Code_Aster
®
Version
3
Titrate:
SDLX02 Piping: Problem of Hovgaard. Spectral analysis
Date:
20/09/99
Author (S):
B. QUINNEZ, L. VIVAN
Key:
V2.05.002-C
Page:
1/12
Manual of Validation
V2.05 booklet: Linear dynamics of the assembled structures
HI-75/96/004 - Ind A
Organization (S):
EDF/IMA/MN, CISI
Manual of Validation
V2.05 booklet: Linear dynamics of the assembled structures
Document: V2.05.002
SDLX02 - Piping: 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
bending, 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).
Code_Aster
®
Version
3
Titrate:
SDLX02 Piping: Problem of Hovgaard. Spectral analysis
Date:
20/09/99
Author (S):
B. QUINNEZ, L. VIVAN
Key:
V2.05.002-C
Page:
2/12
Manual of Validation
V2.05 booklet: Linear dynamics of the assembled structures
HI-75/96/004 - Ind A
1 Problem of reference
1.1 Geometry
=
=
=
=
=
=
=
=
=
=
1
2
34 5
6
7
8
9
10
12
13
14
15
3.69
1.96
2.75
Z
y
X
With
B
11
With
- 0.
- 1.828
- 0.922
B - 0.922
- 0.922
- 0.
·
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 AP
= 0.3
= 13404.106 kg/m
3
(pipe full of water)
1.3 Boundary conditions and loadings
Embedded items 1 and 15 (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)
according to X and y
0.2
2.
2.
0.2
0.2
Acceleration (G)
according to Z
0.1
1.
1.
0.1
0.1
Code_Aster
®
Version
3
Titrate:
SDLX02 Piping: Problem of Hovgaard. Spectral analysis
Date:
20/09/99
Author (S):
B. QUINNEZ, L. VIVAN
Key:
V2.05.002-C
Page:
3/12
Manual of Validation
V2.05 booklet: Linear dynamics of the assembled structures
HI-75/96/004 - Ind A
2 Reference solution
2.1 Method of calculation used for the reference solution
Averages of codes: Lice, ADL, TITUS-T.
Guide validation of the Software packages 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.
Code_Aster
®
Version
3
Titrate:
SDLX02 Piping: Problem of Hovgaard. Spectral analysis
Date:
20/09/99
Author (S):
B. QUINNEZ, L. VIVAN
Key:
V2.05.002-C
Page:
4/12
Manual of Validation
V2.05 booklet: Linear dynamics of the assembled structures
HI-75/96/004 - Ind A
3 Modeling A
3.1 Characteristics of modeling
The curved elements are modelized by elements
“POU_C_T”
(2 elements per elbow).
The right elements are modelized by elements
“POU_D_T”
.
=
=
=
=
=
=
=
=
=
=
3.69
1.96
2.75
Z
y
X
With
B
N100
N200
N400
N500N600
N700N800
N1000
N1200
N1300
N1400
N1500
N1100
N900
N300
3.2 Functionalities tested
Controls
Keys
AFFE_MODELE
GROUP_MA
“MECHANICAL”
“POU_D_T'
“POU_C_T'
[U4.22.01]
CALC_MATR_ELEM
OPTION
“MASS_MECA_DIAG”
[U4.41.01]
MODE_ITER_SIMULT
METHOD
CALC_FREQ
“TRI_DIAG”
DIM_SOUS_ESPACE
NMAX_FREQ
[U4.52.01]
NORM_MODE
NORMALIZES
“TRAN_ROTA”
[U4.64.02]
CALC_ELEM
OPTION
“SIELF_ELGA_DEP”
“EFGE_ELNO_DEPL”
[U4.61.02]
CALC_NO
OPTION
“REAC_NODA”
[U4.61.03]
MODE_STATIQUE
ACCE_UNIF
[U4.52.04]
COMB_SISM_MODAL
EXCIT
CORR_FREQ
MONO_APPUI
TRI_SPEC
SPEC_OSCI
[U4.54.04]
3.3 Characteristics of the mesh
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.
Code_Aster
®
Version
3
Titrate:
SDLX02 Piping: Problem of Hovgaard. Spectral analysis
Date:
20/09/99
Author (S):
B. QUINNEZ, L. VIVAN
Key:
V2.05.002-C
Page:
5/12
Manual of Validation
V2.05 booklet: Linear dynamics of the assembled structures
HI-75/96/004 - Ind A
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 the operator
COMB_SISM_MODAL
)
Displacement
Identification
Reference
Aster
% difference
DEPL
N300
DX
4.847 10
3
4.8449 10
3
0.04
DY
2.192 10
3
2.1914 10
3
0.03
DZ
2,735 10
6
2.7341 10
6
0.03
N500
DX
4.808 10
3
4.8062 10
3
0.04
DY
2.914 10
3
2.131 10
3
0.03
DZ
6.507 10
4
6.5048 10
4
0.03
N700
DX
3.588 10
3
3.5866 10
3
0.04
DY
2.914 10
3
2.9124 10
3
0.05
DZ
8.599 10
4
8.5959 10
4
0.03
N900
DX
2.342 10
3
2.3415 10
3
0.02
DY
2.913 10
3
2.9114 10
3
0.05
DZ
1.027 10
3
1.0261 10
3
0.08
N1100
DX
3.009 10
6
3.007 10
6
0.05
DY
9.375 10
4
9.3711 10
4
0.04
DZ
3.364 10
4
3.3621 10
4
0.06
Code_Aster
®
Version
3
Titrate:
SDLX02 Piping: Problem of Hovgaard. Spectral analysis
Date:
20/09/99
Author (S):
B. QUINNEZ, L. VIVAN
Key:
V2.05.002-C
Page:
6/12
Manual of Validation
V2.05 booklet: Linear dynamics of the assembled structures
HI-75/96/004 - Ind A
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
Y
587.3
587.35
0.01
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
Y
924.7
924.69
0.
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 the operator
COMB_SISM_MODAL
)
Displacement and nodal Reaction
Identification
Reference
Aster
% difference
DEPL
N3
DX
4.847 10
3
4.8469 10
3
0
DY
2.192 10
3
2.1922 10
3
0.01
N7
DX
3.588 10
3.5881 10
0
DY
2.914 10
3
2.9135 10
0.01
DRY
1.436 10
1.4361 10
0.01
REAC_NOD
With
N1
DX
2132.
2131.66
0.02
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
Code_Aster
®
Version
3
Titrate:
SDLX02 Piping: Problem of Hovgaard. Spectral analysis
Date:
20/09/99
Author (S):
B. QUINNEZ, L. VIVAN
Key:
V2.05.002-C
Page:
7/12
Manual of Validation
V2.05 booklet: Linear dynamics of the assembled structures
HI-75/96/004 - Ind A
4.2 Remarks
Values of the spectrum (interpolation).
Mode
1, 2, 3
4
5
6
7
8, 9
Following Acclération
X and y
19.620
8.06148
6.72586
3.38994
3.04168
1.9620
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
Overall dimension memory:
8 megawords
Time CPU To use:
14 seconds
Code_Aster
®
Version
3
Titrate:
SDLX02 Piping: Problem of Hovgaard. Spectral analysis
Date:
20/09/99
Author (S):
B. QUINNEZ, L. VIVAN
Key:
V2.05.002-C
Page:
8/12
Manual of Validation
V2.05 booklet: Linear dynamics of the assembled structures
HI-75/96/004 - Ind A
5 Modeling B
5.1 Characteristics of modeling
The curved elements are modelized by elements
“POU_C_T”
(2 elements per elbow).
The right elements are modelized by elements
“POU_D_T_G”
.
=
=
=
=
=
=
=
=
=
=
3.69
1.96
2.75
Z
y
X
With
B
N100
N200
N400
N500N600
N700N800
N1000
N1200
N1300
N1400
N1500
N1100
N900
N300
5.2 Functionalities tested
Controls
Keys
AFFE_MODELE
GROUP_MA
“MECHANICAL”
“POU_D_T_G”
“POU_C_T'
[U4.22.01]
CALC_MATR_ELEM
OPTION
“MASS_MECA”
[U4.41.01]
MODE_ITER_SIMULT
METHOD
CALC_FREQ
“TRI_DIAG”
DIM_SOUS_ESPACE
NMAX_FREQ
[U4.52.01]
5.3 Characteristics of the mesh
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.
Code_Aster
®
Version
3
Titrate:
SDLX02 Piping: Problem of Hovgaard. Spectral analysis
Date:
20/09/99
Author (S):
B. QUINNEZ, L. VIVAN
Key:
V2.05.002-C
Page:
9/12
Manual of Validation
V2.05 booklet: Linear dynamics of the assembled structures
HI-75/96/004 - Ind A
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
Code_Aster
®
Version
3
Titrate:
SDLX02 Piping: Problem of Hovgaard. Spectral analysis
Date:
20/09/99
Author (S):
B. QUINNEZ, L. VIVAN
Key:
V2.05.002-C
Page:
10/12
Manual of Validation
V2.05 booklet: Linear dynamics of the assembled structures
HI-75/96/004 - Ind A
7 Modeling C
7.1 Characteristics of modeling
The curved elements are modelized by elements
“POU_C_T”
(10 elements per elbow).
The right elements are modelized by elements
“POU_D_T_G”
(10 elements per right beam).
=
=
=
=
=
=
=
=
=
=
3.69
1.96
2.75
Z
y
X
With
B
N1 = P100
P1500 = N6
N2
N3
N4
NS
7.2 Functionalities tested
Controls
Keys
AFFE_MODELE
GROUP_MA
“MECHANICAL”
“POU_D_T_G”
“POU_C_T'
[U4.22.01]
CALC_MATR_ELEM
OPTION
“MASS_MECA”
“MASS_MECA_DIAG”
[U4.41.01]
MODE_ITER_SIMULT
METHOD
CALC_FREQ
“TRI_DIAG”
DIM_SOUS_ESPACE
NMAX_FREQ
[U4.52.01]
7.3 Characteristics of the mesh
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.
Code_Aster
®
Version
3
Titrate:
SDLX02 Piping: Problem of Hovgaard. Spectral analysis
Date:
20/09/99
Author (S):
B. QUINNEZ, L. VIVAN
Key:
V2.05.002-C
Page:
11/12
Manual of Validation
V2.05 booklet: Linear dynamics of the assembled structures
HI-75/96/004 - Ind A
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
Code_Aster
®
Version
3
Titrate:
SDLX02 Piping: Problem of Hovgaard. Spectral analysis
Date:
20/09/99
Author (S):
B. QUINNEZ, L. VIVAN
Key:
V2.05.002-C
Page:
12/12
Manual of Validation
V2.05 booklet: Linear dynamics of the assembled structures
HI-75/96/004 - Ind A
9 Summary of the results and remarks general
Modal calculation:
The results are in conformity with the card of validation.
By refining the mesh (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).