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Code_Aster
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Version
4.0
Titrate:
Stochastic approach for the seismic analysis
Date:
08/02/99
Author (S):
A. DUMOND
Key:
R4.05.02-B
Page:
1/18
Manual of Reference
R4.05 booklet: Seismic analysis
HP-52/96/009/B
Organization (S):
EDF/EP/AMV
Manual of Reference
R4.05 booklet: Seismic analysis
Document: R4.05.02
Stochastic approach for the seismic analysis
Summary:
This document presents a method of calculation probabilistic to determine the response of a subjected structure
with a random excitation of seismic type starting from the interspectres of the excitation at the points of support of
structure. The answer itself is expressed in the form of interspectres.
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Code_Aster
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Version
4.0
Titrate:
Stochastic approach for the seismic analysis
Date:
08/02/99
Author (S):
A. DUMOND
Key:
R4.05.02-B
Page:
2/18
Manual of Reference
R4.05 booklet: Seismic analysis
HP-52/96/009/B
Contents
1 Introduction ............................................................................................................................................ 3
2 Principle of the step ........................................................................................................................ 4
2.1 Position of the problem considered and general principle ....................................................................... 4
2.2 Decomposition of the movement ........................................................................................................ 4
2.3 Decomposition on the modal basis ................................................................................................. 5
2.4 Harmonic answer ...................................................................................................................... 6
3 the random dynamic response ............................................................................................................ 7
3.1 Recall on the spectral concentrations of power [bib2] .................................................................. 7
3.1.1 ............................................................................................................................... Definitions 7
3.1.2 Relations between the DSP and the other characteristics of the signal ............................................. 8
3.2 Equations of motion ......................................................................................................... 8
3.2.1 Stamp “interspectrale-excitation” ......................................................................................... 8
3.2.2 Random dynamic response ................................................................................................ 9
3.3 Application in Code_Aster .................................................................................................... 10
4 Definition of the matrix interspectrale of exiting power .......................................................... 12
4.1 Reading on a file ..................................................................................................................... 12
4.2 Obtaining a interspectre starting from functions of time ........................................................... 13
4.3 Excitations preset or reconstituted starting from existing complex functions ................... 14
4.3.1 Existing complex functions .......................................................................................... 14
4.3.2 White Gaussian noise ............................................................................................................................ 14
4.3.3 White Gaussian noise filtered by KANAI-TAJIMI [bib9] ............................................................................ 14
4.4 Other types of excitation ................................................................................................................ 15
4.4.1 Case of the excitation in imposed forces ................................................................................ 15
4.4.2 Excitation by fluid sources ........................................................................................ 16
4.4.3 Excitation distributed on a function of form ....................................................................... 17
4.5 Applications ................................................................................................................................... 17
5 Bibliography ........................................................................................................................................ 18
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Code_Aster
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Version
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Titrate:
Stochastic approach for the seismic analysis
Date:
08/02/99
Author (S):
A. DUMOND
Key:
R4.05.02-B
Page:
3/18
Manual of Reference
R4.05 booklet: Seismic analysis
HP-52/96/009/B
1 Introduction
Classically the response of a structure subjected to a seismic excitation can be calculated by
two approaches:
·
transitory calculation of dynamics if the excitation is defined by a accélérogramme
(cf [R4.05.01]).
·
calculation by the conventional spectral method if the excitation is defined by a spectrum of answer
of oscillator (SRO) (cf [R4.05.03]).
However a seismic excitation is by random nature. These two methods are not envisaged
initially to hold account of it: in a case it is necessary to reiterate for various excitations the many ones
temporal calculations then to make a statistical average of it (important cost calculation), in the other case one
carry out very conservative assumptions by considering averages (of quadratic type
simple or supplements for example) for the maximum of the answers.
Also it was developed a method of calculation of the probabilistic type, also called “approach
stochastic of the seismic calculation ", based on the calculation of the dynamic response expressed in
interspectres of power starting from the spectral concentrations of power of the excitation. This method
have in particular the advantage of better taking into account the correlations between the excitations to
various supports of the structure.
The discussion of the various advantages of this method can be thorough in the reference
[bib1].
We thus present the principle of the method and the notations retained starting from the steps
conventional, then in third part probabilistic calculation itself.
Finally in fourth part the various methods will be presented to obtain the interspectre
discharger.
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Code_Aster
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Titrate:
Stochastic approach for the seismic analysis
Date:
08/02/99
Author (S):
A. DUMOND
Key:
R4.05.02-B
Page:
4/18
Manual of Reference
R4.05 booklet: Seismic analysis
HP-52/96/009/B
2
Principle of the step
2.1
Position of the problem considered and general principle
A multi-supported structure is in the case of placed, i.e. the structure has m ddl-
supports, each one being subjected to its own excitation (not necessarily equal everywhere). One supposes
that the structure is represented by a model finite elements comprising N ddl. The answer is sought
in a number finished (and low) of L ddl.
It is supposed that the size excitation is of imposed movement type and results in a family
of accélérogrammes
G T
J
()
for each one of the ddl-supports J, j=1, Mr.
The absolute movement of the structure is broken up classically moving of drive
and relative movement.
The calculation of the response in interspectres of power is carried out by modal recombination.
Following this modal calculation, a calculation of dynamic response random breaks up into three
parts:
·
definition of the interspectre of power discharger,
·
calculation of the interspectre of power answer.
These the first two parts are the subject of the control
DYNA_ALEA_MODAL
[U4.56.06].
The restitution of the interspectre of power response on physical basis is carried out with the control
REST_SPEC_PHYS
[U4.80.01].
·
calculation of statistical parameters starting from the interspectre of power result.
This last stage is treated by the control
POST_DYNA_ALEA
[R7.10.01] [U4.76.02].
2.2
Decomposition of the movement
The following decompositions and projections are detailed in the reference material
relating to the resolution by transitory calculation of a seismic calculation [R4.05.01]. We retain any here only
the broad outline.
That is to say
X
has
the vector absolute displacement (of dimension N) of all ddl of the structure.
The total answer known as absolute
X
has
structure is expressed as the sum of a contribution
relative
X
R
and of the contribution of drive
X
E
had with displacements of anchoring (subjected to
accelerations represented by a accélérogramme
()
G T
J
in each the ddl-supports J, j=1, m).
X
X
X
has
R
E
T
T
T
()
()
()
=
+
Are
MR. K
C
,
and
matrices of mass, rigidity and damping of the problem, limited to
ddl not supported.
The equation of the movement is written then in the reference mark related to the relative movement:
MR. X
C X
K X
MR. X
F
!! ()
! ()
()
!! ()
R
R
R
E
ext.
ext.
T
T
T
T
F
+
+
= -
+
vector of the external forces
In general the external forces are null during a calculation of seismic answers.
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Titrate:
Stochastic approach for the seismic analysis
Date:
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Author (S):
A. DUMOND
Key:
R4.05.02-B
Page:
5/18
Manual of Reference
R4.05 booklet: Seismic analysis
HP-52/96/009/B
2.3
Decomposition on the modal basis
The calculation of response in interspectres of power is carried out by modal recombination and is made
call, moving imposed, at a modal base which includes/understands at the same time dynamic modes and
static modes.
That is to say
{}
=
=
I I
N
,
,
1
matrix (N, N) of the dynamic modes calculated for the conservative system
associated, by maintaining the m locked supports.
That is to say
{
}
=
=
J J
m
,
,
1
the matrix (N, m) of the static modes. Mode
J
corresponds to the deformation of
the structure under a unit displacement imposed on the ddl-support J, other ddl-supports being locked.
The imposed displacement of anchorings
()
X
S
T
is connected to
()
X
E
T
by the relation:
()
()
X
X
E
S
T
T
=
.
Components of the acceleration of the points of anchoring
()
!!X
S
T
are the accélérogrammes
()
G T
J
,
j=1, Mr.
One can thus write
!! ()
!!
()
X
X
E
S
J J
J
m
T
T
G T
=
() =
=1
The change of variable is carried out
() ()
X
Q
Q
R
T
T
T
()
,
=
is the vector of the co-ordinates
generalized. By prémultipliant the equation of the movement by
T
,
one obtains - in the absence of forces
external others that the seismic excitation - the equation projected on the basis of dynamic mode:
T
T
T
T
S
T
T
T
T
M Q
C Q
K Q
MR. X
!!()
!()
()
!! ()
+
+
= -
It is supposed that the matrix of damping is a linear combination of the matrices of mass and of
rigidity (assumption of damping of constant Rayleigh on the structure or assumption of Basile
allowing a diagonal damping). The base
,
who orthogonalise matrices
M
K
and
,
orthogonalise thus also the matrix
C
.
Taking into account this assumption, the preceding equation breaks up into N equations scalar
uncoupled in the form:
!!
!
()
Q
Q
Q
p
I
I I I
I
I
ij J
+
+
= -
=
2
2
1
G T
J
m
for i=1, N
Where one noted:
µ
µ
µ
µ
I
I
I
I
I
I
I
I
I
I
I
I
I I
ij
I
J
I
p
modal mass
modal rigidity
the modal pulsation
reduced modal damping
the factor of modal participation of
the support J on the dynamic mode I
=
=
=
=
=
T
T
T
T
K
K












M
K
C
M
2
The solution
()
Q
I
T
of this equation corresponds to the response of the dynamic mode
I
with the whole of
the seismic excitation.
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Stochastic approach for the seismic analysis
Date:
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Author (S):
A. DUMOND
Key:
R4.05.02-B
Page:
6/18
Manual of Reference
R4.05 booklet: Seismic analysis
HP-52/96/009/B
One can still break up the problem by introducing the unknown factor
()
D
ij
T
solution of the equation
differential:
!!
!
()
D
D
D
ij
I I ij
I
ij
J
G T
+
+
=
2
2
, this last equation corresponds to the answer of
dynamic mode
I
with acceleration
G T
J
()
. Relative displacement on the physical basis is expressed
then:
X
p D
R
ij ij
I
J
m
I
N
T
T
()
()
= -
=
=



1
1
Information on the position of the point of support is contained in the factor of modal participation.
2.4 Answer
harmonic
One thus broke up the total response of the structure into a relative contribution and a contribution
differential due to displacements of anchorings such as:
X
X
X
X
X
X
p D
D
D
D
D
has
R
E
E
S
J J
j=
m
R
ij ij
I
J
m
I
N
ij
ij
I I ij
I
ij
J
T
T
T
T
T
G T
T
T
T
G T
()
()
()
!! ()
!!
()
()
()
()
!!
!
()
1
=
+
=
= -
+
+
=


=
=
with
where
is solution of
() =



1
1
2
2
The solution of this last differential equation by the method of the transformation of made Fourier
to intervene modal transfer functions
H
I
()
such as:
H
I
I
I
I I
()
(
)
=
-
+
1
2
2
2
.
One thus obtains:
D
D
ij
I
J
ij
I
J
H
G
H
G
()
().
()
!! ()
().
()
=
= -
and
2
The total harmonic response of the structure results from the preceding formulas by recombination
modal.
!! ()!! ()!! ()
!! ()
() ()
()
X
X
X
X
p H
has
R
E
has
ij J
J
I
J
m
I
N
J J
J
m
G
G
=
+
=
+
=
=
=1
2
1
1



One then reveals the complex matrix (N, m), known as matrix of transfer
H ()
following:
H
p H
()
=
2
() +
where
p
is the matrix of the factors of participation,
H ()
the vector of the modal transfer functions
H
I
()
.
The total response of the structure is worth
()
() ()
()
!!
!!
,
!!
X
H
E
E
has
=
where
is the vector of m lines
constituted of the transforms of Fourier of accelerations
G T
J
()
with the m ddl-supports.
It is seen that this expression determines the response in acceleration. This then forces to integrate
twice the answer to obtain displacement, this problem is presented in [bib4]. One of
additional interests of the method which we propose here is to abstract itself from this difficulty.
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Stochastic approach for the seismic analysis
Date:
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A. DUMOND
Key:
R4.05.02-B
Page:
7/18
Manual of Reference
R4.05 booklet: Seismic analysis
HP-52/96/009/B
3
The random dynamic response
3.1
Recall on the spectral concentrations of power [bib2]
3.1.1 Definitions
That is to say a probabilistic signal defined by its density of probability p
X
(X
1
, T
1
;…;…; X
N
, T
N
). This density of
probability makes it possible to calculate the functions moments of the signal.
Moment of command 1 or hope of the signal:
[]
µ
X
X
X
() E ()
(,)
T
T
p X T dx
X
=
=
-
+
Moments of command 2 or intercorrelation of two signals:
[
]
XY
X
Y
X y
(,) E () ()
(; ,)
T T
T
T
p X T y T dxdy
1 2
1
2
1
2
=
=
-
+
When the signal is stationary, the intercorrelation depends only on
= -
T
T
2
1
.
It is written
[
]
R
T
T
XY
X
Y
() E () (
)
=
-
Spectral concentration of power and interspectre
One defines
S
XY
()
the interspectre of power or density interspectrale of power between two
stationary probabilistic signals by the transform of Fourier of the function of intercorrelation,
what one writes:
S
R
XY
XY
()
() E
=
-
-
+
1
2
I
D
The opposite formula is written:
R
S
XY
XY
()
() E
=
-
+
I
D
S
XY
()
is generally complex and checks the relation of symmetry:
S
S
YX
XY
()
()
=
.
When
X Y
=
,
S
XX
()
be called autospectre power or density
spectral of
power (DSP). This function with the real and always positive property to be.
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Titrate:
Stochastic approach for the seismic analysis
Date:
08/02/99
Author (S):
A. DUMOND
Key:
R4.05.02-B
Page:
8/18
Manual of Reference
R4.05 booklet: Seismic analysis
HP-52/96/009/B
3.1.2 Relations between the DSP and the other characteristics of the signal
Note:
Most of the time, the signal is defined over a limited time, its transform of Fourier does not exist
not, one defines a transform of Fourier then estimated over one period length T by:
“()
()
/
/
X
X T E
D
T
I
T
T
=
-
-
1
2
2
2
.
One then has the following relationships to this estimated transform of Fourier:
[
]
[
]
S
X
Y
S
X
X
XY
XX
()
lim
E
“()” ()
()
lim
E
“()” ()
=
=
+
+
T
T
T
T
T
T
T
T
2
2
Link between the autospectre of power and the power of the signal:
The power of a signal is equal to its variance. For a centered signal, the variance is worth:
X
XX
2
0
=
R
()
.
One thus has:
X
XX
XX
2
0
=
=
-
+
R
S
D
()
()
.
3.2
Equations of motion
The total response of the structure is determined by the relation:
()
() ()
!!
!!
X
H
E
has
=
,
where
()
!!E
is the vector of m lines made up of the excitations represented by the transforms of
Fourier of the accélérogrammes
()
G T
J
with the m ddl-supports,
()
H
is the matrix of transfer defined by
()
()
H
p H
=
+
2
where
p
is the matrix of the factors of participation,
()
H
the vector of the modal transfer functions
()
H
I
base dynamic modes
base static modes
it comprises N lines (= a number of ddl free of the structures) and m columns.
3.2.1 Stamp
“interspectrale-excitation”
NB:
This name “stamps interspectrale-excitation” is abusive: it means “matrix of
density interspectrale of power of the excitation ".
It is supposed that the seismic excitation can be regarded as a stationary signal - taking into account
relationship between times representative - and centered. This makes it possible to use a certain number of
result of the probabilistic analysis. One is interested then in the stationary response of the system to one
stationary excitation.
One notes
()
S
EE
!! !!
the matrix of the interspectres of power corresponding to the excitation. Its data
is clarified in chapter 4.
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Stochastic approach for the seismic analysis
Date:
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A. DUMOND
Key:
R4.05.02-B
Page:
9/18
Manual of Reference
R4.05 booklet: Seismic analysis
HP-52/96/009/B
For memory we recall here that it is calculated starting from transforms of Fourier of
accelerations. It is a matrix (mxm). The ij term corresponds to the interspectre between the signals
!!E
I
and
!!E
J
that is to say still between the transforms of Fourier of the accélérogrammes
G
G
I
J
and
.
3.2.2 Random dynamic response
It was seen that the interspectre power between two probabilistic signals is the transform of Fourier
function of intercorrelation of the two signals. One applies it to the total response of the structure:
S
R
E
D
T
T
E
D
has has
has has
I
has
T
has
I
!! !!
!! !!
()
()
E
!! ()!! (
)
X X
X X
X
X
=
=
-




-
-
+
-
-
+
1
2
1
2
One works then in the temporal field to express the function of intercorrelation of the answer
total
R
T T
has has
!! !!
(, ')
X X
.
One notes
()
H T
the impulse response of the system:
H
H
()
[()]
T
=
-
TF
1
and
!!()
E T
the transform of Fourier reverses exiting DSP:
!!()
[
!!()]
E
E
T
=
-
TF
1
By transform of Fourier relation reverses:
!! ()
()
!!()
X
H
E
has
=
one has
!! ()
*!! ()
()!!(
)
X
H E
H
E
has
R
T
T
U
T U of
=
=
-
R
T T
T
T
R
T T
U
T U of
v
T v FD
R
T T
U
T U
T v
v FD
has has
has has
has has
has
T
has
R
T
R
R R
T
T
!! !!
!! !!
!! !!
(, ') E
!! ()!! (')
(, ') E
()!!(
)
()!!(')
(, ') E
()!!(
)!!(') ()
X X
X X
X X
X
X
H
E
H
E
H
E
E
H
=


=
-
-




=
-
-




One supposes in this analysis the deterministic system, one can thus leave the impulse response
calculation of the expectation. It comes
R
T T
U
T U
T v
v FD of
has has
R R
T
T
!! !!
(, ')
() E!!(
)!!(')] ()
X X
H
E
E
H
=
-
-
[
The excitation is supposed a stationary process, the intercorrelation thus depends only on the variation on
time
= -
T you
:
R
T T U v
T U
T v
R
T T U v
U v
T
!! !!
!! !!
(
'
) E!!(
)!!(')]
()
'
EE
EE
E
E
- - + =
-
-
=
= - - + = - +
[
for
from where
R
T T
U R
v FD of the R
has has
has has
R
T
R
!! !!
!! !!
!! !!
(, ')
()
()
()
()
X X
EE
X X
H
H
=
=
what justifies a posteriori the approach.
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Code_Aster
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Version
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Titrate:
Stochastic approach for the seismic analysis
Date:
08/02/99
Author (S):
A. DUMOND
Key:
R4.05.02-B
Page:
10/18
Manual of Reference
R4.05 booklet: Seismic analysis
HP-52/96/009/B
One now defers this expression in the expression of the spectral concentration of power of
answer:
S
R
D
U R
U v
v
FD of D
has has
has has
I
R R
T
I
!! !!
!! !!
!! !!
()
() E
()
(
)
() E
X X
X X
EE
H
H
=
=
- +
-
-
+
-
-
+
1
2
1
2
By distributing the dummy variables of integration one reveals the transforms of Fourier
respective of
()
(
)
()
H U R
U v
v
T
,
,
!! !!
EE
H
- +
, it comes finally:
S
H
S
H
X X
EE
!! !!
!! !!
()
()
()
has has
T
=
with
H
p H
()
=
2
() +
Taking into account the relations between the transforms of Fourier of displacement, the speed and of
acceleration, one has moreover:
S
H
S
H
S
H
S
H
X X
EE
X X
EE
! !
!! !!
!! !!
()
()
()
()
()
()
has has
has has
T
T
= -
=
1
1
2
4
These relations make it possible to express the response of the structure by the DSP of displacement or of
speed.
Note:
·
According to the expression given to
H ()
, one respectively expresses the DSP of the displacement (of
the speed or of acceleration) total, relative or differential:
absolute movement:
H
p H
()
=
2
() +
relative movement:
H
p H
()
=
2
()
differential movement (IE of drive):
H ()
=
·
It is of use, during a calculation with Code_Aster, to restrict the matrix of the function of
transfer to the lines of the L ddl of observation. This makes it possible to reduce of as much calculations as soon as L
is small in front of N.
3.3
Application in Code_Aster
The whole of the spectral approach for seismic calculation is treated in the control
DYNA_ALEA_MODAL
[U4.56.06]. The data are gathered under three key words factors and a word
single-ended spanner.
The modal base is consisted of the dynamic modes calculated by the control
MODE_ITER_SIMULT
[U4.52.02] or
MODE_ITER_INV
[U4.52.01] stored in a concept of the type
mode_meca
recovered by the key word factor
BASE_MODALE
, on the one hand; calculated static modes
by the control
MODE_STATIQUE
[U4.52.04] stored in a concept of the type
mode_stat
recovered
by the simple key word
MODE_STAT
, in addition. The key word factor
BASE_MODALE
also have
the arguments which make it possible to determine the frequency band or the modes retained for calculation
and corresponding depreciation.
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Code_Aster
®
Version
4.0
Titrate:
Stochastic approach for the seismic analysis
Date:
08/02/99
Author (S):
A. DUMOND
Key:
R4.05.02-B
Page:
11/18
Manual of Reference
R4.05 booklet: Seismic analysis
HP-52/96/009/B
The data corresponding to the excitation are gathered under the key word factor
EXCIT
(cf paragraph [§4]): one specifies there the type of excitation within the meaning of
SIZE
: excitation in
displacement or in effort, nodes
NODE
and components
NOM_CMP
excited, the name of
interspectres or autospectres
INTE_SPEC
, complex functions read beforehand or calculated,
respectively by the operators
LIRE_INTE_SPEC
[U4.56.01] or
CALC_INTE_SPEC
[U4.56.03] and
stored in a table of interspectre of concept
tabl_intsp
who apply in each ddl excited.
Under the key word factor
ANSWER
are the data related to the choice of the discretization.
The control
DYNA_ALEA_MODAL
provides the response in the form of spectral concentration of power
on modal basis. To obtain the restitution of the DSP on physical basis, one will use
REST_SPEC_PHYS
[U4.80.01] which makes it possible to specify the type of size of the answer (displacement or effort), with
“points of observation” (node-component) of the result. In the presence of a response of the type
displacement, one will specify here also if the answer corresponds to absolute displacement, relative or
differential.
REST_SPEC_PHYS
provides a table of interspectres which contains according to the request of the user,
stamp interspectrale in displacement
S
XX
, of speed
S
! !
XX
, or in acceleration
S
!! !!
XX
for one
expression in the absolute reference mark (index has), the relative reference mark (index R) or of drive (index E).
Each preceding “combination” requires a call specific to the control
REST_SPEC_PHYS
.
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Code_Aster
®
Version
4.0
Titrate:
Stochastic approach for the seismic analysis
Date:
08/02/99
Author (S):
A. DUMOND
Key:
R4.05.02-B
Page:
12/18
Manual of Reference
R4.05 booklet: Seismic analysis
HP-52/96/009/B
4 Definition of the matrix interspectrale of power
exciter
The seismic excitation is by nature, we said it, random. Also it can be known not by
its temporal expression but in frequential form by a spectral concentration of power known as
also interspectre.
When there are several supports, they can be excited by identical or different excitations, it
last case is that of the multi-supports.
For m supports, one defines the matrix of density interspectrale of power of command m, or per abuse
language the interspectre of command m, which is a matrix (m
X
m) of complex functions depending on
frequency.
The diagonal terms represent the “auto-” densities spectral of powers - or autospectres-
to the points of excitation, the extra-diagonal terms correspond to the densities interspectrales enters
the excitations in two points of support distinct (each line or column of the matrix represents in
fact a point of support in physical mesh or a mode in modal calculation). By definition of these terms, it
results from it that the matrices of density interspectrales of power handled are square.
(See [bib2] or reference material associated with the control
POST_DYNA_ALEA
[R7.10.01])
We present hereafter the various controls of Code_Aster which make it possible to obtain one
stamp density interspectrale of power.
4.1
Reading on a file
The most elementary way to define a matrix of density interspectrale of power is of
to give, “with the hand”, the values with the various pitches of frequency.
The operator then is used
LIRE_INTE_SPEC
[U4.56.01].
LIRE_INTE_SPEC
reads in a file “interspectre excitation”. The format of the file in which is
consigned the matrix interspectrale is simple: one describes successively the function of each term
matrix interspectrale; for each function, one gives a line by frequency while indicating
frequency, parts real and imaginary of the complex number; or the frequency, the module and the phase
complex number (key word
FORMAT
).
Example of file interspectre excitation (for a matrix reduced in the term):
INTERSPECTRE
DIM = 1
FONCTION_C
I = 1
J = 1
NB_POIN = 4
VALUE =
2.9999 0. 0.
3. 1. 0.
13. 1. 0.
13.0001 0. 0.
FINSF
END
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Code_Aster
®
Version
4.0
Titrate:
Stochastic approach for the seismic analysis
Date:
08/02/99
Author (S):
A. DUMOND
Key:
R4.05.02-B
Page:
13/18
Manual of Reference
R4.05 booklet: Seismic analysis
HP-52/96/009/B
4.2
Obtaining a interspectre starting from functions of time
One can deduce the matrix from density interspectrale of power starting from functions of time. One
use the operator then
CALC_INTE_SPEC
[U4.56.03] in Code_Aster [bib3].
Starting from a list of NR functions of time, this operator allows to calculate the interspectre
NxN power which corresponds to them.
For each term of the matrix interspectrale (NxN) one uses the following step [bib3].
To calculate the interspectre of two signals one uses the relation of Wiener-Khichnine [bib7] which allows
to establish a formula of computation of the spectral concentration of power by the transform of Fourier
finished samples of the signals
()
()
X
y
T
T
and
.
It comes then:
S
F
T
F T
F T
F T
F
T
dt
F T
F
T
dt
T
K
K
K
K
K
I
F
T
K
K
K
I
F
T
xy
X
Y
X
X
X
Y
y
y
X
y
()
lim
E [
(,).
* (,)]
(,) TF [
] ()
() E
(,) TF [
] ()
() E
.
=
=
=
=
=


-
-
1
2
0
2
0
where
are the discrete transforms of Fourier of and of
When one is interested in signals resulting from measurements, one has most of the time only
known signals in a discrete way, in the same way a transitory computation result is a discrete signal.
An approximation of the interspectre of the discrete signals X [N] and y [N] definite on L points spaced of
T,
cut out out of p blocks of Q points is obtained by the relation:
[]
[]
* []
[]
[]
[]
[]
()
()
()
()
/
()
()
/
S
X
Y
X
X
Y
y
xy
I
I
I
p
I
I
I kN Q
N
Q
I
I
I kN Q
N
Q
K
p Q T
K
K
K
T
N E
K
T
N E
=
=
=
=
-
=
-
=
1
1
2
0
2
0
The various blocks can or not overlap. Values
p
Q
and
are with the choice of the user.
This method is that of the periodogram of WELCH [bib8].
Calculation is done on a window which moves on the field of definition of the functions. The user
specify in the control the length of the window of analysis, the shift between two windows of
calculation successive and the number of points per window.
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Code_Aster
®
Version
4.0
Titrate:
Stochastic approach for the seismic analysis
Date:
08/02/99
Author (S):
A. DUMOND
Key:
R4.05.02-B
Page:
14/18
Manual of Reference
R4.05 booklet: Seismic analysis
HP-52/96/009/B
4.3 Excitations preset or reconstituted starting from functions
complexes existing
One can wish to define a matrix of density interspectrale of power in various ways:
·
by a white Gaussian noise: the values are constant
·
according to the analytical formula of useful KANAI-TAJIMI in seismic calculation (filtered white Gaussian noise),
·
or by taking again existing complex functions.
The operator then is used
DEFI_INTE_SPEC
[U4.56.02].
4.3.1 Existing complex functions
It is enough under the key word factor
PAR_FONCTION
to give the name of the function for each pair
of index
NUME_ORDRE_I
,
NUME_ORDRE_J
, corresponding to the higher triangular matrix (in
reason of its hermiticity).
4.3.2 Noise
white
A white Gaussian noise is characterized by a constant value on all the field of definition considered. Under
the key word factor
CONSTANT
, one gives this value (
VALE_R
or
VALE_C
) on the frequency band
[
FREQ_MIN
,
FREQ_MAX
] for each pair of index
INDI_I
,
INDI_J
, corresponding to the matrix
triangular higher (because of its hermiticity). To define the function perfectly, one specifies
the interpolation and prolongations.
4.3.3 White Gaussian noise filtered by KANAI-TAJIMI [bib9]
For a structure pressed on the ground, it is common to take as excitation the spectral concentration
of power of Kanaï-Tajimi. This spectral concentration represents the filtering of a white Gaussian noise by the ground.
The parameters of the formula make it possible to exploit the center frequency and the bandwidth of
spectrum.
The spectrum
G ()
express yourself by the following relation:
G
G
F
G
G
G
G
G
G
G
G
G
G
()
(
)
=
+
-
+
=
4
2
2
2
2
2 2
2
2
2
0
0
4
4
2
own pulsation
modal damping
white sound level before filtering
The user must specify the Eigen frequency
F
G
filter, modal damping
G
and the level of
white Gaussian noise
G
0
(= VALE_R)
before filtering; like as for any function: the interpolation, them
profiles external and the field of definition (frequency band).
By defect a ground running is well represented by the values
F
G
= 2.5 Hz and
G
= 0.6.
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Code_Aster
®
Version
4.0
Titrate:
Stochastic approach for the seismic analysis
Date:
08/02/99
Author (S):
A. DUMOND
Key:
R4.05.02-B
Page:
15/18
Manual of Reference
R4.05 booklet: Seismic analysis
HP-52/96/009/B
Example of use for a white Gaussian noise filtered by KANAI_TAJIMI:
Interex =
DEFI_INTE_SPEC (
DIMENSION: 1
KANAI_TAJIMI: (
NUME_ORDRE_I: 1
indices of the term of the matrix of density
NUME_ORDRE_J: 1
interspectrale of power
FREQ_MOY: 2.5
Eigen frequency
AMOR: 0.6
modal damping
VALE_R: 1
white sound level
Interpol: “FLAX”
linear interpolation
PROL_GAUCHE: “CONSTANT”
prolongation
PROL_DROIT: “CONSTANT”
FREQ_MIN: 0.
field of definition
FREQ_MAX: 200.
NOT: 1.
));
4.4
Other types of excitation
Calculations of the preceding paragraphs were carried out within the framework of the assumption of an excitation
moving imposed on a ddl. With the help of some amendments it is possible to use the same one
approach for an excitation in effort [§4.4.1] or by fluid sources [§ 4.4.2], the aforementioned being
expressed in a finite element [§4.4.3] or on a function of form of the structure [§4.4.4].
In the continuation of this paragraph, one supposes the random excitation known and provided by the user under
the form of a DSP, spectral concentration of power.
4.4.1 Case of the excitation in imposed forces
Under the key word
EXCIT
one has
SIZE
=
EFFO
.
When the excitation with the supports is of type forces imposed, the general equation of the movement is:
()
()
()
MX
CX
KX
F
!!
!
T
T
T
J
J
m
+
+
=
=
1
The response of the structure is then calculated on a basis of dynamic modes
{}
=
=
I I
N
,
,
1
,
these modes being calculated by supposing the free exiting supports. One does not distinguish, in it
case of absolute, relative and differential movement and one does not use static modes.
One defines the factor of modal participation in the form:
P
ij
T I J
I
F
= µ
The transitory, harmonic and random answers have the same expressions as the answers of
relative movement of the excitation multi-support in the general case [§3]. (What corresponds to the absence
static modes). The exiting force is represented in each ddl-support by its DSP in form
of a term are equivalent to
S
EE
!! !!
()
.
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Code_Aster
®
Version
4.0
Titrate:
Stochastic approach for the seismic analysis
Date:
08/02/99
Author (S):
A. DUMOND
Key:
R4.05.02-B
Page:
16/18
Manual of Reference
R4.05 booklet: Seismic analysis
HP-52/96/009/B
4.4.2 Excitation by fluid sources
The fluid sources appear, for example, in the study of a network of pipings. They
correspond to active parts or branches of secondary pipings. They are
generally sources of pressure or sources of flow. These various types of source are
presented hereafter according to their mathematical working and what Code_Aster makes
in each configuration.
These fluid sources are not directly seismic excitations but can be induced by
a seism. The resolution of the mechanical problem calls upon very the methods, because of their
randomness, which justifies their presentation here.
The modeling of the network of piping is supposed to be realized using acoustic beam vibro of
Code_Aster.
The response to fluid sources is calculated within the framework of the response to imposed forces
(cf [§4.4.1]), within this framework one is interested in answers of size of the type “displacement”
(
SIZE
=
DEPL_R
under the key word
ANSWER)
.
The sources of pressure and force, for reasons of modeling of the fluid sources are
represented by dipoles [bib5], it is thus necessary to give two points of application.
Source of flow-volume:
SIZE
=
SOUR_DEBI_VOLU
under the key word
EXCIT
A volume flow rate is expressed in m
3
/S, its spectral concentration of power in (m
3
/S)
2
/Hz.
A source of flow-volume is considered, in the formulation
P
-
elements of
piping with fluid, like an effort imposed on the ddl
node of application of the source
[R4.02.02].
The user provides the DSP of volume flow rate
()
S
vv
, the DSP
()
S
'vv
applied in effort to the ddl
is:
S
S
'()
(
)
()
vv
vv
=
2
where
is the density of the fluid.
Source of flow-mass:
SIZE
=
SOUR_DEBI_MASS
under the key word
EXCIT
A flow-mass is expressed in kg/S, its spectral concentration of power in (kg/S)
2
/Hz.
flow-mass is the product of flow-volume by the density of the fluid.
The user provides the DSP of flow-mass
S
mm
()
, the DSP
()
mm
applied in effort to
ddl
is:
S
S
'
mm
mm
=
2
Source of pressure:
SIZE
=
SOUR_PRESS
under the key word
EXCIT
A source of pressure is applied in Aster in a dipole P
1
P
2
.
For a source of pressure whose DSP is
S
PP
()
, expressed out of AP
2
/Hz, Aster builds one
stamp density interspectrale of power
()
PP
who is applied in force imposed to
the ddl
points P
1
and P
2
.
S
S
'
()
()
PP
PP
S
dx
S
dx
S
dx
S
dx
=


-
-








2
2
2
2
where
S
is the fluid section,
dx
the distance enters the two points P
1
and P
2
.
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Code_Aster
®
Version
4.0
Titrate:
Stochastic approach for the seismic analysis
Date:
08/02/99
Author (S):
A. DUMOND
Key:
R4.05.02-B
Page:
17/18
Manual of Reference
R4.05 booklet: Seismic analysis
HP-52/96/009/B
Source of force:
SIZE
=
SOUR_FORCE
under the key word
EXCIT
The force corresponds simply to the product of the pressure by the fluid section of the tube: F = PS.
It thus is also applied to a dipole P
1
P
2
.
For a source of force whose DSP is
S
FF
()
, expressed in NR
2
/Hz, Aster applies in force
imposed on the ddl
points P
1
and P
2
, (distant of dx), the matrix of density interspectrale of
power
()
FF
such as:
S
S
'
()
()
FF
FF
dx
dx
dx
dx
=


-
-








1
1
1
1
2
2
2
2
4.4.3 Excitation distributed on a function of form
If spectral concentration of power of the excitation
()
E
corresponds to an effort imposed on one
function of form
F
I
,
()
E
give the frequential dependence of the level of the excitation.
The space weighting of the effort is represented in Code_Aster by a field with the nodes which
does not depend on the frequency: key word
CHAM_NO
under the key word factor
EXCIT
. This field with the nodes
is a “assembled vector”. From the theoretical point of view the formalism of calculation is the same one as
previously (excitation in imposed force [§4.4.1]), for a vector of force in second member
equal to
F
I
.
4.5 Applications
These various types of excitation are included in the tests of validation, and are presented by
examples in the report/ratio [bib6]. In particular the excitations of the fluid type are in the test: pipe
subjected to random fluid excitations [V2.02.105] (SDLL105). Excitations on functions of
form are tested in the case test: beam subjected to a random excitation distributed [V2.02.106]
(SDLL106).
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Code_Aster
®
Version
4.0
Titrate:
Stochastic approach for the seismic analysis
Date:
08/02/99
Author (S):
A. DUMOND
Key:
R4.05.02-B
Page:
18/18
Manual of Reference
R4.05 booklet: Seismic analysis
HP-52/96/009/B
5 Bibliography
[1]
P. LABBE and H. NOAH: “Stochastic approach for the seismic design off nuclear power seedling
equipment ". Nuclear Engineering and Design 129 (1991) 367-379
[2]
A. DUMOND Report/ratio EDF DER HP62/95.021B: Post processing of a calculation of mechanics
vibratory under random excitations in Code_Aster. Note reference of the control
POST_DYNA_ALEA
.
[3]
G. JACQUART Report/ratio EDF DER HP61/93.073: Random generations of signals of
spectral concentration data: Note principle and schedule of conditions of ASTER integration.
[4]
Fe. WAECKEL Report/ratio EDF DER HP62/95.017B: Method for calculation by superposition
modal of the seismic response of a multimedia structure.
[5]
P. THOMAS: Taking into account of the acoustic sources in the models of pipings in
mechanics. Bulletin of DER - series A. Thermal Hydraulic Nucléaire n° 2 1991 pp19-36
[6]
C. DUVAL Report/ratio EDF DER HP-61/92.148: Dynamic response under random excitations
in Code_Aster: theoretical principles and examples of use
[7]
BENDAT and PIERSOL: Spectral engineering applications off correlation and analysis. John
Wiley and His 1980
[8]
MARPLE: DIGITAL spectral analysus with applications. Prentice Hall 1987
[9]
Maximum H. TAJIMI A statistical method off determining the response off has building structure
during year earthquake. Proc 2nd world Conf. Earthquake Eng. Tokyo and Kyoto, Japan (1960)
p 751-797