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Thin-Walled Structures 72 (2013) 61–75 Contents lists available at ScienceDirect Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws Numerical buckling analysis of an inflatable beam made of orthotropic technical textiles Thanh-Truong Nguyen a,b,c,d,n, S Ronel a,b,c, M Massenzio a,b,c, E Jacquelin a,b,c, K.L Apedo a,b,c, Huan Phan-Dinh d a Université de Lyon, F-69622 Lyon, France IFSTTAR, UMR_T9406, LBMC, F-69675 Bron, France c Université Lyon 1, Villeurbanne, France d Ho Chi Minh City University of Technology, HoChiMinh City, Vietnam b art ic l e i nf o a b s t r a c t Article history: Received 21 May 2012 Received in revised form 15 May 2013 Accepted 21 June 2013 Available online 17 July 2013 This paper is devoted to the linear eigen and nonlinear buckling analysis of an inflatable beam made of orthotropic technical textiles The method of analysis is based on a 3D Timoshenko beam model with a homogeneous orthotropic woven fabric The finite element model established here involves a threenoded Timoshenko beam element with C0-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 load 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, while in the nonlinear buckling problem, the tangent stiffness matrix includes the effect of changing the geometry as well as the effect of the stress stiffening The nonlinear finite element solutions were investigated by using the straightforward Newton iteration with the adaptive load stepping for tracing the 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 the soundness of the results, a 3D thin-shell finite element model was used for comparison For a further validation, the results were also compared with those from experiments at low inflation pressures & 2013 Elsevier Ltd All rights reserved Keywords: Inflatable beams Orthotropic fabric Inflation pressure Linear eigen buckling Nonlinear buckling Introduction Finite element (FE) analyses of inflated fabric structures present a challenge in that both material and geometric nonlinearities arise due to the nonlinear load/deflection behavior of the fabric (at low loads), pressure stiffening of the inflated fabric, fabric-tofabric contact, and fabric wrinkling on the structure surface In addition to checking fabric loads, the finite element model is used to predict the fundamental mode of the inflated fabric beam In general, the FE analyses of thin-walled structures can be almost performed by many robust FE package such as ABAQUS and ANSYS However, in reality, the built-in shell and membrane elements are not very suitable for describing the inflatable structure applications with an orthotropic fabric The shell elements are quite rigid, while membranes are too flexible n Corresponding author at: Université de Lyon, F-69622, Lyon, France Tel.: +33 72 65 64; fax: +33 72 65 53 54 E-mail addresses: thtruong@hcmut.edu.vn, thtruong@gmail.com (T.-T Nguyen) 0263-8231/$ - see front matter & 2013 Elsevier Ltd All rights reserved http://dx.doi.org/10.1016/j.tws.2013.06.014 This leads to the need for the numerical models with a specific element appropriates for inflated fabric structures In the reviewed literature, only the inflated tensile structures have been addressed and, the response of an inflated lightweight structure to service loads has been examined Papers in this category generally assume homogeneous isotropic and orthotropic material properties for the inflated structure and employ the membrane or thin shell theory to determine the structural response In earlier work, Libai and Givoli [1] derived the equations governing the incremental state of stress in an orthotropic circular membrane tube The membrane in this study was taken to be hyperelastic and was not specified in detail The changes in loading, including uniform internal pressure and longitudinal extension, are regarded as small perturbations on the initial homogeneous state of stress The approach was based on the linearization of the equations about a known homogeneous reference state The rectangular elements with Hermite cubic shape functions were used in conjunction with the variational principles Wielgosz and Thomas [2–4] developed an inflatable beam finite element and 62 T.-T Nguyen et al / Thin-Walled Structures 72 (2013) 61–75 Fig HOWF inflatable beam used it to compute the deflection of hyperstatic beams The element used is a membrane one Then, Bouzidi et al [5] described two finite elements for 2D problems of inflatable membranes: axisymmetric and cylindrical bending The elements built with the hypothesis of large deflections, finite strains and with follower pressure loading The numerical solution is obtained by solving directly the optimization problem formulated by the theorem of the minimum of the total potential energy Also, by employing membrane elements and experimental results, Cavallaro et al [6] showed that pressurized tube structures differ fundamentally from conventional metal and fiber/matrix composite structures This study led to a note that while the plainwoven fabric appeared to be an orthotropic material, the fabric does not behave as a continuum, but rather as a discrete assemblage of individual tows, whose effective material properties depend on the internal pressure of the beam, weave geometry and the contact area of interacting tows Suhey et al [7] presented the finite element model of an inflatable open-ocean-aquaculture cage using membrane elements with assuming that the material is anisotropic The authors used nonlinear elements to model the tension-only behavior of the fabric material in order to calculate the magnitudes of the deflection and the stress at the onset of wrinkling The results were verified by the modified conventional beam theory [8,9] Le van and Wielgosz [10,11] obtained the numerical results with a beam element developed from the earlier work of Fichter [12] and the 3D isotropic fabric membrane finite element In their approach, the governing equations were discretized by the use of the virtual work principle with Timoshenko's kinematics, finite rotations and small strains The linear eigen buckling analysis were carried out through a mesh convergence test using the 3D membrane finite element computations Last recently, Apedo et al [13] investigated linear and nonlinear finite element solutions in bending by discretizing nonlinear equilibrium equations obtained from his previous analytical model in which a homogeneous orthotropic fabric was considered In inflatable structures, with the arising of the local buckling that leads to the formation of the wrinkles, nonlinear problems pose the difficulty of solving the resulting nonlinear equations that result Problems in this category are geometric nonlinearity, in which deformation is large enough that equilibrium equations must be written with respect to the deformed structural geometry Few works have dealt with buckling analysis of inflated structures By means of the total Lagrangian formulation developed by Le van and Wielgosz [10,11], Diaby et al [14] proposed a numerical computation of buckles and wrinkles appearing in membrane structures The bifurcation analysis is carried out without assuming any imperfection in the structure In consideration of an inflatable beam, Davids and Zhang [15] developed a quadratic Timoshenko beam element based on an incremental virtual work principle that accounts for fabric wrinkling via a momentcurvature nonlinearity However, in these studies, the materials were assumed to be isotropic This paper is devoted to the linear eigen and nonlinear buckling analysis of simply supported inflatable beam made of orthotropic technical textiles The method of analysis is based on a 3D Timoshenko beam model with a homogeneous orthotropic woven fabric (HOWF) The finite element model established here uses a three-noded Timoshenko beam element with C0-type continuity for the transverse displacement and quadratic shape functions for the bending rotation as well as the axial displacement The effects of geometric nonlinearities and the inflation pressure on the stability behavior of inflatable beam are assessed: a simply supported beam is studied The influence of the beam aspect ratios on the buckling load coefficient are also pointed out A 3D thin-shell finite element model is then utilized for comparison Finally, the obtained results are also compared with experimental results Governing equations In this section the governing equations of a 3D Timoshenko beam with a HOWF are briefly presented The Green-Lagrange strain measure is used due to the geometrical nonlinearities Fig shows an inflatable cylindrical beam made of a HOWF l0, R0, t0, A0 and I0 represent respectively the length, the fabric thickness, the external radius, the cross-section and the moment of inertia around the principal axes of inertia Y and Z of the beam in the reference configuration which is the inflated configuration A0 and I0 are given by A0 ¼ 2πR0 t ; I0 ¼ A0 R20 ð1Þ ð2Þ where the reference dimensions l0, R0 and t0 depend on the inflation pressure and the mechanical properties of the fabric [16]: l0 ẳ l ỵ pR l 12lt Þ 2Et t ϕ ð3aÞ pR2ϕ ð2−νlt Þ 2Et t 3bị R0 ẳ R ỵ t0 ẳ t 3pRϕ ν 2Et lt ð3cÞ in which lϕ , Rϕ , and t ϕ are respectively the length, the fabric thickness, and the external radius of the beam in the natural state The internal pressure p is assumed to remain constant, which simplifies the analysis and is consistent with the experimental observations and the prior studies on inflated fabric beams and arches [8–12,14,15,17–20] The initial pressurization takes place prior to the application of concentrated and distributed external loads, and is not included in the structural analysis per se M is a point on the current cross-section and G0 the centroid of the current cross-section lies on the X-axis The beam is T.-T Nguyen et al / Thin-Walled Structures 72 (2013) 61–75 undergoing axial loading Two Fichter's simplifying assumptions are applied in the following [12]: the cross-section of the inflated beam under consideration is assumed to be circular and maintains its shape after deformation, so that there are no distortion and local buckling and the rotations around the principal inertia axes of the beam are small and the rotation around the beam axis is negligible Due to the first assumption, the model considers that no wrinkling occurs so that the ovalization problem is not addressed in this paper as done in many previous papers [10,12] 63 Hookean stress-strain relationships S ẳ S0 ỵ ẳ S0 ỵ C E ∂E ð8Þ where S is the inflation pressure prestressing tensor, the second Piola–Kirchhoff tensor is written in the beam coordinate system as SXX SXY SYY S ẳ4 symmetrical SXZ SYZ 9ị SZZ 2.1 Kinematics The material is assumed orthotropic and the warp direction of the fabric is assumed to coincide with the beam axis; thus the weft yarn is circumferential The model can be adapted to the case where the axes are in other directions In this case, an additional rotation may be operated to relate the orthotropic directions and the beam axes This general case is not addressed here because, for an industrial purpose, the orthotropic principal directions coincide with the longitudinal and circumferential directions of the cylinder [21] The displacement components of an arbitrary point MðX; Y; ZÞ on the beam are [22,23] 9 9 u > < X> = > < −YθZ ðXÞ > = < ZθY ðXÞ > = > < uXị > = > vXị uMị ẳ u Y ẳ 4ị ỵ ỵ > > > :u > ; > : ; : wðXÞ > ; > : ; > 0 Z where u X , u Y and u Z are the components of the displacement at the arbitrary point M, while u(X), v(X) and w(X) correspond to the displacements of the centroid G0 of the current cross-section at abscissa X, related to the base ðX; Y; ZÞ; θY ðXÞ and θZ ðXÞ are the rotations of the current section at abscissa X around both principal axes of inertia of the beam, respectively The definition of the strain at an arbitrary point as a function of the displacements is E ẳE ỵE l 5ị nl where El and Enl are the Green-Lagrange linear and nonlinear strains, respectively The nonlinear term Enl takes into account the geometrical nonlinearities The strain field depends on the displacement field as follows: T ∂uX > > > > u ;X u ;X > > > > ∂X > > > > > > > > T > > > > ∂uY > > > > u u > > > > ;Y ;Y > > > > ∂Y > > > > > > > > T > > > > ∂u Z = = < < u u ;Z ;Z Z 6ị E ẳ uX uY ; E ẳ T T l nl > > > > u u ỵ u u ỵ > > X > > > > ;X ;Y ;Y ;X > > ∂Y > > > > > > > ∂uX ∂uZ > T T > > > > > > > > > > Z ỵ X > u ;X u ;Z ỵ u ;Z u ;X > > > > > > > > > ; > : uY ỵ uZ > > > T T ; : u u ỵ u u Z ∂Y ;Y ;Z ;Z ;Y The higher-order nonlinear terms are the product of the vectors that are defined as 9 u u u > > > < X;X > = < X;Y > = < X;Z > = 7ị u ;X ẳ uY;X ; u ;Y ¼ uY;Y ; u ;Z ¼ uY;Z > > > > > :u ; :u ; :u > ; Z;X Z;Y Z;Z 2.2 Constitutive equations In the present work, the Saint Venant–Kirchhoff orthotropic material is used The energy function E ẳ E ị in this case is also known as the Helmholtz free-energy function The components of the second Piola–Kirchhoff tensor S are given by the nonlinear C is the elasticity tensor expressed in the beam axes In general, the inflation pressure prestressing tensor is assumed spheric and isotropic [10] So S ¼ S0 I ð10Þ where I is the identity second-order tensor and S0 ¼ N =A0 is the prestressing scalar The elasticity tensor in the beam axes was transformed from the orthotropic l; t basis (see [16]) c2 C 12 s2 C 12 csC 12 0 C 11 0 c4 C 22 c2 s2 C 22 c3 sC 22 cs C 22 0 s C 22 C ¼6 0 c2 s2 C 22 s C 66 csC 66 symmetrical c2 C 66 ð11Þ where c ¼ cos φ and s ¼ sin φ with ẳ eZ ; nị being the angle between the Z-axis of the beam and the normal of the membrane at the current point (Fig 1) The tensor components are described as a function of the mechanical properties of the HOW fabric: C 11 ẳ El =1lt tl ị; C 12 ẳ El tl =1lt tl ị; C 22 ẳ Et =1lt tl ị; C 66 ẳ Glt and El =νlt ¼ Et =νtl : Finite element formulations 3.1 Linear eigen buckling In case of linear buckling analysis, the beam is subjected to the inflation pressure prestressing S tensor The first step is to load the inflated beam by an arbitrary reference level of external load, fFref g and to perform a standard linear analysis to determine the finite element stresses in the beam In this case, a linear finite element inflatable beam (LFEIB) model is proposed It is desirable to also have a general formula for finite element stress stiffness matrix ½ks and finite element conventional elastic stiffness matrix ½k (before loading) [24] Matrix ½ks , which augments the conventional elastic stiffness matrix ½k, is a function of the element geometry, the displacement field, and the state of membrane stress To introduce the stress stiffness matrix and the bifurcation buckling calculation of a finite element model of an inflatable beam, an analysis based on the energy concept is considered The strain energy of the beam per unit volume is 12S T E Using Eqs (6)– (11) and integrating through the volume of the beam with respect to the cross-sectional area A0 and the length l0, an expression for 64 T.-T Nguyen et al / Thin-Walled Structures 72 (2013) 61–75 the strain energy of a finite inatable beam is Z Ue ẳ fS0 ịT E þ E T Á C Á E gdV ¼ U m ỵ U b V0 12ị where Um is the change in membrane energy and Ub is the strain energy in bending In order to derive the element stiffness matrices for the beam, a displacement eld ẵu ẳ ðu; v; w; θY ; θZ Þ needs to be interpolated within each element For the use of element for inflatable beam, it is noted that the two-noded element often used for Euler– Bernoulli kinematics with Hermite polynomial as shape functions [25], or a higher order element such as the three-noded quadratic beam with reduced integration [11] or the three-noded Timoshenko beam that has quadratic shape functions for transverse displacement and linear shape functions for bending rotation and axial displacement [15,26] In the present analysis, a three-noded Timoshenko beam element with C0-type continuity is used The element has three nodes with five degrees of freedom (d.o f) at each node Nodal d.o.f fdg defines d.o.f vector ⌊uj vj wj θYj θZj ⌋ of an element That is 8 > ∑3j ¼ Nj uj > > > > > u > > > > > > > > > > > > > > > ∑ N v > > > > j¼1 j j > > v > > > = < = < w ẳ j ẳ N j wj ẳ ẵNfdg ð13Þ > > > > > > > > > ∑3 N θ Y > > θY > > > > > j> > > > j¼1 j > > > ; > > :θ > > > > Z ; : ∑j ¼ N j θZ j > where index j in summations runs from to for three-noded element, and ½N the shape function matrix For the chosen element, the shape function matrix which can be found in [27] is ⌊N⌋ ¼ ⌊N N N ẳ 121ị 12 ỵ 1ị ð14Þ e ðð2=l0 ÞX−1Þ, where ξ is simply the dimensionless axial coordinate ẳ with ẵ1; and X is the local coordinate along the beam element e e axis (X∈½0; l0 ), l0 is the element reference length It should be noted that the strain energy component Um is associated with the stress stiffness matrix ½ks of the beam and Ub relates to the conventional elastic stiffness ½k of the beam, as T U m ẳ 12 ẵd ẵks ẵd; T U b ẳ 12ẵd ẵkẵd 15ị By applying the discretization procedure, Eq (12) then becomes T U e ẳ 12fdg ẵk ỵ ẵkref ịfdg 16ị where is the proportionality coefficient such as F ¼ λF ref , with F is the axial load The coefficients in matrices ½k and ½kref are constant and only are dependent on the geometry, material properties and the inflation pressure prestressing conditions acting on the beam In this study, the stiffness matrices are evaluated using the Gauss numerical integration scheme and the assembly of element stiffness matrix for the entire structure leads to the equilibrium matrix equation in global coordinates For the whole beam, the potential energy is simply the summation of the potential energies of the individual finite elements and the whole structural matrices are generated by following the standard FEM assembly procedure Thus, the potential energy for the whole structure can be expressed as U ẳ 12fDgT ẵKẵKref ÞfDg ð17Þ The vector fDg includes the degrees of freedom for the whole beam Because the problem is presumed linear, the conventional stiffness matrix ½K is unchanged by loading Let buckling displacements fδDg take place relative to displacements fDg of the reference configuration The structural equilibrium equations can be obtained by applying the principle of minimum potential energy This gives an eigenvalue problem in the form: ẵK ỵ i ẵKref ịfDg ẳ f0g 18ị Eq (18) is an eigenvalue problem where λi is the eigenvalue of first buckling mode The smallest root λcr defines the smallest level of external load for which there is bifurcation, namely fFgcr ¼ λcr fFgref ð19Þ As the beam is loaded by an arbitrary reference level of external load fFgref , the eigenvector fδDg associated with λcr is the buckling mode The magnitude of fδDg is indeterminate in a linear buckling problem, so that it defines a shape but not an amplitude 3.2 Nonlinear buckling Let us consider geometrically nonlinear behavior of HOWF inflatable beam made of presumed linear elastic material A nonlinear inflatable beam finite element (NLIBFE) model is introduced The total Lagrangian approach is adopted in which displacements refer to the initial configuration, for the description of geometric nonlinearity Accordingly, we can form a tangent stiffness matrix ½KT , which includes the effect of changing geometry as well as the effect of inflation pressure The axial load at ith increment is calculated by ff i g ẳ ff i1 g ỵ iffg 20ị For a given element, the nonlinear equilibrium equation can be formulated as ẵkT fdg ẳ ff i g 21ị where ½kT is the element tangent stiffness matrix, ff i g is the external load increments vector of an element and fΔdg an unknown displacement increment to be solved for After assembling over all the elements in the model, the following equilibrium equation is obtained: ẵKT fDg ẳ fFi g ð22Þ Eq (22) can be solved by an incremental scheme based on the straightforward Newton using nodal load increments fΔFg, with load correction terms and updates of ½KT after each incremental step Here, the model displacement vector fDgi ¼ fDgi1 ỵ fDg, where fDg is the nodal unknown displacement increment at increment step i and fDgi−1 is the nodal beam displacement vector from the previous solution step The equilibrium solution tolerance was taken as ‖fΔDgi ‖ ¼ ðfΔDgTi fΔDgi Þ1=2 ≤0:0001 ð23Þ or ‖fRgi ‖ ¼ ðfRgTi fRgi Þ1=2 0:0001 24ị with fRi g ẳ fRDi1 ịg ẳ ẵKT fΔDi g being the global unbalanced residual force vector from the previous increment As a limit point is approached, displacement increments fΔDg become very large At either a limit point or a bifurcation point, ½KT becomes singular 3.3 Implementation of an iterative algorithm for solving the NLIBFE model In the following section, the iterative procedure using the straightforward Newton–Raphson iteration with adaptive load stepping for solving the nodal displacement incrementation solution fΔDg is summarized Suppose that at increment ði−1Þ, one obtained an approximation fDi−1 g of the solution as the residual is not zero fRDi1 ịg ẳ fFgẵKDi1 ịfDi1 g≠f0g ð25Þ T.-T Nguyen et al / Thin-Walled Structures 72 (2013) 61–75 65 Table Input parameters for modeling NLIBFE model Parameter type Input Physical interpretation Value Material properties El Et Glt νlt Young's modulus in the warp direction Young's modulus in the weft direction In-plane shear modulus Poisson's ratio due to the loading in the l direction and contraction in the t direction Poisson's ratio due to the loading in the t direction and contraction in the l direction See Table νtl Beam geometry (in the natural state) lϕ Rϕ tϕ Length of the inflatable beam External radius of the inflatable beam Thickness of the inflatable beam See Table External load p FX fF i g ninc Inflation pressure Concentrated load in the X-axis Increment load vector Number of load increments 10–200 (kPa) 1500 (N) Model description ne le Number of elements Element length en nn ndof edof gdof m Number Number Number Number Number Number At increment step i, one seeks an approximation fDi g of the solution such that fRDi ịg ẳ fRDi1 ỵ Di Þg≈f0g ð26Þ The algorithm is obtained by using the first-order Taylor series in the vicinity of fDi g ! ∂R fDi g ẳ f0g 27ị fRDi1 ỵ Di ịg ẳ fRDi1 ịg ỵ D D ẳ Di1 The NLIBFE model with linearized and incremental iterative schemes is implemented using the numerical computing package MATLAB At the FE structural level (Algorithm 1), an iterative equation solution is also performed During this structural loop, the incremental-iterative algorithm will be called at each material (Gaussian) point In every loop within an incremental load step ΔF, the beam parameters (Table 1) and the boundary conditions are prescribed, which are the input variables to the global level routine (Algorithm 2) The output from this global level routine is Eq (22) which is solved iteratively in the structural level In the element level subroutine (Algorithm 3), tangent stiffness matrix ½KeT and load vectors (fFeint g and fFeext g) are computed for each element The superscripts (i,k,m) denote the global counter within the current incremental load step, the number of elements and the number of Gauss integration points, respectively After i load step(s), the converged displacement solution fΔDi g at the current load ΔF will be used to provide incremental displacement for the next load step At the material level, the convergence criterion can be defined using Eq (23) or Eq (24) which is expressed in terms of the displacement vectors or the residual vectors, respectively Applications and results In the following section, some representative analyses are carried out and the results are presented It is noted that in all cases under consideration, the convergence study with regard to the number of elements is accomplished before extracting the results 10 lo ne 2ne þ en Á ndof ndof Á nn of nodes per element of nodes in global degrees of freedom per node degrees of freedom per element of global degrees of freedom of Gauss integration points Table Data set for inflatable beam  10−4 0.5 0.14 Natural thickness, t ϕ (m) Correction shear coefficient, ky Natural radius, Rϕ (m) Natural length, lϕ (m) Orthotropic fabric's mechanical properties Young's modulus in the warp direction, El (MPa) Young's modulus in the weft direction, Et (MPa) In-plane shear modulus, Glt (MPa) Poisson's ratio, νlt Poisson's ratio, νtl Material [37] Material [38] 2609 19 300 2994 14 240 1171 0.21 0.18 6450 0.28 0.22 A simply supported beam loaded by a compressive concentrated F is studied The slenderness ratio is ffiλs ¼ L=ρ, where L ¼ μl0 pffiffiffiffiffiffiffiffiffiffiffi is the beam effective length and ρ ¼ I =A0 is the beam radius of gyration The coefficient μ is equal to in the case of a simply supported beam The beam geometry and two materials described in Table are used For each case of material, a range of inflation pressure normalized as pn ¼ p=pcr , in which pcr ¼ ðEl t 3ϕ =4R3ϕ ð1−νlt νtl ÞÞ are considered in the analyses [16,28,29] The pressure values and the associated normalized pressure are listed in Table 4.1 Linear eigen buckling The linear buckling analysis of a simply supported LFEIB under compressive concentrated load is performed to derive the critical load parameters In order to assess the influence of the inflation pressure, the inflatable beam is pressurized by the normalized pressures corresponding to 10 kPa, 50 kPa, 150 kPa and 200 kPa for two cases of material (Table 3) To examine the linear eigen buckling behavior, the normalized linear buckling load coefficient (K lc ¼ 105  scr =Eeq ) proposed by Ovesy and Fazilati [30] is introduced, in which scr is the linear buckling critical stress of the beam 66 T.-T Nguyen et al / Thin-Walled Structures 72 (2013) 61–75 pffiffiffiffiffiffiffiffiffi and Eeq ¼ El Et is the equivalent Young's modulus of the current material [31,32] The beam radius of curvature in this loading case is given by ỵ v2;X ị3=2 v;XX Rb ẳ 28ị Nguyen et al [32] have recently proposed that an expression of deflection v(X) corresponds to the buckling mode shapes: vðXÞ ẳ F p ỵ C 0s B sin Xị F p F ỵ C 0s 29ị where F p ¼ pπR20 is the pressure force due to the inflation pressure, C 0s ¼ 12ky A0 C 66 (with ky ¼ 0.5, a correction of the shear coefficient) and B is an arbitrary constant The quantity Ω in this load case can be written for the fundamental buckling mode as ẳ l0 30ị In this loading case, Rb is determined at X ¼ l0 =2 From Eqs (28) to (30), the radius of curvature of a simply supported HOWF inatable beam becomes F ỵ C Fịl p 0 s Rb ¼ ðF p þ C 0s ÞBπ ð31Þ Table Normalized pressure ðpn Þ for different values of internal pressure ðpÞ used in the study p (kPa) As shown in Fig 2, the convergence studies on the normalized buckling coefficient Klc of LFEIB model described in Section 3.1 reveal that about six elements are sufficient to obtain converged results These results are in a good agreement with those derived by an analytical approach in [32] (see Table 4) It can be seen that the results using the analytical method vary in a larger range than those predicted using the LFEIB model with the given inflation pressures The differences between the results are within 8.67% and 3.50% in the case of materials and 2, respectively Based on the converged results, the variation of normalized buckling coefficient Klc with the change in radius-to-thickness ratio (Rrt) for a HOWF inflatable beam is depicted in Fig The beams are considered to be of m length, 0.14 m radius and varying fabric thicknesses to change the radius-to-thickness ratio Rrt It is noticeable that in both cases of material, the normalized pressure has an increased effect on normalized buckling load coefficient Klc at high ratio Rrt Further, the buckling load coefficient Klc gradient as a function of Rrt depends on the normalized pressure pn: at higher of pn, the gradient of Klc becomes larger The effect of material properties is noticeable only on the magnitude but not on the gradient of Klc These results highlight the importance of the fabric thickness: a thicker Table Buckling coefficient (Klc) comparison between LFEIB results and analytical solutions Normalized buckling coefficient, Klc Material Normalized pressure pn Material Material 10 20 30 40 50 324 648 972 1295 1619 43 85 128 171 214 100 150 200 3238 4858 6477 427 640 854 Analytical model LFEIB model [32] (present) 400 324 1619 4858 6477 305.6 314.9 339.0 351.5 333.3 336.4 344.4 348.4 8.67 6.60 1.58 0.89 43 214 640 854 409.8 412.0 417.5 420.2 424.4 425.2 427.1 428.0 3.50 3.15 2.27 1.84 490 p1 = 10 kPa, p = 324 Material p1 = 10 kPa, p = 43 n Material p2 = 50 kPa, pn = 1619 p3 = 150 kPa, p = 4858 n p2 = 50 kPa, pn = 214 p3 = 150 kPa, p = 640 n 390 n 480 p4 = 200 kPa, pn = 854 l Normalized buckling coefficient, Kc p4 = 200 kPa, pn = 6477 l Normalized buckling coefficient, Kc Difference (%) 380 370 360 350 340 330 470 460 450 440 430 Number of elements 10 420 10 Number of elements Fig Linear eigen buckling: mesh convergence test of normalized linear buckling load coefficient (K lc ¼ 105  scr =Eeq ) for a simply supported LFEIB model T.-T Nguyen et al / Thin-Walled Structures 72 (2013) 61–75 67 450 p1 = 10 kPa, p = 324 Material n n p2 = 50 kPa, p = 214 n n p3 = 150 kPa, pn = 4858 p3 = 150 kPa, pn = 640 p4 = 200 kPa, p = 6477 p4 = 200 kPa, p = 854 n n 445 Normalized buckling coefficient, Kcl Normalized buckling coefficient, Kcl p1 = 10 kPa, p = 43 Material p2 = 50 kPa, p = 1619 440 420 400 380 360 440 435 430 340 425 500 1000 1500 2000 500 Radius−to−thickness ratio, Rrt 1000 1500 2000 Radius−to−thickness ratio, Rrt Fig Linear eigen buckling: normalized buckling load coefficient (K lc ¼ 105  scr =Eeq ) versus radius-to-thickness ratio (Rrt ¼ R0 =t ) for a simply supported LFEIB model 1200 1200 p1 = 10 kPa, p = 324 p1 = 10 kPa, p = 43 n n p2 = 50 kPa, p = 1619 p2 = 50 kPa, p = 214 n 1100 n 1100 p3 = 150 kPa, pn = 4858 p3 = 150 kPa, pn = 640 p4 = 200 kPa, p = 6477 p4 = 200 kPa, p = 854 n l Normalized buckling coefficient, Kc l Normalized buckling coefficient, Kc n 1000 900 800 700 600 500 400 1000 900 800 700 600 500 400 Material 300 10 15 20 Material 25 30 35 Slenderness ratio, λs 300 10 15 20 25 30 35 Slenderness ratio, λs Fig Linear eigen buckling: normalized buckling load coefficient (K lc ¼ 105  scr =Eeq ) versus slenderness ratio (λs ¼ L=ρ) for a simply supported LFEIB model yarn with a higher yarn number will all result a stronger fabric (lower values of Rrt) [33] For inflatable beam made of a stronger fabric, one can increase the beam load-carrying capacity by pressurizing to higher pressures due to its high resistant to the internal pressure Fig clearly shows the large variation in normalized buckling load coefficient Klc with the change in the slenderness ratio λs of HOWF inflatable beam The beam is considered to be of 0.14 m radius and the length varied from 0.5 to m to change the slenderness ratio Here, except for the comments concerning the beam slenderness made above, the effect of normalized pressure pn is only noticeable at low values of slenderness ratio in the case of material The normalized pressure only shows its role more evident at larger of beam radii R0 (lower values of λs ) In the case of high moduli material, the influence of inflation pressure is not noticeable In order to assess the flexural stability that depends on the bending radius ratio, we define the bending radius ratio Rbr as Rb B=2R0 , which is the ratio of the radius of curvature Rb to the outer radius R0 of the beam before bending The normalized coefficient Klc is then plotted against Rbr ratio (Fig 5) Regarding the normalized coefficient Klc, the discrepancies due to the material properties are rather large between the results: the variations of Klc with the change in Rbr ratio between both materials are within 23.95% and 21.08% in the case of lowest and highest normalized pressure, respectively Conversely, the discrepancies due to the effect of normalized pressure are small: the variation of Klc versus Rbr ratio is 3.56% in the case of material and reduced to 0.66% in the case of material These results shown that the inflation pressure has a decreasing effect in the case of high moduli material In other words, the inflation pressure only plays a dominant role when the fabric mechanical properties are poor 68 T.-T Nguyen et al / Thin-Walled Structures 72 (2013) 61–75 nonlinear load parameter at ith increment of axial load is defined by 4.2 Nonlinear buckling of a simply supported NLIBFE model In this section, the nonlinear buckling of a simply supported beam is investigated by the procedure proposed in Section 3.2 The numerical examples contain large deformation analyses of NLIBFE model and illustrate the performance of the derived algorithm Hence, a simply supported NLIBFE model subjected to an axial compressive load F (Fig 6a) is solved for tracing the beam response curves Here are the solutions of transverse displacement These solutions are normalized by the ratio-to-deflection of the beam The critical load calculated in the linear buckling analysis above is appropriate only if there is little or no coupling between membrane deformation and bending deformation This is probably the case for initially straight beams Consider Fig 6b, in which a small initial imperfection is introduced: either a slight initial curvature or a slight eccentricity of the compressive load F With the increasing initial imperfections, the beam implies large displacements rather than buckling Hence, a linear bifurcation analysis may overestimate the actual collapse load Generally, nonlinear analysis is more appropriate, so that the coupling of membrane and bending actions is taken into account from the outset Eventual collapse maybe associated with bifurcation of with reaching a limit point, or collapse might be defined as excessive deflection It was noted that the axial load was divided into 10 increments to calculated the displacement increments The normalized Mat − p1 = 10 kPa, p = 324 440 Mat − p1 = 10 kPa, p = 43 Material Mat − p2 = 50 kPa, p = 1619 K nl c ¼ 10  Fi Eeq A0 The model is made up of the materials and as defined in Table The three-noded quadratic element as given in Eq (14) is used The deflection solutions Dv along the Y axes obtained from the NLIBFE model are considered as the change in the flexionto-radius ratio (Rfr) as Dv =R0 , whereas the axial displacement solutions Du along the X axes are referred to the change in the length-to-radius ratio (Rlr) as Du =R0 For the same normalized pressure and material properties, the smaller values of Rlr and Rfr represent the more stable beam But first we have to determine the limit of validity of this model 4.2.1 Wrinkling loads and maximum deflections: limit of validity for numerical solutions The non-linear model is valid until the onset of wrinkles From the criterion of wrinkling (non-negative principal stress), we deduce the wrinkling load Fw which corresponds to the onset of wrinkles Experimentally, we see that there are always wrinkles before the loss of stability of the beam Thus, the model is theoretically valid until the load Fw (Fig 7) We will see in the next, until a certain pressure level, these first wrinkles have a weak influence Because of this, the model is in practice valid even beyond Fw from a certain pressure level The determination of the load-carrying capacity of an inflatable beam can be achieved by means of the analysis based on the so-called critical bounds, which are the wrinkling, buckling and crushing bound The wrinkling bound and the buckling bound Buckling load coefficient, Kcl Mat − p2 = 50 kPa, p = 214 420 Mat − p3 = 150 kPa, p = 4858 Mat − p3 = 150 kPa, p = 640 Mat − p4 = 200 kPa, p = 6477 400 Mat − p4 = 200 kPa, p = 854 380 360 Material 340 320 0.91 0.915 0.92 0.925 0.93 0.935 Bending radius ratio, Rbr Fig Linear eigen buckling: normalized buckling load coefficient (K lc ¼ 105  scr =Eeq ) versus bending radius ratio (Rbr ¼ Rb B=2R0 ) for a simply supported LFEIB model ð32Þ Fig Wrinkling load: limit of validity of numerical solutions Fig (a) Inflatable beam subjected to compressive axial load F (b) The effect of an initial imperfection T.-T Nguyen et al / Thin-Walled Structures 72 (2013) 61–75 indicate the values of applied compressive load Fw when local buckling (wrinkles) and lateral (global) buckling of the beam walls appear, respectively; while the crushing bound corresponds to the crushing force [11] when the load F reaches the pressure force due to the inflation pressure Fp The beam subjected to the compressive load F reaches in turn the wrinkling bound and buckling bound before the collapse due to stress limitation occurs As for the crushing bound, the correlation between the values of buckling load and crushing load depends on the internal pressure and the slenderness of the beam For a high enough inflation pressure, the crushing load of an inflatable beam is more meaningful in the case of small slenderness and vice versa From the beam response curves traced by solving the simply supported NLIBFE model, the wrinkling bound can be determined by a combined method, which is proposed based on the previous results obtained in the theoretical analysis study done by Apedo et al [16] with the same assumptions of materials and beam geometry To identify the point of the beam where the first wrinkles appear, the buckling test as shown in Section 4.2.3 was performed The beam was loaded until the first wrinkles appear The test shown that the wrinkling phenomenon was initiated at the middle of the beam (X ¼ l0 =2, Y ¼ R0 , Z ¼ 0) corresponding to the first buckling mode The stress criterion was adopted to obtain the wrinkling loads adapted to the present model [7,10,16,17,34,35] With the linearization which is performed, the non-negative principal stress is summarized in the non-negative axial stress SXX as [16] SXX ẳ S0XX ỵ C 11 EXX ỵ c2 C 12 EYY ỵ s2 C 12 EZZ ỵ 2csC 12 EYZ 33ị By linearizing Eq (33) around the prestressed reference configuration and due to in-plane buckling loading conditions (N ¼ F p −F), the following expression is obtained: l0 F p F l0 ; R0 ; ẳ 34ị S1 −R0 C 11 θZ;X A0 2 As the axial stress SXX reaches the maximum value, i.e SXX ¼ 0, the first wrinkles appear The axial load F is then the wrinkling load Fw The rotation expression for an HOWF simply supported inflatable beam and loaded with a compressive load at one end was obtained by Nguyen et al [32]: Z Xị ẳ B cos Xị 35ị Together with Eq (30) applied in this loading case, the wrinkling load of the current model is then Fw ¼ Fp− 2 2π R t C 11 B l0 ð36Þ in which, the arbitrary constant B depends on the internal pressure, the mechanical properties of the fabric and the slenderness ratio (λs ) The wrinkling bound can be obtained indirectly by determining experimentally the constant B 4.2.2 Validation of the NLIBFE model: the reference model In this section, the NLIBFE model is validated by comparing the normalized nonlinear load parameter Knl c with those from a shell finite element model The nonlinear shell finite element (NLSFE) models are developed by using the general-purpose FE package ABAQUS/Standard ABAQUS S4R shell element with reduced integration is selected for the NLSFE model as it is efficient and reliable for large displacements, large rotations and finite membrane strains [36] The S4R element is a four-node element Each node has three displacements and three rotation degrees of freedom Each of the six degrees of freedom uses an independent bilinear interpolation function An element size of 30 mm lateral and longitudinal direction for both the body and the ends of NLSFE beam model The reduced integration is used in order to avoid 69 Fig Global beam axes and local material orientation assigned for a orthotropic fabric shear locking that usually occurs in fully integrated elements, to reduce the time necessary for the analysis but gives about the same results The built of materials and 2, as given in Table The material orthotropic is assigned a local material orientation as shown in Fig The analysis involves three steps in which the initial step specifies the initial conditions for particular nodes or elements The inflation pressure is introduced into the model by a general static procedure in the second step The buckling stresses are then calculated relative to the base state of the beam It is noted that, because geometric nonlinearity was included in the general analysis steps prior to the eigenvalue buckling analysis, the base state geometry is the deformed geometry at the end of the second step In this study, the base state is the pressurized state and the analysis must take into account the nonlinear effects of large deformations and displacements The eigenvalues and their modes are then obtained by a buckling analysis in the linear perturbation procedure in the third step The subspace eigensolver is chosen for 10 eigenvalues requested The description of modeling the reference beam is as follows: The two ends of the beam are constrained dependent on the boundary conditions (Fig 9a) As mentioned above, only the case of simply supported beams are considered The loads were introduced in two steps In the first step, the beam was inflated by an internal pressure p which is normalized as pn Then, a concentrated compressive load was applied at one end of the beam in the second step (see Fig 9b) The beam model was meshed by the automatic meshing capability of the package as shown in Fig 9c The model consists of 1764 elements and 1762 nodes All the elements are S4R, a four-node general-purpose shell element, reduced integration with hourglass control and finite membrane strains The critical stresses are ðλi P i Þ, where λi and Pi are the lowest eigenvalues and the incremental stress pattern, respectively For a better comparison the critical results obtained from the models are normalized to the nonlinear load parameter Knl c Recall that the beam theories developed in this study are valid only when local buckles (wrinkles) of the beam fabric have not yet appeared The local buckles are caused by the insufficient inflation pressure or the very high load increments And it is also noted that the material properties govern directly the buckling coefficient and only slightly affect the reference dimensions, which provide 70 T.-T Nguyen et al / Thin-Walled Structures 72 (2013) 61–75 Based on the properties given in Table 2, a comparison between the NLIBFE model and the NLSFE model (the reference model) is performed as shown in Table It is noticeable that, at high normalized pressure, the minimum difference showed a good agreement between the models However, a poor agreement was demonstrated in a low inflation pressure case, particularly for material due to ABAQUS that could handles the complexity of the highly nonlinear models with its robust solver while in the NLIBFE model, one attempts to treat the nonlinear problem by solving the linearized behavior of the model This leads to the cumulative errors in the solutions between the models for low normalized pressures The maximum difference between the models are 114.74% and 71.84% in the case of materials and 2, respectively, even at the lowest normalized pressure On the other hand, both built-in ABAQUS shell and membrane elements are not very suitable for describing the inflatable structure applications with an orthotropic fabric The shell elements are quite rigid, while membranes are too flexible Therefore, the results from the NLIBFE model are in accord with the results from the NLSFE model only at high normalized pressure Due to the limitation of ABAQUS post-processor, it has not been possible to trace the load–deformation curves Instead of this, in the following section, the beam responses, which are the nonlinear solutions obtained from the NLIBFE, are validated with those from experiments at low inflation pressures Fig (a) Inflatable beam with constrained ends, (b) applied loads on the beam and (c) beam meshing using S4R shell elements Table Comparison between normalized load parameter Knl c of the NLIBFE model and the NLSFE model with various slenderness ratios Material Normalized pressure Normalized load parameter Knl Difference (%) c NLIBFE model NLSFE model 324 648 972 1295 1619 4.25 8.87 13.10 17.10 21.20 15.69 22.65 26.14 29.67 33.23 114.74 87.44 66.46 53.75 44.20 3238 4858 6477 42.30 64.80 87.80 51.23 69.30 87.91 19.10 6.71 0.13 43 85 128 171 214 1.24 2.66 4.32 5.41 6.63 2.63 5.30 7.97 10.63 13.27 71.84 66.33 59.40 65.09 66.73 22.19 25.05 28.05 48.00 15.72 5.71 427 640 854 13.6 21.4 29.7 the expressions for the slenderness ratio of the beam For these reasons, two material cases are needed to be considered and the pressures used spread from 10 to 200 kPa and are normalized (Table 3) 4.2.3 Comparison with the experimental results In this section, the numerical solutions will be validated with the results from experiments A buckling test was conducted on a simply supported HOWF inflatable beam made of material 1, as defined in Table with a 140 mm nominal diameter The beam is a Ferrari type provided by Losberger Company (Dagneux, France) The fabric is a high tenacity polyester scrim with a 0– 901 continuous woven and is coated on both sides with a PVC compound The axial load was applied in a gradual increase and measured by a load cell type ZFA The results were observed by a Vishay Data Acquisition System 5000 (VDAS 5000) The displacement along the beam was recorded by a tachometer Leica TCR 307 The pressure was taken as constant and monitored one time at the beginning of each test From the measured load–deflection curves, the wrinkling load Fw and the buckling load Fb of the beam are also monitored and then compared with the corresponding numerical solutions calculated from NLIBFE model using MATLAB A low range of inflation pressure (p ¼ 10–30 kPa corresponding to pn ¼ 324–972 for the material case 1) is focused in order to compare the nonlinear behaviors between the NLIBFE model and the inflatable beam model from experiments As shown in Figs 10–12, there is an evolution of concordance between the numerical solutions and the experimental results within the given range of pressures As the wrinkling load is reached, the model correctly predicts a gradual softening of the load–deformation response These results highlight the influence of the inflation pressure on the degree of nonlinearity of the inflatable beam behaviors and the position of the wrinkling and limit points In the given range of normalized pressures, the normalized pressures pn ¼324 and 648 are not enough to obtain the stable load–deflection responses (Figs 10 and 11) This leads to the specimens inflated to these pressures exhibited more nonlinearity in the load–deflection response than the specimen with pn ¼972 T.-T Nguyen et al / Thin-Walled Structures 72 (2013) 61–75 71 1800 1200 1100 1600 1000 1400 800 Numerical Experimental Wrinkling first onset (Experimental) Buckling onset (Experimental) 700 600 500 Axial load (N) Axial load (N) 900 1200 Numerical Experimental Wrinkling first onset (Experimental) Buckling onset (Experimental) 1000 800 600 400 300 400 200 200 100 0.5 1.5 2.5 3.5 4.5 x 10−3 Mid−span displacement (mm) Fig 10 Numerical and experimental mid-span deflection curves of a simply supported NLIBFE model with pn ¼ 324 1400 1200 Axial load (N) 1000 Numerical Experimental Wrinkling first onset (Experimental) Buckling onset (Experimental) 800 600 400 200 0.5 1.5 2.5 Mid−span displacement (mm) 0.5 1.5 2.5 3.5 4.5 Mid−span displacement (mm) x 10−3 Fig 12 Numerical and experimental mid-span deflection curves of a simply supported NLIBFE model with pn ¼ 972 normalized pressure (pn ¼324 and 648) and are normalized by two aspect ratios Rlr and Rfr The axial load increments are normalized to the pressure force due to the inflation pressure: we obtain the so-called incremental load ratio K f ¼ F i =F p It should be noted that when Kf is equal to unity, the beam is crushed down Both Rlr and Rfr ratios are plotted versus the incremental load ratio Kf as shown in Figs 14 and 15 It is also shown that in both cases of normalized pressure, the beams made of high moduli fabric (material 2) exhibit more stability (lower values of Rlf and Rfr) The comparison between the beam response curves in two different inputs of normalized pressure also illustrates well that the beams with higher normalized pressures have the larger limits of Rlr and Rfr ratios before crushing than those with lower pressures This is attributed to the fact that once the tows are sufficiently stressed, the inflatable beam possesses flexural stiffness capable of resisting a combination of direct compressive stress and bending 3.5 x 10−3 Fig 11 Numerical and experimental mid-span deflection curves of a simply supported NLIBFE model with pn ¼ 648 4.2.4 Parametric studies of NLIBFE model A parametric study is carried out for studying the influence of normalized pressure on the NLIBFE model by solving Eq (22) At each level of normalized pressure, the corresponding crushing load ðF crush ¼ F p Þ is the upper bound of the axial load applied to the beam The nodal displacements at the middle span of the beam are extracted from the global solution Fig 13 shows the variation of flexion-to-radius ratio Rfr with increments of normalized load parameter Knl c in two cases of material It was also noted that five elements are sufficient to obtain converged results The discrepancy due to the normalized pressure between the results is clearly shown At low pressure pn ¼ 324, the model is unstable and therefore will fail first At higher pressures, the Rfr ratio responses are quasi-linear for low increments of Knl c The curves become nonlinear gradually at higher Knl c The results show that the beam pressurized to higher pressures exhibits a better load-carrying capacity (more stable) In another parametric study, the influence of the fabric properties in conjunction with the effect of the normalized pressure is pointed out Two HOWF inflatable beams made of materials and are considered As mentioned in Section 4, the nonlinear iterative solutions are obtained with two inputs of Algorithm Outline of the algorithm – FE structural level: nonlinear solutions for tracing load–deflection response of NLIBFE model Require: Beam geometry, material properties, external loads, model description (see Table 1) Ensure: Displacement incrementation solutions fΔDi g for tracing load–deflection response of NLIBFE model Initialize F X ¼ 0, fDg ¼ fD0 g, fRg ¼ fFint g−fFext g ¼ f0g Loop over load increments: for i ¼1 to ninc i  F X in which i is the current load Find fΔFg: fΔFg ¼ ninc increment, Call global level routine (Algorithm 2) for computing ½KT , fFext g and fFint g of the beam, Solve nonlinear equation ẵKT fDi gfFext gfFint gị ¼ for fΔDi g, Calculate fDi g ¼ fDi−1 þ ΔDi g, h i1=2 i 〉fΔDi g , Calculate the criterion ∥fΔDi g∥ ¼ 〈ΔD 〈Di 〉fDi g Convergence check for stopping the iteration loop: ∥fΔDi g∥ ≤10−6 , Save the current solution fDi g in the global solution vector fDg end for Display the solution fΔDi g at the maximum deflection node (depends on the boundary conditions) in function of increment load FX 72 T.-T Nguyen et al / Thin-Walled Structures 72 (2013) 61–75 180 80 Material Material 160 70 nl 120 100 80 60 p1 = 10 kPa p2 = 20 kPa p3 = 30 kPa p4 = 40 kPa p5 = 50 kPa p6 = 100 kPa p7 = 150 kPa p8 = 200 kPa 40 20 0 10 60 Normalized load coefficient, Kc Normalized load coefficient, Kc nl 140 50 40 30 p1 = 10 kPa p2 = 20 kPa p3 = 30 kPa p4 = 40 kPa p5 = 50 kPa p6 = 100 kPa p7 = 150 kPa p8 = 200 kPa 20 10 12 Flexio−to−radius ratio, Rfr Flexio−to−radius ratio, Rfr 1 0.9 0.9 0.8 0.8 0.7 0.6 Material Material 0.5 0.4 0.3 Incremental load ratio, Kf Incremental load ratio, Kf Fig 13 Nonlinear buckling: variation of exion-to-radius ratio Rfr ẳ Dv =R0 ị with increasing normalized nonlinear load parameter ðK nl c ¼ 10  F i =ðEeq A0 ÞÞ for a simply supported NLIBFE model 0.7 0.6 Material Material 0.5 0.4 0.3 200 kPa 10 kPa 0.2 0.1 0.2 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Length−to−radius ratio, Rlr 0.1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Length−to−radius ratio, Rlr Fig 14 Nonlinear buckling: variation of length-to-radius ratio ðRlr ¼ Du =R0 Þ with increasing incremental load ratio ðK f ¼ F i =F p Þ for a simply supported NLIBFE model Algorithm Outline of the algorithm – global level: assembly of FE model matrices and vectors Require: Beam geometry, model description (see Table 1), increment load ith Ensure Nonlinear equilibrium equation: ẵKT fDi gfFext gfFint gị ẳ f0g 37ị which fRg ẳ fFext gfFint gị, the complete residual load of the beam Allocate the variables for load vectors and stiffness matrix Declare nodal displacement vector fΔDi g in accordance with the boundary conditions Localize all elements in the global mesh: tabcon ¼[number of element first node d.o.f last node d.o.f] Loop over element number: for k ¼1 to ne Call element level subroutine (numerical integration procedure) (Algorithm 3) for computing the element load vectors (fFeint g, fFeext g) and the element tangent stiffness matrices (½KeT ), Call localization table subroutine for assembling the element load vectors and the element tangent stiffness matrices into the global load vectors fFint g, fFext g and tangent stiffness matrix ½KT for each element loop end for Apply the boundary conditions on the fFint g, fFext g vectors and ½KT matrix 1 0.9 0.9 0.8 0.8 0.7 0.6 Material Material 0.5 0.4 0.3 Incremental load ratio, Kf Incremental load ratio, Kf T.-T Nguyen et al / Thin-Walled Structures 72 (2013) 61–75 0.7 0.6 Material Material 0.5 0.4 0.3 200 kPa 10 kPa 0.2 0.1 73 0.2 0.02 0.04 0.06 0.08 Flexion−to−radius ratio, Rfr 0.1 0.02 0.04 0.06 0.08 Flexion−to−radius ratio, Rfr Fig 15 Nonlinear buckling: variation of the flexion-to-radius ratio ðRfr ¼ Dv =R0 ị with increasing incremental load ratio K f ẳ F i =F p Þ for a simply supported NLIBFE model Algorithm Outline of the algorithm – element level: numerical integration procedure for calculating the element stiffness matrix at the jth element Numerical integration procedure Require: Nodal unknown displacements fΔDi g, element number jth, model description (see Table 1) Ensure: Element stiffness matrix ½KeT , element load vectors fFeint g and fFeext g Loop on 1D Gauss integration m point(s) in the ξ direction: for m ¼1 to Set sampling point location ξ ¼ ξm and associated weight factor Wm, Call shape function subroutine to calculate element matrix ½B and Jacobian operator J, all at point ξm , Calculate product ẵẵBT ẵ int ẵ ext ịẵB W m (see Appendix) and add it to array ½KeT , Calculate element internal load factor fT eint g Á W k and add it to array fFeint g, Calculate element external load factor fT dext g ỵ fT pext gÞ Á W k and add it to array fFeext g end for Shape function subroutine to generate ½B, J and the intermediate matrices at a given point Require: Nodal unknown displacements fΔDi g, element number jth, tabcon, ξ, beam geometry, material properties and external loads (see Table 1) Ensure: Element matrices for computing the external loads (fT pext g, fT dext g), the internal load (fT int e g) and the product ẵẵBT ẵ int ẵ ext ịẵB for computing the element stiffness matrix Calling routine supplies coordinates ξ at which ½B, J and the intermediate matrices are to be calculated At this point: Calculate shape functions fNj g and their ξ derivatives (see Eq (14)), Calculate Jacobian operator J, Calculate element matrix for computing the external load due to dead-loads: fT pext g ẳ ẵTPfGPg fT dext g ¼ fF e ð2ndof Á j−9 : 2ndof Á j ỵ 5ịg where ẵTP, fGPg and fF e g given in Appendix, Calculate element matrices ½A1, ½B1, ½C1, ½D1, ½E1, ½F1, ½H1 for discretization of internal loads acting on the beam (see Appendix), Element products for discretization of internal loads ½Ψ int and external loads ½Ψ ext are obtained, Calculate strain–displacement matrix ½B, Calculate element matrix for computing the internal load: ð38Þ T fT int e g ¼ ½B Á ½A1 B1 C1 D1 E1 F1 H1 Conclusion This study was devoted to the stability analysis of a HOWF inflatable beam based on a finite element model using a threenode Timoshenko beam element with C0-type continuity The element formulation is based on an energy concept on an inflatable beam that accounts for the change in membrane energy and strain energy in bending, which are related to the stiffness matrices of the beam In the linear buckling analysis, a mesh convergence test on the beam critical force showed the importance of the fabric thickness and the slenderness ratio of the beam The results on the buckling coefficient were also in a good agreement with those derived by an analytical approach from the literature Further, the load–deflection response of inflatable beam was traced in the nonlinear buckling analysis A method for computing the nonlinear finite element solutions accounting for the effect of changing geometry as well as the effect of inflation pressure was detailed The parametric studies pointed out that the internal pressure and the fabric properties have a significant cross influence not only on the buckling coefficients, but also on the maximum displacement solutions of an HOWF inflatable beam To check the validity and the soundness of the results, an experiment work was conducted on an inflated beam loaded in a chassis with two supports at two ends The nonlinear finite element model was shown to be capable of accurately predicting 74 T.-T Nguyen et al / Thin-Walled Structures 72 (2013) 61–75 the post-wrinkling beam behavior and the load–deformation response at low inflation pressures This work has provided a numerical approach for buckling analysis of an HOWF inflatable beam using the finite element method The initial results obtained in this work show the need for further analysis which should be addressed in future research: more extensive experiments over a larger range of pressures, loadings and beam slendernesses should be performed The future research will focus on these topics as well as the extension of the arc-length method to allow tracing more precisely the load– deformation response and the complex roots of an inflatable beam and more complex structures such as inflatable fabric arches Acknowledgments This work was supported by Laboratory Design of Structures (Laboratory DDS of GMP, IUT Lyon 1, University of Claude Bernard) and Ho Chi Minh University of Technology – VNUHCM, Vietnam Appendix A The shape functions at nodes and their derivatives: N ẳ 121ị 12 ỵ 1ị N ; ẳ 12 1ị 12 ỵ 1ị The interpolation of displacement vector of a point in the element: 9 8 u;ξ > > N ;ξ Á uðXÞ > > > > > > > > > > > > > N ;ξ Á vðXÞ > > v;ξ > > > > > > > > > > > > > > > > > > > > > w;ξ > N ;ξ Á wðXÞ > > > > > = = < < Y N Y Xị ẳ A:1ị > > > > > > > > > N ;ξ Á θY ðXÞ > > θY;ξ > > > > > > > > > > > > > > > N Á θ ðXÞ > > > > > > > > > θZ > Z > > > > > ; ; > : N Á θ ðXÞ > :θ > ;ξ Z;ξ Z The Jacobian operator, relating natural and global coordinates: Jẳ e l0 A:2ị e with l0 being the length of an element reference The discretization of internal forces is then given by & ' N ẳ N ỵ C 11 A0 u; þ ðu2;ξ þ v2;ξ þ w2;ξ ÞJ J 2 ỵC 11 I Y; ỵ Z; ịJ þ C 12 A0 ðθ2Y þ θ2Z Þ T y ẳ C y ẵv; u; Z ịJZ T z ẳ C z ẵw; u; Y ịJY M y ẳ C 11 I Y; ỵ Ju; ÞJ M z ¼ C 11 I θZ;ξ ð1 ỵ Ju; ịJ Q ẳ I R20 C 11 J θY;ξ θZ;ξ −C 12 θY θZ ị 4( ! N0 1 ỵ C 11 u; ỵ u2; ỵ v2; ỵ w2; ịJ ỵ R20 32Y; ỵ 2Z; J Q ẳ I0 A0 ' ỵ C 12 2Y ỵ 3θ2Z Q ¼ 14C 66 I ð3θY;ξ Z Y Z; ịJ Q ẳ C 66 I ðθY;ξ θY −θZ θZ;ξ ÞJ 4( ! N0 1 u; ỵ v2; ỵ w2; J þ R20 ðθ2Y;ξ þ 3θ2Z;ξ ÞJ þ C 11 u;ξ þ Q ¼ I0 A0 ' þ C 12 ð3θ2Y þ θ2Z Þ Q ẳ C 66 I Y Y; ỵ Z Z; ịJ Q ẳ C 66 I ð3θY θZ;ξ −θY;ξ θZ ÞJ & ! 1 u; ỵ v2; ỵ w2; J J Q ẳ N ỵ A0 C 12 u; ỵ 2 ' 1 ỵ C 22 2Y ỵ 32Z ị ỵ C 12 I 32Y; ỵ 2Z; ÞJ 16 & ! 1 u;ξ þ v2;ξ þ w2;ξ J J Q ¼ N þ A0 C 12 u;ξ þ 2 ' 1 C 22 3Y ỵ 2Z ỵ C 12 I 2Y; ỵ 32Z; J þ 16 1 Q 10 ¼ C 22 A0 θY θZ − C 12 I θY;ξ θZ;ξ J Quantities used in the expression of the discretized virtual works 9 A1 ðξÞ > > ỵ u; ịN ỵ M y Y; ỵ M z Z; T y Z ỵ T z Y > > > > > > > > > > > > > > B ðξÞ > > > > Nv; ỵ T y > > > > > > > > > > > > > > > > C ị > > > > Nw ỵ T ;ξ z > > = > = < > < T z ỵ u; ị ỵ Q Y; ỵ Q Z; ỵ Q Y ỵ Q 10 Z D1 ị ẳ A:3ị > > > > > > > > > > > > M ỵ u ị ỵ Q ỵ Q þ Q θ þ Q θ ðξÞ E y ;ξ Z;ξ Y;ξ Z Y > > > > > > > > > > > > > > > > > > > −T y ỵ u; ị ỵ Q Y; ỵ Q Z; ỵ Q Z ỵ Q 10 Y > > > > > > > F ðξÞ > > > > > ; : M ỵ u ị ỵ Q ỵ Q ỵ Q ỵ Q > ; : H ị z ;ξ Z;ξ Y;ξ Z Y Matrices for calculating the external load due to inflation pressure: JN 1;ξ 0 0 0 0 JN 1;ξ 7 6 0 JN 1;ξ 6 0 N1 7 6 0 0 N1 7 6 JN 0 0 7 2;ξ 7 0 0 JN 2;ξ 7 T 0 JN ½TP ¼ 2;ξ 7 0 0 N 6 0 0 N2 7 7 JN 3;ξ 0 0 6 0 0 JN 3;ξ 7 6 0 JN 3;ξ 7 0 N3 0 0 N3 ẵGPT ẳ ẵ1 u; Y; Jw; −Jv;ξ T The strain–displacement JN 1;ξ 0 JN 1;ξ 6 0 JN 1;ξ 6 0 6 0 6 0 JN 2;ξ 6 0 JN 2;ξ 6 0 JN 2; ẵBT ẳ 6 0 6 0 6 JN 0 3;ξ 6 0 JN 3;ξ 6 0 JN 3;ξ 6 0 0 matrix is given by 0 0 0 0 N1 0 JN 1;ξ 0 N1 0 N2 0 JN 2;ξ 0 0 0 0 N2 0 N3 0 JN 3;ξ 0 N3 0 7 7 7 7 JN1;ξ 7 7 7 7 7 7 JN2;ξ 7 7 7 7 7 JN3;ξ T.-T Nguyen et al / Thin-Walled Structures 72 (2013) 61–75 The matrices concerning internal and external forces for calculating the tangent stiffness, respectively, are ∂A ∂A ∂A ∂A ∂A ∂A ∂A ∂u ;ξ ∂B1 ∂u;ξ 6 ∂C u; 6 D1 ẵ int ẳ 6 ∂u;ξ ∂E1 6 ∂u;ξ ∂F 6 ∂u;ξ ∂H ∂u;ξ and 1 1 1 ∂v;ξ ∂w;ξ ∂θY ∂θY;ξ ∂θZ ∂θ Z;ξ ∂B1 ∂v;ξ ∂B1 ∂w;ξ ∂B1 ∂θY ∂B1 ∂θY;ξ ∂B1 ∂θZ ∂B1 ∂θ Z;ξ ∂C ∂v;ξ ∂C ∂w;ξ ∂C ∂θY ∂C ∂θY;ξ ∂C ∂θZ ∂D1 ∂v;ξ ∂D1 ∂w;ξ ∂D1 ∂θY ∂D1 ∂θY;ξ ∂D1 ∂θZ ∂E1 ∂v;ξ ∂E1 ∂w;ξ ∂E1 ∂θY ∂E1 ∂θY;ξ ∂E1 ∂θZ ∂F ∂v;ξ ∂F ∂w;ξ ∂F ∂θY ∂F ∂θY;ξ ∂F ∂θZ ∂H ∂v;ξ ∂H ∂w;ξ ∂H ∂θY ∂H ∂θY;ξ ∂H1 ∂θZ 0 0 0 0 0 −F p 0 Fp 0 60 6 60 6 ẵ ext ẳ 6 60 6 40 −F p 0 0 0 Fp 0 0 0 0 0 7 7 ∂C ∂θ Z;ξ 7 ∂D1 7 ∂θ Z;ξ ∂E1 7 ∂θ Z;ξ 7 ∂F ∂θ Z;ξ ∂H1 ∂θ Z;ξ 07 7 07 7 07 07 7 05 The element nodal concentrated loads vector is given by F e ẳ ẵF eX1 F eY1 F eZ1 M eY1 M eZ1 F eX2 F eY2 F eZ2 M eY2 M eZ2 F eX3 F eY3 F eZ3 MeY3 M eZ3 T References [1] Libai Avinoam, Givoli Dan Incremental stresses in loaded orthotropic circular membrane tubes I Theory International Journal of Solids and Structures 1995;32(13):1907–25 [2] Wielgosz C, Thomas JC Deflection of inflatable fabric panels at high pressure Thin-Walled Structures 2002;40:523–36 [3] Wielgosz C, Thomas JC An inflatable fabric beam finite element Communications in Numerical Methods in Engineering 2003;19:307–12 [4] Thomas JC, Wielgosz C Deflections of highly inflated fabric tubes Thin-Walled Structures 2004;42:1049–66 [5] Bouzidi R, Ravaut Y, Wielgosz C Finite elements for 2d problems of pressurized membranes Computers and Structures 2003;81:2479–90 [6] Cavallaro PV, Johnson ME, Sadegh AM Mechanics of plain-woven fabrics for inflated structures Composite Structures 2003;61:375–93 [7] Suhey JD, Kim NH, Niezrecki C Numerical modeling and design of inflatable structures—application to open-ocean-aquaculture cages Aquacultural Engineering 2005;33:285–303 [8] Main JA, Peterson SW, Strauss AM Load–deflection behavior of space-based inflatable fabric beams Journal of Aerospace Engineering 1994;7(2):225–38 [9] Main JA, Peterson SW, Strauss AM Beam-type bending of space-based membrane structures Journal of Aerospace Engineering 1995;8(2):120–5 [10] Le van A, Wielgosz C Bending and buckling of inflatable beams: some new theoretical results Thin-Walled Structures 2005;43(8):1166–87 [11] Le van Anh, Christian Wielgosz Finite element formulation for inflatable beams Thin-Walled Structures 2007;45(2):221–36 [12] Fichter WB A theory for inflated thin wall cylindrical beams Technical Report, NASA Technical Note, NASA TND-3466, 1966 75 [13] Apedo KL, Ronel S, Jacquelin E, Bennani A, Massenzio M Nonlinear finite element analysis of inflatable beams made from orthotropic woven fabric International Journal of Solids and Structures 2010;47(16):2017–33 [14] Diaby A, Le-Van A, Wielgosz C Buckling and wrinkling of prestressed membranes Finite Elements in Analysis and Design 2006;42:992–1001 [15] Davids WG, Zhang H Beam finite element for nonlinear analysis of pressurized fabric beam-columns Engineering Structures 2008;30:1969–80 [16] Apedo KL, Ronel S, Jacquelin E, Massenzio M, Bennani A Theoretical analysis of inflatable beams made from orthotropic fabric Thin-Walled Structures 2009;47(12):1507–22 [17] Comer RL, Levy S Deflections of an inflated circular cylindrical cantilever beam AIAA Journal 1963;1(7) [18] Haughton D, McKay B Wrinkling of inflated elastic cylindrical membranes under flexure International Journal of Engineering Science 1996;34 (13):1531–50 [19] Molloy SJ Finite element analysis of a pair of leaning pressurized arch-shells under snow and wind loads PhD thesis, MS thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA; 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Apedo KL, Jacquelin E Analytical buckling analysis of an inflatable beam made of orthotropic technical textiles Thin-Walled Structures 2012;51(0):186–200 [33] Pan Ning Analysis of woven fabric strengths: