Luận án TS: Numerical modeling, buckling analysis inflatable structures - N.T. Truong

University Claude Bernard Lyon 1 École Doctorale MEGA, INSA de Lyon Thesis reference number : 123 − 2012 Numerical modeling and buckling analysis of inflatable structures PhD THESIS Presented and defended publicly on August, 31th 2012 at 10:30 in the Lecture hall 2 − Building A, IUT Lyon 1 − Site Gratte-Ciel A Dissertation submitted in partial fulfilment of the requirements for the degree of Doctor of University Claude Bernard – Lyon 1 (Department of Mechanical Engineering) by NGUYEN Thanh Truong Graduate Committee Chairman : Professor Huan PHAN DINH Ho Chi Minh City University of Technology, Vietnam Reviewers : Professor Anh LE VAN University of Nantes, France Professor Frédéric LEBON University of Marseille, France Advisors : Professor Michel MASSENZIO University Claude Bernard Lyon 1, France Professor Sylvie RONEL University Claude Bernard Lyon 1, France Examinateur : Professor Eric JACQUELIN University Claude Bernard Lyon 1, France Biomechanics and Impact Mechanics laboratory — UMR T 9406 Mis en page avec la classe thloria. Acknowledgments During my time in Lyon, there are a number of people who have supported me both inside and outside the lab. From my early days at IUT Lyon 1, I had many difficulties to adapt a new life in French, which is different deeply from Vietnam. At that time, I received a tremendous amount of help and support from the personnels of IUT Lyon 1 - Gratte Ciel, specially Professor Christian Jardin, Mrs.

Bettina Fenet and Mr. I would like to thank them for all their kindness during my time here. Although being a Ph. student at UCBL has not always been easy and straightfor- ward to me, there has been so much help and support around.

First, I would like to thank my supervisors, Sylvie Ronel and Michel Massenzio for providing me the opportunity to work with them. It is a superb experience to have them as supervisors and learn how to face, think, approach and evaluate problems directly from them. Sylvie Ronel with her keen insight into science of structures has always amazed me. She has a wonderful ability to see and find beautiful things from what looks somewhat boring and unimportant.

I would like to thank her for her consistent support and encouragement in the middle of failures and sometimes slow progresses. Also, Michel Massenzio, who just makes everything in our group much simpler, is also thanked for careful reading and helpful advices throughout this thesis. In particular, I would like to thank Professor Eric Jacquelin for giving me lots of valuable suggestions in research. He has always inspired me to see how to research in each paper with challenging questions that I had never thought of.

Those questions always have guided and helped me gain a solid understanding of my research projects with new insights. Special thanks are due to Professor Le Van Anh and Professor Frédéric Lebon for their input, for reviewing this thesis, and for being members of the graduate committee. I would also like to thank Professor Phan Dinh Huan for being an examiner of this thesis and a member of the graduate committee. I owe him many thanks for teaching me i to be a scientist.

Professor Pham Huy Hoang and Mr. Nguyen Tuan Kiet are gratefully acknowledged in the same way for their advices and encouragement. Komla Lolonyo Apedo has been much more than a colleague and a friend to me over the first two years. Komla and I were on the same research theme at LBMC and worked beside together in a same office.

My work is a development based on his work. Komla’s infectious friendliness, his passion for science and his obsession with understanding have been instrumental in making my life at LBMC joyful and productive. We spent many memorable time wrestling the formulations, explained me to understand how an inflatable beam is. In addition, Komla’s deep understanding of inflatable structures provided a fantastic resource to bounce ideas back and forth several times a day.

I want to single out and thank the people I have worked most closely at Laboratory DDS of GMP, IUT Lyon 1, especially Abdelkrim Bennani and Lagarde Gérard for their availability. I am also grateful for the financial support from the Vietnamese government for this thesis and from LBMC-IFSTTAR/UCBL for my first European Conference in Austria. Losberger Company and specially Mr. Robert Dartois are acknowledged for having provided the material samples and inflatable beams which were very useful for the exper- iments in this thesis.

Parents, to whom I have dedicated this work, have supported and encouraged me as I worked toward this degree. Finally, I would like to thank my girlfriend for all her love and support over the years and for her encouragement and faith in my ability to finish the degree program. Let anyone who has contributed directly or indirectly to the success of this project, finds here my acknowledgments. ii Je dédie cette thèse à mes parents iii Contents List of Figures ix List of Tables xv Notations and conventions xvii GENERAL INTRODUCTION 1 1 Textile fabric composites.

3 3 Stability of inflatable structures .1 Textile structures and textile preforms .2 Classification of textile preforms .1 Unit cell and geometric parameter .2 Stress transfer and characteristics lengths .3 Damage due to tensile loading .3 Prediction of engineering properties using micro-mechanics .4 Prediction of engineering properties using numerical approach .5 Experimental measurement of engineering properties .6 The role of experiments in structural stability. 39 v Contents Chapter 2 EXPERIMENTAL STUDIES 41 2.2 Mechanical behavior of the fabric .3 Fabric tensile testing at our laboratory: Biaxial beam inflation test on fabric beam .1 Analysis of elastic moduli .2 Determination of shear modulus of HOWF composite .4 Experimental buckling of an inflatable beam .2 Experimental buckling test on a simply supported HOWF beam .1 Test set-up and instrumentation .3 Measurement of displacements. 65 Chapter 3 ANALYTICAL BUCKLING ANALYSIS OF AN HOWF INFLA- TABLE BEAM 67 3.3 Virtual work principle .4 Theoretical buckling loads .5 Previous works on the critical load .3 Examples: in-plane buckling for linearized problems .1 Simply supported inflatable beam under compressive concentrated load .2 Cantilever inflatable beam under compressive axial load at the free end .3 Clamped-clamped inflatable beam under compressive axial load .4 Influence of the slenderness ratio on the critical load of an inflatable beam 106 3.5 Wrinkling load for an inflatable beam under a compressive concentrated load109 3. 111 vi Chapter 4 FINITE ELEMENT BUCKLING ANALYSIS OF AN HOWF INFLATABLE BEAM 113 4.2 Finite element formulations .1 Linear eigen buckling .3 Implementation of an iterative algorithm for solving the NLIBFE model .3 Applications and results .1 Linear eigen buckling .2 Nonlinear buckling of a simply supported NLIBFE model .1 Wrinkling loads and maximum deflections: Limit of valid- ity for numerical solutions .2 Validation of the NLIBFE model: the reference model .3 Comparison with the experimental results .4 Parametric studies of NLIBFE model.

145 GENERAL CONCLUSION AND FUTURE WORK 149 Appendices 155 Appendix A Reminders in mechanics and material science 155 A.1 Mechanical properties of composite materials .2 Hyperelasticity: theoretical basis .3 Hyperelasticity and orthotropic materials. Venant-Kirchhoff orthotropic material .4 Thin-walled structures : thin-shells and membranes. 164 Appendix B Theoretical model 171 Appendix C Nonlinear finite element model 175 Bibliography 181 vii Contents viii List of Figures 1 Woven fabrics .1 Classification of textile preforms in the YTF processes (Cox and Flanagan (1996a)).2 Different woven fabric (Source: TexGen).3 Schematic diagrams of (a) warp knitted and (b) weft knitted fabrics.4 Basic weave constructions: (a) plain, (b) twill and (c) 5HS satin weave.5 Schematics of performing and resin injection molding processes.6 Cross-section of an orthogonal 2-D woven fabric along the warp direction.7 Specimens for microscopic observation.8 Fabric pattern and a unit cell of plain woven fabrics.10 Failure of tensile specimens .11 Schematic of damage evolution due to tensile loading.12 Fibrous medium and equivalent continuum.13 Experimental set-ups used to evaluate intra-ply shear properties: (a) Bias- extension set-up and (b) Picture-frame set-up.14 Schematics showing the undeformed (left) and deformed (right) shapes of the specimen in the bias-extension test. 32 ix List of Figures 1.15 A bias-extension test apparatus (Cao et al.16 Biaxial testing machine layout (Quaglini et al.17 Shear frame loaded with fabric specimen (King et al.18 Trellising-shear test apparatus fabricated and used by the research groups (Cao et al.1 Plain weave fabric structure.2 Stress components referred to specimen and material axes.3 Longitudinal stress distribution in thin-walled cylinder.4 Transversal stress distribution in thin-walled cylinder.5 Inflation test layout.6 Schematic diagram of simply supported HOWF inflatable beam and in- strumentation for buckling test.7 Digital Manometer KELLER LEO 1.8 Mounting load cell type ZFA to the structure.9 Simply supported HOWF inflatable beam and VDAS 5000.10 Experimental apparatus of HOWF simply supported inflatable beam for measuring the critical load.11 Inflatable beam with alphabet labels.12 Tachometer LEICA TPS 300.13 The formation of the first wrinkles.14 Crushing test: Experimental curves of axial load versus time.15 Quasi-static axial compression test: Deflections of simply supported HOWF beam for various pressures.16 Progression of first buckling mode of the beam with pressure of 30 kPa.1 HOWF inflatable beam: (a) in natural state and (b) in the reference configuration (inflated state) .2 (a) Fabric local coordinate system, (b) Beam Cartesian coordinate system .3 Uniform pressure on the cylindrical surface.4 Definition of the curvilinear coordinate system.5 Definition of the curvilinear basis at the beam ends.6 Simply supported inflatable beam.7 Critical loads for a simply supported inflatable beam under a compressive concentrated load in the case of materials 1 and 2.8 Fundamental buckling modes on a simply supported inflatable beam under a compressive concentrated load in the case of material 1.9 Fundamental buckling modes on a simply supported inflatable beam under a compressive concentrated load in the case of material 2.10 First three buckling modes on a simply supported HOWF inflatable beam under compressive concentrated load.11 Cantilever inflatable beam.12 Critical loads for a cantilever inflatable beam under a compressive concen- trated load in the case of materials 1 and 2.13 Fundamental buckling modes on a cantilever inflatable beam under a com- pressive concentrated load in the case of material 1.14 Fundamental buckling modes on a cantilever inflatable beam under a com- pressive concentrated load in the case of material 2.15 First three buckling modes on a cantilever inflatable beam under a com- pressive concentrated load.16 Clamped-clamped inflatable beam under a compressive axial load.17 Critical load versus the slenderness ratio of a cantilever inflatable beam under compressive concentrated load in the case of material 1.18 Critical load versus the slenderness ratio of a cantilever inflatable beam under compressive concentrated load in the case of material 2.

108 xi List of Figures 4.1 Linear eigen buckling: mesh convergence test of normalized linear buckling load coefficient (Kcl = 105 × σcr /Eeq ) for a simply supported LFEIB model.2 Linear eigen buckling: normalized buckling load coefficient (Kcl = 105 × σcr /Eeq ) versus radius-to-thickness ratio (Rrt = R0 /t0 ) for a simply sup- ported LFEIB model.3 Linear eigen buckling: normalized buckling load coefficient (Kcl = 105 × σcr /Eeq ) versus slenderness ratio (λs = L/ρ) for a simply supported LFEIB model.4 Linear eigen buckling: normalized buckling load coefficient (Kcl = 105 × σcr /Eeq ) versus bending radius ratio (Rbr = R2Rb B0 ) for a simply supported LFEIB model.5 (a) Inflatable beam subjected to compressive axial load F. (b) The effect of an initial imperfection .6 Wrinkling load: limit of validity of numerical solutions .7 Global beam axes and local material orientation assigned for a orthotropic fabric.8 (a) Inflatable beam with constrained ends, (b) Applied loads on the beam and (c) Beam meshing using S4R shell elements .9 Numerical and experimental mid-span deflection curves of a simply sup- ported NLIBFE model with pn =324.10 Numerical and experimental mid-span deflection curves of a simply sup- ported NLIBFE model with pn =648.11 Numerical and experimental mid-span deflection curves of a simply sup- ported NLIBFE model with pn =972.