horizontal axis) is given by /_ = m :V*^N^KN^KNiK)r Let the value of Ms = iM *-^A„ results the moment of inertia as n-\ /,„.. = m„L„ ±lx+l)> .£<*±2 1 « 1 f n \ V -v=l J (6.18) •> .v=l x The height of the centre of mass of the system including the structure and the gyros with respect to the ground is given by L„ = mL " s m„ m„ m„ mn + ^ 3 Z „ +-^-6La + —10Z,„ + — 1 5 L +... 4 * 9 ° 16 " 25 J m0 m ma m m +—*- + —- + —'- + ——+ . +M. " 4 9 16 25 V 1 MSLM (6.19) Setting the values of Llol from Eq. (6.12) andM s from Eq. (6.14), the above equation is summarized as L„ v=l .v=0 L„ Y « n 1 1 A\ ^X ff *• .v=l v=l AX 1 X— +''Z-r IV -v=l •* J 97 .v=l (6.20) The frequency of the gyros (Eq. 6.11) in revolution per minute (rpm) required for stability of the system is given by / = 60 rM'0' gL™\m* )( 2x3.14x7^ rpm, (6.21) where g is the acceleration due to gravity. Eq. (6.21) can be applied to the experimental set up as described in Section 6.3. 6.2.2. Pressure Control Mechanism The strength and stiffness of inflated beams depend upon the level of pressure inside the beams. When inflated, the beams store energy depending upon the inflation pressure and volume of inflated beam. The process of inflation changes configuration. During the process of inflation, the potential pressure energy of the inflated beam increases from minimum (zero, when not inflated) to a level depending upon internal pressure and volume acquired by the inflated beam (Energy = Pressure x Volume). Considering an inflated multiple-beam structure consists of three inflated beams arranged in such a way that structure stands vertically. The energy stored in inflated multiple-beam comprising three beams is given by ET=plVl+p2V2+p3V3 (6.22) where Ej is total energy stored in the structure comprising three inflated beams due to inflation gas pressure/?/, p2 andps corresponding to their respective volume acquired V\, V? and Vi in each of the three inflated beams A, B and C as shown in Fig. 6.4. 98 The three beams comprising the inflated structure have the same geometrical parameters; length, radius and thickness of the wall material. At first, the levels of pressure and their corresponding volume in three beams are equalized such that the inflated structure stands in a vertical orientation. The total energy of inflated structure is therefore given by ET=3pV (6.23) where p = p\ = P2 - P3 and their corresponding volume be V = V/ = V2 =¥3. Eq. (6.23) can be written as ET=3p(7iR2L) (6.24) where L is length and R is the radius of each inflated beam, when inflated at pressure/?. Figure 6.4 - Top view of the multiple-beam structure comprising inflated beams. The potential pressure energy stored in the inflated structure is limited by the tensile strength of the material chosen and geometrical parameters of the beams (Eq. 5.3). The 99 maximum inflation pressure depends upon tensile stress a, radius R of the beam and thickness t of the wall material. Therefore, the total maximum energy stored in the structure comprising pressurized beams can be generalized as where N is number of beams comprising the inflated structure. Bending in inflated structure can be controlled by varying pressure and hence energy in the inflated columns. A lateral force F acting along y-axis bends the structure along that direction through a lateral displacement d. The bending moment corresponding to applied force should not be more than critical wrinkling moment of the structure comprising inflated fabric beams, such as, (using equation 5.11): Mw 2 muln=m'NnpR(R Fc=m'NnpR(R2 +r2) +r2)/L, (6.26) where Fc is the critical force for a structure of length L and its value is different at different levels of pressure. Work done by the applied lateral load F (F < Fc) causing a lateral displacement d is given as Wexl=Fd. (6.27) External work done during bending is equal to the elastic potential energy stored in the inflated beam (depending upon the stiffness), kinetic energy of the total mass of inflated beam during bending and also energy dissipated in compressing the air in the 100 inflated beams (Cavallaro et al. 2007). In order to restore its vertical orientation, the pressure in the inflated beams comprising the structure should be increased at its appropriate level, so that increase in pressure energy would be sufficient to resist lateral loading by maintaining its strength. For N number of beams, the increase in pressure energy in the inflated multiple-beam structure by increasing the pressure in each column up to a level p (p < PTmm), using Equations (6.23) and (6.24), is given as AE = N(p'-p)V AE = N(AP)TTR2L 57" sin(5Q)M (6.28) String Pulley Inflated Multiple-beam structure Scale Pressure Gauge Air Compressor > , , , ) , > , ) ^^^^^^^^^^^^^/ fc Figure 6.5 - Inflated multiple-beam structure force diagram. 101 The increase in pressure required for vertical restoration of the structure can be explained as follows. Initially, the multiple-beams are inflated, so that structure comprising inflated columns stands vertically. For an applied lateral tip load F (F < Fc), inflated structure has a lateral deflection d and the structure bends through an angle 89 as shown in Fig. 6.5. The structure will remain in its bending state until restored to its vertical orientation by increasing differential pressure inside inflated columns. The pressure ^required to restore the structure orientation can be calculated using force diagram shown in Fig. 6.5. The tension in the inflated beam skin and hence of inflated beam made of strong fabric increases with the increase in pressure. The component of increase in tension ST (increase in pressure^ x area NnR ) by increasing pressure restores the structure back in the presence of lateral load is given as ST sin(S9) = F f d Sp NnR' 2 ^ 2 F ^L +d Sp F NnR1 ^L2+d7 d (6.29) The pressure can be increased in any of the three columns in the presence of lateral loads; however it is advantageous to increase the differential pressure appropriately in inflated column on the lean side. Pressure in the inflated beam on the lean side must be higher than that the other two inflated columns in order to generate restoring bending 102 moments opposite to that generated by an applied load for further maintaining vertical stability and to eliminate hysteresis. 6.2.2.1. Generating Restoring Moments Bending moments caused by external forces can be counter balanced simply by increasing pressure energy in the inflated columns. The inflation pressure is increased in beam C by an amount Ap and lowered by half the amount in other two beams A and B in order to get net force equal to zero for generating bending moment. The upward force acting in the inflated column increases in C and decreases collectively in A and B by the same amount AF (AF = ApnR2). The moment (force x perpendicular distance between lines of action of coupled forces) generated in this case can be estimated as M active = A F X (DtS t a I 1 Ce = Ap{nR2)~. CE> ) (6.30) The differential change in pressure leads to change in the pressure energy in the inflated columns that helps to generate restoring moments in order to maintain attitude of the inflated structure. 6.2.2.2. Pressure Coefficients Multiple-beam structure comprising similar inflated beams have same values of length and radius for each inflated column when pressurized by equal pressure p. It is observed experimentally that increase in pressure in inflated beams indeed result in increase in length and radius of the beams. Results are summarized previously in 103 Table 4.2. These changes in geometrical parameters depend upon axial Young's modulus (Ex) and circumferential Young's modulus (Eg) of the beam fabric material and are small for a material of high elasticity, yet these changes cannot be neglected because of their significant contribution in generating restoring moments in the inflated structure. Inflated beams comprising the structure have same amount of increase in length and radius when pressurized equally and therefore no restoring bending moments are generated in these cases. Considering AL(L'-L,L' is length at pressure a t ; / ) be change in length and AR (R' - R,R' is radius at pressure p ) be the change in radius by increasing pressure by an amount Ap(p' - p). The increase in length and radius and their respective pressure coefficients can be determined as follows. Axial stress (see Equation 5.1) on the wall fabric ApR It . . , AL and axial strain = — L Axial Young's modulus is given as ApRf L Ex=^—\ — \. It I AL ' (6.31) Axial pressure coefficient corresponding to length is given as AL aL = (6.32) . LAp Equations (6.32) in terms of elasticity can be written as 104 a, R 2EJ (6.33) Using equations (6.32) and (6.33), the length of inflated beam comprising the structure at pressure p is given by L' = L\ u (P'-P)R 2Ext j Therefore increase in length of the inflated beam can be written as AL = Ap a L L. (6.34) Setting the value of aL from equation (6.33), the increase in length is given by AL ApRL 2ERt ' (6.35) Again, hoop stress (see equation 5.1) on the tower wall = ApR , ,. , AR and radial strain = . R Circumferential (radial) Young's modulus is given by ApRf R t {AR (6.36) Pressure coefficient of radius is given by ac AR RAp (6.37) 105 Equations (6.37) in terms of elasticity can be written as hRt Using equations (6.37) and (6.38), radius of inflated beam comprising the structure at pressure p is given by R' = R u(p'-p)R -'si J Therefore increase in radius of the inflated beam can be written as AR = ApaRR. Setting the value of aR (6.39) from equation (6.38), increase in radius is given by 2 AR=^^. (6.40) ERt The variation in length and radius affects bending moments and hence flexural rigidity of the inflated multiple-beam structure. 6.3. Experimental Set Up 6.3.1. Experimental Set up for Gyroscopic Control Mechanism The experimental set up is shown Fig 6.6. The set up consists of an inflated beam made of fiber reinforced polyethylene, a motor with flywheel assembly, an air compressor with regulated air supply, and two pressure gauges and a heavy base. Initially, the load-deflection experiment has been investigated by authors without 106 flywheel assembly to investigate behaviour of inflated beams at various pressure levels under applied lateral loads (Zhu et al., 2008). >1.5L <- -• Flywheel assembly String Pulley 2R-" Inflated Tube Scale Pressure Gauge Q Load 75Air Compressor '//////////////// Figure 6.6 - Experimental set up using flywheel assembly. After performing experiment with inflated beam, the motor and flywheel assembly is mounted at the upper end of the inflated beam as shown in Fig. 6.7. The lower end of the inflated beam is rigidly bolted to a heavy base with the regulated air supply. The two air pressure gauges are mounted on the upper plug to observe the inflation pressure. The experiment is conducted at an inflation pressure of 138 kPa (20 psi). Figure 6.7 - Spinning wheel mounted at the top on the inflated beam of length 1 m. 107 At first, the lateral load-deflection data is taken without running the flywheel. Table 6.1 describes the numerical values of the parameters involved in experimentation to study control moments. Table 6.1 -Numerical values of physical quantities involved in gyroscopic control of the inflated beam. Name of physical quantity Numerical value Length of the inflated beam lm Diameter of the inflated beam 0.082 m Wall thickness of the material of beam 0.00122 m Pressure in the beam 138kPa(20psi) Mass of the flywheel 6.5 kg Total mass of the motor and flywheel assembly Hkg Mass of the aluminum plug 0.685 kg Mass of two pressure gauges 0.530 kg Mass of 1 m hose pipe 0.27 kg Radius of the flywheel 0.07 m Frequency of the rotating flywheel 1575 rpm Angular momentum of the flywheel 5.3 kg m" sec" At the upper free end of the beam, the lateral load is applied using a long string (greater than 1.5 times the beam length) and pulley arrangement. The pulley is positioned at the same height relative to the free end to ensure that the applied load will be 108 perpendicular to the beam as shown in Fig. 6.6. The transverse deflection of the upper free end is measured at each applied load immediately after the load is applied in order to minimize the creeping effect. The lateral applied load is then removed and the beam does not return to its original vertical undeformed position. The beam along with mounted assembly is manually restored to its undeformed position and then next load experiment is performed. The same procedure is repeated with flywheel running at an angular speed of 1575 rpm. The accuracy of length and deflection measurement is 0.5 mm, while the accuracy for the pressure and load measure is 7 kPa (1 psi) and 0.01 N, respectively. 6.3.2. Experimental Set up for Pressure Control Mechanism To counter wind loads, a mechanism employing differential change of gas pressure in the inflated columns is designed and developed experimentally. A computer controlled electronic system involving electromechanical valves and pressure sensors are utilized to perform structure control experiments. It is observed that structure comprising inflated columns restores its vertical orientation in the presence of applied lateral loads when level of gas pressure is changed appropriately in the individual inflated columns using control mechanism. Experimental investigations show that the inflated beams behave as Euler beams at an appropriate level of internal gas pressure under load conditions (Zhu et al, 2008). The experimental set up is shown in Figure 6.8 (a). The set up involves an inflated multiple-beam structure of length 2 m comprising three inflated columns braced together using brackets. At upper free end of the inflated structure, a string is attached passing over the pulley. The other end is attached to weight hanger for loading the structure. 109 Figure 6.8(a) - Experimental set up of pressure control mechanism of an inflated 2 m structure. Figure 6.8(c) - Three pressure sensors mounted at top of the structure along with string and pulley arrangement. Figure 6.8(b) - Three inlet channels (Right) and three outlet channels (Left) through the solenoids mounted at the base. Figure 6.8(d) - Computer screen display along with supplies 110 electronics box and power Each inflated column has one inlet channel and one outlet channel controlled by using solenoid control valves as shown in Figure 6.8(b). Air supply to input channels is provided by regulated air compressor. The operation of solenoid control valves is performed by custom designed electronics utilizing a LABVIEW control program. Pressure values in each inflated column are pre-assigned as an input reference pressure. The computer controls pressure in each column and limits it to pre-defined reference pressure values. The pressure sensors mounted at the top of each individual inflated column along with pulley and string arrangement is shown in Figure 6.8(c). The electronics box connected with power supplies and computer screen displaying the values of pressures in each column during operation is shown in Figure 6.8(d). 6.3.2.1. Electronic Circuit for Performing Control Operation cf L-A/W Figure 6.9 - Electronics circuit designed to control solenoids inlet and outlet valves. Ill An electronic circuit is designed for active control mechanism of multiple-beam inflated structure and is shown in Figure 6.9. Each of the separate support inflated columns has its own pressure control circuit. A pressure sensor PX-177, is mounted at each inflated column. Pressure sensors convert the pressure reading from 0-100 psi to 420 mA of current. Pressure sensors are calibrated using electromechanical pressure gauges with an accuracy estimated as 7 kPa (1PSI). The pressure sensors are connected to a 12 V power supply that provides the necessary power for the sensors to function. The output of the pressure sensors is connected to a 200 ohm resistance. The USB controller is then connected in parallel to 200 ohm resistance in order to convert current readings to voltage levels. The USB controller uses the LABVIEW control program to determine the proper output, based on the input from the pressure sensor and the predefined input from the user as explained above. The action taken will be either to signal the input or output valve depending on what is needed to reach the user defined pressure. The output of the USB controller is fed into an inverting 741 op-amp which amplifies the output signal to approximately -5V. The op-amp is biased using +15V / -15V from a dual power supply. The required gain is achieved by using a lkohm resistance and a 1.6kohm resistance, which creates a gain of 1.6. The amplified output is then directed into a unifying gain inverting op-amp which serves to produce a positive voltage of the same magnitude as the input voltage. The final +5V signal is then fed into a 4.7 kohm resistor and then into a transistor. The transistor is connected to a diode and magnetic control valve. The diode serves as a switch to ensure that the magnetic valve (solenoid valve) is only opened when desired. 112 6.4. Experimental Results 6.4.1. Results and Analysis of Gyroscopic Control Mechanism The bending of inflated beam is investigated by considering three different cases. The load-deflection observations are shown in Table 6.2. First, the load-deflection behaviour of the inflated beam is studied. The lateral load applied (FL) causes no wrinkle in the beam as applied load is less than the critical wrinkling load. The inflated beam behaves as an Euler beam up to a critical lateral displacement, which is found to be 0.05 m at a pressure 138 kPa (20 psi) (Zhu et al, 2008). The observed deflections are less than critical lateral displacement and the inflated beam maintains its shape. Table 6.2 - Load - deflection observations using flywheel assembly. Inflated Beam With flywheel mounted at the top With rotating flywheel mounted (not rotating) at the top FL d(m) FT(N) FM~ FL+ FT d, (m) FR(N) FN -FM 0.49 0.002 0.22 0.71 0.003 0.011 0.699 0.003 0.98 0.004 0.44 1.42 0.006 0.022 1.398 0.006 1.47 0.006 0.66 2.13 0.009 0.033 2.097 0.009 1.96 0.008 0.86 2.82 0.011 0.044 2.776 0.011 2.45 0.009 0.97 3.42 0.014 0.049 3.371 0.014 3.92 0.016 1.72 5.64 0.023 0.088 5.552 0.023 - FR dR(m) Second case with flywheel (not rotating) mounted at the top of the inflated beam, the applied lateral load (FL) causes deflection more than the previous case, when flywheel is not mounted. In this case, contribution of transverse component of the weight FT 113 (mg sin 9) of flywheel increases the effective lateral load, which further increases lateral deflection. With flywheel mounted at the top and rotating with a speed of 1575 rpm, the same lateral loads are applied as in the previous cases, which results in the same amount of lateral displacements (d), when flywheel is not rotating. This is because the amount of angular momentum is not sufficiently large to resist bending. A very small amount of torque (force) is required to make a change in the angular momentum of flywheel by bending the vertical axis through a very small angular displacement (or lateral displacement). As an example, an applied lateral load (FL) of 0.49 N causes a lateral displacement (d) of 0.002 m in the inflated beam as shown in Table 2. With the flywheel assembly mounted at the top, the transverse component of weight (FT) of the flywheel at a displacement (d) of 0.002 m is calculated to be 0.22 N, adds into the applied lateral load (0.49 N) increasing the lateral load (total lateral load FM) to 0.71 N. This total lateral load (0.71 N) causes a lateral deflection (d,) of 0.003 m. The force (FR) required for making a change in the angular momentum by bending the tip of the vertical axis through a displacement (d) of 0.002 m is estimated to be 0.011 N (Eq. (6.10)) and can be taken as a resistive force. Therefore by subtracting this small force 0.011 N from a total lateral load (FM) of 0.71 N, the remaining net load (FN) 0.699 N is used to bend the inflated beam along with rotating flywheel (gyro) assembly through a lateral deflection (da) of 0.003 m, and found to be same when flywheel was not rotating. It has been found that resistive force required (FR) utilizing angular momentum is found to be very small to resist 114 bending and also too small to be compared with transverse component of weight (FT) of the flywheel and hence with the total effective lateral load (FM). Experimental investigations show that rotating flywheel resists external applied torque (force moment) on a small scale. Inflated beams bend due to a combined effect of applied external force and gravitational component of weight of the system against the flexural rigidity maintained by air pressure inside the beam with rotating flywheel at the top. It is also determined that critical buckling load of the inflated beam of length 1 m (Effective length = 1x2.4=2.4) at a pressure of 138 kPa (20 psi) using effective elastic modulus (2.6xl0 8 Pa) is 12 kg (see Chapter 4, Section 4.3.4). The designed axial load which includes mass of flywheel assembly, two pressure gauges and aluminum plug is found to be 12.2 kg and is nearly equal to critical buckling load (12 kg), which otherwise should be sufficiently less for maintaining stability of the inflated beam. Therefore, the inflated beam with the existing designed load critically satisfies the Euler buckling load condition. Considering a multiple-beam structure configuration as discussed in Chapter 7, the critical buckling load (with fixed-free end condition) of a 15 km high space tower o made of Kevlar constructed at an altitude of 5 km is estimated as 2.7x10 kg. For gyroscopes to be effective, the angular momentum of the gyro should be sufficiently large to resist bending of the inflated beam. For increasing angular momentum, the mass moment of inertia of the flywheel and its frequency of rotation need to be increased. However, the vibrations corresponding to small lateral displacements due to impulsive impacts on the inflated beam can be controlled easily using gyro assembly 115 for inertial stabilization of the composite system (Higashiyama et al, 1998; Moon et al., 2005). By setting the values of parameters involved in Eq. (6.21), the frequency of the gyro required for free standing of the inflated beam using current experimental set up is found to be approximately 43000 rpm generating an angular momentum of 144.2 kg m s"1, which corresponds to a tip speed of 0.3 km/s of the flywheel which is beyond the scope of the existing experimental set up. A flywheel having sufficiently large radius about 0.367 m would generate the same value of angular momentum (144.2 kg m2 s"1) and would be able to stabilise the structure with the current setups frequency of rotation. While the current experimental setup is incapable of stabilizing the beam, the multi-gyro technique is analyzed to provide a theoretical solution to the problem. The results obtained from theoretical analysis (Eq. 6.21) utilizing single gyro and three gyros are shown in Table 6.3 and Table 6.4. Table 6.3 - Theoretical result for a single gyro system, radius = 100 m, Total mass of the system = 200,000 kg. Height (m) Frequency required (RPM) Tip speed of the gyros (km/s) 50 7.5 0.07 100 21 0.22 150 39 0.40 200 84 0.88 As an example, the radius and mass of gyro are taken as 100 m and 100,000 kg respectively to assess its feasibility to be utilized as a control mechanism for a space 116 tower structure. For a physical and practical approach, 50% of the total mass of system is assumed to contribute in the mass of structure and remaining mass is contributed by gyros giving a total mass of 200,000 kg. Table 6.4 - Number of gyros = 3, radius = 100 m, mass of lower most gyro = 100,000 kg, Total mass of the system = 272,220 kg. Height of lower Maximum Height of Frequency of Tip speed of the most gyro (m) achievable centre of mass gyros height (m) (m) (RPM) 50 300 120 48 0.50 75 450 180 89 0.93 100 600 240 136 1.42 required gyros (km/s) At first, only one gyro is considered for analysis and tip speed of the gyro found for specific heights is shown in Table 6.3. The calculated speed maintains vertical stabilisation of the structure utilizing its angular momentum. In the second case, three gyros are considered for analysis. Again the radius and mass of the lower most gyro have been taken as 100 m and 100,000 kg respectively. The mass of successive gyros decreases in the specified series with increase in height. The total mass of the system consisting of mass of three gyros and mass of structure is found to be 272,220 kg. The height of lowermost gyro determines total achievable height of the tower. The tip speed in different cases is computed and the results are shown in Table 6.4. Tip speeds above 1 km/s are not recommended and are infeasible because of the materials constraints (McNab, 2003). The results show that using multiple-gyro approach, the inertial 117 stabilization of the system can be maintained up to a height of approximately 450 m, an effect that may have utility for the construction of a space tower structure. For a particular height of the system, the stabilization conditions may be maintained by lowering the centre of gravity. In this approach, the mass of the lower most gyro is taken to be maximum and masses of the subsequent gyros decreases in a definite order. With increase in height of the tower, the gyro frequency needs to be increased for maintaining vertical stability due to angular momentum of the rotating gyros. The decrease in mass of subsequent gyros advantageously decreases rotational inertia about the horizontal axis. Simultaneously, the rotational inertia about the vertical axis also decreases, which is undesirable. Gyros radius is a significant factor because Eq. (6.21) involves its square. Mass is also an important factor in determining the mass moment of inertia of the system. It has also been seen that frequency of rotating gyros is found to be same when value of lowermost mass is replaced by another mass value. It is due to the fact that final result (Eq. (6.21)) is mass independent. Impulsive impacts on the system may have limited bending effect resulting in small lateral deflections. The inertial stabilization of the system consisting of a rotating flywheel mounted on the inflated beam mitigates such vibrations and may be effective under impulsive loads (Higashiyama et ah, 1998; Moon et al., 2005). To enhance inertial stabilization, the designed structure made of high elastic fabric must be inflated to a sufficient level and the angular momentum of gyro control machinery must be sufficiently large. The analysis is conducted by taking ground level as a reference point, 118 however in the appropriate stable configuration of a space tower; the installation of gyros at appropriate locations in the structure can effectively be used to mitigate dynamic wind induced loads due to its characteristic of high angular momentum stabilization. 6.4.2. Demonstration and Analysis of Pressure Control Mechanism The control program operates in three stages. Each stage is pre-assigned by entering the values of pressure for each inflatable column comprising the multiple-beam structure. The applied lateral load bends the inflated structure when inflated beams are pressurized at an appropriate level in a particular stage and also when pressure in the beams gradually decreases in each stage. In the next stage, when level of pressure is increased, the inflated structure restores its vertical orientation. Pressure level can be increased in all the beams simultaneously or in the single beam along lean side of the structure. The values of pressure in each stage are assigned in such a way that the inflated structure restores its vertical stability in the presence of applied lateral load. While restoring vertical stability, the level of pressure in the beam (Beam 2, see Figure 6.8(b) and Figure 6.8(c)) along lean side of structure is assigned more value in order to enhance restoring effects by generating bending moments along with increase in pressure energy level of the structure. A lateral load of 0.40 kg is applied when each column is inflated at a pressure of 13.8 kPa (2 psi) resulting in a lateral deflection of 0.025 m. In order to restore the structure in its vertical orientation, the pressure in column 2 is increased to a level of 79 kPa (11.5 psi), while keeping same level of pressure (2 psi) in other two inflated columns. Increasing pressure by 9.5 psi (65.5 kPa) restores the structure vertical 119 orientation. This inflation differential pressure is found to be in close agreement with theoretical analysis (Equation 6.29), which calculates a pressure requirement (Sp) of 67 kPa (9.7 psi) for vertical restoration of the inflated structure. Further increase in pressure would generate bending moments in the structure. Further analysis can be conducted using different values of lateral loads and selecting appropriate values of pressure in each inflated columns. It has been found that structural restoring performance increases when pressure is increased in a beam element on lean side of the structure as might be expected. This helps mitigate hysteresis effects caused by lateral loads that generate bending moments in the structure. Active control pressure requirements to maintain vertical restoration of a 15 km space tower in the presence of wind loads is estimated in Chapter 7. The successful performance of active control mechanism of inflated multiple-beam structure is demonstrated and video recorded. In addition to pressure control mechanism as discussed above, the length and radius of the inflated beam changes as discussed in Section 6.2.2.2. The variation in length and radius affects bending moments and hence flexural rigidity of the inflated multiple-beam structure. The effect depends upon pressure coefficient of length and radius and is discussed as follows. The worst case wind load acting on a 15 km tower constructed at an altitude of 5 km is estimated at 9.3xl0 7 N. For control purpose, a 10% of the worst case wind load must be assumed to apply on the tower in 10 seconds. The lateral load deflection being directly proportional to the wind loads is estimated to be 30 m. The energy required per second to control the tower would be approximately 2.8xl0 7 Joule, equivalent to a power of 37,500 120 HP. An inefficient factor of 50% may be considered for control purpose. High pressure tanks may be utilized instead of compressors to provide rapid transient response. The experimentally observed change in length for a 2 m structure at a pressure of 138 kPa (20psi) is 0.018 m and is nearly equal to theoretically predicted value of 0.016 m (Chapter 4, Section 4.2.3) for a structure made of polyethylene reinforced fabric (Axial Young's Modulus = 277 MPa). The change in radius of inflated tube (Circumferential Young's Modulus = 242 MPa) as observed experimentally is 0.0008 m and is consistent with the theoretical predicted value of 0.0007 m. For a 15 km structure comprising cells of thickness 0.012 m and radius 0.5 m made of boron (Elasticity 450 GPa, p min = 28600 kPa (286 bar), p max = 83300 kPa (833 bar)), the estimated limiting value of increase in length using equation (6.35) would be 38 m and utilizing Kevlar (Elasticity 131 GPa, p min = 21500 kPa (215 bar), p max = 85700 kPa (857 bar)) for the same configuration, the estimated value of increase in length would be 153 m. The estimated increase in radius (using equation (6.40)) of inflated cell made of Boron is 0.0025 m (increase of 0.5%) and that of Kevlar is 0.01 m (increase of 2%). The flexural rigidity of the inflated structure fabricated from Boron and Kevlar increases by 1.6% and 0.65% respectively. These predicted values are maximum corresponding to maximum limiting pressure inside the inflated beams and can also be calculated at different pressures utilizing same or different configurations of the structure. The potential application of this effect can be utilized in changing the attitude of a high altitude observatory installed at the top of a space tower. 121 6.5 Thermal effects of sun light and UV radiation Kevlar maintains its strength between a temperature range of -196 C to +160 C. However ultraviolet spectrum (UV) in sunlight can degrade and decomposes Kevlar causing it to lose its strength. Therefore outer surface of the structure requires UV protection. Acrylic coatings or other UV resistant materials can be utilized to protect outer skin from UV radiations. However thermal coefficient of expansion of Kevlar is 54x10"6 K"1 and space tower accessing 20 km altitude constructed on a mountain of height 5 km would be safe from thermal effects due to sun light in a temperature range of about -55°C to + 25°C. 6.6. Conclusion Theoretical and experimental analysis of active control mechanisms found to have significant useful in controlling the attitude of inflatable structures. It is clear from the investigations that pressure control mechanism can effectively control the attitude of the structure, where as wind induced vibrations can be controlled by the spinning gyroscopes. An appropriate active control mechanism similar to that designed and developed by the author can potentially be used to control pneumatically supported space towers. The maximum inflation speed is limited by the speed of sound. To control the structure, the maximum reaction force required from the active control system is 1.3x10 N (Quine et ai, 2009). Considering pumping of hydrogen gas in order to control the tower, the estimated volume pumping rate would be 4.2x105 m /s. For helium gas and air, the volume pumping rate would be approximately 2.1xl0 5 m/s and 2.4x10 m / s respectively. The pumping rate can be adjusted utilizing lines of appropriate sizes. 122 CHAPTER 7 FEASIBILITY OF FREE-STANDING INFLATABLE SPACE TOWER 7.1 Introduction The chapter describes the theory and analysis for the construction of a thin walled inflatable space tower of 15 km vertical extent constructed at an altitude of 5 km in an equatorial location on Earth using gas pressure to access altitudes more than 20 km as proposed by Quine, Seth and Zhu (2009). The suborbital tower of 20 km height would provide an ideal surface mounting point where the geostationary orbital space tether could be attached without experiencing the atmospheric turbulence and weathering in the lower atmosphere. Kevlar is chosen as an example material in most of the computations due to its compatibility in space environment. The Euler beam theory is employed to the inflatable cylindrical beam structure. The critical wrinkling moment of the inflated beam and the lateral wind load moments are taken into account as the key factors for design guidelines. A comparison between single inflatable cylindrical beam and inflatable multiple-beam structures is presented in order to consider the problems involving control, repair and stability of the inflated space tower. An appropriate number of thin wall gas cells are utilized for construction guidelines of a 15 km high space tower that would remain stable in wind loads. For enhancing load bearing capacity of the tower and for availability of more surface area at the top, the non-tapered inflatable structure design is chosen for the basic analysis, however further analysis can be performed with tapered structures. 123 Following a brief introduction in Section 7.1, Section 7.2 presents a selection of currently available suitable space material for the construction of a space tower. A designed practical structural concept is described in Section 7.3. The theory of inflatable structures is presented in Section 7.4. Maximum limiting gas pressure and load capacity of space towers of different heights utilizing different inflation gas pressure are presented in this section. Critical bending moments and wind load bending moments causing lateral deflection in space towers of various heights are discussed in Section 7.5. Inflatable multiple-beams being essential part for the construction of space towers is presented in Section 7.6. The cost of proposed space elevator structure of 15 km in height constructed at 5 km altitude is estimated in Section 7.7. Subsequently, the chapter is concluded in Section 7.8. 7.2 Material Selection Current industry widely produces artificial a = (3.0 - 6.0) GPa and density p = 1200 - 2000 kglm\ fibers having tensile strength Kevlar 49 {a = 3.6 GPa, p=1440 kg/m ) is selected as an example material for most of the computations and analysis due to its suitable mechanical properties to the inflatable beam structure and its compatibility in space environment. The other fabrics can also be chosen to further elaborate the analysis. Among the man made fibers, the organic fiber Kevlar is a highly flexible composite with high strength, low cost and impact resistance. Past space structures have already utilized Kevlar as a design material as discussed in Chapter 3. The inflated structure is subject to possible punctures caused by high velocity space debris. However, 124 the high tensile strength of Kevlar resists punctures. The interwoven design of Kevlar resists ripping, tearing, and cracking (Tanner et al., 1989). Today, there are three standard grades of Kevlar available: Kevlar 29, Kevlar 49, and Kevlar 149. Among them, the Kevlar 49 is dominant in structural composites because of its higher modulus. The Kevlar 29 is used in composites when higher toughness, damage tolerance, or ballistic stopping performance is desired. An ultrahigh-modulus fiber, Kevlar 149 is also available. The Table 7.1 shows the mechanical properties of Kevlar for the different grades (Standard test method for tensile properties, ASTM; Magat et al., 1980). Table 7.1 - Mechanical properties of Kevlar 1440 Tensile Modulus GPa 83 Tensile Strength GPa 3.6 Tensile Elongation % 4.0 49 (High Modulus) 1440 131 3.6-4.1 149 (Ultrahigh Modulus) 1470 179 3.4 2.8 2.0 Grade Density kg.m"3 29 (High Toughness) Recently, Carbon nanotube (CNT) technology has developed rapidly. CNTs are a new form of carbon, formed in laboratories and are not commercially available as yet on bulk scale. However, the strength of CNT combined with its low density makes it important when considering the design of a space elevator. If CNTs prove to be mass manufacturable, this may be an ideal material for the construction of a future space elevator. Theoretical density of a pure carbon nanotube is 1300 kg/m3 and its strength (tensile yield stress) can be as high as 300 GPa, although the NIAC Phase 1 report 125 (Edwards, 2003) uses the more conservative value of 130 GPa for calculation. Even this more conservative figure is an order of magnitude higher than the strength of conventional engineering materials and could be ideal for building the space elevator tower. Although CNTs are the strongest material discovered so far, the material is not taken into account for the current analysis due to its non-availability on mass scale. In contrast, Kevlar 49 is a commercially available material, also found successful in past space missions and is chosen for the current analysis. The maximum characteristic length — of a space tower (Arvind, 2007), depends Pg upon tensile strength and density of the material chosen and varies for different materials. Using Kevlar 49, the characteristic length is found to be 255 km and, for (CNT) carbon fiber (cr = 130 GPa, p = 1300 kg I m 3 ), the material available in laboratory, the value can be as high as 10200 km. 7.3 Practical Structural Concept Quine, Seth and Zhu (2009) proposed a device comprising a pneumatically pressurized core structure consisting of compartments arranged in segments with equipment decks or pods. The compartments are constructed using conventional high stress materials and pressurized with a gas mixture of low atomic mass such as hydrogen or helium. An inertial stabilization maintains the attitude of the structure with respect to the planet surface using a variety of methods including pressure balancing and angular momentum stabilization (Quine et al., 2009). A 7.0 m scale model of a structure similar to Elevator A as shown in Figure 7.1 is developed. This 1:2000 test model comprises 126 three 0.082 m diameter cores braced at intervals of 1.0 m as shown in Figure 5.9 in Chapter 5. The overall diameter of the structure is 0.34 m. Constructed of laminated polyethylene (Young's Modulus of the material measured as 280 MPa) with an average wall thickness of 0.00122 m, the structure has a total mass of 17 kg excluding the base support and is freestanding when pressurized with air above 48,000 Pa (7 PSI). Pod Main Pod Main Pod Tapered Segments *—^Segments Pod B Figure 7.1 - Core structure configurations A, B and C (Quine et al, 2009). 7.4 Theory of Inflatable Structure in the Atmosphere The value of the air or gas pressure in a column varies with the altitude as follows (Bolonkin, 2003): 127 P = Poe I RJ ) , (7-1) where/? is the pressure at a height z and p0 is the pressure at the planet's surface. Ra is the gas constant, ju is the molecular mass, T is the temperature of the air or gas in Kelvin, and g (9.8m.sec~2) is the acceleration due to gravity, respectively. The typical value of the gas constant i s i ^ = 8.314 Joule.K~lmol~l. The typical value of molecular mass for air is ju = 28.96 * 10~3 kg.mol~l( 28.96 gm.mol~{), whilst for helium, ju = 4*10"3 kg.moF1 (4 gm.mol'1) and for hydrogen ju = 2 *\0~ikg.mor](2 gm.mol'1). To keep the pressure constant in the tower, the whole tower must be divided into multiple sections of certain height. The pressure in all the sections can be kept the same with the help of pressure sensors in each section and by using gas compressors and pressure regulators for the respective sections at ground. If height of each section is taken as 25-100 m, then using Eq. (7.1), the percentage change in the pressure due to altitude is 0.3-1.4% for air, 0.040.2% for helium and 0.02-0.1% for hydrogen, which is small and therefore can be neglected and taken as constant throughout the height of the cylindrical tower for a particular R/t ratio (radius of beam/thickness of wall). For the given range of tensile strength, the variation of maximum pressure with R/t ratio computed using pnmx = R It (see Eq. (5.3) in Chapter 5) and is shown in Fig. 7.2. The air or gas density varies with the pressure and temperature. At standard temperature and pressure (STP), density for air ispa =l.29kg/mJ, 128 for helium pa = 0.1787 kg/m3 and for hydrogen pa = 0.0898 kglmi. The average temperature ranging from ground to 20 km atmosphere is 242 K (U.S. Standard Atmosphere 1976). The density p of the internal gas corresponds to internal pressure p at STP. The density of air at different pressures and temperatures can be found as P* MP RgT (7.2) Let p'g be the density of the gas at a pressure/?'. Therefore the values of densities of hydrogen, helium and air at different pressures can be computed using Eq. (7.2) as (7.3) P x 10 Q. P CD Q. O 500 Figure 7.2 materials. 1000 1500 2000 2500 3000 R/t Ratio 3500 4000 4500 5000 Maximum limiting gas pressure at various R/t ratios using different 129 The Eq. (7.3) simply shows that the density of the gas is directly proportional to its pressure. Here p is the pressure at STP and p is the pressure corresponding to given tensile strength of the material and R/t ratio, see Eq. (5.3). The Eq. (7.3) can be written as 4=^ a (7.4) P And the results of computations by using Eq. (7.4) for hydrogen, helium and air using Kevlar are shown in Figure 7.3. For R/t=1150, Hydrogen density = 2.8 kg/rrr 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 R/t Ratio Figure 7.3 - Variation of gas density with R/t ratio using Kevlar 49 (p = 1440 kg I m 3 ). The inflated cylindrical tower is divided into many sections and each section has the same gas pressure. To determine the value of maximum optimal attainable height, the load of each section and hence of the entire tower due to the presence of internal gas must 130 be taken into account in addition to the weight of the tower. Thus, the maximum attainable height H for particular R/t ratio is given as nRp' = InRtHpg + nR'Hp'gg . where p'g is the density of the gas/or air to be used at internal gas pressure p'. Setting the value of p = pmax from Eq. (5.3) into the above equation leads to the maximum attainable height of the tower and is given by • " max height" / r, \ \' •~>) • 2 PS + \— P'gg and Eq. (7.2) corresponding to density/?^of the gas/or air to be used at internal gas pressure/)' becomes Pl= or MCT RgT(R/t) —p^=-^—. t * RgT (7.6) The quantity —p'a measured in kg/m3, in Eq. (7.6) is constant for a particular material and for a particular gas to be used. By using Eq. (7.6) in Eq. (7.5), the maximum attainable height is given by ma\_lieighl f \ \'-') [RJj 131 For hydrogen, helium and air inflatable space towers made of Kevlar 49 ( a = 3.6 GPa, p = 1440 kg Im1), the maximum attainable height for all R/t ratios is found to be 57 km, 37 km and 7 km respectively. The result is independent of R/t ratio. The limiting payload capacity per unit cross-sectional area of the inflated tower for a particular R/t ratio, density p'a of air/or gas and the height H is given by W payload _ capacity =D r 2Hpg R/t (7.8) + Hp'g with the condition that p < pmm . To calculate maximum payload capacity of the tower of height H, for a given R/t ratio, put the value of pressure p given in (p - pmm) corresponding to its maximum limit as Eq. (5.3). The maximum pay load capacity per unit cross-sectional area then expressed as W max_ payload _ capacity a R/t 2Hpg] R/t J + Hp\g (7.9) The maximum load capacity for hydrogen, helium and air tower is computed for the given height at different R/t ratios and the result of the computation is shown in Figure 7.4. 132 x 10 500 1000 1500 2000 2500 3000 R/t Ratio 3500 4000 4500 5000 Figure 7.4 - Maximum load capacity per unit cross-section area for particular tower height and gas at different R/t ratios using Kevlar 49 ( p = 1440kg I m1). The payload capacity of the tower based on the internal pressure is numerically equal to the tension in the walls. To provide stability for the tower, the tension in the wall should be sufficiently large after placing the payload. The load on the tower contributes in compressive stress in the walls; however, the wall of the tower should always be under tension instead of compression. Therefore, the weight Wload of the payload should always be less than the tension in the wall. The tension in the wall after the contribution of the payload is given by T = nR2p' - InRHtpg - nR2Hp[g ~ Wload h ' (7.10) For T' > 0 , the value of the internal gas pressure with payload should be as follows: 133 The pressure p should never exceed />maxfor a particular R/t ratio (see Eq. (5.3), Chapter 5). The maximum payload capacity of a 15 km high space tower having radius 115 m and thickness of wall 0.1 m utilizing hydrogen as a pressurization gas is estimated at 1.0 x 10 n N. However, Euler buckling load for a 15 km single wall space tower made of Kevlar is calculated to be 2.7 x 10 9 N (270,000 toone) using fixed - guided (Le=L) conditions and 6.9 x 10 N (69000 tonne) for fixed-free (Le=2L) end conditions (Chapter 4, Section 4.3.4). 7.5 Critical Bending Moment The value of the critical wrinkling moment is given by M„ = mnp'R1 = 0Anp'R\ Setting p'= pmax = (7.12) from Eq. (5.3), the value of the critical bending moment for R/t an inflated beam and hence for the tower can be computed for different values of radii at different inflated pressures and hence for different R/t ratios as M „ = ^ ^ . R/t (7.13) The result of the computation for the given R/t ratio at different radii is shown in Figure 7.5. 134 x 10 R/t=1000 R/t=1150 R/t=1500 Q — R/t=2000 * — R/t=5000 100 110 120 130 140 150 Radius 160 170 180 190 fc 200 Figure 7.5 - Variation of critical wrinkling bending moment with radius (m) at various pressures corresponding to different R/t ratios using Kevlar 49. 7.5.7 Bending Moments due to Wind Loads The dominant critical live loads acting on the structure are the wind loads. These loads vary in intensity depending on the building's geographic location, structural height and shape. Wind is a phenomenon of great complexity and can apply loads to structures from unexpected directions because of the many flow situations arising from the interaction of wind with structure and topology. Some structures, particularly those that are tall or slender, for example bridges, respond dynamically to the effects of wind. The best known structural collapse due to wind was the Tacoma Narrows Bridge which occurred in 1940 at a wind speed of only about 19m/s. It collapsed due to the wind induced vibration resonating in coupled torsional and flexural modes (Mendis, 2007). To 135 minimize the wind load affects, the most suitable geographic location for the space elevator tower structure is the equator, which offers an excellent location on scientific grounds for a high-altitude astronomical observing stations. Located at or near the equator, hurricanes and cyclones occurring within 10 degrees of the equator are nearly absent (Fesena, 2006). The wind load exerts a drag force acting on a structure as given by (Talay, 1975) Fd=^CdPfv2.S, (7.14) where pf is the air density, v is the airflow velocity, S is the surface area projected to the airflow (also known as the body frontal area), and Cd is the drag coefficient, respectively. The value of Cd is a dimensionless number that depends upon the Reynolds number, air turbulence, air viscosity, surface roughness and shape of the structure. For cylindrical shape typically Cd =1.2 (Talay, 1975). The density p1 of air is not constant and varies with the altitude (U.S. Standard Atmosphere 1976). Wind velocity v is also not constant, but varies with altitude in a very complex manner which is difficult to predict. The geographical location of the structure under consideration is at equator and therefore the wind speed profile with altitude determined by Equatorial Atmospheric Radar (EAR) can be used in computation (Fukao et al. 2003; Equatorial Atmosphere Radar Observation Data, version 02.0212). It is found that Zonal winds are much stronger than the meridional and vertical winds. 136 The body frontal area for the cylindrical surface is S = h * D , where h is the length of the cylindrical surface on which air strikes and D is its diameter as shown in Figure 7.6. D I • * - * • > Wind - - •" F - -X i 1 il L >. y Figure 7.6 - Cylindrical section of the inflated beam exposed to wind. The bending moment at the base of the structure resulting from the wind load is given by M. (7.15) F *-y> where;; is the vertical distance from the ground to the forceF d . Substituting Eq. (7.14) into above equation leads to the wind load moment M »M =^C<IP/V (7.16) S y -- 137 Since the values of density of air and wind velocity are functions of altitude y. Therefore, the total moment due to the wind load acting along the entire length of the tower can be calculated by integrating the Eq. (7.16) from ground jy = 0 to height y = H such as Mwind_lolal = X -Cd.s\"pf{y).v\y).dy. (7.17) The bending moment due to wind load must be less than the critical wrinkling moment of the inflated beam (Mw = m.7vp'Rl, see Eq. (7.12)) for the survival of structure, otherwise the structure would certainly start to fail. Although wind direction is not always perpendicular to the tower, but it is taken to be normal to the surface to evaluate maximum wind load bending moment as shown in the Figure 7.7. To evaluate maximum possible estimated wind load bending moment conservatively, the wind flow is assumed to be in one direction and the tower is divided into four sections (regions) each of length 5000 m. The average values of wind velocities and atmospheric densities are taken for each section. For estimation of maximum possible wind load bending moment, all the terms are taken positive and the total wind load moment is given by the sum of the components. ^H,w_,0M, = 2 X - J V (7-18) 138 15fo« J FA 20 km •Kegion4 Region 3 ^ is Re gz'o/72 i At 5 km altitude 5 km Wind load Fx = 0 Region! Sea Level Figure 7.7 - Inflatable 15 km high space tower at an altitude of 5 km accessing 20 km altitude under different wind loads in various regions. 1 where Fd. = —Cdpfyj 2 .S is the average force acting on the tower in the i th region at an altitude;;.. By substituting area S=2R.h and drag coefficient Crf = 1.2, the value of drag force in the i th region becomes Fdj = Cdpf yj .R.h, where subscript / is used for the section or region chosen above the earth's surface for the respective average values of 139 force, density and velocity in each region as shown in the Figure 7.7. The average values of density of atmosphere and wind velocity in all the selected four regions along the height of tower is shown in Table 7.2. The average value of wind velocity is maximum in region 3 (10-15 km) and contributes most in the calculation of wind load bending moment. 4 Mmnd total =l.2^Pfy,2.R.h.yi = \2Rh{pfv{2y\+pfi v22y2+PfVi2yi+PfA vA2y4). (719) Table 7.2 - Average wind velocity and atmospheric density Region Range (km) Average wind velocity (m/s) Average atmospheric density 1 0-5 5.5 0.9 2 5-10 6.5 0.54 3 10-15 25 0.28 4 15-20 12 0.11 The maximum possible bending moment due to the wind load for different radii is estimated by using recent (2007 data) zonal wind velocity profile measured by EAR (Equatorial Atmosphere Radar (EAR) Observation Data, version 02.0212). A 20 km high tower anchored at sea level would experience additional wind load due to region 1, which otherwise is absent for a 15 km high tower constructed at 5 km altitude and will not experience any wind load due to region 1. The result for different heights of towers by using Eq. (7.19) is also shown in Figure 7.8. A comparison of critical bending moment (Figure 7.5) to wind load bending moment (7.8) indicates that maximum wind load 140 bending moment is 1/61 of critical bending moment for a tower of height 15 km constructed at an altitude of 5 km. Therefore a 15 km inflatable space tower having radius more than 115m and thickness 0.1m anchored at an altitude of 5 km, would never reach its critical stage of bending. However the stability of the proposed structure can be enhanced utilizing other multiple beam configurations by increasing its flexural rigidity. x 10 2.4 1 2.2 * • height-10km heiylil-15km X 2 ~> V 1.8 (115, 1.1e12) s * 1.6 > ~J* 1.4 . 1.2 V*l... „>; ? ,•+"•' : .Jf' 1 0.8 : 0.6 .._ .^C..*i' - xj*'* 0.4 50 100 150 Radius 200 250 Figure 7.8 - Wind load variation with radius of the tower for different heights. The lateral tip deflection of the tower due to wind load can be estimated using following formula (Gore, 2006): «=^<*-r). (7.20) 141 where Fd is the transverse force acting at a distance y from the fixed end (the ground). L is the length of the tower (see Figure 7.8). Substituting / = nR^t into Eq. (7.20) leads to the lateral deflection as d=^4w^-y). (7.21) dnR'Et The lateral tip displacement of tower of length L is found by summing the deflections caused by individual transverse loads by using Eq. (7.21) as 4 4 FH y , Yd * total Substituting Fdj =Cdpfyi =Y-^-(3L-y,). .R.h into the above equation, the tip displacement of the tower is given by d >°'"<=l i3L y ] 6*R*Et --- (7.22) Angle of inclination (degrees) or the angle through which the tower deviates is also *iven by $ tan" fd.total L 180^ 7T , 142 (7.23) 1200 200 250 350 Radius (m) Figure 7.9 - Variation of lateral tip displacement of the tower of different heights and thickness 0.1 m, with radius of the tower. The lateral tip displacement and angle of inclination due to wind loads for different heights and radii by using Equations (7.22) and (7.23) is computed and the result of computations for wall thickness t = 0.1m is shown in Figures 7.9 and 7.10. For a 15 km high space tower (constructed at an altitude of 5 km) made of Kevlar having radius 115 m, the lateral displacement is 837 m (angle of inclination 3.2°). A further increase in radius to 150 m would result in a significant reduction in lateral displacement (526 m) corresponding to an angle of inclination (1.9 ) that can be controlled utilizing active control mechanism. The space towers of similar geometrical parameters fabricated from Boron (Elasticity = 450 GPa), would perform better by reducing a lateral deflection to 243 m (angle of inclination=0.93°) for a 115 radius and similarly for a radius of 150 m, 143 the lateral deflection would be 143 m (angle of inclination = 0.54 ). A 20 km high inflatable space tower having radius 150 m and thickness 0.1 m utilizing Kevlar anchored at sea level at an equatorial location on Earth would be deflected through a lateral displacement of 1700 m (angular inclination 4.8°) (Seth, Quine and Zhu, 2009). Therefore construction of space towers at an altitude of 5 km would be more advantageous in terms of its stability along with control mechanism as proposed by Quine, Seth and Zhu (2009). The stability and control can further be enhanced by utilizing multiple-beam structure that would significantly reduce lateral deflection under wind loads. 100 150 200 250 Radius (m) 300 350 Figure 7.10 - Variation of angle of inclination of the tower of different heights and thickness 0.1 m, with radius of the tower. 144 The analysis suggests that geometrical parameters along with mechanical properties of the fabric materials are significantly important to increase flexural rigidity (EI) of the space tower structures. An appropriate combination of geometrical parameters and material could be utilized in constructing a multiple beam structure to maintain vertical stability of the structure by increasing its flexural rigidity. 7.5.2 Bending Moments due to Dead Load Contribution of the Tower Live wind loads causes lateral movement of the tower and during the lateral movement, the bending moment due to the dead load of the tower comes into effect. The dead load of the tower is numerically equal to its weight, which further depends upon the height of the tower, the nature of cover material and internal gases used in the tower. The bending moment due to the dead load depends upon the lean angle and height of center of gravity with respect to the Earth surface. The tower is considered to be highly stable if the lateral movement of the tower under wind load is negligible. For this reason, the best results obtained in the analysis correspond to the minimum values of the lateral tip displacement under the wind load for given values of radius of the tower and thickness of the cover material for a particular height of the tower. The damping effect may also be achieved actively using a high pressure line-and-vent network system or passively by allowing support gas to vent from compartment to compartment along a connecting line network. For the primary bending moment, the force component exerted perpendicular to the core structure is mg sin 6. Consequently, for a building lean angle of 1.0°, the force is 1/57 of the weight force (Quine et al. 2009). As an example, a 15 km high space tower having radius 115 m and thickness 0.1 m made of Kevlar and pressurized with hydrogen 145 would have an estimated weight of 3.2 x 10 10 N (weight of material = 1.5 x 10 10 N, weight of hydrogen at high pressure =1.7 x 101 N). If the attitude error of the core structure can be maintained at less than 0.1°, the maximum reaction force required from the active control system is estimated as 5.6xl0 7 N (or 5600 tonnes of mass equivalent force). However the value of maximum reaction force changes by changing geometrical parameters corresponding to different configurations of space tower structures. 7.6 Inflated Multiple-Beam Structure The lateral movement of the tower is undesirable as it worsens the situation due to dead load contribution of the tower in the bending moment in addition to the wind load. Inflatable multiple-beam structures are highly recommended because the structure can be actively controlled by the differential change of internal gas pressure in the inflatable beams. Moreover, the multiple-beam structure can be designed easily for zero tip displacement using inflatable beams of certain radii and separated symmetrically from each other. The stability and control issue of the inflated tower can be solved by using multiple inflated beam structure similar to that shown in Figure 5.9 in Chapter 5. Based on engineering design guidelines, a number of permutations involving geometrical parameters of structure can be analysed for building a space tower structure with an equivalent radius more than 150 m. A 15 km high space tower having radius (r) 115 m utilizing multiple-beam structure comprising approximately 2085 appropriately positioned ((outer circle, radius 114.5 m) 700+695+690 (inner circle, radius 112.4 m)) inflated tubes each having radii (R) 0.5 m and thickness (t) 0.012 m made of Kevlar arranged in concentric circles as shown in Figure 7.11 can be constructed that would have 146 an estimated equivalent wall (te = (number of inflated tubes in a wall)Rt/r). thickness of 0.10 m The inner and outer walls of the structure have radii equal to 112 m and 115 m respectively. The lateral deflection of the structure is again estimated as 3.2° because equivalent geometrical parameters of composite structure have the same values as that of a 15 km high space tower with radius 115 m and thickness 0.1 m. Consider a composite arrangement of concentric cylindrical walls comprising inflated beams deployed horizontally at radii 30 m, 60 m, 90 m, 120 m and 150 m from centre of the tower structure. The number of inflated beams of radii (R) 0.5 m and thickness (t) 0.012 m at respective radii (r) are 533, 1085, 1637, 2190, and 2740. A total number (8185) of inflated beams is divided into three sections such that each section consists of approximately 2728 inflated beams. One of the cylindrical walls of the structure comprising concentric inflated walls is shown in Figure 7.11. Figure 7.11 - A section of structure configuration of a 15 km high space tower comprising inflated gas cells each of radii 0.5 m and thickness 0.012 m. 147 Each wall comprising the inflated tubes is about 3.0 m wide. However, the equivalent thickness of fabric material utilizing in inflated vertically deployed tubes comprising each wall is again estimated as 0.1 m. As a result each concentric 3.0 m wide wall made of inflated fabric tubes has a material thickness of (teq) 0.1 m. The inflated sections can be deployed by stacking the sections vertically utilizing methodologies described previously (Quine et al., 2009). The equivalent radius (Chapter 5, Eq. (5.13)) of the composite tower structure (Figure 7.11) made of Kevlar is estimated as 182 m and the corresponding lateral deflection is 300 m (angular inclination 1.1°, Figures 7.9 - 7.10). The space tower structure made of Kevlar having equivalent radius 182 m is quite stable with an angular deviation of 1.1° (lateral deflection of 300 m) under a maximum predicted wind load moments. The critical buckling load (with fixed-free end condition) of the multiple-beam structure made of Kevlar as shown in Figure 7.11 is estimated as 2.7x10 kg (270,000 tonne). A similar structure fabricated from Boron as proposed by Quine, Seth and Zhu (2009) would further enhance stability by significantly reducing lateral deflections (88 m with angular deviation of 0.3 ) under maximum wind load. The critical buckling load of the structure made of Boron is estimated as 9.3x10 kg (930,000 tonne). A composite inflated multiple-beam tower structure with proposed geometrical parameters can be controlled using an active control mechanism. However other configurations of the tower can be analysed to optimise the utility of the space tower. To analyse control mechanism experimentally, an inflated multiple-beam structure comprising three beams is constructed that can be controlled utilizing an active control 148 mechanism as described in Chapter 6. The experimental results obtained can be scaled to a space tower accessing 20 km altitude. For estimating pressure required for active control of tower to maintain vertical orientation, consider the composite structure configuration as shown in Figure 7.11. For a 15 km high space tower with a radius 150 m constructed at an altitude of 5 km, the maximum wind load moment (1.4 xl0 1 2 Nm) causes a lateral deflection of about 300 m in the composite tower structure corresponding to an equivalent radius of 182 m, when inflated at an appropriate level of pressure. An increase in pressure of 2100 kPa (21 bar, equation 6.29) in each beam is required in an inflated composite section along lean side of the structure in order to maintain balance in wind induced lateral deflection of 300 m (Figure 7.9). The wind shear frequencies can be estimated from wind load impacts causing oscillation in the concentric wall structure comprising inflated beams having stiffness due I0 to pressurized gas in the tubes. The estimated wind load moment of 1.4x10 "Nm exerts a wind load impact of 9.3xl0 7 N on a 15 km high tower standing on the top of a mountain that can cause a lateral deflection of approximately 300 m. The stiffness of the structure is found to be about 3.1x10 N/m. Assuming estimated mass of composite structure comprising concentric walls made of inflated tubes inflated with low atomic gas is 3.2x10 kg, the oscillation frequencies due to wind load is estimated to be 0.002 s"1. Similarly, a space tower having a radius 115 m comprising 2085 inflated columns each of radius 0.5 m and thickness 0.012 m, with a lateral deflection of 837 m under wind load can be controlled by increasing the pressure to 2300 kPa (23 bar) in each of the 695 columns along the lean side of the structure. However such structure configurations seem 149 difficult to build having a large internal diameter span of approximately 224 m. A structure configuration utilizing multiple concentric walls of inflated columns at appropriate distances from the centre (internal diameter span approximately 60 m) appears more plausible. A zero lateral tip displacement can advantageously be achieved by making space tower slightly tapered by appropriate gradual decrements in pressure in the inflated columns radially towards the centre of tower structure. In practice, a space tower would likely utilize an appropriate combination of air, helium and hydrogen (above 10 km) as its inflation gases. Advantageously, the central runnel similar to the structure tunnel as shown in Figure 7.12 can be utilized for carrying payloads. The structure as proposed by Quine, Seth and Zhu (2009) can be constructed comprising an appropriate number of such tunnels for emergency exits and security purposes. Figure 7.12 - The vertical space through the centre of inflated multi-beam structure. 150 7.7 Cost of Space Elevator Structure The estimated cost of construction of a space elevator is CAD $7.7 billion including the labour, construction material and pressurization gases assuming that gas production and material manufacturing plants are provided on site. Any such estimates are rough order of magnitude (ROM) without a detailed cost study. It is anticipated that the scale of the project will bring along technological efficiencies in the manufacture of the columns that may reduce capital cost. Such a structure would appear expensive however, for comparison; United Arab Emirates (UAE) is currently constructing a zero-carbon city (Masdar City) for 50,000 inhabitants at an estimated cost of $15- $30 billion USD. 7.8 Conclusion Theory and computation presented here shows the feasibility of an inflatable space tower using example material Kevlar 49. Space towers of height 15 km and 20 km require effective radius of at least 150 m. The primary reason for instability is the wind load in region 3 (10-15 km), which contributes most due to high wind speed in that region. The stability and control factors can potentially be increased by using multiplebeam structure. The stability of the multiple-beam structure depends upon the geometry (radius of each beam and the interspacing between the beams) of the structure in addition to the internal gas pressure. A 15 km high space tower with radius 150 m constructed at an altitude of 5 km comprising appropriate number of sections of inflated columns arranged circumferentially symmetrical along horizontal is found to have stable configuration. The load bearing capacity of inflated space towers is significantly large in 151 comparison to balloons and airships, which is limited by their material of construction and size and also due to atmospheric and weathering conditions. The attitude of the inflated multi-beam structure can also be guided actively by differential change of pressure in the inflated columns. The concentric cylindrical wall structure is desirable because drag coefficient due to wind load is less as compared to other configurations. However, other configurations such as continuous tapering the structure or changing the diameter at appropriate altitudes can also be analyzed with respect to atmospheric wind load conditions to stabilize the structure further in order to reduce wind load effects. 152 CHAPTER 8 SUMMARY AND FUTURE WORK 8.1 Summary The pneumatic space structure technology has a potential to replace existing construction methodologies for the construction of large space structure systems. Inflated space towers with sufficient load carrying capacity can be designed using latest inflatable technology involving various design strategies. Inflatable technology being light weight, cost effective and compatible with the space environment is most suitable for large space structural systems. Author analysed and validated the proposed practical concept (Quine et al, 2009) for the construction of a device to access altitudes above 20 km that is realisable utilising current material technologies (Seth et al., 2009). The tower can be utilized as a platform for various scientific and space missions or as an elevator to carry payloads and tourists. Suborbital towers will also likely be required for the construction of a geostationary space tether as they would provide an ideal surface mounting point where the orbital tether could be attached without experiencing the atmospheric turbulence and weathering in the lower atmosphere. Experimental investigations of inflated single beams along with inflated multiplebeam structures provide guidelines based on critical lateral wrinkling loads as a basis for the construction of a space tower. Euler's critical load predicts maximum sustainable axial load in order to avoid buckling of inflated tower. A space tower for accessing an altitude of 20 km can be constructed using current strong materials such as Kevlar. 153 Compared to solid structures, the pneumatic structures can be easily controlled by making use of an active control mechanism comprising gyroscopic stabilization and differential change of pressure in inflated columns. However pressure control mechanism is more effective and desirable in controlling the attitude of the structure as compared to gyroscopic stabilization. The results obtained after experimental investigation of inflatable cylindrical cantilevered fabric beams are useful in constructing inflated space tower. The theoretical analysis and successful demonstration of inflated multiple beam structures confirms the feasibility of inflatable space tower using example material Kevlar 49. Space towers of height 15 km and 20 km require an effective radius at least of 150 m for maintaining stability. The primary reason for instability is the wind load in a region of 10-15 km above the sea level, which contributes most due to high wind speed in that region. The stability and control factors can potentially be increased by using multiple-beam structure. The attitude of the inflated multiple-beam structure can also be guided actively by differential change of pressure in the inflated columns. The analysis is conducted using non-tapered structure; however an appropriate tapering of the inflated structure may be advantageous. Extensive theoretical and experimental investigations are required to analyze other structure configurations. 154 8.2 Future Work The following theoretical and experimental investigations are recommended for further analysing inflatable structures for the construction of a space tower: • Load-deflection behaviour of tapered inflated structures using cylindrical inflatable beams can be experimentally studied and the stability factors can be analysed that can be compared with non-tapered structure with and without the use of gyros. • Experimental investigations of damping of the oscillating vertical inflated beams fixed at the lower end at different pressures using different lengths can be investigated. • Hysteresis effect in the inflated fabric beam at different pressures under different loading and unloading conditions is an important parameter to study. • Response of an inflated beam with its lower end mounted on the vibrating base (shaker) using vibration test facility can be studied. The test is important as it simulates the behaviour of the structure under seismic affects that can be compared with steel reinforced concrete skyscrapers. • The behaviour of an inflated structure consisting of three inflated beams under compressive loads hanging from the top centre for a particular length to diameter ratio at different pressures can be studied more precisely using load cells to analyse load carrying capacity of non-tapered and tapered inflated structures. • Torsional effect due to wind loads on large inflatable space structures is an important factor and must be analysed in order to construct a space tower. 155 • The resonance induced by the dynamic complex wind loads needs to be investigated. • The response time of pressure control mechanism under dynamic wind load impacts needs to be investigation. • A finite element analysis of the inflated structures can be conducted to develop a mathematical simulation model of the inflated space tower using different structural configurations (non-tapered and tapered) under different lateral and axial loading and unloading conditions. 8.3 Future Innovation The utilization of space towers for generating electrical power may have significant importance. Analytical investigations can be conducted in order to enhance the use of green renewable energy sources with the added benefit of a pollution free environment. World's largest Horizontal Axis Wind Turbine (HAWT) is an E-126 manufactured by Enercon. The E-126 has a rotor diameter of 126 m generates power up to 7 MW at an average wind speed of 8.5 m/s. The complete upper assembly including turbine house, hub and blades weighs 650 tonnes. The biggest section is the hub with the steel blades installed, weighing 303 tonnes. As an example, an appropriately designed turbine similar to E-126 would generate 180 MW power at a wind speed of 25 m/s. In the region 10- 15km (average velocity = 25 m/s, density = 0.28 kg/mJ) above the earth at equator, the wind energy is available at approximately 100% of time. The number of wind turbines that can be deployed on the proposed 15 km structure is estimated as 100, deploying 4 turbines at each level with a 156 pair of turbines at diagonally opposite ends. The total weight of 100 turbines would be approximately 65,000 tonnes. Euler buckling load for a 15 km (for fixed-free end conditions; Le = 2L) tower made of Kevlar is estimated as 270,000 tonne. The total continuous power generation utilizing 100 turbines is estimated as 4GW. This might further be increased to 18 GW by employing a more comprehensive design. For comparison, the peak power consumption of Ontario on March 22, 2010 is recorded as 18 GW. In addition, the light weight inflatable wind turbines can be developed to harness wind energy. The average wind load acting on a E-126 turbine at an altitude high above the Earth in a range of about 7 km to 20 km where wind is more consistent with an average wind speed of 25 m/s and average density of about 0.3 kg/m is estimated at 1130000 N (113 tonne). The wind load acting on 100 such turbines could be approximately 11300 tonne. The turbines can be appropriately braced with tower in a comprehensive design. The tower would be capable to support the lateral load without any additional bracing utilizing tethers because kinetic energy associated with the wind loads would be converted into rotational energy associated with the turbines. The wind turbines of appropriate sizes can be installed at different altitudes by analysing atmospheric conditions at high altitudes based upon topographical locations. Further horizontal axis wind turbines rotating at appropriate speeds could serve as gyroscopes and therefore gyroscopic effect of rotating wind turbines could further enhance the stability the space tower. 157 Space towers can also be utilized in solar updraft technology by installing wind turbines in the path of hot air flow created artificially by green house effect. Atmospheric static electricity can also be harvested extending the idea of Plauson to capture electricity at high altitudes using balloon (Power from air, Feb-Mar 1922) to space towers accessing high altitudes made of Kevlar. The main advantage of insulating towers made of Kevlar or other strong polymer fabric is that tower can be deployed in such a way that charge collected by tower skin does not flow into the ground plane. Therefore tower can be made insulated from ground that could serve as a giant capacitor to store electricity. Pneumatic towers made of Kevlar can be easily built at any location on Earth for the purpose of harvesting green energy sources. The outer skin of the tower can be sprinkled with zinc amalgam and radium to get pointed ends for increasing surface charged density to enhance capturing capacity of the tower. Since atmospheric electricity increases with altitude, an appropriate increase in surface area of a tapered 20 km high space tower can generate an electrical power more than 1 GW without the use of any moving machinery parts. The power generated at different altitudes can be transmitted on ground utilizing appropriately rated power cables. High power transmission lines are usually operated at approximately 110 kV. Assuming a transmission of 1GW power per cable from an average height of 10 km and utilizing aluminum as a conductor, the line would weigh approximately in a range of 1000 kg. 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[3] Quine, B.M., Seth, R.K., Zhu, Z.H., "A Free Standing Space Elevator Structure: A Practical Alternative to Space Tether," Acta Astronautica. Volume 65, Issues 3-4, pp. 365-375,2009. [4] Zhu, Z.H., Seth, R. K., Quine, B. M., "Experimental Investigation of Inflatable Cylindrical Cantilevered Beams," JP Journal of Solids and Structures, 2, 95-110, 2008. Paper Submitted: [1] Seth, R.K., Quine, B.M., and Z.H. Zhu, "Active control mechanism for a pneumatically supported space tower," Acta Astronautica, Submitted May 2010, Ref: AA-D-10-00277. [2] Seth, R.K., Zhu, Z.H., and B.M. Quine, "Experimental investigation of inflated multiple-beam structures for building space towers," J. Smart Materials and Structures, Submitted April 2010, Ref: SMS/353733/PAP/249971. Conference Oral Presentations: [1] The research paper has been presented recently on August 14 , 2010 in 2010 Space Elevator Conference organised by Microsoft Corporation, International Space 173 Elevator Consortium and Leeward Space Foundation held at Microsoft Conference Centre in Redmond, Washington. [2] The paper has been presented on June 10' , 2010 in CAP (Canadian Association of Physicist) Congress 2010 held at University of Toronto. 174

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