Buckling analysis of inflatable composite beams

ABSTRACT This thesis presents a numerical modeling and an experimental program approach to investigate the buckling behavior of inflatable beams made from woven fabric composite materials In the numerical study, the Isogeometric Analysis (IGA) is utilized to analyze the bucking response of inflatable beams subject to axial compressive load and predict the critical load at which the first wrinkle occurs In the numerical model, Timoshenko’s kinematics principle is used to build a 3D model of inflating orthotropic beams In this modeling process, geometrical nonlinearity is considered by using the energy concept that accounts for the change in membrane and strain energies when the beams are bent By using Lagrangian and virtual work principles, nonlinear equilibrium equations were derived These equations are then discretized by using NURBS basis functions inherited from the IGA approach to derive the global nonlinear equation The well-known Newton-Raphson algorithm is then used to solve the nonlinear equation The numerical results are then calibrated with the experimental one It was found that a good agreement between IGA predictions and test results is achieved The numerical model could be used for other parametric studies to investigate the influences of material and geometrical parameters on the buckling behavior of inflatable beams In the experiment study, the mechanical properties of the woven fabric composite material used in the fabrication of inflatable beams are determined and the biaxial buckling test is carried out The experimental studies are performed under various inflation pressures to characterize the orthotropic mechanical properties and the nonlinear buckling behaviors Load versus deflection curve of inflating beams beam with different air pressures obtained from the experiments are illustrated., and the first wrinkles of the beams when buckling happens are also monitored Therefore, the maximum load-carrying capacity of the inflating beam with respect to the appearance of the first wrinkle is totally found In addition, the critical buckling load is determined through distinct load cases Then, the discrepancy is evaluated among the proposed orthotropic and isotropic models in the literature vi Contents BIOGRAPHY I ORIGINALITY STATEMENT IV ACKNOWLEDGEMENTS V ABSTRACT VI CONTENTS VII NOTATIONS AND CONVENTIONS X LIST OF FIGURE XVI LIST OF TABLE XX CHAPTER 1: INTRODUCTION 1.1 Background information .1 1.2 The motivation of the thesis 1.3 The objectives and scope of the study 1.4 Methodology 1.5 Outline of the thesis 1.6 Original contributions of the thesis .5 1.7 Significances of the thesis .5 CHAPTER 2: OVERVIEW OF FIBROUS COMPOSITE MATERIALS AND LITERATURE REVIEW 2.1 An overview of fibrous composite materials and practical applications of inflating composite structures 2.2 Literature review on the analysis of inflatable beams .12 2.2.1 Analytical approach .12 2.2.2 Numerical approach .14 2.2.3 Isogeometric Analysis 21 2.3 Conclusions 22 CHAPTER 3: THEORETICAL FORMULATIONS 23 3.1 Continuum-based governing equations of stability problems of inflating beams 23 vii 3.1.1 Mathematical description of inflating beams 24 3.1.2 Theoretical formulation .26 3.1.2.1 Kinematic relations .26 3.1.2.2 Constitutive equations 27 3.1.3 Virtual work principle 29 3.2 Conclusion 38 CHAPTER 4: IGA-BASED BUCKLING ANALYSIS OF INFLATING COMPOSITE BEAMS 40 4.1 A brief review on isogeometric analysis .40 4.1.1 The concept of isogeometric analysis 40 4.1.2 Disadvantages of IGA 42 4.1.3 B-Spline .43 4.1.3.1 Knot Vector 43 4.1.3.2 B-Spline Basis Functions .44 4.1.3.3 B-Spline Curves 45 4.1.4 NURBS Curves 46 4.1.5 NURBS-based IGA 48 4.1.6 NURBS-based elements for IGA .49 4.1.7 Isogeometric analysis versus classical finite element in the analysis 54 4.2 IGA-based formulations for the buckling problems of inflating composite beams 55 4.2.1 Linear eigen buckling 55 4.2.2 Nonlinear buckling 57 4.2.3 Implementation of an iterative algorithm in solving nonlinear model 61 4.3 Numerical examples 63 4.3.1 Linear buckling analysis 63 4.3.1.1 Simply-supported beam 64 4.3.1.2 Fixed – Free beam 66 4.3.2 Nonlinear analysis 67 4.3.2.1 Simply-supported beam 71 viii 4.3.2.2 Fixed-free beam 73 4.4 Conclusions 76 CHAPTER 5: BUCKLING EXPERIMENTS OF INFLATING BEAMS 78 5.1 Introduction 78 5.2 Material properties and selection of fabrics 78 5.2.1 The woven fabric materials 79 5.2.1 Testing equipment 83 5.2.2 Mechanical properties of woven fabric composites 84 5.3 Test of joint’s durable strength 89 5.3.1 Glued joint PVC cm 90 5.3.2 Glued joint PVC 1cm thermal .92 5.3.3 Glued joint PVC cm thermal 93 5.3.4 Glued joint PVC 2.5 cm with thermal attachment .95 5.4 Inflatable beam specimens 96 5.5 Buckling test set-up .98 5.6 Experimental results and discussion 107 5.6.1 Load vs displacement u relation of the beam at pressure 107 5.6.1.1 Beams inflated with different air pressures 117 5.6.1.2 Comparison of beams at pressure p = 80 kPa 120 5.6.2 Load vs displacement v relation of the beam at pressure 121 5.6.2.1 Beams inflated with different air pressures 131 5.6.2.2 Comparison of beams at pressure p = 80 kPa 134 5.7 Comparison between experimental and IGA numerical methods 135 5.8 Conclusion .138 CHAPTER 6: CONCLUSIONS AND FURTHER STUDIES 140 6.1 Conclusions .140 6.2 Further studies 141 REFERENCES 143 LIST OF PUBLICATIONS 148 ix Notations and conventions Abbreviations CAD Computer Aided Design FEA Finite Element Analysis IGA Isogeometric Analysis FGM Functionally graded material 3D Three dimensional NURBS Non-Uniform Rational B-splines DOF Degree of Freedom OPCWFC Orthogonal plain classical woven fabric composite FE Finite element FEM Finite elementingmethod HOWF Homogeneous orthotropic woven fabric LFEIB Linear finite element inflating beam NLIBFE Nonlinear inflating beam finite element Plain weave fabric testing (x, y) Axes of symmetry of plain weave fabric specimen (L, T) Axes of symmetry of textile fabric, which are assumed to represent the axes of material symmetry (L: warp direction; T: weft direction) Textile composite parameters and mechanical properties t Fabric thickness El Warp effective Young’s modulus Et Weft effective Young’s modulus Glt In-plane effective shear modulus x  lt Poisson ratio due to the loading in the warp and contraction in the weft  tl Poisson ratio due to the loading in the weft and contraction in the warp Constants c cos  s sin  Structural mechanics of inflating beams Coordinates systems l, t, n Warp, weft, normal directions of the fabric (X, Y, Z) Cartesian coordinates ( e X , eY , eZ ) Unit vectors of the cartesian coordinates (  = eZ , n ) Angle Mechanical properties E Young modulus of the isotropic fabric G Shear modulus of the isotropic fabric E * = Eto' Membrane elastic modulus of the isotropic fabric G* = Gto' Membrane shear modulus of the isotropic fabric  The Poisson ratio of the isotropic fabric El Modulus of elasticity in the warp direction of the orthotropic fabric Et Modulus of elasticity in the weft direction of the orthotropic fabric Glt In-plane shear modulus of the orthotropic fabric  lt Poisson ratio relative to the contraction of the warp under weft traction  tl Poisson ratio relative to the contraction of the weft under warp traction xi Eeq Equivalent Young’s modulus of the orthotropic fabric Geq Equivalent shear modulus of the orthotropic fabric  eq Equivalent Poisson ratio of the orthotropic fabric elt Level of orthotropy glt Parameter to measure the difference between Glt and Geq C, C loc Elasticity tensors Internal forces N Axial force Ty , TZ Shear force along y and z axes M y, MZ Moments around y and z axes Beam geometry l The natural length of the inflating beam R The natural radius of the inflating beam t Natural thickness of the inflating beam l The natural length of the inflating beam lo Reference length of the inflating beam Ro Reference radius of the inflating beam to Reference thickness of the inflating beam Ao The reference cross-section area of the inflating beam Io Reference moment of inertia of the inflating beam Loads F Compressive concentrated load FX , FY , FZ Components of concentrated loads fx , f y , fz Components of the distributed load MY , M Z Components of bending moments xii Fw Wrinkling load Fcrushing Crushing load Pressure, pressure forces p Inflation pressure pn Normalized inflation pressure Fp = p Ro2 Pressure force No Axial force due to the inflation pressure Coefficients k , k y , kz Shear correction coefficients Kinematics u Displacements field u Axial displacement v, w Deflections along Y and Z axes Y , Z Rotations around Y and Z axes Tensors E Green-Lagrange tensor EXX , EXY EXZ , EYY Components of E EYZ , EZZ E Virtual Green-Lagrange tensor  EXX ,  EXY  EXZ ,  EYY Components of  E  EYZ ,  EZZ S Second Piola-Kirchhoff tensor S XX , S XY S XZ , SYY Components of S xiii SYZ , SZZ S o Inflation pressure prestressing tensor Functions and constants E Strain energy function  Wint Internal virtual work  Wext External virtual work  Wextd External virtual work of the service load  Wextp External virtual work of the pressure load Qi , i = 10 Quantities depending on the initial geometry of the cross-section Matrix and tensors for finite element formulations d  Nodal d.o.f D Beam d.o.f  D Buckling displacements k  Element conventional elastic stiffness matrix  K  Element initial stress stiffness matrix K  Beam conventional elastic stiffness matrix K ref Beam initial stress stiffness matrix KT Beam tangent stiffness matrix Di Beam displacement vector at increment step i D Nodal unknown displacement increment R Beam residual load Fint  Internal load vector of the beam Fext  External load vector of the beam N  Shape function matrix xiv  B Strain-displacement matrix Aspect ratios Linear eigen buckling K cl Normalized linear buckling load coefficient s Slenderness ratio of the beam Rrt Radius-to-thickness ratio Rbr Bending radius ratio Nonlinear buckling K cl Normalized nonlinear load parameter Kf Incremental load ratio Rlr Length-to-radius ratio R fr Flexion-to-radius ratio xv 5.7 Comparison between experimental and IGA numerical methods Table 5.27 Comparison of beams at pressure p = 80 kPa Beam Beam Beam v (mm) P (N) v (mm) P (N) v (mm) P (N) 28 300 26 300 66 300 30 350 28 350 68 350 33 570 34 570 72 570 40 950 43 950 83 950 52 1380 61 1380 91 1380 69 1860 77 1860 99 1860 83 2300 93 2300 108 2300 99 2670 104 2670 115 2670 119 3320 123 3320 133 3320 128 2670 134 2670 142 2670 133 2300 142 2300 151 2300 141 950 148 950 160 1380 5.7 Comparison between experimental and IGA numerical methods Figure 5.52 and Figure 5.53 compare the experimental results and numerical results obtained from IGA In general, it is seen that the results obtained from experiments and those from IGA are somewhat similar in the structural response of inflating beams For the beams with low pressure (e.g 20 kPa, 40 kPa), it can be seen that the experimental results and modeling results are not in good agreement The experimental ultimate buckling strength of the beam with internal pressure p = 20 kPa is around 600 kPa, while the corresponding results obtained from the numerical study are around 1750 kPa For an inflatable beam with internal pressure is 40 kPa, the results obtained from the numerical model are around 1500 kN, while the experimental results are roughly 1900 kN However, if the pressure in the beam increases, the prediction of the IGA model becomes close to the experimental results (e.g The results for an inflatable beam with the internal pressure being 80 kPa, the 135 CHAPTER 5: BUCKLING EXPERIMENTS OF INFLATING BEAMS buckling strength obtained experimentally is around 2600, the results from numerical model is close to experimental results) In addition to the ultimate strength, it is observed that the loading paths obtained from numerical and experimental results are different The loading path in the numerical model follows a bifurcation trend, where the lateral and axial displacements slightly increase with the elevation of axial compressive load The displacements increase rapidly when the compressive load is close to the buckling load In the experimental model, it can be seen that the displacement increases linearly with the increase of compressive load The displacement and applied compressive load are in accending relations until the peak load is captured Beyond the peak load, the displacement still increases but the magnitude of the compressive load is reduced In general, the agreement between experimental and numerical results can be confirmed in some aspects, e.g the ultimate strength However, there are differences between experimental and numerical results in load-displacement relations and the ultimate strength for beams with low internal pressure These differences can be explained as follows: 136 5.7 Comparison between experimental and IGA numerical methods Figure 5.52 IGA prediction vs Experimental results, in axial displacement u with air pressure 20 kPa, 40 kPa, 60 kPa, and 80 kPa - In the experimental process, while we inflate and conduct experiments at low pressures, the beam is not tense enough so that it can keep the beam firm at this time We can just put the sensors in at this time and it creates settlement on the beam body At the same time, the sensor has not received the result of compressive force during the compression process - The formation according to “u” changes that make the beam radius increase The result was that we can see the initial stages of experiments, the sensors often earlier receive the results on the diagrams However, when increasing the pump pressure in the beam, we observe that the numerical and experimental results are converged 137 CHAPTER 5: BUCKLING EXPERIMENTS OF INFLATING BEAMS Figure 5.53 IGA prediction vs Experimental results, in transverse displacement v with air pressure 20 kPa, 40 kPa, 60 kPa, and 80 kPa - The shortage of real material information and errors in experimental procedures might cause significant errors in the experimental results - The numerical simulation does not account for the failure of the material, which might be the main failure reason in the case of low pressure inflating beams - Material models used in the numerical approach might not be appropriate for the use of composite fabric material, this needs further comprehensive investigations 5.8 Conclusion In this chapter, an experimental program was conducted in detail to investigate the buckling response of HOWF inflating beams First of all, some tests are conducted to find out the properties of the material Then the buckling tests are successfully 138 5.8 Conclusion conducted with different air pressure The results of axial load-deflection are recorded and then compared with numerical predictions based on Isogeometric Analysis The experiment results show that the strength of inflating beams increases with the rise of air pressure This is consistent with those obtained from the numerical prediction in the previous chapter In addition, it is found out that the numerical models only show acceptable predictions for the inflating beams with high pressure 139 CHAPTER 6: CONCLUSIONS AND FURTHER STUDIES CHAPTER 6: CONCLUSIONS AND FURTHER STUDIES 6.1 Conclusions In this study, a numerical modeling technique and an experimental program are conducted to investigate the stability behavior of inflating beams made from composite materials The numerical modeling is done using the Isogeometric Analysis method, which involves developing beam models based on Timoshenko's beam theory The governing equations are constructed using the whole Lagrange technique, which considers both the membrane and bending effects at the same time The study found that only the mechanical characteristics El and Glt interfere overtly in the solution of critical load through C11 and C66, whereas Et intervenes implicitly through the reference dimensions of the beam, thanks to the orthotropic character of the model Due to the disparity of mechanical properties in the yarn directions, only the amount of orthotropy of the fabric creates notable variations in the buckling behavior of the inflated beam The discrepancies in the models analyzed are also due to how the constitutive equations are established The material is believed to be hyperelastic and isotropic, and it follows the Saint Venant-Kirchhoff rule, which solely considers SXX and SYY In the Hookean stress-strain relationship, the Young modulus E is also employed directly All components of the second Piola-Kirchhoff tensor are taken into account in this model To develop the numerical model, the NURBS basis functions of the IGA approach are utilized to describe the governing equations and develop the global equations Both linear and nonlinear buckling analyses are carried out In the nonlinear buckling analysis, the well-known Newton-Raphson technique is adopted to trace the buckling curves Validation and various parametric studies are conducted to show the reliability of the approach and study the influence of internal pressure in the beams 140 6.2 Further studies In the experimental study, the material properties of fabric composite material are firstly investigated Then, the buckling tests are carried out to study the behavior of inflating beams with different air pressure Experimental results are also compared with those obtained from the numerical modeling approach Some major conclusions are drawn from this study could be summarized as follows: - It is shown that the common analytical approach is widely employed to analyze the structural responses of inflating structures Additionally, a great portion of published studies only focused on isotropic materials The studies on orthotropic materials and numerical approaches are still limited This is the main motivation for this research - A numerical approach based on IGA was successfully developed to investigate the stability of inflating beams - The results obtained from the IGA approach are in good agreement with those from traditional FEM In addition, it was found out that the IGA-based approach has a better convergence rate than FEM - From the numerical modeling and experimental results, it is seen that the stability strength of inflating beams increases with the level of the internal pressure - The prediction of the proposed IGA-based numerical model is more reliable in cases that pressure is high, for cases with low pressure, the prediction shows a similar prediction trend with experimental results but the predicted strength is smaller than experimental results - When it comes to the comparison of numerical and experimental results, differences between the two approaches are captured in some cases The differences could be raised due to the shortage of real material information and errors in experimental procedures might cause significant errors in the experimental results, hyper-elastic assumptions, and the ignorance of cross-section deformation 6.2 Further studies The thesis has achieved certain results; however, there are still problems unresolved related to the selection of materials, air-beams producing and measuring 141 CHAPTER 6: CONCLUSIONS AND FURTHER STUDIES methods Therefore, this study could be expanded to those with infilled-gas beams and other inflating structures Different loading conditions and different shapes of the inflating structures could be considered to be investigated in the future For the numerical model, the problems could be extended to those which also considered the failure of composite materials Other modeling techniques, e.i using 3d shell model, could be used as an alternative approach to investigate the response of the inflating beams, especially when the local responses are of interest As the results showed that there are inconsistencies between numerical and experimental results for low-pressure 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ASTM International, “ASTM D638: The Definitive Guide To Plastic Tensile Testing;” Accessed: Nov 22, 2021 [62] ASTM International, “ASTM D4833: Puncture Resistance of Geomembranes and Related Products.” [63] ASTM International, “ASTM D1004 :Tear Resistance of Plastic Film and Sheeting.” [64] ASTM International, “ASTM D1603: Standard Test Method for Carbon Black Content in Olefin Plastics.” 147 List of publications List of Publications Parts of this dissertation have been published in international journals, national journals, or presented in conferences These papers are: • Articles in the international scientific journal T Le-Manh, Q Huynh-Van, Thu D Phan, Huan D Phan, H NguyenXuan “Isogeometric nonlinear bending and buckling analysis of variable thickness composite plate structures” Composite Structures 2017, Pages 818-826 • International Conference Phan Thi Dang Thu, Phan Dinh Huan and Nguyen Thanh Truong “Effect parametric to properties of a 2D orthogonal plain classical woven fabric composite” International Conference on Engineering Mechanics and Automation (ICEMA), Ha Noi city 2014 - ISBN: 978-604-913-367-1, pages 509-517 • National Conference Phan Thi Dang Thu, Phan Dinh Huan and Nguyen Thanh Truong “Biaxial beam inflation test on orthotropic fabric beam”; National Conference on Solid Mechanics, Ho Chi Minh city 2013 - ISBN: 978-604-913-213-1, pages 1169-1176 Nguyen Thanh Truong, Phan Dinh Huan, Phan Thi Dang Thu “Discretizing an analytical inflating beam model by the shell membrane finite element” National Conference on Solid Mechanics, Ho Chi Minh city 2013 - ISBN: 978-604-913-213-1, pages 1221-1228 Phan Thi Dang Thu, Le Manh Tuan, Nguyen Xuan Hung, Nguyen Thanh Truong “Geometrically nonlinear behavior of composite beams of variable fiber volume fraction in isogeometric analysis” National Conference on Solid Mechanics, Da Nang city 2015 - ISBN: 978-604-82-2028-0, Pages: 1404-1409 Thu Phan-Thi-Dang, Tuan Le-Manh, Giang Le-Hieu, Truong NguyenThanh “Buckling of cylindrical inflating composite beams using isogeometric analysis” Proceedings of the National Conference on science and technology in 148 List of publications mechanics IV, Ho Chi Minh City 2015, Viet Nam - ISBN: 978-604-73-3691-3, Pages 821-826 Phan Thi Dang Thu, Nguyen Thanh Truong, Phan Dinh Huan “Mơ hình dầm composite phi tuyến chịu uốn” National Scientific Conference on Composite Materials and Structures, Nha Trang city 2016 - ISBN: 976-604-82-2026-6, Page 699-706 Phan Thi Dang Thu, Nguyen Thanh Truong, Phan Dinh Huan, Le Dinh Tuan “Biaxial experiments for determining material properties and joint strength of textile plain-woven fabric composites” National Conference on Solid Mechanics, Ha Noi city 2017 - ISBN: 978-604-913-722-8, Page 1174-11 149 ... IGA-BASED BUCKLING ANALYSIS OF INFLATING COMPOSITE BEAMS 40 4.1 A brief review on isogeometric analysis .40 4.1.1 The concept of isogeometric analysis 40 4.1.2 Disadvantages of. .. nolinear analysis of axisymmetric inflatable beams was implemented in the work of Eslabbagh [30] The proposed model in that study could simulate the performance of inflatable and inflatable beams, ... dedicated to linear eigen analysis and nonlinear buckling analysis of inflating beams made of orthotropic materials when using isogeometric analysis The influences of geometric nonlinearities