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Titrate:
SSLV139 - Flambement of a circular plate subjected to a force
Date:
12/05/04
Author (S):
X. DESROCHES, C. REZETTE Key
:
V3.04.139-A Page:
1/10

Organization (S): EDF-R & D/AMA, DeltaCAD

Handbook of Validation
V3.04 booklet: Linear statics of the voluminal systems
Document: V3.04.139

SSLV139 - Flambement of a circular plate
subjected to a compressive force uniformly
distributed on its contour

Summary:

This test represents a calculation of stability of a circular plate subjected to a compressive force
uniformly distributed on its contour. We determine the critical load leading to buckling
rubber band of Euler as well as the associated modal deformation.

This test validates modeling AXIS_FOURIER for linear buckling with meshs QUAD8 and TRIA6
(circumferential mode equal to zero), and modeling AXIS with meshs QUAD8.

The critical load obtained is compared with an analytical reference solution.

Handbook of Validation
V3.04 booklet: Linear statics of the voluminal systems
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
SSLV139 - Flambement of a circular plate subjected to a force
Date:
12/05/04
Author (S):
X. DESROCHES, C. REZETTE Key
:
V3.04.139-A Page:
2/10

1
Problem of reference

1.1 Geometry

Z
Y
Fcr
Y
X
Z.
X
With
B

·
Thickness of the plate: H = 0.0005 m
·
Radius of the plate: R = 0.115 m

1.2
Properties of material

The properties of material constituting the plate are:

11

E =2.110 Pa
Young modulus
= 3
.
0


Poisson's ratio

1.3
Boundary conditions and loadings

Boundary conditions:
·
on the contour of the plate (not B): displacement following Y=0 and rotation around Z = 0

Loading: one applies a compressive force uniformly distributed F to external contour
plate.

1.4 Conditions
initial

Without object.

Handbook of Validation
V3.04 booklet: Linear statics of the voluminal systems
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
SSLV139 - Flambement of a circular plate subjected to a force
Date:
12/05/04
Author (S):
X. DESROCHES, C. REZETTE Key
:
V3.04.139-A Page:
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2
Reference solution

2.1
Method of calculation

The value of the critical load is given in [bib1] by the following expression:

68
.
14
D
Fcr =

2
R

with: D the bending stiffness of the plate (in N.m) defined by the following expression:


3
=
Eh
D

1
(
12
2
-)

This critical load is associated a circumferential mode equal to 0.

2.2
Sizes and results of reference

For the characteristics given, the critical load is worth:

Cr
F = 2668.315 NR/m

2.3
Uncertainties on the solution

Analytical solution

2.4 References
bibliographical

[1]
S.P. TIMOSHENKO, J.M. MANAGES: Theory of elastic stability, second edition, DUNOD
(1966)
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal systems
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
SSLV139 - Flambement of a circular plate subjected to a force
Date:
12/05/04
Author (S):
X. DESROCHES, C. REZETTE Key
:
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3 Modeling
With

3.1
Characteristics of modeling

Modeling AXIS_FOURIER (QUAD8): 3 degrees of freedom per node (DX, DY, DZ)

Y
D
C
X
With
B
Z


Limiting conditions:
·
group meshs AD: DX = 0, DZ = 0
·
group meshs BC: DY = 0, DZ = 0

Characteristics of the discretization
·
Sides AB and CD: 460 elements
·
Sides AD and BC: 4 elements

3.2
Characteristics of the grid

A number of nodes: 6449
A number of meshs: 1840 (QUAD8)

3.3 Functionalities
tested

Commands


AFFE_MODELE
AFFE
MODELING = “AXIS_FOURIER”
AFFE_CHAR_MECA DDL_IMPO
GROUP_MA
FORCE_CONTOUR
FX
MACRO_ELAS_MULT
CAS_CHARGE
MODE_FOURIER = 0
CALC_MATR_ELEM
OPTION = “RIGI_MECA”


OPTION = “RIGI_GEOM”


MODE_FOURIER = 0
MODE_ITER_SIMULT

METHOD = “SORENSEN”


TYPE_RESU = “MODE_FLAMB”

CALC_FREQ
OPTION = “SMALLER”
NORM_MODE

= “TRAN NORMALIZES”

3.4
Sizes tested and results

Identification
Reference
Aster %
Difference
Pressure criticizes (n=0)
8.4935 105 Pa
8.4847 105 Pa
­ 0.104
Displacement DY with the node D
1 9.99985
10­1 ­ 0.001
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal systems
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
SSLV139 - Flambement of a circular plate subjected to a force
Date:
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Author (S):
X. DESROCHES, C. REZETTE Key
:
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3.5 Remarks

·
The pressure criticizes Pcr of reference, used in the command file, was obtained with
to leave the referred critical load in the paragraph [§2.2]:

F
2
P
Cr
=
= 849350.94N/m
Cr
2 H


·
The standardization of the clean mode for largest of the components of translation
imply a value of reference equal to 1 for displacement DY to node D.

Handbook of Validation
V3.04 booklet: Linear statics of the voluminal systems
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
SSLV139 - Flambement of a circular plate subjected to a force
Date:
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X. DESROCHES, C. REZETTE Key
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4 Modeling
B

4.1
Characteristics of modeling

Modeling AXIS (QUAD8): 2 degrees of freedom per node (DX, DY)

Y
D
C
X
With
B
Z


Limiting conditions:
·
group meshs AD: DX = 0
·
group meshs BC: DY = 0

Characteristics of the discretization
·
Sides AB and CD: 460 elements
·
Sides AD and BC: 4 elements

4.2
Characteristics of the grid

A number of nodes: 6449
A number of meshs: 1840 (QUAD8)

4.3 Functionalities
tested
.
Commands


AFFE_MODELE
AFFE
MODELING = “AXIS”
AFFE_CHAR_MECA DDL_IMPO
GROUP_MA
FORCE_CONTOUR
FX
MACRO_ELAS_MULT CAS_CHARGE

CALC_MATR_ELEM
OPTION = “RIGI_MECA”


OPTION = “RIGI_GEOM”
MODE_ITER_SIMULT

METHOD = “SORENSEN”


TYPE_RESU = “MODE_FLAMB”

CALC_FREQ
OPTION = “SMALLER”
NORM_MODE

= “TRAN NORMALIZES”

4.4
Sizes tested and results

Identification
Reference
Aster %
Difference
Pressure criticizes (n=0)
8.4935 105 Pa
8.4847 105 Pa
­ 0.104
Displacement DY with the node D
1 9.99985
10­1 ­ 0.001
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal systems
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
SSLV139 - Flambement of a circular plate subjected to a force
Date:
12/05/04
Author (S):
X. DESROCHES, C. REZETTE Key
:
V3.04.139-A Page:
7/10

4.5 Remarks

·
The pressure criticizes Pcr of reference, used in the command file, was obtained with
to leave the referred critical load in the paragraph [§2.2]:

F
2
P
Cr
=
= 849350.94N/m
Cr
2 H


·
The standardization of the clean mode for largest of the components of translation
imply a value of reference equal to 1 for displacement DY to node D.

Handbook of Validation
V3.04 booklet: Linear statics of the voluminal systems
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
SSLV139 - Flambement of a circular plate subjected to a force
Date:
12/05/04
Author (S):
X. DESROCHES, C. REZETTE Key
:
V3.04.139-A Page:
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5 Modeling
C

5.1
Characteristics of modeling

Modeling AXIS_FOURIER (TRIA6): 3 degrees of freedom per node (DX, DY, DZ)

Y
D
C
X
With
B
Z


Limiting conditions:
·
group meshs AD: DX = 0, DZ = 0.
·
group meshs BC: DY = 0, DZ = 0.

Characteristic of the discretization
·
Sides AB and CD: 690 elements
·
Sides AD and BC: 6 elements

5.2
Characteristics of the grid

A number of nodes: 17 964
A number of meshs: 8.280 (TRIA6)

5.3 Functionalities
tested
.
Commands


AFFE_MODELE
AFFE
MODELING = “AXIS_FOURIER”
AFFE_CHAR_MECA DDL_IMPO
GROUP_MA
FORCE_CONTOUR
FX
MACRO_ELAS_MULT
CAS_CHARGE
MODE_FOURIER = 0
CALC_MATR_ELEM
OPTION = “RIGI_MECA”


OPTION = “RIGI_GEOM”


MODE_FOURIER = 0
MODE_ITER_SIMULT

METHOD = “SORENSEN”


TYPE_RESU = “MODE_FLAMB”

CALC_FREQ
OPTION = “SMALLER”
NORM_MODE

= “TRAN NORMALIZES”

5.4
Sizes tested and results

Identification
Reference
Aster %
Difference
Pressure criticizes (n=0)
8.4935 105 Pa
8.6943 105 Pa
2.364
Displacement DY with the node D
1 9.99986
10­1 ­ 0.001
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal systems
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
SSLV139 - Flambement of a circular plate subjected to a force
Date:
12/05/04
Author (S):
X. DESROCHES, C. REZETTE Key
:
V3.04.139-A Page:
9/10

5.5 Remarks

·
The pressure criticizes Pcr of reference, used in the command file, was obtained with
to leave the referred critical load in the paragraph [§2.2]:

F
2
P
Cr
=
= 849350.94N/m
Cr
2 H


·
The standardization of the clean mode for largest of the components of translation
imply a value of reference equal to 1 for displacement DY to node D.
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal systems
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
SSLV139 - Flambement of a circular plate subjected to a force
Date:
12/05/04
Author (S):
X. DESROCHES, C. REZETTE Key
:
V3.04.139-A Page:
10/10

6
Summary of the results

The results obtained are very satisfactory for meshs QUAD8 independently of
modeling used (AXIS or AXIS_FOURIER): uncertainties on the critical pressure
0.104% do not exceed.

However, it will be noted that modeling AXIS_FOURIER is definitely less precise with
meshs TRIA6 that with meshs QUAD8.

This test with license to test and compare modelings AXIS and AXIS_FOURIER in buckling
linear of Euler of a circular mean structure subjected to an external force of compression
uniformly distributed on its contour.

Handbook of Validation
V3.04 booklet: Linear statics of the voluminal systems
HT-66/04/005/A

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