Code_Aster
®
Version
4.0
Titrate:
SDLL10 Beam of rectangular section variable
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.010-C
Page:
1/6
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
Organization (S):
EDF/IMA/MN
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
V2.02.010 document
SDLL10 - Rectangular beam of section
variable (embed-embedded)
Summary:
This plane problem consists in seeking the frequencies and the modes of vibration of a mechanical structure
composed of an embed-embedded beam whose surface of the cross-section varies exponentially. This test
of Mechanics of the Structures corresponds to a dynamic analysis of a linear model having one
linear behavior. It includes/understands only one modeling.
Via this problem, one tests the element of beam in bending of variable Timoshenko of section
as well as the calculation of the frequencies and modes of vibration by the method of Lanczos. One also tests
functionality “normalizes to 1.” at the point of maximum amplitude in translation " of the modes of vibration.
By using a fine space discretization, the results obtained are in concord with the results
analytical given in guide VPCS.
Code_Aster
®
Version
4.0
Titrate:
SDLL10 Beam of rectangular section variable
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.010-C
Page:
2/6
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
1
Problem of reference
1.1 Geometry
y, v
B
O
H
Z, W
With
H
B
L
X, U
y
Z
H
B
O
y
Z
H
B
1
B
1
initial
finale
Length of the beam: L = 0.6 m
Constant thickness: H = 0.01 m
Rectangular section:
Initial cross-section:
Variation of the section:
width:
B
0
= 0.03 m
B = B
0
E
2
X
with
= 1.
surface:
With
0
= 3.10
4
m
2
WITH = A
0
E
2
X
moment of inertia:
Iz
0
= 0.25 10
8
m
4
Iz = Iz
0
E
2
X
Co-ordinates of the points (m):
With
B
X
0.
0.6
y
0.
0.
1.2
Material properties
E
= 2.10
11
AP
= 0.3
= 7.800. kg/m
3
1.3
Boundary conditions and loadings
Embedded points A and b:, U = v = 0.,
= 0.
Code_Aster
®
Version
4.0
Titrate:
SDLL10 Beam of rectangular section variable
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.010-C
Page:
3/6
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
2
Reference solution
2.1
Method of calculation used for the reference solution
The reference solution is that given in card SDLL10/89 of the guide VPCS which presents
method of calculation in the following way:
The pulsation
I
is given by the roots of the equation:
1
-
cos rL
()
CH SSL
()
+
S
2
-
R
2
2rs HS SSL
()
sin rL
()
=
0
with:
(
)
I
I
Z
I
I
I
With
E I
R
S
4
0
2
0
2
2
2
2
2
2
0
=
=
+
=
>
;
;
if
Components of translation
mode
F
I
X
()
are then:
()
()
()
()
()
()
()
()
()
(
)
I
X
X
E
X-ray
sx
rL
SSL
R
SSL
S
rL S
X-ray
R
sx
=
-
+
-
-
-
cos
CH
cos
CH
HS
sin
sin
HS
2.2
Results of reference
the first 4 Eigen frequencies and normalized clean modes with 1 for the largest component in
translation.
2.3
Uncertainty on the solution
Analytical solution.
2.4 References
bibliographical
[1]
Working group Analyzes Dynamic. Committee of Validation of the Software packages of Calculation of
Structure. French company of the Mechanics (1988).
Code_Aster
®
Version
4.0
Titrate:
SDLL10 Beam of rectangular section variable
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.010-C
Page:
4/6
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
3 Modeling
With
3.1
Characteristics of modeling
Element of beam
POU_D_T
(Right Beam of Timoshenko)
y
B
With
X
Cutting: beam AB: 120 meshs SEG2 of section variable.
Limiting conditions:
in all the nodes
DDL_IMPO
(ALL:“YES”, DZ: 0., DRX: 0., DRY: 0. )
with the nodes
ends
(NODE: (AB) DX: 0., DY: 0., DRZ: 0. )
Names of the nodes:
Not A
X = 0.
= N1
X = 0.1
= N21
X = 0.2
= N41
X = 0.3
= N61
X = 0.4
= N81
X = 0.5
= N101
Not B
X = 0.6
= N121
3.2
Characteristics of the mesh
A number of nodes:
121
A number of meshs and types:
120 SEG2
3.3 Functionalities
tested
Controls
Keys
AFFE_CARA_ELEM
BEAM
“GENERAL”
NET
[U4.24.01]
AFFE_CHAR_MECA
DDL_IMPO
ALL
[U4.25.01]
NODE
AFFE_MATERIAU
ALL
[U4.23.02]
AFFE_MODELE
“MECHANICAL”
“POU_D_T'
ALL
[U4.22.01]
DEFI_MATERIAU
ELAS
[U4.23.01]
MODE_ITER_SIMULT
METHOD
“TRI_DIAG”
[U4.52.02]
CALC_FREQ
OPTION
“PLUS_PETITE”
NMAX_FREQ
NORM_MODE
NORMALIZES
“TRAN”
[U4.64.02]
Code_Aster
®
Version
4.0
Titrate:
SDLL10 Beam of rectangular section variable
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.010-C
Page:
5/6
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
4
Results of modeling A
4.1 Values
tested
Command of the mode
clean
Frequency
Reference
Aster
% difference
% tolerance
1
143.303
145.924
1.598
1.6
2
396.821
398.511
0.426
0.45
3
779.425
777.115
0.296
0.3
4
1289.577
1278.011
0.897
0.9
Clean modes of
Aster
were normalized to 1. at the point of maximum amplitude in translation like
in the reference.
Clean mode F
I
(X) normalized to 1 at the point of maximum amplitude
I
X = 0.1
X = 0.2
X = 0.3
X = 0.4
X = 0.5
Reference
Aster
% difference
% tolerance
1
0.2349
0.2363
0.583
0.6
0.6962
0.6970
0.119
0.15
0.98960
0.9895
0.
0.1
0.8505
0.8516
0.132
0.15
0.3507
0.3529
0.631
0.7
Reference
Aster
% difference
% tolerance
2
0.4653
0.4670
0.37
0.4
0.7558
0.7555
0.041
0.1
0.
2.910
4
2.910
4
1.10
3
0.9232
0.9226
0.063
0.1
0.6941
0.6971
0.435
0.45
Reference
Aster
% difference
% tolerance
3
0.6278
0.6290
0.192
0.2
0.1969
0.1952
0.89
0.9
0.7783
0.7782
0.014
0.1
0.2406
0.2377
1.226
1.23
0.9366
0.9387
0.228
0.25
Reference
Aster
% difference
% tolerance
4
0.666
0.6656
0.081
0.1
0.4832
0.4840
0.18
0.2
0.
4.610
4
4.610
4
1.10
3
0.5901
0.5919
0.31
0.35
0.9937
0.9928
0.089
0.1
4.2 Remarks
Calculations carried out by:
MODE_ITER_SIMULT METHOD: “TRI_DIAG”
OPTION: “PLUS_PETITE”
NMAX_FREQ: 4
Contents of the file results:
the first 4 Eigen frequencies, clean vectors.
4.3 Parameters
of execution
Version: 3.02.21
Machine: CRAY C90
System:
UNICOS 8.0
Overall dimension memory:
8 megawords
Time CPU To use:
9.8 seconds
Code_Aster
®
Version
4.0
Titrate:
SDLL10 Beam of rectangular section variable
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.010-C
Page:
6/6
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
5
Summary of the results
Suitable modeling (frequencies and modes suitable for less than 2%) with a fine mesh.
A calculation carried out on a coarse mesh (12 meshs) shows more important variations with
reference solution. This is especially due to the way in which the modes are normalized.