Code_Aster
®
Version
4.0
Titrate:
SDLL09 Vibration of a slim beam of rectangular section
Date:
07/01/98
Author (S):
B. QUINNEZ, A. PENET
Key:
V2.02.009-A
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
V2.02.009 document
SDLL09 - Vibration of a slim beam
of variable rectangular section (embed-free)
Summary:
This plane problem consists in seeking the frequencies of vibration of a free fixed beam with section
rectangular variable. This test comprises only one modeling.
The variation of section of the beam is either homothetic, or nonhomothetic. Characteristics of
section of the beam are given according to meshs' in two different ways:
·
section and inertias,
·
height and width.
This problem thus makes it possible to thus test the element of beam with variable section for a prismatic structure
that the calculation of the frequencies of vibration by iterations opposite. In addition, in the operator
AFFE_CARA_ELEM
[U4.24.01], the remanence of certain key words is tested.
The results obtained are in concord with those given in guide VPCS.
Code_Aster
®
Version
4.0
Titrate:
SDLL09 Vibration of a slim beam of rectangular section
Date:
07/01/98
Author (S):
B. QUINNEZ, A. PENET
Key:
V2.02.009-A
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
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
Case 1
Case 2
height:
H
O
= 0.04 m
= 0.04 m
H
1
= 0.01 m
width:
B
O
= 0.04 m
= 0.05 m
B
1
= 0.01 m
surface:
With
O
= 1.6 10
3
m
2
= 2.10
3
m
2
With
1
= 1.10
4
m
2
inertia:
lz
O
= 2.1333 10
7
m
4
= 2.6667 10
7
m
4
Iz
1
= 8.3333 10
10
m
4
Co-ordinates of the points (m):
With
B
X
0.
1.
y
0.
0.
Z
0.
0.
1.2
Material properties
E
= 2.10
11
AP
= 7.800 kg/m
3
1.3
Boundary conditions and loadings
Not a: embedded U = v = 0
= 0.
Code_Aster
®
Version
4.0
Titrate:
SDLL09 Vibration of a slim beam of rectangular section
Date:
07/01/98
Author (S):
B. QUINNEZ, A. PENET
Key:
V2.02.009-A
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 SDLL09/89 of the guide VPCS which presents
method of calculation in the following way:
Exact calculation by numerical integration of the differential equation of the bending of the beams (Theory
of Euler-Bernouilli).
2
E I
Z
2
v
X
2
X
2
= -
With
2
v
T
2
where
I
Z
and
With
vary with the X-coordinate.
One obtains:
F
I
=
1
2
I
,
()
H
1
L
2
E
12
with:
=
H
0
H
1
=
4
=
B
0
B
1
=
4
or
5
1
2
3
4
5
= 4
23.289
73.9
165.23
299.7
478.1
= 5
24.308
75.56
167.21
301.9
480.4
2.2
Results of reference
the first 5 clean modes of bending.
2.3
Uncertainty on the solution
Semi-analytical solution.
2.4 References
bibliographical
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:
SDLL09 Vibration of a slim beam of rectangular section
Date:
07/01/98
Author (S):
B. QUINNEZ, A. PENET
Key:
V2.02.009-A
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
Modeling: Elements of beam
POU_D_E
y
With
B
X
Cutting:
beam AB: 30 meshs SEG2 of section variable
15 meshs in “General section”
15 meshs in “Rectangular section”
Limiting conditions:
in all the nodes
at end A
DDL_IMPO: (ALL:“YES” DZ: 0., DRX: 0., DRY: 0. )
(Node: WITH DX: 0., DY: 0., DRZ: 0. )
Names of the nodes:
Not A = N100
Not B = N200
3.2
Characteristics of the mesh
Mesh:
A number of nodes: 31
A number of meshs and types: 30 SEG2
3.3 Functionalities
tested
Controls
Keys
AFFE_CARA_ELEM
BEAM
“GENERAL”
NET
[U4.24.01]
AFFE_CARA_ELEM
BEAM
“RIGHT-ANGLED”
NET
[U4.24.01]
AFFE_CHAR_MECA
DDL_IMPO
[U4.25.01]
AFFE_MODELE
“MECHANICAL”
“POU_D_E”
[U4.22.01]
DEFI_MATERIAU
ELAS
[U4.23.01]
MODE_ITER_INV
OPTION
“ADJUSTS”
[U4.52.01]
Test of the remanence of a key word (
SECTION
and
CARA
).
Code_Aster
®
Version
4.0
Titrate:
SDLL09 Vibration of a slim beam of rectangular section
Date:
07/01/98
Author (S):
B. QUINNEZ, A. PENET
Key:
V2.02.009-A
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
Identification
Reference
Aster
% difference
frequency in HZ
Case 1 H
O
/H
1
= 4 B
O
/B
1
= 4
homothetic
bending 1
54.18
54.1354
0.08
bending 2
171.94
171.7122
0.13
bending 3
384.40
383.8764
0.14
bending 4
697.24
696.1877
0.15
bending 5
1112.28
1110.4727
0.16
Cases 2 H
O
/H
1
= 4 B
O
/B
1
= 5
nonhomothetic
bending 1
56.55
56.4984
0.09
bending 2
175.19
175.5306
+0.19
bending 3
389.01
388.3426
0.17
bending 4
702.36
700.9879
0.20
bending 5
1117.63
1115.4275
0.20
4.2 Remarks
Calculations by
MODE_ITER_INV (…
CALC_FREQ: (FREQ: (53. , 1150.))) ;
4.3 Parameters
of execution
Version: NEW3.03.15
Machine: CRAY C90
Overall dimension memory:
8 MW
Time CPU To use:
7.72 seconds
Code_Aster
®
Version
4.0
Titrate:
SDLL09 Vibration of a slim beam of rectangular section
Date:
07/01/98
Author (S):
B. QUINNEZ, A. PENET
Key:
V2.02.009-A
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
Good establishment of the element of non-prismatic beam with a fine mesh.
A coarser modeling would be sufficient.