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Code_Aster
®
Version
5.0
Titrate:
SDND103 Post subjected to an axial dynamic stress
Date:
30/08/01
Author (S):
Fe WAECKEL, L. VIVAN
Key
:
V5.01.103-B
Page:
1/6
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-62/01/012/A
Organization (S):
EDF/RNE/AMV, CS IF















Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
V5.01.103 document



SDND103 - Post subjected to a stress
axial dynamics




Summary

It is a question of calculating the response of a post subjected to an unspecified seismic loading. The post is
modelized by a system mass-arises not deadened, its connection with the ground by a non-linearity of the type
effort-displacement.

The discrete element in traction and compression, the calculation of the clean modes and the calculation of the answer are tested
transient by modal recombination with taking into account of a non-linearity of the effort-displacement type.
initial speed is taken nonnull and the loading is of acceleration type imposed on the ground.

The results obtained are in very good agreement with the results of reference which are results
analytical.
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Code_Aster
®
Version
5.0
Titrate:
SDND103 Post subjected to an axial dynamic stress
Date:
30/08/01
Author (S):
Fe WAECKEL, L. VIVAN
Key
:
V5.01.103-B
Page:
2/6
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-62/01/012/A
1
Problem of reference
1.1 Geometry
The system consists of a post resting on the ground and subjected to a seismic stress. It is
modelized by a mass, its connection with the ground by a spring K
0
of which the relation of behavior
translated a non-linearity of the effort-displacement type.
L
X
Y
K
0
Characteristics of the post:
length: L = 2 m;
section: S = 0,3 m
2
.
1.2
Properties of materials
Mass post: m = 450 kg.
Stiffness within the competence of connection: K
0
= 10
5
NR/Mr.
1.3
Boundary conditions and loadings
Boundary conditions
Only authorized displacements are the translations according to axis X: Dy = dz = 0.
The corrective force
F
C
had with nonthe linearity of the ground is defined by the following relation:
()
F X
F
F X
C
=
-
(X
)
X
()
threshold
threshold
with, if
X >
threshold
X
,
F X
X
X
()
K
X
.
=
-




0
0
1
.
One takes
X
threshold
= 10
- 6
m, K
0
= 10
5
NR/m and
X
0
= 0,1 Mr.
One thus imposes under the key word
RELA_EFFO_DEPL
of the operator
DYNA_TRAN_MODAL
function:
()
[
]
F X
X X
X
C
threshold
=
-
K
X
.
0
0
.
Loading
The ground is subjected to an acceleration
(T) in direction X, built so that it
displacement of the system mass-arises is sinusoidal
()
X
has
T
= .sin
with
has
= 0,01 and
=/4.
1.4 Conditions
initial
In the initial state, the system is released of its position of balance with a speed v
0
: with T = 0, dx (0) = 0,
v
0
= dx/dt (0) =
a.
.
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Code_Aster
®
Version
5.0
Titrate:
SDND103 Post subjected to an axial dynamic stress
Date:
30/08/01
Author (S):
Fe WAECKEL, L. VIVAN
Key
:
V5.01.103-B
Page:
3/6
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-62/01/012/A
2
Reference solution
2.1
Method of calculation used for the reference solution
This test is developed in detail in the reference [bib1].
The fundamental equation of dynamics, moving relative of the system mass-arises by
report/ratio on the ground is written:
()
()
&&
X
K X
m X
T
+
=
.
For a displacement of the form
()
()
X
has
T
X
has
T
=
= -
.sin
&&
sin
and
2
, one obtains from
the equation of the movement the form of the accélérogramme:
()
()
()
T
has
T
K
m
has
T
X
=
-
+
-




sin
sin
2
0
0
1
.
The fundamental frequency
F
0
oscillator not deadened is worth
F
K
m
0
0
1
2
=
.

2.2
Results of reference
Fundamental frequency
F
0
oscillator not deadened.
Displacements relating to moments 2, 6, 10, 14 and 18 seconds.

2.3
Uncertainty on the solution
No if one calculates the integral of Duhamel analytically [bib2].

2.4 References
bibliographical
[1]
P. LALUQUE, P. LABBE, S. PETETIN and A. TIXIER: Seismic response of a ship
jet engine PWR1300 by taking account of separation enters the foundation and the ground. Note
SEPTEN TA83.06 (May 1984).
[2]
J.S. PRZEMIENIECKI: Theory off matrix structural analysis. New York, Mac Graw-Hill, 1968,
p. 351-357.
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Code_Aster
®
Version
5.0
Titrate:
SDND103 Post subjected to an axial dynamic stress
Date:
30/08/01
Author (S):
Fe WAECKEL, L. VIVAN
Key
:
V5.01.103-B
Page:
4/6
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-62/01/012/A
3 Modeling
With
3.1
Characteristics of modeling
The system mass-arises is modelized by a discrete element
DIS_T
.

X
Y
K
0
m
Numerical data:
for the system mass-arises: m = 450 kg
for the ground:
K
0
= 10
5
NR/m
for non-linearity:
X
0
= 0,1 m; has = 0,01 and
=/4.
Temporal integration is carried out with the algorithm of Euler or the algorithm of Devogelaere and one
no times of 0,02 second. Calculations are filed all the pitches of time.
A reduced damping is considered
I
no one for the whole of the calculated modes.

3.2
Characteristics of the mesh
The mesh consists of a node and a mesh of the type
POI1.

3.3 Functionalities
tested
Controls
Keys Doc. V5
FORMULATE
[U4.31.05]
CALC_FONC_INTERP
[U4.32.01]
“MECHANICAL” AFFE_MODELE GROUP_MA
“DIS_T'
[U4.41.01]
AFFE_CHAR_MECA DDL_IMPO
[U4.44.01]
DISCRETE AFFE_CARA_ELEM
NET
M_T_D_N
[U4.42.01]
GROUP_MA
K_T_D_L
MODE_ITER_SIMULT CALC_FREQ
PLUS_PETITE
[U4.52.03]
CALC_CHAR_SEISME MONO_APPUI
[U4.63.01]
MACRO_PROJ_BASE
[U4.63.11]
DYNA_TRAN_MODAL ETAT_INIT
[U4.53.21]
RELA_EFFO_DEPL
POST_DYNA_MODA_T RESU_GENE
[U4.84.02]
RELA_EFFO_DEPL
REST_BASE_PHYS
[U4.63.21]
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Code_Aster
®
Version
5.0
Titrate:
SDND103 Post subjected to an axial dynamic stress
Date:
30/08/01
Author (S):
Fe WAECKEL, L. VIVAN
Key
:
V5.01.103-B
Page:
5/6
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-62/01/012/A
4
Results of modeling A
4.1 Values
tested
One checks the Eigen frequency of the oscillator as well as displacements relative of node NO1 to
various moments (for the algorithm of integration EULER).
Frequency (Hz)
Reference
Code_Aster Error
(%)
2,37254
2,37254
0

Relative displacement of node NO1 with the numerical algorithm of integration of Euler:
Time (S)
Reference
Code_Aster Error
(%)
2 0,01
9,99988E03
0,001
6 ­ 0,01
­ 9,99985E03
0,002
10 0,01
9,99990E03
0,001
14 ­ 0,01
­ 9,99985E03
0,001
18 0,01
9,99987E03
0,001

Relative displacement of node NO1 with the numerical algorithm of integration of Devogelaere:
Time (S)
Reference
Code_Aster Error
(%)
2 0,01
9,99991E03
8,88E06
6 ­ 0,01
­ 9,99981E03
­ 0,002
10 0,01
9,99992E03
7,72E06
14 ­ 0,01
­ 9,99988E03
­ 0,001
18 0,01
9,99982E03
­ 0,002

4.2 Parameters
of execution
Version: STA5.02
Machine: SGI Origin 2000
Time CPU to use: 3,6 seconds
background image
Code_Aster
®
Version
5.0
Titrate:
SDND103 Post subjected to an axial dynamic stress
Date:
30/08/01
Author (S):
Fe WAECKEL, L. VIVAN
Key
:
V5.01.103-B
Page:
6/6
Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-62/01/012/A
5
Summary of the results
One notes a very good agreement with the analytical solution (error lower than 0,01%).