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Titrate:
SDLS503 - Vibrations of inflection of a sandwich beam

Date:
01/10/01
Author (S):
P. MASSIN, F. LEBOUVIER Key
:
V2.03.503-A Page:
1/12

Organization (S): EDF/MTI/MN, DeltaCAD

Handbook of Validation
V2.03 booklet: Linear dynamics of the hulls and plates
Document: V2.03.503

SDLS503 - Vibrations of inflection 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.
Handbook of Validation
V2.03 booklet: Linear dynamics of the hulls and plates
HI-75/01/010/A

Code_Aster ®
Version
5.0
Titrate:
SDLS503 - Vibrations of inflection of a sandwich beam

Date:
01/10/01
Author (S):
P. MASSIN, F. LEBOUVIER Key
:
V2.03.503-A Page:
2/12

1
Problem of reference

1.1 Geometry

Z, W
y, v
L
D
X, U
C
B
Core
h1
H2
L = 1.0 m
D = 0.1m
With
H = 0.1m
1
D
H = 0.05m
2


1.2
Properties of material

Coatings:
E = 4. 1010 Pa
G = 4. 109 Pa
= 0.3
= 2000 kg/m3
X
xz
xz
1
Core:
E = 4. 107 Pa
G = 1.5. 107 Pa
= 0.3
= 50 kg/m3
X
xz
xz
2
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
Handbook of Validation
V2.03 booklet: Linear dynamics of the hulls and plates
HI-75/01/010/A

Code_Aster ®
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Titrate:
SDLS503 - Vibrations of inflection of a sandwich beam

Date:
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:
V2.03.503-A Page:
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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
2
T
2
X

y
v
+ T =
=

I
S

X
y
T 2 X
T 2

v
M
I.E.(internal excitation)
Z
=
=
-
Z
T
K GS
X
y
X
Z

These relations make it possible to write the equation of the movement of dynamic inflection 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:
1


(+ has)

1 - has 2
2
1

1
sin (X) = 0 with X
2
2
2
=
+

2
2
2

2 + r2






and
I 2 2l
I
S I.E.(INTERNAL EXCITATION)
2 =
; R 2 =
; has =

I.E.(internal excitation)
S l2
K I GS
The solutions of the equation at the Eigen frequencies are written then: X = N (n=1,2,3,…)
2

2.2
Results of reference

the first 5 frequencies and clean modes of inflection 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]:
= = 0.
y
Z

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 - Edition Hermes.
[2]
CIEAUX J.M.: Dynamic inflection of the composite beams with orthotropic phases; Validity of
quasi-static field, thesis of the university Paul Sabatier Toulouse III, 1988.
Handbook of Validation
V2.03 booklet: Linear dynamics of the hulls and plates
HI-75/01/010/A

Code_Aster ®
Version
5.0
Titrate:
SDLS503 - Vibrations of inflection of a sandwich beam

Date:
01/10/01
Author (S):
P. MASSIN, F. LEBOUVIER Key
:
V2.03.503-A Page:
4/12

3 Modeling
With

3.1
Characteristics of modeling

Z, W
Modeling DKT (TRIA3)
C
B
D
y, v
With
9.48°
X, U
- Plate located in the plan x= 0.33
- Boundary conditions: Dimensioned AB and CD: u=0

3.2
Characteristics of the grid

A number of nodes: 22
A number of meshs and type: 20 TRIA3

3.3 Functionalities
tested

Commands Key word
factor
Key word
AFFE_MODELE
AFFE
MODELISATION: “DKT”
AFFE_CARA_ELEM HULL
EPAIS
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
:“AJUSTE”
FREQ: (5. 8500.)

Handbook of Validation
V2.03 booklet: Linear dynamics of the hulls and plates
HI-75/01/010/A

Code_Aster ®
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Titrate:
SDLS503 - Vibrations of inflection of a sandwich beam

Date:
01/10/01
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P. MASSIN, F. LEBOUVIER Key
:
V2.03.503-A Page:
5/12

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

Obstruction memory: 16 megabytes
Time CPU To use: 3.37 seconds

Handbook of Validation
V2.03 booklet: Linear dynamics of the hulls and plates
HI-75/01/010/A

Code_Aster ®
Version
5.0
Titrate:
SDLS503 - Vibrations of inflection of a sandwich beam

Date:
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:
V2.03.503-A Page:
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5 Modeling
B

5.1
Characteristics of modeling
Z, W
Modeling DKT (QUAD4)
C
B
D
y, v
With
9.48°
X, U
- Plate located in the plan x= 0.33
- Boundary conditions: Dimensioned AB and CD: u=0

5.2
Characteristics of the grid

A number of nodes: 22
A number of meshs and type: 10 QUAD4

5.3 Functionalities
tested

Commands Key word
factor
Key word
AFFE_MODELE
AFFE
MODELISATION: “DKT”
AFFE_CARA_ELEM HULL
EPAIS
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
:“AJUSTE”
FREQ: (5. 8500.)
Handbook of Validation
V2.03 booklet: Linear dynamics of the hulls and plates
HI-75/01/010/A

Code_Aster ®
Version
5.0
Titrate:
SDLS503 - Vibrations of inflection of a sandwich beam

Date:
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Author (S):
P. MASSIN, F. LEBOUVIER Key
:
V2.03.503-A Page:
7/12

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

Obstruction memory: 16 megabytes
Time CPU To use: 3.53 seconds

Handbook of Validation
V2.03 booklet: Linear dynamics of the hulls and plates
HI-75/01/010/A

Code_Aster ®
Version
5.0
Titrate:
SDLS503 - Vibrations of inflection of a sandwich beam

Date:
01/10/01
Author (S):
P. MASSIN, F. LEBOUVIER Key
:
V2.03.503-A Page:
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7 Modeling
C

7.1
Characteristics of modeling

Modeling DST (TRIA3)
Z, W
C
B
D
y, v
With
9.48°
X, U
- Plate located in the plan x= 0.33
- Boundary conditions: Dimensioned AB and CD: u=0

7.2
Characteristics of the grid

A number of nodes: 22
A number of meshs and type: 20 TRIA3

7.3 Functionalities
tested

Commands Key word
factor
Key word
AFFE_MODELE
AFFE
MODELISATION: “DST”
AFFE_CARA_ELEM HULL
EPAIS
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
:“AJUSTE”
FREQ: (5. 500.)

Handbook of Validation
V2.03 booklet: Linear dynamics of the hulls and plates
HI-75/01/010/A

Code_Aster ®
Version
5.0
Titrate:
SDLS503 - Vibrations of inflection of a sandwich beam

Date:
01/10/01
Author (S):
P. MASSIN, F. LEBOUVIER Key
:
V2.03.503-A Page:
9/12

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

Obstruction memory: 16 megabytes
Time CPU To use: 2.84 seconds

Handbook of Validation
V2.03 booklet: Linear dynamics of the hulls and plates
HI-75/01/010/A

Code_Aster ®
Version
5.0
Titrate:
SDLS503 - Vibrations of inflection of a sandwich beam

Date:
01/10/01
Author (S):
P. MASSIN, F. LEBOUVIER Key
:
V2.03.503-A Page:
10/12

9 Modeling
D

9.1
Characteristics of modeling

Modeling DST (QUAD4)
Z, W
C
B
D
y, v
With
9.48°
X, U
- Plate located in the plan x= 0.33
- Boundary conditions: Dimensioned AB and CD: u=0

9.2
Characteristics of the grid

A number of nodes: 22
A number of meshs and type: 10 QUAD4

9.3 Functionalities
tested

Commands Key word
factor
Key word
AFFE_MODELE
AFFE
MODELISATION: “DST”
AFFE_CARA_ELEM HULL
EPAIS
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
:“AJUSTE”
FREQ: (5. 500.)
Handbook of Validation
V2.03 booklet: Linear dynamics of the hulls and plates
HI-75/01/010/A

Code_Aster ®
Version
5.0
Titrate:
SDLS503 - Vibrations of inflection of a sandwich beam

Date:
01/10/01
Author (S):
P. MASSIN, F. LEBOUVIER Key
:
V2.03.503-A Page:
11/12

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

Obstruction memory: 16 megabytes
Time CPU To use: 3.21 seconds

Handbook of Validation
V2.03 booklet: Linear dynamics of the hulls and plates
HI-75/01/010/A

Code_Aster ®
Version
5.0
Titrate:
SDLS503 - Vibrations of inflection of a sandwich beam

Date:
01/10/01
Author (S):
P. MASSIN, F. LEBOUVIER Key
:
V2.03.503-A Page:
12/12

11 Summary of the results

Modeling DKT is not adapted to model 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 4th and 5th frequency with mesh TRIA3 is significant. One
finer grid should make it possible to improve the results by having the best
representation of the last modes.
Handbook of Validation
V2.03 booklet: Linear dynamics of the hulls and plates
HI-75/01/010/A

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