An exact solution to calculate the length and
radius of an orthotropic inflatable beam - a
theoretical application to the determination
of the material coefficients
Quang Tung Nguyen, Jean-Christophe Thomas, Anh Le Van1
1
LUNAM Université, University of Nantes, GeM – Research Institute in Civil and
Mechanical Engineering, UMR CNRS 6183, , 2, rue de la Houssinière, BP 92208,
44322 Nantes Cedex 3, France
Abstract
Studying the behaviour of inflatable elements demands to know their geometry at
the end of the inflation. When the tensions in the walls of the fabric are important,
the radius and the length are not linear functions of the pressure. In this work, the
geometrical properties of an inflated tube are calculated in the case of finite
deformations. The material is supposed to be linear orthotropic, following a SaintVenant Kirchhoff hyperelastic law. A system of non linear equations is obtained,
which allows to calculate the radius, the length and the rotation of the cross section
of the cylinder. The results are compared to finite elements simulations and are used
to estimate the material coefficients by experiment.
Keywords: Inflatable, orthotropic fabric, material coefficient
1
Introduction
Inflatable structures are more and more used, through an important variety of
structures that are built nowadays. These modern structures are light, easy to deploy,
they have a low storage volume and are not too expensive to manufacture.
Inflatable structures are often made of architectural fabrics made of woven yarns,
with a coating which ensures the airtighness. The bearing capacities come from the
internal pressure which increases the stiffness of the beam. The strength of material
dedicated to inflatable elements gives analytical formula that can be used to predict
1
the deflections of inflatable structures. This problem has been studied bat several
authors (for example [1], [2]).
All these studies focus on the behaviour of the inflated beam between the inflated
state, and the deformed state after external loading. The use of strength of material
formulae to calculate for example the deflection of an inflated beam demands to
know precisely its geometrical properties (radius and length) at the end of the
inflation. Linear formulations already exist, easily written with the assumption of
little deformations. When the tubes are highly pressurized, the assumption of little
deformations can no more been used.
We assume in this work to consider the fabric like an orthotropic material, and focus
on the geometrical non linearities. The purpose of this study is to accurately
calculate the radius, length and rotations of the sections at the end of the inflation of
a tube. This work generalizes the case of orthotropic beams for which the angle
between the warp direction and the axis of the beam (orthotropic angle) is zero [3].
2
Calculation of the length, the radius and the rotations of the
sections of an inflated cylinder
The theory has been written in finite deformations. The first step of the
demonstration is the development of the equations for a thick-walled tube. It is
necessary to distinguish two configurations. The reference configuration
corresponds to the unpressurized tube and the current configuration corresponds to
the tube at the end of inflation. A, B, L and H are respectively the inner and outer
radius, length and thickness of the fabric in the reference configuration. These
quantities become a, b, l and h in the current configuration.
R, Θ, Z are the coordinates of the particle in the reference cylindrical base B=(er(Θ),
eθ(Θ), ez), and r, θ, z are its coordinates in the current cylindrical base b=(er(θ),
eθ(θ), ez). ez is parallel to the longitudinal axis of the beam. The position of the
particle is then X = Rer()+ Zez in the reference configuration and x = rer()+ zez in
the current configuration.
Figure 1.
Geometry of the tube in the reference configuration
We consider here an orthotropic material. The orthotropy basis is denoted (en, eℓ, et)
in the reference configuration, where en= er(Θ). α corresponds to the orientation
angle of the fabric. It is the angle between eℓ and ez, so between the warp direction
and the axis of the beam.
2
The relations between the coordinates are: r  r(R)     (R, Z) z  z(Z) . The
deformation gradient tensor can then be calculated in terms of , kr=dr/dR and
kθ=r/R. E=(FT.F-I)/2 allows to obtain the Green strain tensor. The second PiolaKirchoff stress tensor  is in relation with E via the compliance matrix: E=C: ,
which can be logically written in the orthotropy basis (eℓ, et, ez). Among all the
coefficients of the compliance matrix, use will be made in this work of:
C 

1
1
1
, Ctt  C tt   t , C tttt 
and Cttt 
E
E
Et
G t
where Eℓ and Et are the Elasticity moduli in the warp and weft direction of the
fabric, Gℓt is the shear coefficient, and ℓt is the Poisson's ratio. All the calculus are
expressed in the reference basis B, and a change-of-basis is used to pass from the
reference basis B to the orthotropy basis. This matrix makes appear the orientation
angle of the fabric . For example, the following terms of the constitutive law in the
reference matrix can then be expressed (here, s=sin , and c=cos ).
C  C s 4  Cttttc 4  (Ctt  2Ctt )s 2c2
Czz  (C  Ctttt  Ctt )s 2c2  Ctt (s 4  c4 )
C zzzz  C c 4  C tttts 4  (Ctt  2Ctt )s 2c 2
Czz  4(C  Ctttt  Ctt )s 2c2  Ctt (c2  s 2 ) 2
Cz  2(C s3c  Cttttc3s)  2Ctt (s3c  c3s)  Ctt (c3s  s3c)
C zzz  2(C c3s  C tttts3c)  2Ctt (s3c  c3s)  Ctt (c3s  s3c)
The last point of the first step is the computation of  from the Cauchy stress tensor
 with the relation: =J.F-1..F, where the matrix of  is expressed in the current
basis b. The change of basis makes appear here the angle .
The second step is the development of the equations for a thin-walled tube. It is
assumed that the thickness H is very small compared to the initial radius A. Some
assumptions can then be done. The first one is that kr and kθ are constant through the
thickness, so they do not depend on R. The second assumption is that the rotation
angle (R,Z) depends linearly on Z: (Z)=kZ. The third one is that the final
coordinate z depends also linearly on Z: z=kzZ. Calculating kθ, kz and kwill then
allow to obtain the radius, the length and the rotation of any cross section in the
inflated state.
Under these assumptions, the Green Lagrange strain matrix can be simplified:
 2

0
0
k r  1

1
2
mat(E; B) 
0
k 1
rkk  

2
2
2 2
 0

rk
k
k

r
k

1
z

 


Formula 1.
Green strain matrix in the case of a thin cylinder
3
The fourth assumption is that all the Cauchy stress components except rr are
constant in the thickness and that rr <<and rr << zz. It also can be shown
the rrzz
2
2
The relationship dx / dX  N.(2E  I).N gives kr =h/H. The use of the boundary
condition on the end cross section and the use of the equilibrium equations leads to:
 
pa pA k 
pa pA k 

and zz 

2h 2H k r
h
H kr
The components of the stress tensors are then completely expressed in term of the
unknows kθ, kz and k
0
0

mat(; b)  0  
0
0
Formula 2.
0 

0 
 zz 


0

mat (; B)  0


0

0
r 2 k r k 2
k k z

 zz 
krkz
 
k
rk r k 
 zz
kz


0

rk r k 


 zz 
kz

krk

 zz 
kz

Cauchy and PK2 stress matrix for a thin cylinder
The constitutive law reduces to:
 E   C Czz

 
 E zz    C zz C zzzz
2 E   C
 z   z Czzz
Formula 3.
Cz    


C zzz    zz 


Czz   z 
Constitutive law of the orthotropic material
Finally, by replacing the terms of formula 1. and formula 2. in formula 3. leads to
the following system of three non linear equations, with r=kA.
 pAC (A 2 k 2 k 2  2k 2z )  Czz k 2  Cz Ak2 k   / H


(k 2z  A 2 k 2 k 2  1)k z  pAC zz (A 2 k 2 k 2  2k 2z )  C zzzz k 2  C zzzAk2 k   / H


2
2 2 2
2
2
2 

2Ak k  k z
 pA Cz (A k  k   2k z )  Czzz k   Czz Ak k  / H


(k 2  1)k z
Formula 4.
Non linear system of equations
The nonlinear system can easily be solved by means of Newton-Raphson’s iterative
scheme. Once kz, kand kare obtained, the final geometry of the inflated tube is
4
known – final length ℓ = kzL, final radius a = kA – as well as the rotation of the
tube  (Z) = kZ.
3
Evolution of the geometric properties
In this part, we present the evolution of the geometric properties in relation with the
pressure and with the orientation angle of the fabric. The material coefficients used
here have been estimated on a real tube (see part 5).
E ℓH
EtH
GℓtH
ℓt
284 kPam
315 kPam
4.09 kPam
0.287
Table 1.
Material coefficients used for the simulations
For these simulations, the tube has a reference radius A=0.0765 m and a reference
length L=1.0 m. The orthotropy angles range from 0 to 180 ° in increments of 15 °.
The inflation pressure pass from 0 to 60 kPa by increments of 10 kPa.
3.1
Radius
The figure 2. presents the evolution of the final radius related with the orientation
angle of the fabric and with the pressure. The first important point is that the
variation of the radius depends on the angle , and the difference between the peaks
and the lows can be consequent. For example at p=60 kPa, the maximum of the
radius is a= 0.0831m for angle  and the minimum is a=0.0775m for angle
The fact that the maximum of is not exactly for comes from the
fact that the fabric is unbalanced. The second important point is that in every cases,
the radius increases.
Figure 2.
Evolution of the final radius of the tube a
5
3.2
Length
Figure 3.
Evolution of the final length ℓ of the tube
The figure 3. presents the evolution of the final length ℓ related with the orientation
angle of the fabric and with the pressure. As in the case of the radius, the variation
of the length depends on the angle . As can be seen, the tube can stretch or retract.
The difference between the peaks and the lows can also be consequent. For example
at p=60 kPa, the maximum of the length is ℓ=1.0036 m for angle  and the
minimum is ℓ=0.9248 m for angle In this particular case, the difference is
7.52%. It is important to notice that the length can decrease. Another important
point is that the initial length L can be kept in the case of specific orientation of the
fabric: in this case, ℓ=L for and 
3.3
Rotation of the sections
The figure 4. presents the evolution of the rotations of the cross section of the tube
related with the orientation angle of the fabric and with the pressure. In this case,
L=1m. We note that the sections can rotate. The rotation angle (L) can be positive
or negative, depending on the orientation angle of the fabric . The values of this
rotation can also be important. At p=60 kPa for instance, the minimum value of the
cross section rotation is (L)=-55.35° for and the maximum value is
(L)=54.08° for  There is no rotation of the cross section for and for
(this is valid for all orientations of the fabric). In the particular case of this
unbalanced fabric, the cross sections do not rotate for 
6
Figure 4.
4
Evolution of the tip rotation angle (L)
Comparison with 3D code results
The results of the non linear system of equations (formula 5) have been compared
with a the results of a 3-D code dedicated to non linear structural analysis.
Figure 5. Typical shape of the tube in the reference and actual
configurations
The figure 5. presents a typical shape that can be obtained with the code. The
inflation pressure is p=60 kPa and the rotation angle of the fabric is For the
simulations, the reference radius is A=0.0765m.
Specific comparisons were made between the results obtained with the finite
element code and those of the nonlinear system. These comparisons are shown in
Figures 6-7. To avoid any inaccuracies due to the fact that the tube end is a spherical
shape, the change in length and the rotation section are taken for Z = 1 m in the 3D
code results.
7
For all these curves, the continuous lines correspond to the results of the nonlinear
system, and the points correspond to the results of the finite element code. It is clear
that, whether for radius, length or rotations of the cross section, there is an excellent
agreement between the results of both types of computation. This shows the
accuracy of the nonlinear system of equations (Formula 5).
Figure 6. Comparison between the non linear system results and the
finite element results for the radius and the length at Z=1m
Figure 7. Comparison between the non linear system results and the
finite element results for the rotation of the sections at Z=1m.
5
Application to the determination of material coefficient
In this part, we present a first attempt to identify the material coefficients by
experiment. These experiments have been conducted on three inflated tubes of the
same geometry but with different orthotropy angles (1, 2, 3) made of Ferrari 302.
The membrane-membrane connection are made by high-frequency welding. The
tube is closed at one end, and just adjusted to the size of a metallic sleeve on which
it is engaged and held by means of flange metal at the other end. Changes in radius
8
and length are measured by displacement sensors. The rotation of the section is
measured using a graduated sector pointed to by a laser
Figure 8.
Experimental device for an inflatable tube
Due to the viscoelasticity of the material, it was imposed to respect sufficient
waiting times so that the measures were fully stabilized. From the different measures
for the three tubes made of the same material (but with different orientations of the
fabric), the identification of the material coefficients were carried out using a least
squares method based on the difference between the theoretical and the
experimental results. Among the values to identify, we added the angle of the fabric.
Figure 9-10. present the experimental and theoretical results after identification of
the material coefficients for the radius and the length. The continuous lines
correspond to the experiments, and the square symbols correspond to the results of
the non linear system. The identified orientation angles are 1=3.2°, 2=31.3° and
3=45.2°. The material coefficients identified are in Table 1. One can see that there
is a good agreement between the result which shows that this method of
identification can be an interesting way to identify the material coefficients.
3
1
2
2
1
Figure 9.
3
Experiments on an inflatable tube: radius and length
9
1
3
1
Figure 10. Experiments on an inflatable tube: rotation of the sections
6
Conclusion
In this work, the inflation of the inflatable orthotropic tubes undergoing finite
deformations is studied. The Saint-Venant Kirchhoff hyperelastic law is used and a
system of three non-linear equations is derived. The solution of the problem
amounts to determine the inflated geometry of the tube by solving this system of
equation. The influence of the orientation of the fabric on the inflated geometry is
then studied. The results of the non linear system of equations are compared to those
of a 3D code dedicated to inflatable structures and show the accuracy of these
method. By application, these results can be used to determine the material
coefficients (Young's moduli, shear coefficient and Poisson's ratio) of an orthotropic
fabric by the measure of geometrical properties as function of the pressure and the
orthotropic angle.
Acknowledgments:
The authors would like to thank the companies SERGE FERRARI and ATC Air et
Toiles Concept for their support in the manufacture of the inflatable tubes.
References
[1] LE VAN A., WIELGOSZ C., Bending and buckling of inflatable beams:
Some new theoretical results. Thin-Walled Structures, Vol. 43, 2005,
pp.1166-1187.
[2] APEDO K.L., RONEL S., JACQUELIN E., MASSENZIO M.,
BENNANI A, Theoretical analysis of inflatable beams made from
orthotropic fabric. Thin-Walled Structures, Vol. 47, 2009, pp.15071522.
[3] THOMAS J.-C., LE VAN A., An exact solution for inflated orthotropic
membrane tube, Thin-Walled Structures, in revision.
10