Code_Aster
®
Version
5.0
Titrate:
SDLS503 - Vibrations of bending of a sandwich beam
Date:
01/10/01
Author (S):
P. MASSIN, F. LEBOUVIER
Key
:
V2.03.503-A
Page:
1/12
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and plates
HI-75/01/010/A
Organization (S):
EDF/MTI/MN, DeltaCAD
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and plates
Document: V2.03.503
SDLS503 - Vibrations of bending of a beam
sandwich
Summary:
This test represents a calculation in modal analysis of a sandwich beam simply supported. This test allows
to validate:
·
modeling finite elements DKT with meshs QUAD4 and TRIA3,
·
modeling finite elements DST with meshs QUAD4 and TRIA3,
·
the taking into account of rigidity in transverse shearing,
·
the taking into composite material account.
The frequencies and the modes obtained are compared with an analytical reference solution.
Code_Aster
®
Version
5.0
Titrate:
SDLS503 - Vibrations of bending of a sandwich beam
Date:
01/10/01
Author (S):
P. MASSIN, F. LEBOUVIER
Key
:
V2.03.503-A
Page:
2/12
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and plates
HI-75/01/010/A
1
Problem of reference
1.1 Geometry
Z, W
y, v
L = 1.0 m
D = 0.1m
H
1
= 0.1m
H
2
= 0.05m
L
H
2
H
1
D
X, U
D
C
With
B
Core
1.2
Properties of material
Coatings:
E
X
= 4. 10
10
AP
G
xz
= 4. 10
9
AP
xz
= 0.3
1
= 2000 kg/m
3
Core:
E
X
= 4. 10
7
AP
G
xz
= 1.5. 10
7
AP
xz
= 0.3
2
= 50 kg/m
3
Coefficient of shearing K: 1/K=110.8
The Poisson's ratios are identical:
xz
=
xy
=
yz
1.3
Boundary conditions and loadings
C.L. : the beam rests simply on the with dimensions AB and CD.
1.4 Conditions
initial
Without object
Code_Aster
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Titrate:
SDLS503 - Vibrations of bending of a sandwich beam
Date:
01/10/01
Author (S):
P. MASSIN, F. LEBOUVIER
Key
:
V2.03.503-A
Page:
3/12
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and plates
HI-75/01/010/A
2
Reference solution
2.1
Method of calculation used for the reference solution
Calculation is carried out starting from the relations of dynamic balance and behavior [bib2] pointed out
hereafter:
M
X
T
I
T
X
y
+
=
2
2
T
X
S
v
T
y
=
2
2
M
I.E.(internal excitation)
X
Z
Z
=
T
K GS
v
X
y
Z
=
-
These relations make it possible to write the equation of the movement of dynamic bending transverse v (X, T). One
the equation at the Eigen frequencies obtains after having associated the boundary conditions.
The equation at the Eigen frequencies is written:
sin (
)
X
2
0
=
with
(
)
X
has
has
R
2
2
2
2
2
2
1
2
1
2
1
2
1
=
+
+
-
+
and
2
2 2
=
I
L
I.E.(internal excitation)
;
R
I
S L
2
2
=
;
has
S I.E.(INTERNAL EXCITATION)
K I GS
=
The solutions of the equation at the Eigen frequencies are written then:
X
N
2
=
(n=1,2,3,…)
2.2
Results of reference
the first 5 frequencies and clean modes of bending associated.
·
Frequency mode 1:
64.476 Hz
·
Frequency mode 2:
131.918 Hz
·
Frequency mode 3:
198.734 Hz
·
Frequency mode 4:
265.383 Hz
·
Frequency mode 5:
331.963 Hz
2.3
Uncertainties on the solution
The reference solution is calculated within the framework of the assumptions of the theory of the beams [bib2]:
y
Z
=
= 0
.
2.4 References
bibliographical
[1]
VPCS: Software package of composite structural analysis; Examples of validation. Review of
composites and of advanced materials, Volume 5 - number except series/1995 - Hermes Edition.
[2]
CIEAUX J.M.: Dynamic bending of the composite beams with orthotropic phases; Validity of
quasi-static field, thesis of the university Paul Sabatier Toulouse III, 1988.
Code_Aster
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Version
5.0
Titrate:
SDLS503 - Vibrations of bending of a sandwich beam
Date:
01/10/01
Author (S):
P. MASSIN, F. LEBOUVIER
Key
:
V2.03.503-A
Page:
4/12
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and plates
HI-75/01/010/A
3 Modeling
With
3.1
Characteristics of modeling
Modeling DKT (TRIA3)
X, U
y, v
Z, W
9.48°
- Plate located in the plan x= 0.33
- Boundary conditions: Dimensioned AB and CD: u=0
With
B
D
C
3.2
Characteristics of the mesh
A number of nodes: 22
A number of meshs and type: 20 TRIA3
3.3 Functionalities
tested
Controls Key word
factor
Key word
AFFE_MODELE
AFFE
MODELING: “DKT”
AFFE_CARA_ELEM HULL
THICK
DEFI_MATERIAU
ELAS_ORTH
E_L E_T G_LT G_LN
NU_LT NU_LN RHO
CALC_MATR_ELEM OPTION
“RIGI_MECA”
“MASS_MECA”
MODE_ITER_INV CALC_FREQ
OPTION
:“ADJUSTS”
FREQ: (5. 8500.)
Code_Aster
®
Version
5.0
Titrate:
SDLS503 - Vibrations of bending of a sandwich beam
Date:
01/10/01
Author (S):
P. MASSIN, F. LEBOUVIER
Key
:
V2.03.503-A
Page:
5/12
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and plates
HI-75/01/010/A
4
Results of modeling A
4.1 Values
tested
Identification Reference Aster %
difference
Frequency mode 1
64.476
277.449
330.
Frequency mode 2
131.918
1105.83
738.
Frequency mode 3
198.734
2473.80
1.14E3
Frequency mode 4
265.383
4363.97
1.54E3
Frequency mode 5
331.963
6753.904
1.93E3
4.2 Remarks
In the table of results, we deferred the frequencies whose modes are identical to
modes of reference.
·
the effects of transverse shearing are neglected in modeling “DKT”,
·
the Aster results are much higher than the results of reference,
·
appearance of a mode of membrane enters modes 2 and 3 and between modes 5 and 6 of
reference.
4.3 Parameters
of execution
Version:
NEW 5.04.17
Machine:
SGI-Origin2000 R12000
Overall dimension memory: 16 megabytes
Time CPU To use: 3.37 seconds
Code_Aster
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Version
5.0
Titrate:
SDLS503 - Vibrations of bending of a sandwich beam
Date:
01/10/01
Author (S):
P. MASSIN, F. LEBOUVIER
Key
:
V2.03.503-A
Page:
6/12
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and plates
HI-75/01/010/A
5 Modeling
B
5.1
Characteristics of modeling
Modeling DKT (QUAD4)
X, U
y, v
Z, W
9.48°
- Plate located in the plan x= 0.33
- Boundary conditions: Dimensioned AB and CD: u=0
With
B
D
C
5.2
Characteristics of the mesh
A number of nodes: 22
A number of meshs and type: 10 QUAD4
5.3 Functionalities
tested
Controls Key word
factor
Key word
AFFE_MODELE
AFFE
MODELING: “DKT”
AFFE_CARA_ELEM HULL
THICK
DEFI_MATERIAU
ELAS_ORTH
E_L E_T G_LT G_LN
NU_LT NU_LN RHO
CALC_MATR_ELEM OPTION
“RIGI_MECA”
“MASS_MECA”
MODE_ITER_INV CALC_FREQ OPTION
:“ADJUSTS”
FREQ: (5. 8500.)
Code_Aster
®
Version
5.0
Titrate:
SDLS503 - Vibrations of bending of a sandwich beam
Date:
01/10/01
Author (S):
P. MASSIN, F. LEBOUVIER
Key
:
V2.03.503-A
Page:
7/12
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and plates
HI-75/01/010/A
6
Results of modeling B
6.1 Values
tested
Identification Reference
Aster %
difference
Frequency mode 1
64.476
277.788
331
Frequency mode 2
131.918
1111.225
742.
Frequency mode 3
198.734
2500.930
1.16E3
Frequency mode 4
265.383
4449.073
1.52E3
Frequency mode 5
331.963
6960.324
2.00E3
6.2 Remarks
In the table of results, we deferred the frequencies whose modes are identical to
modes of reference.
·
the effects of transverse shearing are neglected in modeling “DKT”,
·
the Aster results are much higher than the results of reference,
·
appearance of a mode of membrane enters modes 2 and 3 and between modes 5 and 6 of
reference.
6.3 Parameters
of execution
Version:
NEW 5.04.17
Machine:
SGI-Origin2000 R12000
Overall dimension memory: 16 megabytes
Time CPU To use: 3.53 seconds
Code_Aster
®
Version
5.0
Titrate:
SDLS503 - Vibrations of bending of a sandwich beam
Date:
01/10/01
Author (S):
P. MASSIN, F. LEBOUVIER
Key
:
V2.03.503-A
Page:
8/12
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and plates
HI-75/01/010/A
7 Modeling
C
7.1
Characteristics of modeling
Modeling DST (TRIA3)
X, U
y, v
Z, W
9.48°
- Plate located in the plan x= 0.33
- Boundary conditions: Dimensioned AB and CD: u=0
With
B
D
C
7.2
Characteristics of the mesh
A number of nodes: 22
A number of meshs and type: 20 TRIA3
7.3 Functionalities
tested
Controls Key word
factor
Key word
AFFE_MODELE
AFFE
MODELING: “DST”
AFFE_CARA_ELEM HULL
THICK
DEFI_MATERIAU
ELAS_ORTH
E_L E_T
G_LT G_LN
NU_LT NU_LN
RHO
CALC_MATR_ELEM OPTION
“RIGI_MECA”
“MASS_MECA”
MODE_ITER_INV CALC_FREQ OPTION
:“ADJUSTS”
FREQ: (5. 500.)
Code_Aster
®
Version
5.0
Titrate:
SDLS503 - Vibrations of bending of a sandwich beam
Date:
01/10/01
Author (S):
P. MASSIN, F. LEBOUVIER
Key
:
V2.03.503-A
Page:
9/12
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and plates
HI-75/01/010/A
8
Results of modeling C
8.1 Values
tested
Identification Reference
Aster %
difference
Frequency mode 1
64.476
64.573
0.150
Frequency mode 2
131.918
133.987
1.568
Frequency mode 3
198.734
206.046
3.679
Frequency mode 4
265.383
282.875
6.591
Frequency mode 5
331.963
365.919
10.229
8.2 Parameters
of execution
Version:
NEW 5.04.17
Machine:
SGI-Origin2000 R12000
Overall dimension memory: 16 megabytes
Time CPU To use: 2.84 seconds
Code_Aster
®
Version
5.0
Titrate:
SDLS503 - Vibrations of bending of a sandwich beam
Date:
01/10/01
Author (S):
P. MASSIN, F. LEBOUVIER
Key
:
V2.03.503-A
Page:
10/12
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and plates
HI-75/01/010/A
9 Modeling
D
9.1
Characteristics of modeling
Modeling DST (QUAD4)
X, U
y, v
Z, W
9.48°
- Plate located in the plan x= 0.33
- Boundary conditions: Dimensioned AB and CD: u=0
With
B
D
C
9.2
Characteristics of the mesh
A number of nodes: 22
A number of meshs and type: 10 QUAD4
9.3 Functionalities
tested
Controls Key word
factor
Key word
AFFE_MODELE
AFFE
MODELING: “DST”
AFFE_CARA_ELEM HULL
THICK
DEFI_MATERIAU
ELAS_ORTH
E_L E_T
G_LT G_LN
NU_LT NU_LN
RHO
CALC_MATR_ELEM OPTION
“RIGI_MECA”
“MASS_MECA”
MODE_ITER_INV CALC_FREQ
OPTION
:“ADJUSTS”
FREQ: (5. 500.)
Code_Aster
®
Version
5.0
Titrate:
SDLS503 - Vibrations of bending of a sandwich beam
Date:
01/10/01
Author (S):
P. MASSIN, F. LEBOUVIER
Key
:
V2.03.503-A
Page:
11/12
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and plates
HI-75/01/010/A
10 Results of modeling D
10.1 Values
tested
Identification Reference
Aster %
difference
Frequency mode 1
64.476
64.595
0.184
Frequency mode 2
131.918
131.495
- 0.320
Frequency mode 3
198.734
196.861
- 0.942
Frequency mode 4
265.383
260.247
- 1.935
Frequency mode 5
331.963
320.409
- 3.480
10.2 Parameters
of execution
Version:
NEW 5.04.17
Machine:
SGI-Origin2000 R12000
Overall dimension memory: 16 megabytes
Time CPU To use: 3.21 seconds
Code_Aster
®
Version
5.0
Titrate:
SDLS503 - Vibrations of bending of a sandwich beam
Date:
01/10/01
Author (S):
P. MASSIN, F. LEBOUVIER
Key
:
V2.03.503-A
Page:
12/12
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and plates
HI-75/01/010/A
11 Summary of the results
Modeling DKT is not adapted to modelize this case-test, the errors are very important.
Formulation DKT does not take into account transverse shearing contrary to modeling
DST. For this type of example, where the structure makes up of a composite material and relatively
thick (H/L =0.1), it is preferable to use DST modeling.
The results obtained with DST are:
·
satisfactory for the first 3 frequencies with mesh TRIA3 and for the 5 first
frequencies for mesh QUAD4 with a better precision for mesh QUAD4,
·
the error of 10% for the 4
ičme
and 5
ičme
frequency with mesh TRIA3 is significant. One
finer mesh should make it possible to improve the results by having the best
representation of the last modes.