Code_Aster ®
Version
4.0
Titrate:
SDLV100 Vibration of a slim beam of rectangular section
Date:
31/01/00
Author (S):
D. Key GIRARDOT
:
V2.04.100-A Page:
1/8

Organization (S): EDF/EP/AMV
Handbook of Validation
V2.04 booklet: Linear dynamics of the voluminal structures
Document: V2.04.100

SDLV100 - Vibration of a slim beam
of variable rectangular section (embed-free)

Summary:

The studied structure is a beam out of free steel embedded with rectangular variable section modelled by
voluminal elements. One is interested in his Eigen frequencies in inflection. The same problem is dealt with in
modeling beam in the case test SDLL09.

This problem makes it possible to test voluminal elements MECA_HEXA20 and MECA_PENTA15 in modal analysis.
It also makes it possible to test option MASS_MECA_DIAG of calculation of the matrices of mass diagonalized for
voluminal modelings.

The reference solution is a numerical solution obtained using the computer code by finite elements
The SAMCEF software for similar modelings. The results obtained are also in concord with
semi-analytical results given in guide VPCS.

Handbook of Validation
V2.04 booklet: Linear dynamics of the voluminal structures
HT-62/01/012/A

Code_Aster ®
Version
4.0
Titrate:
SDLV100 Vibration of a slim beam of rectangular section
Date:
31/01/00
Author (S):
D. Key GIRARDOT
:
V2.04.100-A Page:
2/8

1
Problem of reference

1.1 Geometry

y, v
L
y
y
bo
b1
B
H
H
O
h1
H
With
1
Z
O
X, U
Z
Z, W
b1
bo
Rectangular sections


Length of the beam:

L = 1 m

Rectangular section:


Initial cross-section
Final cross-section
height:
Ho = 0.04 m
h1 = 0.01 m
width:
bo = 0.04 m
b1 = 0.01 m
surface:
Ao = 1.6 10­3 m2
A1 = 1.10­4 m2
inertia:
lzo = 2.1333 10­7 m4 Iz1 = 8.3333 10­10 m4

Co-ordinates of the points (in meters)


WITH B
X
0. 1.
y
0. 0.
Z
0. 0.

1.2
Properties of steel

E = 2.1011 Pa

= 7.800 kg/m3

1.3
Boundary conditions and loadings

Not a: embedded U = v = Z = 0

Handbook of Validation
V2.04 booklet: Linear dynamics of the voluminal structures
HT-62/01/012/A

Code_Aster ®
Version
4.0
Titrate:
SDLV100 Vibration of a slim beam of rectangular section
Date:
31/01/00
Author (S):
D. Key GIRARDOT
:
V2.04.100-A Page:
3/8

2
Reference solution

2.1
Method of calculation used for the reference solution

The reference solution is obtained using the computation software by finite elements the SAMCEF software for
identical modelings but with elementary matrices of mass coherent.

One points out the analytical solution given in card SDLL09/89 of guide VPCS. The equation
differential in inflection of the beam considered, in theory of Euler-Bernoulli is written (Théorie
of Euler-Bernoulli):


2
2
v
E I

Z

2
X
2 v
= - A

2
X
2
T

where Iz and A vary with the X-coordinate.

The Eigen frequencies are then of the form:

1
H
E
F
1
I
=
I
(,)

2
L2

12

H
B
with
=
0
= 4
and
=
0
= 4.
H
B
1
1

For this value of and, the first values of the continuation (I) are:


1
2
3
4
5
= 4
23.289 73.9 165.23 299.7 478.1

2.2
Results of reference

The results of reference selected are the first 5 Eigen frequencies of the modes of inflection.

2.3
Uncertainty on the solution

Analytical solution in theory of beam of Bernoulli, and numerical solution the SAMCEF software.

2.4 References
bibliographical

[1]
H.H. MABIE, C.B. ROGERS, Transverse vibrations off double-tapered cantilever beams -
Newspaper off the Acoustical Society off America, n° 51, p. 1771-1774 (1972).

Handbook of Validation
V2.04 booklet: Linear dynamics of the voluminal structures
HT-62/01/012/A

Code_Aster ®
Version
4.0
Titrate:
SDLV100 Vibration of a slim beam of rectangular section
Date:
31/01/00
Author (S):
D. Key GIRARDOT
:
V2.04.100-A Page:
4/8

3 Modeling
With

3.1
Characteristics of modeling

Elements of volume MECA_HEXA20



Discretization:
beam AB: 30 meshs HEXA20
(1 mesh in the section)

Boundary conditions:


·
in all the nodes
DDL_IMPO: (TOUT:“YES” DZ: 0.)
·
at end A (group of G_1 nodes) (GROUP_NO: G_1 DX: 0., DY: 0)

3.2
Characteristics of the grid

Grid:
A number of nodes: 368
A number of meshs and type: 30 HEXA20

3.3 Functionalities
tested

Commands


Keys
AFFE_CHAR_MECA DDL_IMPO


[U4.25.01]
“MECHANICAL” AFFE_MODELE
“3D”

[U4.22.01]
DEFI_MATERIAU ELAS


[U4.23.01]
CALC_MATR_ELEM OPTION
“MASS_MECA_DIAG”

[U4.41.01]
MODE_ITER_INV


[U4.52.01]

Handbook of Validation
V2.04 booklet: Linear dynamics of the voluminal structures
HT-62/01/012/A

Code_Aster ®
Version
4.0
Titrate:
SDLV100 Vibration of a slim beam of rectangular section
Date:
31/01/00
Author (S):
D. Key GIRARDOT
:
V2.04.100-A Page:
5/8

4
Results of modeling A

4.1 Values
tested

Identification Solution
beam
Reference
Aster %
difference
analytical
The SAMCEF software
The Aster-SAMCEF software

frequency
in HZ
in HZ





coherent matrix





inflection 1
54.18
56.84
56.85
0.0176%
inflection 2
171.94
180.0
180.08
0.0444%
inflection 3
384.40
401.0
401.23
0.0574%
inflection 4
697.24
723.2
724.02
0.1134%
inflection 5
1112.28
1145.41
1147.51
0.1833%










stamp diagonal





inflection 1
54.18
56.84
56.78
­ 0.1033%
inflection 2
171.94
180.00
179.57
­ 0.2419%
inflection 3
384.40
401.00
399.24
­ 0.4408%
inflection 4
697.24
723.20
718.69
­ 0.6273%
inflection 5
1112.28
1145.41
1136.01
­ 0.8273%

4.2 Parameters
of execution

Version: NEW4.03.06



Machine: CRAY C90



Obstruction memory: 8 MW,
time CPU User: 32.08 seconds.

Handbook of Validation
V2.04 booklet: Linear dynamics of the voluminal structures
HT-62/01/012/A

Code_Aster ®
Version
4.0
Titrate:
SDLV100 Vibration of a slim beam of rectangular section
Date:
31/01/00
Author (S):
D. Key GIRARDOT
:
V2.04.100-A Page:
6/8

5 Modeling
B

5.1
Characteristics of modeling

Elements of volume MECA_PENTA15




Discretization:
beam AB: 60 meshs PENTA15
(2 meshs in the section)

Boundary conditions:


·
in all the nodes
DDL_IMPO: (TOUT:“YES” DZ: 0.)
·
at end A (group of G_1 nodes) (GROUP_NO: G_1 DX: 0., DY: 0)

5.2
Characteristics of the grid

Grid:
A number of nodes: 368
A number of meshs and type: 60 PENTA15

5.3 Functionalities
tested

Commands


Keys
AFFE_CHAR_MECA DDL_IMPO


[U4.25.01]
“MECHANICAL” AFFE_MODELE
“3D”

[U4.22.01]
DEFI_MATERIAU ELAS


[U4.23.01]
CALC_MATR_ELEM OPTION
“MASS_MECA_DIAG”

[U4.41.01]
MODE_ITER_INV


[U4.52.01]

Handbook of Validation
V2.04 booklet: Linear dynamics of the voluminal structures
HT-62/01/012/A

Code_Aster ®
Version
4.0
Titrate:
SDLV100 Vibration of a slim beam of rectangular section
Date:
31/01/00
Author (S):
D. Key GIRARDOT
:
V2.04.100-A Page:
7/8

6
Results of modeling B

6.1 Values
tested

Identification Solution
beam
Reference
Aster %
difference
semi-analytical
The SAMCEF software
ASTER-SAMCEF

frequency
in HZ
in HZ






consistent matrix





inflection 1
54.18
56.84
56.82
­ 0.038%
inflection 2
171.94
180.00
179.96
­ 0.022%
inflection 3
384.40
401.00
400.93
­ 0.018%
inflection 4
697.24
723.20
723.41
0.029%
inflection 5
1112.28
1145.41
1146.41
0.088%










stamp diagonal





inflection 1
54.18
56.84
56.76
­ 0.149%
inflection 2
171.94
180.00
179.51
­ 0.272%
inflection 3
384.40
401.00
399.25
­ 0.437%
inflection 4
697.24
723.20
719.
­ 0.583%
inflection 5
1112.28
1145.41
1140.
­ 0.740%

6.2 Parameters
of execution

Version: NEW4.03.06
Machine: CRAY C90
Obstruction memory: 8 MW,
time CPU User: 47.09 seconds.

Handbook of Validation
V2.04 booklet: Linear dynamics of the voluminal structures
HT-62/01/012/A

Code_Aster ®
Version
4.0
Titrate:
SDLV100 Vibration of a slim beam of rectangular section
Date:
31/01/00
Author (S):
D. Key GIRARDOT
:
V2.04.100-A Page:
8/8

7
Summary of the results

The differences between the results of Aster calculations and the SAMCEF software with coherent masses are lower than
0.2%.

Differences between the computation results Aster with diagonal masses and the SAMCEF software with masses
coherent remain lower than 1%.

These results are in conformity so that one could wait, and validate in a reliable way them
calculations of Eigen frequencies in Aster by MODE_ITER_INV and operator CALC_MATR_ELEM in
coherent masses as in diagonal masses.

Handbook of Validation
V2.04 booklet: Linear dynamics of the voluminal structures
HT-62/01/012/A

Outline document