Code_Aster
®
Version
3
Titrate:
SDLS01 Plates square thin, free or embedded at the edge
Date:
24/08/99
Author (S):
G. ROUSSEAU, C. VARE, J. PELLET
Key:
V2.03.001-C
Page:
1/12
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/96/013 - Ind A
Organization (S):
EDF/EP/AMV, EDFIMA/MN
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
V2.03.001 document
SDLS01 - Thin, free or embedded square plate
at the edge
Summary:
The applicability of this case test relates to the dynamics of the structures, and more particularly
modal calculation and the harmonic calculation of answer.
For modal calculation, it is a question of calculating the clean modes of bending of a thin square plate in
two configurations:
·
Plate embedded on an edge,
·
Free plate.
The plate is with a grid in triangular elements to which elements DKT are affected.
Four different modelings are tested:
·
Modal calculation Arêtes of the plate directed according to axes' of the reference mark,
·
Modal calculation unspecified Orientation of the plate and harmonic response for the plate
embedded,
·
Modal calculation by conventional and cyclic dynamic sousstructuration,
·
Modal calculation following a condensation of Guyan.
The results of reference of modal calculations result from analytical calculations. They validate on the one hand
tools for creation of the matrices of mass and rigidity, as well as the operators of
conventional and cyclic dynamic under-structuring implemented in
Code_Aster
. In addition, it
case test validates modal calculation following a condensation of Guyan (condensation of the matrix of
mass).
Code_Aster
®
Version
3
Titrate:
SDLS01 Plates square thin, free or embedded at the edge
Date:
24/08/99
Author (S):
G. ROUSSEAU, C. VARE, J. PELLET
Key:
V2.03.001-C
Page:
2/12
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/96/013 - Ind A
1
Problem of reference
1.1 Geometry
With
D
B
C
I
H
J
K
G
y, v
has
C
has
X, U
With
D
B
T
Dimensioned = 1.m has
thickness T = 0.01 m
Co-ordinates of the points (in m):
With
B
C
D
G
H
I
J
K
X
0.
1.
1.
0.
0.5
0.25
0.75
0.75
0.25
y
0.
0.
1.
1.
0.5
0.25
0.25
0.75
0.75
Z
0.
0.
0.
0.
0.
0.
0.
0.
0.
1.2
Material properties
E
AP
kg m
=
=
=
2 110
0 3
7800
11
3
.
.
/
1.3
Boundary conditions and loadings
Case 1: dimensioned embedded AB
for any point
P
y
P
such as
=
0
:
U
v
W
X
y
Z
= = =
=
=
=
0
0
.
.
Case 2: free plate
1.4 Conditions
initial
Without object for the modal analysis
Code_Aster
®
Version
3
Titrate:
SDLS01 Plates square thin, free or embedded at the edge
Date:
24/08/99
Author (S):
G. ROUSSEAU, C. VARE, J. PELLET
Key:
V2.03.001-C
Page:
3/12
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/96/013 - Ind A
2
Reference solution
2.1
Method of calculation used for the reference solution
The reference solution is that given in card SDLS01/89 of the guide VPCS which presents
method of calculation in the following way:
The formulation of Mr. V. BARTON, for a plate of with dimensions A, conduit with:
(
)
F
has
And
I
I
I
=
-
=
1
2
12
1
1 2
2
2
2
2
,…
with, for a Poisson's ratio
=
0 3
.
:
1°: Plate embedded on a side
2°: Free plate
I
I
2
I
I
2
1
3.492
1 to 6
0.
2
8.525
7
13.49
3
21.43
8
19.79
4
27.33
9
24.43
5
31.11
10
35.02
6
54.44
11
35.02
(6 modes of solid body at null frequency).
This reference solution applies to the thin sections such as:
T has
/
.
<
0 1
.
Coefficients
I
are establish by development limited on the modal deformations of a network of
cross beams (beam encastréelibre and beam librelibre).
2.2
Results of reference
Case 1: the first 6 clean modes
Case 2: the first 11 clean modes
2.3
Uncertainty on the solution
Semianalytic solution.
2.4 References
bibliographical
[1]
Mr. V. BARTON Vibrations off rectangular and skew cantilever punts. Journal off Applied
Mechanics, flight 18, p. 129134 (1951)
Code_Aster
®
Version
3
Titrate:
SDLS01 Plates square thin, free or embedded at the edge
Date:
24/08/99
Author (S):
G. ROUSSEAU, C. VARE, J. PELLET
Key:
V2.03.001-C
Page:
4/12
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/96/013 - Ind A
3 Modeling
With
3.1
Characteristics of modeling A
Modeling DKT
D
C
With
B
K
J
I
y
X
G
H
Names of the nodes:
Points
With = N1
B = N78
C = N145
D = N80
G = N65
H = N17
I = N73
J = N121
K = N71
Limiting conditions:
Case 1 in all the nodes on the side AB:
DDL_IMPO: (GROUP_NO: AB DX:0., DY:0., DZ:0., DRX:0., DRY:0., DRZ:0.)
Cases 2 none
3.2
Characteristics of the mesh
A number of nodes: 145
A number of meshs and types: 256 TRIA3
3.3 Functionalities
tested
Controls
Keys
AFFE_CARA_ELEM
HULL
ALL
[U4.24.01]
AFFE_CHAR_MECA
DDL_IMPO
GROUP_NO
[U4.25.01]
AFFE_MODELE
“MECHANICAL”
“DKT”
ALL
[U4.22.01]
DEFI_MATERIAU
ELAS
[U4.23.01]
MODE_ITER_SIMULT
“TAPE”
[U4.52.01]
CALC_ELEM
OPTION
“EPOT_ELEM_DEPL”
“ECIN_ELEM_DEPL”
[U4.61.02]
Code_Aster
®
Version
3
Titrate:
SDLS01 Plates square thin, free or embedded at the edge
Date:
24/08/99
Author (S):
G. ROUSSEAU, C. VARE, J. PELLET
Key:
V2.03.001-C
Page:
5/12
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/96/013 - Ind A
4
Results of modeling A
4.1 Values
tested
Frequency (Hz)
Clean mode
Reference
Aster
% difference
Tolerance
1°: Plate embedded on a side
1
8.7266
8.6718
0.63
2
21.3042
21.2904
0.06
3
53.5542
53.0992
0.85
1. 10
2
4
68.2984
67.9269
0.54
5
77.7448
77.4294
0.40
6
136.0471
135.7635
0.21
Aster
epot
=
ecin
1
1.4796 10
4
2
1.7331 10
4
3
4.3802 10
4
4
3.7367 10
4
5
5.4956 10
4
6
1.3483 10
5
2°: Free plate
7
33.7119
33.6839
0.08
8
49.4558
48.9362
1.05
9
61.0513
60.5849
0.76
1.1 10
2
10
87.5160
87.0993
0.48
11
87.5160
87.0993
0.48
Aster
epot
=
ecin
7
2.2396 10
4
8
4.7270 10
4
9
7.2453 10
4
10
1.4974 10
5
11
1.4974 10
5
4.2 Remarks
MODE_ITER_SIMULT
OPTIONS: “TAPE”
FREQ: (8., 140.)
CASE 1
FREQ: (32., 90.)
CASE 2
Contents of the file results:
1°:
the first 6 Eigen frequencies, clean vectors and modal parameters
deformation energy and kinetic energy of the 6 modes.
2°:
5 Eigen frequencies, clean vectors and modal parameters (F > 0)
deformation energy and kinetics of the 5 modes.
4.3 Parameters
of execution
Version: 3.04.05
Machine: CRAY C98
System:
UNICOS 8.0
Overall dimension memory:
8 megawords
Time CPU To use:
18.91 seconds
Code_Aster
®
Version
3
Titrate:
SDLS01 Plates square thin, free or embedded at the edge
Date:
24/08/99
Author (S):
G. ROUSSEAU, C. VARE, J. PELLET
Key:
V2.03.001-C
Page:
6/12
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/96/013 - Ind A
5 Modeling
B
5.1
Characteristics of modeling B
Modeling DKT with mesh identical to modeling A.
Rotation of the plate such as side AB is on the line
3
4
y
X
=
With
H
K
J
I
G
C
B
D
X
y
Names of the nodes:
Points
With = N1
B = N78
C = N145
D = N80
G = N65
H = N17
I = N73
J = N121
K = N71
Limiting conditions:
Case 1 in all the nodes on the side AB:
DDL_IMPO: (GROUP_NO: AB DX:0., DY:0., DZ:0., DRX:0., DRY:0., DRZ:0.)
Case 2: none
Harmonic answer:
Nodal force point C (N145):
Fz = 98100
Material
: AMOR_ALPHA: 0.1 AMOR_BETA: 0.1
5.2
Characteristics of the mesh
A number of nodes: 145
A number of meshs and types: 256 TRIA3
5.3 Functionalities
tested
Controls
Keys
AFFE_CARA_ELEM
HULL
ALL
[U4.24.01]
AFFE_CHAR_MECA
DDL_IMPO
GROUP_NO
[U4.25.01]
AFFE_MODELE
'MECHANICAL
“DKT”
ALL
[U4.22.01]
DEFI_MATERIAU
ELAS
AMOR_ALPHA
AMORBETA
[U4.23.01]
MODE_ITER_SIMULT
“TAPE”
[U4.52.01]
DYNA_LINE_HARM
[U4.54.02]
CALC_ELEM
OPTION
EFGE_ELNO_DEPL
SIGM_ELNO_DEPL
[U4.61.02]
Code_Aster
®
Version
3
Titrate:
SDLS01 Plates square thin, free or embedded at the edge
Date:
24/08/99
Author (S):
G. ROUSSEAU, C. VARE, J. PELLET
Key:
V2.03.001-C
Page:
7/12
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/96/013 - Ind A
6
Results of modeling B
6.1 Values
tested
The values of the Eigen frequencies are identical to those of modeling A.
Harmonic answer:
FREQ:
50 Hz
NODE:
N145
NET:
M255
Reference
Aster
3.03.15
Aster
3.05.16
% difference
DEPL “DZ”
2.90290E02
5.20606E02
2.90290E02
5.20606E02
0.0
DEPL “DRX”
2.52920E02
9.44717E02
2.52920E02
9.44717E02
0.0
QUICKLY “DZ”
1.63553E+01
9.11973E+00
1.63553E+01
9.11973E+00
0.0
QUICKLY “DRX”
2.96792E+01
7.94573E+00
2.96792E+01
7.94573E+00
0.0
ACCE “DZ”
2.86505E+03
5.13817E+03
2.86505E+03
5.13817E+03
0.0
ACCE “DRX”
2.49622E+03
9.32398E+03
2.49622E+03
9.32398E+03
0.0
“EFGE_ELNO_DEPL” “MXX”
1.14053E+01
1.45539E+03
1.14053E+01
1.45539E+03
0.0
“EFGE_ELNO_DEPL” “MYY”
1.10224E+01
1.31441E+03
1.10224E+01
1.31441E+03
0.0
“EFGE_ELNO_DEPL” “MXY”
1.03148E+01
3.55382E+02
1.03148E+01
3.55382E+02
0.0
“EFGE_ELNO_DEPL” “QX”
3.66163E+02
3.77331E+03
3.66163E+02
3.77331E+03
0.0
“EFGE_ELNO_DEPL” “QY”
3.14676E+02
2.06813E+03
3.14676E+02
2.06813E+03
0.0
“SIGM_ELNO_DEPL” “SIXZ”
5.49245E+04
5.65997E+05
5.49245E+04
5.65997E+05
0.0
“SIGM_ELNO_DEPL” “SIYZ”
4.72014E+04
3.10219E+05
4.72014E+04
3.10219E+05
0.0
6.2 Remarks
MODE_ITER_SIMULT
OPTIONS: “TAPE”
FREQ: (8., 140.)
CASE 1
FREQ: (32., 90.)
CASE 2
Contents of the file results:
1°:
the first 6 Eigen frequencies, clean vectors and modal parameters.
2°:
the first 11 Eigen frequencies, clean vectors and modal parameters.
3°:
displacement DZ DRX with the N145 node
efforts generalized and forced M255 mesh
6.3 Parameters
of execution
Version: 3.03.10
Machine: CRAY C90
System:
UNICOS 8.0
Overall dimension memory:
8 megawords
Time CPU To use:
15.35 seconds
Code_Aster
®
Version
3
Titrate:
SDLS01 Plates square thin, free or embedded at the edge
Date:
24/08/99
Author (S):
G. ROUSSEAU, C. VARE, J. PELLET
Key:
V2.03.001-C
Page:
8/12
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/96/013 - Ind A
7 Modeling
C
7.1
Characteristics of modeling C
G
In the 2 cases, the plate is cut out in 4 parts of equal size. Each sousstructure
considered is with a grid in triangles to which elements of plate DKT are affected.
Case 1: Plate embedded on an edge
The structure is studied using the method of sousstructuration conventional with interfaces of the type
CRAIG_BAMPTON. The modal base used for each sousstructure is made up of 25 modes
clean and of the constrained modes associated the interfaces.
Case 2: Free plate
The structure is studied using the method of sousstructuration cyclic with interfaces of the type
HARMONIC CRAIG_BAMPTON and taking into account of the specificity of the node of the axis (not G).
base modal used for the basic sector is made up of 25 clean modes and the modes
harmonics associated with the interfaces.
7.2
Characteristics of the mesh
A number of nodes: 121
A number of meshs and types: 200 TRIA3
7.3 Functionalities
tested
Controls
Keys
MACR_ELEM_DYNA
MACR_ELEM_DYNA
BASE_MODALE
OPTION
“RITZ”
[U4.55.05]
DEFI_MODELE_GENE
DEFI_MODELE_GENE
DEFI_MODELE_GENE
DEFI_MODELE_GENE
DDEFI_MODELE_GENE
SOUS_STRUC
SOUS_STRUC
SOUS_STRUC
CONNECTION
CONNECTION
NAME
MACR_ELEM_DYNA
ANGL_NAUT
SOUS_STRUC_1
INTERFACE_1
SOUS_STRUC_2
INTERFACE_2
[U4.55.06]
NUME_DDL_GENE
MODELE_GENE
[U4.55.07]
ASSE_MATR_GENE
ASSE_MATR_GENE
NUME_DDL_GENE
OPTION
“MASS_GENE”
“RIGI_GENE”
[U4.55.08]
DEFI_SQUELETTE
DEFI_SQUELETTE
MODELE_GENE
SOUS_STRUC
NAME
[U4.75.01]
REST_BASE_PHYS
REST_BASE_PHYS
SKELETON
TOUT_ORDRE
[U4.64.01]
MODE_ITER_CYCL
MODE_ITER_CYCL
MODE_ITER_CYCL
CONNECTION
CALCULATION
CALCULATION
CENTER
TOUT_DIAM
OPTION
“YES”
“CENTER”
[U4.52.03]
DEFI_SQUELETTE
DEFI_SQUELETTE
MODE_CYCL
SECTOR
[U4.75.01]
Code_Aster
®
Version
3
Titrate:
SDLS01 Plates square thin, free or embedded at the edge
Date:
24/08/99
Author (S):
G. ROUSSEAU, C. VARE, J. PELLET
Key:
V2.03.001-C
Page:
9/12
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/96/013 - Ind A
8
Results of modeling C
8.1 Values
tested
Command of
Frequency (Hz)
clean mode I
Reference
Aster
% difference
Tolerance
1°: Plate embedded on a side
1
8.7266
8.6419
0.97
2
21.3042
21.2253
0.37
3
53.5542
52.9693
1.09
1.25 10
2
4
68.2984
67.5444
1.10
5
77.7448
77.3966
0.45
6
136.0471
134.5785
1.08
2°: Free plate
7
33.7119
33.6808
0.09
8
49.4558
48.9785
0.96
9
61.0513
60.6739
0.62
1. 10
2
10
87.5160
87.0662
0.51
11
87.5160
87.0662
0.51
8.2 Parameters
of execution
Version: 3.05.13
Machine: CRAY C98
System:
UNICOS 8.0
Overall dimension memory:
16 megawords
Time CPU To use:
112.81 seconds
Code_Aster
®
Version
3
Titrate:
SDLS01 Plates square thin, free or embedded at the edge
Date:
24/08/99
Author (S):
G. ROUSSEAU, C. VARE, J. PELLET
Key:
V2.03.001-C
Page:
10/12
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/96/013 - Ind A
9 Modeling
D
9.1
Characteristics of modeling D
DKT
+ sousstructuration of GUYAN
H
K
J
C
D
NR
O
With
B
M
y
L
I
G
X
Limiting conditions: Free plate
Condensation of the matrices of mass and rigidity on the nodes:
(A, B, C, D, G, H, I, J, K, L, M, NR, O).
9.2
Characteristics of the mesh
A number of nodes: 145
A number of meshs and types: 256 TRIA3
9.3 Functionalities
tested
Controls
Keys
AFFE_CARA_ELEM
HULL
ALL
[U4.24.01]
AFFE_CHAR_MECA
DDL_IMPO
GROUP_NO
[U4.25.01]
AFFE_MODELE
'MECHANICAL
“DKT”
ALL
[U4.22.01]
DEFI_MATERIAU
ELAS
[U4.23.01]
MODE_ITER_SIMULT
“TAPE”
[U4.52.01]
MACR_ELEM_STAT
RIGI_MECA
[U4.44.01]
MACR_ELEM_STAT
MACR_MECA
[U4.44.01]
Code_Aster
®
Version
3
Titrate:
SDLS01 Plates square thin, free or embedded at the edge
Date:
24/08/99
Author (S):
G. ROUSSEAU, C. VARE, J. PELLET
Key:
V2.03.001-C
Page:
11/12
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/96/013 - Ind A
10
Results of modeling D
10.1 Values
tested
Command of
Frequency (Hz)
clean mode I
Reference
Aster
% difference
Tolerance
2°: Free plate
7
33.7119
33.8758
0.48
8
49.4558
49.5240
0.14
1.1 10
2
9
61.0513
61.6240
0.94
10.2 Remarks
One seeks to calculate the first 3 nonnull Eigen frequencies of the problem of the free plate
on its edges.
If one condenses the matrices on the only nodes:
(A, B, C, D, G, H, I, J, K)
The precision of the frequencies is not whereas 2%.
To obtain the results wanted with the awaited precision (1%), it is necessary to add the points (L, M, NR, O).
Code_Aster
®
Version
3
Titrate:
SDLS01 Plates square thin, free or embedded at the edge
Date:
24/08/99
Author (S):
G. ROUSSEAU, C. VARE, J. PELLET
Key:
V2.03.001-C
Page:
12/12
Manual of Validation
V2.03 booklet: Linear dynamics of the hulls and the plates
HI-75/96/013 - Ind A
11
Summary of the results
·
Modelings A and b:
Precision on the Eigen frequencies ² 1% until the sixth mode of bending.
·
Modeling C:
In sousstructuration, the quality of the results could be improved by the use of a mesh
of finer sousstructure.
·
Modeling D:
To obtain an accuracy of 1% on the Eigen frequencies, it is necessary to condense too
on the 4 nodes mediums of the edges L, M, NR and O.