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Titrate:
SSLV109 - Full cylinder in pressure nonuniform mode 1
Date:
23/09/02
Author (S):
X. DESROCHES
Key
:
V3.04.109-C
Page:
1/16
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A
Organization (S):
EDF/AMA
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
Document: V3.04.109
SSLV109 - Full cylinder in nonuniform pressure
mode 1
Summary:
This test validates all the elements of Fourier (triangles and quadrangles of degrees 1 and 2) in elasticity.
functionalities are as follows:
·
variable pressure in space,
·
imposed displacements,
·
matrices of rigidity Fourier mode 1,
·
stresses with the nodes Fourier mode 1,
·
recombination of Fourier on displacements and stresses (modeling A),
·
transverse isotropic material (modeling F).
The test has a quadratic analytical solution in displacements.
The interest of the test lies in:
·
the comparison between solution calculated and analytical solution on the various finite elements,
·
the comparison between the results and Code PERMAS on elements TRIA6 (modeling A).
Code_Aster
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Titrate:
SSLV109 - Full cylinder in pressure nonuniform mode 1
Date:
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X. DESROCHES
Key
:
V3.04.109-C
Page:
2/16
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A
1
Problem of reference
1.1 Geometry
R = 1
p = R
L = 12
With
B
D
C
Z
R
The modelized field is ACDB (plane
= 0).
1.2
Material properties
E = 72 NR/m
2
= 0.3
1.3
Boundary conditions and loadings
()
()
()
()
U With
U With
U With
U AB
R
Z
Q
Z
=
=
=
=
0
0
p
p rR
=
cos
with
p
=
1.
and
R
=
1
applied in
Z
=
12
1.4 Conditions
initial
Without object for the static analysis.
Code_Aster
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Titrate:
SSLV109 - Full cylinder in pressure nonuniform mode 1
Date:
23/09/02
Author (S):
X. DESROCHES
Key
:
V3.04.109-C
Page:
3/16
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A
2
Reference solution
2.1
Method of calculation used for the reference solution
(
)
()
()
(
)
()
()
(
)
()
(
)
()
U R Z
U R Z
U R Z
M
I.E.(internal excitation) Z
p
ER R
U R Z
v R Z
v R Z
p
I.E.(internal excitation) rz
U R Z
W R Z
W R Z
M
I.E.(internal excitation) Z
p
ER R
R
Z
,
,
cos
,
,
,
cos
,
,
,
sin
,
=
=
+
=
= -
=
-
=
-
with
with
with
2
2
2
2
2
2
2
2
2
All the stresses are null except
()
zz
R Z
p
R R
,
= -
.
The data were selected in such way that
()
()
u.a.
U
L
=
=
0
1
,
.
Displacements are thus written here:
()
()
()
U R Z
Z
R
v R Z
R Z
W R Z
Z
R
,
,
,
=
+
= -
=
-
2
2
2
2
144
480
72
144
480
and:
()
zz
R Z
R
,
= -
2.2
Results of reference
U v W
U U U
zz
R
Z
,
,
in
R
0., 0.5, 1.
Z 0., 6., 12.
in R
0., Z 6., 12.,
45
=
=
=
=
=
°
2.3
Uncertainty on the solution
Analytical solution.
2.4 References
bibliographical
[1]
PERMAS-HS. Axisymmetric Continued with arbitrary loads. Stuttgart 1985. INTES publication
n°224 p 42 - 49.
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Titrate:
SSLV109 - Full cylinder in pressure nonuniform mode 1
Date:
23/09/02
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X. DESROCHES
Key
:
V3.04.109-C
Page:
4/16
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A
3 Modeling
With
3.1
Characteristics of modeling
Number of the nodes:
With = N1
B = N3
C = N13
D = N15
E = N7
F = N8
G = N9
With
B
D
C
F
E
locked node
face
locked
in Z
G
X (R)
y (Z)
Limiting conditions:
DDL_IMPO
:
(
NODE
:
With
DX = 0.
DY = 0.
DZ = 0.)
face
AB
(
GROUP_NO
:
AB
DY = 0.)
Pressure on the face CD:
PRES_REP
(
GROUP_MA
:
Boils
NEAR
:
p
)
p being defined by
AFFE_CHAR_MECA_F
by p (X) = X
3.2
Characteristics of the mesh
A number of nodes: 15
A number of meshs and types: 4 TRIA6, 1SEG3 on segment CD
3.3 Functionalities
tested
Controls
AFFE_MODELE
MECHANICS
“AXIS_FOURIER”
ALL
DEFI_FONCTION
NOM_PARA
“X”
AFFE_CHAR_MECA_F
DDL_IMPO
NODE
PRES_REP
GROUP_MA
CALC_MATR_ELEM
“RIGI_MECA”
MODE_FOURIER
CALC_CHAM_ELEM
“SIGM_ELNO_DEPL”
MODE_FOURIER
COMB_CHAM_NO
COMB_FOURIER
COMB_CHAM_ELEM
COMB_FOURIER
3.4 Remarks
The number of the mode of Fourier not affecting the loading, the key word
MODE_FOURIER
is not
necessary in the control
CALC_VECT_ELEM
(by defect, it is regarded as being worth 0).
Code_Aster
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Version
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Titrate:
SSLV109 - Full cylinder in pressure nonuniform mode 1
Date:
23/09/02
Author (S):
X. DESROCHES
Key
:
V3.04.109-C
Page:
5/16
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A
4
Results of modeling A
4.1 Values
tested
Node Size
Reference
Aster %
difference
B
U
2.0833 10
3
2.0835
10
3
9.10
3
v
0.
DDL_IMPO
W
2.0833 10
3
2.0834 10
3
3.10
3
zz
1. 1.000001
6.10
4
E
U 0.25
0.250001
5.10
4
v
0.
4.10
6
W
0.25
0.250000
2.10
4
zz
0.
1.10
4
F
U 0.250521
0.250522
4.10
4
v
0.04166 0.041668 3.10
3
W
0.0249479
0.0249479
6.10
- 3
zz
0.5 0.50005 1.10
2
G
U 0.252083
0.252084
3.10
4
v
0.083333
0.083333 7.10
3
W
0.247917 0.247916
1.10
4
zz
1. 1.
9.10
3
C
U 1. 1.0006
6.10
3
v
0.
5.10
5
W
1.
1.00006
6.10
3
zz
0.
1.1 10
3
D
U 1.00208
1.00215
7.10
3
v
0.16666 0.166691 1.5 10
2
W
0.99791 0.997981
1.9 10
2
zz
1. 0.999 0.1
4.2 Remarks
The analytical solution is found with a precision < 0.02% for displacements and < 0.1%
for the stresses.
With a numerical formula of integration at 6 points of GAUSS (instead of 3) to calculate the stiffness,
the relation with 10 would be found
10
close (like PERMAS).
Code_Aster
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Titrate:
SSLV109 - Full cylinder in pressure nonuniform mode 1
Date:
23/09/02
Author (S):
X. DESROCHES
Key
:
V3.04.109-C
Page:
6/16
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A
5 Modeling
B
5.1
Characteristics of modeling
Number of the nodes:
With = N1
B = N3
C = N13
D = N15
E = N7
F = N8
G = N9
With
B
D
C
F
E
locked node
face
locked
in Z
G
X (R)
y (Z)
Limiting conditions:
DDL_IMPO
:
(
NODE
:
With
DX = 0.
DY = 0.
DZ = 0.)
face
AB
(
GROUP_NO
:
AB
DY = 0.)
Pressure on the face CD:
PRES_REP
(
GROUP_MA
:
Boils
NEAR
:
p
)
p being defined by
AFFE_CHAR_MECA_F
by p (X) = X
5.2
Characteristics of the mesh
A number of nodes: 15
A number of meshs and types: 2 QUAD8, 1 SEG3 on segment CD
5.3 Functionalities
tested
Controls
AFFE_MODELE
MECHANICS
“AXIS_FOURIER”
ALL
DEFI_FONCTION
NOM_PARA
“X”
AFFE_CHAR_MECA_F
DDL_IMPO
NODE
PRES_REP
GROUP_MA
CALC_MATR_ELEM
“RIGI_MECA”
MODE_FOURIER
CALC_CHAM_ELEM
“SIGM_ELNO_DEPL”
MODE_FOURIER
5.4 Remarks
The number of the mode of Fourier not affecting the loading, the key word
MODE_FOURIER
is not
necessary in the control
CALC_VECT_ELEM
(by defect, it is regarded as being worth 0).
Code_Aster
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Version
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Titrate:
SSLV109 - Full cylinder in pressure nonuniform mode 1
Date:
23/09/02
Author (S):
X. DESROCHES
Key
:
V3.04.109-C
Page:
7/16
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A
6
Results of modeling B
6.1 Values
tested
Node Size Reference
Aster %
difference
B
U
2.0833 10
3
2.0833
10
3
10.10
11
v
0.
DDL_IMPO
W
2.0833 10
3
2.0833 10
3
10.10
11
zz
1. 1 10.10
11
E
U 0.25
0.25
10.10
11
v
0.
3.9 10
14
W
0.25
0.25
10.10
11
zz
0.
4.10
12
F
U 0.250521
0.250521
10.10
11
v
0.04166 0.04166 10.10
11
W
0.0249479
0.0249479
10.10
11
zz
0.5 0.5 10.10
11
G
U 0.252083
0.252083
10.10
11
v
0.08333 0.08333 10.10
11
W
0.247917 0.247917
10.10
11
zz
1. 1. 10.10
10
C
U 1. 1. 10.10
11
v
0.
4.3.10
14
W
1.
1
10.10
11
zz
0.
6.10
12
D
U 1.00208
1.00208
10.10
11
v
0.16666 0.16666 10.10
11
W
0.99791 0.99791
10.10
11
zz
1. 1. 10.10
11
6.2 Remarks
The analytical solution is found with 10 or 11 significant digits.
Code_Aster
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Titrate:
SSLV109 - Full cylinder in pressure nonuniform mode 1
Date:
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X. DESROCHES
Key
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V3.04.109-C
Page:
8/16
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A
7 Modeling
C
7.1
Characteristics of modeling
Number of the nodes:
With = N1
B = N3
C = N13
D = N15
E = N7
F = N8
G = N9
With
B
D
C
F
E
locked node
face
locked
in Z
G
X (R)
y (Z)
Limiting conditions:
DDL_IMPO
:
(
NODE
:
With
DX = 0.
DY = 0.
DZ = 0.)
face
AB
(
GROUP_NO
:
AB
DY = 0.)
Pressure on the face CD:
PRES_REP
(
GROUP_MA
:
Boils
NEAR
:
p
)
p being defined by
AFFE_CHAR_MECA_F
by p (X) = X
7.2
Characteristics of the mesh
A number of nodes:15
A number of meshs and types: 2 QUAD9, 1 SEG3 on segment CD
7.3 Functionalities
tested
Controls
AFFE_MODELE
MECHANICS
“AXIS_FOURIER”
ALL
DEFI_FONCTION
NOM_PARA
“X”
AFFE_CHAR_MECA_F
DDL_IMPO
NODE
PRES_REP
GROUP_MA
CALC_MATR_ELEM
“RIGI_MECA”
MODE_FOURIER
CALC_CHAM_ELEM
“SIGM_ELNO_DEPL”
MODE_FOURIER
7.4 Remarks
The number of the mode of Fourier not affecting the loading, the key word
MODE_FOURIER
is not
necessary in the control
CALC_VECT_ELEM
(by defect, it is regarded as being worth 0).
Code_Aster
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Version
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Titrate:
SSLV109 - Full cylinder in pressure nonuniform mode 1
Date:
23/09/02
Author (S):
X. DESROCHES
Key
:
V3.04.109-C
Page:
9/16
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A
8
Results of modeling C
8.1 Values
tested
Node Size
Reference
Aster %
difference
B
U
2.0833 10
3
2.0833
10
3
10.10
11
v
0.
DDL_IMPO
W
2.0833 10
3
2.0833 10
3
10.10
11
zz
1. 1 10.10
10
E
U 0.25 0.25 10.10
11
v
0.
3.9 10
14
W
0.25
0.25
10.10
11
zz
0.
4.10
12
F
U 0.250521
0.250521
10.10
11
v
0.04166
0.04166
10.10
11
W
0.0249479
0.0249479
10.10
11
zz
0.5 0.5 10.10
11
G
U 0.252083
0.252083
10.10
11
v
0.08333
0.08333
10.10
11
W
0.247917
0.247917
10.10
11
zz
1. 1. 10.10
11
C
U 1.
1.
10.10
11
v
0.
4.3.10
14
W
1.
1.
10.10
11
zz
0.
6.10
12
D
U 1.00208
1.00208
10.10
11
v
0.16666
0.16666
10.10
11
W
0.99791
0.99791
10.10
11
zz
1. 1. 10.10
11
8.2 Remarks
The analytical solution is found with 10 or 11 significant digits.
Code_Aster
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Version
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Titrate:
SSLV109 - Full cylinder in pressure nonuniform mode 1
Date:
23/09/02
Author (S):
X. DESROCHES
Key
:
V3.04.109-C
Page:
10/16
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A
9 Modeling
D
9.1
Characteristics of modeling
Number of the nodes:
With = N1
B = N1129
C = N1369
D = N2169
E = N141
F = N705
G = N1269
With
B
D
C
F
E
locked node
face
locked
in Z
y (Z)
X (R)
G
Limiting conditions:
DDL_IMPO
:
(
NODE
:
With
DX = 0.
DY = 0.
DZ = 0.)
face
AB
(
GROUP_NO
:
AB
DY = 0.)
Pressure on the face CD:
PRES_REP
(
GROUP_MA
:
Boils
NEAR
:
p
)
p being defined by
AFFE_CHAR_MECA_F
by p (X) = X
9.2
Characteristics of the mesh
A number of nodes: 2169
A number of meshs and types: 1920 QUAD4, 8 SEG2 on segment CD
9.3 Functionalities
tested
Controls
AFFE_MODELE
MECHANICS
“AXIS_FOURIER”
ALL
DEFI_FONCTION
NOM_PARA
“X”
AFFE_CHAR_MECA_F
DDL_IMPO
NODE
PRES_REP
GROUP_MA
CALC_MATR_ELEM
“RIGI_MECA”
MODE_FOURIER
CALC_CHAM_ELEM
“SIGM_ELNO_DEPL”
MODE_FOURIER
9.4 Remarks
The number of the mode of Fourier not affecting the loading, the key word
MODE_FOURIER
is not
necessary in the control
CALC_VECT_ELEM
(by defect, it is regarded as being worth 0).
Code_Aster
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Version
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Titrate:
SSLV109 - Full cylinder in pressure nonuniform mode 1
Date:
23/09/02
Author (S):
X. DESROCHES
Key
:
V3.04.109-C
Page:
11/16
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A
10 Results of modeling D
10.1 Values
tested
Node Size
Reference
Aster %
difference
B
U
2.0833 10
3
2.0919
10
3
0.41
v
0.
DDL_IMPO
W
2.0833 10
3
2.0674 10
3
0.76
zz
1. 1.00974
0.97
E
U 0.25 0.2498 0.07
v
0.
2.7.10
10
W
0.25
0.2498 0.07
zz
0. 0.0090
0.90
F
U 0.250521
0.250347
0.07
v
0.04166
0.04164
0.07
W
0.0249479
0.0249317 0.06
zz
0.5 0.51005
2.01
G
U 0.252083
0.251911
0.07
v
0.083333
0.083273
0.07
W
0.247917
0.247752 0.07
zz
1. 1.0103
1.03
C
U 1.
0.99927
0.07
v
0.
2.3.10
7
W
1.
0.99928 0.07
zz
0. 0.0088
D
U 1.00208
1.001357
0.07
v
0.16666
0.16653
0.08
W
0.99791
0.997208 0.06
zz
1. 1.0027
0.27
10.2 Remarks
To obtain a precision of about 1% on the stresses, it is necessary to modelize
structure very finely (8 elements radially and 240 axially).
Code_Aster
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Titrate:
SSLV109 - Full cylinder in pressure nonuniform mode 1
Date:
23/09/02
Author (S):
X. DESROCHES
Key
:
V3.04.109-C
Page:
12/16
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A
11 Modeling
E
11.1 Characteristics of modeling
Number of the nodes:
With = N1
B = N2421
E = N121
F = N1331
G = N2541
With
B
E
locked node
face
locked
in Z
F
X (R)
y (Z)
G
Limiting conditions:
DDL_IMPO
:
(
NODE
:
With
DX = 0.
DY = 0.
DZ = 0.)
face
AB
(
GROUP_NO
:
AB
DY = 0.)
Pressure on face EG:
PRES_REP
(
GROUP_MA
:
Boils
NEAR
:
p
)
p being defined by
AFFE_CHAR_MECA_F
by p (X) = X
11.2 Characteristics of the mesh
A number of nodes: 2541
A number of meshs and types: 4800 TRIA3, 20 SEG2 on segment EG
11.3 Functionalities
tested
Controls
AFFE_MODELE
MECHANICS
“AXIS_FOURIER”
ALL
DEFI_FONCTION
NOM_PARA
“X”
AFFE_CHAR_MECA_F
DDL_IMPO
NODE
PRES_REP
GROUP_MA
CALC_MATR_ELEM
“RIGI_MECA”
MODE_FOURIER
CALC_CHAM_ELEM
“SIGM_ELNO_DEPL”
MODE_FOURIER
11.4 Remarks
To decrease the number of nodes, one modelized the structure for y
6.
The precision on the results is nevertheless less than for elements QUAD4.
Code_Aster
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Titrate:
SSLV109 - Full cylinder in pressure nonuniform mode 1
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X. DESROCHES
Key
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Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A
12 Results of modeling E
12.1 Values
tested
Node Size
Reference
Aster %
difference
B
U
2.0833 10
3
2.0228
10
3
2.90
v
0.
DDL_IMPO
W
2.0833 10
3
2.1143 10
3
1.49
zz
1. 1.0078
0.78
E
U 0.25 0.24949 0.20
v
0.
2.4 10
5
W
0.25
0.24950 0.20
zz
0.
5.3 10
2
F
U 0.250521
0.25003 0.19
v
0.04166 0.04161
0.13
W
0.249479 0.24900 0.19
zz
0.5 0.49738
0.52
G
U 0.252083
0.251605
0.19
v
0.083333 0.083157
0.21
W
0.247917 0.247449 0.19
zz
1. 0.9814
1.86
12.2 Remarks
The precision on displacements is lower than 3%, that on the stresses lower than 2%.
On this example, the TRIA3 converge definitely less quickly than the QUAD4 towards the solution
exact.
Code_Aster
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Titrate:
SSLV109 - Full cylinder in pressure nonuniform mode 1
Date:
23/09/02
Author (S):
X. DESROCHES
Key
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Page:
14/16
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A
13 Modeling
F
13.1 Characteristics of modeling
Number of the nodes:
With = N1
B = N3
C = N13
D = N15
E = N7
G = N9
With
B
D
C
E
locked node
face
locked
in Z
G
X (R)
y (Z)
N4
N2
N12
N14
N12
Limiting conditions:
DDL_IMPO
:
(
NODE
:
With
DX = 0.
DY = 0.
DZ = 0.)
face
AB
(
GROUP_NO
:
AB
DY = 0.)
Pressure on the face CD:
PRES_REP
(
GROUP_MA
:
Boils
NEAR
:
p
)
p being defined by
AFFE_CHAR_MECA_F
by p (X) = X
13.2 Characteristics of the mesh
A number of nodes: 15
A number of meshs and types: 2 QUAD8, 1SEG3 on segment CD
13.3 Functionalities
tested
Controls
AFFE_MODELE
MECHANICS
“AXIS_FOURIER”
ALL
DEFI_MATERIAU
ELAS_ORTH
DEFI_FONCTION
NOM_PARA
“X”
AFFE_CHAR_MECA_F
DDL_IMPO
NODE
PRES_REP
GROUP_MA
CALC_MATR_ELEM
“RIGI_MECA”
MODE_FOURIER
CALC_CHAM_ELEM
“SIGM_ELNO_DEPL”
MODE_FOURIER
13.4 Remarks
The number of the mode of Fourier not affecting the loading, the key word
MODE_FOURIER
is not
necessary in the control
CALC_VECT_ELEM
(by defect, it is regarded as being worth 0).
Code_Aster
®
Version
5.0
Titrate:
SSLV109 - Full cylinder in pressure nonuniform mode 1
Date:
23/09/02
Author (S):
X. DESROCHES
Key
:
V3.04.109-C
Page:
15/16
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A
14 Results of modeling F
14.1 Values
tested
Node Size
Reference
Aster %
difference
N2
U 2.6041666
2.6041666
1.10
9
W
2.6041666 2.6041666
1.10
9
With
zz
0.
1.10
13
B
zz
1. 1. 3.10
10
N4
U 0.0625 0.0625 4.10
10
W
0.0625
0.0625
4.10
10
E U
0.25
0.25
3.10
10
W
0.25
0.25
3.10
10
zz
0.
1. 10
14
G v
0.083333 0.083333
2.10
10
zz
1. 1. 2.10
11
N10 U
0.5625 0.5625 2.10
10
W
0.5625 0.5625 2.10
10
N12 v
0.125 0.125 1.10
10
C U
1. 1. 1.10
10
W
1. 1. 1.10
10
zz
0.
7.10
14
N14 v
0.083333 0.083333 9.10
11
D v
0.166666 0.166666 9.10
11
zz
1. 1. 1.10
11
Code_Aster
®
Version
5.0
Titrate:
SSLV109 - Full cylinder in pressure nonuniform mode 1
Date:
23/09/02
Author (S):
X. DESROCHES
Key
:
V3.04.109-C
Page:
16/16
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/02/001/A
15 Summary of the results
The elements of command 2 give the analytical solution.
The elements of command 1 converge slowly towards the solution and require very fine mesh.
Times calculations remain however reasonable.