Code_Aster
®
Version
4.0
Titrate:
SDLL101 Vibration of a beam with prestressing
Date:
01/09/99
Author (S):
B. QUINNEZ
Key:
V2.02.101-B
Page:
1/8
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/051 - Ind A
Organization (S):
EDF/IMA/MN
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
V2.02.101 document
SDLL101 - Vibration of a beam with
prestressed
Summary:
This plane problem consists in seeking the frequencies of vibration of a mechanical structure made up of one
hurled beam, of circular section, under voltage embed-slide. This test of Mechanics of the Structures
corresponds to a dynamic analysis of a linear model having a linear behavior. This test comprises
two modelings.
In the first modeling, one tests the element of beam of Timoshenko subjected to a prestressing, it
calculation of geometrical rigidity and the calculation of the Eigen frequencies by the method of Lanczos. In
the second modeling, one tests the element of beam of Euler-Bernouilli subjected to a prestressing, the calculation of
geometrical rigidity and the calculation of the Eigen frequencies by the method of Bathe and Wilson.
The results obtained are in concord with the results of guide VPCS. One notices a shift towards
high frequencies of vibration when prestressing in the beam increases.
Code_Aster
®
Version
4.0
Titrate:
SDLL101 Vibration of a beam with prestressing
Date:
01/09/99
Author (S):
B. QUINNEZ
Key:
V2.02.101-B
Page:
2/8
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/051 - Ind A
1
Problem of reference
1.1 Geometry
B
P
With
P
Full circular section
diameter D: 0.01 m
Length of the beam
L: 2 m
1.2
Material properties
E = 2 10
11
NR/m
2
= 0.3
= 7.800. kg/m
3
1.3
Boundary conditions and loadings
·
Beam pose-posed,
·
4 loadings are studied P = 0., P = 10., P = 100., P = 1.000. NR
Code_Aster
®
Version
4.0
Titrate:
SDLL101 Vibration of a beam with prestressing
Date:
01/09/99
Author (S):
B. QUINNEZ
Key:
V2.02.101-B
Page:
3/8
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/051 - Ind A
2
Reference solution
2.1
Method of calculation used for the reference solution
The equation of vibration of a prestressed beam is:
I.E.(internal excitation)
Z
4
y
X
4
+
P
2
y
X
2
= -
S
2
y
X
2
prestressed traction if P > 0, of compression if P< 0, and led to the Eigen frequencies of bending
(assumption of Euler-Bernoulli)
F
I
=
I
2
2 L
2
1
+
PL
2
I.E.(internal excitation)
Z
I
2
2
1/2
I.E.(internal excitation)
Z
S
1/2
I
=
1,2,3,….
2.2
Results of reference
the first 5 Eigen frequencies.
2.3
Uncertainty on the solution
Analytical solution (assumption of the beams of Euler-Bernouilli).
2.4 References
bibliographical
[1]
Robert D. BLEVINS Formulated for natural frequency and shape mode - 1979 p.144 (formula
8.20 rectified).
Code_Aster
®
Version
4.0
Titrate:
SDLL101 Vibration of a beam with prestressing
Date:
01/09/99
Author (S):
B. QUINNEZ
Key:
V2.02.101-B
Page:
4/8
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/051 - Ind A
3 Modeling
With
3.1
Characteristics of modeling
Elements of beam
POU_D_T
(Right Beam of Timoshenko)
B
With
X
y
Cutting: 10 elements of beam
node a: translations in X and y locked
node b: translation in y locked.
Note:
The force P applied out of B generates a reaction - P in A.
3.2
Characteristics of the mesh
A number of nodes:
21
A number of meshs and types:
20 SEG2
3.3 Functionalities
tested
Controls
Keys
AFFE_CARA_ELEM
BEAM
“CIRCLE”
[U4.24.01]
AFFE_CHAR_MECA
FORCE_NODALE
NODE
[U4.25.01]
DDL_IMPO
AFFE_MATERIAU
ALL
[U4.23.02]
AFFE_MODELE
“MECHANICAL”
“POU_D_T'
“ALL”
[U4.22.01]
DEFI_MATERIAU
ELAS
[U4.23.01]
CALC_MATR_ELEM
OPTION
SIEF_ELGA
“RIGI_GEOM”
[U4.41.01]
CALC_VECT_ELEM
OPTION
CHAR_MECA
[U4.41.02]
CALC_CHAM_ELEM
DEPL
[U4.61.01]
OPTION
SIEF_ELGA_DEPL
MODE_ITER_SIMULT
METHOD
“TRI_DIAG”
[U4.52.02]
CALC_FREQ
OPTION
NMAX_FREQ
“PLUS_PETITE”
COMB_MATR_ASSE
[U4.53.01]
Code_Aster
®
Version
4.0
Titrate:
SDLL101 Vibration of a beam with prestressing
Date:
01/09/99
Author (S):
B. QUINNEZ
Key:
V2.02.101-B
Page:
5/8
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/051 - Ind A
4
Results of modeling A
4.1 Values
tested
Pre stress/command of the mode
clean
Reference
Aster
% difference
P = 0 1
4.97137
4.9711
0.01
2
19.8851
19.8829
0.01
3
44.7414
44.7320
0.02
4
79.5403
79.5203
0.02
5
124.2818
124.2706
0.01
P = 10 1
5.0728
5.0727
0.0
2
19.9874
19.9852
0.01
3
44.8439
44.8345
0.02
4
79.6429
79.6229
0.03
5
124.3844
124.3732
0.01
P = 100 1
5.9090
5.9088
0.00
2
20.8860
20.8839
0.01
3
45.7561
45.7467
0.02
4
80.5600
80.5400
0.03
5
125.3037
125.2923
0.01
P = 1.000 1
11.2577
11.2576
0.00
2
28.3462
28.3442
0.01
3
54.0370
54.0281
0.02
4
89.2134
89.1935
0.02
5
134.1511
134.1379
0.01
4.2 Parameters
of execution
Version: 3.02.21
Machine: CRAY C90
System:
UNICOS 8.0
Overall dimension memory:
8 megawords
Time CPU To use:
10 seconds
Code_Aster
®
Version
4.0
Titrate:
SDLL101 Vibration of a beam with prestressing
Date:
01/09/99
Author (S):
B. QUINNEZ
Key:
V2.02.101-B
Page:
6/8
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/051 - Ind A
5 Modeling
B
5.1
Characteristics of modeling
Elements of beam
POU_D_E
(Beam of Euler-Bernouilli)
B
With
X
y
Cutting: 19 elements of beam
node a: translations in X and y locked
node b: translation in y locked.
Note:
The force P applied out of B generates a reaction - P in A.
5.2
Characteristics of the mesh
A number of nodes:
21
A number of meshs and types:
20 SEG2
5.3 Functionalities
tested
Controls
Keys
AFFE_CARA_ELEM
BEAM
“CIRCLE”
[U4.24.01]
AFFE_CHAR_MECA
FORCE_NODALE
NODE
[U4.25.01]
DDL_IMPO
AFFE_MATERIAU
ALL
[U4.23.02]
AFFE_MODELE
“MECHANICAL”
“POU_D_T'
“ALL”
[U4.22.01]
DEFI_MATERIAU
ELAS
[U4.23.01]
CALC_MATR_ELEM
OPTION
“RIGI_GEOM”
[U4.41.01]
SIEF_ELGA
CALC_VECT_ELEM
OPTION
CHAR_MECA
[U4.41.02]
CALC_CHAM_ELEM
DEPL
[U4.61.01]
OPTION
SIEF_ELGA_DEPL
MODE_ITER_SIMULT
METHOD
“JACOBI”
[U4.52.02]
CALC_FREQ
OPTION
“PLUS_PETITE”
NMAX_FREQ
COMB_MATR_ASSE
[U4.53.01]
Code_Aster
®
Version
4.0
Titrate:
SDLL101 Vibration of a beam with prestressing
Date:
01/09/99
Author (S):
B. QUINNEZ
Key:
V2.02.101-B
Page:
7/8
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/051 - Ind A
6
Results of modeling B
6.1 Values
tested
Pre stress/command of the mode
clean
Reference
Aster
% difference
P = 0 1
4.97137
4.9713
0.00
2
19.8851
19.8853
0.00
3
44.7414
44.7439
0.01
4
79.5403
79.5574
0.02
5
124.2818
124.3594
0.06
P = 10 1
5.0728
5.0728
0.00
2
19.9874
19.9876
0.00
3
44.8439
44.8464
0.01
4
79.6429
79.6599
0.02
5
124.3844
124.4619
0.06
P = 100 1
5.9090
5.9090
0.00
2
20.8860
20.8862
0.00
3
45.7561
45.7585
0.01
4
80.5600
80.5768
0.02
5
125.3037
125.3807
0.06
P = 1.000 1
11.2577
11.2577
0.00
2
28.3462
28.3463
0.00
3
54.0370
54.0391
0.00
4
89.2134
89.2287
0.02
5
134.1511
134.2234
0.05
6.2 Parameters
of execution
Version: 3.02.21
Machine: CRAY C90
System:
UNICOS 8.0
Overall dimension memory:
8 megawords
Time CPU To use:
11 seconds
Code_Aster
®
Version
4.0
Titrate:
SDLL101 Vibration of a beam with prestressing
Date:
01/09/99
Author (S):
B. QUINNEZ
Key:
V2.02.101-B
Page:
8/8
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/051 - Ind A
7
Summary of the results
The results obtained are in concord with the results of reference. It is noticed well that them
frequencies of vibration increase when prestressing increases.