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default
Code_Aster
Titre : Éléments MEMBRANE et GRILLE_MEMBRANE
Responsable : DE SOZA Thomas
Date : 05/09/2012 Page : 1/4
Révision
Clé : R3.08.07
:
f0e4562ed57b
Elements MEMBRANE and GRILLE_MEMBRANE
Summary:
This document describes the formulation and the establishment in Code_Aster elements MEMBRANE and
GRILLE_MEMBRANE. Elements MEMBRANE allow to model a linear behavior of unspecified membrane. Elements
GRILLE_MEMBRANE are finite elements more specifically dedicated to the representation of steel reinforcements
in a solid mass (for applications of standard Génie Civil reinforced concrete). The main features of these
elements are the following ones:
•
•
•
elements of membrane, without rigidity of torsion;
pas de degrees of freedom of rotation, but on the other hand not of possibility of offsetting;
geometrical support surface (triangle, quadrangle; linear or quadratic);
Elements GRILLE_MEMBRANE allow to model a non-linear behavior of the bars of reinforcement, which is not
the case of the elements MEMBRANE.
Warning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in part
and is provided as a convenience.
Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)
Version
default
Code_Aster
Titre : Éléments MEMBRANE et GRILLE_MEMBRANE
Responsable : DE SOZA Thomas
1
Date : 05/09/2012 Page : 2/4
Révision
Clé : R3.08.07
:
f0e4562ed57b
Introduction
Modeling MEMBRANE allows to represent the mechanical behavior of a possibly anisotropic membrane.
Let us specify that it is currently limited to linear behaviors. It makes it possible to model elements of
structure whose rigidity of inflection is negligible.
Elements GRILLE_MEMBRANE allow to represent the possibly non-linear behavior of bars of
reinforcement in a reinforced concrete structure. The principal constraint is that the bars of
reinforcement must be periodically distributed on a surface, and directed all in the same direction. Let
us specify however that bars of cross reinforcements can be modelled by superposition of two
modelings GRILLE_MEMBRANE (see further).
Elements of the type GRILLE_MEMBRANE come to supplement the possibilities of modeling of
reinforcement in Code_Aster, in complement of modeling GRILLE_EXCENTRE. One presents below the
differences between modelings GRILLE_MEMBRANE and GRILLE_EXCENTRE.
It is pointed out that modeling GRILLE_EXCENTRE is based on a kinematics of hull DKT with only one
sleep in the thickness [R3.07.03]. This base DKT implies the presence of degrees of freedom of
rotation to the nodes of the elements GRILLE_EXCENTRE: if it allows the concept of offsetting, it is
useless when one does not need offsetting (in this case it weighs down the useless model of way,
because not only the degrees of freedom of rotation lengthen the vector of unknown factors, but it is
necessary moreover block one considerable number of these degrees of freedom by double multiplier
of Lagrange). Modeling GRILLE_MEMBRANE is a modeling based on a “surface” kinematics, it does not
require other degrees of freedom that usual displacements (on the other hand, obviously, this
modeling does not make it possible to use the concept of offsetting).
Modeling GRILLE_EXCENTRE, based on DKT, requires geometric standards of support of type triangle
or linear quadrangle; modeling GRILLE_MEMBRANE is developed starting from the geometrical
supports surface triangle or quadrangle, linear or quadratic.
For the two types of modeling, on the other hand, only a direction of reinforcement is available by finite
elements. That makes it possible to model any type of reinforcement to several directions, by
superimposing an element by direction; the cost of calculation generated by these duplications is weak:
no the duplication of the degrees of freedom (thus constant cost of inversion of matrix), duplication of
the elementary computation functions (but they remain simple, of reduced number in 3D – elements of
surfaces against elements of volume – and elementary calculations for structures with great numbers
of degrees of freedom are of weak cost compared to cost of inversion).
2
Formulation of the elements of MEMBRANE
For an element of membrane, the deformation energy can be put in the form:
=
1
 : ds
2∫
with  the membrane constraint and  membrane deformation.
The only difficulty is to obtain an expression of the type
 =BU nodal .
For that, a little differential geometry should be used. One leaves the expression of the derivative
contravariante:
∇ u=
∂u i
=ui∣j a i⊗a j =u i∣ j g jk ai ⊗a k
∂ξ j
Warning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in part
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Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)
Version
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Code_Aster
Titre : Éléments MEMBRANE et GRILLE_MEMBRANE
Responsable : DE SOZA Thomas
with
{ j }
Date : 05/09/2012 Page : 3/4
Révision
Clé : R3.08.07
:
f0e4562ed57b
an acceptable parameter setting of surface and
a i=
∂x
.
∂i
while noting a the natural base (nonorthogonal, only 3ème vector, normal on the surface, is
normalized) plan of the reinforcement and g metric the contravariante associated with this base (cf
[R3.07.04] .pour more details).
One then defines the direction of reinforcement by the normalized vector e 1 (that one supplements,
{e i } ) and one calls
R the operator of passage such
as a i =R e p . One notes in Greek the indices taking only the values in  1,2  , and one obtains:
∂u i   jk
  =  ∇ u    =
⋅a Ri R k g
∂ j
for the facility of the talk in an orthonormal base
p
i

By definition of

R : R13=0 and by definition of g : g 13=g 23 =0 . One thus obtains:

  =  ∇ u    =
If one notes now

∂u     
⋅a R  R  g

∂
B the derivative of the functions of form at the point of Gauss considered, it comes:
   

  =R  R g B
 n  a i U in
from where it B sought.
   

Bin =R  R g B
 n  a i
From B , there are then all the classical expressions of the deformation, the nodal forces and the
tangent matrix which are written:
 =BU
F =∫ BT 
T ∂
K =∫ B
B
∂
3
Formulation of the elements of GRILLE_MEMBRANE
For a tablecloth of uniaxial reinforcement, the deformation energy can be put in the form:
=
1
S   ds
2∫
with S the section of reinforcement per unit of length,
(scalar).
 the constraint (scalar) and  deformation
One seeks to obtain an expression of the type =BU nodal .
By taking again the approach of the preceding section, one shows this time that:
Warning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in part
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Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)
Version
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Code_Aster
Titre : Éléments MEMBRANE et GRILLE_MEMBRANE
Responsable : DE SOZA Thomas
= ∇ u 11=
By introducing the derivative

Date : 05/09/2012 Page : 4/4
Révision
Clé : R3.08.07
:
f0e4562ed57b

∂u  1 1 
⋅a R R g
∂ 
B functions of form at the point of Gauss considered, it comes:
 βn  a i U in
=R1 R1 g   B
from where it B sought. It will be noted that it has the shape of a vector, due to the scalar nature of
the required deformation.
Bin =R 1 R 1 g   B  n  a  i
From B , one finds all the classical expressions of the deformation, the nodal forces and the tangent
matrix which are written:
=BU
F =∫ BT 
∂
K =∫ BT
B
∂
It will be noted that in fact the laws of behavior 1D are used to obtain the constraint starting from the
deformation. All the laws of behavior available in 1D are usable. Failing this, one can also use the laws
3D, thanks to the method De Borst.
4
Description of the versions of the document
Version
Aster
7.4
9.5
11.3
Author (S)
Organization (S)
P.Badel
EDF-R&D/AMA
J.M.Proix
EDF-R&D/AMA
Mr. David
EDF-R&D/AMA
Description of the modifications
Initial text
Modification of GRID in GRILLE_EXCENTRE
Addition of modeling MEMBRANE and minor modifications
Warning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in part
and is provided as a convenience.
Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)