Code_Aster
®
Version
6.0
Titrate:
SSLA100 - Infinite cylinder subjected to a field of voluminal forces
Date:
30/11/01
Author (S):
B. RIOU
Key
:
V3.06.100-A
Page:
1/10
Manual of Validation
V3.06 booklet: Linear statics of the axisymmetric structures
HM-77/01/149/A
Organization (S):
EDF/ERMEL/PEL
Manual of Validation
V3.06 booklet: Linear statics of the axisymmetric structures
Document: V3.06.100
SSLA100 - Infinite cylinder subjected to a field of
voluminal and surface forces
Summary:
This test of linear quasi-static mechanics makes it possible to validate the assignment of a loading of field of
forces, surface or voluminal.
The studied structure is cylindrical. The fields with the nodes of voluminal and surface density of forces are
read in a file with the Ideas format. For the voluminal loading, the field read varies quadratically in
function of the distance to the axis; for the surface loading, the field read corresponds to an internal pressure.
Three modelings of the same problem are carried out:
·
modeling 3D;
·
axisymmetric modeling 2D;
·
modeling 2D plane deformations;
The reference solution is analytical.
Code_Aster
®
Version
6.0
Titrate:
SSLA100 - Infinite cylinder subjected to a field of voluminal forces
Date:
30/11/01
Author (S):
B. RIOU
Key
:
V3.06.100-A
Page:
2/10
Manual of Validation
V3.06 booklet: Linear statics of the axisymmetric structures
HM-77/01/149/A
1
Problem of reference
1.1 Geometry
Z
y
X
Selected geometrical dimensions are as follows:
·
height
= 0.5 m;
·
interior radius
= 1 m;
·
external radius = 1.2 Mr.
1.2
Properties of material
The cylinder consists of a homogeneous material which follows a law of elastic behavior linear:
·
E = 10 AP;
·
= 1 kg/m
3
;
·
= 0.3.
1.3
Boundary conditions and loadings (cf [Figure 1.3-a])
The voluminal force considered is radial, it varies in a quadratic way with the radius: F
V
=
.r ²
with
= 1 NR/m
3
.
The surface force considered is applied to the internal wall of the cylinder, perpendicular to
wall (is equivalent to an internal pressure imposed on the cylinder): F
S
(R = R
int
) = 1N/m ².
The boundary conditions make it possible to be placed on the assumption of the plane deformations on one
section of the cylinder: vertical displacements locked on the sections high and low of the cylinder.
Note:
For modeling 3D, the suppression of the clean modes is ensured by the conditions of
plane 2D applied to the low section of the cylinder. This type of boundary conditions allows
to obtain an axisymmetric in displacement, directly comparable solution with
analytical solution.
Code_Aster
®
Version
6.0
Titrate:
SSLA100 - Infinite cylinder subjected to a field of voluminal forces
Date:
30/11/01
Author (S):
B. RIOU
Key
:
V3.06.100-A
Page:
3/10
Manual of Validation
V3.06 booklet: Linear statics of the axisymmetric structures
HM-77/01/149/A
F
S
Uz = 0
F
V
Uz = 0
Uz = 0
F
S
Uz = 0
F
V
F
V
U
Y
= 0
F
S
U
X
= 0
U
Y
= 0
U
X
= 0
Modeling 3D:
Axisymmetric modeling 2D:
Modeling plane 2D:
Appear 1.3-a: Boundary conditions and loadings
Code_Aster
®
Version
6.0
Titrate:
SSLA100 - Infinite cylinder subjected to a field of voluminal forces
Date:
30/11/01
Author (S):
B. RIOU
Key
:
V3.06.100-A
Page:
4/10
Manual of Validation
V3.06 booklet: Linear statics of the axisymmetric structures
HM-77/01/149/A
2
Reference solution
2.1
Method of calculation used for the reference solution
The problem of linear static mechanics axisymmetric considered can be solved in manner
analytical. One solves independently the response to the stress forces voluminal and forces surface
to summon them then.
Voluminal force quadratic F
V
(R) =
R ²
One considers the equilibrium equations in cylindrical co-ordinates:
R
R
rz
R
R
Z
R
R
zz
rz
Z
R
Z
R
R
Z
R
F
R
Z
R
R
F
Z
R
R
R
F
+
+
+
-
+
=
+
+
+
+
=
+
+
+
+
=
1
0
1
2
0
2
0
who are simplified being given axial symmetry
in:
R
R
R
R
R
F
+
-
+
= 0
By using the law of behavior then the relations deformation-displacements, one leads to
the following differential equation:
U
U
R
U
R
F
E
V
''
'
²
(
)
(
) (
)
+ -
+
-
+
-
=
1
1
1 2
0
The voluminal force applied is of the type: F
V
=
.r ²
The solution of the differential equation is written then:
U
C
R
R
E
C R
= -
-
+
-
-
+
1
4
2
2
1
1 2
15
1
(
) (
)
(
)
éq
2.1-1
Two constants of integrations C
1
and C
2
are given thanks to the boundary conditions:
(
)
(
)
int
R
R
ext.
=
=
0
0
One obtains:
C
E
R R
R
R
R
R
C
E
R
R
R
R
R
R
ext.
ext.
ext.
ext.
ext.
ext.
1
2
2
3
3
2
2
2
3
2
3
3
2
2
4 3
1
2
15
1
1
1 2
4 3
1
15
= -- +
-
-
=
+
-
--
-
-
-
int
int
int
int
int
int
(
)
(
) (
)
(
)
Surface force standard pressure F
S
(R
int
) = P
The problem to be solved is of comparable nature, but with a voluminal force applied null: F
V
= 0
= 0.
Code_Aster
®
Version
6.0
Titrate:
SSLA100 - Infinite cylinder subjected to a field of voluminal forces
Date:
30/11/01
Author (S):
B. RIOU
Key
:
V3.06.100-A
Page:
5/10
Manual of Validation
V3.06 booklet: Linear statics of the axisymmetric structures
HM-77/01/149/A
The solution in displacement [éq 2.1-1] is written then:
U
C
R
C R
= -
+
1
2
2
, having to observe the conditions:
(
)
(
)
int
R
P
R
ext.
= -
=
0
What gives:
U
P
E
R
R
R
R
R
R
ext.
ext.
= +
-
+ -
1
1 2
2
2
2
2
int
int
(
)
éq
2.1-2
2.2
Results of reference
Numerical application:
·
height
= 0.5 m;
·
interior radius
= 1 m;
·
external radius
= 1.4 m;
·
E
= 10 AP;
·
= 1 kg/m
3
;
·
= 0.3;
·
= 1 NR/m
5
;
·
P
= 1 NR/m ².
by injecting the numerical values in the solutions [éq 2.1-1] and [éq 2.1-2] one finds afterwards
summation:
U
m
U
m
(. )
.
(. )
.
10
0 52130982
14
0 44203108
=
=
2.3
Uncertainties on the solution
Null (analytical reference solution).
Code_Aster
®
Version
6.0
Titrate:
SSLA100 - Infinite cylinder subjected to a field of voluminal forces
Date:
30/11/01
Author (S):
B. RIOU
Key
:
V3.06.100-A
Page:
6/10
Manual of Validation
V3.06 booklet: Linear statics of the axisymmetric structures
HM-77/01/149/A
3 Modeling
With
3.1
Characteristics of modeling
The cylinder is modelized in voluminal elements 3D:
3.2
Characteristics of the mesh
The cylinder is represented by a regular mesh of quadratic elements with 20 nodes containing:
·
8 elements;
·
96 nodes.
The mesh contains 1 only element in the radial and vertical direction and 8 cuttings on
circumference.
3.3 Functionalities
tested
Controls Key word
factor
Key word
LIRE_RESU NOM_CHAM
FVOL_3D
LIRE_RESU NOM_CHAM
FSUR_3D
AFFE_CHAR_MECA EVOL_CHAR
4
Results of modeling A
4.1 Values
tested
Identification Moments Reference
Aster %
difference
U
X
in P1
1
0.52130982
0.52097
6.54 10
2
%
U
X
in P2
1
0.44203108
0.44178
5.74 10
2
%
4.2 Parameters
of execution
Version: 6.01.19
Machine: Origin 2000
Overall dimension memory: 16 Mo
Time CPU To use: 2.64 seconds
P1
P2
Code_Aster
®
Version
6.0
Titrate:
SSLA100 - Infinite cylinder subjected to a field of voluminal forces
Date:
30/11/01
Author (S):
B. RIOU
Key
:
V3.06.100-A
Page:
7/10
Manual of Validation
V3.06 booklet: Linear statics of the axisymmetric structures
HM-77/01/149/A
5 Modeling
B
5.1
Characteristics of modeling
A longitudinal section of the cylinder is modelized in voluminal elements 2D, while considering
the assumption of axisymetry.
5.2
Characteristics of the mesh
The cylinder is represented by a regular mesh of quadratic elements with 8 nodes containing:
·
4 elements;
·
21 nodes.
The mesh contains 2 cuttings in the radial direction and 2 cuttings in the vertical direction.
5.3 Functionalities
tested
Controls Key word
factor
Key word
LIRE_RESU NOM_CHAM
FVOL_2D
LIRE_RESU NOM_CHAM
FSUR_2D
AFFE_CHAR_MECA
EVOL_CHAR
6
Results of modeling B
6.1 Values
tested
Identification Moments Reference
Aster %
difference
U
X
in P1
1
0.52130982
0.52129
4.07 10
3
%
U
X
in P2
1
0.44203108
0.44202
3.95 10
3
%
6.2 Notice
Modeling more powerful than the 3D because 2 cuttings in the radial direction and not of discretization
circonférencielle.
6.3 Parameters
of execution
Version: 6.01.19
Machine: Origin 2000
Overall dimension memory: 16 Mo
Time CPU To use: 1.89 seconds
P1
P2
Code_Aster
®
Version
6.0
Titrate:
SSLA100 - Infinite cylinder subjected to a field of voluminal forces
Date:
30/11/01
Author (S):
B. RIOU
Key
:
V3.06.100-A
Page:
8/10
Manual of Validation
V3.06 booklet: Linear statics of the axisymmetric structures
HM-77/01/149/A
7 Modeling
C
7.1
Characteristics of modeling
A transverse section of the cylinder is modelized in voluminal elements 2D, while considering
the assumption of the plane deformations.
7.2
Characteristics of the mesh
The cylinder is represented by a regular mesh of quadratic elements with 8 nodes containing:
·
8 elements;
·
40 nodes.
The mesh contains 1 only cutting in the radial direction and 8 cuttings in the vertical direction
(like the 3D).
7.3 Functionalities
tested
Controls Key word
factor
Key word
LIRE_RESU NOM_CHAM
FVOL_2D
LIRE_RESU NOM_CHAM
FSUR_2D
AFFE_CHAR_MECA
EVOL_CHAR
8
Results of modeling C
8.1 Values
tested
Identification Moments Reference
Aster %
difference
U
X
in P1
1
0.52130982
0.52131
6.76 10
2
%
U
X
in P2
1
0.44203108
0.44204
5.74 10
2
%
8.2 Remarks
Modeling of performance very close to the 3D because same discretizations circonférencielle and radial.
8.3 Parameters
of execution
Version: 6.0
Machine: Origin 2000
Overall dimension memory: 16 Mo
Time CPU To use: 1.89 seconds
P1
P2
Code_Aster
®
Version
6.0
Titrate:
SSLA100 - Infinite cylinder subjected to a field of voluminal forces
Date:
30/11/01
Author (S):
B. RIOU
Key
:
V3.06.100-A
Page:
9/10
Manual of Validation
V3.06 booklet: Linear statics of the axisymmetric structures
HM-77/01/149/A
9
Summary of the results
The results obtained by the code_Aster are very close to the analytical solution, in spite of
very coarse mesh.
Modelings 3D and 2D plane give further information very close because they present them
same discretizations circonférencielle and radial. Axisymmetric modeling 2D is more
powerful because it presents 2 cuttings in the radial direction and not of discretization
circonférencielle.
Code_Aster
®
Version
6.0
Titrate:
SSLA100 - Infinite cylinder subjected to a field of voluminal forces
Date:
30/11/01
Author (S):
B. RIOU
Key
:
V3.06.100-A
Page:
10/10
Manual of Validation
V3.06 booklet: Linear statics of the axisymmetric structures
HM-77/01/149/A
Intentionally white left page.