Numerical modeling and buckling analysis of inflatable structures

´ Ecole Doctorale MEGA, INSA de Lyon University Claude Bernard 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 − Building A, IUT Lyon − Site Gratte-Ciel A Dissertation submitted in partial fulfilment of the requirements for the degree of Doctor of University Claude Bernard – Lyon (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 Professor Fr´ed´eric LEBON University of Nantes, France University of Marseille, France Advisors : Professor Michel MASSENZIO Professor Sylvie RONEL University Claude Bernard Lyon 1, France 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 - Gratte Ciel, specially Professor Christian Jardin, Mrs Bettina Fenet and Mr Benoit Thomas I would like to thank them for all their kindness during my time here Although being a Ph.D student at UCBL has not always been easy and straightforward 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 experiments 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 mes parents iii iv Contents List of Figures ix List of Tables xv Notations and conventions xvii GENERAL INTRODUCTION 1 Textile fabric composites 2 Inflatable structures 3 Stability of inflatable structures Objectives Thesis Outline Chapter BACKGROUND 1.1 1.2 Textile structures and textile preforms 1.1.1 Context 1.1.2 Classification of textile preforms Microscopic observation 16 1.2.1 Unit cell and geometric parameter 16 1.2.2 Stress transfer and characteristics lengths 20 1.2.3 Damage due to tensile loading 21 1.3 Prediction of engineering properties using micro-mechanics 24 1.4 Prediction of engineering properties using numerical approach 26 1.5 Experimental measurement of engineering properties 29 1.6 The role of experiments in structural stability 39 v Contents Chapter EXPERIMENTAL STUDIES 41 2.1 Introduction 43 2.2 Mechanical behavior of the fabric 44 2.3 2.2.1 Engineering constants 44 2.2.2 Strain measurement 47 Fabric tensile testing at our laboratory: Biaxial beam inflation test on fabric beam 48 2.4 2.5 2.3.1 Analysis of elastic moduli 48 2.3.2 Determination of shear modulus of HOWF composite 51 Experimental buckling of an inflatable beam 53 2.4.1 Introduction 53 2.4.2 Experimental buckling test on a simply supported HOWF beam 54 2.4.2.1 Test set-up and instrumentation 54 2.4.2.2 Boundary conditions 56 2.4.2.3 Measurement of displacements 59 Conclusion 65 Chapter ANALYTICAL BUCKLING ANALYSIS OF AN HOWF INFLATABLE BEAM 67 3.1 Introduction 69 3.2 Theoretical background 71 3.3 3.2.1 Kinematics 73 3.2.2 Constitutive equations 75 3.2.3 Virtual work principle 77 3.2.4 Theoretical buckling loads 85 3.2.5 Previous works on the critical load 89 Examples: in-plane buckling for linearized problems 90 3.3.1 Simply supported inflatable beam under compressive concentrated load 92 3.3.2 Cantilever inflatable beam under compressive axial load at the free end 99 3.3.3 vi Clamped-clamped inflatable beam under compressive axial load 101 3.4 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 3.6 Conclusion 111 Chapter FINITE ELEMENT BUCKLING ANALYSIS OF AN HOWF INFLATABLE BEAM 113 4.1 Literature review 115 4.2 Finite element formulations 118 4.2.1 Linear eigen buckling 118 4.2.2 Nonlinear buckling 121 4.2.3 Implementation of an iterative algorithm for solving the NLIBFE model 122 4.3 Applications and results 124 4.3.1 Linear eigen buckling 128 4.3.2 Nonlinear buckling of a simply supported NLIBFE model 132 4.3.2.1 Wrinkling loads and maximum deflections: Limit of validity for numerical solutions 135 4.4 4.3.2.2 Validation of the NLIBFE model: the reference model 137 4.3.2.3 Comparison with the experimental results 140 4.3.2.4 Parametric studies of NLIBFE model 142 Conclusion 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 155 A.2 Hyperelasticity: theoretical basis 157 A.3 Hyperelasticity and orthotropic materials 160 A.3.1 Orthotropic materials 160 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Titre de la thèse : “Modélisation numérique et analyse du flambement des structures gonflables en textiles techniques orthotropes ” L’objectif principal de cette thèse est de modéliser en flambement des poutres pressurisées en tissu souple homogène orthotrope (THO) composite Cette thèse comporte parties La première détaille les études expérimentales qui ont été menées sur des poutres gonflables plusieurs niveaux de pression afin de caractériser les propriétés mécaniques du matériau et le comportement en flambement de la structure Deux études expérimentales ont été effectuées: un essai de gonflement pour déterminer les propriétés mécaniques orthotropes en sollicitations bi-axiales Les caractéristiques du matériau obtenues ont été utilisées pour les modèles éléments finis de la poutre gonflable en THO Les seconds essais soumettent en compression une structure cylindrique pressurisées jusqu’au flambement, en accordant une attention particulière la détection des plis Dans une deuxième partie, une approche analytique a été envisagée afin d’étudier le flambement ainsi que le comportement d’une poutre gonflable orthotrope soumise des charges de compression uniformes dans des conditions aux limites différentes Afin d’apprécier la stabilité de telles structures, il est nécessaire d’évaluer la charge critique associée différentes configurations de pression Dans un premier temps, un modèle 3D gonflables poutre orthotrope basé sur la cinématique de Timoshenko a été présenté brièvement: les non-linéarités (rotation finie, les forces suiveuses) ont été incluses dans ce modèle Les équations d’équilibre non-linéaires ont été obtenues partir de la forme Lagrangienne Totale du principe des travaux virtuels: les équations linéarisées ont ensuite été obtenues La résolution de ces équations linéarisées a conduit l’expression analytique de la charge critique de flambement Cette charge critique a été étudiée pour différents cas de charge avec différentes conditions aux limites Nous avons pu évaluer l’écart entre notre modèle orthotrope et les modèles isotropes issus de la littérature, ainsi que l’influence de la pression de gonflage et des propriétés mécaniques de tissu sur la valeur de la charge critique Les modes de flambement ont également été déterminés Pour vérifier la limite de validité des résultats, la charge d’apparition des plis a également fait l’objet d’une étude pour chacun des cas La dernière partie est consacrée une étude linéaire des valeurs propres et une analyse non-linéaire du flambement de la poutre gonflable en textiles orthotropes techniques La méthode d’analyse est basée sur un modèle 3D poutre de Timoshenko en THO Le modèle éléments finis établi ici implique un élément poutre de Timoshenko trois-noeuds avec une continuité de type C pour le déplacement transversal et des fonctions de forme quadratiques pour la rotation de flexion et le déplacement axial Dans l’analyse linéaire du flambement, un test de convergence du maillage sur la force critique de la poutre a été réalisé par la résolution du problème aux valeurs propres La matrice de raideur dans ce cas est généralement admise ne pas être fonction des déplacements, tandis que dans le problème non-linéaire de flambement, la matrice de raideur tangente tient compte des non linéarités géométriques et des forces suiveuses dues la pression Le modèle éléments finis non-linéaire a été développé en utilisant la procédure itérative de Newton−Raphson avec incréments de chargement adaptatif permettant le tracé pas pas de la réponse charge-déflexion de la poutre Afin d’évaluer l’incidence des non-linéarités géométriques et de la pression de gonflage sur la stabilité du comportement de structures pressurisées, une poutre appuyée simple a été étudiée L’influence des ratios géométriques de la poutre sur le coefficient de charge de flambement a également été soulignée Pour vérifier la validité et la solidité des résultats, la poutre a également été modélisée sur le code ABAQUS en 3D avec des éléments coque mine Pour une validation supplémentaire, les résultats obtenus ont également été comparés ceux obtenus lors d’expérimentation des pressions de gonflage faibles Mots-clés: Poutre gonflable, tissu orthotrope, essais en gonflement, essais en flambement, cinématique de Timoshenko, charge critique, charge des plis, force suiveuse, pression de gonflage, modèle éléments finis, Newton−Raphson, réponse charge-déflexion, modèle de coque mince 3D Abstract The main goals of this thesis are to modeling and to perform the buckling study of inflatable beams made from homogeneous orthotropic woven fabric (HOWF) composite Three main scenarios were investigated in this thesis The first is the experimental studies which were performed on HOWF inflatable beam in various inflation pressures for characterizing the orthotropic mechanical properties and buckling behaviors Two types of experimental studies were reported in this scenario: the biaxial testing with beam inflation test for determining the orthotropic mechanical properties which were used as material inputs for finite element models of HOWF inflatable beam; and the buckling tests was involved to obtain the buckling behavior and monitor the initiation of the first wrinkles of the beam In the second scenario, an analytical approach was considered to study the buckling and the behavior of an inflatable orthotropic beam subjected to uniform compression loads under different boundary conditions In order to assess the stability of inflatable structures, it is necessary to evaluate the critical load of the inflatable components in their pressurized configurations A 3D inflatable orthotropic beam model based on the Timoshenko’s kinematics was briefly introduced: the nonlinearities (finite rotation, follower forces) were included in this model The nonlinear equilibrium equations were derived from the total Lagrangian form of the virtual work principle: the linearized equations were then obtained By solving these linearized equations, an analytical expression of the critical buckling load was obtained This critical buckling load was investigated through several load cases with several boundary conditions The discrepancy due to the orthotropic character between the present model and the isotropic models found in the literature was evaluated, as well as the influence of the inflation pressure and the fabric mechanical properties on the value of critical load The buckling mode shapes were also determined To check the limit of validity of the results, the wrinkling load was also presented in every case The last scenario is devoted to the linear eigen and non-linear buckling analysis of inflatable beam made of orthotropic technical textiles The method of analysis was based on a 3D Timoshenko beam model with a homogeneous orthotropic woven fabric The finite element model established here involves a three-noded Timoshenko beam element with C −type continuity for the transverse displacement and quadratic shape functions for the bending rotation and the axial displacement In the linear buckling analysis, a mesh convergence test on the beam critical force was carried out by solving the linearized eigenvalue problem The stiffness matrix in this case is generally assumed not to be a function of displacements, whilst in the non-linear buckling problem, the tangent stiffness matrix includes the effect of changing geometry as well as the effect of stress stiffening The nonlinear finite element solutions were investigated by using the straightforward Newton iteration with adaptive load stepping for tracing load−deflection response of the beam To assess the effect of geometric nonlinearities and the inflation pressure on the stability behavior of inflatable beam, a simply supported beam was studied The influence of the beam aspect ratios on the buckling load coefficient was also pointed out To check the validity and soundness of the results, a 3D thin-shell finite element model was utilized for comparison For further validation, the obtained results were also compared with those from experiments at low inflation pressures Keywords: Inflatable beam, orthotropic fabric, inflation test, buckling test, Timoshenko’s kinematics, critical load, wrinkling load, follower force, inflation pressure, finite element model, Newton−Raphson, load-deflection response, 3D thin shell model ... Natural length of the inflatable beam lo Reference length of the inflatable beam xix Notations and conventions Ro Reference radius of the inflatable beam to Reference thickness of the inflatable. .. force along y and z axes My , Mz Moments around y and z axes Beam geometry lφ Natural length of the inflatable beam Rφ Natural radius of the inflatable beam tφ Natural thickness of the inflatable. .. and are more durable during handling Inflatable structures Of the large family of textile structures, coated woven fabrics have attracted the most serious interest in the aerospace industry and