Code_Aster ®
Version
6.0
Titrate:
SSLA100 - Infinite Cylindre subjected to a field of voluminal forces
Date:
30/11/01
Author (S):
B. RIOU Key
:
V3.06.100-A Page:
1/10

Organization (S): EDF/ERMEL/PEL

Handbook of Validation
V3.06 booklet: Linear statics of the axisymmetric structures
Document: V3.06.100

SSLA100 - Infinite Cylindre 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.
Handbook of Validation
V3.06 booklet: Linear statics of the axisymmetric structures
HM-77/01/149/A

Code_Aster ®
Version
6.0
Titrate:
SSLA100 - Infinite Cylindre subjected to a field of voluminal forces
Date:
30/11/01
Author (S):
B. RIOU Key
:
V3.06.100-A Page:
2/10

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 Pa;
·
= 1 kg/m3;
·
= 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: FV = .r ²
with = 1 NR/m3.
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): FS (R = Rint) = 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 blocked 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.
Handbook of Validation
V3.06 booklet: Linear statics of the axisymmetric structures
HM-77/01/149/A

Code_Aster ®
Version
6.0
Titrate:
SSLA100 - Infinite Cylindre subjected to a field of voluminal forces
Date:
30/11/01
Author (S):
B. RIOU Key
:
V3.06.100-A Page:
3/10

Modeling 3D:
FS
Uz = 0
FV
Uz = 0
Axisymmetric modeling 2D:
Uz = 0
FS
FV
Uz = 0
Modeling plane 2D:
FV
UY = 0
FS
UX = 0
UX = 0
UY = 0

Appear 1.3-a: Conditions in the limits and loadings
Handbook of Validation
V3.06 booklet: Linear statics of the axisymmetric structures
HM-77/01/149/A

Code_Aster ®
Version
6.0
Titrate:
SSLA100 - Infinite Cylindre subjected to a field of voluminal forces
Date:
30/11/01
Author (S):
B. RIOU Key
:
V3.06.100-A Page:
4/10

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 FV (R) = R ²

One considers the equilibrium equations in cylindrical co-ordinates:


1

-
R
R
rz
R

+
+
+
+
=

F
0
R
R

R
Z
R

1

Z


R

R

+
+
+ 2
+ F = 0 which is simplified being given axial symmetry
R
Z


R
R




zz
rz


Z

R
+
+
+ 2
+
= 0

F
Z
R

Z
R
R

-
R
R

in:
+
+
=

Fr 0
R
R

By using the law of behavior then the relations deformation-displacements, one leads to
u'
U
F
the following differential equation: U
V
'' +
-
+
= 0
R
R ²
E (1 -)
(1 +) (1 - 2)
The voluminal force applied is of the type: fV= .r ²

The solution of the differential equation is written then:

- C
1
(+) 1
(- 2) r4
U =
1 -
+ C R
éq
2.1-1
R
15th 1
(-
2
2
)
The two constants of integrations c1 and c2 are given thanks to the boundary conditions:

(
)
int
R
= 0



(R)

ext. = 0

4 - 3 2 1 + R2 R2 (R3 - R3)

C
int ext.
int
ext.
1 =




1 -
15
E
R2 - R2
One obtains:
ext.

int

1
(+) 1
(- 2) 4 - 3
R3 - R3
C
=


(R3 - R2
int
ext.
2


)
E
ext.
1 - 15
int R2 - R2

ext.
int
Surface force standard pressure FS (Rint) = P

The problem to be solved is of comparable nature, but with a voluminal force applied null: fV= 0
= 0.
Handbook of Validation
V3.06 booklet: Linear statics of the axisymmetric structures
HM-77/01/149/A

Code_Aster ®
Version
6.0
Titrate:
SSLA100 - Infinite Cylindre subjected to a field of voluminal forces
Date:
30/11/01
Author (S):
B. RIOU Key
:
V3.06.100-A Page:
5/10

- C
The solution in displacement [éq 2.1-1] is written then: U =
1 + C R, having to observe the conditions:
R
2
2

(
)
int
R
= - P


(R)

ext. = 0

What gives:

1 +
R2
R2


U
P
int
ext.
=


+ 1
(- 2) R
éq
2.1-2
E
R2 - R2int R
ext.




2.2
Results of reference

Numerical application:

·
height
= 0.5 m;
·
interior radius
= 1 m;
·
external radius
= 1.4 m;
·
E
= 10 Pa;
·

= 1 kg/m3;
·

= 0.3;
·

= 1 NR/m5;
·
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 (.
1)
0 =.
0 52130982m


U (.
14) =.
0 44203108m


2.3
Uncertainties on the solution

Null (analytical reference solution).
Handbook of Validation
V3.06 booklet: Linear statics of the axisymmetric structures
HM-77/01/149/A

Code_Aster ®
Version
6.0
Titrate:
SSLA100 - Infinite Cylindre subjected to a field of voluminal forces
Date:
30/11/01
Author (S):
B. RIOU Key
:
V3.06.100-A Page:
6/10

3 Modeling
With

3.1
Characteristics of modeling

The cylinder is modelled in voluminal elements 3D:

P2
P1


3.2
Characteristics of the grid

The cylinder is represented by a regular grid of quadratic elements with 20 nodes containing:

·
8 elements;
·
96 nodes.

The grid contains 1 only element in the radial and vertical direction and 8 cuttings on
circumference.

3.3 Functionalities
tested

Commands 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
UX in P1
1
0.52130982
0.52097
6.54 10­ 2%
UX in P2
1
0.44203108
0.44178
5.74 10­ 2%

4.2 Parameters
of execution

Version: 6.01.19

Machine: Origin 2000

Obstruction memory: 16 Mo
Time CPU To use: 2.64 seconds
Handbook of Validation
V3.06 booklet: Linear statics of the axisymmetric structures
HM-77/01/149/A

Code_Aster ®
Version
6.0
Titrate:
SSLA100 - Infinite Cylindre subjected to a field of voluminal forces
Date:
30/11/01
Author (S):
B. RIOU Key
:
V3.06.100-A Page:
7/10

5 Modeling
B

5.1
Characteristics of modeling

A longitudinal section of the cylinder is modelled in voluminal elements 2D, while considering
the assumption of axisymetry.

P1
P2


5.2
Characteristics of the grid

The cylinder is represented by a regular grid of quadratic elements with 8 nodes containing:

·
4 elements;
·
21 nodes.

The grid contains 2 cuttings in the radial direction and 2 cuttings in the vertical direction.

5.3 Functionalities
tested

Commands 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
UX in P1
1
0.52130982
0.52129
4.07 10­ 3%
UX 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

Obstruction memory: 16 Mo
Time CPU To use: 1.89 seconds
Handbook of Validation
V3.06 booklet: Linear statics of the axisymmetric structures
HM-77/01/149/A

Code_Aster ®
Version
6.0
Titrate:
SSLA100 - Infinite Cylindre subjected to a field of voluminal forces
Date:
30/11/01
Author (S):
B. RIOU Key
:
V3.06.100-A Page:
8/10

7 Modeling
C

7.1
Characteristics of modeling

A transverse section of the cylinder is modelled in voluminal elements 2D, while considering
the assumption of the plane deformations.
P1
P2


7.2
Characteristics of the grid

The cylinder is represented by a regular grid of quadratic elements with 8 nodes containing:

·
8 elements;
·
40 nodes.

The grid contains 1 only cutting in the radial direction and 8 cuttings in the vertical direction
(like the 3D).

7.3 Functionalities
tested

Commands 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
UX in P1
1
0.52130982
0.52131
6.76 10­ 2%
UX 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

Obstruction memory: 16 Mo
Time CPU To use: 1.89 seconds
Handbook of Validation
V3.06 booklet: Linear statics of the axisymmetric structures
HM-77/01/149/A

Code_Aster ®
Version
6.0
Titrate:
SSLA100 - Infinite Cylindre subjected to a field of voluminal forces
Date:
30/11/01
Author (S):
B. RIOU Key
:
V3.06.100-A Page:
9/10

9
Summary of the results

The results obtained by the code_Aster are very close to the analytical solution, in spite of
very coarse grids.

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.

Handbook of Validation
V3.06 booklet: Linear statics of the axisymmetric structures
HM-77/01/149/A

Code_Aster ®
Version
6.0
Titrate:
SSLA100 - Infinite Cylindre subjected to a field of voluminal forces
Date:
30/11/01
Author (S):
B. RIOU Key
:
V3.06.100-A Page:
10/10

Intentionally white left page.
Handbook of Validation
V3.06 booklet: Linear statics of the axisymmetric structures
HM-77/01/149/A

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