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Titrate:
Seismic response by spectral method
Date:
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Author (S):
J.R. LEVESQUE, L. VIVAN, D. SELIGMANN, Y. PONS Key: R4.05.03-B Page
: 1/34
Organization (S): EDF/AMA, CS IF, EDF/TESE
Handbook of Référence
R4.05 booklet: Seismic analysis
Document: R4.05.03
Seismic response by spectral method
Summary:
The study of the response of a structure under the effect of imposed movements of seismic type, with one
single imposed movement (mono support) or multiple (multi supports) is possible in transitory analysis (time
history). One will refer to the note [R4.05.01].
For studies of dimensioning, one can be interested only in one estimate of the induced maximum efforts
by the stresses, to evaluate the safety margin with payments of construction, without resorting to
a transitory analysis.
The spectral method is based on the concept of spectrum of oscillator of a accélérogramme of seism. One
detail the method of development of this spectrum of answer available in operator CALC_FONCTION
[U4.32.04].
It is shown how this spectrum of oscillator can be used to evaluate one raising of the answer in
relative displacement of a simple oscillator. This approach is justified if one does not wish to know the history of
displacements and of the efforts, while limiting themselves to the analysis of the inertial effects.
The spectral method uses general notions of the method of modal recombination [R5.06.01].
One describes the various rules of combination usable to obtain one raising realistic but conservative
the maximum response of the structure. These methods are available in operator COMB_SISM_MODAL
[U4.84.01].
Handbook of Référence
R4.05 booklet: Seismic analysis
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Seismic response by spectral method
Date:
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Count
matters
1 Concept of spectrum of oscillator ............................................................................................................... 4
1.1 Imposed movement defined by a accélérogramme A (T) ................................................................ 4
1.2 Spectrum of oscillator of a accélérogramme .................................................................................... 5
1.2.1 Spectrum of oscillator in relative displacement .......................................................................... 5
1.2.2 Spectrum of oscillator in pseudo relative speed .................................................................... 6
1.2.3 Spectrum of oscillator in pseudo absolute acceleration ........................................................... 7
1.3 Determination of the spectrum of oscillator ............................................................................................ 7
1.4 Representation and use of the spectra of oscillators ................................................................. 8
1.4.1 Representation tri logarithmic curve ............................................................................................ 8
1.4.2 Use of the spectra of oscillators ..................................................................................... 8
1.5 Spectra of oscillators used for studies .............................................................................. 9
1.5.1 Spectrum of ground of design and checking of the buildings ................................................... 9
1.5.2 Spectrum of floor of checking of the equipment ......................................................... 10
2 seismic Response by modal recombination ................................................................................... 10
2.1 Recalls of the formulation .............................................................................................................. 10
2.1.1 Single imposed movement: mono support ............................................................................ 11
2.1.2 Multiple imposed movement: multi supports .......................................................................... 12
2.1.3 Summary ................................................................................................................................ 13
2.2 Response in modal base .............................................................................................................. 13
2.2.1 Temporal response of a modal oscillator ......................................................................... 13
2.2.2 Modal factor of participation in mono support .................................................................... 14
2.2.3 Modal factor of participation in multi supports .................................................................... 14
3 seismic Response by spectral method .......................................................................................... 15
3.1 Spectral response of a modal oscillator in mono support ............................................................ 15
3.2 Spectral response of a modal oscillator in multi supports ........................................................... 16
3.3 Generalization with other sizes ............................................................................................. 16
4 Rules of combination of the modal answers .................................................................................. 17
4.1 Direction of the seism and directional answer ............................................................................... 17
4.2 Choice of the modes suitable to combine ........................................................................................... 17
4.2.1 Expression of the modal deformation energy .................................................................. 17
4.2.2 Expression of the modal kinetic energy ............................................................................ 18
4.2.3 Conclusion ............................................................................................................................ 19
4.3 Static correction by pseudo-mode ........................................................................................... 19
4.3.1 Mono support ........................................................................................................................... 19
4.3.2 Multi supports .......................................................................................................................... 20
4.4 General information on the rules of combination ................................................................................... 20
4.4.1 Arithmetic combination .................................................................................................... 20
4.4.2 Combination in absolute value ........................................................................................... 21
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Seismic response by spectral method
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4.4.3 Quadratic combination simple ......................................................................................... 21
4.5 Establishment of the directional response in mono support ......................................................... 21
4.5.1 Combined response of the modal oscillators ...................................................................... 21
4.5.1.1 Summon absolute values ................................................................................. 22
4.5.1.2 Quadratic combination simple (CQS) ................................................................. 22
4.5.1.3 Quadratic combination supplements (CQC) ............................................................ 22
4.5.1.4 Combination of ROSENBLUETH ........................................................................... 23
4.5.1.5 Combination with rule of the 10% ........................................................................... 23
4.5.2 Contribution of the static correction of the neglected modes ................................................. 23
4.6 Establishment of the directional response in multi supports ......................................................... 23
4.6.1 Calculation of the total answer ................................................................................................ 23
4.6.2 Separate calculation of the components primary education and secondary of the answer .............................. 24
4.6.3 Combined response of the modal oscillators ...................................................................... 24
4.6.4 Contribution of the pseudo mode .............................................................................................. 24
4.6.5 Contribution of the movements of drive .................................................................... 24
4.6.6 Combination of the directional answers of supports ........................................................... 25
4.7 Combination of the directional answers .................................................................................. 25
4.7.1 Quadratic combination .................................................................................................... 25
4.7.2 Combination of NEWMARK ............................................................................................... 25
4.8 Warning on the combinations ............................................................................................. 26
4.9 Practical lawful .............................................................................................................. 26
4.9.1 Partition of the primary and secondary components of the answer ................................... 26
4.9.2 Method of the spectrum envelope ............................................................................................ 27
5 Bibliography ........................................................................................................................................ 28
Appendix 1
Transitory response of a simple oscillator deadened .................................................. 29
Appendix 2
Movement imposed of a system on a D.D.L. in translation ................................... 31
Appendix 3
Movement imposed not periodical of a system on a D.D.L. ................................. 33
Handbook of Référence
R4.05 booklet: Seismic analysis
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Titrate:
Seismic response by spectral method
Date:
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1
Concept of spectrum of oscillator
Spectral method for the study of the answer of a structure under the effect of imposed movements
of seismic type is based on the concept of spectrum of oscillator of a accélérogramme of seism.
1.1
Imposed movement defined by a accélérogramme A (T)
For a movement imposed S of the seismic type, one can deal with the problem in absolute displacement
X or in relative displacement X such as: X = X + S. Les general equations of the movement of one
simple oscillator are written then:
Absolute movement
Relative movement
X
S
X
X
m &+ X
C & + kx = - S
m
K
& + & +
= & +
X
m
X
C
kX
S
C
ks
&
m
C
One retains the formulation starting from the relative movement for two principal reasons:
·
the seismic analysis of the structures uses the constraints induced by the inertial effects of
seism, constraints calculated starting from the structural deformations which are expressed with
to leave relative displacements;
·
the characterization of the signal of excitation can be reduced in this case to the accélérogramme
seism s=
& (
With T), size provided directly by the seismographs. Signals of
displacement S and speed &s are in general not available in the bases of
data geotechnics.
For the determination of the response of a simple oscillator to an imposed movement and the notations
conventional one will refer to appendix 2 [R4.05.03 Annexe 2].
The reduced equation is in this case, if the seism is defined by a accélérogramme (
With T), acceleration
absolute applied to the base:
x+
& 2 X
éq
1.1-1
0 &+
2 X = - S
0
& = - (
With T)
The solution of this problem is the integral of DUHAMEL presented at appendix A [éq A3.3-1]:
T
X (T) 1
=
(
With)
- T
0 (
)
E
sin (T
-) D F
= (,
With,
-
éq
1.1-2
0)
0
0
0
=
1 -
0
(2
0
)
Handbook of Référence
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Seismic response by spectral method
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1.2
Spectrum of oscillator of a accélérogramme
The concept of spectrum of oscillator was introduced initially to compare between them the effects of
different accélérogrammes. The spectrum of FOURIER of a signal (
With T) informs about its contents
frequential. The response of a mechanical system to a movement imposed on the base largely depends
dynamic characteristics of this system: Eigen frequencies and reduced damping (.
0)
Appendix A details this aspect.
If one wishes to know the maximum value of the response of a simple oscillator to the parameters
(, A, one must evaluate the integral of DUHAMEL which provides the response of the oscillator [éq 1.1-2] to
0)
an excitation imposed on the base.
Accélérogramme
0.7
0.6
0.4
0.2
0.0
0.2
Absolute acceleration
0.4
0.6
0.7
0
3
6
9
12
15
18
Time
Appear 1.2-a: Accélérogramme
1.2.1 Spectrum of oscillator in relative displacement
From the integral of DUHAMEL, one can define the spectrum of oscillator of a accélérogramme
(
With T) like the function of the maximum values of relative displacement X (T) F
= (,
With, for each
0)
value of (, by recalling that '
=
1 -
.
0
(2
0
)
0)
Srox (,
With, = X T
0)
() max
T
X (T) 1
=
(
With) - T
0 (
)
E
sin (T
-) D F
= (,
With,
-
0)
0
0
0
One notes, on the figure [Figure 1.2.1-a] that beyond a certain frequency (35 Hz here), known as
cut-off frequency of the spectrum, it does not have there significant dynamic amplification: displacement
relative is null.
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Spectrum of oscillator (relative Déplacement)
1
0.1
10
Increasing depreciation
from 0.01 to 0.20
10 2
Depreciation
103
1%
5%
Relative displacement
10%
20%
10 4
10 5
1e05
10 0
10 1
Frequency
Appear 1.2.1-a: Specter of oscillator in relative displacement
1.2.2 Spectrum of oscillator in pseudo relative speed
For structures with weak reduced damping < 2
.
0 =
%
20, for which it is
acceptable to assimilate 0 and 0 one usually uses the spectrum pseudo speed defined by:
Srox& (,
With, = Srox,
With, = X T
0)
0
(
0)
0
() max
Pseudo speed is the value the speed which gives a value of the kinetic energy of the mass
oscillator equal to that of the maximum deformation energy of the spring:
1
1
1
1
E = m X & T 2
() = m Srox & A
2
,
= m2
. X T 2
()
= K X T 2
()
= E
C
()
[
(
0)]
0
p
max
max
2
2
2
2
Spectrum of oscillator (Pseudo relative speed)
0.5
Increasing depreciation
from 0.01 to 0.20
I
ve
1
10
Depreciation
relat
E
1%
5%
10%
20%
Pseudo 102 saw
0.004
10 0
10 1
Frequency
Appear 1.2.2-a: Specter of oscillator in pseudo relative speed
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1.2.3 Spectrum of oscillator in pseudo absolute acceleration
In the same way for a weak reduced damping one can define the spectrum of pseudo acceleration definite
by:
Srox (&,
With, = Srox,
With, = X T
0)
2
0
(
0)
2
0
() max
Spectrum of oscillator (Pseudo absolute acceleration)
4
Increasing depreciation
from 0.01 to 0.20
10 0
Depreciation
1%
5%
10%
20%
Pseudo absolute acceleration
1
10
0.1
1
100
10
Frequency
Appear 1.2.3-a: Specter of oscillator in pseudo absolute acceleration
The interest of this spectrum of pseudo acceleration lies in the fact that S R OX (
&,
With, is good
0)
approximation of the maximum of absolute acceleration X& (T). Indeed, at the moment when relative displacement
is maximum, relative speed is cancelled and the reduced equation is written X & + 0 + 2
X
= - S what us
0
max
&
show that
2
2
X&
= x&+ s&
= X
= Srox,
With, = S R OX &,
With,
0
max
0
(
0)
(
0)
max
max
For this reason, this spectrum of oscillator is called spectrum pseudo absolute acceleration.
The asymptote of this high frequency spectrum (acceleration at null period) corresponds to the answer
of a clean high frequency oscillator, i.e. very rigid. In this case, the mass tends to follow
completely the imposed movement of the base. This asymptote thus corresponds to acceleration
maximum
(
With T)
imposed movement (ground or not of fixing of the oscillator). It is
max
attack in practice starting from the cut-off frequency of the spectrum. For this reason, one says that one
accélérogramme is fixed, for example, on 0.15 G, when its maximum amplitude and its spectrum
of oscillator of pseudo absolute acceleration at null period are equal to 0.15 G.
1.3
Determination of the spectrum of oscillator
Determination of the spectrum of oscillator of a accélérogramme (
With T) is available in the operator
CALC_FONCTION [U6.62.04] with key word SPEC_OSCI: it is obtained by numerical integration of
the equation of DUHAMEL by the method of NIGAM [R5.05.01]. This command provides the spectrum of
pseudo absolute acceleration and, on request, the spectrum pseudo speed or the spectrum of
relative displacement.
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1.4
Representation and use of the spectra of oscillators
1.4.1 Representation tri logarithmic curve
The spectra of response of oscillator are usually represented by tri graphs
logarithmic curves which make it possible to read on only one graph the three sizes: relative displacement,
pseudo relative speed, pseudo absolute acceleration.
This representation is obtained by tracing the spectrum pseudo relative speed Srox & in
co-ordinates log-log such as log Srox & = F (log) on which one defers two graduations
0
complementary to ± 45° if the scale of the graduations logarithmic curves is the same one on the two axes:
·
a graduation logarithmic curve with +45° to measure relative displacements
log Srox = log (Srox & = log Srox & + log
0
)
0
·
a graduation logarithmic curve with to measure absolute accelerations
Srox
log Srox = log
&
&
= log Srox & - log 0
0
10
Speed m/s 10
103
1
Displacement m
F = 3 Hz
Ý
X f= 1
= 3 Hz
m/s
1
X & = 1 m/s
X = Ý
X = 1 = 0.05305
With
x&
1
C 1
1
C
6
0 2
E
X =
=
= 05305
.
0
breadth
10
rat
6
io
Ý
N
X = Ý
X = 6 = 18.849
ms2
X & = X = 6 =
849
.
18
101
10
102
1
10
Frequency Hz
3
1
10
3 Hz
Appear 1.4.1-a: Représentation tri logarithmic curve
1.4.2 Use of the spectra of oscillators
To evaluate the maximum response of a modal oscillator (
,
with a accélérogramme (
With T), one
I
I)
use the spectrum of pseudo absolute acceleration.
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It is represented in Code_Aster, by a tablecloth made up of several functions
Srox & = F (freq) with = cte
N
.
One uses a linear interpolation on the damping reduced for < <
because amplification
N
I
N 1
+
X
1
dynamics with resonance for = (that is to say = 1) is equal to m =
[éq A2.2-3].
0
S
2
0
I
The variation of the module of the response in the vicinity of resonance also justifies an interpolation
logarithmic curve for < <
. The spectrum of oscillator must be represented with one
m
I
m 1
+
discretization in sufficiently fine frequency to limit the effects of the interpolation.
1.5
Spectra of oscillators used for studies
For the studies of industrial facilities, such as the nuclear thermal power stations, the seismic analysis
conduit to establish several models:
·
a model of the civil engineering of design of the buildings to determine:
-
accidental stresses for the calculation of the frameworks of these buildings;
-
movements imposed on the points of fixing of the equipment (reactor vessel,
supports of the networks of pipings, electrical equipment boxes.) at various levels of
buildings;
·
models of study of checking of each equipment subjected to the imposed movements
amplified by the dynamic behavior of the buildings.
1.5.1 Spectrum of ground of design and checking of the buildings
This stage, the equipment is known only like inertial overloads and one can
to admit that they do not bring any rigidity to the building. The structures in this case are subjected to one
spectrum of ground.
The frequential contents of a spectrum of oscillator reflect that of the accélérogramme used and are thus
“marked” by the properties of the ground instead of recording. To work out the spectrum of ground at the stage
project, it is thus recommended to establish the spectra of oscillators for several accélérogrammes
and to build a spectrum envelope which smoothes anti resonances.
Spectrum of oscillator of a accélérogramme and the spectrum of associated ground
10 0
1
Representation
tri logarithmic curve
10 1
Damping 1%
102
Pseudo_vitesse_relative
0.001
10 3
1
101
10 0
10
10 2
Frequency
Appear 1.5.1-a: Specter of ground for a project
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Note:
In many cases one does not know the rotational movement imposed by the seism,
since the accélérogrammes of known seisms result from recordings of sismo-
graphs, sensors with a degree of freedom of translation.
1.5.2 Spectrum of floor of checking of the equipment
The study of the dynamic behavior of the equipment subjected to the movements imposed by
structure support at the points of supports is possible starting from the accélérogrammes of answer in these
points, results of the transitory analysis of the behavior of the building: these accélérogrammes, known as of
floor, make it possible to build spectra of floors.
For a checking of the equipment, one can limit oneself to a spectral analysis starting from the spectra
of floor and the differential displacements imposed on the supports.
The spectra of floor are representative of the dynamic amplification brought by the structure
support: a smoothing of the spectrum can be useful to take into account uncertainty on the position
Eigen frequencies of the building, but one will take care to preserve realistic margins, since it
spectrum of ground is already one raising of the seismic stress. The spectrum of oscillator must be
represented with a discretization in sufficiently fine frequency “to collect” resonances of
structure.
Note:
Techniques of direct determination of the spectra of floors, starting from the spectrum of ground and
modes of the structures were developed [bib1], but are not available
currently in Code_Aster.
2
Seismic response by modal recombination
2.1
Recalls of the formulation
The spectral method of seismic analysis is based on the formulation of the dynamic response
transient by modal recombination presented in the documents “Méthodes of RITZ in
linear and nonlinear dynamics " [R5.06.01] and “seismic Analyze by direct method or
modal recombination " [R4.05.01].
Let us summarize the principles of the step detailed in the note [R4.05.01] for a structure
represented in form discretized by the matric system:
DRIVEN & + CU & + KU = F (T)
éq
2.1-1
Notations moving absolute
U represents all the components of the movement (ddl of structure and ddl subjected to one
X
imposed movement): one separates them in the form U = S. describing Les operators
K
K xs
C
cxs
m
mxs
structure become: K =
C =
M =
K
K
C
C
m
m
sx
S
sx
S
sx
S
The problem moving relative of the structure compared to the supports with the decomposition
Absolute movement = relative Mouvement + Mouvement of drive results in introducing it
change of variable U = U + E.
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Notations moving relative
U represents all the components of the movement (ddl of structure and ddl subjected to one
X
movement imposed) which is written, with the preceding partition, U = 0. The partition of
components of drive gives, by expressing the drive of the ddl of structure like one
be
combination liéaire of the movements imposed on the supports, E = S
Assumption
It is supposed that no force of excitation is applied to D.D.L. of structure what reduces it
0
second member F (T), with the same partition with F = R
The passage of the absolute movement to the relative movement can be written by introducing the operator of
passage:
X
X be
X
I E
U =
U.E.
with
S = + = 0 + S =
S
= 0 I
The system [éq. 2.1-1] takes the general form then:
x&
x&
X
0
T
+ T C
+ T K
= T
M
éq
2.1-2
s&
s&
S
R
2.1.1 Single imposed movement: mono support
The movement of drive corresponds then to a movement of solid body: the vector
of drive in any point of the structure can express like a linear combination R S
components of displacement imposed on the center of gravity of the foundation, where the R are
stamp modes of bodies rigid of the structure reduced to the ddl of structure, which leads to:
X
X
R S
U = S = 0
+ S
I
=
R
0
I
The properties of the modes of rigid body Cf. [R4.05.01] lead to:
K 0
C 0
m
m + m
R
T K =
T C =
T M
xs
= T
0 0
0 0
m + m
m
R
sx
R
what makes it possible well to uncouple the system [éq 2.1-2]
m x&+c x&+k X = - m (
+ m
m
1
R
xs) S
& éq
2.1.1-1
In this case the transformation highlights well the inertial effect of the seismic loading.
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2.1.2 Multiple imposed movement: multi supports
This situation corresponds to a discrete number of points of connection of the structure in supports
subjected to different imposed displacements. In this case, the diagonalisation of the term of rigigity
is acquired while imposing:
K
K 0
xs
S
=
of or K + K
= 0
that is to say = - K - 1 K
éq 2.1.2-1
K
K
I
R
S
xs
S
xs
sx
S
The matrix S gathers 6. nappuis static modes for the models of structures and 3 times the number
supports for the models of continuous mediums. Each static mode
1
S = -
-
K
K
J
xs is a mode
J
of fastener, corresponding to a unit displacement imposed on a component of support, the others
null components, and being produced by operator MODE_STATIQUE [U4.52.14].
The change of reference mark can then be expressed by:
X
X
X
S S
U = S = 0
+ S
X
I
S
= 0 I
S 1
S 2
m
m
S + m
T M =
xs
TS m+ msx
ms
Concerning the terms of damping, decoupling is acquired only if damping is
proportional to rigidity, usually allowed assumption, but was not necessary with the modes
rigid.
This makes it possible well to uncouple the system [éq 2.1-2]
MX & + cx & + kx = m
- (
m m
1
-
+
S
xs) S &
éq
2.1.2-2
This formulation must be interpreted like the decomposition of the movement of the structure in one
movement of drive corresponding to an instantaneous static deformation (displacement
differential of the supports) and a relative movement corresponding to the inertial effects around this
new static deformation.
This interpretation is in conformity with the classification of the stresses defined by the rules of
construction (ASME, RCC-M):
·
the constraints induced by the relative movement are, as for the statical stresses,
primary constraints (effects of inertia),
·
constraints induced by differential displacements of the supports which are they classified
in secondary constraints.
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2.1.3 Summary
The equations [éq 2.1.1-1] and [éq 2.1.2-2] lead to the general form
mz & + cz & + kz -
= m (+ m m
1
X
xs) S & = -
S
O
m &
éq
2.1.3-1
The mxs terms correspond under the terms of coupling of the matrix of mass with the degrees of
freedom of support: this fraction of the total mass is very weak and it is justified to neglect it. Let us recall
that this term is indeed null for the models of structures whose matrix of mass is
diagonal: models masses - arises, models with elements of the type “lumped farmhouse”.
In this case, one obtains the simplified formulas:
·
mono support: O =
where
R
R are the six modes of solid body
·
multi supports: O =
where
S
S are 6 N supports modes of fastener
The second member - m O is built by operator CALC_CHAR_SEISME [U4.63.01].
2.2
Response in modal base
2.2.1 Temporal response of a modal oscillator
If the studied structure is represented by its spectrum of low frequency real clean modes
in embedded base, solution of (
2
K -
M
) = 0 or of (
2
K - m) = 0 one can introduce one
new transformation X = Q and the system of equations [éq 2.1.3-1] is written, by using the matrix
modal factors of participation P
T C
2
T Mo
Q & +
Q
éq
2.2.1-1
T
& + Q = -
S
T
& = - Ps&
m
m
Assumption:
For industrial studies concerned with the seismic analysis by spectral method, one limits oneself
with the case of damping proportional, known as of RAYLEIGH, for which one can diagonaliser it
T C
term
=
2. Damping is then represented by a modal damping
T m
I,
possibly different for each clean mode [R4.05.01].
Each clean mode, characterized by the parameters (,
I
I) is compared to a simple oscillator
whose behavior is represented in the general case by
Q & + Q & + 2
2
Q = - PS éq
2.2.1-2
I
I
I
I
I
I
() &i
Let us recall that the &&s are accelerations of drive.
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2.2.2 Modal factor of participation in mono support
When the movement of drive is single, [éq 2.2.1-2] becomes
Q & + Q & + 2
2
Q = - p S
éq
2.2.2-1
I
I
I
I
I
I
I &
with
T
T
Mo m
I
I
p
R
=
=
éq
2.2.2-2
I
T
m
µ
I
I
I
where µi is the generalized modal mass, which depends on the standardization of the clean mode. Let us state
some properties of the factors of modal participation pi in the case of rigid modes of
translation, but extensible with the modes of rotation.
·
A mode X-ray, that we will note X, to recall that components in the direction
X are unit, belongs to the space of dimension NR degrees of freedom of which NR
clean modes constitute a base in which =
X
I I.
I
From the properties of orthogonality of the clean modes T m = µ
I
I
I ij, one identifies them
coefficients I with the factors of modal participation piX in direction X and
= p
X
iX
I
éq
2.2.2-3
I
·
Moreover T m
X
X =
T
m masses total structure what leads to:
2
p µ
T
T
2
2
m
=
p p
m
p µ and m
p µ or
éq 2.2.2-4
X
X
=
iX
jX
J
I
=
iX
I
T
iX I iX I =1
ij
ij
ij
ij
MT
Modal parameter pi depends on the standard of the clean mode and is accessible, for each mode
X
clean in the concept result of the type mode_meca [U5.01.23] under name FACT_PARTICI_DX; of
even p2iX I
µ, independent of the standard, is accessible under the name of MASS_EFFE_UN_DX.
2.2.3 Modal factor of participation in multi supports
For a multiple imposed movement, [éq 2.2.1-2] becomes:
Q & + Q & + 2
2
Q = -
p S
éq
2.2.3-1
I
I
I
I
I
I
ij &j
J
with
T
T
Mo m
I
I
S J
p =
=
éq
2.2.3-2
ij
T
m
µ
I
I
I
where µi is the generalized modal mass, which depends on standardization on the clean mode and the pij
can be regarded as factors of participation relating to mode I and a direction J
of imposed movement of a support.
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As previously, one can establish [bib4] the two properties
= p and T m = p2 µ
S J
ij
I
S J
S J
ij
I
éq
2.2.3-3
I
I
One makes, at this stage, no assumption of dependence between the various pij terms. Let us recall
that the components &sj express the acceleration of drive applied to a direction of support J.
The factors of pij participation are not built independently and only appear
like intermediate variables in command COMB_SISM_MODAL [U4.84.01].
3
Seismic response by spectral method
The spectral method is an approximate technique of evaluation of the maximum of the answer of
structure starting from the maxima of answer of each modal oscillator read on the spectrum
of oscillator of the excitation.
3.1
Spectral response of a modal oscillator in mono support
The maximum response in relative displacement of a modal oscillator (,
I
I) for a direction X is
determined by reading on a spectrum of oscillator of pseudo absolute acceleration Cf. [§1.4.2] the value
pseudo absolute acceleration has
= Sro X & (A
2
iX
X
, I, I
) and while dividing by I from where:
Sro x&X (A, I, I
) has
Q
iX
iX max =
=
2
2 éq
3.1-1
I
I
The contribution of this oscillator to the relative displacement of the structure for the component xk
depends on the factor on participation and component K
I in physical space:
has
xk
K
= p Q
K
= p
iX
iX max
I
iX
iX max
I
iX
I
2
éq
3.1-2
and the contribution to the pseudo absolute acceleration &xk is of the same &xk
K
max =
p has
iX
I
iX
iX.
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3.2
Spectral response of a modal oscillator in multi supports
One proceeds in the same manner to determine, starting from the value read &S jX on the spectrum
of oscillator of pseudo absolute acceleration associated &sj, the contribution of the support J in
direction X:
Sro s&j (A, I
, I
) S&
Q
jX
iX max J =
=
2
2
éq
3.2-1
I
I
The expression of the contribution of this oscillator to the relative displacement of the structure for
component xk in physical space and for a movement imposed J becomes:
S&
xk
K
= p
Q
K
= p
jX
iX max J
I
ijX
iX max J
I
ijX
I
2 éq
3.2-2
3.3
Generalization with other sizes
Note:
The method of spectral analysis is strictly limited to the sizes depending linearly on
displacements in linear elasticity: generalized strains, stresses, efforts, nodal forces,
reactions of supports.
In particular it cannot apply to equivalent sizes of deformation or of
constraints (Von Mises).
For each Rk size, component of a field by elements it is possible to calculate
modal component R K
I associated with the clean mode I what leads to
has
Rk
= rk p Q
= rk p
iX
iX max
I
iX
iX max
I
iX
I
2
éq
3.3-1
or
S&
Rk
= rk p
Q
= rk p
jX
iX max J
I
ijX
iX max J
I
ijX
I
2 éq
3.3-2
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4
Rules of combination of the modal answers
To evaluate one raising of the answer R of the structure, one must now combine the answers
modal Rkimax defined previously. Several levels of combination are necessary:
·
combination of the clean modes selected,
·
static correction by pseudo mode,
·
effect of the excitations different applied to groups from supports,
·
combination according to the directions of excitation seism.
4.1
Direction of the seism and directional answer
Various considerations result in separately studying the seismic behavior according to each
direction of space:
·
for the study of a building on a ground, the accélérogramme of the vertically imposed movement
is different from that describing the horizontal movement, him even different according to two
orthogonal directions of space;
·
for the study of equipment, the spectra of floor differ significantly according to
three directions of space, since they integrate the participations of various modes of
building (inflection of floors, inflection or torsion of the framework.).
This resulted in establishing a directional modal answer X-ray starting from spectra of oscillator
different and from factors of modal participation established in each direction X representative one from
directions of reference mark GLOBAL of definition of grid (X, Y or Z) or a particular direction
defined explicitly by the user.
4.2
Choice of the modes suitable to combine
To correctly represent the modes of deformation likely to be excited by
imposed movement, it would be necessary to know all the clean modes of frequency lower than
cut-off frequency of the spectrum, beyond which there is no dynamic amplification
significant. This condition can prove to be difficult to fill for the complex structures having one
large numbers of clean modes.
The size of the modal base necessary must thus be evaluated to make sure that no mode having one
important contribution in the internal efforts and the constraints was not omitted in each
studied direction.
4.2.1 Expression of the modal deformation energy
1
The deformation energy associated with each clean mode U
T
I =
xi
K X
max
I
2
max can be expressed
for a particular direction
2
2
1
has
1
has
1 a2
U
= p iX
T
K = p
iX
2
iX
µ =
p2
iX
I
µ
2
I
I
iX
2
I
I
2
iX
I
éq
4.2.1-1
2
2
2
I
I
I
This expression corresponds to an excitation mono support and can extend to the case from the multi supports.
The classification of the modes with decreasing deformation energies makes it possible not to retain
systematically, for a general study of the structure, modes which do not produce
significant deformations. On the other hand, for the study of the effect of the stresses in a zone
particular of the structure, it will be necessary to use the “local” modes which can be detected
by an analysis of the distribution of the deformation energy on groups of mesh.
Let us note that one does not have an estimate of the total deformation energy to quantify
the error made by being unaware of certain modes.
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4.2.2 Expression of the modal kinetic energy
1
The kinetic energy associated each clean mode is written V
T
I =
ximax m xi
who gives
2 &
& max
1
has 2
1 a2
V
= p
iX
T
iX
m =
p2
iX
iX
I
I
µ
2
iX
I
éq
4.2.2-1
2
I
2 I
The expression [éq 4.2.2-1] utilized effective modal mass p2iX I
µ defined in [§2.2] what
allows to state the criterion of office plurality of the unit effective modal masses [éq 2.2.2-4]
Criterion of office plurality of the effective modal masses
The quality of a modal base, the point of view of the representation of the inertial properties of
structure, is evaluated by cumulating, for this direction, the unit effective modal masses
modes available. A threshold of admissibility of 95% of the total mass is usually allowed.
The same criterion can apply partially in the case of an excitation multi supports with N
T
N
modes while comparing
m
and
p2
µ
S J
S J
ij
I.
I
Estimate of the error made with an incomplete modal base
The criterion of office plurality of the effective modal masses cannot always be satisfied. Indeed one
limit in general at a modal base of N clean modes with N modes << NR ddl. For
rigid foundations, the spectrum of the Eigen frequencies necessary usually exceeds the frequency
of cut of the spectrum of oscillator.
From the expression [éq 4.2.2-1] one can write the total kinetic energy in the form
N
NR
V = V + V
X
iX
iX
1
1
who allows to express the absolute error from [éq 3.1-1]
NR
NR
2
2
has
has
NR
(+
2
iX
2
N 1) X
2
V = V =
p µ
p
X
iX
µ
2
iX
I
2
iX
I
N 1
+
N 1
+
I
I
N 1
+
while noting has
= Sro X
(
)
1
& (A
N
X
X
, m
in,
+
N
+1) the value read on the spectrum of pseudo acceleration
absolute for
N
n+1 and the modal damping weakest min likely to give
the raising amplitude. If the maximum frequency of the base fn exceeds the cut-off frequency then
has
= has
= (
With T)
(n+)
1 X
nX
max. Ceci gives one raising of the absolute error
1 a2
NR
2
(
1)
1 A
N
n+ X
(n+1) X
V
2
2
µ
µ
2
p
=
m -
2
p
X
iX
I
T
iX
I
éq
4.2.2-2
2
2
I
n+1
I
1
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4.2.3 Conclusion
The sizes allowing for choice of the modes necessary to each analysis are available in
Code_Aster (operator POST_ELEM with options MASS_INER, ENER_POT and ENER_CIN and
modal parameters FACT_PARTICI_DX and MASS_EFFE_UN_DX in the concept result of the type
mode_meca [U5.01.23]).
No criterion of automatic admissibility is currently programmed and the sizes
N
T m and p2 µ
Sj
Sj
ij
I, necessary to the checking of the criterion for an excitation multi supports,
1
are not printed.
4.3
Static correction by pseudo-mode
4.3.1 Mono
support
The evaluation of one raising of the response to a seismic excitation requires, as suggests it
preceding analysis, a correction by a term representing the static contribution of the modes
clean neglected.
If one subjects the structure to a quasi-static constant acceleration in direction X, the answer
aX is solution of K = m I
aX
X
, without dynamic amplification. The field of displacement
aX of the nodes of the structure subjected to a constant acceleration in each direction is
product by operator MODE_STATIQUE [U4.52.14] with key word PSEUDO_MODE.
By breaking up this deformation on the basis of clean mode one obtains cf [§2.2.2]
NR
NR
NR p
K
= m p from where = k-1 m p
iX
=
aX
iX
I
aX
iX
I
2 I
1
1
1
I
This makes it possible to introduce a pseudo-mode cX, for each direction, while withdrawing from the mode quasi
statics has X the static contributions of the modes used I
N p
=
iX
-
cX
aX
éq
4.3.1-1
1 2
I
I
N
The expression [éq 4.3.1-1] is homologous with the term m -
p µ
2
T
iX
I of [éq 4.2.2-2] and the pseudo one
1
mode makes it possible to supplement the incomplete base of clean modes to introduce a correction of
static effects of the neglected modes. The contribution of the pseudo mode is the value read on the spectrum
pseudo absolute acceleration has
= Sro X
(
)
& (A
N
X
X
, m
in, N
),
+
+
for
N
1
1
N
+1 and
the modal damping weakest min.
Correction to be brought to relative displacements and the sizes which result (efforts from them
generalized, forced, reactions of supports) in excitation mono support is then K
K
X
= has
cX
cX
(n+)
1 X
in accordance with the conditions of estimate of the error Cf. [§ 4.2.2].
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For the evaluation of the correction of absolute acceleration one obtains:
K
N
K
X & = -
cX
X
piX I has (n+) 1X
1
4.3.2 Multi
supports
In excitation multi supports, the formulation of the pseudo mode and its contribution take again the principle
precede.
The field of displacement
K 1
-
=
m
nodes of the structure subjected to an acceleration
ajX
SjX
unit of the support J in direction X is produced by operator MODE_STATIQUE [U4.52.14] with
key word PSEUDO_MODE.
The correction to be brought to relative displacements and the sizes which result some writes then
for the support J in direction X:
N P
X
ijX
I
cjX = cjX has (n+)
1 jX with
=
-
cjX
ajX
2
1
I
For absolute acceleration, the correction is written:
N
&
=
-
cjX
X
SjX
I ijX
P
(n+) 1 has jX
1
4.4
General information on the rules of combination
Rules of combination or office plurality of the various components, modal or directional,
are multiple and more or less complexes to be implemented.
One presents the methods “natural” from the point of view of their aptitude required one raising realistic
stresses induced in a structure represented by a base of real clean modes resulting
of a model in linear elasticity, raising estimated without transitory analysis for a size of
Gk component, which one will name Gkmax. For the continuation the suffix max indicates the estimate of
maximum value attack during the seismic excitation, by being unaware of the moment when it was reached) and
the index R applies to clean modes, pseudo modes, directions of supports,…
Note:
Whatever the method of combination used, the value of a component obtained by
combination cannot be used as data to calculate a new size: for example calculation
of one raising of a differential displacement between two points must be calculated mode by mode,
then compound.
4.4.1 Combination
arithmetic
Gk
= Gk
max
R max
R
It is not usable since the spectral method disregards moment when values
maximum are reached in two directions or for two different modes. No relation of
phase, and thus of sign, does not exist between the contributions to combine. They is thus available only
in the case of differential displacements in multi support.
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4.4.2 Combination in absolute value
Gk
= Gk
max
R max
R
In an obvious way, it can provide an upper limit, since it supposes that all them
contributions reach their maximum at the same moment with the same sign. Too much penalizing, it is
available, but unusable industrially.
4.4.3 Simple quadratic combination
This method is also known under denomination SRSS (Square Root off Sum off Squares).
2
Gkmax = (Gkrmax)
R
Assumption:
The assumption which justifies this method of combination can be stated:
the probable maximum of the energy stored in the structure is the sum of the maxima
probable of the energy stored on each mode and each component
directional of the seism, i.e. with respect to energy, the clean modes and them
components of the seism are uncoupled. They is similar to the rule of addition of the variables
random Gaussian and with null average.
Validity of this assumption, which will be discussed for each particular case of use of this
method of combination, is not established and various proposals were presented to obtain
a better approximation whenever it is put at fault cf [§3.4.1.2] following.
In addition, one will be able to refer to [bib3] for a criticism of this approach, in particular of sound
aptitude to consider a maximum probable of the deformations and constraints, but the approach
alternative which it evokes was the subject of any development in Code_Aster.
4.5
Establishment of the directional response in mono support
The directional answer, previously definite, is obtained by simple quadratic combination of
two terms which we will discuss:
R = R2 + R2
X
m
C
Rm answer combined of the modal oscillators
Rc contribution of the static correction of the neglected modes (pseudo mode)
The assumptions justifying the method of quadratic combination simple, on this level, do not seem
not to have to be called into question [bib1]. To simplify the notations one notes Rm instead of RmX,…
4.5.1 Combined response of the modal oscillators
The response of the Rm structure, in a direction of seism, is obtained by one of the combinations
possible of the contributions of each clean mode taken into account for this direction.
The number of possible methods proves simply the difficulty in releasing a justification
sufficient to guarantee a conservative and realistic estimate. If simple quadratic combination
(SRSS or CQS) is evoked by all, one will retain [bib1] that it is often put at fault and one
he will prefer the complete quadratic combination (CQC). The other methods are available for
possible comparisons.
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4.5.1.1 Summon absolute values
This combination corresponds to an assumption of complete dependence of the oscillators associated with
each mode clean and led to a systematic overvaluation of the answer:
N
R =
R
m
I
I
4.5.1.2 Simple quadratic combination (CQS)
By considering that the contribution of each modal oscillator is a random variable
independent, an estimate of the maximum answer, for the component of xkmax displacement,
can be obtained by simple quadratic combination of the contributions of each mode from where, for
an excitation mono support:
N
2
N
2
xk
= xk
K
max
I max
=
(p Q
I
I I max)
éq
4.5.1.2-1
I
I
Generally, for any size IH associated with a modal oscillator (,)
I
I.
N
R = R2
m
I
I
It constitutes a good approximation of reality when the spectrum of oscillator defining it
seism is with broad frequency band and where the clean modes of the structure are quite separate them
from/to each other and are located inside or in the vicinity of this tape. It is in particular put in
defect if clean modes are at close frequencies or for distant modes
peak of excitation. [bib2]. The other methods of combination of the modal answers try
to correct this point.
4.5.1.3 Quadratic combination supplements (CQC)
The quadratic combination supplements (established by DER KIUREGHIAN [bib5]) makes a correction to
the preceding rule by introducing coefficients of correlation depending on depreciation and
distances between close clean modes:
R = R R
m
I I
I
I
1 2
1
2
I
I
1
2
with the coefficient of correlation:
8
(+)
I J I
J
I I
J
J
I
J
ij =
2
(
- 2 2
) +
4 2
(
+ 2) + 4 2
(
+ 2
)
2 2
I
J
I J I
J
I
J
I
J
I
J
or by introducing the report/ratio of pulsation or frequencies between two modes =/
J
I
8
(+)
I J
I
J
ij = 1
(- 2 2
) +
4 1
(+ 2) +
4 2 2
(
+ 2
I J
I
J)
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and for constant
8 2 1
(+)
ij =
1
(- 2 2
) + 4 2 1
(+ 2) + 8 2 2
4.5.1.4 Combination of ROSENBLUETH
This rule (proposed by E. ROSENBLUETH and J. ELORDY [bib6]) introduced a correlation enters
modes, different from that of method CQC. The answers of the oscillators are combined by
double nap (Double Sum Combination):
Rm = R R
1
I 2i 1i 2i
1
I
2
I
It requires an additional data, the duration S of the “strong” phase of the seism. The coefficient of
correlation is then:
- 1
“-”
2
2
I
J
= 1+
where '
1 2 ' and '
ij
“+”
I
I
I
I
I
=
-
=
+ S
I
I
J
J
I
4.5.1.5 Combination with rule of the 10%
The close modes (of which the frequencies different from less than 10%) are initially combined by
summation of the absolute values. The values resulting from this first combination are then
combined quadratically (simple quadratic combination). This method was proposed by
American payment U.S. Nuclear Regulatory Commission (Regulatory Guide 1.92 - Février 1976)
to attenuate the conservatism of the method of nap of the absolute values. It remains at fault
for structures with an own frequency spectrum dense and should not be used any more.
4.5.2 Contribution of the static correction of the neglected modes
The contribution of the pseudo mode Cf. [§4.3.1] can be combined quadratically because independence
with the contributions of the modes of vibration is not disputed.
4.6
Establishment of the directional response in multi supports
4.6.1 Calculation of the total answer
In this case, one retains like command of the combinations a step similar to that retained for
the excitation mono support without that being completely justified.
One establishes the directional answers for each movement &sj applied: one will note RjX it
result of this combination. To obtain this directional answer a news will thus be needed
stage of combination by taking account of the dependence or the independence of the &sj.
The flow diagram for treatment becomes:
·
for each movement imposed &sj calculation of the directional answers
R
= R2 + R2 + R2
jX
mj
cj
ej
Rmj answer combined of the modal oscillators
Rcj contribution of the static correction of the neglected modes (pseudo mode)
Rej contribution of the movement of drive of the support J
·
combination of the answers R jX
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4.6.2 Separate calculation of the components primary education and secondary of the answer
Each component is the subject of a similar separate processing. This step is adapted to the post
processing RCC-M in force for the seismic analysis of pipings [§ 4.9]:
primary component RIX (inertial answer):
·
for each movement imposed &sj calculation of the directional answers
R
= R2 + R2
I J X
mj
cj
Rmj answer combined of the modal oscillators
Rcj contribution of the static correction of the neglected modes (pseudo mode)
·
combination of answers IH J X
secondary component RII (quasi static answer):
·
combination of the Rej answers
4.6.3 Combined response of the modal oscillators
The response of the Rmj structure, in a direction of seism, is obtained by one of the combinations
possible of the contributions of each clean mode taken into account for this direction.
The selection criterion of the method of combination of the contributions of the modes is the same one as for
an excitation mono support and one will use method CQC preferentially.
4.6.4 Contribution of the pseudo mode
The corrective term by pseudo mode Cf. [§4.3.2] can be combined quadratically.
4.6.5 Contribution of the movements of drive
The movement of drive of the structure not being uniform, one can add a term with calculation
directional answer. This is not necessary if one chooses to consider this contribution
statics like a specific loading case inducing of the secondary constraints. This term is
defined starting from the maximum relative displacement which cannot be known starting from the only spectra of
pseudo absolute acceleration of the supports.
Rej = J
S
J max
Sj
static mode for the support J
jmax maximum relative displacement of the support J compared to a support of reference
(for which J max = 0)
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4.6.6 Combination of the directional answers of supports
This stage is obligatory, but the choice of the method of combination of the directional answers
remain very open. Indeed the assumption of independence of the &s J strongly depends on the clean modes
structure support of the studied equipment. An analysis of the studied system is necessary for
to gather the supports by groups: for examples for a piping connecting two buildings, the group
supports of the building 1, that of building 2 and finally that of the intermediate stanchions.
For each group one will be able to choose one of the three methods:
Combinaiso quadratiqu
N
E:
R =
2
X
RjX
Linear Combinaiso
N
:
R =
X
RjX
Combinaiso
value
in
N
:
absolute
R =
X
RjX
The rule of combination can be the same one for all the supports or differentiated according to the supports or
groups of supports defined by an occurrence of the key word factor COMB_MULT_APPUI. In this case
total answer is obtained by:
2
2
R =
R2 +
R +
R
X
Q X
L
has
Q
L
has
where Q supports combined quadratically, L linearly combined supports, has supports combined in
absolute value.
4.7
Combination of the directional answers
Two rules of combination of the directional answers are available.
4.7.1 Combination
quadratic
This combination corresponds to the assumption of strict independence of the answers in each
direction cf [§ 3.3.3]. Let us recall that this rule of combination does not have any geometrical significance,
although the three directions of analysis are orthogonal.
R = R2 + R2 + R2
X
Y
Z
The assumptions justifying the method of quadratic combination simple, on this level, do not seem
not to have to be called into question [bib3], but this method is not used.
4.7.2 Combination of NEWMARK
This rule of empirical combination is most usually used and in general leads to
estimates slightly stronger than the preceding one. It supposes that when one of the answers
directional is maximum, the other are with most equal to the 4/10 their contributions maximum
respective. For each direction I (X, Y, Z), one calculates the 8 values:
R = ± R ± 0 4
, R ± 0 4
, R
L
X
Y
Z
What leads, by circular shift, with 24 values and R = Max (Rl)
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4.8
Warning on the combinations
Several remarks are essential to warn the user on the way of using the methods
of combination and sizes combined in a note of study.
Notice 1:
If one wishes to use arithmetic combinations (direction) and combinations
quadratic (modes) the quadratic office pluralities must be always carried out in the last.
Notice 2:
Any quadratic combination applies only to the sizes for which in values
instantaneous the office plurality has the direction of a sum: combination of the components of displacement,
or effort generalized or of constraint of each clean mode.
The modal or directional quadratic combination cannot thus apply to intensities of
constraint (constraint principal, of Von Mises, Tresca).
Notice 3:
The results of a combination, whatever the rule of office plurality, should not be used as data
to calculate other sizes: for example a differential displacement between two points (or
a deformation) can be calculated only starting from modal differential displacements that one
combine then.
A fortiori the generalized efforts and the constraints can be calculated only mode by mode
before any combination and not starting from inertias deduced from the fields from acceleration
obtained by combination of modal accelerations.
4.9 Practical
lawful
4.9.1 Partition of the primary and secondary components of the answer
The various supports of a line of piping can be animated different movements. One
even section of piping can be left again on different buildings, levels or
different equipment. It thus undergoes a multiple excitation. This results in two types of
loading [§ 2.1.2]:
·
an excitation whose frequential contents vary from one support to another and who constitutes one
primary education loading according to classification RCC-M,
·
of Déplacements Différentiels Sismiques (DDS) inducing a state of stress by
displacements imposed on the supports and classified like secondary.
The generalized moments resulting from these 2 loadings intervene separately in
inequations of dimensioning RCC-M and on several levels. Thus, for pipings of levels
2 and 3, the DDS not taken into account with the inertial seismic loading in inequation 10, are
cumulated with the cases of displacements of thermal anchoring of origin in inequation 7.
In the sight of a post deepened processing RCC-M, It is thus necessary to have the components
primary education and secondary of the seismic answer.
In a more general way, the method of combination of the answers of supports can differ according to
whether one treats the case of the inertial or differential components. Moreover the number of support
concerned with these two summations can not be equal. One is caused to often to impose
overall differential movements even for supports associated with spectra users
different. In addition, of the DDS formulated in rotation are sometimes to consider. They cannot be
associated an inertial loading (limited to the translations).
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Code_Aster thus proposes two processing:
·
Determination of the total answer:
The contributions inertial and static of drive are cumulated during calculation of
directional answers of support [§4.6].
· Partition of the primary and secondary components of the total answer:
The two preceding contributions are not cumulated any more during the calculation of the answers
directional and are the subject of 2 independent processing:
The inertial component is obtained by removing the term of drive Re J in
calculation of the total answer [§ 4.6].
The static component is given by combining the terms of drive
defined under key word DEPL_MULT_APPUI. Methods of combination of these
loading cases DDS are indicated in key word COMB_DEPL_APPUI.
4.9.2 Method of the spectrum envelope
Even if pipings are subjected to a multiple seismic excitation, the current practice is of
to be reduced to the calculation of a structure mono-supported while preserving the loading cases DDS.
This simplified step implies to define a single spectrum by direction for all the supports of
piping. For each direction, one adopts a spectrum then “wraps” various spectra
with the supports. The spectra retained for the horizontal directions X and Y are identical.
In almost the whole of the cases, this method is generating of “margin of dimensioning”.
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5 Bibliography
[1]
“Paraseismic Engineering” collective Ouvrage - Presses of the E.N.P.C. (1985)
[2]
“State of the art as regards dynamic calculation of the structures” Jacques BETBEDER-MATIBET
in Paraseismic Génie (1985)
[3]
“The problem of the superposition of the clean modes in vibration. Use of the modes
clean of deformation of the elements. “ Vincent GUILLOT in Paraseismic Génie (1985)
[4]
“Calculation of the structures subjected to multiple excitations” Jean
RIGAUDEAU -
Pierre SOLLOGOUB in Paraseismic Génie (1985)
[5]
“A response spectrum method for random vibrations” DER KIUREGHIAN in Report
UCB/EERC - 80/15 Berkeley (1980)
[6]
“Response off linear systems to some transient disturbances” ROSENBLUETH, ELORDY in
Proceedings, Fourth World conference one earthquake engineering-Santiago off Chile (1969)
[7]
“Shorts communication: replacement for the SRSS method in seismic analysis” DER has
KIUREGHIAN, WILSON, BAYO in “Earthquake structural engineering and dynamics”, flight 9
(1980)
[8]
Transitory seismic analysis [R4.05.01] - J.R. LEVESQUE, Francoise WAECKEL
[9] Operator
COMB_SISM_MODAL [U4.84.01] E1 index
[10] Operator
CALC_FONCTION [U4.32.04] F1 index
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Transitory appendix 1 Réponse of a deadened simple oscillator
A1.1 Vibration forced of a system with a D.D.L. in translation
For a simple oscillator of rigidity K, mass m and viscous damping C, the equation of the movement
is form:
K
C
MX & + cX & + kX = 0
m
F X
X
for which the traditional notations are:
K
the own pulsation of the system not deadened:
0 = m
critical damping:
C criticize = 2 m0
the amortissementréduit:
C
C
=
=
(expressed as a percentage damping criticizes)
C criticizes 2 m 0
the own pulsation of the deadened system:
'=
2
0
0
(1 -)
F
the static deflection for a force F:
0
0
St = K
the reduced frequency:
= 0
reduced equation of the movement:
X & + 2 X & + 2
0
0 X = 0
The total response to a harmonic excitation of the form F (T) = F cos (T
0
) is the sum:
·
of a free answer X T
L () deadened oscillatory general solution where X l0 and
0
are given
by the initial conditions
X (T) = X
- T
E
0 Co (
S T
L
l0
0 + 0
)
·
of a forced answer X
T
F () permanent particular solution X (T) = X
cos (T
F
f0
-)
F
C
X
0
=
= arctg
f0
2
éq A1.1-1
K - m2 2 + (c)2
K - m
(
)
who is written in reduced form:
X
K X
f0
f0
1
2
=
=
= arctg
éq A1.1-2
F
St
0
(
2
2
2
1 - 2) + (
2)
1 -
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Harmonic response of a system to 1 ddl: modulate
30
Modulate
28
displacement
X relative/X static 24
function of
reduced frequency
20
16
Depreciation
20%
12
10%
05%
Relative amplitude
02%
8
01%
4
0
0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Reduced frequency
Harmonic response of a system to 1 ddl: phase
180
Phase of
displacement
160
X relative/X static 140
120
100
Depreciation
20%
Phase 80
10%
05%
60
02%
01%
40
20
0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Reduced frequency
Appear A1.1-a: Réponse of an oscillator in imposed force (module and phase)
The response to a harmonic excitation of the form F T = F
J T
()
0 E
with a forced answer solution is written
(J T)
particular permanent X (T) = X
F
f0 E
F
C
X
0
=
= arctg
f0
2
éq A1.1-3
K - m2 2 + (c)2
K - m
(
)
who is written in reduced form:
K X f0
1
2
=
H
2
(J) = arctg
éq A1.1-4
F
2
0
1 - + J 2
1 -
where H (J) is the harmonic answer complexes of a simple oscillator
1
H (J) = (
2
1 - 2) + (
2) 2
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Appendix 2 Mouvement imposed of a system on a D.D.L. in
translation
A2.1 Mouvement absolute of a system with a D.D.L.
For a simple oscillator of rigidity K, mass m and viscous damping C, the equation of the movement
absolute is form:
MX & + (
C X & - s&) + K (X - S) = 0
K
C
S
MX & + cX & + kX = ks + cs&
m
X & + 2 X
2
2
0 & +
X =
S + 2
S
0
0
0 &
F X
X
The response forced to a harmonic imposed movement of the form S (T) = S cos (T
0
) is form
X (T) = X
cos (T
m
m0
- -
1
2) nap of two terms of answer, particular solutions permanent:
·
term induced by the excitation in displacement X
cos (T
d0
- D
)
K S
C
X
0
=
= arctg
d0
D
2
K - m2 2 + (c)2
K - m
(
)
·
term induced by the excitation of speed X
cos (T
vo
- v
)
C S
C
X
0
=
= arctg
v0
v
2
(K - m2) 2 + (c)2
K - m
what leads to a total forced answer
K 2 + C 2
X (T) = X cos (T - -)
()
S
[
cos T
m
m
- -
1
2
0
1
2
(K - m2) 2 + (c)2]
(
)
from where the reduced form of the absolute amplitude:
2
X
1 + 2
2
1
m
(
)
=
= arctg
= arctg
S
2
1
2
2
0
[1 (- 2 2
) + (2)]
1 -
2
If the movement imposed on the base is expressed in form S T complexes =
(S J T
() Re 0 E), the relative amplitude
or transmissibility can be written starting from the harmonic answer complexes of an oscillator simple H (J)
X m =
2
1 + (2) H (J)
éq A2.1-1
s0
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A2.2 Mouvement relating of a system to a D.D.L.
The problem of the response to an imposed movement can be dealt with in relative displacement of the mass by
report/ratio at the base by posing X = X - S.
The equation of the relative movement for a harmonic imposed movement of the form S (T) = S cos (T
0
) is
then of the form m X & + C X & + K X = - m S & or in reduced form:
&
X + 2
&x + 2 X = - &s = 2
S
(
cos T
0
0
0
) éq A2.2-1
The relative forced answer is then, for a permanent solution X
(T
m0 cos -),
m2 S
C
X
0
=
= arctg
m0
2
éq A2.2-2
(K - m2 2
) + (c)2
K - m
who is written in reduced form:
X
2
m0
=
éq A2.2-3
s0
(- 2 2
) + () 2
1
2
Harmonic response of a system to 1 ddl: relative module
24
21
Modulate displacement
X relative/X static
18
function of
the reduced frequency
15
12
Depreciation
20%
9
10%
5%
2%
6
1%
Relative amplitude
3
0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Reduced frequency
Appear A2.2-a: Réponse of an oscillator moving imposed (module of relative displacement)
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Appendix 3 Mouvement imposed not periodical of a system on one
D.D.L.
The problem dealt with previously was limited to a periodic imposed movement. For an excitation not
periodical, of variable amplitude with time, being exerted for one finished length of time.
Impulse A3.1 Réponse
The simplest form is the unit impulse force, which applied to a rest mass front
the application of the impulse (X = X = for T < or T = -
& 0
0
0) can be written
~
t+ T
F = lim F dt = F.dt = 1 = m &X (T =)
0 - m &X (T = -
0) = m &X0
T
T
0
1
The initial conditions are then noted X (T =)
0 = X = 0 and X & (T =)
0 = X
0
&0 =
m
The general equation of the response in free vibration of a system to a D.D.L.
X & + X
X T
- T
0
() = E
X cost
0
0 0
+
sin T
L
0
0
0
0
G (T) of a system then becomes the impulse response to a D.D.L.
- T
0
E
X (T) = G (T) =
sin T
L
éq A3.1-1
m
0
0
~
F
For an impulse nonunit F = F.T
initial speed is &X0 = and the answer becomes
m
~
F - T
E
0
~
X (T) =
sin T = F G (T
L
)
m
0
éq A3.1-2
0
If the impulse force is applied to one unspecified moment the answer is
~
X T = F G (T
L ()
-)
A3.2 Réponse in unspecified forced vibration
The force of excitation F (T) can be broken up into a series of impulses of variable amplitude F ()
applied to the moment during a time. If the 0 response to one moment T is obtained by
T
X (T) =
F () G (T -
) D
0
and while replacing by the expression of the impulse response [éq A.3-2] one obtains the equation of convolution
for a system at rest at moment 0 of the form
1
T
X (T) =
F () - 0 (T -)
E
sin 0 (T -) D
éq A3.2-1
m 0 0
known under the name of Intégrale of DUHAMEL
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A3.3 Réponse moving unspecified imposed
For an analysis moving relative represented by [éq An2-2]
&
X + 2
2
2
0
&x + X = -
0
&s = s0
(
cos T)
the integral of DUHAMEL becomes
1 T
X (T) =
S (
&) - 0 (T -)
E
sin 0 (T -) D
éq
A3.3-1
0 0
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