SDND102 - Seismic response of a system mass-arises nonlinear multimedia

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Titrate:

SDND102 - Seismic response of a multimedia system

Date

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Author (S):

G. DEVESA, Fe WAECKEL, E. BOYERE

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V5.01.102-C

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Organization (S):

EDF-R & D/AMA

Manual of Validation
V5.01 booklet: Nonlinear dynamics of the discrete systems
V5.01.102 document

SDND102 - Seismic response of a system
mass-arises nonlinear multimedia

Summary

The problem consists in analyzing the response of a mechanical structure, modelized 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.

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Titrate:

SDND102 - Seismic response of a multimedia system

Date

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Author (S):

G. DEVESA, Fe WAECKEL, E. BOYERE

Key:

V5.01.102-C

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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).

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 K

shock

= 5,76 10

NR/Mr.

As for problem 2 (shock of two deformable structures), it is worth K

shock

= 2,88 10

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:

(T) = sin

with

= 20.

and the point C with an acceleration

(T) = - sin

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. 10

Mr. As for problem 2,

the masses are separated from the play J = 2 J = 10

Mr.

With

2= -

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Titrate:

SDND102 - Seismic response of a multimedia system

Date

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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 F

, standardized associated clean vectors

compared to the modal mass

and static modes

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.

The function is calculated

diff

defined 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.

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Titrate:

SDND102 - Seismic response of a multimedia system

Date

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G. DEVESA, Fe WAECKEL, E. BOYERE

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3 Modeling

With

3.1

Characteristics of modeling

The systems mass-arises are modelized by discrete elements with 3 degrees of freedom

DIS_T

Modeling of problem 1:

Appear 3.1-a: Modeling of a system mass-arises impacting a rigid wall

The node no1 is subjected to an imposed acceleration

(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 the aforementioned

very far (cf [Figure 3.1-a]).
From where:

the origin of the obstacle

ORIG_OBST

: ( 1. 0. 0.) ;

and corresponding play

play

1.1005

Modeling of problem 2:

Appear 3.1-b: Modeling of two systems mass-arises which are entrechoquent

Node NO1 is subjected to an imposed acceleration

(T), node NO4 with

(T) = 

(T). One calculates

the relative displacement of nodes NO2 and NO3, their displacement of drive and their displacement
absolute.

2= - 1

dist_1

dist_2

NO2

NO1

NO3

NO4

Yloc

Zloc

orig_

no2

no1

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Titrate:

SDND102 - Seismic response of a multimedia system

Date

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Key:

V5.01.102-C

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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 the 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 = 10

mm (cf [Figure 3.1-

B]).

Temporal integration is carried out with the algorithm of Euler and a pitch of time of 2,5. 10

calculations are filed all the 8 pitches of time.
A reduced damping is considered

from 7% for the whole of the calculated modes.

3.2

Characteristics of the mesh

One calls

model

the mesh associated with the problem made up of a system mass-arises butting

against a fixed wall and

bichoc

that which is associated problem 2.

Mesh associated with the model

model:

a number of nodes: 2;
a number of meshs and types: 1

DIS_T.

Mesh associated with the model

bichoc:

a number of nodes: 4;
a number of meshs and types: 2

DIS_T.

3.3 Functionalities

tested

Controls

AFFE_MODELE GROUP_MA

“MECHANICAL”

“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

BANDAGE

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

METHOD

EULER

REST_BASE_PHYS MULT_APPUI

“YES”

RECU_FONCTION RESU_GENE

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Titrate:

SDND102 - Seismic response of a multimedia system

Date

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Key:

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Results of modeling A

4.1

Values tested of modeling A

The function is calculated

diff

defined 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

(problem 2)

Aster

Absolute error

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

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Titrate:

SDND102 - Seismic response of a multimedia system

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HT-66/03/008/A

5 Modeling

5.1

Characteristics of modeling

The systems mass-arises are modelized, as in modeling A, by a discrete element with
3 degrees of freedom

DIS_T

Modeling of problem 1:

Appear 5.1-a: Modeling of a system mass-arises impacting a rigid wall

Node NO1 is subjected to an imposed acceleration

(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

control

DEFI_MATERIAU

The thickness of matter surrounding the node of shock in the direction considered is specified by
the operand

DIST_1

control

DEFI_MATERIAU

. In the treated case, one chooses

DIST_1

= 0.4495

and

PLAY

= 0.45 so that at the initial moment, the node of shock and the obstacle are separated from the play J = 5. 10

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

charge

starting from the operand

VECT_ASSE

order

AFFE_CHAR_MECA

One uses the diagram of integration of NEWMARK of

DYNA_NON_LINE

with a pitch of time of 10

and default settings.

Modeling of problem 2:

Appear 5.1-b: Modeling of two systems mass-arises which are entrechoquent

Play

dist_1

NO2

NO1

elm1

2= - 1

dist_1

dist_2

NO2

NO1

NO3

NO4

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Node NO1 is subjected to an imposed acceleration

(T), node NO4 with

(T) = 

(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 the key word

RIGI_NOR

law of behavior

DIS_CONTACT

control

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 the operands

DIST_1

and

DIST_2

control

DEFI_MATERIAU

. In the treated case, one

chooses

DIST_1

DIST_2

= 0.4495 so that at the initial moment, the two nodes of shock are separate

play J = 2. J = 10

m (cf [Figure 5.1-a]).

The seismic loading, due to imposed displacements of anchorings (node NO1 and NO4, is
calculated by the operator

CALC_CHAR_SEISME

. A concept is created

charge

starting from the operand

VECT_ASSE

control

AFFE_CHAR_MECA

Temporal integration is carried out with the algorithm of Newmark and a pitch of time of 10

calculations are filed all the 8 pitches of time.
A reduced damping is considered

from 7% for the whole of the calculated modes (key word

AMOR_MODAL

of the operator

DYNA_NON_LINE

5.2

Characteristics of the mesh

Mesh associated with the model

bichoc

consists of 4 nodes and 3 meshs of the type

DIS_T

5.3 Functionalities

tested

Controls

AFFE_MODELE GROUP_MA

“MECHANICAL”

“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

PLAY

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

“YES”

COMP_INCR

DIS_CHOC

RECU_FONCTION SIEF_ELGA

DEPL

DEPL_ABSOLU

CALC_FONCTION MAX

COMB

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Results of modeling B

6.1

Values tested of modeling B

The function is calculated

diff

defined 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

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7 Modeling

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 modelized, 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:

Play

dist_1

NO2

NO1

elm1

Appear 7.1-a: Modeling of a system mass-arises impacting a rigid wall

Node NO1 is subjected to an imposed acceleration

(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

DIS_CONTACT

control

DEFI_MATERIAU

The thickness of matter surrounding the node of shock in the direction considered is specified by
the operand

DIST_1

control

DEFI_MATERIAU

. In the treated case, one chooses

DIST_1

= 0.4495

and

PLAY

= 0.45 so that at the initial moment, the node of shock and the obstacle are separated from the play J = 5. 10

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

charge

starting from the operand

VECT_ASSE

order

AFFE_CHAR_MECA

One uses the diagram of integration of explicit NEWMARK of type DIFFERENCES CENTREES with
a pitch of time of 10

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 mesh

The mesh associated with the model consists of 2 nodes, of a mesh

SEG2

of type

DIS_T

and of one

specific mesh

POI1

of type

DIS_T

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Titrate:

SDND102 - Seismic response of a multimedia system

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7.3 Functionalities

tested

Controls

AFFE_MODELE GROUP_MA

“MECHANICAL”

“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

PLAY

AFFE_CHAR_MECA DDL_IMPO GROUP_NO
VECT_ASSE

MODE_STATIQUE DDL_IMPO

CALC_CHAR_SEISME MONO_APPUI

“YES”

DYNA_NON_LINE AMOR_MODAL

MODE_STAT

COMP_INCR

DIS_CHOC

DYNA_TRAN_EXPLI AMOR_MODAL

PROJ_MODAL

COMP_INCR

DIS_CHOC

RECU_FONCTION

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SDND102 - Seismic response of a multimedia system

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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

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SDND102 - Seismic response of a multimedia system

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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 temporal integration clarifies (

DYNA_TRAN_EXPLI

) and implicit (

DYNA_NON_LINE