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Titrate:
Finite elements of voluminal hulls
Date:
14/04/05
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X. DESROCHES
Key
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R3.07.04-B
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HT-66/05/002/A
Organization (S):
EDF-R & D/AMA
Manual of Reference
R3.07 booklet: Machine elements on average surface
R3.07.04 document
Finite elements of voluminal hulls
Summary:
With an aim of supplementing the library of finite elements of plate plans [R3.07.03] currently available
in Code_Aster (DKT, DST, Q4G…), one proposes to introduce two finite elements of voluminal hull or
three-dimensional [bib1]. This new modeling COQUE_3D [U1.12.03] makes it possible to carry out calculations of
structures hull of an unspecified form with a better approximation of the geometry and
kinematics.
One will limit oneself to the framework of linear kinematics. One thus remains in small displacements and small
deformations. No restriction is made on the type of behavior in plane stresses.
The two elements which are introduced are the quadratic element quadrangle Hétérosis with 9 nodes and sound
triangular equivalent with 7 nodes. The formulation of the continuous problem is done in Cartesian co-ordinates, it
who allows to avoid explicit calculations of the curvatures. These two elements have as an agent the element
linear of hull with 3 nodes presented in the document [R3.07.02].
These two new elements are validated on existing case-tests of plate, and on three new cases
tests of hull developed in the documentation of validation and whose main conclusions are
presented briefly here.
This note also presents in appendix how to take into account the anisotropy of materials.
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Count
matters
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4.9.1
Elementary discretization of the work of the forces and external couples being exerted on
4.9.2
Elementary discretization of the work of the forces and external couples being exerted on
Code_Aster
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Finite elements of voluminal hulls
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1 Introduction
One introduces into Code_Aster two finite elements of hull voluminal with shearing
transverse (the quadrange with 9 nodes MEC3QU9H and the triangle with 7 nodes MEC3TR7H) in calculation of
structures hull of an unspecified form. To represent this type of structures, one used until
present with Code_Aster of the elements of plate with plane breakages which induced bendings
parasites and of the too restrictive hulls of revolution on the type of structure [R3.07.02].
development was carried out for isotropic materials with linear kinematics. They cannot
thus to be used that within the framework of small displacements and small deformations. This
formulation can be extended to anisotropic materials [Appendix 1] and to nonlinear kinematics
[R3.07.05].
For the resolution of chained thermomechanical problems, one must use the elements before
stop thermal hull with 7 and 9 nodes described in [R3.11.01].
One develops hereafter the mechanical continuous problem by describing the kinematics of hull of the type
Hencky-Mindlin-Naghdi (assumption of the cross-sections or plane) supplemented by a distortion
transverse and the law of thermo behavior elastoplastic. Thanks to a parameter of penalization
one can pass from a theory with shearing to a theory without shearing. One presents then
the selected finite elements which are isoparametric quadratic elements making it possible to have one
fine representation of a curved geometry and good estimates of the stresses. The interpolation and
the method of integration are also described.
One validates finally the development on some cases of test.
The nonlinear kinematics of these hulls is treated in the reference material [R3.07.05].
Code_Aster
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Finite elements of voluminal hulls
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2 Formulation
2.1
Geometry of the hull
For the elements of voluminal hull
a surface of reference is defined
, or surfaces
average, left (of curvilinear co-ordinates
1 2
for example) and a thickness
()
H
1 2
measured according to the normal on the average surface. This thickness must be small compared to
other dimensions (extensions, radii of curvature) of the structure to be modelized. The figure [Figure 2.1-
has] below illustrates our matter.
E
2
2
, X
2
O
E, X
1
1
E
3
3
, X
N,
3
1
H
Thickness H < L, B, R
1
, R
2
Solid 3D
X
Y
Z
H
L
B
R
1
R
2
Appear 2.1-a
The position of the points of the hull is given by the curvilinear co-ordinates
()
1 2
surface
average
and front elevation
3
compared to this surface.
(
)
O
K
, E
is the total Cartesian reference mark,
associated axes
()
X
K
.
Code_Aster
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Titrate:
Finite elements of voluminal hulls
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2.1.1 Geometrical description of average surface
Local natural base and bases Cartesian local
That is to say
P
an unspecified point of the average surface of reference
, one a:
(
)
COp
E
=
X
K
K
0 1 2
,
The vectors are defined
has
natural local base of the tangent plan in
P
with
, attached to
P
by:
has
COp
COp
=
=
,
and the unit normal is defined
N
by:
N
has
has
has
has
=
1
2
1
2
3
is the variable of position in the thickness associated with
N
.
(
)
has has
has
1
2
3
,
,
constitute the natural base attached to
P
.
The curvilinear frame of reference
()
1 2
not being inevitably orthogonal, the base
()
has
is not
thus not inevitably orthogonal (and even orthonormée). A local base is thus defined
orthonormée
T
K
as follows:
T
has
has
T
N T
T
N
1
1
1
2
1
3
=
=
=
,
and one notes
(
)
S S
1
2
,
the frame of reference associated with
(
)
T T
1
2
,
.
Calculation of the tensor of curvature
The tensor of curvature is related to the variation of the normal on
. It is defined by its components
mixed:
N
A.c.
,
= -
or by its components covariantes:
C
N has
N has
= -
=
.
,
,
.
. This tensor is symmetrical
since
has
has
,
,
=
. Its trace
tr
C
is the average curvature and its determinant the curvature
Gaussian.
Code_Aster
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Titrate:
Finite elements of voluminal hulls
Date:
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2.1.2 Description of the geometry of the hull
That is to say
Q
an unspecified point of
, volume of the hull thickness
H
considered constant, one a:
OQ COP PQ COP
N
+
+
=
+
3
2 H
where
[]
3
1 1
-
,
.
1
2
3
2
,
,
H
constitute a curvilinear frame of reference of
.
One can also write
OQ
according to its components
()
X
K
in the total base
()
E
K
:
OQ
E
=
X
K
K
Local natural base, bases orthonormée local and tensor metric
As for
P
, one defines the natural base of space 3D
()
G
K
attached to
Q
by:
G
OQ
has
N
G
G
G
G
G
N
=
=
+
=
=
3
3
1
2
1
2
2
H
,
,
Like
()
G
K
is not inevitably orthogonal, one defines a orthonormée local base
()
T
K
like
follows:
T
G
G
T
N T
T
N
1
1
1
2
1
3
=
=
=
,
,
and one notes
()
X
K
the frame of reference associated with
()
T
K
.
One will call
()
T
K
the local orthonormée base, and
()
X
K
co-ordinates in this base
orthonormée local.
By definition, one a:
T
OQ
E
E
K
K
J
K
J
K
J
J
X
X
X
T
=
=
=
~
~
Code_Aster
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Titrate:
Finite elements of voluminal hulls
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with
X
X
T
J
K
K
J
~
=
components of
()
T
K
in the total base
()
E
J
. (They are also them
components of the matrix of passage of
()
T
K
with
()
E
J
since the matrix of passage is
orthogonal. Thus if
T
E
K
K
J
J
T
=
one has too
E
T
K
jk
J
T
=
).
The metric tensor is defined
G
associated
Q
by its components deduced from the scalar products
vectors of the local orthonormée base:
G
ij
I
J
=
T T
.
This tensor
G
the identity is worth
Id
.
2.1.3 Notice
The figures [Figure 2.1.3-a] and [Figure 2.1.3-b] illustrate the geometrical magnitudes mentioned
above.
2
1
N
P
T has
1
,
1
T
2
has
2
Appear 2.1.3-a
P
T
1
Q
T
1
3
N
1
Appear 2.1.3-b
It should be noted that two local orthonormées bases, that associated average surface
()
T
K
and
the other with the volume of the hull
()
T
K
are confused only when the curvature is null. In it
cases the elements of hull are comparable to elements of plate
Code_Aster
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Titrate:
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2.2
Theory of the plates and the hulls
These elements are based on the theory of the plates and the hulls according to which:
2.2.1 Kinematics
2.2.1.1 Field of displacement
The cross-sections which are the sections perpendicular to average surface remain right;
material points located on a normal at not deformed average surface remain on a line
in the deformed configuration. It results from this approach that the fields of displacement vary
linearly in the thickness of the hull.
If one notes
Q'
, the position of
Q
after deformation, one a:
()
OQ
OQ QQ
OQ U
'
'
=
+
=
+
Q
where the field of displacement chosen, corresponding to the kinematics of Hencky-Mindlin, is written:
()
()
()
()
U
U
N
Q
P
H
P
P
=
+
=
3
2
0
with
.
where
()
U P
and
()
P
are respectively the vector displacement and the vector rotation of
P
, projection
of
Q
on the average surface of the hull. The fact that
()
P.
N
=
0
indicate that one does not take in
count in this kinematics rotations of the hull around its normal.
Notation:
One notes ~ the quantities expressed in the local Cartesian bases
()
T
K
or
()
T
K
for the points
P
and
Q
respectively. It results from it that:
·
the vector three-dimensional displacement
U
can be written
U
T
=
~
U
K K
or
U
E
=
U
K K
,
where it is expressed respectively in its local orthonormée base or the base
Cartesian total,
·
the vector displacement of average surface
U
can be written
U
T
=
~u
K K
or
U
E
=
U
K
K
according to whether it is expressed in its local orthonormée base or in the base
Cartesian total,
·
the vector rotation of average surface is written
=
~
T
in its orthornormée base
local.
being the rotation of the normal
N
(on average surface), one writes too
=
N
with
, vector rotation of average surface, such as
=
~
T
. The equivalence of both
formulations shows that
~
~, ~
~
1
2
2
1
=
= -
.
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2.2.1.2 Expression of the three-dimensional deformations
The tensor of deformation is calculated in the local orthonormée Cartesian base
()
T
K
. It is defined
like the half-difference of the metric tensors associated the local orthonormées bases afterwards and
before deformation. The metric tensor associated this base in the not-deformed state is simply
identity
Id
, while the metric tensor of the deformed state is
G
T T
ij
I
J
=
“. ”
with
T
OQ
'
'
~
K
K
X
=
.
Components of the tensor of deformation in
()
T
K
are thus given by:
~
~
~
~
~
~
~
~
~
~
=
+
=
+
1
2
1
2
3
3
3
U
X
U
X
U
X
U
X
The equations above are linear relations deformation-displacements. Variables of
displacement are the components
~
U
K
.
Components
~
kl
tensor
~
can also express itself according to the components in
total reference mark
U
X
p
m
. Indeed as in the total reference mark
=
ij
ij
kl
E
E
T
T
T
T
I
J
ki L
J
K
L
K
L
T T
=
=
~
one thus deduces from it immediately that
:
~
kl
ij
=
T T
ki L J
.
()
E
K
and
()
T
K
are the bases contravariantes associated
()
E
K
and
()
T
K
respectively such as:
E E
I
ij
.
J
=
and T T
I
ij
.
J
=
. Like the bases
()
E
K
and
()
T
K
are
orthonormées, their associated bases contravariantes are confused with themselves. Thus of
even manner that one had
T
E
K
K
J
J
T
=
one finds
T
E
K
K
J J
T
=
.
If one notes
T
E
T
=
T
ki
K
I
then
T T
T
T
=
=
:
~
~
kl
K
L
. For the continuation one indicates by
~
the form of the tensor of the deformations in the local reference mark orthonormé and by
the expression of
even tensor in the total reference mark. The relation of passage of the one with the other is given above in
term of tensors.
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Note:
Terms
T
kj
the terms of curvature of the hull contain
.
One notes in the relations deformation-displacements that the component
~
33
is not
determined by kinematics. This is to be associated the assumption nullity stresses
transverse normals
~
33
= 0 justified by the behavior of the hulls.
In the literature (see for example [bib3]), the modeling of the hulls by the approach based on
curvilinear components
~
U
K
displacement reveals explicitly the sizes of
curvature on the level of the form of the tensor of deformation [bib5]. Like, in general,
geometry of the hull is not known explicitly, one must thus determine
numerically the geometrical characteristics which are the vectors
G has
,
,… and them
curvatures
C
. With the finite element method it is necessary to derive them twice
functions of form (see page 20 of [bib5] and [R3.07.02]) to calculate them
C
. This can return
their vague calculation according to the family of the functions of selected form. The made error depends on
these last (linear, quadratic, cubic polynomials….) and becomes independent of
refinement of the mesh. A formulation utilizing derivation first functions of
form (calculation of slopes) does not present this disadvantage. Thus the consequent error with calculations
terms of curvature in a formulation based on the curvilinear approach does not decrease
with the refinement of the mesh whereas for the formulation described above it becomes small
by increasing the number of finite elements. Within sight of the preceding observations, approach known as
curvilinear was not followed.
2.2.2 Law of behavior
The behavior of the hulls is a behavior 3D in “plane stresses”. It binds the components
stresses and deformations, in the form of vectors, in the local orthonormée base.
transverse stress
~
33
bus is null regarded as negligible compared to the others
components of the tensor of the stresses (assumption of the plane stresses). The law of behavior
most general is written then as follows:
~
~
~
~
~
~ (,)
~
~
~
~
~
~
~
µ
11
22
12
13
23
11
11
22
22
12
1
2
=
-
-
C
HT
HT
where
~ (,)
C
µ
is the local matrix of behavior in plane stresses and
µ
represent the unit
variables intern when the behavior is nonlinear.
For behaviors where the transverse distortions are uncoupled from the deformations from
membrane and of bending,
~ (,)
C
µ
puts itself in the form:
~
~
~
C
H
H
=
0
0
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where
~ (,)
H
µ
is a matrix of behavior of membrane-bending 3x3 and
~ (,)
H
µ
a matrix of
transverse behavior of distortion 2x2. The two phenomena being uncoupled one can too
to write the behavior in the form:
~
~
~ (,) ~~
µ
MF
MF
=
C
with:
~
~
~
~
~ (,)
~
~
~
~
~
~ (,) ~
µ
µ
MF
MF
=
=
-
-
=
11
22
12
11
11
22
22
12
H
H
HT
HT
and
~
~
~
~ (,) ~~
~ (,) ~
µ
µ
=
=
=
13
23
1
2
H
H
One will remain from now on within the framework of this assumption.
For an isotropic homogeneous linear behavior elastic, one has as follows:
~
(
)
(
)
C
= -
-
-
-
E
v
v
v
v
K
v
K
v
1
1
0
0
0
1
0
0
0
0 0 1 2
0
0
0 0
0
1
2
0
0 0
0
0
1
2
2
where
K
is factor of transverse correction of shearing whose significance is given in
reference material of the elements of plate [R3.07.03], and [bib4] for more details. It
coefficient is worth 5/6 for a theory of the Reissner type and 1 within the framework of the theory of
Hencky-Mindlin. Lastly, if one chooses
K
very large, one brings back oneself to a theory of the type Coils-Kirchhoff.
One neutralizes the transverse distortion by penalization of associated energy by taking K = 10
6
H/R
(H being the thickness of the hull and R its average radius of curvature).
Always in the isotropic case, the two only nonnull components of
~
HT
are
~
iith
for i=1,2,
such as:
~
(
)
iith
ref.
T T
=
-
where
is the thermal expansion factor and
T T
ref.
-
the difference in supposed temperature
known.
Note:
One does not describe the variation thickness nor that of the transverse deformation
~
33
that one
can however calculate by using the preceding assumption of plane stresses. In addition
no restriction is made on the type of behavior in plane stresses which one can
to represent.
Same manner as
T T
=
:
~
one can deduce some
(
):
:
~
T
T
T
=
=
MF
MF
MF
and
(
):
:
~
T T
T
=
=
, which makes it possible to find
~
MF
and
~
starting from the tensor of
deformations in the total reference mark.
Code_Aster
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3
Principle of virtual work
3.1
Work of deformation
In 3D the expression of the work of deformation is written:
W
FD
C
FD
P PC
P P
FD
C
FD
FD
ij ij
H
H
S
ij ijkl kl
H
H
S
rs ir ks ijkl kp lq pq
H
H
S
rs rspq pq
H
H
S
ij ij
H
H
S
def
/
/
/
/
/
/
/
/
/
/
(~ ~)
(~ ~ ~)
(
~
)
(
)
(
)
=
=
=
=
=
-
-
-
-
-
2
2
2
2
2
2
2
2
2
2
It is checked that this expression is invariant compared to the base in which the tensors are
expressed. One chooses for the continuation of this document all to express in the local base
()
T
K
in
knowing that one passes from the local tensor of behavior to the total tensor of behavior by
relation
C
P PC
P P
rspq
ir ks ijkl K
p
L
Q
=
~
.
The general expression of the work of deformation 3D for the element of hull is worth:
W
FD
FD
FD
FD
H
H
S
H
H
S
MF
MF
H
H
S
H
H
S
def
/
/
/
/
/
/
/
/
(~ ~)
(~~ ~)
(~ ~ ~)
(~ ~ ~)
=
=
=
+
-
-
-
-
2
2
2
2
2
2
2
2
C
H
H
where
S
is average surface and the position in the thickness of the hull varies between H/2 and +h/2. It
in the expression of the work of deformation a contribution of deformation appears in
membrane-bending and a contribution of transverse shearing strain.
3.1.1 Energy interns elastic hull
It is expressed in the following way:
int
[
(~
~
~ ~)
(~
(~
~))]
=
-
+
+
+
+
+
1
2
1
2
2
112
22
2
11 22
12
2
12
22
E
G
K
FD
S
where K is the factor of correction in transverse shearing defined in paragraph 2 and
G
E
=
+
2 1
(
)
.
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3.1.2 Expression of the resulting efforts
One notes:
NR
=
=
-
+
NR
NR
NR
dz
H
H
11
22
12
11
22
12
2
2
~
~
~
/
/
;
M
=
=
-
+
M
M
M
zdz
H
H
11
22
12
11
22
12
2
2
~
~
~
/
/
;
T
=
=
-
+
T
T
dz
H
H
1
2
13
23
2
2
~
~
/
/
.
NR
NR
NR
11
22
12
,
,
are the generalized efforts of membrane (in NR/m);
M
M
M
11
22
12
,
,
are the generalized efforts of bending or moments (in NR);
T T
1
2
,
, are the generalized efforts of shearing or sharp efforts (in NR/m);
The expression of the resulting efforts that one gives here is an approximate expression which does not hold
count curvature of the hull (cf p.316 of [bib3]). The error made on these efforts is then in
H
R
2
/
where
1/R
is the average curvature. When the hull becomes plane, expressions given
above are exact and the significance of the resulting efforts can be found in [R3.07.03].
We will not develop more this aspect in addition documented well in [bib3] because the theory of
hull used here does not rest on a resulting generalized deformations formulation/efforts but
on three-dimensional/forced a deformations formulation.
3.2
Work of the forces and couples external
The work of the forces being exerted on the voluminal hull is expressed in the following way:
W
FD
dS
dzds
H
H
S
S
H
H
C
ext.
=
+
+
-
+
-
+
F U
F U
F U
v
S
C
.
.
.
/
/
/
/
2
2
2
2
where
F F F
v
S
C
,
,
are the voluminal, surface efforts and of contour being exerted on the hull,
respectively.
C
is the part of the contour of the hull on which efforts of contour
F
C
are
applied.
has) Loads given in the total reference mark:
With the kinematics of [§2.2.1], one determines as follows:
W
F U
C
dS
U
ds
F U
C
T
T
dS
U
T
T
ds
F U
C
T
T
dS
ds
I I
I I
S
I I
I I
C
I I
I
I
I
S
I I
I
I
I
C
I I
I
I
I
S
C
ext.
=
+
+
+
=
+
-
+
+
-
=
+
-
+
+
(
)
(
)
(
(~
~
))
(
(~
~
))
(
(~
~
))
(
)
2 1
1 2
2 1
1 2
2 1
1 2
U
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·
where are present on the hull:
F
F
F
1
2
3
,
,
:
surface forces acting along the axes of the reference mark
Cartesian total
F
dz
I
H
H
=
+
-
+
F E
F E
v
I
S
I
.
.
/
/
2
2
where them
E
I
are the vectors of the total Cartesian base.
C C C
1
2
3
,
,
:
surface couples acting around the axes of the reference mark
total.
C
Z
dz H
I
H
H
=
±
-
+
F E
F E
v
I
S
I
.
.
/
/
2
2
2
where them
E
I
are the vectors of the total Cartesian base.
·
and where are present on the contour of the hull:
1
2
3
,
:
linear forces acting along the axes of the reference mark
Cartesian total.
I
H
H
dz
=
-
+
F E
C
I
.
/
/
2
2
where them
E
I
are the vectors of the total Cartesian base.
1
2
3
,
,
:
linear couples acting around the axes of the total reference mark.
I
H
H
Z
dz
=
-
+
F E
C
I
.
/
/
2
2
where them
E
I
are the vectors of the total Cartesian base.
Note:
One notes too
and
linear distributions of force and moment applied to
contour of the finite element.
b) Loads given in the local reference mark:
One has then:
-
+
+
-
+
=
+
+
+
+
+
=
=
=
=
=
C
I
I
I
S
I
I
I
C
I
I
I
S
I
I
I
ds
U
T
dS
C
C
U
T
F
ds
U
T
dS
C
C
U
T
F
W
)
~
~
~
(
)
~
~
~
~
~
(
)
~
~
~
(
)
~
~
~
~
(
1
2
2
1
3
1
1
2
2
1
3
1
2
2
1
1
3
1
2
2
1
1
3
1
ext.
Expressions of
~, ~, ~
F
F
F
1
2
3
and
~, ~, ~
C C C
1 2
3
are the analogues of the expressions obtained for
F
F
F
1
2
3
,
,
and
C C C
1
2
3
,
,
by replacing them
E
I
by
T
I
.
Note:
For the couple
C
, the contribution
~
C
3
associated
N
is null in theory of hull.
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3.3
Work of the inertias
Work due to the quantities of acceleration is written:
W
ac
=
OQ OQ
.
'.
'FD
where
is the density.
It is supposed that
OQ
.
'
, the vector of acceleration of the point
Q
'
is following form:
[
]
OQ
E
E
.
K
'
&&
X
=
+
U
K
K
K
0
where one neglected the forces of Coriolis and the correction of metric in the thickness.
One notes
&&
U
D U
dt
K
K
=
2
2
, and
is the uniform vector of rotation of the total reference mark
(
)
O
K
, E
(by
report/ratio with a Galiléen reference mark which in the same beginning
O
that the total reference mark).
One expresses
in the total base
()
E
K
:
=
K K
E
For virtual displacement
OQ
'
, one a:
OQ
E
'
=
U
K
K
Work due to the quantities of acceleration becomes then:
(
)
[
]
W
U
U
X
FD W
W
ac
K
K
K
K
K
K
farmhouse
ac
hundred
ac
=
+
=
+
E
E
E
&&
0
with:
W
U U FD
mass
ac
K
K
=
&&
and:
(
)
[
]
W
U
X
FD
hundred
ac
K
K
K
K
=
E
E
0
3.4
Principle of virtual work
For a static loading, he is written in the following way:
W
W
ext.
def
=
where
W
ext.
is the sum
various elementary work, corresponding to the various loadings.
In harmonic dynamics (calculations of clean modes), the principle of virtual work
give:
W
W
ext.
farmhouse
ac
+
=
0
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4
Numerical discretization of the variational formulation
exit of the principle of virtual work
4.1 Introduction
This chapter is devoted to the discretization of the various terms of energy introduced into the chapter
precedent. The choice of framework HENCKY-MINDLIN-NAGHDI to describe the kinematics of hull,
presented at the paragraph [§2] led to expressions of the deformations where the derivative are limited
with command 1, contrary to the model of LOVE-KIRCHHOFF. One can thus use a finite element
of a nature limited while ensuring conformity (see p.110 [bib7]).
The degrees of freedom are 3 displacements in the total reference mark and 2 rotations in local reference mark.
The selected elements are isoparametric quadrangles or triangles. The quadrangle is
represented below. The quadrangles give the best results (see p.202 [bib8]).
better choice consists in taking for these elements of the quadratic functions of interpolation (see
p.224 of [bib8]) in order to modelizing the effects of membrane correctly, of bending and shearing.
According to the results based on many case-tests of the literature, the best alternative is it
quadratic isoparametric quadrangle, which makes it possible to have a fine representation of a geometry
curve and of good estimates of the stresses. One chooses among the elements with functions
quadratic the element hétérosis (Q9H) whose displacements are approached by the functions
of interpolation of the Sérendip element and rotations by the functions of the element of Lagrange
(cf Annexe3). This choice is justified hereafter.
3
3
2
1
P
1
2
4
5
6
7
8
1
3
2
3
1
2
4
5
6
7
8
1
1
= -
2
1
= -
3
1
=
Appear 4.1-a: Representations of the isoparametric quadrangle
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The figure [Figure 4.1-b] summarizes the three families of elements previously named.
Sérendip element
Hétérosis element
Element of Lagrange
U
K
, ~
~
Appear 4.1-b: Families of finite elements for the isoparametric quadrangle
Risks of bloquage or locking of membrane or shearing appear when
the thickness of the hull becomes small compared to its radius of curvature and that functions
of interpolation are of a too low nature. To solve them a selective numerical integration is used
[bib6]. For certain types of boundary conditions (embedding) with the Sérendip element it
locking persists in spite of selective integration. Moreover, for the element of Lagrange, this type
of integration leads to singularities in the matrix of rigidity. The element Hétérosis Q9H with
selective integration does not encounter the problems mentioned and seems being more
powerful for the modeling of the very thin hulls (see p.224 [bib8]). It should be noted that this
element has a mode of deformation without associated energy if it is used only. This mode
disappears when one uses more than two elements [bib7].
For the elements triangle, the element Hétérosis T7H is essential for the same reasons but proves
definitely less powerful (see paragraph 5 concerning the validation).
One decides to carry out all calculations of discretization in the total Cartesian base.
4.2
Discretization of the geometrical terms
Co-ordinates
X
k0
of a point P of average surface
are interpolated by the functions of form
in the following way:
X
NR X
K
I
ik
I
Nb
0
1 0
1
1
=
=
()
where the number
Nb1
and functions of form
()
NR
I 1
depend on the type of element chosen, and
X
ik
0
are them
co-ordinates with node I of the element.
The vectors covariants
has
(attached to the point P) are then given by:
has
E
=
=
NR
X
I
I
Nb
ik
K
()
1
1
1
0
The calculation of the vectors is avoided
T
K
because components
T
kj
the sizes of curvature contain
whose calculation is often vague like it was shown in the paragraph [§2.2.1].
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In order to avoid the presence of the terms of curvature, one writes:
N
N
I
=
=
NR
I
I
Nb
()
1
1
1
where
N
I
is the normal vector with the nodes of the element.
4.3
Discretization of the field of displacement
One adopts the following writing for displacement at the point Q:
(
)
U
E
T
T
K
1
2
=
+
-
=
=
NR
U
NR
H
I
ik
I
Nb
I
I
I
I
I
I
I
Nb
()
()
~
~
1
1
1
3
2
2
1
1
2
2
where them
T
are evaluated with the nodes, and where it is observed that the functions of interpolation
NR
I ()
2
and them
numbers
Nb2
for rotations
~
are a priori different from those used for displacements
U
K
.
By expressing them
T
I
according to their components in the total Cartesian base, one obtains:
(
)
U
E
E
=
+
+
=
=
NR
U
NR
H
T
T
I
ik K
I
Nb
I
I
I
I K
I
I K
I
Nb
K
()
()
~
~
1
1
1
3
2
2 1
1 2
1
2
2
One calculates then the various elementary terms, in order to obtain the complete discretized formulation.
In the continuation one uses the convention of summation of Einstein, while having with the spirit that the number
interpolations is
Nb1
for
X
U
K
K
0
,
N
, and
Nb2
for
~
,
T
.
4.3.1 Element Hétérosis Q9H
With this element, the number of interpolations for the geometry
()
X
k0
, N
and displacements
U
K
is
Nb1=8 (nodes nodes and mediums on the sides), while the number of interpolations for
T
and
rotations
~
is Nb2=9 (nodes nodes and mediums on the sides + barycentre). The number of degrees
of freedom total of the element is thus Nddle=3x8+2x9=42.
Functions of interpolation NR
I (1)
and NR
I (2)
respectively for the geometry and displacements, and for
rotations, can be found for example in [bib2] and are quoted in appendix 2.
The elementary vector of displacement can be put in the following form:
(
)
~
, ~, ~,…,
, ~, ~,…, ~, ~
,
Q
E
11
I
I
I
I
I
U, U, U
U, U, U
I
=
=
12
13
11
12
1
2
3
1
2
91
92
1 8
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4.3.2 Element triangle T7H
With this element Nb1=6 (nodes nodes and mediums on the sides) and Nb2=7 (nodes nodes and mediums
sides + barycentre). The number of degrees of freedom total of the element is Nddle=3x6+2x7=32.
6 functions of interpolation
NR
I ()
1
who are conventional can be found in [bib2] and are
quoted in appendix 4. On the other hand 7
NR
I ()
2
are much less and their expressions are
data in Appendix 3.
The elementary vector of displacement can be put in the following form:
(
)
~
, ~, ~,…,
, ~, ~,…, ~, ~
,
Q
E
11
I
I
I
I
I
U, U, U
U, U, U
I
=
=
12
13
11
12
1
2
3
1
2
71
72
1 6
4.3.3 Notice
One notices on the level of the elementary vector
~
Q
E
the presence of terms associated with the local base and
at the total base.
4.4
Discretization of the field of deformation
The field of deformation is expressed like the symmetrized gradient of the field of displacement:
= =
+
S U
U
U
1
2 (
)
T
Like:
[]
U
NR
Q
E
()
() ~
X
X
=
one thus has:
=
U
NR
X Q
E
()
~
where
NR
gather the functions of form
NR
I ()
1
and
NR
I ()
2
and matrices of passage
T
I K
,
X
is
the reverse of the jacobien
J
and
~
Q
E
is the vector of the degrees of freedom to the nodes (translations
U
K
and
rotations
~
).
Taking into account these relations and of
~
=
T
T
, one obtains the components of the tensor of
deformation in the local reference mark:
~
~ ~
=
B Q
E
where
~
B
is the matrix of interpolation of
~
, such as:
()
~
B T
T S J
NR
=
-
1
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Note:
If one takes again the expression of
(
)
U
E
E
U
U
()
() +
()
X
~
~
X
X
()
()
=
+
+
=
=
=
NR
U
NR
H
T
T
I
ik
K
I
Nb
I
I
I
I K
I
I K
I
Nb
K
T
R
1
1
1
3
2
2 1
1 2
1
2
2
one notices
that the terms of membrane are contained in the first part
U
T
()
X
of
U ()
X
and that them
terms of bending are contained in the second part
U
R
()
X
of
U ()
X
. Terms of
transverse shearing come from the two contributions. One obtains as follows:
~
~ ~
~
~
~
~
~ ~
m
m
F
F
=
=
=
B Q
B Q
B Q
E
E
E
where
()
()
()
~
~
[
]
~
B
T SJ
NR
B
T SJ
NR
B
T SJ
NR
1
m
MF
F
MF
H
=
=
=
-
-
-
1
1
3
2
1
2
by simple decomposition of the expression
~
~ ~
=
B Q
E
. One calls
membrane part of the deformation projection on the membrane-bending part of the field of
deformation room of the symmetrized gradient of the translations in the total reference mark. Part is called
bending of the deformation projection on the membrane-bending part of the field of deformation
room of the symmetrized gradient of rotations in the total reference mark. One calls transverse distortion
projection on the shearing part of the local field of deformation of the gradient symmetrized of
total displacement.
4.5
Stamp rigidity
The principle of virtual work is written in the following way:
W
W
ext.
def
=
that is to say still
THE U.K.U
U F
T
T
=
in matric form where
K
is the matrix of rigidity coming from the assembly
in the total reference mark of the whole of the elementary matrices of rigidity. At the elementary level
discretization of the work of deformation is written with the preceding notations:
W
D D
D
def
el
and
T
E
With
and
E
E
R
=
=
-
~
~ ~ ~ det
~
~
~ ~
Q
B C B
J
Q
Q
K Q
1
2
3
1
1
where
With
R
is the area of reference of the element.
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4.5.1 Decomposition of the elementary matrices
This matrix of rigidity includes/understands three contributions due to the deformations of membrane, of bending
and of transverse distortion. One has as follows:
~
~
~
~
K
K
K
K
E
m
E
Fe
E
=
+
+
with:
~
~
~ det
;
~
~
~ det
;
~
~
~ det
.
K
B H B
J
K
B H B
J
K
B H B
J
m
E
m
T
m
With
Fe
ft
F
With
E
T
With
D D
D
D D
D
D D
D
R
R
R
=
=
=
-
-
-
1
2
3
1
1
1
2
3
1
1
1
2
3
1
1
4.5.2 Assembly of the elementary matrices
The principle of virtual work for the whole of the elements is written:
W
W
def
def
E
E
nbelem
T
=
=
=
1
U KU
where
U
is the whole of the degrees of freedom of the discretized structure
and
K
comes from the assembly of the elementary matrices.
4.5.2.1 Degrees of freedom
The process of assembly of the elementary matrices implies that all the degrees of freedom are
expressed in the total reference mark. In the total reference mark, the degrees of freedom are the three
displacements compared to the three axes of the total Cartesian reference mark and three rotations compared to
these three axes. One thus uses, for the degrees of freedom of rotation, of the matrices of passage of
identify local orthonormé
T
with the total reference mark for each element.
4.5.2.2 Rotations
fictitious
Rotation compared to the normal with the hull is not a true degree of freedom. To ensure
compatibility between the passage of the local reference mark to the total reference mark, one thus adds a degree of freedom
additional room of rotation to the hull which is that corresponding to rotation compared to
normal on the average surface of the element. This implies an expansion of the blocks of dimension (5,5)
matrix of local rigidity in blocks of dimension (6,6) by adding a line and a column
agent with this rotation. These additional lines and these columns are a priori null. One
then carry out the passage of the matrix of local rigidity extended to the matrix of total rigidity.
In the preceding transformation, one was satisfied to add rotations compared to
normals on the surface of the elements without modifying the deformation energy. The contribution to energy
brought by these additional degrees of freedom is indeed null and no rigidity is to them
associated.
The matrix of total rigidity thus obtained presents the risk however to be noninvertible. For
to avoid this nuisance it is allowed to allot a small rigidity to these degrees of freedom
additional on the level of the matrix of widened local rigidity. Practically, one chooses it between 10
6
and 10
3
time the diagonal minor term of the matrix of rigidity of local rotation. The user can
to choose this multiplicative coefficient
COEF_RIGI_DRZ
itself in
AFFE_CARA_ELEM
; by defect it
is worth 10
5
.
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4.6
Stamp of mass
The virtual work of the effects of inertia can be expressed in the form:
W
Q
Q
farmhouse
ac
=
& & (). (D
U
U
)
It is supposed that the deformations and displacements remain sufficiently small so that the normal
on the average surface of the hull remains unchanged.
With these assumptions, we can write the field of virtual displacement:
U
U
N
(
,)
(
,)
,)
,)
Q
P
H
) (
) (
(
(
1
1
1
1
2
3
2
3
2
2
2
=
+
and the field of acceleration:
& & (
,)
&& (
,)
& &,)
,)
U
U
N
Q
P
H
) (
) (
(
(
1
1
1
1
2
3
2
3
2
2
2
=
+
In this expression, we neglected the gyroscopic terms.
4.6.1 Discretization
displacement for the matrix of mass
At the point
Q
, one takes as interpolation of the field of displacement:
U (
,
,)
,
)
,
)
Q) (
(
(
1
1
1
2
3
1
1
1
2
1
2
3
3
2
1
2
2
3
2
3
1
2
1
1
2
3
2
0
0
0
=
-
-
-
-
=
=
NR
U
U
U
H
NR
N
N
N
N
N
N
I
I
Nb
I
I
I
I
I
Nb
I
I
I
I
I
I
I
I
I
For the field of acceleration, the interpolation is written:
& & (
,
,)
,
)
&&
&&
&&
,
)
&&
&&
&&
U Q) (
(
(
1
1
1
2
3
1
1
1
2
1
2
3
3
2
1
2
2
3
2
3
1
2
1
1
2
3
2
0
0
0
=
-
-
-
-
=
=
NR
U
U
U
H
NR
N
N
N
N
N
N
I
I
Nb
I
I
I
I
I
Nb
I
I
I
I
I
I
I
I
I
We rewrite the two preceding equations in the matric form:
U
NR U
(
,)
Q
E
) (
1 2
3
=
& & (
,)
&&
U
Naked
Q
E
) (
1
2
3
=
where
NR
is the matrix of interpolation, whose expression is:
-
-
-
-
-
-
-
-
=
=
0
0
0
2
0
0
0
2
1
0
0
0
1
0
0
0
1
21
22
21
23
22
23
2 2
3
1
,
1
1
2
1
3
2
3
2
3
1
Nb
Nb
Nb
Nb
Nb
Nb
Nb
Nb
I
I
I
I
I
I
I
I
I
N
N
N
N
N
N
NR
H
N
N
N
N
N
N
NR
H
NR
NR
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The vector
U
E
is the elementary nodal vector of displacements in the total reference mark which is put
in the following form:
(
)
1
,
1
,
,
,…,
,
,
,
,…,
,
,
,
23
22
21
3
2
1
3
2
1
13
12
11
13
12
11
Nb
I
U
,
U
,
U
U
,
U
,
U
Nb
Nb
Nb
I
I
I
I
I
I
E
=
=
U
4.6.2 Stamp of elementary mass
With the preceding notations, the virtual work of the effects of inertia is put in the matric form
following:
W
farmhouse
inertia
and
E E
=
U M u&&
with
M
E
the matrix of coherent mass which can be expressed in the form:
M
NR NR
J
E
E
=
T
det ((
D D
D
3
1
2
3
))
It is important to note that because of the curvature, a coupling of the terms of translation with those
of rotation is possible (indeed,
det (())
J
3
is not constant in the thickness).
4.6.3 Assembly of the elementary matrices of mass
The assembly of the matrices of mass follows same logic as that of the matrices of rigidity.
degrees of freedom are the same ones and one finds the processing specific to normal rotations to
surface hull. Although the matrix of coherent mass is built in the total reference mark, it
remain singular compared to the rotation of the normal in each node. We need
to supply this matrix on the basis of the variational form:
W
m
inertia
E
I
I
Nb
I
I
I
N
N
N
=
=
1
2
(
)
&&
where
m
E
is selected constant by element and calculated according to the formula:
m
Cm
E
=
max
m
max
being the major term due to rotations (in the local reference mark of the element) on the diagonal
matrix
M
E
. It is thus to note that with this intention it was necessary to bring back the contribution
the rotations initially expressed in the total reference mark of the element, the local reference mark of
the element by change of reference mark.
For modal calculations utilizing at the same time the calculation of the matrix of rigidity and that of
stamp of mass, it is necessary to take a mass on the degree of normal rotation on the surface of the hull
being worth C time the diagonal minor term of the matrix of mass for the terms of rotation in
identify local, where C is worth between 10
6
and 10
3
. One chooses to confuse the values of this coefficient with
those of
COEF_RIGI_DRZ
for the equivalent operation on the matrix of rigidity. By defect
C
is worth
thus 10
5
. That makes it possible to inhibit, during a modal analysis, the modes being able to appear on
additional degree of freedom of rotation around the normal on the surface of the hull.
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4.7
Numerical integration for elasticity
4.7.1 Integration
surface
For the element Hétérosis Q9H the bending part of the matrix of stiffness is integrated classically
with 9 points of Gauss while the parts membrane and shearing are obtained by integration
reduced with 4 points by Gauss.
For element T7H, by analogy with Q9H, the matrix of stiffness is obtained with 7 points
of integration of Hammer for the bending part and 3 points of integration of Hammer for the parts
shearing and membrane.
Cordonnées of the points
Weight
1
1
1 3
1 3
=
=
/;
/
9/80
2
2
=
=
has
has
;
has
= +
6
15
21
With =
155
15
2400
+
3
3
1 2
= -
=
has
has
;
With
4
4
1 2
=
= -
has
has
;
With
5
5
=
=
B
B
;
B
has
=
-
4 7
/
31/240 - A
6
6
1 2
= -
=
B
B
;
31/240 - A
7
7
1 2
=
= -
B
B
;
31/240 - A
y
D D
(,)
=
-
0
1
0
1
I
I
I
I
N
y (,)
=
1
Normal numerical formulas of integration on triangle T7H (Hammer)
X-coordinates of the points
Weight
Ordinates of the points
Weight
µ
1
3 5
= -
/
5/9
1
3 5
= -
/
5/9
2
0
=
8/9
2
0
=
8/9
3
3 5
= +
/
5/9
2
3 5
= +
/
5/9
y
D D
(,)
=
-
-
1
1
1
1
J
N
I
J
I
J
I
N
y
=
=
1
1
µ
(,
)
Normal numerical formulas of integration 3x3 on quadrangle Q9H (Gauss)
The principle of reduced integration consists in evaluating the membrane and shearing strains
at the points of reduced integration and to extrapolate them at the points of conventional integration. This returns to
to suppose that these deformations are bilinear on element Q9H and linear on the T7H. Them
functions of form chosen to make this extrapolation are related conventional to form
bilinear of the quadrangle with 4 nodes for the Q9H and linear of the triangle with 3 nodes for the being worth T7H
1 at the points of reduced integration.
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For more details on the principle of reduced or selective integration, one can refer to [bib6].
Cordonnées of the points
Weight
1
1
1 6
1 6
=
=
/;
/
1/6
2
2
2 3
1 6
=
=
/;
/
1/6
3
3
1 6
2 3
=
=
/;
/
1/6
y
D D
(,)
=
-
0
1
0
1
I
I
I
I
N
y (,)
=
1
Numerical formulas of integration reduced on triangle T7H (Hammer)
For the elements quadrangle an integration of Gauss 2x2 is used.
Cordonnées of the points
Weight
1
1
1
3
1
3
=
=
/
;
/
1
2
2
1
3
1
3
=
= -
/
;
/
1
3
3
1
3
1
3
= -
=
/
;
/
1
3
3
1
3
1
3
= -
= -
/
;
/
1
y
D D
(,)
=
-
-
1
1
1
1
I
I
I
I
N
y (,)
=
1
Reduced numerical formulas of integration 2x2 on quadrangle Q9H (Gauss)
4.7.2 Integration in the thickness
Integration in the thickness is made with three points for the two elements.
Cordonnées of the points
Weight
1
1
= -
1/3
2
0
=
4/3
3
1
= +
1/3
y
D
()
=
-
1
1
I
I
I
N
y ()
=
1
Formulate numerical integration in the thickness in elasticity
4.8
Numerical integration for plasticity
The principle of surface integration remains the same one as in elasticity, but the initial thickness is
divided into NR identical layers thicknesses. There are three points of integration per layer. Points
of integration are located in higher skin of layer, in the middle of the layer and in lower skin
of layer. For NR layers, the number of points of integration is of 2N+1. One advises to use of 3
with 5 layers in the thickness for a number of points of integration being worth 7, 9 and 11 respectively.
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For rigidity, one calculates for each layer, in plane stresses, the contribution to the matrices of
rigidity of membrane, bending and transverse distortion. These contributions are added and
assemblies to obtain the matrix of total tangent rigidity.
For each layer, one calculates the state of the stresses (
11
,
22
,
12
) and the whole of the variables
interns, in the middle of the layer and in skins higher and lower of layer, from
local plastic behavior and of the local field of deformation (
11
,
22
,
12
).
The positioning of
points of integration enables us to have the rightest estimates, because not extrapolated, in skins
lower and higher of layer, where it is known that the stresses are likely to be maximum.
plastic behavior does not include/understand for the moment the terms of transverse shearing which
are treated in an elastic way, because transverse shearing is uncoupled from the behavior
membrane in plane stresses.
Cordonnées of the points
Weight
1
1
= -
1/3
2
0
=
4/3
3
1
= +
1/3
y
D
()
=
-
1
1
I
I
I
N
y ()
=
1
Formulate numerical integration for a layer in the thickness in plasticity
Note:
One already mentioned with [§2.2.2] that the value of the coefficient of correction in shearing
transverse for the elements of plate and hull was obtained by identification of
elastic complementary energies after resolution of balance 3D. This method is not
more usable in elastoplasticity and the choice of the coefficient of correction in shearing
transverse is posed then. The transverse terms of shearing are thus not affected
by plasticity and are treated elastically, for want of anything better. If one places oneself in
theory of Coils-Kirchhoff for a value of this coefficient of 10
6
H/R (H being the thickness of
the hull and R its average radius of curvature) transverse terms of shearing
become negligible and the approach is more rigorous.
4.9
Discretization of elementary work for the loadings
4.9.1 Discretization
elementary of the work of the forces and external couples being exerted
on average surface
According to the paragraph [§3.2], one recalls that one has for these efforts and couples:
W
dS
ext.
S
=
+
(
)
F u.a.
where
S
is the average surface of the hull.
For the first term of this expression one has as follows:
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4.9.1.1 Loads given in the total reference mark
W
F NR U
C NR
T
T
H
D D
ext.
K
I
ik
K
J
J
J K
J
J K
With
R
=
+
-
°
[
(~
~
)] det
1
2
2 1
1 2
1
2
2
J
with
det
det (
)
J
J
° =
=
3
0
4.9.1.2 Loads given in the local reference mark
W
F NR T
NR U
C T NR
T
T
H
D D
ext.
J J K
I
ik
K
J
J
J K
J
J K
With
R
=
+
-
°
[
(~
~
)] det
1
1
2
2 1
1 2
1
2
2
J
4.9.2 Discretization
elementary of the work of the forces and external couples being exerted
on contour
According to the paragraph [§3.2], one recalls that one has for these efforts and couples:
W
ds
ext.
C
=
+
(
)
U
where
C
is the average contour of the hull.
and
linear distributions of force and moment
applied to the contour of the hull in the total reference mark.
The discretization gives then:
W
NR U
NR
T
T
ds
ext.
K
I
ik
K
J
J
J K
J
J K
C
=
+
-
[
(~
~
)]
1
2
2 1
1 2
4.9.3 Discretization of the term of gravity
One has for this term:
W
FD
G U
FD
G NR U
NR
T
T
FD
E
E
E
K
K
K
I
ik
J
J
J K
J J K
pes
Q
Q
=
=
=
+
-
G U ()
()
[
(~
~
)]
1
3
2
2 1
1 2
2
That is to say:
W
G NR U FD
pes
K
I
ik
E
=
1
by supposing negligible the second term of expression Ci
above.
4.9.4 Discretization of the term of pressure
It is supposed that the pressure p is applied to average surface
hull. One has then:
W
ep
P dS
ep
P D D
near
With
With
R
R
=
=
N U
has
has
U
()
(
) ()
1
2
1
2
where e=
±
1 according to whether p is applied in internal or external skin.
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Like
has
E
=
has
K K
, this is still written:
W
epN U v D D
near
I
ik K
With
R
=
1
1
2
where
(
)
v
v
v
J J
J J
J J
J J
J J
J J
J
J
ij
ij
1
2
3
12 23
13 22
13 21
11 23
11 22
12 21
3
0
=
-
-
-
=
=
°
°
°
°
°
°
°
°
°
°
°
°
°
,
.
4.9.5 Discretization of the terms of centrifugal inertia
One adds with the expression of the field of accelerations of the paragraph [§4.6] the corresponding term
with the accelerative forces centrifuges if the total reference mark
(
)
O
K
,
E
is in uniform rotation
by
report/ratio with a Galiléen reference mark which in the same beginning
O
that the total reference mark. The expression of the field
accelerations becomes as follows:
& & (
,)
&& (
,)
& &,)
,)
[
]
U
U
N
COp
Q
P
H
) (
) (
(
(
1
1
1
1
2
3
2
3
2
2
2
=
+
+
where one neglected the forces of Coriolis and the correction of metric in the thickness.
One expresses
in the total base (
E
K
):
=
K K
E
.
By taking again the expression of:
W
Q
Q D
inertia
=
& & (). (
U
U
)
the contribution of the terms is identified
of centrifugal inertia:
W
U
X
FD
hundred
E
inertia
K K
K K
=
E
E
[
(
)]
0
by neglecting the terms of rotation
in virtual displacement. The terms of mass are unchanged compared to [§4.6].
Like one a:
=
=
X
X
X E
K
K
p
p
K
K
p
K
qpk
Q
0
0
0
E
E
E
E
where
E
qpk
is the permutation of Lévi-Strauss.
One also writes:
(
=
X
E
E
X
K
K
qpk
srq
R
p
K
K
0
0
E
E
)
From where it results from it that:
W
U NR
E
E
X NR
D D D
hundred
inertia
is
I
qpk srq
R
p
jk
J
Ar
=
-
()
()
det
1
0
1
1
2
3
1
1
J
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4.9.6 Taking into account of the loadings of thermal dilation
One treats only the case where elastic thermo characteristics
E
,
,
depend only on
average temperature
T
in the thickness. Moreover, the material is thermo isotropic rubber band
homogeneous in the thickness.
The variational formulation of work due to thermal dilations is written:
3
2
1
1
1
3
2
1
HT
1
1
HT
HT
det
)
(
~
~
~
det
~
~
~
~
)
~
(
~
~
D
D
D
T
T
D
D
D
FD
W
ref.
With
T
E
With
T
E
R
R
E
J
Id
C
B
Q
J
C
B
Q
C
-
=
=
-
-
=
+
-
+
-
T
T
The temperature is represented by the model of thermics to three fields according to [R3.11.01]:
(
)
()
()
()
()
()
()
T
T
P
T
P
T
P
m
S
I
,
.
.
.
3
1
3
2
3
3
3
=
+
+
,
with:
()
P
J
3
: three polynomials of LAGRANGE in the thickness:
]
[
- +
1 1
,
:
()
()
()
(
)
()
(
)
P
P
P
1 3
3
2
2
3
3
3
3 3
3
3
1
2 1
2 1
= -
=
+
= -
-
;
;
;
From the representation of the temperature above, one obtains:
·
the average temperature in the thickness:
()
(
)
() () ()
(
)
T
T
D
T
T
T
m
S
I
,
=
=
+
+
-
+
1
2
1
6 4
3
3
1
1
;
·
the average variation in temperature in the thickness:
()
(
)
()
()
$
,
T
T
D
T
T
S
I
=
=
-
-
+
3
3
3
3
1
1
;
Thus the temperature can be written in the following way:
(
)
() ()
(
)
T
T
T
T
,
$
.
/
~
,
3
3
3
2
=
+
+
such as:
(
)
(
)
~
,
;
~
,
T
T
3
1
1
3
3
1
1
0
0
-
+
-
+
=
=
.
If the temperature is indeed closely connected in the thickness one has,
~
T
=
0
.
It is necessary to evaluate the three-dimensional thermal stresses, in each point of integration
in the thickness. These stresses of thermal origin withdrawn from the mechanical stresses
usual are calculated at the points of integration in the thickness by:
(
)
~
.
$. /
ther
ref.
E T T
T
= -
-
+
1
2
2
3
Code_Aster
®
Version
7.4
Titrate:
Finite elements of voluminal hulls
Date:
14/04/05
Author (S):
X. DESROCHES
Key
:
R3.07.04-B
Page
:
31/42
Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
4.9.7 Assembly
The variational formulation of the work of the efforts external for the unit of the elements is written
then:
W
W
E
E
nbelem
T
ext.
ext.
U F
=
=
=
1
where
U
is the whole of the degrees of freedom of the discretized structure and
F
comes from the assembly of the vectors forces elementary.
As for the matrices of rigidity, the process of assembly of the vectors forces elementary
imply that all the degrees of freedom are expressed in the total reference mark. In the total reference mark,
the degrees of freedom are three displacements compared to the three axes of the total Cartesian reference mark
and three rotations compared to these three axes. Matrices of passage of the reference mark are thus used
room with the total reference mark for rotations of each element.
Note:
The external efforts can also be defined in the reference mark user. One then is used
stamp passage of the reference mark user towards the local reference mark of the element to have the expression
of these efforts in the local reference mark of the element and to deduce the vector from it elementary room forces
corresponding. For the assembly one passes then from the local reference mark of the element to the total reference mark.
5 Validation
To judge relevance of thick the hull formulation, few examples of application
according to relate to as well linear statics as the calculation of clean modes. Three new cases
tests relating to the two finite elements described in the preceding parts were integrated in
Code_Aster. They come to enrich the case-tests by the elements of plate already present in
environment of Code_Aster. Most of these case-tests were indexed in [bib10].
The three new case-tests, two in statics plus one in dynamics, are conventional examples
of validation drawn from [bib3]. Reference solutions, analytical or numerical, resulting from [bib3]
are compared with the numerical results given by Code_Aster. For more information on these
case-tests, one will refer to the documentation of validation indicated in reference.
Code_Aster
®
Version
7.4
Titrate:
Finite elements of voluminal hulls
Date:
14/04/05
Author (S):
X. DESROCHES
Key
:
R3.07.04-B
Page
:
32/42
Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
5.1
Case test in linear statics
5.1.1 Static case test n° 1
The first case test is that of a cylindrical panel subjected to its own weight [V3.03.107].
This test makes it possible to highlight effects of membrane more important than those of bending. It
allows to measure the performance of the elements hulls compared to elements DKT or DKQ of which
the interpolation out of membrane is linear.
5.1.2 Static case test n° 2
The second case test is that of a helicoid hull subjected to two concentrated types of loading
[V3.03.108].
The helicoid shape of the hull makes it possible to study the geometrical representation of the finite elements.
The concentrated loadings can be:
·
in the plan: the influence due to the effects of membrane is then not important and it
behavior dominating is that due to the bending,
·
except plan: the effects of membrane affect the behavior of the hull.
5.2
Case test in dynamics
This case test is a simplified model of vane of compressor, which is in fact a cylindrical panel
[V2.03.102].
This test highlights the performances of the elements in dynamic behavior by the data
frequencies and clean modes.
The frequencies and clean modes of the vane are experimental values which are used as results
of reference.
6 Chaining
thermomechanics
6.1 Description
For the resolution of chained thermomechanical problems, one must use for thermal calculation
finite elements of thermal hull [R3.11.01] whose field of temperature is recovered like
input datum of Code_Aster for mechanical calculation. It is necessary thus that there is compatibility between
thermal field given by the thermal hulls and that recovered by the mechanical hulls. It
the last is defined by the knowledge of the 3 fields
TEMP_SUP
,
TEMP
and
TEMP_INF
given in skins
lower, medium and higher of hull.
The table below indicates compatibilities between the elements of mechanical hull and hull
thermics.
Modeling
THERMICS
Net
Finite element
to use with Mesh Element
finished
Modeling
MECHANICS
HULL QUAD9
THCOQU9
//////////////
QUAD9 MEC3QU9H COQUE_3D
HULL TRIA7
THCOTR7///////////////
TRIA7 MEC3TR7H COQUE_3D
Code_Aster
®
Version
7.4
Titrate:
Finite elements of voluminal hulls
Date:
14/04/05
Author (S):
X. DESROCHES
Key
:
R3.07.04-B
Page
:
33/42
Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
Note:
·
The nodes of the thermal elements of hulls and mechanical hulls must
to correspond. The mesh for thermics and mechanics will thus have the same number
and the same type of meshs.
·
The elements of thermal hulls surface are treated like plane elements by
projection of the initial geometry on the level defined by the first 3 nodes.
The thermomechanical chaining is also possible if one knows by experimental measurements
variation of the field of temperature in the thickness of the structure or certain parts of
structure. In this case one works with a card of temperature defined a priori; the field of
temperature is not given any more by the three values
TEMP_INF
,
TEMP
and
TEMP_SUP
thermal calculation
obtained by
EVOL_THER
. It can be much richer and contain an arbitrary number of points
of discretization in the thickness of the hull. The operator
DEFI_NAPPE
allows to create of such
profiles of temperatures starting from the data provided by the user. These profiles are affected by
order
AFFE_CARTE
(cf the case-test HSNS100B). It will be noted that it is not necessary for
mechanical calculation that the number of points of integration in the thickness is equal to the number of points
of discretization of the field of temperature in the thickness. The field of temperature is
automatically interpolated at the points of integration in the thickness of the elements of hulls.
6.2 Case-test
The case-tests for the thermomechanical chaining enters of the thermal elements of hulls and of
mechanical elements of hulls are the HPLA100C (elements MEC3QU9H) and HPLA100D (elements
MEC3TR7H). It is about a heavy thermoelastic hollow roll in uniform rotation [V7.01.100]
subjected to a phenomenon of thermal dilation where the fields of temperature are calculated with
THER_LINEAIRE
by a stationary calculation.
Z
R
R
E
R
I
F
B
D
With
C
R
Z
J
H
+
Interior radius
R
I
= 19.5 mm
External radius
R
E
= 20.5 mm
Not F
R = 20.0 mm
Thickness
H = 1.0 mm
Height
L = 10.0 mm
y
Z
X
K
L
M
NR
P
Q
Thermal dilation is worth:
()
()
(
) (
)
(
)
T
T
T
T
T
T R R H
ref.
S
I
S
I
-
=
+
+
+
-
0 5
2
.
.
/
with:
·
T
C
T
C
T
C
S
I
ref.
=
°
= -
°
=
°
0 5
0 5
0
.
,
.
,
.
·
T
C
T
C
T
C
S
I
ref.
=
°
=
°
=
°
01
01
0
.
,
.
,
.
One tests the stresses, the efforts and bending moments in L and Mr. the results of reference are
analytical. One obtains very good results whatever the type of element considered.
Code_Aster
®
Version
7.4
Titrate:
Finite elements of voluminal hulls
Date:
14/04/05
Author (S):
X. DESROCHES
Key
:
R3.07.04-B
Page
:
34/42
Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
7
Establishment of the elements of hull in Code_Aster
7.1 Description
These elements (of names
MEC3TR7H
and
MEC3QU9H
) are pressed on meshs
TRIA7
and
QUAD9
curves.
These elements are not exact with the nodes and it is necessary to net with several elements to obtain
correct results.
7.2
Introduced use and developments
These elements are used in the following way:
MA = CREA_MAILLAGE (MESH: MAILINI
MODI_MAILLE: (OPTION:“QUAD8_9”
ALL:“YES”)…)
One calls upon a routine
MODI_MAILLE
of amendment of the mesh to pass from the elements
quadrangles with 8 nodes with the elements quadrangles with 9 nodes or many elements triangles to 6
nodes with the elements triangles with 7 nodes.
AFFE_MODELE (MODELING: “COQUE_3D”…)
for the triangle and the quadrangle
One calls upon the routine
INI080
for the position of the points of Hammer and Gauss on the surface of
the corresponding hull and weights.
AFFE_CARA_ELEM (HULL:(THICKNESS:“EP”
ANGL_REP: ('
''
')
COEF_RIGI_DRZ: “CTOR”)
To make postprocessings (forced, generalized efforts,…) in a reference mark chosen by
the user who is not the local reference mark of the element, one defines the X1 direction of the reference mark user
like the projection of a direction of reference D on surface
element. This direction of
reference D is chosen by the user who defines it by two nautical angles in the total reference mark.
The normal NR on the surface of the element fixes the second direction at the point of observation concerned.
vector product of two vectors previously definite Y1=N
X1 makes it possible to define the local trihedron
in which will be expressed the generalized efforts representing the state of stresses. The user
will have to take care that the selected reference axis is not found parallel with the normal of some
elements of hull. By defect, the direction of reference D is axis X of the total reference mark of definition of
mesh.
The value
CTOR
corresponds to coefficent that the user can introduce for the processing of the terms
of rigidity and mass according to normal rotation on the surface of the hull. This coefficient must be
sufficient small not to disturb the energy balance of the element and not too small so that
the matrices of rigidity and mass are invertible. A value of 10
5
is put by defect.
ELAS: (E:NAKED Young:
ALPHA:
. RHO:
. )
For an elastic thermo behavior isotropic homogeneous in the thickness one uses the key word
ELAS
in
DEFI_MATERIAU
where the coefficients are defined
E
, Young modulus,
,
coefficient of
Poisson,
,
thermal expansion factor and
RHO
density
.
AFFE_CHAR_MECA (DDL_IMPO: (
DX:. DY:. DZ:. DRX:. DRY:. DRZ:. DDL
of hull in the total reference mark.
FORCE_COQUE: (FX:. FY:. FZ:. MX:. MY:. MZ:. )
. They are the efforts
surface on elements of hull. These efforts can be given in the total reference mark or
in the reference mark user defined by
ANGL_REP
.
FORCE_NODALE: (FX:. FY:. FZ:. MX:. MY:. MZ:. )
. They are the efforts
hull in the total reference mark.
Code_Aster
®
Version
7.4
Titrate:
Finite elements of voluminal hulls
Date:
14/04/05
Author (S):
X. DESROCHES
Key
:
R3.07.04-B
Page
:
35/42
Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
7.3
Calculation in linear elasticity
The matrix of rigidity and the matrix of mass (respectively options
RIGI_MECA
and
MASS_MECA
)
are integrated numerically in
TE0401
and
TE0406
, respectively
.
Calculation takes account of
fact that the terms corresponding to the DDL of rotation of hull are expressed in the local reference mark of
the element. A matrix of passage makes it possible to pass from the local DDL to the total DDL.
Elementary calculations (
CALC_ELEM
) currently available correspond to the options:
·
EPSI_ELNO_DEPL
and
SIGM_ELNO_DEPL
who provide the strains and the stresses
with the nodes in the reference mark user of the element in lower skin, with semi thickness and in
higher skin of hull, the position being specified by the user. Calculation is carried out in
TE0410.
One stores these values in the following way: 6 components of deformation or
stresses,
·
EPXX EPYY EPZZ EPXY EPXZ EPYZ
or
SIXX SIYY SIZZ SIXY SIXZ SIYZ
,
·
EFGE_ELNO_DEPL
: who gives the efforts generalize by element with the nodes from
displacements:
NXX, NYY, NXY, MXX, MYY, MXY, QX, QY
. This option is calculated
in
TE0410
,
·
SIEF_ELGA_DEPL
: who gives the stresses by element to the points of Gauss in
identify local element starting from displacements:
SIXX, SIYY, SIZZ, SIXY, SIXZ,
SIYZ
. This option is calculated in
TE0410
,
·
EPOT_ELEM_DEPL
: who gives the elastic energy of deformation per element from
displacements. This option is calculated in
TE0401
,
·
ECIN_ELEM_DEPL
: who gives the kinetic energy by element. This option is calculated in
TE0401
,
Finally it
TE0416
calculate also the option
FORC_NODA
of calculation of the nodal forces for the operator
CALC_NO
.
7.4
Plastic design
The matrix of rigidity is also integrated numerically, by layers, in
TE0414
. One calls upon
the option of calculation
STAT_NON_LINE
in which one defines in the level of the nonlinear behavior it
a number of layers to be used for numerical integration. All laws of plane stresses
available in Code_Aster can be used.
STAT_NON_LINE (….
COMP_INCR: (RELATION:''
COQUE_NCOU:“A NUMBER OF LAYERS”)
….)
Elementary calculations (
CALC_ELEM
) currently available correspond to the options:
·
EPSI_ELNO_DEPL
who provides the deformations by element to the nodes in the reference mark
user starting from displacements, in lower skin, with semi thickness and in skin
higher of hull. This option is calculated in
TE0410
,
·
SIGM_ELNO_COQU
who allows to obtain the stress field in the thickness by element
with the nodes for a given layer and a position requested (in lower skin, with
medium or in higher skin of layer). These values are given in the reference mark
user. This option is calculated in
TE0415
,
·
SIEF_ELNO_ELGA
who allows to obtain the efforts generalized by element with the nodes in
the reference mark user. This option is calculated in
TE0415
,
·
VARI_ELNO_ELGA
who calculates the field of internal variables and the stresses by element
with the nodes for all the layers, in the local reference mark of the element. This option is
calculated in
TE0415
.
Code_Aster
®
Version
7.4
Titrate:
Finite elements of voluminal hulls
Date:
14/04/05
Author (S):
X. DESROCHES
Key
:
R3.07.04-B
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:
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Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
8 Conclusion
The finite elements of hull curves which we describe here are used in the structural analyzes
thin curves whose thickness report/ratio over characteristic length is lower than 1/10. Two
finite elements of voluminal hull being pressed on quadrangular and triangular meshs were
introduced into Code_Aster. They were selected with a quite particular aim: to be able to represent one
complete behavior of curved structures whereas until now one could only use
elements with plane breakages which induced parasitic bendings and required to refine them
mesh.
It is elements for which strains and stresses in the plan of the element
vary linearly with the thickness of the hull. Selected kinematics is a kinematics hull
of Hencky-Mindlin-Naghdi type allowing to utilize the transverse energy of shearing.
distortion associated with transverse shearing is constant in the thickness of the element.
variable correction on the coefficient K of shearing transverse offers a flexibility in use
allowing to pass from the theory of HENCKY-MINDLIN-NAGHDI for k=1, with that of REISSNER
for k=5/6 and with that of LOVE_KIRCHHOFF (for very mean structures) if a value is chosen
K equalizes with
10
6
×
H L
/
H being the thickness and L a characteristic distance (radius of curvature
means, area of application of the loads….). As in this last case, one uses a method of
penalization to make small the terms of shearing transverse, one can, if a value is taken
K too important, to make singular the numerical system. In this case, it is necessary to decrease the value of
k.
The default value of K is 5/6. It is generally used when the structure to be netted has one
thickness report/ratio over characteristic length ranging between 1/20 and 1/10. For thicknesses more
weak where the transverse distortion becomes low one can want to use a value of k=
10
6
×
H L
/
(to be able to make comparisons with elements of plate DKT for example). When
transverse distortion is nonnull, the elements of hull do not satisfy the equilibrium conditions
3D and boundary conditions on the nullity of stresses shear transverse on the faces
higher and lower of hull, compatible with a constant transverse distortion in
the thickness of the hull. It results from it thus that on the level from the behavior a coefficient from 5/6 for
a homogeneous hull corrects the usual relation between the stresses and the transverse distortion of
way to ensure the equality enters energies of shearing of the model 3D and the model of hull to
constant distortion. In this case, the arrow
~
U
3
has as an interpretation average transverse displacement
in the thickness of the hull and not the displacement of the average surface of the hull.
For structures low thickness in order to avoid the phenomena of blocking, one uses
under-integration reduced for the parts membrane and shearing of the matrix of rigidity. The choice
on the finite elements went on the elements quadrangle Hétérosis Q9H and triangle T7H. Indeed,
among the finite elements with quadratic functions of interpolation, the performance of the Hétérosis element
Q9H is known. It is in particular higher than that of the elements Sérendip Q9S or the elements
of Lagrange Q9. This performance rests however on the selective integration of the element with
reduced integration of the terms of membrane and shearing on the one hand, and normal integration of
terms of bending in addition. By analogy with Q9H, one took the finite element T7H like element of
triangular form. However, as far as possible, one will use the Q9H rather than the T7H which is
definitely less powerful.
The non-linear behaviors in plane stresses are available for these elements. One
announce however that the stresses generated by the transverse distortion are treated
elastically, for want of anything better. Indeed the rigorous taking into account of a transverse shearing
constant not no one on the thickness and the determination of the correction associated on rigidity with
shearing compared to a model satisfying the equilibrium conditions and the boundary conditions
are not possible and thus return the use of these elements, when transverse shearing
is nonnull, rigorously impossible in plasticity. Rigorously, for behaviors not
linear, it would thus be necessary to use these elements within the framework of the theory of Coils-Kirchhoff.
Elements corresponding to the machine elements exist in thermics; chainings
thermo mechanics is thus available with finite elements of thermal hulls to 7 and 9
nodes. Extensions of the preceding formulation presented in appendix allow also the catch
in account of the anisotropy of materials and kinematic non-linearity. This second extension
is operational in Code_Aster and is the subject of a reference material [R3.07.05].
Code_Aster
®
Version
7.4
Titrate:
Finite elements of voluminal hulls
Date:
14/04/05
Author (S):
X. DESROCHES
Key
:
R3.07.04-B
Page
:
37/42
Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
9 Bibliography
[1]
S. Ahmad, Will go B.M., O.C. Zienkiewicz, “Analysis off thick and thin Shell structures by curved
finite elements “, IJNME, Vol.2, p.419-451,1970.
[2]
J.L. Batoz, G. Dhatt, “Modeling of the structures by finite elements”, Volume 1, Solids
rubber bands - Hermès, Paris, 1990.
[3]
J.L. Batoz, G. Dhatt, “Modeling of the structures by finite elements”, Volume 3, Hulls -
Hermès, Paris, 1992.
[4]
B. Bui, “shearing in the plates and hulls: modeling and calculation”, Note
HI-71/7784, 1992.
[5]
D. Bui, “Modeling of the hulls thicknesses average by an approach 3D “degenerated””,
Note EDF-DER HI-74/95/013, 1992.
[6]
E.Carnoy, G. Laschet, “Elements of isoparametric hull”, LTAS, Report/ratio SF-108,
November 1992.
[7]
T.J.R. Hughes, “The Finite Method Element”, Prentice-Hall, 1987.
[8]
J.F. Imbert, “Analyzes strutures by finite elements”, 3rd edition - Cepaduès Editions,
1992.
[9]
E. Lorentz “a non-linear relation of behavior hyperelastic”, Note EDF-DER
HI-74/95/011/0.
[10]
P. Massin, “Functionalities available for the elements of hulls and plates in
Code_Aster “, Note EDF-DER HI-74/97/027/0.
[11]
O.C. Zienkiewicz, “The finite elements method”, 3nd edition - Mc Graw-Hill 1977.
[12]
R3.07.02:F. Voldoire, C. Sevin, “thermoelastic Hulls axisymmetric and 1D”, Manual
of reference of Code_Aster.
[13]
R3.07.03: P. Massin, “Elements of plate DKT, DST, DKQ, DSQ and Q4G”, Manual of
reference of Code_Aster.
[14]
R3.07.05: P. Massin, Mr. Al Mikdad, “Finite elements of voluminal hull into nonlinear
geometrical “, Manual of reference of Code_Aster.
[15]
R3.11.01: P. Massin, F. Voldoire, S. Andrieux, “Model of thermics for the hulls
thin ", Manual of reference of Code_Aster.
[16]
V2.03.102: P. Massin, A. Laulusa, “free Vibrations of a vane of compression”, Manual
of validation of Code_Aster.
[17]
V3.03.107: P. Massin, D. Bui, A. Laulusa, “cylindrical Panel subjected to its own weight”,
Manual of validation of Code_Aster.
[18]
V3.03.108: P. Massin, D. Bui, A. Laulusa, “helicoid Hull under concentrated loadings”,
Manual of validation of Code_Aster.
[19]
V7.01.100: P. Massin, F. Voldoire, “heavy thermoelastic Hollow roll in rotation
uniform “, Manual of validation of Code_Aster.
Code_Aster
®
Version
7.4
Titrate:
Finite elements of voluminal hulls
Date:
14/04/05
Author (S):
X. DESROCHES
Key
:
R3.07.04-B
Page
:
38/42
Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
Appendix 1 Extension to anisotropic materials not
programmed
It is considered that the hull consists of an orthotropic material, axes of orthotropism
~
~
X
K
associated the base
K
K
. The law of behavior in these axes is written:
~
~
~
~
~
~
(
)
(
) (
)
6 1
6.6 6 1
X
K
X
X
=
S
where
~
S
is the matrix of flexibility of the component
K
.
Are
~
and
~
, tensors of strain and stresses in the axes
~
X
K
, one a:
~
~
~
~
~
~
=
=
T
T
Q Q
Q Q
where
[
]
Q
T T T
=
1
2
3
,
,
/K
K
(
Q
ij
=
T K
I
J
.
) is the matrix of the cosine Directors of
T
K
in the base
K
K
.
In vectorial form, one a:
~
~ ~~
~
~~~
=
=
T
T
where components of
~
T
are defined according to those of
Q
.
Conversely, one a:
~
~
~ ~
~
~
~ ~
=
=
- 1
- 1
T
T
therefore, one obtains:
~
~ ~~ ~ ~
=
-
TS T
K
1
that one writes:
~
~ ~
=
S
K
To be coherent with the assumption of plane stress
~
33
0
=
, one writes:
~
~
~
(
)
(
) (
)
R
X
Kr
X
R
X
5 1
5 5 5 1
=
S
Code_Aster
®
Version
7.4
Titrate:
Finite elements of voluminal hulls
Date:
14/04/05
Author (S):
X. DESROCHES
Key
:
R3.07.04-B
Page
:
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Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
with the symbol R like tiny room, which gives:
~
~ ~
,
~
~
=
R
K R
C
C
S
K
Kr
=
-
1
that one récrit by omitting the symbol
R
,
~
~ ~
=
C
K
The elastic deformation energy
W
el
is:
W
D D
D
el
T
E
T
Ar
E
=
-
1
2
1
1
1
2
3
~
~ ~ ~ det
~
Q
B C B
J
Q
K
If the hull consists of Nc layers, each layer being regarded as a component
K
, then:
W
D
D
D
el
T
E
T
Ar
E H
E H
K
E
K
Nc
K
K
=
-
+
=
1
2
2
2
1
2
3
1
~
~ ~ ~ det
~
Q
B C B
J
Q
where
E
K
-
and
E
K
+
are the X-coordinates of the limits lower and higher of the layer
K
of thickness
E
E
E
K
K
K
=
-
+
-
, with
E
1
-
= - H/2 and
E
Nc
+
= H/2.
While posing:
[
]
3
3
3
11
=
+
+
-
+
-
E
H
E
E
H
K
K
K
,
,
one a:
(
)
W
E
H
D
D
D
el
T
E
K
T
Ar
E
K
Nc
=
-
=
1
2
1
1
1
2
3
1
2
3
1
~
~ ~ ~ det
,
,
~
Q
B C B
J
Q
K
In the same way, for work due to thermal dilations
W
HT
, one a:
(
)
~
~
,
,
,
HT
K
K
K
K
T
T
T
=
1
2
3
0 0 0
where them
ik
are the expansion factors thermal of the layer
K
in the axes of orthotropism (
~
~
X
K
).
With the relation:
~
~ ~~
HT
K
HT
K
=
T
one obtains:
(
)
W
D
D
D
HT
T
E
Ar
K
HT
K
=
-
-
~
~
~ ~
det
Q
B
C
J
T
1
1
1
2
3
That is to say:
W
E
H
D
D
D
HT
T
E
K
H
Nc
Ar
K
HT
K
=
=
-
~
~ ~ ~ det
Q
B C
J
T
1
1
1
1
2
3
Code_Aster
®
Version
7.4
Titrate:
Finite elements of voluminal hulls
Date:
14/04/05
Author (S):
X. DESROCHES
Key
:
R3.07.04-B
Page
:
40/42
Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
Appendix 2 Functions of form for element Q9H
These functions are given on page 174 of [bib8].
A2.1 Functions of form for the translations
8 functions of the shape of incomplete Lagrange of the element quadrangle Q9H [A2.2-a Figure] for
the interpolation of displacements
U
K
are:
·
(
)
NR
I
I
I
I
I
I
()
,
(
) (
) (
)
,
1
1
2
1 1
2 2
1 1
2 2
1
4
1
1
1
1 2 3 4
=
- +
+
+
+
=
·
(
)
NR
I
I
I
()
,
(
) (
)
,
1
1 2
12
2 2
1
2 1
1
5 7
=
-
+
=
·
(
)
NR
I
I
I
()
,
(
) (
)
,
1
1
2
22
1 1
1
2 1
1
6 8
=
-
+
=
with:
1
1
1
1
1 8 4
0
5 7
1
2 6 3
I
I
I
I
I
I
= -
=
=
=
= +
=
;
;
.
and
2
2
2
1
1 5 2
0
6 8
1
3 7 4
I
I
I
I
I
I
= -
=
=
=
= +
=
;
;
.
.
A2.2 Functions of form for rotations
9 functions of the shape of Lagrange of the element quadrangle Q9H [A2.2-a Figure] for the interpolation of
rotations
~
are:
(
)
NR
NR
NR
I
I
I
()
,
()
()
2
1
2
1
2
=
where
NR
I
R I
()
Pr
Pr
P
P
Pi
=
-
-
for p=1,2 and where
R
described the whole of both
nodes aligned with node I in the direction
P
.
One a:
1
1
1
1
1 8 4
0
5 7
1
2 6 3
I
I
I
I
I
I
= -
=
=
=
= +
=
;
;
.
and
2
2
2
1
1 5 2
0
6 8
1
3 7 4
I
I
I
I
I
I
= -
=
=
=
= +
=
;
;
.
.
1
(0,0)
1
2
3
4
5
6
2
(0,0)
1
2
3
4
5
6
7
1
2
8
7
8
9
(0,1)
(0,1)
(1,0)
(1,0)
Appear A2.2-a: Degrees of freedom for the translations and rotations of the element quadrangle Q9H
Code_Aster
®
Version
7.4
Titrate:
Finite elements of voluminal hulls
Date:
14/04/05
Author (S):
X. DESROCHES
Key
:
R3.07.04-B
Page
:
41/42
Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
Appendix 3 Functions of form for element T7H
A3.1 Functions of form for the translations
6 functions of form of triangular element T7H [A3.2-a Figure] for the interpolation of displacements
U
K
are given on page 175 of [bib8]:
·
()
NR
11
1 2
2
2
2
1
()
,
(
)
=
-
·
(
)
NR
21
1 2
2
1
()
,
(
)
=
-
·
(
)
NR
31
1 2
1
1
2
1
()
,
(
)
=
-
·
()
NR
41
1 2
2
4
()
,
=
·
()
NR
51
1 2
1
4
()
,
=
·
()
NR
61
1 2
1 2
4
()
,
=
where:
= - -
1
1
2
A3.2 Functions of form for rotations
7 functions of form of triangular element T7H [A3.2-a Figure] for the interpolation of rotations
~
are:
·
(
) (
)
NR
NR
12
1
2
2
2
72
2
1
1
9
()
()
,
=
- +
·
(
) (
) (
)
[
]
NR
NR
22
1
2
1
2
1
2
72
1
2 1
1
1
9
()
()
,
= - -
- -
- +
·
() (
)
NR
NR
32
1 2
1
1
72
2
1
1
9
()
()
,
=
- +
·
(
)
(
)
NR
NR
42
1
2
2
1
2
72
4
1
4
9
()
()
,
=
- -
-
·
(
)
(
)
NR
NR
52
1
2
1
1
2
72
4
1
4
9
()
()
,
=
- -
-
·
(
)
NR
NR
62
1
2
1 2
72
4
4
9
()
()
,
=
-
with:
·
(
)
(
)
NR
72
1
2
1 2
1
2
27
1
()
,
=
- -
(0,0)
1
2
3
4
5
6
1
2
(0,0)
1
2
3
4
5
6
7
1
2
(0,1)
(0,1)
(1,0)
(1,0)
Appear A3.2-a: Degrees of freedom for the translations and rotations of the element triangle T7H
Code_Aster
®
Version
7.4
Titrate:
Finite elements of voluminal hulls
Date:
14/04/05
Author (S):
X. DESROCHES
Key
:
R3.07.04-B
Page
:
42/42
Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
Intentionally white left page.