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Titrate:
SDLX02 Tuyauterie: Problem of Hovgaard. Spectral analysis
Date:
20/09/99
Author (S):
B. QUINNEZ, L. VIVAN
Key:
V2.05.002-C Page:
1/12
Organization (S): EDF/IMA/MN, CISI
Handbook of Validation
V2.05 booklet: Linear dynamics of the assembled structures
Document: V2.05.002
SDLX02 - Tuyautery: Problem of Hovgaard.
Spectral analysis
Summary:
The three-dimensional problem consists firstly, to seek the modes of vibration of a mechanical structure
composed of a curved beam embed-embedded (problem of Hovgaard), secondly, to analyze the answer
of this structure subjected to a spectrum of acceleration. This test of mechanics of the structures corresponds to one
analyze dynamic of a linear model (assembled structure) having a linear behavior. It includes/understands
three modelings.
Via this problem, one tests the element of beam of Timoshenko (right beam or curve) in
inflection, the calculation of the clean modes by the method of Lanczos, the calculation of the static modes and calculation
of a spectral response of a structure subjected to a spectrum of acceleration (one tests also the interpolation of
spectrum).
The results obtained are in concord with the results of reference (compilation of results obtained by
other software packages).
Handbook of Validation
V2.05 booklet: Linear dynamics of the assembled structures
HI-75/96/004 - Ind A
Code_Aster ®
Version
3
Titrate:
SDLX02 Tuyauterie: Problem of Hovgaard. Spectral analysis
Date:
20/09/99
Author (S):
B. QUINNEZ, L. VIVAN
Key:
V2.05.002-C Page:
2/12
1 Problem of reference
1.1 Geometry
- 0.
- 0.922
2.75
With - 1.828
B - 0.922
=
Z
- 0.922
- 0.
=
4
3 5
=
6
=
=
With
7
8
9
3.69
B
10
2
12
11
y
13
=
=
14
=
15
=
=
1.96
X
1
· diameter external of the pipe: 0.185 m
· thickness of the pipe: 6.12 m
· radius of curvature of the elbows: 0.922 m
1.2 Material properties
E = 1.658 E + 11 Pa
= 0.3
= 13404.106 kg/m3 (pipe full of water)
1.3 Boundary conditions and loadings
Items 1 and 15 embedded (U = v = W = X = y = Z = 0).
Loading: without object for the modal analysis.
For the spectral analysis: definition of a spectrum of acceleration to the supports for a damping of
2%.
Frequency (Hz)
1
10
30
100
10000
Acceleration (G)
0.2
2.
2.
0.2
0.2
according to X and y
Acceleration (G)
0.1
1.
1.
0.1
0.1
according to Z
Handbook of Validation
V2.05 booklet: Linear dynamics of the assembled structures
HI-75/96/004 - Ind A
Code_Aster ®
Version
3
Titrate:
SDLX02 Tuyauterie: Problem of Hovgaard. Spectral analysis
Date:
20/09/99
Author (S):
B. QUINNEZ, L. VIVAN
Key:
V2.05.002-C Page:
3/12
2 Reference solution
2.1 Method of calculation used for the reference solution
Averages of codes: Lice, ADL, TITUS-T.
Guide validation of Progiciels of structural analysis - AFNOR - 1990 (for modal calculation).
values provided in the card under are estimated and were corrected thereafter in 1992. However,
they were preserved for calculations with matrix of diagonal mass.
2.2 Results of reference
Modal calculation:
the first 9 Eigen frequencies.
Spectral answer:
displacement of the nodes N3 N5 and N7, N9, N11.
Reaction of supports to the nodes N1, N15.
Generalized efforts of the nodes N3, N7, N11.
2.3 Uncertainty on the solution
About 1% on the first 5 modes.
Between 1 and 2,5% for modes 6 to 9.
2.4 Bibliographical references
[1] Guide Technical VPCS AFNOR - 1990
[2] W. HOVGAARD “dimensional Stress in three pipe bens”, Trans off ASME vol. 57, FSP 75-
12 P 401-416.
Handbook of Validation
V2.05 booklet: Linear dynamics of the assembled structures
HI-75/96/004 - Ind A
Code_Aster ®
Version
3
Titrate:
SDLX02 Tuyauterie: Problem of Hovgaard. Spectral analysis
Date:
20/09/99
Author (S):
B. QUINNEZ, L. VIVAN
Key:
V2.05.002-C Page:
4/12
3 Modeling A
3.1 Characteristics of modeling
The curved elements are modelled by elements “POU_C_T” (2 elements per elbow).
The right elements are modelled by elements “POU_D_T”.
2.75
=
Z
=
N400
N500
=
N600
N300
=
=
With
N700N800
N900
3.69
B N1000
N200
N1100
N1200
y
=
N1300
=
N1400
=
N1500
=
=
1.96
X
N100
3.2 Functionalities tested
Commands
Keys
AFFE_MODELE
GROUP_MA
“MECANIQUE”
“POU_D_T'
[U4.22.01]
“POU_C_T'
CALC_MATR_ELEM
OPTION
“MASS_MECA_DIAG”
[U4.41.01]
MODE_ITER_SIMULT
METHODE
“TRI_DIAG”
[U4.52.01]
CALC_FREQ
DIM_SOUS_ESPACE
NMAX_FREQ
NORM_MODE
NORME
“TRAN_ROTA”
[U4.64.02]
CALC_ELEM
OPTION
“SIELF_ELGA_DEP”
[U4.61.02]
“EFGE_ELNO_DEPL”
CALC_NO
OPTION
“REAC_NODA”
[U4.61.03]
MODE_STATIQUE
ACCE_UNIF
[U4.52.04]
COMB_SISM_MODAL
EXCIT
MONO_APPUI
[U4.54.04]
TRI_SPEC
SPEC_OSCI
CORR_FREQ
3.3 Characteristics of the grid
A number of nodes:
15
A number of meshs and types:
10 POU_D_T
4 POU_C_T
3.4 Remarks
The modes are normalized in the following way: larger component (degree of freedom of
translation or rotation) with one.
The total answer is obtained by quadratic combination of the directions of the excitations.
Handbook of Validation
V2.05 booklet: Linear dynamics of the assembled structures
HI-75/96/004 - Ind A
Code_Aster ®
Version
3
Titrate:
SDLX02 Tuyauterie: Problem of Hovgaard. Spectral analysis
Date:
20/09/99
Author (S):
B. QUINNEZ, L. VIVAN
Key:
V2.05.002-C Page:
5/12
4 Results of modeling A
4.1 Values tested
Frequencies of the structure (matrix of complete mass).
Eigen frequencies
Reference
Aster
% difference
Mode 1
10.39
10.4488
0.566
2
20.02
19.9067
0.566
3
25.45
25.3369
0.444
4
48.32
46.8806
2.979
5
52.60
51.7708
1.576
6
84.81
82.4334
2.802
7
87.16
84.1737
3.426
8
129.31
130.138
0.640
9
131.69
131.691
0
Frequencies of the structure (matrix of mass diagnonale).
Eigen frequency
Reference
Aster
% difference
10.18
10.198
0.18
2
19.54
19.544
0.02
3
25.47
25.368
0.40
4
48.09
74.763
0.68
5
52.86
52.509
0.66
6
75.94
75.131
1.07
7
80.11
79.512
0.75
8
122.34
120.785
1.27
9
123.15
121.480
1.36
Spectral answer: one does not take account of the correction of the frequencies due to damping
(option CORR_FREQ with not in operator COMB_SISM_MODAL)
Displacement
Identification
Reference
Aster
% difference
DEPL
N300
DX
4.847 103
4.8449 103
0.04
DY
2.192 103
2.1914 103
0.03
DZ
2,735 106
2.7341 106
0.03
N500
DX
4.808 103
4.8062 103
0.04
DY
2.914 103
2.131 103
0.03
DZ
6.507 104
6.5048 104
0.03
N700
DX
3.588 103
3.5866 103
0.04
DY
2.914 103
2.9124 103
0.05
DZ
8.599 104
8.5959 104
0.03
N900
DX
2.342 103
2.3415 103
0.02
DY
2.913 103
2.9114 103
0.05
DZ
1.027 103
1.0261 103
0.08
N1100
DX
3.009 106
3.007 106
0.05
DY
9.375 104
9.3711 104
0.04
DZ
3.364 104
3.3621 104
0.06
Handbook of Validation
V2.05 booklet: Linear dynamics of the assembled structures
HI-75/96/004 - Ind A
Code_Aster ®
Version
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Titrate:
SDLX02 Tuyauterie: Problem of Hovgaard. Spectral analysis
Date:
20/09/99
Author (S):
B. QUINNEZ, L. VIVAN
Key:
V2.05.002-C Page:
6/12
Nodal reaction
Identification
Reference
Aster
% difference
REAC
N100
DX
2132
2130.81
0.06
DY
1241
1240.24
0.06
DZ
564.6
564.33
0.05
DRX
2352
2351.27
0.03
DRY
4746
4744.27
0.04
DRZ
937.3
936.92
0.04
N1500
DX
1653
1652.24
0.05
DY
3354
3352.19
0.05
DZ
893.7
893.34
0.04
DRX
170.8
170.74
0.03
DRY
1668
1667.71
0.02
DRZ
4903
4900.62
0.05
Generalized efforts
Identification
Reference
Aster
% difference
EFGE
N300
NR
559.9
559.86
0.01
VY
430.8
430.75
0.01
VZ
914.9
914.88
0.
MT
932.5
932.50
0.
MF
587.3
587.35
0.01
Y
MFZ
620.4
620.36
0.01
N700
NR
162.5
1624.83
0.01
VY
1367.
1367.04
0.
VZ
225.4
225.38
0.01
MT
170.6
170.64
0.03
MF
924.7
924.69
0.
Y
MFZ
2150
2150.29
0.01
Spectral answer: one takes account of the correction of the frequencies due to damping (option
CORR_FREQ with yes in operator COMB_SISM_MODAL)
Displacement and Réaction nodal
Identification
Reference
Aster
% difference
DEPL
N3
DX
4.847 103
4.8469 103
0
DY
2.192 103
2.1922 103
0.01
N7
DX
3.588 10
3.5881 10
0
DY
2.914 103
2.9135 10
0.01
DRY
1.436 10
1.4361 10
0.01
REAC_NOD
N1
DX
2132.
2131.66
0.02
With
DY
1241.
1240.94
0.02
DZ
564.6
564.56
0
N15
DRX
170.8
170.81
0
DRY
166.8
1668.38
0.02
DRZ
4903.
4902.58
0
Handbook of Validation
V2.05 booklet: Linear dynamics of the assembled structures
HI-75/96/004 - Ind A
Code_Aster ®
Version
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Titrate:
SDLX02 Tuyauterie: Problem of Hovgaard. Spectral analysis
Date:
20/09/99
Author (S):
B. QUINNEZ, L. VIVAN
Key:
V2.05.002-C Page:
7/12
4.2 Remarks
Values of the spectrum (interpolation).
Mode
1, 2, 3
4
5
6
7
8, 9
Following Acclération
19.620
8.06148
6.72586
3.38994
3.04168
1.9620
X and y
Acceleration according to Z
9.810
4.03074
3.36293
1.69497
1.52084
0.9810
4.3 Parameters of execution
Version: 3.04.7
Machine: CRAY C90
System:
UNICOS 8.0
Obstruction memory:
8 megawords
Time CPU To use:
14 seconds
Handbook of Validation
V2.05 booklet: Linear dynamics of the assembled structures
HI-75/96/004 - Ind A
Code_Aster ®
Version
3
Titrate:
SDLX02 Tuyauterie: Problem of Hovgaard. Spectral analysis
Date:
20/09/99
Author (S):
B. QUINNEZ, L. VIVAN
Key:
V2.05.002-C Page:
8/12
5 Modeling B
5.1 Characteristics of modeling
The curved elements are modelled by elements “POU_C_T” (2 elements per elbow).
The right elements are modelled by elements “POU_D_T_G”.
2.75
=
Z
=
N400
N500
=
N600
N300
=
=
With
N700N800
N900
3.69
B N1000
N200
N1100
N1200
y
=
N1300
=
N1400
=
N1500
=
=
1.96
X
N100
5.2 Functionalities tested
Commands
Keys
AFFE_MODELE
GROUP_MA
“MECANIQUE”
“POU_D_T_G”
[U4.22.01]
“POU_C_T'
CALC_MATR_ELEM
OPTION
“MASS_MECA”
[U4.41.01]
MODE_ITER_SIMULT
METHODE
“TRI_DIAG”
[U4.52.01]
CALC_FREQ
DIM_SOUS_ESPACE
NMAX_FREQ
5.3 Characteristics of the grid
A number of nodes:
15
A number of meshs and types:
10 POU_D_T_G
4 POU_C_T
5.4 Remarks
The modes are normalized in the following way: larger component (degree of freedom of
translation or rotation) with one.
Handbook of Validation
V2.05 booklet: Linear dynamics of the assembled structures
HI-75/96/004 - Ind A
Code_Aster ®
Version
3
Titrate:
SDLX02 Tuyauterie: Problem of Hovgaard. Spectral analysis
Date:
20/09/99
Author (S):
B. QUINNEZ, L. VIVAN
Key:
V2.05.002-C Page:
9/12
6 Results of modeling B
6.1 Values tested
Frequencies of the structure (matrix of complete mass).
Eigen frequency
Reference
Aster
% difference
Mode 1
10.39
10.4488
0.566
2
20.02
19.9067
0.566
3
25.45
25.3369
0.444
4
48.32
46.8806
2.979
5
52.60
51.7708
1.576
6
84.81
82.4334
2.802
7
87.16
84.1737
3.426
8
129.31
130.138
0.641
9
131.69
131.691
0
Handbook of Validation
V2.05 booklet: Linear dynamics of the assembled structures
HI-75/96/004 - Ind A
Code_Aster ®
Version
3
Titrate:
SDLX02 Tuyauterie: Problem of Hovgaard. Spectral analysis
Date:
20/09/99
Author (S):
B. QUINNEZ, L. VIVAN
Key:
V2.05.002-C Page:
10/12
7 Modeling C
7.1 Characteristics of modeling
The curved elements are modelled by elements “POU_C_T” (10 elements per elbow).
The right elements are modelled by elements “POU_D_T_G” (10 elements per right beam).
2.75
N3
=
Z
=
N2
=
=
=
With
N4
3.69
B
NS
y
=
=
P1500 = N6
=
=
=
1.96
X
N1 = P100
7.2 Functionalities tested
Commands
Keys
AFFE_MODELE
GROUP_MA
“MECANIQUE”
“POU_D_T_G”
[U4.22.01]
“POU_C_T'
CALC_MATR_ELEM
OPTION
“MASS_MECA”
[U4.41.01]
“MASS_MECA_DIAG”
MODE_ITER_SIMULT
METHODE
“TRI_DIAG”
[U4.52.01]
CALC_FREQ
DIM_SOUS_ESPACE
NMAX_FREQ
7.3 Characteristics of the grid
A number of nodes:
51
A number of meshs and types:
30 POU_D_T_G
20 POU_C_T
7.4 Remarks
The modes are normalized in the following way: larger component (degree of freedom of
translation or rotation) with one.
Handbook of Validation
V2.05 booklet: Linear dynamics of the assembled structures
HI-75/96/004 - Ind A
Code_Aster ®
Version
3
Titrate:
SDLX02 Tuyauterie: Problem of Hovgaard. Spectral analysis
Date:
20/09/99
Author (S):
B. QUINNEZ, L. VIVAN
Key:
V2.05.002-C Page:
11/12
8 Results of modeling C
8.1 Values tested
Frequencies of the structure (matrix of complete mass).
Eigen frequency
Reference
Aster
% difference
Mode 1
10.39
10.41
0.170
2
20.02
20.02
0.002
3
25.45
25.52
0.289
4
48.32
48.47
0.314
5
52.60
52.68
0.152
6
84.81
84.89
0.1
7
87.16
87.29
0.153
8
129.31
130.22
0.704
Frequencies of the structure (matrix of diagonal mass).
Reference
Aster
% difference
10.39
10.39
0.058
20.02
20.02
0.089
25.45
25.51
0.265
48.32
48.51
0.392
52.60
52.70
0.192
84.81
84.63
0.215
87.16
87.14
0.021
129.31
129.77
0.357
Handbook of Validation
V2.05 booklet: Linear dynamics of the assembled structures
HI-75/96/004 - Ind A
Code_Aster ®
Version
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Titrate:
SDLX02 Tuyauterie: Problem of Hovgaard. Spectral analysis
Date:
20/09/99
Author (S):
B. QUINNEZ, L. VIVAN
Key:
V2.05.002-C Page:
12/12
9 Summary of the results and remarks general
Modal calculation:
The results are in conformity with the card of validation.
By refining the grid (modeling C) one obtains correct results.
Spectral answer:
The results are in conformity with the results of reference (the error is lower than the thousandths).
Handbook of Validation
V2.05 booklet: Linear dynamics of the assembled structures
HI-75/96/004 - Ind A