DSpace at VNU: Numerical buckling analysis of an inflatable beam made of orthotropic technical textiles

DSpace at VNU: Numerical buckling analysis of an inflatable beam made of orthotropic technical textiles tài liệu, giáo á...

Numerical buckling analysis of an in flatable beam made of orthotropic

technical textiles

Thanh-Truong Nguyena,b,c,d,n, S Ronela,b,c, M Massenzioa,b,c, E Jacquelina,b,c,

K.L Apedoa,b,c, Huan Phan-Dinhd

a

Université de Lyon, F-69622 Lyon, France

b

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

a r t i c l e i n f o

Article history:

Received 21 May 2012

Received in revised form

15 May 2013

Accepted 21 June 2013

Available online 17 July 2013

Keywords:

Inflatable beams

Orthotropic fabric

Inflation pressure

Linear eigen buckling

Nonlinear buckling

a b s t r a c t 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 Thefinite element model established here involves a three-noded 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

1 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-to-fabric contact, and fabric-to-fabric wrinkling on the structure surface

In addition to checking fabric loads, thefinite 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

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 homo-geneous state of stress The approach was based on the lineariza-tion of the equalineariza-tions 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

Contents lists available atScienceDirect

Thin-Walled Structures

0263-8231/$ - see front matter & 2013 Elsevier Ltd All rights reserved.

n Corresponding author at: Université de Lyon, F-69622, Lyon, France.

Tel.: +33 4 72 65 64; fax: +33 4 72 65 53 54.

E-mail addresses: thtruong@hcmut.edu.vn ,

used it to compute the deflection of hyperstatic beams.

The element used is a membrane one Then, Bouzidi et al [5]

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

plain-woven fabric appeared to be an orthotropic material, the fabric

does not behave as a continuum, but rather as a discrete

assem-blage of individual tows, whose effective material properties

depend on the internal pressure of the beam, weave geometry

and the contact area of interacting tows

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 membranefinite 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

et al [13]investigated linear and nonlinear finite element

solu-tions in bending by discretizing nonlinear equilibrium equasolu-tions

obtained from his previous analytical model in which a

homo-geneous 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

assum-ing 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

moment-curvature 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) Thefinite 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

ratios on the buckling load coefficient are also pointed out A 3D thin-shellfinite element model is then utilized for comparison Finally, the obtained results are also compared with experimental results

2 Governing equations

In this section the governing equations of a 3D Timoshenko

strain measure is used due to the geometrical nonlinearities

Fig 1shows 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

A0and I0are given by

I0¼A0R2

where the reference dimensions l0, R0 and t0 depend on the

inflation pressure and the mechanical properties of the fabric[16]:

l0¼ lϕþpRϕlϕ

R0¼ Rϕþ pR

2 ϕ

t0¼ tϕ−3pR2Eϕ

in which lϕ, Rϕ, and tϕ 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

Fig 1 HOWF inflatable beam.

T.-T Nguyen et al / Thin-Walled Structures 72 (2013) 61–75 62

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

deforma-tion, 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 thefirst 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]

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

cylin-der[21]

The displacement components of an arbitrary point MðX; Y; ZÞ

on the beam are[22,23]

uðMÞ ¼

uX

uY

uZ

8

>

>

9

>

>¼

uðXÞ 0 0

8

>

>

9

>

>þ

ZθYðXÞ 0 wðXÞ

8

>

>

9

>

>þ

−YθZðXÞ vðXÞ 0

8

>

>

9

>

where uX, uY and uZare 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; θYðXÞ and θZðXÞ 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 ¼ E

lþ E

where El and Enl are the Green-Lagrange linear and nonlinear

strains, respectively The nonlinear term Enltakes into account the

geometrical nonlinearities The strain field depends on the

dis-placementfield as follows:

E

l¼

∂u X

∂X

∂u Y

∂Y

∂u Z

∂Z

∂u X

∂Yþ∂u Y

∂X

∂u X

∂Z þ∂u Z

∂X

∂u Y

∂Z þ∂u Z

∂Y

8

>

>

>

>

>

>

>

>

9

>

>

>

>

>

>

>

>

; Enl¼

1uT

;Xu;X

1uT

;Yu;Y

1uT

;Zu;Z

1uT

;Xu;Yþ1uT

;Yu;X

1uT

;Xu;Zþ1uT

;Zu;X

1uT

;Yu;Zþ1uT

;Zu;Y

8

>

>

>

>

<

>

>

>

>

:

9

>

>

>

>

=

>

>

>

>

;

ð6Þ

The higher-order nonlinear terms are the product of the vectors

that are defined as

u;X¼

uX ;X

uY ;X

uZ ;X

8

>

>

9

>

>; u;Y¼

uX ;Y

uY ;Y

uZ ;Y

8

>

>

9

>

>; u;Z¼

uX ;Z

uY ;Z

uZ ;Z

8

>

>

9

>

2.2 Constitutive equations

In the present work, the Saint Venant–Kirchhoff orthotropic

material is used The energy functionΦE¼ ΦðEÞ 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

Hookean stress-strain relationships

S ¼ S0þ ∂Φ

∂E ¼ S

where

 S0is the inflation pressure prestressing tensor,

coordinate system as

S ¼

SYY SYZ

2 6

3

 C is the elasticity tensor expressed in the beam axes.

In general, the inflation pressure prestressing tensor is assumed spheric and isotropic[10] So

S0¼ S0

where I is the identity second-order tensor and S0¼ N0=A0is the prestressing scalar The elasticity tensor in the beam axes was transformed from the orthotropic l; t basis (see[16])

C ¼

c4C22 c2s2C22 c3sC22 0 0

s2C66 csC66

2 6 6 6 6 6

3 7 7 7 7 7 ð11Þ where c ¼ cos φ and s ¼ sin φ with φ ¼ ðeZ; nÞ 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:

C11¼ El=ð1−νltνtlÞ; C12¼ Elνtl=ð1−νltνtlÞ;

C22¼ Et=ð1−νltνtlÞ; C66¼ Glt and El=νlt¼ Et=νtl:

3 Finite element formulations 3.1 Linear eigen buckling

In case of linear buckling analysis, the beam is subjected to the

inflation pressure prestressing S0

tensor Thefirst step is to load the inflated beam by an arbitrary reference level of external load,

fFrefg 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 forfinite element stress stiffness matrix ½ks and finite element conventional elastic stiffness matrix

½k (before loading)[24] Matrix ½ks, which augments the conven-tional elastic stiffness matrix ½k, is a function of the element

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 is1STE Using Eqs.(6)–

(11)and integrating through the volume of the beam with respect

to the cross-sectional area A and the length l, an expression for

the strain energy of afinite inflatable beam is

Ue¼1

2

Z

V 0

fðS0ÞT

where Umis the change in membrane energy and Ubis the strain

energy in bending

In order to derive the element stiffness matrices for the beam,

a displacementfield ½u ¼ ðu; v; w; θY; θZÞ 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

Timoshenko beam that has quadratic shape functions for

trans-verse displacement and linear shape functions for bending

rota-tion 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 withfive degrees of freedom (d.o

f) at each node Nodal d.o.f fdg defines d.o.f vector ⌊uj vjwj θYjθZj⌋

of an element That is

u

v

w

θY

θZ

8

>

>

<

>

>

:

9

>

>

=

>

>

;

¼

∑3

j ¼ 1Njuj

∑3

j ¼ 1Njvj

∑3

j ¼ 1Njwj

∑3

j ¼ 1NjθY j

∑3

j ¼ 1NjθZ j

8

>

>

>

<

>

>

>

:

9

>

>

>

=

>

>

>

;

where index j in summations runs from 1 to 3 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

whereξ is simply the dimensionless axial coordinate ξ ¼ ðð2=le

0ÞX−1Þ, withξ∈½−1; 1 and X is the local coordinate along the beam element

axis (X∈½0; le

0), le

0is 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 Ubrelates to the conventional

elastic stiffness ½k of the beam, as

By applying the discretization procedure, Eq.(12)then becomes

Ue¼1fdgT

whereλ is the proportionality coefficient such as F ¼ λFref, 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

matrices are generated by following the standard FEM assembly

procedure Thus, the potential energy for the whole structure can

be expressed as

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

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:

Eq.(18)is an eigenvalue problem whereλiis the eigenvalue of first buckling mode The smallest root λcrdefines the smallest level

of external load for which there is bifurcation, namely

As the beam is loaded by an arbitrary reference level of external load fFgref, the eigenvector fδDg associated with λcris 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

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

For a given element, the nonlinear equilibrium equation can be formulated as

where ½kT is the element tangent stiffness matrix, ffig is the

unknown displacement increment to be solved for After assem-bling over all the elements in the model, the following equilibrium equation is obtained:

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 fDgi¼ fDgi −1þ fΔDg,

increment step i and fDgi−1is the nodal beam displacement vector from the previous solution step The equilibrium solution toler-ance was taken as

‖fΔDgi‖ ¼ ðfΔDgT

or

‖fRgi‖ ¼ ðfRgT

with fRig ¼ fRðDi −1Þg ¼ ½KTfΔDig being the global unbalanced resi-dual 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 solu-tion fΔDg is summarized Suppose that at increment ði−1Þ, one obtained an approximation fDi −1g of the solution as the residual is not zero

T.-T Nguyen et al / Thin-Walled Structures 72 (2013) 61–75 64

At increment step i, one seeks an approximation fDig of the

solution such that

The algorithm is obtained by using thefirst-order Taylor series in

the vicinity of fDig

fRðDi −1þ ΔDiÞg ¼ fRðDi −1Þg þ ∂R

∂D

 

D ¼ Di−1

The NLIBFE model with linearized and incremental iterative

schemes is implemented using the numerical computing package

MATLAB At the FE structural level (Algorithm1), an iterative equation

solution is also performed During this structural loop, the

incremental-iterative algorithm will be called at each material

(Gaus-sian) point In every loop within an incremental load step ΔF, the

beam parameters (Table 1) and the boundary conditions are

pre-scribed, which are the input variables to the global level routine

(Algorithm2) 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 ½Ke and load

vectors (fFeintg and fFe

extg) are computed for each element The super-scripts (i,k,m) denote the global counter within the current

incre-mental load step, the number of elements and the number of Gauss

integration points, respectively

After i load step(s), the converged displacement solution fΔDig

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

4 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

A simply supported beam loaded by a compressive concen-trated F is studied The slenderness ratio isλs¼ L=ρ, where L ¼ μl0

is the beam effective length andρ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiI0=A0

p

is the beam radius of

simply supported beam The beam geometry and two materials described inTable 2are used For each case of material, a range

pcr¼ ðElt3

ϕ=4R3

ϕð1−νltνtlÞÞ are considered in the analyses[16,28,29] The pressure values and the associated normalized pressure are listed inTable 3

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 (Klc¼ 105

 scr=Eeq) proposed by Ovesy and Fazilati[30]is intro-duced, in whichs is the linear buckling critical stress of the beam

Table 1

Input parameters for modeling NLIBFE model.

direction and contraction in the t direction

direction and contraction in the l direction

n e

Table 2 Data set for inflatable beam.

Material 1 [37] Material 2 [38] Orthotropic fabric's mechanical properties

Young's modulus in the warp direction, E l (MPa)

Young's modulus in the weft direction, E t (MPa)

and Eeq¼ ffiffiffiffiffiffiffiffiffi

ElEt

p

is the equivalent Young's modulus of the current

material[31,32] The beam radius of curvature in this loading case

is given by

Rb¼ð1 þ v

2

;XÞ3 =2

Nguyen et al.[32]have recently proposed that an expression of

deflection v(X) corresponds to the buckling mode shapes:

vðXÞ ¼ Fpþ C0

s

Fp−F þ C0

s

B

where Fp¼ pπR2

is the pressure force due to the inflation pressure,

C0

s¼1kyA0C66(with ky¼0.5, a correction of the shear coefficient)

and B is an arbitrary constant The quantityΩ in this load case can

be written for the fundamental buckling mode as

l0

ð30Þ

In this loading case, Rbis determined at X ¼ l0=2 From Eqs.(28)

to (30), the radius of curvature of a simply supported HOWF

inflatable beam becomes

Rb¼ðFpþ C0

s−FÞl0

ðFpþ C0

sÞBπ



As shown inFig 2, the convergence studies on the normal-ized buckling coefficient Kl

cof LFEIB model described inSection

converged results These results are in a good agreement with those derived by an analytical approach in[32](seeTable 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 1 and 2, respectively

Based on the converged results, the variation of normalized buckling coefficient Kl

c with the change in radius-to-thickness ratio (Rrt) for a HOWF inflatable beam is depicted inFig 3 The beams are considered to be of 3 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 Kl

cat high ratio Rrt Further, the buckling load coefficient Kl

cgradient as a function of Rrtdepends on the normalized pressure pn: at higher of pn, the gradient of Kl

c becomes larger The effect of material properties is noticeable

c These results highlight the importance of the fabric thickness: a thicker

Table 3

Normalized pressure ðp n Þ for different values of internal pressure ðpÞ used in

the study.

330 340 350 360 370 380 390 400

Number of elements

420 430 440 450 460 470 480 490

Number of elements

p3 = 150 kPa, p

p3 = 150 kPa, p

Fig 2 Linear eigen buckling: mesh convergence test of normalized linear buckling load coefficient (K l ¼ 10 5  s =E ) for a simply supported LFEIB model.

Table 4 Buckling coefficient (K l

c ) comparison between LFEIB results and analytical solutions Material Normalized

pressure

Normalized buckling coefficient, K l

c Difference (%) Analytical model

[32]

LFEIB model (present)

T.-T Nguyen et al / Thin-Walled Structures 72 (2013) 61–75 66

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 4clearly shows the large variation in normalized buckling

load coefficient Kl

c with the change in the slenderness ratio λs

0.14 m radius and the length varied from 0.5 to 3 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 1 The normalized pressure only shows its

role more evident at larger of beam radii R0(lower values ofλs) In

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

RbB=2R0, which is the ratio of the radius of curvature Rb to the

coefficient Kl

cis then plotted against Rbrratio (Fig 5) Regarding the normalized coefficient Kl

c, the discrepancies due to the mate-rial 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

Kl

cversus Rbrratio is 3.56% in the case of material 1 and reduced

to 0.66% in the case of material 2 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

340 360 380 400 420 440

Radius−to−thickness ratio, Rrt Radius−to−thickness ratio, Rrt

p1 = 10 kPa, p

n = 324 p2 = 50 kPa, p

n = 1619 p3 = 150 kPa, p

n = 4858 p4 = 200 kPa, p

n = 6477

425 430 435 440 445

450

p1 = 10 kPa, p

n = 43 p2 = 50 kPa, p

n = 214 p3 = 150 kPa, p

n = 640 p4 = 200 kPa, p

n = 854

Fig 3 Linear eigen buckling: normalized buckling load coefficient (K l

c ¼ 10 5  s cr =E eq ) versus radius-to-thickness ratio (R rt ¼ R 0 =t 0 ) for a simply supported LFEIB model.

300 400 500 600 700 800 900 1000 1100 1200

Slenderness ratio, λs Slenderness ratio, λs

p1 = 10 kPa, p

n = 324 p2 = 50 kPa, p

n = 1619 p3 = 150 kPa, p

n = 4858 p4 = 200 kPa, p

n = 6477

300 400 500 600 700 800 900 1000 1100

1200

p1 = 10 kPa, p

n = 43 p2 = 50 kPa, p

n = 214 p3 = 150 kPa, p

n = 640 p4 = 200 kPa, p

n = 854

Fig 4 Linear eigen buckling: normalized buckling load coefficient (K l

c ¼ 10 5  s cr =E eq ) versus slenderness ratio (λ s ¼ L=ρ) for a simply supported LFEIB model.

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 inSection 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

displace-ment 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

bifurca-tion 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

nonlinear load parameter at ith increment of axial load is defined by

Knlc ¼ 106 Fi

EeqA0

ð32Þ The model is made up of the materials 1 and 2 as defined in

Table 2 The three-noded quadratic element as given in Eq (14)

is used The deflection solutions Dvalong the Y axes obtained from the

flexion-to-radius ratio (Rfr) as Dv=R0, whereas the axial displacement solutions

Dualong 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 Rlrand Rfrrepresent 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 Fwwhich 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, thesefirst wrinkles have a weak influence Because of this, the model is in practice valid even beyond Fwfrom 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

320

340

360

380

400

420

440

Bending radius ratio, Rbr

Mat 1 − p1 = 10 kPa, p = 324 Mat 2 − p1 = 10 kPa, p = 43 Mat 1 − p2 = 50 kPa, p = 1619 Mat 2 − p2 = 50 kPa, p = 214 Mat 1 − p3 = 150 kPa, p = 4858 Mat 2 − p3 = 150 kPa, p = 640 Mat 1 − p4 = 200 kPa, p = 6477 Mat 2 − p4 = 200 kPa, p = 854 Material 2

Material 1

Fig 5 Linear eigen buckling: normalized buckling load coefficient (K l

c ¼ 10 5 

s cr =E eq ) versus bending radius ratio (R br ¼ R b B=2R 0 ) for a simply supported

LFEIB model.

Fig 6 (a) Inflatable beam subjected to compressive axial load F (b) The effect of an initial imperfection.

Fig 7 Wrinkling load: limit of validity of numerical solutions T.-T Nguyen et al / Thin-Walled Structures 72 (2013) 61–75

68

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

wrinkles appear, the buckling test as shown inSection 4.2.3was

performed The beam was loaded until thefirst 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

thefirst 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

SXXas[16]

SXX¼ S0

XXþ C11EXXþ c2C12EYYþ s2C12EZZþ 2csC12EYZ≤0 ð33Þ

By linearizing Eq.(33) around the prestressed reference con

(N0¼ Fp−F), the following expression is obtained:

S1

l0

2; R0; 0

¼Fp−F

A0 −R0C11θZ ;X l0

2

 

As the axial stress SXXreaches the maximum value, i.e SXX¼ 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]:

Together with Eq.(30)applied in this loading case, the wrinkling

load of the current model is then

Fw¼ Fp−2π2R

2t0

l0

in which, the arbitrary constant B depends on the internal

pressure, the mechanical properties of the fabric and the

slender-ness 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 Knlc 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

inte-gration is selected for the NLSFE model as it is efficient and reliable

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

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 1 and 2, as given inTable 2 The material orthotropic is assigned a local material orientation as shown inFig 8

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 displace-ments 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:

boundary conditions (Fig 9a) As mentioned above, only the case of simply supported beams are considered

 The loads were introduced in two steps In thefirst step, the beam was inflated by an internal pressure p which is normal-ized as pn Then, a concentrated compressive load was applied

at one end of the beam in the second step (seeFig 9b)

consists of 1764 elements and 1762 nodes All the elements are S4R, a four-node general-purpose shell element, reduced

strains

The critical stresses are ðλiPiÞ, where λi and Piare 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 Fig 8 Global beam axes and local material orientation assigned for a orthotropic fabric.

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)

between the NLIBFE model and the NLSFE model (the reference model) is performed as shown inTable 5 It is noticeable that, at high normalized pressure, the minimum difference showed a good agreement between the models However, a poor agree-ment was demonstrated in a low inflation pressure case, parti-cularly for material 1 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 1 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

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

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 inTable 2with 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 70–7901 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 displace-ment 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 corre-sponding numerical solutions calculated from NLIBFE model using MATLAB

A low range of inflation pressure (p¼10–30 kPa corresponding

to pn¼324–972 for the material case 1) is focused in order to compare the nonlinear behaviors between the NLIBFE model and

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 –defor-mation 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 normal-ized pressures pn¼324 and 648 are not enough to obtain the stable load–deflection responses (Figs 10and 11) This leads to the specimens inflated to these pressures exhibited more non-linearity in the load–deflection response than the specimen with

pn¼972

Fig 9 (a) Inflatable beam with constrained ends, (b) applied loads on the beam

and (c) beam meshing using S4R shell elements.

Table 5

Comparison between normalized load parameter K nl

c of the NLIBFE model and the NLSFE model with various slenderness ratios.

Material Normalized pressure Normalized load parameter K nl

c Difference (%) NLIBFE model NLSFE model

T.-T Nguyen et al / Thin-Walled Structures 72 (2013) 61–75 70