Code_Aster ®
Version
7.2
Titrate:
SDND102 - Seismic Réponse of a multimedia system
Date
:
08/10/03
Author (S):
G. DEVESA, Fe WAECKEL, E. BOYERE
Key: V5.01.102-C Page:
1/14
Organization (S): EDF-R & D/AMA
Handbook of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
V5.01.102 document
SDND102 - Seismic Réponse of a system
mass-arises nonlinear multimedia
Summary
The problem consists in analyzing the response of a mechanical structure, modelled by two systems
mass-arises not deadened, subjected to a seismic loading of harmonic type, with possibility of shock.
One tests the discrete element in traction and compression, the calculation of the clean modes and the static modes, it
calculation of the transitory response by nonlinear modal recombination of a structure subjected to one
accélérogramme (modeling A) as well as the calculation of the direct transitory seismic response of a structure
nonlinear (modeling B).
This case test is also used to validate a calculation with explicit resolution on accelerations and shock (modeling C)
by comparing the results resulting from DYNA_NON_LINE and DYNA_TRAN_EXPLI.
The results obtained are in very good agreement with the results of reference.
Handbook of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-66/03/008/A
Code_Aster ®
Version
7.2
Titrate:
SDND102 - Seismic Réponse of a multimedia system
Date
:
08/10/03
Author (S):
G. DEVESA, Fe WAECKEL, E. BOYERE
Key: V5.01.102-C Page:
2/14
1
Problem of reference
1.1 Geometry
One compares the seismic response of a system mass-arises with a degree of freedom which can impact
a fixed wall (problem 1) with that of two systems mass-arises identical being able
entrechoquer and subjected to the same seismic stress (problem 2).
With
K
m
B
C
X
K
m
m K
X
1
1
2= -
Problem 1
Problem 2
1.2
Material properties
Stiffness of the springs: K = 98696 NR/Mr.
Specific mass: m = 25 kg.
For problem 1 (impact on a rigid wall), the normal rigidity of shock is worth Kchoc = 5,76 107 NR/Mr.
As for problem 2 (shock of two deformable structures), it is worth Kchoc = 2,88 107 NR/Mr.
In both cases, the damping of shock is null.
1.3
Boundary conditions and loadings
Boundary conditions
Only authorized displacements are the translations according to axis X.
The points A, B and C are embedded: dx = Dy = dz = 0.
Loading
The points of anchoring A and B are subjected to an acceleration according to direction X: 1 (T) = sin T
with = 20. s1 and the point C with an acceleration 2 (T) = - sin T.
1.4 Conditions
initial
In both cases, the systems mass-arises are initially at rest:
with T = 0, dx (0) =0, dx/dt (0) = 0 in any point.
For problem 1, the mass is separated from the fixed wall of the play J = 5. 104 Mr. Quant au problem 2,
the masses are separated from the play J = 2 J = 103 Mr.
Handbook of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-66/03/008/A
Code_Aster ®
Version
7.2
Titrate:
SDND102 - Seismic Réponse of a multimedia system
Date
:
08/10/03
Author (S):
G. DEVESA, Fe WAECKEL, E. BOYERE
Key: V5.01.102-C Page:
3/14
2
Reference solution
2.1
Method of calculation used for the reference solution
It is a question of comparing the response of a symmetrical system consisted two systems mass-arises
identical to the response of a system mass-arises. Two problems, explained in detail in
reference [bib2], are requested by same the accélérogramme.
One calculates the Eigen frequencies initially fi, the standardized associated clean vectors
compared to modal mass Ni and the static modes of the system (analytical values). One
calculate then the generalized response of the system multimedia while solving analytically
the integral of Duhamel [bib1]. Lastly, one restores on the physical basis the relative displacement of the nodes
of shock what allows us, after having calculated the field of displacements of drive, of
to calculate the field of absolute displacements.
One calculates the definite function diff as being the difference between absolute displacement of the node
shocking on a mobile obstacle and that of the node shocking on a fixed obstacle. It is checked that it is
quite null for various moments.
2.2
Results of reference
Displacements relating and absolute to the nodes of shock.
2.3
Uncertainty on the solution
Comparison between two equivalent modelings.
2.4 References
bibliographical
[1]
J.S. PRZEMIENIECKI: Structural Theory off matrix analysis New York, Mac Graw - Hill, 1968,
p. 351-357.
[2]
Fe. WAECKEL: Use and validation of the developments carried out to calculate
seismic response of multimedia structures - HP52/96.002.
Handbook of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-66/03/008/A
Code_Aster ®
Version
7.2
Titrate:
SDND102 - Seismic Réponse of a multimedia system
Date
:
08/10/03
Author (S):
G. DEVESA, Fe WAECKEL, E. BOYERE
Key: V5.01.102-C Page:
4/14
3 Modeling
With
3.1
Characteristics of modeling
The systems mass-arises are modelled by discrete elements with 3 degrees of freedom DIS_T.
Modeling of problem 1:
Yloc
Y
I
J
Zloc
K
X
orig_
m
no1
no2
1
Appear 3.1-a: Modélisation of a system mass-arises impacting a rigid wall
The node no1 is subjected to an imposed acceleration 1 (T). One calculates the relative displacement of the node
no2, its displacement of drive and its absolute displacement.
An obstacle of the type PLAN_Z (two parallel plans) is retained to simulate the impact of the system
mass-arises on a rigid wall. The normal in the plan of shock is axis Z, NORM_OBST: (0. 0.
1.). Not to be obstructed by the rebound of the oscillator on the symmetrical level, one pushes back this one
very far (cf [Figure 3.1-a]).
From where:
· the origin of obstacle ORIG_OBST: ( 1. 0. 0.) ;
· and play corresponding play: 1.1005
Modeling of problem 2:
Y
dist_1
dist_2
J
K
K
X
m
m
NO1
NO2
NO3
NO4
1
2= - 1
Appear 3.1-b: Modélisation of two systems mass-arises which are entrechoquent
Node NO1 is subjected to an imposed acceleration 1 (T), node NO4 to 2 (T) = 1 (T). One calculates
the relative displacement of nodes NO2 and NO3, their displacement of drive and their displacement
absolute.
Handbook of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-66/03/008/A
Code_Aster ®
Version
7.2
Titrate:
SDND102 - Seismic Réponse of a multimedia system
Date
:
08/10/03
Author (S):
G. DEVESA, Fe WAECKEL, E. BOYERE
Key: V5.01.102-C Page:
5/14
The conditions of shock between the two systems mass-arises are simulated by an obstacle of the type
BI_PLAN_Z (plane obstacle between two mobile structures). The normal in the plan of shock is selected
according to axis Z, that is to say NORM_OBST: (0. 0. 1.).
The thicknesses of matter surrounding the nodes of shock in the direction considered are specified
by operands DIST_1 and DIST_2. In the treated case, one chooses DIST_1 = DIST_2 = 0.4495 for
that at the initial moment, the two nodes of shock are separated from the play J = 2 J = 103 mm (cf [Figure 3.1-
B]).
Temporal integration is carried out with the algorithm of Euler and a step of times of 2,5. 104s. Les
calculations are filed all the 8 steps of time.
One considers a reduced damping of 7% for the whole of the calculated modes.
3.2
Characteristics of the grid
One calls model the grid associated with the problem made up of a system mass-arises butting
against a fixed wall and bichoc that which is associated problem 2.
Grid associated with the model model:
a number of nodes: 2;
a number of meshs and types: 1 DIS_T.
Grid associated with the model bichoc:
a number of nodes: 4;
a number of meshs and types: 2 DIS_T.
3.3 Functionalities
tested
Commands
AFFE_MODELE GROUP_MA
“MECANIQUE”
“DIS_T'
DISCRETE AFFE_CARA_ELEM GROUP_MA M_T_D_N
GROUP_MA
K_T_D_L
AFFE_CHAR_MECA DDL_IMPO
GROUP_NO
MACRO_MATR_ASSE
MODE_ITER_SIMULT METHOD
JACOBI
CALC_FREQ
BANDE
NORM_MODE NORMALIZES
MASS_GENE
MODE_STATIQUE DDL_IMPO
CALC_CHAR_SEISME MONO_APPUI
MULTI_APPUI
MACRO_PROJ_BASE
DEFI_OBSTACLE PLAN_Z
BI_PLAN_Z
DYNA_TRAN_MODAL EXCIT
MULT_APPUI “YES”
AMOR_REDUIT
METHODE
EULER
REST_BASE_PHYS MULT_APPUI
“OUI”
RECU_FONCTION RESU_GENE
Handbook of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-66/03/008/A
Code_Aster ®
Version
7.2
Titrate:
SDND102 - Seismic Réponse of a multimedia system
Date
:
08/10/03
Author (S):
G. DEVESA, Fe WAECKEL, E. BOYERE
Key: V5.01.102-C Page:
6/14
4
Results of modeling A
4.1
Values tested of modeling A
One calculates the definite function diff as being the difference between absolute displacement of the node
NO2 and that of the node no2. And it is checked that it is quite null for various moments.
Time (S)
Reference
Aster
Absolute error
0,1 0,0
5,8884E-07
5,89E-07
0,3 0,0
1,8891E-06
1,89E-06
0,5 0,0
1,5586E-07
1,56E-07
0,7 0,0
1,8213E-06
1,82E-06
1 0,0
1,7231E-06
1,72E-06
One also tests the value of the absolute displacement of node NO2 for various moments.
Time (S)
Reference
Aster
Absolute error
(problem 2)
0,05 3,58082E-04
3,5808E-04
1,71E-10
0,156 1,22321E-04 1,2232E-04
4,72E-10
0,25 1,8876E-04 1,8876E-04
1,96E-11
0,4 1,89772E-04
1,8977E-04
1,22E-10
0,5 6,84454E-05
6,8445E-05 4,72E-11
0,8 1,11982E-04
1,1198E-04
1,71E-11
0,9 1,20103E-04
1,2010E-04
1,37E-10
1 1,07178E-04
1,0718E-04
3,31E-10
One represents Ci below the pace of displacements relating and absolute to node NO2:
Absolute displacements
Relative displacements
Handbook of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-66/03/008/A
Code_Aster ®
Version
7.2
Titrate:
SDND102 - Seismic Réponse of a multimedia system
Date
:
08/10/03
Author (S):
G. DEVESA, Fe WAECKEL, E. BOYERE
Key: V5.01.102-C Page:
7/14
5 Modeling
B
5.1
Characteristics of modeling
The systems mass-arises are modelled, as in modeling A, by a discrete element with
3 degrees of freedom DIS_T.
Modeling of problem 1:
Y
Play
dist_1
K
m
NO1
NO2
1
elm1
Appear 5.1-a: Modélisation of a system mass-arises impacting a rigid wall
Node NO1 is subjected to an imposed acceleration 1 (T). One calculates the relative displacement of the node
NO2, its displacement of drive and its absolute displacement.
An element of the type DIST_T on a mesh POI1 is retained to simulate the impact of the beam on one
rigid wall: the possible shocks between the beam and the obstacle are taken into account as being
forces intern with this element. One affects to him a nonlinear behavior of type shock (stiffness) via
law of behavior DIS_CONTACT of command DEFI_MATERIAU.
The thickness of matter surrounding the node of shock in the direction considered is specified by
operand DIST_1 of command DEFI_MATERIAU. In the treated case, one chooses DIST_1 = 0.4495
and JEU = 0.45 so that at the initial moment, the node of shock and the obstacle are separated from the play J = 5. 104
mm (cf [Figure 5.1-a]).
The seismic loading, due to imposed displacements of node NO1, is calculated by the operator
CALC_CHAR_SEISME. One creates then a concept charges starting from operand VECT_ASSE of
order AFFE_CHAR_MECA.
One uses the diagram of integration of NEWMARK of DYNA_NON_LINE with a step of time of 103 S
and default settings.
Modeling of problem 2:
Y
dist_1
dist_2
J
K
K
X
m
m
NO1
NO2
NO3
NO4
1
2= - 1
Appear 5.1-b: Modélisation of two systems mass-arises which are entrechoquent
Handbook of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-66/03/008/A
Code_Aster ®
Version
7.2
Titrate:
SDND102 - Seismic Réponse of a multimedia system
Date
:
08/10/03
Author (S):
G. DEVESA, Fe WAECKEL, E. BOYERE
Key: V5.01.102-C Page:
8/14
Node NO1 is subjected to an imposed acceleration 1 (T), node NO4 to 2 (T) = 1 (T). One calculates
displacements relative and absolutes of nodes NO2 and NO3, their displacement of drive and them
absolute displacement.
The possible shocks between the two beams are taken into account as being internal forces with one
element with two nodes. One assigns to this element a nonlinear behavior of type shock (stiffness)
via key word RIGI_NOR of the law of behavior DIS_CONTACT of command DEFI_MATERIAU.
The normal direction of contact is the local axis X of the discrete element with two nodes.
The thicknesses of matter surrounding the nodes of shock in the direction considered are specified
by operands DIST_1 and DIST_2 of command DEFI_MATERIAU. In the treated case, one
DIST_1 chooses = DIST_2 = 0.4495 so that at the initial moment, the two nodes of shock are separate
play J = 2. J = 103 m (cf [Figure 5.1-a]).
The seismic loading, due to imposed displacements of anchorings (node NO1 and NO4, is
calculated by operator CALC_CHAR_SEISME. One creates a concept charges starting from the operand
VECT_ASSE of command AFFE_CHAR_MECA.
Temporal integration is carried out with the algorithm of Newmark and a step of time of 103 S. Les
calculations are filed all the 8 steps of time.
One considers a reduced damping of 7% for the whole of the calculated modes (key word
AMOR_MODAL of operator DYNA_NON_LINE).
5.2
Characteristics of the grid
The grid associated with the model bichoc consists of 4 nodes and 3 meshs of the type DIS_T.
5.3 Functionalities
tested
Commands
AFFE_MODELE GROUP_MA
“MECANIQUE”
“DIS_T'
DISCRETE AFFE_CARA_ELEM
GROUP_MA M_T_D_N
GROUP_MA
K_T_D_L
DEFI_MATERIAU DIS_CONTACT
DIST_1
DIST_2
JEU
AFFE_CHAR_MECA DDL_IMPO GROUP_NO
VECT_ASSE
MODE_STATIQUE DDL_IMPO
CALC_CHAR_SEISME MULTI_APPUI
DYNA_NON_LINE AMOR_MODAL
MODE_STAT
EXCIT
MULT_APPUI
“OUI”
COMP_INCR
DIS_CHOC
RECU_FONCTION SIEF_ELGA
DEPL
DEPL_ABSOLU
CALC_FONCTION MAX
COMB
Handbook of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-66/03/008/A
Code_Aster ®
Version
7.2
Titrate:
SDND102 - Seismic Réponse of a multimedia system
Date
:
08/10/03
Author (S):
G. DEVESA, Fe WAECKEL, E. BOYERE
Key: V5.01.102-C Page:
9/14
6
Results of modeling B
6.1
Values tested of modeling B
One calculates the definite function diff as being the difference between absolute displacement of the node
NO2 and that of the node no2. And it is checked that it is quite null for various moments.
Time (S)
Reference
Aster
Absolute error
0,1 0,0
1,7144E-17
1,71E-17
0,2 0,0
5,1386E-16
5,14E-16
0,3 0,0
5,1365E-16
5,14E-16
0,4 0,0
2,1570E-15
2,16E-15
0,5 0,0
2,7105E-19
2,71E-19
One also tests the maximum value of the force of impact to node NO2.
Type of impact
Reference
Aster
Relative error
against a rigid wall
6,29287E+02
6,29292E+02
7,21E-06
between two mobile structures 6,29287E+02 6,29292E+02 7,21E-06
Handbook of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-66/03/008/A
Code_Aster ®
Version
7.2
Titrate:
SDND102 - Seismic Réponse of a multimedia system
Date
:
08/10/03
Author (S):
G. DEVESA, Fe WAECKEL, E. BOYERE
Key: V5.01.102-C Page:
10/14
7 Modeling
C
7.1
Characteristics of modeling
Modeling C is before a whole test of DYNA_TRAN_EXPLI, whose results are compared
with DYNA_NON_LINE.
The systems mass-arises are modelled, as in modeling A, by a discrete element with
3 degrees of freedom DIS_T. Only modeling with a degree of freedom is tested.
Modeling of the problem:
Y
Play
dist_1
K
m
NO1
NO2
1
elm1
Appear 7.1-a: Modélisation of a system mass-arises impacting a rigid wall
Node NO1 is subjected to an imposed acceleration 1 (T). One calculates the relative displacement of the node
NO2, its displacement of drive and its absolute displacement.
An element of the type DIST_T on a mesh POI1 is retained to simulate the impact of the beam on one
rigid wall: the possible shocks between the beam and the obstacle are taken into account as being
forces intern with this element. One affects to him a nonlinear behavior of type shock (stiffness) via
law of behavior DIS_CONTACT of command DEFI_MATERIAU.
The thickness of matter surrounding the node of shock in the direction considered is specified by
operand DIST_1 of command DEFI_MATERIAU. In the treated case, one chooses DIST_1 = 0.4495
and JEU = 0.45 so that at the initial moment, the node of shock and the obstacle are separated from the play J = 5. 104
mm (cf [Figure 5.1-a]).
The seismic loading, due to imposed displacements of node NO1, is calculated by the operator
CALC_CHAR_SEISME. One creates then a concept charges starting from operand VECT_ASSE of
order AFFE_CHAR_MECA.
One uses the diagram of integration of explicit NEWMARK of type DIFFERENCES CENTREES with
a step of time of 103 S. calculation by DYNA_TRAN_EXPLI is carried out in modal space,
non-linearity being due to the shock and thus resident local.
7.2
Characteristics of the grid
The grid associated with the model consists of 2 nodes, of a mesh SEG2 of the type DIS_T and one
specific mesh POI1 of the type DIS_T.
Handbook of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-66/03/008/A
Code_Aster ®
Version
7.2
Titrate:
SDND102 - Seismic Réponse of a multimedia system
Date
:
08/10/03
Author (S):
G. DEVESA, Fe WAECKEL, E. BOYERE
Key: V5.01.102-C Page:
11/14
7.3 Functionalities
tested
Commands
AFFE_MODELE GROUP_MA
“MECANIQUE”
“DIS_T'
DISCRETE AFFE_CARA_ELEM
GROUP_MA M_T_D_N
GROUP_MA
K_T_D_L
DEFI_MATERIAU DIS_CONTACT
DIST_1
DIST_2
JEU
AFFE_CHAR_MECA DDL_IMPO GROUP_NO
VECT_ASSE
MODE_STATIQUE DDL_IMPO
CALC_CHAR_SEISME MONO_APPUI
“OUI”
DYNA_NON_LINE AMOR_MODAL
MODE_STAT
COMP_INCR
DIS_CHOC
DYNA_TRAN_EXPLI AMOR_MODAL
PROJ_MODAL
COMP_INCR
DIS_CHOC
RECU_FONCTION
Handbook of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-66/03/008/A
Code_Aster ®
Version
7.2
Titrate:
SDND102 - Seismic Réponse of a multimedia system
Date
:
08/10/03
Author (S):
G. DEVESA, Fe WAECKEL, E. BOYERE
Key: V5.01.102-C Page:
12/14
8
Results of modeling C
8.1
Values tested of modeling C
Calculation is non-linear because of the shock and one does not have analytical solution. One thus tests
calculation on values of not-regression on displacement according to X of node NO2.
Time (S)
Reference
Aster
Relative error
0,1 15,6520E-3
15,6520E-3 <1E-3%
0,2 51,4832E-3
51,4832E-3 <1E-3%
0,3 28,1291E-3
28,1291E-3 <1E-3%
0,4 44,9343E-3
44,9343E-3 <1E-3%
0,5 37,7508E-3
37,7508E-3 <1E-3%
One compares absolute displacements resulting from DYNA_TRAN_EXPLI with those given by
DYNA_NON_LINE.
Handbook of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-66/03/008/A
Code_Aster ®
Version
7.2
Titrate:
SDND102 - Seismic Réponse of a multimedia system
Date
:
08/10/03
Author (S):
G. DEVESA, Fe WAECKEL, E. BOYERE
Key: V5.01.102-C Page:
13/14
9
Summary of the results
The results obtained with Code_Aster are in conformity with those awaited (error lower than
thousandths).
On this example, direct nonlinear calculation is much more expensive in calculating times, of one
factor 20, that that on modal basis.
Modeling C shows that one obtains many similar results with a method
of explicit temporal integration (DYNA_TRAN_EXPLI) and implicit (DYNA_NON_LINE).
Handbook of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
HT-66/03/008/A
Code_Aster ®
Version
7.2
Titrate:
SDND102 - Seismic Réponse of a multimedia system
Date
:
08/10/03
Author (S):
G. DEVESA, Fe WAECKEL, E. BOYERE
Key: V5.01.102-C Page:
14/14
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Handbook of Validation
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