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Finite elements for 2D problems of pressurized
membranes
Rabah Bouzidi, Yannick Ravaut, Christian Wielgosz
To cite this version:
Rabah Bouzidi, Yannick Ravaut, Christian Wielgosz. Finite elements for 2D problems of
pressur-ized membranes. Computers and Structures, Elsevier, 2003, 81, pp.2479 - 2490.
�10.1016/S0045-7949(03)00308-0�. �hal-01403833�
Finite elements for 2D problems of pressurized membranes
Rabah
Bouzidi, Yannick Ravaut, Christian Wielgosz
Laboratoire de Génie Civil de Nantes, GéM, Institut de Recherches en Génie Civil et Mécanique, Saint-Nazaire, 2 rue de la Houssinière, BP 92208, 44322 Nantes Cedex 3, France
This paper presents theoretical and numerical developments of finite elements for axisymmetric and cylindrical bending problems of pressurized membranes. The external loading is mainly a normal pressure to the membrane and the developments are made under the assumptions of follower forces, large displacements and finite strains. The nu-merical computing is carried out in a different way that those used by the conventional finite element approach which consists in solving the non-linear system of equilibrium equations in which appears the stiffness matrix. The total potential energy is here directly minimized, and the numerical solution is obtained by using optimization algorithms. When the derivatives of the total energy with respect to the nodal displacements are calculated accurately, this approach presents a very good numerical stability in spite of the nil bending rigidity of the membrane. Our numerical models show a very good accuracy by comparisons to analytical solutions and experimental results.
Keywords: Finite element; Finite strains; Large deflections; Energy minimization; Pressurized membranes; Follower forces
1. Introduction
The finite element method is widely used to solve a lot of engineering problems. Its formulation leads to the resolution of a non-linear system of equations. In the case of membrane structures, which present a singular stiffness matrix due to the loss of bending stiffness, the numerical solution obtained by using shell elements cannot be straightforward because the convergence of the iterating process is difficult to reach. The numerical method, which is built in this paper, is based on a direct minimization of the total potential energy to compute the state of equilibrium of pressurized membranes.
The literature on the finite element analysis of the pressurized membrane structures is sparse. We can quote the papers of Main et al. [8], Kawabata and Ishii
[7]. Bonet [2] presented a finite element analysis of closed membrane structures that contain a constant mass fluid such as air. The membrane formulation presented by the author avoids the need for local co-ordinate axes by using the isoparametric finite element plane as a material reference configuration. Full linearization of the internal pressure forces and the resulting additional term in the tangent operator are derived. General finite element membrane analyses have been presented by Argyris et al. [1], Gruttman and Taylor [5], De Souza Neto et al. [10], Oden and Sato [9].
In the first section of this paper, we will come back on the main analytical results concerned with inflated axi-symmetrical pressurized membranes. The second section is devoted to the construction of the axisymmetric finite element. Numerical results are compared with analytical ones and show the good accuracy of our finite elements. In the third section, a cylindrical bending finite element is developed, and the numerical solution is compared to experimental results on inflatable flat panels inflated at high pressure. Once again a good accuracy between ex-perimental and numerical results is obtained.
2. Analytical solutions for inflated circular membranes The main bibliographical results are about circular inflated membranes clamped at their rim (Fig. 1). The membrane is supposed to have large deflections and is not pre-tensioned at its initial state (Henchy’s problem [6]). This bibliographical recall is mainly based on Fichter’s paper [4]. Campbell [3] proposed a solution to Hencky’s problem by taking into account the case of a pre-tensioned circular membrane.
2.1. Governing equations
The equilibrium equation depends on the assumption made on the effects of the external pressure. If the pressure remains vertical during the inflation, Hencky [6] shows that the radial equilibrium leads to:
Nhh¼
d
drðrNrrÞ ð1Þ
When the pressure remains orthogonal to the membrane during the inflation, Fichter [4] shows that this equation is rewritten as: Nhh¼ d drðrNrrÞ pr dw dr ð2Þ
The lateral equilibrium leads to the following equation: Nrr
dw dr ¼
pr
2 ð3Þ
The behavior is assumed linear and elastic Nhh mNrr¼ Ehehh
Nrr mNhh¼ Eherr
ð4Þ The strain–displacement relation takes into account the non-linear terms: err¼ du drþ 1 2 dw dr 2 ehh¼ u r ð5Þ
Only the large deflection are here taken into account. The other quadratic terms of the strains are supposed to be small.
2.2. Hencky’s theory
Hencky [6] searched the solution for the a-dimen-sional deflection WðqÞ and stress resultant N ðqÞ in the form of powers series where q represents the a-dimen-sional radius. NðqÞ ¼1 4 pa E 2=3X1 0 b2nq2n ð6Þ Nomenclature
a radius of the circular membrane a2n, b2n Hencky’s power series coefficients
Dij terms of the elasticity matrix
er
!, e!z unit vectors
E, m elasticity modulus and Poisson’s ratio E¼ Eh membrane modulus (product of the
elastic-ity modulus by the membrane thickness) fXi, fYi components of the nodal external load
F nodal external load h thickness of the membrane E full Green–Lagrange tensor H displacement gradient tensor
K linear stiffness tensor i.e. Hessian matrix of the total potential energy
L, M third and fourth order displacement deriv-ative of the total potential energy
l0, l initial length and elongation for the
cylin-drical bending finite element
n2m, w2m Fichter’s power series coefficients
N dimensionless meridian stress resultant (Nrr=
Ehin Hencky’s problem, Nrr=pain Fichter’s
approach)
Nrr, Nhh radial and lateral stresses resultants, Nrr¼
rrrh, Nhh¼ rhhh
p inflating pressure
r radius
u radial displacement
w, W ¼ w=a deflection, a-dimensional deflection UF external force work for the cylindrical
bending finite element Up pressure work
W strain energy P potential energy
q a-dimensional radial co-ordinate, q¼ r=a rrr, rhh radial and lateral stresses
WðqÞ ¼ pa E
1=3X1 0
a2nð1 q2nþ2Þ ð7Þ
By using Eqs. (1), (4) and (5), the following equations are obtained: q d dq d dqðqNÞ þ N þ1 2 dW dq 2 ¼ 0 ð8Þ NdW dq ¼ 1 2 pa E q ð9Þ
Fichter [4] shows that by taking into account the boundary condition wðaÞ ¼ 0 in the preceding equa-tions, it is possible to calculate the Hencky’s coefficients a2n, b2nby expanding the coefficients of the power series.
The values until n¼ 10 are given in Table 1.
All the coefficients are related to b0. Considering the
boundary condition uðaÞ ¼ 0 in Eqs. (1)–(8), Fichter [4] finds that b0 is related to the Poisson’s ratio as follows:
ð1 mÞb0þ ð3 mÞb2þ ð5 mÞb4þ ð7 mÞb6þ ¼ 0
ð10Þ b0 depends only on the Poisson’s ratio, therefore, all the
other coefficients also only dependent on this ratio. 2.3. Fichter’s theory
Fichter [4] has also searched the solution in the form of powers series: NðqÞ ¼X 1 0 n2mq2m ð11Þ WðqÞ ¼X 1 0 w2mð1 q2mþ2Þ ð12Þ
By using the same way as Hencky’s approach, one can find that [4]: N2 q2d 2 N dq2 þ 3qdN dq 1 2q 3dN dqþ 3þ m 2 q2N þ1 8 q2E pa ¼ 0 ð13Þ NdW dq ¼ 1 2q ð14Þ
The coefficients n2m, w2mare once again obtained from
the boundary conditions and can be found in [4]. Note that the undetermined coefficient n0 depends now both
on m and on pa=E.
Fig. 2 shows the lateral deflection obtained for each of these analytical methods for three levels of the pres-sure. The characteristics of the considered membrane are: membrane modulus E¼ 311 488 Pa m, Poisson
ratio m¼ 0:34, radius a ¼ 0:1425 m. A difference be-tween the deflection shapes appears for medium and high pressure levels (250 and 400 kPa). The differences between these two formulations come from the radial component of the pressure, which was neglected in Hencky’s problem [6]. When the pressure is supposed to follow the shape, the deflection is more similar to a spherical one.
3. An axisymmetrical membrane finite element
In this section, we present the development of an axi-symmetrical finite element for the membranes. The de-velopments are made under the assumptions of large deflections, finite strains and follower forces due to the applied pressure. We use the total Lagrangian formu-lation to represent the kinematics of the deflections. They are described by the mid-surface in the plane containing the meridian curve. We will present the the-oretical developments and its numerical implementation. 3.1. Theoretical foundations
At its initial position the membrane is circular and flat so that, the geometry is defined by only the radius
Table 1
Coefficients of the power series a0¼ 1=b0 b0 a2¼ 1=2b40 b2¼ 1=b20 a4¼ 5=9b70 b4¼ 2=3b50 a6¼ 55=72b100 b6¼ 13=18b80 a8¼ 7=6b130 b8¼ 17=18b110 a10¼ 205=108b160 b10¼ 37=27b140 a12¼ 17 051=5292b190 b12¼ 1205=567b170 a14¼ 2 864 485=508 032b220 b14¼ 219 241=63 504b200 a16¼ 103 863 265= 10 287 648b25 0 b16¼ 6 634 069=1 143 072b230 a18¼ 27 047 983= 1 469 664b28 0 b18¼ 51 523 763=5 143 824b260 a20¼ 42 367 613 873= 1 244 805 408b31 0 b20¼ 998 796 305= 56 582 064b29 0
value. The displacement vector between the current position and the initial shape is defined by:
U !
¼ uðrÞe! þ wðrÞer !z ð15Þ
The radial u and lateral w components are assumed to depend only on the radius r. Fig. 3 shows the initial and the current positions of the membrane and the dis-placement vector. The disdis-placement gradient tensor is written in cylindrical co-ordinates and takes into ac-count the simplifications due to the axisymmetrical configuration: H¼ u;r 0 0 0 u=r 0 w;r 0 0 2 4 3 5 ð16Þ
The Green–Lagrange tensor in the case of finite strains is defined by: E¼1 2 Hþ H T þ HTH ð17Þ Only three components of the strain tensor have non-null values Err¼ u;rþ ðu2;rþ w 2 ;rÞ=2 Ehh¼ u=r þ ðu=rÞ 2 =2 Erz¼ w;r=2 ð18Þ
One easily identifies the contributions of the three hy-pothesis made in continuum mechanics: small strains and deflectionsðErr u;r; Ehh u=rÞ, finite strains ðu2;r=2;
ðu=rÞ2=2Þ, and large deflections ðw;r=2; w2;r=2Þ.
The hypothesis of finite strains and so the use of the strain tensor of Green–Lagrange implies that the stress tensor will be the second tensor of Piola–Kirchoff. The material is supposed to be isotropic and its constitutive law is: rrr rhh rrz 8 < : 9 = ;¼ Drr Drh 0 Drh Dhh 0 0 0 Drz 2 4 3 5 EEhhrr Erz 8 < : 9 = ; ð19Þ
The strain energy is given by: Wd¼ 1 2 Z Z X Z r r r r : EdX ¼1 2 Z Z X Z DrrE2rr þ DhhE2hhþ 2DrhErrEhh þ 2DrzE2rz dX ð20Þ
We can summarize this result by defining the following energies components: Wd rr¼ 1 2 Z 2p 0 dh Z h 0 dz Z r2 r1 DrrE2rrrdr ð21Þ Wd hh¼ 1 2 Z 2p 0 dh Z h 0 dz Z r2 r1 DhhE2hhrdr ð22Þ Wd rh¼ 1 2 Z 2p 0 dh Z h 0 dz Z r2 r1 2DrhErrEhhrdr ð23Þ Wdrz¼1 2 Z 2p 0 dh Z h 0 dz Z r2 r1 2DrzErz2rdr ð24Þ
The total strain energy is then: Wd¼ Wdrrþ W d hhþ W d rhþ W d rz ð25Þ
We use linear shape functions for the lateral and radial deflections: uðrÞ ¼ r2 r r2 r1 r r1 r2 r1 u1 u2 ð26Þ wðrÞ ¼ r2 r r2 r1 r r1 r2 r1 w1 w2 ð27Þ Their derivatives with regards to the r parameter lead to: u;r¼ 1 r2 r1 1 r2 r1 u1 u2 ¼u2 u1 r2 r1 ð28Þ w;r¼ 1 r2 r1 1 r2 r1 w1 w2 ¼w2 w1 r2 r1 ð29Þ These two quantities are constant within an elementary volume, so that the elementary strain energies Wd
ijcan be
written as (for a finite element between the radii r1and r2):
Wdrr¼ ph Z r2 r1 DrrE2rrrdr ¼ ph Z r2 r1 Drrðu;rþ u2;r=2þ w 2 ;r=2Þ 2 rdr ð30Þ Wdhh¼ ph Z r2 r1 DhhEhh2rdr ¼ ph Z r2 r1 Dhh u=rþ ðu=rÞ2 2 !2 rdr ð31Þ
Wdrh¼ ph Z r2 r1 2DrhErrEhhrdr ¼ ph Z r2 r1 2Drh u;rþ u2 ;r 2 þ w2 ;r 2 ! u rþ ðu=rÞ2 2 ! rdr ð32Þ Wd rz¼ ph Z r2 r1 2DrzE2rzrdr¼ ph Z r2 r1 2Drzðw;r=2Þ 2 rdr ð33Þ The two terms u;rand w;rare independent of r, and will
be considered as constant during the integration. There-fore we will keep this notation in the following results: Wd rr¼ phDrrðu;rþ u2;r=2þ w 2 ;r=2Þ 2 ðr2 2 r 2 1Þ=2 ð34Þ Wd hh¼ phDhh ðu1r2 " u2r1Þð2u;rþ 3u2;rþ u 3 ;rÞ þ ðr2 2 r 2 1Þ u2 ;r 2 þ u3 ;r 2 þ u4 ;r 8 ! þ u1r2 u2r1 r2 r1 2 ln r2 r1 1þ 3u;rþ 3u2 ;r 2 ! u1r2 u2r1 r2 r1 3 ð1 þ u;rÞ r1 r2 r1r2 1 8 u1r2 u2r1 r2 r1 4 r2 1 r 2 2 r2 1r22 # ð35Þ Wd rh¼ phDrhðu;rþ u2;r=2þ w 2 ;r=2Þ ðu;r " þ u2 ;r=2Þðr 2 2 r 2 1Þ þ 2ðu1r2 u2r1Þð1 þ u;rÞ þ u1r2 u2r1 r2 r1 2 ln r2 r1 # ð36Þ Wd rz¼ phDrzw2;rðr 2 2 r 2 1Þ=4 ð37Þ
Note: a singularity arises when one of the radii of the finite element is equal to zero. This singularity disappears when we assume that the strains and stresses cannot be infinite when r¼ 0. We will therefore simply write that Wdhh¼ Wd
rrand W d
rh¼ 0 when the radius vanishes.
3.1.1. Pressure work
We have to calculate the work of the applied pres-sure p:
Up¼
Z
S
pU! !dS ð38Þ
Which can be written as follows: Up¼
Z r2þu2
r1þu1
2ppU!!r drn ð39Þ
Here n! is the normal to the surface dS Up¼ 2pp Z r2þu2 r1þu1 r2 r r2 r1 r r1 r2 r1 w1 w2 rdr ð40Þ
And for an element in its deformed configuration: Up¼ 2pp r2 r1 w2 w1 3 ððr2 h þ u2Þ3 ðr1þ u1Þ3Þ þ r2w1 r1w2 2 ððr2þ u2Þ2 ðr1þ u1Þ2Þ i ð41Þ The formulation is written on the deformed state of membrane and takes into account the real effects of the pressure. The pressure work is exactly written for the complete structure.
3.1.2. Theorem of the potential energy
The potential energy is defined as the difference be-tween the strain energy and the work of the external loads (here the pressure work). We must evaluate the potential energy for each finite element modelling the membrane and then add all these elementary energies to obtain the complete potential energy of the structure: P¼X
i
ðWi UiÞ ð42Þ
Here i is the finite element index. The displacement so-lution is obtained by minimizing the total potential energy with respect to each node component. For each element defined between the two nodes 1i and 2i, four non-linear equations have to be solved:
oP ou1i ¼ 0; oP ow1i ¼ 0; oP ou2i ¼ 0; oP ow2i ¼ 0 ð43Þ
We can show that this system of non-linear equations involves until third degrees of displacement components. It can be written in a more usual way as used in finite element method:
K U þ L U U þ M U U U ¼ F ð44Þ where K is a second order tensor (Hessian matrix of P, or stiffness matrix), L, M third and fourth order tensors, UT¼ fu1i; w1i; u2i; w2ig is the vector of the nodal
dis-placements of one element and F is the vector of the external load applied to the membrane. We give here the components of the stiffness matrix K. L and M tensors are given in Appendix A. The stiffness tensor K can be decomposed as two stiffness tensors, one depending on the mechanical characteristics of the membrane K0 and
K¼ K0þ Kp ¼ K11 2ppr1 K13 0 2ppr1 K22 0 K22 K13 0 K33 2ppr2 0 K22 2ppr2 K22 2 6 6 4 3 7 7 5 ð45Þ With K11¼ ph r2 r1 r2 2 r21 r2 r1 ðDrr þ Dhhþ 2DrhÞ þ 2r2 r2 r2 r1 lnr2 r1 2 Dhh 4r2Drh ð46Þ K13¼ ph r2 r1 r 2 2 r 2 1 r2 r1 ðDrrþ Dhhþ 2DrhÞ þ 2 r1 þ r2 r1r2 r2 r1 lnr2 r1 Dhhþ 2ðr1þ r2ÞDrh ð47Þ K22¼ ph r2 r1 1 2 r2 2 r21 r2 r1 Drz ð48Þ K33¼ ph r2 r1 r2 2 r 2 1 r2 r1 ðDrr þ Dhhþ 2DrhÞ þ 2r1 r1 r2 r1 lnr2 r1 2 Dhh 4r1Drh ð49Þ 3.2. Numerical results 3.2.1. Mesh sensitivity
Many computations with different meshes are com-pared in order to evaluate the solution convergence. Figs. 4 and 5 show the mesh sensitivity of the central deflection and the potential energy for a loading pres-sure equal to 250 kPa. Fig. 4 shows the evolution of the central deflection as a function of the number of axi-symmetric finite elements, compared to analytical solu-tions. The deflection is quite stable beyond 25 elements
and the numerical solution seems to be an average of the two analytical solutions.
3.2.2. Comparison between Fichter’s analytical model and the finite element solution
Several numerical tests were performed to validate our numerical model. We have not found any analytical reference with a complete model including finite strains, large deflections and follower pressure. We will therefore compare our numerical results with Fichter’s theoretical solution [4]. The main difference between these two so-lutions comes from the hypothesis of finite strains. Fig. 6 shows the different deflections obtained by the two models for three levels of the pressure: 100, 250 and 400 kPa. The mechanical properties of the membrane are E¼ 311 488 Pa m for the membrane modulus, m ¼ 0:34
for the Poisson’s ratio and a¼ 0:1425 m for the radius. At medium pressure (100 kPa), the deflections given by the two solutions are almost identical because the strains remain small. When the pressure is higher (400 kPa), a difference between the deflection shapes is quite appar-ent. This pressure value gives high membrane stresses, and the finite strains imply a less spherical shape, but a higher central deflection. The finite element results differ from the analytical model, which is less accurate for
Fig. 4. Mesh sensitivity on the central deflection (p¼ 250 kPa).
Fig. 5. Mesh sensitivity on the potential energy (p¼ 250 kPa).
Fig. 6. Deflections for the axisymmetric finite element and Fichter’s solution.
higher pressure levels. The main differences appear on the curvature of the membrane. Fig. 7 shows the radial displacements of the nodes of the discretized structure for pressures varying from 100 to 400 kPa. Even if these displacements are not very high (max of 4.53 mm for 400 kPa) for this size of the membrane, it is important to know their level.
3.2.3. Relative error distribution
We define the relative error between the numerical and Fichter’s analytical solution by:
e¼wFEðrÞ wFichterðrÞ wFichterðr ¼ 0Þ
ð50Þ where wFEis the deflection obtained with the finite
ele-ment and wFichter the deflection given by Fichter’s
solu-tion.
The differences increase with the pressure, as shown in Fig. 8. The maximum difference does not stand in the middle of the membrane but near the rim. Fichter’s deflection shows a more spherical shape than the nu-merical solution according to the error distribution. The maximum relative error reaches 7.3% for a pressure of 400 kPa. These results show that finite strains have an
influence when the level of pressure increase and that they cannot be neglected.
4. A finite element for cylindrical bending membrane 4.1. Formulation
The formulation of another finite element based on the same energetic approach is now displayed. This element is devoted to cylindrical bending inflatable membranes. It also takes into account finite strains, large deflections and follower forces. The element is defined between the nodes 1 and 2 in theðx; yÞ plane, and has a unit width along the z-axis. The co-ordinates are namedðx1; y1Þ and ðx2; y2Þ (Fig. 9).
4.1.1. Strain energy
The strain energy is defined by: Wd¼ 1 2 Z S EE2 xxdS ð51Þ
Let us denote by l0the initial length and by l the length
after deformations: l0¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx2 x1Þ2þ ðy2 y1Þ2 q l¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx2þ u2 x1 u1Þ 2 þ ðy2þ v2 y1 v1Þ 2 q ð52Þ
The square strain is: E2xx¼ðl l0Þ
2
l2 0
ð53Þ The strain energy is be obtained by integration along the element: Wdi¼ 1 2E Z l0 0 E2xxdx¼1 2E E2 xxl0 ð54Þ 4.1.2. Pressure work
The pressure work is equal to the product of the applied pressure by the volume generated by the dis-placements of the element:
Fig. 9. Midsurface of the cylindrical bending finite element. Fig. 7. Radial displacements for the axisymmetric finite
ele-ment.
Fig. 8. Relative error between the axisymmetric finite element and Fichter’s solution.
Up¼
p
2ðx2þ u2 x1 u1Þðy2þ v2þ y1þ v1Þ ð55Þ The work of the applied forces on the structure nodes is defined by the dot product of the applied forces by the displacement vector:
UF ¼ F U ¼ fX1u1þ fY1v1þ fX2u2þ fY2v2 ð56Þ
4.1.3. Potential energy
The total potential energy of one element is the dif-ference between the strain energy and the work of ex-ternal forces applied to the element:
P¼ W Up UF ð57Þ
The total potential energy of the structure is calculated by adding the elementary potential energy of each ele-ment.
4.2. Comparison with experimental results
The validation of this cylindrical bending finite ele-ment is done by the comparison with experiele-mental tests made on double-layered pressurized panels.
4.2.1. Double-layered panel
The tested panels are prototypes constructed by TISSAVEL Inc, and made of two parallel-coated woven fabrics. The upper and lower layers are linked with flexible yarns [11]. The yarn density is strong enough to ensure the flatness of the pressurized structure. Fabrics are made with high strength polyester material. The behavior of the panel depends on the inflation pressure, which generates the pre-stress of the fabrics and of the yarns.
4.2.2. Experimental tests
Panels have been tested like beams submitted to bending loads and with various boundary conditions. Fig. 10 shows the experimental device we used to test the panels. Two kinds of boundary conditions have been used: isostatic conditions (simply supported at the two ends), and hyperstatic conditions (clamped at one end and simply supported at the other end). Analytical so-lutions are available in [12] for the isostatic case.
4.2.3. Numerical results
The cylindrical bending finite element is used to model the upper and lower layer of the panel, and high stiffness elements are used to take into account the ver-tical yarns linking the two layers. Computations have been done using a membrane stiffness modulus equal to E¼ 650 000 Pa m evaluated by an inflation test [12].
The length between the supports is 1.6 m for both de-vices. First, we pressurize the panels to make them strong and then we apply a concentrated load F at the middle of the panel. Fig. 11 shows comparisons for the isostatic case between the numerical model and the ex-periments for three levels of pressure and for three val-ues of the load F . Fig. 12 shows the comparisons for the hyperstatic case. These two comparisons show the ac-curacy of the cylindrical bending finite element. Nu-merical and experimental comparisons for a higher hyperstatic case are available in [13].
5. Conclusions
We have described two finite elements for 2D prob-lems of inflatable membranes: axisymmetric and cylin-drical bending one. The elements are built with the
Fig. 12. Numerical and experimental results for the hyperstatic case.
Fig. 10. The panel experimental device.
Fig. 11. Numerical and experimental results for the isostatic case.
hypothesis of large deflections, finite strains and with follower pressure loading. A new formulation is used to construct these two elements: the numerical solution is obtained by solving directly the optimization problem formulated by the theorem of the minimum of the total potential energy. The stiffness matrix appears as the Hessian matrix of the energy and depends explicitly on the inflation pressure. These two elements are compared with analytical and experimental results, and in both cases, numerical results are close to reference results, which prove the accuracy of the proposed elements. The energy formulation and the solution techniques for more general cases are straightforward. The numerical tech-nique is only limited by the size of the optimization problem. A further modelling with a 3D triangular finite element based on the same energetic formulation is in progress, and we would be able to predict the deflections of 3D pressurized membrane structures.
Appendix A
We give here the expressions of the components of the L and M tensors, which appear in the equilibrium equations (Eq. (44)). First, we define the following constants: a¼u1r2 u2r1 r2 r1 b¼u2 u1 r2 r1 c¼w1r2 w2r1 r2 r1 d¼w2 w1 r2 r1 a1¼ 1 r2 r1 lnr2 r1 ð3Dhhþ DrhÞ 3 r1 Dhh b1¼ 3 r 2 2 r21 r2 r1 Drr 2 þDhh 2 þ Drh þ 3r2ðDhhþ DrhÞ c1¼ r2 r2 r1 lnr2 r1 ð6Dhh þ 2DrhÞ 6ðDhhþ DrhÞ d1¼ 1 2 r2 2 r 2 1 r2 r1 ðDrrþ DrhÞ þ r2Drh a2¼ 1 r2 r1 lnr2 r1 ð3Dhh þ DrhÞ 3 r2 Dhh b2¼ 3 r2 2 r 2 1 r2 r1 Drr 2 þDhh 2 þ Drh 3r1ðDhhþ DrhÞ c2¼ r1 r2 r1 lnr2 r1 ð6Dhhþ 2DrhÞ þ 6ðDhhþ DrhÞ d2¼ 1 2 r2 2 r 2 1 r2 r1 ðDrr þ DrhÞ r1Drh A¼r 2 2 r21 r2 r1 ðDrrþ DrhÞ
Then, the components of L tensor can be written as:
Lijk¼ ph ðr2 r1Þ 2lijk with l111¼ r22a1þ b1 r2d1 l112¼ pðr2 r1Þ h ð2r1 r2Þ l113¼12½2r1r2a1 2b1 ðr2þ r1Þc1 l114¼ pðr2 r1Þ h r1 l121¼ pðr2 r1Þ h ð2r1 r2Þ l122¼ d1 l123¼ 0 l124¼ d1 l131¼12½2r1r2a1 2b1 ðr2þ r1Þc1 l132¼ 0 l133¼ r12a1þ b1 r1d1 l134¼ 0 l141¼ pðr2 r1Þ h r1 l142¼ d1 l143¼ 0 l144¼ d1
l211¼ 1 2 4pðr2 r1Þ h r1 r2 2 l212¼12½A 2r2Drh l213¼ 0 l214¼12½A þ 2r2Drh l221¼12½A 2r2Drh l222¼ 0 l223¼12½A þ 2r1Drh l224¼ 0 l231¼ 0 l232¼12½A þ 2r1Drh l233¼ 1 2 2pðr2 r1Þ h r2 l234¼12½A 2r1Drh l241¼12½A þ 2r2Drh l242¼ 0 l243¼12½A 2r1Drh l244¼ 0 l311¼ r22a2þ b2 r2d2 l312¼ 0 l313¼12½2r1r2a2 2b2 ðr2þ r1Þc2 l314¼ 0 l321¼ 0 l322¼ d2 l323¼ pðr2 r1Þ h r2 l324¼ d2 l331¼12½2r1r2a2 2b2 ðr2þ r1Þc2 l332¼ pðr2 r1Þ h r2 l333¼ r21a2þ b2 r1d2 l334¼ pðr2 r1Þ h ð2r2 r1Þ l341¼ 0 l342¼ d2 l343¼ pðr2 r1Þ h ð2r2 r1Þ l344¼ d2 l411¼ 1 2 2pðr2 r1Þ h r1 l412¼12½A 2r2Drh l413¼ 0 l414¼12½A þ 2r2Drh l421¼12½A 2r2Drh l422¼ 0 l423¼12½A þ 2r1Drh l424¼ 0 l431¼ 0 l432¼12½A þ 2r1Drh l433¼ 1 2 4pðr2 r1Þ h r2 r1 2 l434¼12½A 2r1Drh l441¼12½A þ 2r2Drh l442¼ 0 l443¼12½A 2r1Drh l444¼ 0
Also, We define the following constants for the M ten-sor:
a3¼ Dhh 1 r1r2 þ1 2 r1þ r2 r2 1r2 b3¼ 1 2 r2 2 r 2 1 r2 r1 ðDrrþ Dhhþ 2DrhÞ þ r2ðDhhþ DrhÞ c3¼ 1 r2 r1 lnr2 r1 ð3Dhhþ DrhÞ þ 3 r1 Dhh d3¼ r2 r2 r1 lnr2 r1 ð3Dhh þ DrhÞ 3ðDhhþ DrhÞ e3¼ r2 r2 r1 lnr2 r1 1 Drh f3¼ 1 2 r2 2 r 2 1 r2 r1 ðDrrþ DrhÞ þ r2Drh g3¼ 1 r2 r1 lnr2 r1 Drh h3¼ 1 2 r2 2 r 2 1 r2 r1 ðDrrþ DrhÞ k3¼ 2Drh l3¼ 1 2 r2 2 r 2 1 r2 r1 Drr a4¼ Dhh 1 r1r2 1 2 r1þ r2 r1r22 b4¼ 1 2 r2 2 r 2 1 r2 r1 ðDrr þ Dhhþ 2DrhÞ r1ðDhhþ DrhÞ c4¼ 1 r2 r1 lnr2 r1 ð3Dhh þ DrhÞ 3 r2 Dhh d4¼ r1 r2 r1 lnr2 r1 ð3Dhhþ DrhÞ þ 3ðDhhþ DrhÞ e4¼ 1 r1 r2 r1 lnr2 r1 Drh f4¼ 1 2 r2 2 r21 r2 r1 ðDrr þ DrhÞ r1Drh g4¼ 1 r2 r1 lnr2 r1 Drh h4¼ 1 2 r2 2 r 2 1 r2 r1 ðDrrþ DrhÞ k4¼ 2Drh l4¼ 1 2 r2 2 r21 r2 r1 Drr
Then, the components of M are given by: Mijkl¼ Mijlk¼ Miljk¼ Milkj¼ Mikjl¼ Miklj
M1111¼ ph ðr2 r1Þ3 ðr3 2a3 b3 r22c3þ r2d3Þ M1211¼ M1411¼ ph ðr2 r1Þ 3 2pðr2 r1Þ2 3 M1333¼ ph ðr2 r1Þ 3ðr 3 1a3 b3 r12c3þ r1d3Þ M1311¼ 1 3 ph ðr2 r1Þ 3 3r1r22a3þ 3b3þ ð2r1r2þ r22Þc3 ðr1þ 2r2Þd3 M1331¼ 1 3 ph ðr2 r1Þ 3 3r 2 1r2a3 3b3 ð2r1r2þ r21Þc3 þ ðr2þ 2r1Þd3 M1221¼ M1441¼ 1 3 ph ðr2 r1Þ 3ðr2e3 f3Þ M1223¼ M1443¼ 1 3 ph ðr2 r1Þ 3ðr1e3þ f3Þ M1421¼ 1 3 ph ðr2 r1Þ 3ðr2e3þ f3Þ M1423¼ 1 3 ph ðr2 r1Þ3 ðr1e3 f3Þ M2111¼ M2333¼ ph ðr2 r1Þ3 2 3 pðr2 r1Þ 2 h l3 ! M2311¼ M2133¼ ph ðr2 r1Þ 3l3 M2211¼ M2411¼ 1 3 ph ðr2 r1Þ3 ðr2 2g3þ h3 r2k3Þ M2233¼ M2433¼ 1 3 ph ðr2 r1Þ 3ðr 2 1g3þ h3 r1k3Þ
M2231¼ M2431 ¼1 6 ph ðr2 r1Þ3 ½2r1r2g3þ 2h3 ðr1þ r2Þk3 M3111¼ ph ðr2 r1Þ3 ðr3 2a4 b4 r22c4þ r2d4Þ M3433¼ M3233¼ ph ðr2 r1Þ3 2pðr2 r1Þ 2 3 M3333¼ ph ðr2 r1Þ 3ðr 3 1a4 b4 r21c4þ r1d4Þ M3311¼ 1 3 ph ðr2 r1Þ3 3r1r22a4þ 3b4þ ð2r1r2þ r22Þc4 ðr1þ 2r2Þd4 M3331¼ 1 3 ph ðr2 r1Þ 3 3r 2 1r2a4 3b4 ð2r1r2þ r12Þc4 þ ðr2þ 2r1Þd4 M3221¼ M3441¼ 1 3 ph ðr2 r1Þ 3ðr2e4 f4Þ M3223¼ M3443¼ 1 3 ph ðr2 r1Þ 3ðr1e4þ f4Þ M3421¼ 1 3 ph ðr2 r1Þ3 ðr2e4þ f4Þ M3423¼ 1 3 ph ðr2 r1Þ3 ðr1e4 f4Þ M4111¼ M4333¼ ph ðr2 r1Þ3 2 3 pðr2 r1Þ 2 h l4 ! M4311¼ M4133¼ ph ðr2 r1Þ3 l4 M4211¼ M4411¼ 1 3 ph ðr2 r1Þ 3ðr 2 2g4þ h4 r2k4Þ M4233¼ M4433¼ 1 3 ph ðr2 r1Þ 3ðr 2 1g4þ h4 r1k4Þ M4231¼ M4431 ¼1 6 ph ðr2 r1Þ 3½2r1r2g4þ 2h4 ðr1þ r2Þk4 References
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