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Code_Aster
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Version
7.2
Titrate:
Voluminal element of hull SHB8
Date:
04/08/04
Author (S):
J.M. PROIX, S. BAGUET, A. COMBESCURE
Key
:
R3.07.07-A
Page
:
1/18
Manual of Reference
R3.07: Machine elements on average surface
HT-66/04/002/A
Organization (S):
EDF-R & D/AMA, INSA-LYON















Manual of Reference
R3.07 booklet: Machine elements on average surface
Document: R3.07.07




Code_Aster: Voluminal element of hull SHB8


Summary:

We present in this document the theoretical formulation of element SHB8PS and its establishment
numerical for non-linear incremental analyzes implicit (great displacements, small rotations,
small deformations).

It is about a three-dimensional cubic element with 8 nodes with a called privileged direction thickness.
Thus, it can be used to represent mean structures while correctly taking into account them
phenomena through the thickness (bending, elastoplasticity), thanks a numerical integration to 5 points of
Gauss in this privileged direction.

In order to reduce the calculating time considerably and to draw aside various likely blockings
to appear, this element under-is integrated. It requires consequently a mechanism of stabilization in order to
to control the modes of deformation to null energy (modes of Hourglass).

In addition to its cost of relatively weak calculation and its good performances in elastoplasticity, this element
have another advantage. Since it is based on a three-dimensional formulation and that it only has
degrees of freedom of translation, it is very easy to couple it with voluminal elements 3D, which is
very useful in systems where voluminal hulls and elements must cohabit.
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Titrate:
Voluminal element of hull SHB8
Date:
04/08/04
Author (S):
J.M. PROIX, S. BAGUET, A. COMBESCURE
Key
:
R3.07.07-A
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:
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R3.07: Machine elements on average surface
HT-66/04/002/A
Count
matters
1
Introduction ............................................................................................................................................ 3
2
Kinematics of the element ...................................................................................................................... 4
3
Variational formulation ...................................................................................................................... 4
4
Discretization ......................................................................................................................................... 5
4.1
Discretization of the field of displacement ....................................................................................... 5
4.2
Operator gradient discretized .......................................................................................................... 5
4.3
Stamp rigidity ............................................................................................................................ 7
4.4
Stamp geometrical rigidity K
.................................................................................................. 8
4.5
Stamp pressure K
p
..................................................................................................................... 9
5
Stabilization of the element ..................................................................................................................... 10
5.1
Motivations ..................................................................................................................................... 10
5.2
Modes of “Hourglass” ................................................................................................................... 11
5.3
Stabilization of the type “Assumed Strain Method” ............................................................................ 11
6
Strategy for non-linear calculations ............................................................................................... 13
6.1
Geometrical non-linearities .......................................................................................................... 13
6.2
Small displacements ...................................................................................................................... 14
6.3
Forces of stabilization ................................................................................................................... 14
6.4
Plasticity ......................................................................................................................................... 15
7
Establishment of element SHB8 in Code_Aster ......................................................................... 15
7.1
Description ..................................................................................................................................... 15
7.2
Use ....................................................................................................................................... 15
7.2.1
Mesh ................................................................................................................................ 15
7.2.2
Modeling ......................................................................................................................... 15
7.2.3
Material ................................................................................................................................ 16
7.2.4
Boundary conditions and loading .................................................................................. 16
7.2.5
Calculation in linear elasticity ................................................................................................... 16
7.2.6
Calculation in linear buckling .............................................................................................. 16
7.2.7
Calculation in geometrical nonlinear “elasticity” .................................................................. 16
7.2.8
Plastic nonlinear calculation ................................................................................................ 16
7.3
Establishment ................................................................................................................................... 17
7.4
Validation ....................................................................................................................................... 17
8
Bibliography ........................................................................................................................................ 18
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Code_Aster
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Version
7.2
Titrate:
Voluminal element of hull SHB8
Date:
04/08/04
Author (S):
J.M. PROIX, S. BAGUET, A. COMBESCURE
Key
:
R3.07.07-A
Page
:
3/18
Manual of Reference
R3.07: Machine elements on average surface
HT-66/04/002/A
1 Introduction
Many recent work proposed to use a voluminal formulation for the structures
thin. Two main families of methods, which rest all on the introduction of a field of
postulated deformation (“assumed strain”), emerge. Methods of the first family
consist in using a conventional numerical integration with an adequate control of all them
modes of blocking and locking (volume, transverse shearing, membrane). Methods of
the second family consist in under-integrating the elements to remove blockings and controlling
the modes of Hourglass which rise from this under-integration (see [bib3] [bib4]). Both
approaches were studied in details in the case of an elastic behavior. On the other hand, very little
work treats elastoplastic case.
The element presented here rests on an under-integrated formulation especially developed for
elastoplastic behavior of the structures in bending. The basic idea consists first of all with
to make sure that there are sufficient points of Gauss in the thickness to represent it correctly
phenomenon of bending, then to calculate rigidities of stabilization in an adaptive way according to the state
plastic of the element. That represents an unquestionable improvement compared to the formulations
conventional for the forces of stabilization, because these last rest on an elastic stabilization
who becomes too rigid when the effects of plasticity dominate the response of the structure.
Element SHB8 is a continuous three-dimensional cube with eight nodes, in which a direction
privileged, called thickness, was selected. It can thus be used to modelize the structures
thin and to take into account the phenomena which develop in the thickness within the framework
mechanics of the continuous mediums three-dimensional. Since this element is under
integrated, it exhibe of the modes of Hourglass which must be stabilized. We chose the method of
stabilization introduced by Belytschko, Bindeman and Flanagan [bib3] [bib4]. This element and this
method of stabilization were already implemented in an explicit formulation by Abed-Meraim
and Combescure [bib2]. This documentation describes the formulation of this element, its implementation
numerical for the prediction of elastic and elastoplastic structural instabilities, like sound
establishment in Code_Aster. For the non-linear problems, an incremental formulation
implicit of Newton-Raphson type is used [R5.03.01]. The equilibrium equations are solved by
method of Lagrangian the update. The control of the increments of load and displacement is
based on a method of piloting close to the algorithm to Riks [bib5]. Implementation the numerical
of this element within a non-linear framework was proposed by Legay and Combescure in [bib1].
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Code_Aster
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Titrate:
Voluminal element of hull SHB8
Date:
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Author (S):
J.M. PROIX, S. BAGUET, A. COMBESCURE
Key
:
R3.07.07-A
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R3.07: Machine elements on average surface
HT-66/04/002/A
2
Kinematics of the element
Element SHB8 is a hexahedron with 8 nodes. The five points of integration are selected along
direction
in the reference mark of the local co-ordinates. The shape of the element of reference as well as
points of integration are represented on [Figure 2-a].
5
4
1
8
7
6
5
1
3
4
2
2
3
Appear 2-a: Geometry of the element of reference and points of integration
This element is isoparametric and has the same linear interpolation and same kinematics
that hexaèdraux elements with 8 standard nodes.


3 Formulation
variational
The formulation used for the construction of element SHB8PS differs from a formulation
conventional simply by the choice of a postulated deformation
&
, therefore of an operator gradient
discretized, allowing to avoid the induced modes parasitized by under integration.
Thus, the variational principle is written:
()
()
0
:
,
=
-
=
ext.
V
F
U
FD
v
&
&
&
where
represent the total virtual power,
variation,
v
the field speed,
u&
speeds
nodal,
&
the rate of postulated deformation (assumed strain misses),
the stress of Cauchy,
V
updated volume and
ext.
F
external forces.
The discretized equations thus require the only interpolation speed
v
and of the rate of
postulated deformation
&
in the element. We now will build element SHB8PS to be left
of this equation. The complete developments and the demonstrations concerning this element are
explained in details in [bib2].
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Code_Aster
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Titrate:
Voluminal element of hull SHB8
Date:
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Author (S):
J.M. PROIX, S. BAGUET, A. COMBESCURE
Key
:
R3.07.07-A
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R3.07: Machine elements on average surface
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4 Discretization
4.1
Discretization of the field of displacement
Space co-ordinates
I
X
element are connected to the nodal co-ordinates
II
X
by means of
isoparametric functions of forms
I
NR
by the formulas:
8
1
(
)
(
)
I
II
I
I
II
I
X
X NR
NR
X
=
=
,
=
,
In the continuation, and except contrary mention, one will adopt the convention of summation for the indices
repeated. Indices in small letters
I
vary from one to three and represent the directions of
space co-ordinates. Those in capital letters
I
vary from one to eight and correspond to the nodes of
the element.
The same functions of forms are used to define the field of displacement of the element
I
U
in
function of nodal displacements
II
U
:
(
)
I
II
I
U
U NR
=
,
Trilinear isoparametric functions of form are chosen:
1
(
)
(1
) (1
) (1
)
8
[1 1]
1
8
I
I
I
I
NR
I
,
=
+
+
+


, -,
=,
These functions of form transform a unit cube in space
(
)
,
in a hexahedron
unspecified in space
1
2
3
(
)
X X X
,
.
4.2
Operator discretized gradient
The gradient
I J
U
,
field of displacement is a function of displacement
II
U
node
I
in
direction
I
:
I J
II
I J
U
NR
,
,
= U
The linear tensor of deformation is given by the symmetrical part of the gradient of displacement:
1 (
)
2
ij
I J
J I
U
U
,
,
=
+
Let us introduce the three vectors
I
B
, derived from the functions of form at the points of Gauss
K
P
:
0
0
()
K
T
I
K
I
NR
P
X
=, =, =
=
B
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Titrate:
Voluminal element of hull SHB8
Date:
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Author (S):
J.M. PROIX, S. BAGUET, A. COMBESCURE
Key
:
R3.07.07-A
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R3.07: Machine elements on average surface
HT-66/04/002/A
Also let us introduce the following vectors:
1
2
3
4
1
2
3
(
1
1
1
1
1
1 1 1
)
(
1
1
1
1
1
1 1 1
)
(
1
1
1 1
1 1 1
1)
(
1
1 1
1 1
1 1
1)
(
1 1
1 1
1
1 1
1)
(
1 1
1
1
1 1 1
1)
(
1
1 1
1
1
1 1 1
)
(
1
1
1
1 1
1 1 1
)
T
T
T
T
T
T
T
T
=
=
-
-
-
-
=
-
-
-
-
=
-
-
-
-
=
-
-
-
-
=
-
-
-
-
=
-
-
-
-
=
-
-
-
-
S
H
H
H
H
X
X
X
Three vectors
T
I
X
the nodal co-ordinates of the eight nodes represent. Four vectors
T
H
the functions represent respectively
1
H
,
2
H
,
3
H
and
4
H
for each of the eight nodes, which are
defined by:
1
2
3
4
H
H
H
H
=
=
=
=
Let us introduce finally the four following vectors:
(
)




-
=
=
3
1
.
8
1
J
J
J
T
B
X
H
H
The gradient of the field of displacement can be now written in the form (without any
approximation [bib3]):
4
1
T
T
T
T
I J
J
J
I
J
J
I
U
,
,
,
=
=
+
.
=
+
.
B
H
U
B
H
U
Or, in the form of vector:
X X
y y
Z Z
S
X y
y X
X Z
Z X
y Z
Z y
U
U
U
U
U
U
U
U
U
,
,
,
,
,
,
,
,
,
=
+
+
+
U

with
I
U
nodal displacement in the direction
I
. The symmetrical operator gradient (noted
S
)
discretized connecting the tensor of deformation to the vector of nodal displacements
.
S
=
U B U
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Code_Aster
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Titrate:
Voluminal element of hull SHB8
Date:
04/08/04
Author (S):
J.M. PROIX, S. BAGUET, A. COMBESCURE
Key
:
R3.07.07-A
Page
:
7/18
Manual of Reference
R3.07: Machine elements on average surface
HT-66/04/002/A
takes the matric form then:
0
0
0
0
0
0
0
0
0
T
T
X
X
T
T
y
y
T
T
Z
Z
T
T
T
T
y
y
X
X
T
T
T
T
Z
Z
X
X
T
T
T
T
Z
Z
y
y
H
H
H
H
H
H
H
H
H
,
,
,
,
,
,
,
,
,
+
+
+
=
+
+
+
+
+
+
B
B
B
B
B
B
B
B
B
B
The detailed formulation was presented by Belytschko in [bib3].
4.3
Stamp rigidity
The matrix of rigidity of the element is given by:
E
T
E
D
=
K
B C B
The five points of integration considered are on the same vertical line. Their co-ordinates
are
(
)
,
and their weights of integration are the roots of the polynomial of Gauss-Legendre:
P (1)
0 0
1
0 91
=.
1
0 24
=.
P (2)
0 0
2
0 54
=.
2
0 48
=.
P (3)
0 0
0
0 57
.
P (4)
0 0
2
-
2
P (5)
0 0
1
-
1
Thus, the expression of rigidity
E
K
is:
5
1
() ()
()
()
T
E
J
J
J
J
J
J
=
=
K
B
C B
where
()
J
J
is Jacobien, calculated at the point of Gauss
J
, of the transformation the configuration enters
unit of reference and an arbitrary hexahedron. The elastic matrix of behavior
C
chosen with
following form:
2
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
E
µ
µ
µ
µ
µ
+
+
=
C
where
E
is the Young modulus,
the Poisson's ratio,
(
)
µ
+
= 1
2
E
the modulus of rigidity and
2
1
-
= E
the coefficient of modified Lamé. This law is specific to element SHB8. It resembles
with that which one would have in the case of the assumption of the plane stresses, put aside the term (3,3). One
can note that this choice involves an artificial anisotropic behavior.
This choice makes it possible to satisfy all the tests without introducing blocking.
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Titrate:
Voluminal element of hull SHB8
Date:
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Author (S):
J.M. PROIX, S. BAGUET, A. COMBESCURE
Key
:
R3.07.07-A
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R3.07: Machine elements on average surface
HT-66/04/002/A
4.4
Stamp geometrical rigidity K
By introducing the quadratic deformation
Q
E
:
1 3
1
2
Q
K I K J
K
E
U U
,
,
=,
=
one can define this matrix of geometrical rigidity by:
0
0
(
)
(
)
T
Q
T
D
D
.
. =
:
,
=
:
.
the U.K.U
E U U
U
U
In order to express this matrix in discretized space, let us introduce the operators gradient
quadratic discretized
Q
B
such as:
()
(() ())
T
T
J
Q
.
.
.
+
+
+
,
=
=
+
+
+
+
+
+
Q
Q
Q
Q
xx
xx
yy
zz
Q
Q
Q
Q
yy
zz
xy
yx
Q
Q
Q
Q
Q
zz
xy
yx
xz
zx
J
J
Q
Q
Q
Q
Q
Q
xy
yx
xz
zx
yz
zy
Q
Q
Q
Q
xz
zx
yz
zy
Q
Q
yz
zy
E
E
E
U B
U U
E
E
E
E
E
E
E
E
E
E U
U
E
E
E
E
E
E
E
E
E
E
E
E
()
()
()
()
()
()
()
()
()
()
()
()
()
()
T
J
J
T
T
T
J
J
J
T
T
T
J
J
J
T
T
T
J
J
J
T
T
J
J
T
J
.
.
.

.
.
.
.
.
.


.
.
.
.
.
.


.
.
.
.
.
.


.
.
.
.


.
.

Q
Q
yy
zz
Q
Q
Q
yy
zz
xy
Q
Q
Q
zz
xy
xz
Q
Q
Q
xy
xz
yz
Q
Q
xz
yz
Q
yz
B
U U B
U
U B
U U B
U U B
U
U B
U U B
U U B
U
U B
U U B
U U B
U
U B
U U B
U
U B
U















Various terms
Q
ij
B
are given by the following equations:
0
0
()
0
0
0
0
J
.
=
.
.
T
X
X
Q
T
xx
X
X
T
X
X
B B
B
B B
B B
0
0
()
0
0
0
0
J
.
=
.
.
T
y
y
Q
T
yy
y
y
T
y
y
B B
B
B B
B B
0
0
()
0
0
0
0
J
.
=
.
.
T
Z
Z
Q
T
zz
Z
Z
T
Z
Z
B B
B
B B
B B
0
0
()
0
0
0
0
J
. +.
=
. +.
. +.
T
T
X
y
y
X
Q
T
T
xy
X
y
y
X
T
T
X
y
y
X
B B
B B
B
B B
B B
B B
B B
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Code_Aster
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Titrate:
Voluminal element of hull SHB8
Date:
04/08/04
Author (S):
J.M. PROIX, S. BAGUET, A. COMBESCURE
Key
:
R3.07.07-A
Page
:
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R3.07: Machine elements on average surface
HT-66/04/002/A
0
0
()
0
0
0
0
J
. +.
=
. +.
. +.
T
T
X
Z
Z
X
Q
T
T
xz
X
Z
Z
X
T
T
X
Z
Z
X
B B
B B
B
B B
B B
B B
B B
0
0
()
0
0
0
0
J
. +.
=
. +.
. +.
T
T
y
Z
Z
y
Q
T
T
yz
y
Z
Z
y
T
T
y
Z
Z
y
B B
B B
B
B B
B B
B B
B B

With these notations, the geometrical matrix of rigidity
K
at the point of Gauss
J
is given by:
()
()
()
()
()
()
()
J
xx
J
J
yy
J
J
zz
J
J
=
.
+
.
+
.
Q
Q
Q
xx
yy
zz
K
B
B
B
()
()
()
()
()
()
xy
J
J
xz
J
J
yz
J
J
+
.
+
.
+
.
Q
Q
Q
xy
xz
yz
B
B
B

and geometrical rigidity of the element stamps it is given by:
5
1
() ()
()
J
J
J
J
J
=
=
K
K
4.5
Stamp pressure K
p
The following compressive forces are present in the tangent matrix via the matrix
P
K
, because them
following external forces depend on displacement. The following compressive forces are written:
1
0
0
0
0
[()]
()
T
T
p
dS
p
F
dS
p
p
F
-
.
=
.
=
-
.
P
N U
det
U N
F
K U
U
() 1
F
= +
U
U
by using the notations:
·
0
(
)
T
X
y
Z
N N N
=
,
N
, normal on the surface external of the element in the configuration of
reference
·
I
B
%
, vector of dimension 4, drift of the functions of form to the 4 nodes of the face of
the element charged in pressure
·
0
S
surface of the face charged in pressure
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Titrate:
Voluminal element of hull SHB8
Date:
04/08/04
Author (S):
J.M. PROIX, S. BAGUET, A. COMBESCURE
Key
:
R3.07.07-A
Page
:
10/18
Manual of Reference
R3.07: Machine elements on average surface
HT-66/04/002/A
The preceding formulation leads to a not-symmetrical matrix. It is known that one can nevertheless
to use a symmetrical formulation if the external forces due to the pressure derive from a potential.
It is the case if the compressive forces do not work on the border of the modelized field. One
thus consider that the symmetrical part of the matrix is enough. The symmetrized matrix takes the form
following:
0
0
0
0
0
0
0
0
T
T
T
T
X
y
X
Z
T
T
T
T
X
y
X
Z
T
T
T
T
X
y
X
Z
T
T
T
T
X
y
X
Z
T
T
T
T
y
Z
y
X
T
T
T
T
y
Z
y
X
T
T
T
T
y
Z
y
X
T
y
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
S
N
N
N
N
N
-
-
-
-
-
-
-
-
-
-
-
-
=
-
-
y
X
Z
X
y
X
Z
X
y
X
Z
X
y
X
Z
X
y
Z
y
X
y
Z
y
X
P
y
Z
y
X
X
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
K
B
B
B
B
B
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
0
0
0
0
0
T
T
T
y
Z
X
T
T
T
T
Z
X
Z
y
T
T
T
T
Z
X
Z
y
T
T
T
T
Z
X
Z
y
T
T
T
T
Z
X
Z
y
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
-
-
-
-
-
-
-
-
-
-
y
Z
y
y
X
Z
Z
X
Z
y
Z
X
Z
y
Z
X
Z
y
Z
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
It is a matrix
(12 12)
×
, that it is necessary to multiply by displacements of the 4 nodes of the face on
which one applies a pressure.


5
Stabilization of the element
5.1 Motivations
The under-integration of element SHB8 (5 points of Gauss only) aims at reducing considerably
calculating time (gradient displacement, law of behavior,…). It also makes it possible to draw aside
the various blockings met in implementation the numerical of the finite elements.
However, this under-integration does not have only advantages: it introduces unfortunately
parasitic modes associated a null energy (mode of Hourglass or sand glass). In statics, that
can lead to a singularity of the matrix of total stiffness for certain boundary conditions.
In transitory dynamics, on the other hand, that led to modes of sand glass which will deform it
unrealistic mesh of way and which ends up exploding the solution. This deficiency of the matrix
of stiffness, due to under-integration, must thus be compensated while adding to elementary rigidity
a matrix of stabilization. The core of the new rigidity, thus obtained, must be reduced to only
modes corresponding to the rigid movements of solids.
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Titrate:
Voluminal element of hull SHB8
Date:
04/08/04
Author (S):
J.M. PROIX, S. BAGUET, A. COMBESCURE
Key
:
R3.07.07-A
Page
:
11/18
Manual of Reference
R3.07: Machine elements on average surface
HT-66/04/002/A
5.2
Modes of “Hourglass”
Since the points of integration are on the same vertical line (privileged direction),
the derivative of the functions
3
H
and
4
H
cancel themselves in these points. The operator discretized gradient is thus
tiny room to:
2
1
2
1
2
1
2
2
1
1
2
2
1
1
2
2
1
1
0
0
0
0
0
0
0
0
0
T
T
X
X
T
T
y
y
T
T
Z
Z
T
T
T
T
y
y
X
X
T
T
T
T
Z
Z
X
X
T
T
T
T
Z
Z
y
y
H
H
H
H
H
H
H
H
H
,
=
,
=


,
=



,
,

=
=



,
,
=
=



,
,
=
=
+
+
+
=
+
+
+
+
+
+
B
B
B
B
B
B
B
B
B
B




















The modes of Hourglass are modes of displacement to null energy, i.e they check
0
=
Drunk
.
Six modes, others that rigid modes of solids, which check this equation are:
3
4
3
4
3
4
0
0
0
0
0
0
0
0
0
0
0
0
H
H
H
H
H
H
5.3
Stabilization of the type “Assumed Strain Method”
In this approach, inspired of work of Belytschko, Bindeman and Flanagan [bib3] [bib4], them
derived
I
B
functions of form are not calculated at the points of Gauss but are realized on
the element:
1
^
(
)
1 2 3
E
T
I
I
D
I
V
,
=
,
,
=,
NR
B
Thus, the new operator discretized gradient can be written:
^
^
= +
stab
B B B
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Titrate:
Voluminal element of hull SHB8
Date:
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Author (S):
J.M. PROIX, S. BAGUET, A. COMBESCURE
Key
:
R3.07.07-A
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:
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R3.07: Machine elements on average surface
HT-66/04/002/A
The expression of
^
stab
B
is given by:
4
3
4
3
4
4
^
0
0
^
0
0
^
^
0
0
0
0
0
0
0
0
0
0
0
T
X
T
y
T
Z
H
H
H
,
=
,
=
,
=
stab
B
and that of the vectors
^
by:
3
1
1
^
^
(
)
8
T
J
J
J
=
=
-
.
H
H X B
The new matrix of rigidity becomes:
^
^
^
^
E
E
E
E
T
T
T
T
D
D
D
D
=
+


+
+


stab
stab
stab
stab
stab
K
B C B
B C B
B
C B
B
C B
K
144424443
The last term of the preceding equation (
stab
K
) is enough to stabilize the element. One can thus reduce
the matrix of rigidity stabilized with:
E
=
+
stab
K K
K
^
^
E
T
D
=
stab
stab
stab
K
B
C B
The many cases which were studied showed that it is enough to calculate the diagonal terms of
stamp stabilization
II
stab
K
,
1 2 3
I
=,
, which is given by:
$ $
$ $
11
11
3
3
4
4
1
(
2) [
]
3
T
T
H
µ
=
+
+
stab
K
$ $
$ $
22
22
3
3
4
4
1
(
2) [
]
3
T
T
H
µ
=
+
+
stab
K
$ $
33
33
4
4
3
T
H E
=
stab
K
Coefficients
II
H
themselves are given by the following equation, in which there is not
summation on the repeated indices:
1
3
T
T
J
J
K
K
II
T
I
I
H
=
X X
X X
X X
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Titrate:
Voluminal element of hull SHB8
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Author (S):
J.M. PROIX, S. BAGUET, A. COMBESCURE
Key
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R3.07.07-A
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R3.07: Machine elements on average surface
HT-66/04/002/A
6
Strategy for non-linear calculations
6.1 Non-linearities
geometrical
One treats here the case of great displacements, but weak rotations (see further) and small
deformations. One adopts for that an updated Lagrangian formulation.
Into nonlinear we seek to write balance between internal forces and force external at the end of
the increment of load (identified by index 2):
extr
F
F
2
int
2
=
The expression of the internal forces is written:
2
int
2
2
2
T
F
FD
=
B
In the preceding equation the operator
2
B
is the operator allowing to pass from displacement to
linear deformation calculated on the geometry at the end of the pitch, the stress
2
is the stress of
Cauchy at the end of the pitch and integration is made on volume
2
deformed at the end of the pitch.
We chose this updated Lagrangienne formulation.
The element available to date in Aster is programmed in small rotations. Indeed the increment of
deformation is calculated by using only the linear deformation:
()
()
(
)
U
U
E
+
=
T
1
1
2
1
The operator gradient is calculated on the geometry of beginning of pitch. This writing of the deformation is
limited to small rotations (<5 degrees).
One can without difficulty of extending the formulation to great rotations by including in the deformation them
terms of second command:
()
()
() ()
(
)
1
1
1
1
1
.
2
T
T
E
=
+ +
U
U
U
U
In elasticity, the law of behavior is written:
'E
=
C
where
C
is the matrix of Hooke. Let us notice that for the SHB8 this matrix is a matrix
orthotropic transverse which is written in the axes of the lamina:
[]
2
0
0
0
0
2
0
0
0
0
0
0
0 0
0
'
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
E
µ
µ
µ
µ
µ
µ
µ
+
+
=
C
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Voluminal element of hull SHB8
Date:
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J.M. PROIX, S. BAGUET, A. COMBESCURE
Key
:
R3.07.07-A
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R3.07: Machine elements on average surface
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The formula allowing to calculate the stress of Cauchy
2
starting from the stress of
Piola Kirchoff II
2
is:
()
2
1
2
2
1
1
det
T
F
F
F
F
I
=
+


=


= +

U

The combination of the four last equations with the expression of the internal forces gives
formulation of the element in great deformations into Lagrangian updated.
Let us notice that this updated Lagrangienne formulation is completely equivalent to
total Lagrangienne formulation for which the internal forces are written:
(
)
0
int
2
2
0
()
T
NL
F
FD
=
+
B B
U
In this case all integrations are made on the initial geometry
0
the stress
2
used is
the stress of Piola Kirchoff II. This last method is probably preferable when it
mesh becomes deformed significantly and thus makes it possible to deal with the problems into large
deformations but requires the development of the operator
()
NL
B
U
.
The increment of deformation in Lagrangian total is expressed on the initial geometry of the structure.
()
()
() ()
(
)
0
0
0
0
1
.
2
T
T
E
=
+ +
U
U
U
U
The combination of the two preceding equations gives the formulation of the element into large
deformations in linear behavior material.
6.2 Small
displacements
In the case of small displacements one confuses geometry in beginning and end of pitch, stress of
Cauchy and of Piola Kirchoff II, moreover one uses the linear expression of the deformations.
6.3
Forces of stabilization
The forces of stabilization make it possible to avoid the modes of sand glass and are added in the calculation of
residues to balance the contribution of the matrix of stiffness of stabilization to the first member.
forces of stabilization
stab
F
, to add to the internal forces
int
2
F
, are written:
=
stab
stab
F
K
U
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Titrate:
Voluminal element of hull SHB8
Date:
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J.M. PROIX, S. BAGUET, A. COMBESCURE
Key
:
R3.07.07-A
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R3.07: Machine elements on average surface
HT-66/04/002/A
For reasons of effectiveness, one chooses not to assemble again
stab
K
to calculate
stab
F
in
end of pitch, but rather to build
stab
F
from
^
stab
B
that one calculated previously. One must
for that to place itself in the reference frame corotationnel of medium of pitch suggested in [bib3]. For this
reason, one does not obtain an exact expression of
stab
F
, and some additional iterations are
generally necessary to converge. These some iterations are however unimportant by
report/ratio at the cost of calculation saved while not assembling
stab
K
.
6.4 Plasticity
The elastoplastic behavior of Von Mises, with isotropic work hardening, is calculated of each one of
5 points of integration. One thus uses quite simply the formulas and the usual programming of
plasticity with a three-dimensional state of stresses, but the linear matrix of behavior
'
C
is
orthotropic. We must quite simply slightly modify the usual algorithm of flow
elastoplastic three-dimensional by replacing the usual matrix of Hooke
C
by the matrix of
transverse orthotropic behavior
'
C
.
We must find the stress at the end of the pitch which checks balance. In great displacements it
problem is written:
(
)
p
C
-
+
=
'
1
2
This equation is solved as soon as the increment of plastic deformation is known. This deformation
is determined while forcing the final stress to be plastically acceptable. This method is
completely similar to the usual three-dimensional method except that there is no explicit solution
with this problem if one uses for example the effective approximation of the radial return to calculate
solution. We chose to solve this nonlinear problem by a method of Newton.

7
Establishment of element SHB8 in Code_Aster
7.1 Description
This element is pressed on the voluminal meshs 3D
HEXA8
.
7.2 Use
This element is used in the following way:
7.2.1 Mesh
Check the good orientation of the faces of the indicated elements (compatibility with the direction
privileged) by using ORIE_SHB8 of operator MODI_MAILLAGE.
7.2.2 Modeling
To affect modeling
SHB8
with the meshs
HEXA8
indicated.
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Voluminal element of hull SHB8
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J.M. PROIX, S. BAGUET, A. COMBESCURE
Key
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R3.07.07-A
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R3.07: Machine elements on average surface
HT-66/04/002/A
7.2.3 Material
For a homogeneous isotropic elastic behavior in the thickness one uses the key word
ELAS
in
DEFI_MATERIAU
where one defines the coefficients E, NAKED modulus Young and, Poisson's ratio.
To define a plastic behavior the key word is used
TRACTION
in
DEFI_MATERIAU
where one
the name of a traction diagram defines. Only this type of definition is available for the moment.
7.2.4 Boundary conditions and loading
DDL of volume 3D in the total reference mark. One imposes the boundary conditions on the ddl of volume
3D (AFFE_CHAR_MECA/DDL_IMPO), and efforts in the total reference mark (FORCE_NODALE).
They are the efforts in the total reference mark.
One defines the efforts of pressure distributed on the faces of the element (under key word PRES_REP). One
will have taken care to define meshs of skin as a preliminary
QUAD4
and to direct them suitably
outgoing normals with these meshs of skin using the control
MODI_MAILLAGE key word ORIE_PEAU_3D
7.2.5 Calculation in linear elasticity
Order
MECA
_STATIQUE
The options of postprocessing available are
SIEF_ELNO_ELGA
and
EQUI_ELNO_SIGM
.
7.2.6 Calculation in linear buckling
The option
RIGI_MECA_GE
being activated in the catalog of the element, it is possible to carry out one
conventional calculation of buckling after assembly of the matrices of elastic and geometrical rigidity.
7.2.7 Calculation in geometrical nonlinear “elasticity”
One chooses behavior ELAS under key word COMP_INCR of STAT_NON_LINE, into small
deformations (“SMALL”) or in great deformations “GREEN” under the key word DEFORMATION.
Strategy used being based on the use of a matrix of tangent rigidity during
iterations (reactualization at the beginning of pitch only), one will take care not to use another option
that that which is activated by defect, namely REAC_ITER = 0 pennies NEWTON.
Numerical integration in the thickness is carried out with 5 points of Gauss, just like in not
linear hardware.
7.2.8 Plastic nonlinear calculation
Only the criterion of Von Mises is available to date (RELATION = “VMIS_ISOT_TRAC” under
COMP_INCR). One defines the mode of calculation of the deformations as in the case of elasticity not
linear (DEFORMATION = “GREEN” or “SMALL”).
Strategy used being based on the use of a matrix of tangent rigidity during
iterations (reactualization at the beginning of pitch only), one will take care not to use another option
that that which is activated by defect, namely REAC_ITER = 0 pennies NEWTON.
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Voluminal element of hull SHB8
Date:
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J.M. PROIX, S. BAGUET, A. COMBESCURE
Key
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R3.07.07-A
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:
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R3.07: Machine elements on average surface
HT-66/04/002/A
7.3 Establishment
Options
RIGI_MECA
,
RIGI_MECA_GE
,
FORC_NODA
,
FULL_MECA
,
RIGI_MECA_TANG
,
RAPH_MECA
,
SIEF_ELGA_DEPL
,
SIEF_ELNO_ELGA
were activated in the catalog
gener_shb 3d_3.cata
. They direct all calculation towards
te0520.f
, then towards
shb8.f
.
The forces of stabilization of the element require the storage of a vector of size 12 for each
not Gauss. We chose to store these terms like additional components of
stress field.
No development was necessary for the compressive forces distributed and the forces of
following pressures. Indeed, these loadings are pressed on meshs of skin identical to those
voluminal elements 3D.

7.4 Validation
The tests validating this element are, in version 7.2 of Code_Aster:
· SSLS108 C and D: beam bored in bending, test allowing to check the absence of blocking
[V3.03.108],
· SSLS105 C: hemisphere doubly pinch [V3.03.105] conventional test to check
convergence of the element,
· SSLS123 a: sphere under external pressure [V3.03.123] to validate the loadings of
pressure and the orthotropic behavior particular to this element,
· SSLS124 A and thin b: section in bending with various twinges, to delimit the field
of use of the element [V3.03.124]. The results are correct (less than 1% with the solution
analytical) for reports/ratios of twinge (thickness/width) going from 1 to 5 10
­ 3
,
· SSLS125 a: buckling (modes of Euler) of a free cylinder under external pressure [V3.03.125]
this test makes it possible to validate the geometrical nature of rigidity,
· SSNS101 A, B and C: insulation breakdown of a cylindrical roof [V6.03.101]. This test makes it possible to validate it
geometrical nonlinear calculation and elastoplasticity,
· SSNS102 a: buckling of a hull with stiffeners in great displacements and pressure
following [V6.03.102].
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Titrate:
Voluminal element of hull SHB8
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J.M. PROIX, S. BAGUET, A. COMBESCURE
Key
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Manual of Reference
R3.07: Machine elements on average surface
HT-66/04/002/A
8 Bibliography
[1]
LEGAY A. and COMBESCURE A.: “Elastoplastic stability analysis off shells using the
physically stabilized finite element SHB8PS ", International Newspaper for Numerical Methods and
Engineering, vol. 57, 1299-1322, 2003
[2]
ABED-MERAIM F. and COMBESCURE A.: “SHB8PS has new adaptive assumed strain
continuum mechanics Shell element for impact analysis “, Computers and Structures, vol. 80,
791-803, 2002
[3]
BELYTSCHKO T. and BINDEMAN L.P.: “Assumed strain stabilization off the eight node
hexahedral elements ", Methods Computer in Applied Mechanics and Engineering, vol. 105,
225-260, 1993
[4]
FLANAGAN D.P. and BELYTSCHKO T.: “A uniform strain hexahedron and equilateral with
orthogonal hourglass control ", International Newspaper for Numerical Methods and Engineering,
Vol. 17, 679-706, 1981
[5]
RIKS E.: “Incremental Year approach to the solution off snapping and buckling problems”,
International Newspaper off Solids and Structures, vol. 15, 524-551, 1979