Code_Aster ®
Version
4.0
Titrate:
Short SDLL01 Poutre on simple supports
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.001-C Page:
1/8
Organization (S): EDF/IMA/MN
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
V2.02.001 document
SDLL01 - Short Poutre on simple supports
Summary:
This two-dimensional problem consists in seeking the frequencies of vibration of a mechanical structure
composed of a beam in simple supports at its two ends. This case test of Mécanique of Structures
corresponds to a dynamic analysis of a linear model having a linear behavior. One studies
the influence of the position of the points considered as points of supports (points on neutral fiber or points
offset at the base of the beam) compared to neutral fiber of a thick beam.
This test which comprises only one modeling, makes it possible to test part of the functionalities which
concern the beams of Timoshenko, the connections rigid and the search for Eigen frequencies by iterations
opposite.
Results obtained, either with the points of supports on neutral fiber, or with the points of offset supports
are compared with results VPCS. In the second configuration, the reference solution is an average
results of several software packages.
When the points of supports are offset, one observes a coupling between the various modes of
traction and compression and of inflection.
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
Code_Aster ®
Version
4.0
Titrate:
Short SDLL01 Poutre on simple supports
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.001-C Page:
2/8
1
Problem of reference
1.1 Geometry
y
y
y
With
With
B
B
H
X
Z
X
C
D
X, U
B
L
L
Rectangular cross-section:
height:
H = 0.2 m
width:
B = 0.1 m
surface:
With = 2.102 m2
inertia:
lz = 6.667 105
shearing:
Ay = Az = 1.17692
torsion:
Jx = 0.45776042 104
Length of the beam
L: 1. m
Co-ordinates of the points (m):
With
B
C
D
X
0.
1.
0.
1.
y
0.
0.
0.1
0.1
1.2
Material properties
E = 2.1011 Pa
= 0.3
= 7.800. kg/m3
1.3
Boundary conditions and loadings
Problem 1:
Not A
U = v = 0.
Not B
v = 0.
Problem 2:
Not C
U = v = 0.
Not D
v = 0.
1.4 Conditions
initial
Without object for the modal analysis.
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
Code_Aster ®
Version
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Titrate:
Short SDLL01 Poutre on simple supports
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.001-C Page:
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2
Reference solution
2.1
Method of calculation used for the reference solution
The reference solution is that given in card SDLL01/89 of the guide VPCS which presents
method of calculation in the following way:
Problem 1: Analytical calculation
The equation of inflection of the nonslim beams gives the formulation of Timoshenko, in
superimposing the effects of the pure bending, deformations of shearing action and the inertia of
rotation.
The Eigen frequencies of reference are determined by a digital simulation of this
equation, independent of any software package.
The Eigen frequencies in traction and compression are given by:
E
2 -
I
(I) 1
F =
with
=
I = 1 2
,…
I
2 L
I
2
Problem 2:
The problem not having an analytical solution, the solution is established by average of several
software package: model of Timoshenko with effect of the deformations of shearing action and inertia of
rotation.
The modes of inflection and traction and compression are coupled.
2.2
Results of reference
Problem 1: the first 6 clean modes.
Problem 2: the first 5 clean modes.
2.3
Uncertainty on the solution
Problem 1: analytical solution.`
Problem 2: ± 0.1%
2.4 References
bibliographical
[1]
S.P. TIMOSHENKO, D.H. YOUNG, W. WEAVER. Problems vibrations in Engineering.
New York: Wiley & Sons, 4° edition, p. 415 (1974).
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
Code_Aster ®
Version
4.0
Titrate:
Short SDLL01 Poutre on simple supports
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.001-C Page:
4/8
3 Modeling
With
3.1
Characteristics of modeling
One uses the element of right beam of Timoshenko: POU_D_T
y
y
B
B
With
With
X
X
C
D
Problem 1:
Cutting:
beam AB: 40 meshs SEG2
Limiting conditions:
in all the nodes
DDL_IMPO: (GROUP_NO: AB DZ: 0., DRX:0, DRY: 0.)
in a:
(NOEUD: WITH DX: 0., DY: 0. )
in b:
(NOEUD: B DY: 0. )
Problem 2:
Cutting:
beam AB: 40 meshs SEG2
2 rigid elements AC, BC: 2 meshs SEG2
Limiting conditions:
in all the nodes
DDL_IMPO: (TOUT:“YES” DZ: 0., DRX:0, DRY: 0.)
out of C:
(NOEUD: C DX: 0., DY: 0. )
in D:
(NOEUD: D DY: 0. )
Names of the nodes:
Not A = N100
Not C = N300
Not B = N200
Not D = N400
3.2
Characteristics of the grid
A number of nodes:
43
A number of meshs and types:
42 SEG2
3.3 Functionalities
tested
Commands
Keys
AFFE_CARA_ELEM
POUTRE
“GENERALE”
TOUT
[U4.24.01]
GROUP_MA
“RECTANGLE”
GROUP_MA
AFFE_CHAR_MECA
DDL_IMPO
TOUT
[U4.25.01]
GROUP_NO
NOEUD
AFFE_MATERIAU
GROUP_MA
[U4.23.02]
AFFE_MODELE
“MECANIQUE”
“POU_D_T'
TOUT
[U4.22.01]
GROUP_MA
DEFI_MATERIAU
ELAS
[U4.23.01]
MODE_ITER_INV
CALC_FREQ
OPTION
“AJUSTE”
[U5.23.01]
FREQ
3.4 Remarks
Definition of the rigid beams AC and data base:
· Section: Hy = 0.2, Hz = 0.2.
· Material: E = 2.1016, = 0.
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
Code_Aster ®
Version
4.0
Titrate:
Short SDLL01 Poutre on simple supports
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.001-C Page:
5/8
4
Results of modeling A
4.1 Values
tested
Frequency (Hz)
Clean mode
Reference
Aster
% difference
Problem 1
inflection 1
431.555
431.8916
0.078
traction 1
1265.924
1266.0056
0.006
inflection 2
1498.295
1500.7635
0.165
inflection 3
2870.661
2873.5344
0.100
traction 2
3797.773
3799.9692
0.058
inflection 4
4377.837
4370.8206
0.160
Problem 2
1
392.8 ±2.7%
394.4774
0.427
coupling 2
922.2 ±5.7%
922.6072
0.044
inflection 3
1592.0 ±2.9%
1638.2311
2.903
traction 4
2629.2 ±5.7%
2778.7000
5.686
compression 5
3126.2 ±4.3%
3261.6699
4.333
4.2 Remarks
Calculations carried out by:
Problem 1:
MODE_ITER_INV OPTION:LIST_FREQ “ADJUSTS”: (430. , 4500. )
Problem 2:
MODE_ITER_INV OPTION:LIST_FREQ “ADJUSTS”: (380. , 3300. )
Contents of the file results:
Problem 1:
the first 6 Eigen frequencies, clean vectors and modal parameters.
Problem 1:
the first 5 Eigen frequencies, clean vectors and modal parameters.
4.3 Parameters
of execution
Version: 3.02.21
Machine: CRAY C90
System:
UNICOS 8.0
Obstruction memory:
8 Megawords
Time CPU To use:
9.7 seconds
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
Code_Aster ®
Version
4.0
Titrate:
Short SDLL01 Poutre on simple supports
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.001-C Page:
6/8
5 Modeling
B
5.1
Characteristics of modeling
POU_D_TG
y
y
B
B
With
With
X
X
C
D
Problem 1:
Cutting:
beam AB: 40 meshs SEG2
Limiting conditions:
in all the nodes
DDL_IMPO: (GROUP_NO: AB DZ: 0., DRX:0, DRY: 0.)
in a:
(NOEUD: WITH DX: 0., DY: 0. )
in b:
(NOEUD: B DY: 0. )
Problem 2:
Cutting:
beam AB: 40 meshs SEG2
2 rigid elements AC, data base: 2 meshs SEG2
Limiting conditions:
in all the nodes
DDL_IMPO: (TOUT:“YES” DZ: 0., DRX:0, DRY: 0.)
out of C:
(NOEUD: C DX: 0., DY: 0. )
in D:
(NOEUD: D DY: 0. )
Names of the nodes:
Not A = N100
Not C = N300
Not B = N200
Not D = N400
5.2
Characteristics of the grid
A number of nodes:
43
A number of meshs and types:
42 SEG2
5.3 Functionalities
tested
Commands
Keys
AFFE_CARA_ELEM
POUTRE
“GENERALE”
TOUT
[U4.24.01]
GROUP_MA
“RECTANGLE”
GROUP_MA
AFFE_CHAR_MECA
DDL_IMPO
TOUT
[U4.25.01]
GROUP_MA
NOEUD
AFFE_MATERIAU
GROUP_MA
[U4.23.02]
AFFE_MODELE
“MECANIQUE”
“POU_D_TG”
TOUT
[U4.22.01]
GROUP_MA
DEFI_MATERIAU
ELAS
[U4.23.01]
MODE_ITER_INV
“AJUSTE”
[U4.52.01]
5.4 Remarks
Definition of the rigid beams AC and data base:
· Section: Hy = 0.2, Hz = 0.2.
· Material: E = 2.1016, = 0.
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
Code_Aster ®
Version
4.0
Titrate:
Short SDLL01 Poutre on simple supports
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.001-C Page:
7/8
6
Results of modeling B
6.1 Values
tested
Frequency (Hz)
Clean mode
Reference
Aster
% difference
Problem 1
inflection 1
431.555
431.8916
0.078
traction 1
1265.924
1266.0056
0.006
inflection 2
1498.295
1500.7635
0.165
inflection 3
2870.661
2873.5344
0.100
traction 2
3797.773
3799.9692
0.058
inflection 4
4377.837
4370.8206
0.160
Problem 2
1
392.8 ±2.7%
394.4774
0.427
coupling 2
922.2 ±5.7%
922.6072
0.044
inflection 3
1592.0 ±2.9%
1638.2311
2.903
traction 4
2629.2 ±5.7%
2778.7000
5.686
compression 5
3126.2 ±4.3%
3261.6699
4.333
6.2 Remarks
Calculations carried out by:
Problem 1:
ITERATIONS_INVERSES OPTION:LIST_FREQ “ADJUSTS”: (430. , 4500. )
Problem 2:
ITERATIONS_INVERSES OPTION:LIST_FREQ “ADJUSTS”: (380. , 3300. )
Contents of the file results:
Problem 1:
the first 6 Eigen frequencies, clean vectors and modal parameters.
Problem 1:
the first 5 Eigen frequencies, clean vectors and modal parameters.
6.3 Parameters
of execution
Version: 3.06
Machine: CRAY C90
System:
UNICOS 8.0
Obstruction memory:
8 Megawords
Time CPU To use:
1.1875E+1 seconds
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
Code_Aster ®
Version
4.0
Titrate:
Short SDLL01 Poutre on simple supports
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.001-C Page:
8/8
7
Summary of the results
The problem without eccentricity is correctly dealt with.
With eccentricity, the problem is dealt with with a dispersion from 3 to 6% by various software packages. Aster
remain in this fork.
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A