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
Organization (S): EDF/IMA/MN
Handbook 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 tension embed-slide. This test of Mécanique of 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.
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/051 - Ind A
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
1
Problem of reference
1.1 Geometry
With
B
P
P
Full circular section
diameter D: 0.01 m
Length of the beam
L: 2 m
1.2
Material properties
E = 2 1011 NR/M2
= 0.3
= 7.800. kg/m3
1.3
Boundary conditions and loadings
· Beam pose-posed,
· 4 loadings are studied P = 0., P = 10., P = 100., P = 1.000. NR
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/051 - Ind A
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
2
Reference solution
2.1
Method of calculation used for the reference solution
The equation of vibration of a prestressed beam is:
4y
2y
2y
I.E.(internal excitation)
+ P
= - S
Z x4
x2
x2
prestressed traction if P > 0, of compression if P< 0, and led to the Eigen frequencies of inflection
(assumption of Euler-Bernoulli)
1/2
1/2
I.E.(internal excitation)
F = i2
Z
I =
I
1 +
PL2
1,2,3,….
2 L2
I.E.(INTERNAL EXCITATION) I2 2 S
Z
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 Formulas for natural frequency and shape mode - 1979 p.144 (formula
8.20 rectified).
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/051 - Ind A
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
3 Modeling
With
3.1
Characteristics of modeling
Elements of beam POU_D_T (Poutre right of Timoshenko)
y
With
B
X
Cutting: 10 elements of beam
node a: translations in X and y blocked
node b: translation in y blocked.
Note:
The force P applied out of B generates a reaction - P in A.
3.2
Characteristics of the grid
A number of nodes:
21
A number of meshs and types:
20 SEG2
3.3 Functionalities
tested
Commands
Keys
AFFE_CARA_ELEM
POUTRE
“CERCLE”
[U4.24.01]
AFFE_CHAR_MECA
FORCE_NODALE
NOEUD
[U4.25.01]
DDL_IMPO
AFFE_MATERIAU
TOUT
[U4.23.02]
AFFE_MODELE
“MECANIQUE”
“POU_D_T'
“TOUT”
[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
METHODE
“TRI_DIAG”
[U4.52.02]
CALC_FREQ
OPTION
NMAX_FREQ
“PLUS_PETITE”
COMB_MATR_ASSE
[U4.53.01]
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/051 - Ind A
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
4
Results of modeling A
4.1 Values
tested
Pre constraint/command of the mode
Reference
Aster
% difference
clean
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
Obstruction memory:
8 megawords
Time CPU To use:
10 seconds
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/051 - Ind A
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
5 Modeling
B
5.1
Characteristics of modeling
Elements of beam POU_D_E (Poutre d' Euler-Bernouilli)
y
With
B
X
Cutting: 19 elements of beam
node a: translations in X and y blocked
node b: translation in y blocked.
Note:
The force P applied out of B generates a reaction - P in A.
5.2
Characteristics of the grid
A number of nodes:
21
A number of meshs and types:
20 SEG2
5.3 Functionalities
tested
Commands
Keys
AFFE_CARA_ELEM
POUTRE
“CERCLE”
[U4.24.01]
AFFE_CHAR_MECA
FORCE_NODALE
NOEUD
[U4.25.01]
DDL_IMPO
AFFE_MATERIAU
TOUT
[U4.23.02]
AFFE_MODELE
“MECANIQUE”
“POU_D_T'
“TOUT”
[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
METHODE
“JACOBI”
[U4.52.02]
CALC_FREQ
OPTION
“PLUS_PETITE”
NMAX_FREQ
COMB_MATR_ASSE
[U4.53.01]
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/051 - Ind A
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
6
Results of modeling B
6.1 Values
tested
Pre constraint/command of the mode
Reference
Aster
% difference
clean
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
Obstruction memory:
8 megawords
Time CPU To use:
11 seconds
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/051 - Ind A
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
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.
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/051 - Ind A