Code_Aster ®
Version
4.0
Titrate:
SDLL10 Poutre of rectangular section variable
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.010-C Page:
1/6
Organization (S): EDF/IMA/MN
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
V2.02.010 document
SDLL10 - Rectangular Poutre of section
variable (embed-embedded)
Summary:
This plane problem consists in seeking the frequencies and the modes of vibration of a mechanical structure
composed of an embed-embedded beam whose surface of the cross-section varies exponentially. This test
of Mécanique of Structures corresponds to a dynamic analysis of a linear model having one
linear behavior. It includes/understands only one modeling.
Via this problem, one tests the element of beam in inflection of variable Timoshenko of section
as well as the calculation of the frequencies and modes of vibration by the method of Lanczos. One also tests
functionality “normalizes to 1.” at the point of maximum amplitude in translation " of the modes of vibration.
By using a fine space discretization, the results obtained are in concord with the results
analytical given in guide VPCS.
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
Code_Aster ®
Version
4.0
Titrate:
SDLL10 Poutre of rectangular section variable
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.010-C Page:
2/6
1
Problem of reference
1.1 Geometry
y, v
L
y
y
bo A
B
H
H
H
H
X, U
Z
Z
b1
bo
b1
initial
finale
Z, W
Length of the beam: L = 0.6 m
Constant thickness: H = 0.01 m
Rectangular section:
Initial cross-section:
Variation of the section:
width:
b0 = 0.03 m
B = b0 e2x with = 1.
surface:
A0 = 3.104 m2
With = A0 e2x
moment of inertia:
Iz0 = 0.25 108 m4
Iz = Iz0 e2x
Co-ordinates of the points (m):
With
B
X
0.
0.6
y
0.
0.
1.2
Material properties
E = 2.1011 Pa
= 0.3
= 7.800. kg/m3
1.3
Boundary conditions and loadings
Embedded points A and b:, U = v = 0., = 0.
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
Code_Aster ®
Version
4.0
Titrate:
SDLL10 Poutre of rectangular section variable
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.010-C Page:
3/6
2
Reference solution
2.1
Method of calculation used for the reference solution
The reference solution is that given in card SDLL10/89 of the guide VPCS which presents
method of calculation in the following way:
The pulsation is given by the roots of the equation:
I
1 - cos rL
() CH (SSL) + s2 - r2 HS (SSL) sin (rL) = 0
2rs
with:
To 2
4
0
I
2
2
2
2
2
2
=
;
=
+
;
=
if
>
I
R
I
S
I
(I) 0
E Iz0
Components of translation of the mode F ()
I X are then:
cos
- CH
()
X
rL
SSL
X = E
(
cos X-ray) - C (
H sx)
()
()
+
sin
-
I
(S X-ray R HS sx)
R
(
HS SSL)
- S
(
sin rL)
()
(
2.2
Results of reference
the first 4 Eigen frequencies and normalized clean modes with 1 for the largest component in
translation.
2.3
Uncertainty on the solution
Analytical solution.
2.4 References
bibliographical
[1]
Working group Analyze Dynamique. Committee of Validation of Progiciels de Calcul of
Structure. French company of Mécaniciens (1988).
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
Code_Aster ®
Version
4.0
Titrate:
SDLL10 Poutre of rectangular section variable
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.010-C Page:
4/6
3 Modeling
With
3.1
Characteristics of modeling
Element of beam POU_D_T (Poutre right of Timoshenko)
y
B
With
X
Cutting: beam AB: 120 meshs SEG2 of section variable.
Limiting conditions:
in all the nodes
DDL_IMPO
(TOUT:“YES”, DZ: 0., DRX: 0., DRY: 0. )
with the nodes
ends
(NOEUD: (AB) DX: 0., DY: 0., DRZ: 0. )
Names of the nodes:
Not A
X = 0.
= N1
X = 0.1
= N21
X = 0.2
= N41
X = 0.3
= N61
X = 0.4
= N81
X = 0.5
= N101
Not B
X = 0.6
= N121
3.2
Characteristics of the grid
A number of nodes:
121
A number of meshs and types:
120 SEG2
3.3 Functionalities
tested
Commands
Keys
AFFE_CARA_ELEM
POUTRE
“GENERALE”
MAILLE
[U4.24.01]
AFFE_CHAR_MECA
DDL_IMPO
TOUT
[U4.25.01]
NOEUD
AFFE_MATERIAU
TOUT
[U4.23.02]
AFFE_MODELE
“MECANIQUE”
“POU_D_T'
TOUT
[U4.22.01]
DEFI_MATERIAU
ELAS
[U4.23.01]
MODE_ITER_SIMULT
METHODE
“TRI_DIAG”
[U4.52.02]
CALC_FREQ
OPTION
“PLUS_PETITE”
NMAX_FREQ
NORM_MODE
NORME
“TRAN”
[U4.64.02]
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
Code_Aster ®
Version
4.0
Titrate:
SDLL10 Poutre of rectangular section variable
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.010-C Page:
5/6
4
Results of modeling A
4.1 Values
tested
Command of the mode
Frequency
Aster
% difference
% tolerance
clean
Reference
1
143.303
145.924
1.598
1.6
2
396.821
398.511
0.426
0.45
3
779.425
777.115
0.296
0.3
4
1289.577
1278.011
0.897
0.9
The clean modes of Aster were normalized to 1. at the point of maximum amplitude in translation like
in the reference.
Clean mode Fi (X) normalized with 1 at the point of maximum amplitude
I
X = 0.1
X = 0.2
X = 0.3
X = 0.4
X = 0.5
Reference
0.2349
0.6962
0.98960
0.8505
0.3507
Aster
1
0.2363
0.6970
0.9895
0.8516
0.3529
% difference
0.583
0.119
0.
0.132
0.631
% tolerance
0.6
0.15
0.1
0.15
0.7
Reference
0.4653
0.7558
0.
0.9232
0.6941
Aster
2
0.4670
0.7555
2.9104
0.9226
0.6971
% difference
0.37
0.041
2.9104
0.063
0.435
% tolerance
0.4
0.1
1.103
0.1
0.45
Reference
0.6278
0.1969
0.7783
0.2406
0.9366
Aster
3
0.6290
0.1952
0.7782
0.2377
0.9387
% difference
0.192
0.89
0.014
1.226
0.228
% tolerance
0.2
0.9
0.1
1.23
0.25
Reference
0.666
0.4832
0.
0.5901
0.9937
Aster
4
0.6656
0.4840
4.6104
0.5919
0.9928
% difference
0.081
0.18
4.6104
0.31
0.089
% tolerance
0.1
0.2
1.103
0.35
0.1
4.2 Remarks
Calculations carried out by:
MODE_ITER_SIMULT METHOD: “TRI_DIAG”
OPTION: “PLUS_PETITE”
NMAX_FREQ: 4
Contents of the file results:
the first 4 Eigen frequencies, clean vectors.
4.3 Parameters
of execution
Version: 3.02.21
Machine: CRAY C90
System:
UNICOS 8.0
Obstruction memory:
8 megawords
Time CPU To use:
9.8 seconds
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A
Code_Aster ®
Version
4.0
Titrate:
SDLL10 Poutre of rectangular section variable
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.010-C Page:
6/6
5
Summary of the results
Suitable modeling (frequencies and modes suitable for less than 2%) with a fine grid.
A calculation carried out on a coarse grid (12 meshs) shows more important variations with
reference solution. This is especially due to the way in which the modes are normalized.
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A