Code_Aster
®
Version
4.0
Titrate:
SDLV100 Vibration of a slim beam of rectangular section
Date:
31/01/00
Author (S):
D. GIRARDOT
Key
:
V2.04.100-A
Page:
1/8
Manual of Validation
V2.04 booklet: Linear dynamics of the voluminal structures
HT-62/01/012/A
Organization (S):
EDF/EP/AMV
Manual 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 modelized by
voluminal elements. One is interested in his Eigen frequencies in bending. The same problem is dealt with in
modeling beam in the case test SDLL09.
This problem makes it possible to test the voluminal elements
MECA_HEXA20
and
MECA_PENTA15
in modal analysis.
It also makes it possible to test the 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.
Code_Aster
®
Version
4.0
Titrate:
SDLV100 Vibration of a slim beam of rectangular section
Date:
31/01/00
Author (S):
D. GIRARDOT
Key
:
V2.04.100-A
Page:
2/8
Manual of Validation
V2.04 booklet: Linear dynamics of the voluminal structures
HT-62/01/012/A
1
Problem of reference
1.1 Geometry
y, v
B
O
H
O
Z, W
With
B
1
H
1
B
L
X, U
y
Z
H
O
B
O
y
Z
H
1
B
1
Rectangular sections
Length of the beam:
L = 1 m
Rectangular section:
Initial cross-section
Final cross-section
height:
H
O
= 0.04 m
H
1
= 0.01 m
width:
B
O
= 0.04 m
B
1
= 0.01 m
surface:
With
O
= 1.6 10
3
m
2
With
1
= 1.10
4
m
2
inertia:
lz
O
= 2.1333 10
7
m
4
Iz
1
= 8.3333 10
10
m
4
Co-ordinates of the points (in meters)
WITH B
X
0. 1.
y
0. 0.
Z
0. 0.
1.2
Properties of steel
E
= 2.10
11
AP
= 7.800 kg/m
3
1.3
Boundary conditions and loadings
Not a: embedded U = v = Z = 0
Code_Aster
®
Version
4.0
Titrate:
SDLV100 Vibration of a slim beam of rectangular section
Date:
31/01/00
Author (S):
D. GIRARDOT
Key
:
V2.04.100-A
Page:
3/8
Manual of Validation
V2.04 booklet: Linear dynamics of the voluminal structures
HT-62/01/012/A
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 bending of the beam considered, in theory of Euler-Bernoulli is written (Theory
of Euler-Bernoulli):
2
2
2
2
2
2
E I
v
X
X
With
v
T
Z
= -
where
I
Z
and
With
vary with the X-coordinate.
The Eigen frequencies are then of the form:
()
F
H
L
E
I
I
=
1
2
12
12
,
with
=
=
=
=
H
H
B
B
0
1
0
1
4
4
and
.
For this value of
and
, 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 bending.
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).
Code_Aster
®
Version
4.0
Titrate:
SDLV100 Vibration of a slim beam of rectangular section
Date:
31/01/00
Author (S):
D. GIRARDOT
Key
:
V2.04.100-A
Page:
4/8
Manual of Validation
V2.04 booklet: Linear dynamics of the voluminal structures
HT-62/01/012/A
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
·
at end A (group of G_1 nodes)
DDL_IMPO: (ALL:“YES” DZ: 0.)
(GROUP_NO: G_1 DX: 0., DY: 0)
3.2
Characteristics of the mesh
Mesh:
A number of nodes: 368
A number of meshs and type: 30 HEXA20
3.3 Functionalities
tested
Controls
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]
Code_Aster
®
Version
4.0
Titrate:
SDLV100 Vibration of a slim beam of rectangular section
Date:
31/01/00
Author (S):
D. GIRARDOT
Key
:
V2.04.100-A
Page:
5/8
Manual of Validation
V2.04 booklet: Linear dynamics of the voluminal structures
HT-62/01/012/A
4
Results of modeling A
4.1 Values
tested
Identification Solution
beam
analytical
Reference
The SAMCEF software
Aster %
difference
The Aster-SAMCEF software
frequency
in HZ
in HZ
coherent matrix
bending 1
54.18
56.84
56.85
0.0176%
bending 2
171.94
180.0
180.08
0.0444%
bending 3
384.40
401.0
401.23
0.0574%
bending 4
697.24
723.2
724.02
0.1134%
bending 5
1112.28
1145.41
1147.51
0.1833%
stamp diagonal
bending 1
54.18
56.84
56.78
0.1033%
bending 2
171.94
180.00
179.57
0.2419%
bending 3
384.40
401.00
399.24
0.4408%
bending 4
697.24
723.20
718.69
0.6273%
bending 5
1112.28
1145.41
1136.01
0.8273%
4.2 Parameters
of execution
Version: NEW4.03.06
Machine: CRAY C90
Overall dimension memory: 8 MW,
time CPU To use: 32.08 seconds.
Code_Aster
®
Version
4.0
Titrate:
SDLV100 Vibration of a slim beam of rectangular section
Date:
31/01/00
Author (S):
D. GIRARDOT
Key
:
V2.04.100-A
Page:
6/8
Manual of Validation
V2.04 booklet: Linear dynamics of the voluminal structures
HT-62/01/012/A
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
·
at end A (group of G_1 nodes)
DDL_IMPO: (ALL:“YES” DZ: 0.)
(GROUP_NO: G_1 DX: 0., DY: 0)
5.2
Characteristics of the mesh
Mesh:
A number of nodes: 368
A number of meshs and type: 60 PENTA15
5.3 Functionalities
tested
Controls
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]
Code_Aster
®
Version
4.0
Titrate:
SDLV100 Vibration of a slim beam of rectangular section
Date:
31/01/00
Author (S):
D. GIRARDOT
Key
:
V2.04.100-A
Page:
7/8
Manual of Validation
V2.04 booklet: Linear dynamics of the voluminal structures
HT-62/01/012/A
6
Results of modeling B
6.1 Values
tested
Identification Solution
beam
semi-analytical
Reference
The SAMCEF software
Aster %
difference
ASTER-SAMCEF
frequency
in HZ
in HZ
consistent matrix
bending 1
54.18
56.84
56.82
0.038%
bending 2
171.94
180.00
179.96
0.022%
bending 3
384.40
401.00
400.93
0.018%
bending 4
697.24
723.20
723.41
0.029%
bending 5
1112.28
1145.41
1146.41
0.088%
stamp diagonal
bending 1
54.18
56.84
56.76
0.149%
bending 2
171.94
180.00
179.51
0.272%
bending 3
384.40
401.00
399.25
0.437%
bending 4
697.24
723.20
719.
0.583%
bending 5
1112.28
1145.41
1140.
0.740%
6.2 Parameters
of execution
Version: NEW4.03.06
Machine: CRAY C90
Overall dimension memory: 8 MW,
time CPU To use: 47.09 seconds.
Code_Aster
®
Version
4.0
Titrate:
SDLV100 Vibration of a slim beam of rectangular section
Date:
31/01/00
Author (S):
D. GIRARDOT
Key
:
V2.04.100-A
Page:
8/8
Manual of Validation
V2.04 booklet: Linear dynamics of the voluminal structures
HT-62/01/012/A
7
Summary of the results
The differences between the results of calculations Aster 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 the operator
CALC_MATR_ELEM
in
coherent masses as in diagonal masses.