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Titrate:
Axisymmetric thermoelastic hulls and 1D
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R3.07.02-B
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Manual of Reference
R3.07 booklet: Machine elements on average surface
HI-75/00/006/A
Organization (S):
EDF/MTI/MN, RNE/AMV
Manual of Reference
R3.07 booklet: Machine elements on average surface
R3.07.02 document
Numerical modeling of the mean structures:
axisymmetric thermoelastoplastic hulls
and 1D
Summary:
One presents a numerical formulation for the modeling of the structures at average surface of geometry
particular:
·
hulls with symmetry of revolution around the axis 0y,
·
invariant hulls with unspecified section along the axis 0z.
One describes the isotropic thermoelastoplastic case completely, within the framework of the theories of
LOVE-KIRCHHOFF and of HENCKY-MINDLIN-REISSNER, as well as the various studied loadings, for
the selected isoparametric finite element.
The examples of validation suggested show qualities of the finite element.
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Titrate:
Axisymmetric thermoelastic hulls and 1D
Date:
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Key:
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R3.07 booklet: Machine elements on average surface
HI-75/00/006/A
Contents
1 Introduction ............................................................................................................................................ 3
2 continuous Problem ................................................................................................................................... 3
2.1 Description of the geometry, kinematics .............................................................................. 4
2.2 Thermoelastoplastic balance ..................................................................................................... 8
3 Formulation of the finite element. Discretization ........................................................................................ 13
3.1 Description of the selected finite element ................................................................................................ 13
3.1.1 Motivations ........................................................................................................................... 13
3.1.2 General presentation of the element ...................................................................................... 14
3.1.3 Transformations finite element/finite element of reference ..................................................... 14
3.1.4 Surface numerical integration ......................................................................................... 15
3.1.5 Numerical integration in the thickness ............................................................................... 16
3.2 Formulation of the elementary terms ........................................................................................... 17
3.2.1 Mass, center of gravity, stamps inertia .......................................................................... 17
3.2.2 Stamp of mass ................................................................................................................. 18
3.2.3 Second member of centrifugal force .................................................................................... 19
3.2.4 Second member of gravity ............................................................................................. 19
3.2.5 Second member of distributed loads ................................................................................. 19
3.3 Calculation of the strains and the stresses .................................................................................. 20
4 Validation - Case test ............................................................................................................................. 21
4.1 Roll under pressure interns ...................................................................................................... 21
4.2 Plate circular embedded under uniform pressure [V3.03.100] ................................................. 25
4.3 Axisymmetric modal analysis of a thin spherical envelope [V2.03.007] ............................ 30
5 Conclusion ........................................................................................................................................... 31
6 Bibliography ........................................................................................................................................ 32
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Titrate:
Axisymmetric thermoelastic hulls and 1D
Date:
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Key:
R3.07.02-B
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Manual of Reference
R3.07 booklet: Machine elements on average surface
HI-75/00/006/A
1 Introduction
One is interested in what follows to the mechanical modeling of mean structures to average surface
of particular geometry:
·
hulls with symmetry of revolution around the axis
0y
,
·
hulls with invariant unspecified sections along the axis
0z
.
More particularly, one limits oneself if mechanical parameters (materials, loadings)
are independent of a direction of space (the circumference for the hulls of revolution, the axis
0z
for the hulls
C_PLAN
and
D_PLAN
).
For the resolution of chained thermomechanical problems, one must use before the finite element
of thermal hull describes in [R3.11.01] according to case's in its axisymmetric version, or its version
plane invariant according to
0z
.
One gives hereafter first of all a progress report on the description of the mechanical model: kinematics, law of
thermoelastoplastic behavior. Then one presents the selected finite element, the interpolation and
method of integration.
One gives finally some numerical results of application, by comparison with solutions
analytical.
2 Problem
continuous
The geometry is defined in a unidimensional way:
·
by the meridian line in the plan
()
0xy
for a hull of revolution,
·
by the section of the hull in the plan
()
0xy
for an invariant hull in
Z
.
In this last case, by analogy with the two-dimensional problems, one considers two cases:
·
the case “forced plane”, i.e. that of a free hull according to the direction
0z
, or
that of an arc in the plan
0xy
,
·
the case “plane deformations”, i.e. when displacements according to the direction
0z
are null.
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Axisymmetric thermoelastic hulls and 1D
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R3.07 booklet: Machine elements on average surface
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2.1
Description of the geometry, kinematics
One considers a hull of revolution of axis
0y
, or an invariant hull according to the axis
0z
. For all
two, average surface is defined by the curve
=
AB
in the plan
0xy
:
is a meridian line
for the hull of revolution, or the section for the invariant hull according to
0z
.
Z
y
X
O
Appear 2.1-a: Hull of revolution
y
T
!!
N
B
·
E
y
E
Z
E
X
m
S
X
With
O
Appear Meridian 2.1-b:
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Titrate:
Axisymmetric thermoelastic hulls and 1D
Date:
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Key:
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R3.07 booklet: Machine elements on average surface
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Z
X
y
O
Appear 2.1-c: Hull with invariant section according to
0z
The curve
=
AB
is parameterized by the curvilinear X-coordinate
S
. One will note the derivative partial
S
by:
, S
.
In a point
m
of
one defines the local reference mark (
N
,
T
,
E
Z
) by:
T
Om
Om
=
,
,
S
S
;
N
T
E
=
Z
.
One notes also the angle
such as:
N
E
E
cos
sin
=
+
X
y
.
Curvature of
is defined by:
1
R
S
S
,
,
= -
=
N T
.
In the case of the hull of revolution, the position on the parallel passing by
m
is noted
.
tangent vector on this parallel is
E
. For the meridian line located in the plan
0xy
,
=
0
and
E
E
= -
Z
. The radius of curvature of the parallel in
m
is:
R
R
=
cos
where
R
is the X-coordinate
X
point
m
of
.
On the other hand, for an invariant hull according to
Z
this parallel is a right generator, directed according to
E
Z
,
of null curvature.
The transformations kinematics of the hull are defined by displacement
U
point
m
surface average, like by rotation
S
normal
N
at the point
m
. The vector
U
can be
expressed in local base:
()
() ()
()
()
U
T
N
S
S
S
S
S
U
W
.
.
=
+
.
Code_Aster
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Titrate:
Axisymmetric thermoelastic hulls and 1D
Date:
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Key:
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R3.07 booklet: Machine elements on average surface
HI-75/00/006/A
Or in Cartesian base:
()
()
()
U
E
E
S
X
X
y
y
U.S.
U.S.
=
+
.
Deformations of the hull associated with this transformation
(
)
U,
S
are determined by:
·
a membrane tensor of deformation
E
,
·
a tensor of variation of curvature
K
,
·
a vector of deformation of distortion tranverse
.
This last appears in the theory of hulls of HENCKY-MINDLIN-NAGHDI and not in that of
COIL. According to displacement
U
and of rotation
S
, these sizes are expressed (cf [bib1]):
Case
Hull of revolution
Invariant hull according to
0z
U
expressed in
base local
(
N
,
T
,
E
Z
)
E
U
W
R
S
S
,
=
+
E
U
W
R
S
S
,
=
+
E
R
=
1
(
-
U
sin
+
W
cos
)
K
S
S S
,
=
K
S
S S
,
=
K
R
S
sin
= -
S
S
S
W
U
R
,
=
+
-
S
S
S
W
U
R
,
=
+
-
U
expressed in
base total
(
)
E
E
E
X
y
Z
,
E
U
S
y S
,
=
cos
-
U
X S
,
sin
E
U
R
X
=
E
U
S
y S
,
=
cos
-
U
X S
,
sin
K
S
S S
,
=
K
S
S S
,
=
K
R
S
sin
= -
S
S
X S
U
,
=
+
cos
,
+
U
y S
sin
S
S
X S
U
,
=
+
cos
,
+
U
y S
sin
Note:
Change of direction of the curvilinear X-coordinate
S
do not modify the values of:
S
S
E
E
,
,
but the sign changes of
,
U W R K
K
S
.
Within the framework of the theory of COILS, the condition
S
=
0
(the normals with the hull remain it afterwards
deformation) results in a direct relation between rotations
S
and the slope
W
S
,
.
components of the tensor variation of curvature are according to displacement in the local base:
K
W
U
R
U R
R
S
S
S
S
,
,
,
= -
+
-
2
K
R
W
U
R
S
sin
,
=
-
Code_Aster
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Titrate:
Axisymmetric thermoelastic hulls and 1D
Date:
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Key:
R3.07.02-B
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R3.07 booklet: Machine elements on average surface
HI-75/00/006/A
If displacement is expressed in total base:
(
)
K
R U
U
U
U
S
X S
y S
X S
y S
sin cos
cos
sin
,
,
,
,
=
-
-
-
1
(
)
K
R
U
U
X S
y S
=
+
sin
cos sin
,
,
It is noticed that the expression of the variations of curvature according to displacement in theory of
COIL is rather intricate and that it utilizes derivative second. If one is required
interpolation conforms i.e. here
C
1
, this requires the use of finite elements of high degree.
Tensors
E K
,
allow to express the three-dimensional deformation
in the thickness.
On [Figure 2.1-d], one indicates by
X
3
the position in the thickness
-
H H
2 2
,
compared to fiber
average, at the point
m
, of curvilinear X-coordinate
S
on
.
S
R
!!
T
X
3
!!
N
+ H
2
-
H
2
m
Appear 2.1-d
In a point thickness, displacement is expressed in total reference mark:
()
()
()
(
)
()
()
()
(
)
U
E
E
(,
)
.
sin.
.
cos
.
S X
U.S.
S X
S
U.S.
S X
S
X
S
X
y
S
y
3
3
3
=
-
+
+
Code_Aster
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Titrate:
Axisymmetric thermoelastic hulls and 1D
Date:
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Key:
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R3.07 booklet: Machine elements on average surface
HI-75/00/006/A
In order to take account of the variation of metric in the thickness (due to the curvature of surface
average), one defines the functions:
()
()
S
X
X
R
X
X
R
3
3
3
3
1
1
;
.cos
=
+
=
+
For a sufficiently thin hull, this correction is negligible:
S
;
1
1
In practice this correction carried out if
MODI_METRIQUE:“YES”
in
AFFE_CARA_ELEM
[U4.42.01] is useless if the reports/ratios
H R
and
H R
, when they exist, are lower than
115
.
In theory of HENCKY-MINDLIN-NAGHDI, the components of the tensor of deformation
are:
()
(
)
()
(
)
()
S
S
S
S
sx
S
S
S X
E
X K
S X
E
X K
S X
,
,
,
3
3
3
3
3
1
1
1
2
3
=
+
=
+
=
(only in the case hull of revolution)
2.2 Balance
thermoelastoplastic
It is considered that the material constitutive of the hull is thermoelastoplastic isotropic. One makes
the usually allowed assumption that the transverse normal stress is null:
X X
3 3
0
. The law of
behavior most general is written then:
11
22
1
1111
1122
2211
2222
11
11
11
22
22
1
3
3 3
3
0
0
0
0
X
X X
HT
HT
X
C
C
C
C
C
=
-
-
where
C (,)
µ
components
C
ijkl
is the local matrix of behavior in plane stresses and
µ
represent the whole of the internal variables when the behavior is nonlinear. In the continuation
index 1 makes reference to the curvilinear X-coordinate and 2 with
or Z. With the three-dimensional deformations
defined above, one associates the components of the tensor then forced
:
·
in the case of a hull of revolution:
S
ssss
S
S
HT
S
HT
S
S
S
HT
HT
sx
ssx X
sx
C
C
C
C
C
(
)
(
)
(
)
(
)
=
-
+
-
=
-
+
-
=
3
3 3
3
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Axisymmetric thermoelastic hulls and 1D
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R3.07 booklet: Machine elements on average surface
HI-75/00/006/A
·
in the case invariant hull according to the direction
Z
and free in
Z
(“plane stresses”):
S
ssss
sszz zzss
zzzz
S
S
HT
zz
sx
ssx X
sx
C
C
C
C
C
=
-
-
=
=
(
) (
)
0
3
3 3
3
·
in the case invariant hull according to the direction
Z
and locked in
Z
(“plane deformations”):
S
ssss
S
S
HT
zz
zzss
S
S
HT
sx
ssx X
sx
C
C
C
(
)
(
)
=
-
=
-
=
3
3 3
3
One fires the expression from it from the elastic energy of deformation, which one will deduce the matrix from rigidity in
function of the kinematics of hull seen in the paragraph [§2.1]:
·
in the case hull of revolution:
[
]
(
)
W
él
ssss S
S
S
S
ssx X
sx
H
H
S
3
C
C
C
C
C
rdsd dx
=
+
+
+
+
+
-
-
1
2
2
1
2
2
2
2
2
0
2
3 3
3
(
)
.
/
/
·
in the case invariant hull according to
Z
, in “plane stresses”:
W
él
ssss
sszz zzss
zzzz
S
ssx X
sx
H
H
S
3
C
C
C
C
C
dsdx
3 3
3
(
)
.
/
/
=
-
+
-
1
2
2
2
2
2
2
·
in the case invariant hull according to
Z
, in “plane deformations”:
[
]
W
él
ssss S
ssx X
sx
H
H
S
3
C
C
dsdx
3 3
3
=
+
-
1
2
2
2
2
2
2
.
/
/
Note:
In thermoelasticity, if one notes
E
the YOUNG modulus and
the Poisson's ratio, one a:
{}
C
E
v
C
E
v
I J
C
E
v
iiii
iijj
X X
= -
= -
= +
1
1
1 2
1
2
2
11
3 3
;
(,)
;
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Titrate:
Axisymmetric thermoelastic hulls and 1D
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Key:
R3.07.02-B
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R3.07 booklet: Machine elements on average surface
HI-75/00/006/A
The following sizes are defined:
·
the membrane rigidity of a hull of revolution:
[]
C
C
C
C
C
dx
ij
S
I
J
H
H
ssss
S
S
3
.
/
/
=
+
-
-
1
2
2
; who is worth:
Eh
1
1
1
2
-
in elasticity and absence of correction of metric in the thickness;
·
the rigidity of coupling membrane-bending of a hull of revolution:
[]
B
X
C
C
C
C
dx
ij
S
I
J
H
H
ssss
S
S
3
.
.
=
+
-
-
3
2
2
1
, which is null in elasticity and in
absence of correction of metric in the thickness;
·
the rigidity of bending of a hull of revolution:
[]
D
X
C
C
C
C
dx
ij
32
S
I
J
H
H
ssss
S
S
3
.
.
=
+
-
-
1
2
2
, which is worth:
Eh
3
2
12 1
1
1
(
)
-
in elasticity and absence of correction of metric in the thickness;
·
the transverse rigidity of distortion of a hull of revolution:
G
C
dx
sx
S
S
H
H
ssx X
3
3
3 3
1
2
2
2
.
=
+
-
-
, which is worth:
Eh
1
+
in elasticity and absence of correction of metric in the thickness.
For an invariant hull according to the direction
Z
, one considers in these expressions only the terms
ij
S
=
; moreover one must replace there
(
)
S
+
-
1
by
S
: the coefficients thus are defined
C
B
D
S
D
S
D
S
D
,
and
C
B
D
S
C
S
C
S
C
,
for the case, respectively, plane deformations or of
plane stresses. In elasticity, coefficients
C
B
D
S
C
S
C
S
C
,
, are the products of the coefficients
C
B
D
S
D
S
D
S
D
,
by
1
2
-
. Lastly, the coefficient of transverse rigidity of distortion
G
sx
3
is
identical for three modelings to the correction of metric near.
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Axisymmetric thermoelastic hulls and 1D
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R3.07 booklet: Machine elements on average surface
HI-75/00/006/A
One can thus express elastic energy according to the tensors of deformations of hull:
E K
,
by:
·
for a hull of revolution:
[
(
)
W
él
S
S
S
S
S
S
S
S
S
S
S
S
S
S
sx
S
C E
B E K
D K
C E
B E K
D K
E.E.C.
B
E K
E K
D K
K
G
R ds D
=
+
+
+
+
+
+
+
+
+
+
1
2
2
2
2
2
2
2
2
2
0
2
2
3
(
)
.
.
·
for an invariant hull according to
Z
in “plane stresses”:
W
él
S
C
S
S
C
S
S
S
C
S
sx
S
C E
B E
K
D K
G
ds
.
.
=
+
+
+
1
2
2
2
2
2
2
3
·
for an invariant hull according to
Z
in “plane deformations”:
W
él
S
D
S
S
D
S
S
S
D
S
sx
S
C E
B E
K
D K
G
ds
.
.
=
+
+
+
1
2
2
2
2
2
2
3
For these expressions, it is necessary to add the potential associated with the thermal stresses, which will be one
contribution to the second member (whom one will express below in total reference mark):
·
in the case hull of revolution:
()
(
)
(
)
(
)
(
)
[
]
L
HT
ref.
ssss
S
S
S
H
H
T T
C
C
C
C
rd dx ds
V
=
-
+
+
+
-
/
/
2
2
0
2
3
expression which for an isotropic elastic behavior becomes:
()
(
)
L
V
HT
ref.
X
X S
y S
S S
S
H
H
E
T T
v
R
v
v
X
R
rd dx ds
=
-
-
-
+
+
-
-
sin
cos
sin
,
,
,
/
/
1
3
2
2
0
2
3
·
in the case invariant according to
Z
in “plane stresses”:
()
(
)
L
V
HT
ref.
ssss
sszz zzss
zzzz
S
H
H
T T
C
C
C
C
dx ds
=
-
-
-
/
/
2
2
3
expression which for an isotropic elastic behavior becomes:
()
(
)
(
)
[
]
L
V
HT
ref.
X S
y S
S S
H
H
E T T
v
v
X
dx ds
=
-
-
+
+
-
sin cos
,
,
,
/
/
3
2
2
3
Code_Aster
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Version
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Titrate:
Axisymmetric thermoelastic hulls and 1D
Date:
06/12/00
Author (S):
P. MASSIN, F. VOLDOIRE, C.SEVIN
Key:
R3.07.02-B
Page:
12/32
Manual of Reference
R3.07 booklet: Machine elements on average surface
HI-75/00/006/A
·
in the case invariant according to
Z
in “plane deformations”:
()
(
)
[
]
L
V
HT
ref.
ssss S
H
H
T T
C
dx ds
=
-
-
/
/
2
2
3
expression which for an isotropic elastic behavior becomes:
()
(
)
(
)
L
V
HT
ref.
X S
y S
S S
H
H
E
T T
v
v
X
dx ds
=
-
-
-
+
+
-
sin
cos
,
,
,
/
/
1
3
2
2
3
In these three expressions, one deliberately neglected the correction of metric in the thickness
(terms in
S
,
seen for rigidity). Moreover the temperature
T
who appears is defined by
thermal model of hull with three fields (cf [R3.11.01]):
(
)
()
()
()
T S X
T
S
X
H
T S xh
X
H
T S xh
X
H
m
S
I
,
.
3
3 2
3
3
3
3
1
2
1
2
1
=
-
+
+
+
-
+
From the whole of these expressions, one deduces the tensors from generalized efforts
NR
and
M
(efforts
normal and bending moments) associated the generalized deformations
E
and
K
by the principle of
virtual work. They are related to the tensor of the stresses
three-dimensional by:
NR
dx
H
H
/
/
=
-
3
2
2
M
X
dx
H
H
.
/
/
=
-
3
3
2
2
(where one neglected the variations of metric in the thickness).
Note:
Transverse energy of shearing
The model of hull presented above, said HENCKY-MINDLIN-NAGHDI, rests on one
kinematic assumption: parameters
W
and
S
indicate the normal displacement of the point
m
average surface
and the rotation of the normal vector
N
.
One also frequently finds the model known as of REISSNER which rests on an assumption
statics of the distribution of stresses shear transverse. Parameters
kinematics deduced
W
and
S
in this model are weighted averages in
the thickness of normal displacement and local rotations. If one wishes to place oneself in it
tally, it is enough to affect the coefficient
= 5/6 at the end of transverse energy of shearing
(in
s2
). (cf [bib7], [bib9]).
Lastly, if one wants, for a thin hull, to be located within the framework of the model of
LOVE-KIRCHHOFF, one can neutralize the energy of shearing with a great value of
(which penalizes the condition
S
=
0
), for example 10
6
H R
/
, where
H
is the thickness and
R
a characteristic radius of curvature or a distance characteristic of the loadings:
(cf [bib 2]). In practice the user can inform the value of
under the key word
A_CIS
of
the control
AFFE_CARA_ELEM
[U4.42.01].
Code_Aster
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Titrate:
Axisymmetric thermoelastic hulls and 1D
Date:
06/12/00
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Key:
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R3.07 booklet: Machine elements on average surface
HI-75/00/006/A
3
Formulation of the finite element. Discretization
3.1
Description of the selected finite element
3.1.1 Motivations
The choice of framework HENCKY-MINDLIN-NAGHDI to describe the kinematics of hull, presented to
paragraph [§2], led to expressions of the deformations where the derivative are limited to command 1,
contrary to the model of LOVE-KIRCHHOFF. This offers the advantage of being able to use an element
finished of a nature limited while ensuring conformity. The natural choice is the element of LAGRANGE P2,
isoparametric, which makes it possible to have a fine representation of a curved geometry and the maid
estimates of the stresses.
The degrees of freedom are of course displacements and rotations.
As it is known as previously, the model of LOVE-KIRCHHOFF can be recovered by penalization
for a parameter
very large affecting the transverse energy of shearing.
This formulation joined the category of the finite elements of hulls known as “degenerated”, i.e. built
by injecting the kinematics of hull in elements of three-dimensional continuous mediums:
cf [bib10].
As for all the finite elements of hulls, of the particular aspects must be analyzed: the catch
in account of the rigid modes and risks of blocking of membrane or shearing.
In the case of the axisymmetric hull of revolution, there is only one rigid mode: translation according to
the axis of symmetry
0y
.
On the other hand, in the case of the invariant hull according to the direction
0z
, there are three rigid modes: two
translations in the plan
()
X y
0
and rotation around
0z
.
So that the finite element is powerful, it is necessary that the approximations retained for
description of displacement ensure an exact representation of the state of null deformations (mode
rigid). In practice, as the concept of rigid mode is defined compared to the total reference mark one decides
thus to describe displacements in total base
(
)
E
E
X
y
,
, in which rigid modes
(functions closely connected) are represented by the selected interpolation.
As for the risks of blocking out of membrane and transverse shearing, usual processing
consist in a selective numerical integration (cf [bib2]), but the practice reveals that these
phenomena seldom appear for the hulls of revolution.
Code_Aster
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Titrate:
Axisymmetric thermoelastic hulls and 1D
Date:
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3.1.2 General presentation of the element
The selected element of reference is quadratic, isoparametric with three nodes and three degrees of freedom
by node. These degrees of freedom are:
U
U
X
y
,
:
components of displacement
U
in total reference mark,
S
:
rotation around
E
Z
normal
N
.
See [Figure 3.1.2-a].
This element is a generalization of the element of plane beam curved. It is well adapted to
discretization of the hulls with meridian curvature
R
variable, cf [bib2].
U
X
U
y
·
·
·
N.1
N.3
N.2
- 1
0
+1
!!
T
N
Appear 3.1.2-a: Element of reference
The functions of form (basic) are the polynomials of LAGRANGE:
()
()
()
“
; “; ”
NR
NR
NR
1
2
3
2
1
2
1
2
1
=
- +
=
+
= -
3.1.3 Transformations finite element/finite element of reference
y
2
y
3
y
1
X
1
X
2
X
3
X
y
N2
N3
N1
·
·
·
N1
N3
N2
- 1
0
+1
The geometry is interpolated using the co-ordinates
(
)
X
y
I
I
,
of the three nodes
NR NR NR
1
3
2
,
,
:
()
() ()
()
X
X NR
y
y NR
I
I
I
I
I
I
“
;
“
=
=
=
=
1
3
1
3
Code_Aster
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Titrate:
Axisymmetric thermoelastic hulls and 1D
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In the same way using the ddl
(
)
U
U
X
y
S
I
I
I
,
on the nodes, one a:
()
()
()
()
()
()
U
U
NR
U
U
NR
NR
X
I
X
I
y
I
y
I
S
I
S
I
I
I
I
“
;
“
“
=
=
=
=
=
=
1
3
1
3
1
3
One also needs the jacobien of the transformation:
()
()
() ()
m
ds
D
X
y
,
,
=
=
+
2
2
And of the vectors of the local base:
()
()
(
)
()
()
(
)
T
E
E
N
E
E
,
,
,
,
=
+
=
-
1
1
m
X
y
m
y
X
X
Z
X
Z
Finally:
()
()
cos
;
sin
,
,
=
= -
y
m
X
m
The meridian curvature is obtained by:
()
()
(
)
1
1
3
R
D
ds
m
X
y
y
X
.
.
.
.
,
,
,
,
,
= -
=
-
N T
Because of the P2 interpolation, the derivative second which appears below express with aid
co-ordinates of the three nodes by:
X
X
X
X
y
y
y
y
,
,
.
.
=
+
-
=
+
-
1
2
3
1
2
3
2
2
3.1.4 Surface numerical integration
For numerical integrations along the element one uses a numerical formula of integration with
four points of GAUSS, single for all the terms to be integrated. This formula reveals them
blockings mentioned in the paragraph [§3.1.1] in the event of extremely localized plasticization. One
thus advise to avoid the use of these elements in plasticity for the moment. The formula
of numerical integration at four points of Gauss will be replaced later on by a formula with
two points of Gauss supposed to avoid these nuisances.
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Titrate:
Axisymmetric thermoelastic hulls and 1D
Date:
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Key:
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R3.07 booklet: Machine elements on average surface
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3.1.5 Numerical integration in the thickness
For an elastic behavior, insofar as it is admitted that one limits oneself to characteristics
uniform rubber bands in the thickness, rigidities
[] [] []
C
B
D
ij
ij
ij
,
and
G
sx
3
defined in the paragraph
[§2.2] are calculated exactly.
For a non-linear behavior, one subdivides the initial thickness in NR layers thicknesses
identical numbered in the direction of the normal to the average surface of the element. For each
sleep one uses three points of integration. The points of integration are located in higher skin of
sleep, 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 from 3 to 5 layers in the thickness for a number of
points of integration being worth 7, 9 and 11 respectively.
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
For a thermoelastic behavior, one uses integration, by layer in the thickness
-
+
H
H
2
2
,
described previously in the non-linear field, of the thermomechanical terms seen
in the paragraph [§2.2]. It is then necessary to use
STAT_NON_LINE
with a behavior
rubber band.
Note:
One already mentioned with [§2.2]. and in [R3.07.04] that the value of the coefficient of correction in
transverse shearing for the elements of 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.
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Titrate:
Axisymmetric thermoelastic hulls and 1D
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3.2
Formulation of the elementary terms
3.2.1 Mass, center of gravity, stamps inertia
In the case of the hulls of revolution, the mass is worth:
(
)
/
/
S
H
H
dx rd ds
hrd ds
H R ds
+
-
=
=
-
1
2
3
2
2
0
2
02
where
is the presumedly constant density of the element.
The position of the center of inertia is given in the Oxyz reference mark of [§2.1] by:
X
y
yr ds H
R
R
rds
R ds
Z
G
G
G
=
=
+
+
=
0
12
1
0
2
sin
cos
The terms of the matrix of inertia compared to O in the Oxyz reference mark of [§2.1] have then for
expression:
I
H X
y
H
X
y
rds
I
hx
H
X
rds
I
H X
y
H
X
y
rds
xx O
yy O
zz O
/
/
/
(
)
(sin
cos
cos
sin)
(cos
cos)
(
)
(sin
cos
cos
sin)
=
+
+
+
+
+
=
+
+
=
+
+
+
+
+
2
2
12
2
2
2
12
2
2
2
12
2
2
2
2
3
2
2
2
3
2
2
2
3
2
2
where
=
+
1
R
R
cos
.
In the case of invariant hulls according to
0z
, the mass is worth:
S
H
H
dx ds
H
ds
3
2
2
-
=
/
/
.
The position of the center of inertia is defined in the Oxy reference mark of [§2.1] by:
X
X ds H
R ds
ds
y
y ds H
R ds
ds
G
G
=
+
=
+
2
2
12
12
cos
sin
Code_Aster
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Titrate:
Axisymmetric thermoelastic hulls and 1D
Date:
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Key:
R3.07.02-B
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R3.07 booklet: Machine elements on average surface
HI-75/00/006/A
The terms of the matrix of inertia compared to O in the Oxyz reference mark of [§2.1] have then for
expression:
I
hy
H
y
ds
I
I
hxy H
X
y
ds
I
hx
H
X
ds
I
H X
y
H
X
y
xx O
xy O
yx O
yy O
zz O
/
/
/
/
/
(sin
sin)
(sin cos
sin
cos)
(cos
cos)
(
)
(
cos
sin)
=
+
+
=
=
+
+
+
=
+
+
=
+
+
+
+
2
3
2
3
2
3
2
2
2
3
12
2
12
12
2
12 1 2
2
ds
where
=
1
R
.
Terms in
H
2
12
for the centers of inertia and
H
3
12
for the matrices of inertia are not taken in
count in the programming. That amounts neglecting the variation of metric with the curvature in
the calculation of these terms.
3.2.2 Stamp of mass
The term:
() ()
v S X
v S X rdx D ds
H
H
,
.
,
/
/
3
3
3
2
2
0
2
-
, of kinetic energy is treated while considering
density
constant in the thickness and the correction of metric due to the curvature
negligible. The intégrande is burst in three terms:
·
(
)
H U U
U U
X
X
y
y
.
.
+
kinetic energy of
translation
·
H
S
S
3
12
.
kinetic energy of
rotation
·
(
)
(
)
(
)
H
U
U
U
U
X S
X S
y S
y S
3
12 sin
cos
-
+
+
+
kinetic energy of
coupling, with:
=
+
1
R
R
cos
for the case axisymmetric hull of revolution.
=
1
R
for the case invariant hull according to
0z
(moreover in this case the integral disappears
rd
0
2
).
Code_Aster
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Titrate:
Axisymmetric thermoelastic hulls and 1D
Date:
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Key:
R3.07.02-B
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HI-75/00/006/A
3.2.3 Second member of centrifugal force
In the case of the hulls of revolution, a vector rotation is considered:
.
=
2
E
y
, carried by
the axis of revolution. The term of the second corresponding member is:
(
)
22
3
3
2
2
0
2
22 2
0
2
.
.
sin
.
/
/
R U
X
dx rd ds
H
R U D ds
X
S
H
H
X
-
=
-
(one neglects the correction of metric in the thickness).
In the case of invariant hulls according to
0z
, a vector rotation is considered:
.
=
3
E
Z
,
perpendicular in the plan of the section
.
The second member is then:
(
)
H
X U
y U
ds
X
y
.
.
32
+
3.2.4 Second member of gravity
In the case of the hulls of revolution, gravity is directed according to
E
y
.
The second member is:
gh U rd ds
y
0
2
In the case of invariant hulls according to
0z
, the aforementioned is directed in the plan
X y
G
G
X
X
y
y
0:
G
E
E
=
+
.
The second member is:
(
)
H G
G
ds
X
X
y
y
.
.
E
E
+
3.2.5 Second member of distributed loads
These distributed loads can be two forces in the plan
()
X y
0
and couples it
M
Z
carried by the axis
0z
. The two forces, which one considers that they are applied to average surface
, will be able
to be provided in total reference mark
(
)
E
E
X
y
,
or room
(
)
T N
,
. The second member is:
(
)
F U
F U
M
rd ds
X
X
y
y
Z
S
+
+
0
2
(in invariant hull according to
Z
, the integral
rd
0
2
disappears).
Note:
The specific actions are treated as nodal forces where they are applied,
since they work in the ddl of the finite element.
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Axisymmetric thermoelastic hulls and 1D
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3.3
Calculation of the strains and the stresses
After resolution, one with the possibility with the operator
CALC_ELEM
[U4.81.01] to calculate with the nodes them
elementary fields according to:
·
generalized deformations
E
K
,
: option
DEGE_ELNO_DEPL
,
·
three-dimensional deformations
on average fiber and in skins internal and external
(with or without correction of curvature): option
EPSI_ELNO_DEPL
,
·
three-dimensional stresses
on average fiber and in skins internal and external
(with or without correction of curvature): option
SIGM_ELNO_DEPL
in linear elasticity,
·
generalized efforts
NR
M
,
(with or without correction of curvature): option
EFGE_ELNO_DEPL
in linear elasticity.
These values with the nodes are obtained by extrapolation starting from the values at the points of GAUSS of
the element, according to the exposed method in [bib4] [R3.06.03].
Lastly, one can have also the values
NR
M
,
at the points of GAUSS of the element: option
SIEF_ELGA_DEPL
in linear elasticity.
No postprocessing of stresses or generalized efforts is for the moment available for
nonlinear behaviors materials.
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Axisymmetric thermoelastic hulls and 1D
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4
Validation - Case test
One considers hereafter, to judge capacities of this formulation, some examples of application
(cf [bib10]).
4.1
Roll under internal pressure
One studies a vertical roll subjected to an internal pressure
p
constant on the part
y
<
0
, and null
on
y
>
0
: to see [Figure 4.1-a].
L/2
+ L/10
R
C
B
2
X
With
- L/10
- L/2
B
B
1
p
Figure 4.1 - a: Rolls under axisymmetric pressure
The radius is:
R
= 4 m, the thickness
T
= 0.25 m, the length
L
= 10 Mr. Celle-ci is selected so that
effects edge free in
y
= ±
L/2
are negligible on the solution (into axisymmetric, L must
to check:
1
2
3
L
Rt
>
=
3 m here).
The material is elastic
(
)
E
AP
.
=
=
1
0 3
.
The boundary conditions are:
p
NR m
/
=
1
2
, vertical displacement in
With
no one.
One chooses the solution obtained by model LOVE-KIRCHHOFF.
To reach it numerically, one takes as coefficient of shearing:
=
10
6
, to inhibit
distortions
S
. The analytical solution is:
for
()
(
)
y
U y
P
D
E
y
y
P
D E
y
y
X
y
S
y
=
-
=
-
0
8
2
8
4
3
:
()
(
cos
),
cos
sin
for
()
(
)
y
U y
P
D E
y
y
P
D E
y
y
X
y
S
y
=
=
+
-
-
0
8
8
4
3
:
()
cos
,
cos
sin
with
D
=
And
3
12 1
-
2
()
, 4
4
=
And
DR.
2
.
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Axisymmetric thermoelastic hulls and 1D
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R3.07 booklet: Machine elements on average surface
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The efforts generalized are
(
)
sin
=
0
:
()
()
NR
And
R U y
M
y
p E
y
X
S
X
y
;
sin
=
=
=
-
4
2
The three-dimensional stresses are:
;
,
'
:
=
+
=
NR
T
M
X
T
M X
T
D where
3
S
S 3
12
12
3
3
for
y
y X
Pr
T
E
y
X
T
y
y X
Pr
T
X
T
E
y
3
y
3
S
3
3
y
(,
)
cos
sin
(,
)
.
sin
=
-
+
-
=
-
0:
1
2
2
3
1
3
1
2
2
for
y
y X
Pr
T
E
y
X
T
y
y X
Pr
T
X
T
E
y
3
y
3
S
3
3
y
(,
)
cos
sin
(,
).
sin
=
-
-
=
-
-
-
0:
2
2
3
1
3
1
2
2
For a regular mesh of one hundred meshs and two hundred nodes, one finds:
Reference
Aster
% difference
Displacement
U
X
Not A
63.9488
63.922
0.042
Not B
32.000
32.005
0.015
Not C
0.05120
0.08755
Rotation
S
Not A
0.06583
0.04057
Not B
41.133
41.165
0.078
Normal effort
NR
Not B
2.0000
2.0003
0.015
Not B
1
(to L/10)
3.84429
3.8442
0.002
Moment
M
S
Not B
1
4.01497 10
2
4.013 10
2
0.05
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Appear 4.1-b: Arrow of the cylinder under pressure
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Appear 4.1-c: Rotation of the cylinder under pressure.
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Appear 4.1-d: Bending moments axial cylinder under pressure
4.2
Plate circular embedded under uniform pressure [V3.03.100]
The plate of radius is considered
R
= 1 m, thickness
T
= 0,1 m (see [Figure 4.2-a] below)
embedded on its circumference.
p
y
p
D
With
X
R/2
R
0
Appear 4.2-a
The material is elastic
(
)
E
AP
.
.
=
=
1
0 3
. The pressure is:
p
NR m
.
/
=
1
2
.
The boundary conditions are: in
0:
. ,
S
=
0
in
With
U
U
X
y
S
:
. ,
=
=
=
0
0
.
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One is interested in the solutions of the models of REISSNER
=
5
6
and of LOVE-KIRCHHOFF (one
will take
=
10
6
).
The analytical solution is for the arrow:
()
U X
Pr
D
X
R
X
R
y
= -
-
- +
4
2
2
64
1
1
.
with
(
)
D
And
T
R
if
;
;
=
-
=
-
=
=
3
2
2
12 1
16
5
1
1
5
6
0
for the solution
LOVE-KIRCHHOFF.
The distortion is indeed:
()
S
X
Pr
D
X
= -
2
16
2
.
Rotation
S
is:
()
S
X
Pr
D X
X
R
=
-
2
2
16
1
.
The variations of curvature are
(
)
sin
= +
1
:
K
X
Pr
D
X
R
K
X
Pr
D
X
R
S
()
()
= -
-
= -
-
2
2
2
2
16
1 3
16
1
The bending moments are
(
)
sin
= +
1
:
(
)
(
)
(
)
(
)
M
X
Pr
X
R
M
X
Pr
X
R
S
()
()
=
+ - +
=
+
- +
2
2
2
2
16
3
1
16
1 3
1
The stresses are written:
S
S
S
X X
E
X K
X
K
X
X X
E
X K
X
K
X
(,
)
[
()
()]
(,
)
[
()
()]
3
2
3
3
2
3
1
1
=
-
+
=
-
+
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One notices independence in
rotation, variations of curvature and moments
bending. In the center
0
plate:
(
)
(
)
U
Pr
D
M
M
Pr
K
K
Pr
D
y
S
S
()
()
()
()
()
.
0
64
1
0
0
16
1
0
0
16
4
2
2
= -
+
=
= -
+
=
= -
,
,
S
T
T
E T Pr
D
(,
/)
(,
/)
0
2
0
2
1
2 16
2
±
=
±
=
-
#
.
It is noticed that one is in compression in higher skin of plate.
With embedding
()
()
With
M
R
Pr
M
R
Pr
S
:
;
=
=
2
2
8
8
.
S
U
y
Appear 4.2-b: Arrow, rotation of an embedded circular plate
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For a regular mesh of 10 meshs (21 nodes) one finds:
Reference
Aster
% difference
Displacement
U
y
Not
D
=
5
6
101.827
101.7769
0.049
LOVEKIRCHHOFF
95.9765
95.0395
0.978
Not
0
=
5
6
178.424
178.368
0.031
LOVEKIRCHHOFF
170.625
169.761
0.507
Rotation
S
Not
D
=
5
6
256.001
0.024
LOVEKIRCHHOFF
255.94
257.123
0.462
Variation of curvature
K
S
Not
D
=
56
173.406
1.60
LOVEKIRCHHOFF
170.625
162.765
4.61
Variation of curvature
K
Not
D
=
56
514.001
0.024
LOVEKIRCHHOFF
511.875
512.242
0.46
Moment
M
S
Not
0
=
56
0.08125
0.081751
+0.617
LOVEKIRCHHOFF
0.081394
0.18
Not
With
=
56
0.125
0.12373
1.02
LOVEKIRCHHOFF
0.10717
14.3
Moment
M
Not
0
=
56
0.08125
0.081751
0.617
LOVEKIRCHHOFF
0.081394
0.18
Not
With
=
56
0.03750
0.037121
1.01
LOVEKIRCHHOFF
0.032146
14.3
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It is noticed that solution LOVE-KIRCHHOFF (
=
10
6
) is less quite approximate than that by
REISSNER
=
5
6
at the variations of curvature and the time bending. On the other hand, them
displacements and rotations are well calculated.
These differences are due to the relative thickness of this plate, with respect to the coarseness of the mesh
chosen. The figures hereafter show the comparison of the solutions analytical and numerical, in
case LOVE-KIRCHHOFF, on mesh of 10 and 100 elements.
-
K
K
S
Figure 4.2 - C: Variations of curvature of an embedded circular plate
The layout of the variations of curvature
K
K
S
and
illustrate the fact that these two components are not
not approximate in the same manner: first is linear since derived from a function of form
P2, while second is constant per pieces.
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4.3 Axisymmetric modal analysis of a thin spherical envelope
[V2.03.007]
One considers a sphere, of average radius
R
m
= 2.5 m, thickness
T
= 0.10 Mr.
The material is elastic (E = 200000 MPa,
= 0,3), of density
= 7800 kg/m
3
.
X
R
y
Appear 4.3-a: Sphere
One studies his axisymmetric free vibrations within framework LOVE-KIRCHHOFF
(
)
=
10
6
.
One uses a mesh made up of 40 meshs and 81 nodes. One is interested in the frequencies included/understood
between 220 and 375 Hz. Compared to the reference solution [V2.03.007] one finds like 5 first
frequencies:
N°
1
2
3
4
5
Reference
237.25
282.85
305.2
324.2
346.8
Aster
237.32
282.78
304.95
323.7
346.2
Table 4.3-a: Frequencies of the axisymmetric modes
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5 Conclusion
The finite elements that we propose were selected with a quite particular aim: structural analysis
thin axisymmetric, or of orthogonal sections of infinite hulls with independence in
direction
Z
, with the concern of obtaining a good precision on the membrane and flexional solution all
by having a part of establishment and not too expensive.
The choice of the degrees of freedom allows a good representation of the boundary conditions. Moreover,
this displacement formulation and rotation lead to elements of smaller degree: elements
are P2 out of membrane and P2 in bending. It appears that they are easy to handle and that their formulation
allows to use a structure of pre and post simple processor, favors considerable for
to carry out rather fine mesh (unidimensional) and to easily display the results (on
a simple curve). Selected kinematics: formulation of HENCKY-MINDLIN-NAGHDI, in
displacements and rotations of average surface makes it possible to utilize the energy of shearing
transverse (interesting for the hulls average thickness).
This energy can be affected of a factor of correction
: if one wants to place oneself in theory of
REISSNER, it is enough to choose
/
=
5 6
instead of 1 (but of course, the arrow
W
and rotations
in this theory only weighted averages in the thickness are). Moreover, the formulation of
hull of LOVE-KIRCHHOFF (for the very mean structures) can be simulated by penalization of
condition of nullity of the transverse distortion, by choosing a factor
=
×
10
6
H L
,
H
being
the thickness and
L
a characteristic distance (radius of curvature, area of application of the loads…).
The non-linear behaviors in plane stresses are available for these elements. One announces
however that the stresses generated by the transverse distortion are treated elastically, fault
of better. Indeed, the taking into account of a transverse shearing constant not no one on the thickness and
determination of the correction associated on rigidity with shearing compared to a model
satisfying the boundary conditions are not possible and thus return the use of these
elements, when transverse shearing is nonnull, rigorously impossible in plasticity. In
any rigor, for nonlinear behaviors, it would thus be necessary to use these elements in
tally of the theory of Coils-Kirchhoff.
Elements corresponding to the machine elements exist in thermics; chainings
thermomechanical are thus available with finite elements of thermal hulls to three nodes
described in [R3.11.01] according to case's in its axisymmetric version, or its invariant plane version according to
0z
.
In the case-test treated, the phenomena of blocking did not appear. Decomposition of
the deformation energy will make it possible, where necessary, to integrate in a selective way the terms
persons in charge for blocking, such an amendment not having to raise particular difficulties. One
more detailed study must of course be undertaken on this subject, as for the numerical methods to use
to avoid this blocking when the thickness becomes low.
The possible developments are:
·
anisotropy in order to be able to treat the multi-layer hulls,
·
problems of buckling,
·
decomposition in Fourier series to study nonaxisymmetric problems of
hulls of revolution,
·
the taking into account a variable thickness…
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6 Bibliography
[1]
B. ALMROTH - D. BRUSH: Buckling off bars, punts and shells. Mc Graw-Hill 1975.
[2]
J.L. BATOZ - G. DHATT: Modeling of the structures by finite elements. Volume 3 Hulls.
Hermès 1992.
[3]
D. BUI - F. VOLDOIRE: Presentation of a finite element of cylindrical hull P2 out of membrane
and Morley in bending. Note EDF-DER-MMN, HI 71/6715, of the 10.10.90.
[4]
X. DESROCHES: Calculation of the stresses to the nodes by a local method of smoothing by
least squares. Note EDF-DER-MMN of the 20.01.92 [R3.06.03].
[5]
G. DHATT - G. TOUZOT: A presentation of the method of the elements finis.2ème edition.
Maloine SA 1984.
[6]
GREEN - ZERNA: Theoretical elasticity. Univ. Oxford 1954.
[7]
TIMOSHENKO and WOINOWSKY-KRIEGER: Plates and hulls. Béranger 1961.
[8]
F. VOLDOIRE: Formulation and numerical evaluation of an elastoplastic model of hull
axisymmetric enriched. Note EDF-DER-MMN, HI-73/7518, of the 04.02.92.
[9]
D. BUI: Shearing in the plates and the hulls: modeling and calculation. Note
EDF-DER-MMN, HI-71/7784, of the 20.02.92.
[10]
S. ANDRIEUX - F. VOLDOIRE: Models of hulls. Applications in linear statics. School
from Numerical Summer CEA-EDF-INRIA of Analysis 1992.