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Seismic response by transitory analysis
Date:
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Organization (S): EDF/EP/AMV
Handbook of Référence
R4.05 booklet: Seismic analysis
Document: R4.05.01
Seismic response by transitory analysis
Summary
The methods most frequently used for the seismic analysis of the structures are the methods
spectral and transitory methods.
The transitory methods (direct linear or not, by modal synthesis) make it possible to calculate the answer of
structures under the effect of imposed seisms: single excitation (identical of each point of anchoring of
structure) or multiple and to take into account their possible nonlinear behavior.
With regard to the spectral methods, one calculates the maximum answer, for each mode of vibration,
of each point of anchoring. The maximum response of the whole of the structure is then determined by
combination of the maximum answers of the modes. This type of analysis is clarified in the documentation of
reference [R4.05.03].
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Contents
1 seismic Behavior of a structure ............................................................................................... 3
1.1 ........................................................................................................................................ Definitions 3
2 seismic Response of a system to a degree of freedom ......................................................................... 3
3 seismic Response of a system to several degrees of freedom ............................................................. 5
3.1 Equations of the movement in the absolute reference mark ............................................................................ 5
3.2 Equations of the movement in the relative reference mark .............................................................................. 6
3.2.1 Decomposition of the absolute movement ................................................................................... 6
3.2.2 Simple excitation or multiple .................................................................................................. 7
3.2.3 Modeling of damping ............................................................................................ 9
3.2.4 Fundamental equation of dynamics ............................................................................... 9
3.3 Calculation of the seismic loading ...................................................................................................... 9
3.4 Loading of incidental the wave type .............................................................................................. 10
4 transitory seismic Response by modal synthesis ............................................................................ 11
4.1 Description of the method ............................................................................................................. 11
4.2 Choice of the modal base ............................................................................................................... 12
4.3 Calculation of the dynamic response of the structure studied by modal synthesis ............................ 12
4.4 Taking into account of the modes neglected by static correction ....................................................... 13
4.5 Taking into account of the character multi - supported of a structure ..................................................... 14
4.6 Post processing ............................................................................................................................ 14
5 direct transitory seismic Answer ................................................................................................... 15
5.1 Taking into account of a damping are equivalent to modal damping .................................. 15
5.2 Taking into account of a stress multi supports with restitutions of the relative fields and absolus15
6 Interaction ground-structure ....................................................................................................................... 16
6.1 Impedance of a foundation ........................................................................................................... 16
6.2 Taking into account of a modal damping calculated according to the rule of the RCC-G ............................. 17
6.3 Distribution of the stiffnesses and damping of ground ......................................................................... 18
6.4 Taking into account of an absorbing border ................................................................................. 19
7 Bibliography ........................................................................................................................................ 20
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1
Seismic behavior of a structure
1.1 Definitions
The analysis of the seismic behavior of a structure consists in studying its response to a movement
imposed: an acceleration, in its various supports. Imposed acceleration is a temporal signal
(T) called accélérogramme (cf [Figure 1.1-a]).
Appear 1.1-a: Accélérogramme LBNS
The seismic movement considered in calculation is is a real accélérogramme known and read by
operator LIRE_FONCTION [U4.21.08] is a synthetic accélérogramme calculated directly in
the code, for example with procedure FORMULE [U4.21.11].
2
Seismic response of a system to a degree of freedom
That is to say a simple oscillator made up of a mass m connected to a fixed point by a spring K and one
damping device C which can move in only one direction X (cf [Figure 2-a]). This oscillator with one
degree of freedom is subjected to a accélérogramme (T) horizontal in its support (not A).
locate galiléen
support
K
m
Xe
With
C
xr
teststemxà
Appear simple oscillating 2-a: subjected to a seismic stress
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Displacements of the oscillator are measured or calculated, that is to say in a relative reference mark related to point a:
relative displacement xr, is in an absolute reference mark (Ra): absolute displacement teststemxà. Displacement
absolute teststemxà breaks up into a uniform displacement of drive in translation Xe and one
relative displacement xr:
X = X + X
has
R
E
éq 2-1
One deduces from it by derivation the relation between accelerations:
!
X =
+
with (T) = X
has
!
X
(T)
R
! E!
éq 2-2
The mass is subjected to a horizontal force of recall which is proportional to relative displacement:
F = - K. X
R
R and with a horizontal force of damping presumedly proportional to the relative speed:
F = - C.X
v
!r.
The equation of the movement of the mass is written then: - K. X - C.X! = Mr. X
R
R
! has
! .
Maybe, taking into account the equations [éq 2-1] and [éq 2-2]:
Mr. X! + c.x! + k.x = - Mr. (T) = p (T
R
R
R
)
éq 2-3
Note:
The study of the seismic response of an oscillator to a degree of freedom in the relative reference mark consists
thus in the study of the response of an oscillator to a force (
p T) of an unspecified form. The solution
equation of motion [éq 2-3] is then provided by the integral of Duhamel:
1
T
X =
p () .e-.(T) .sin
.
.

[D (T -)]D
R
m D 0
with:
p (T) = - Mr. (T)
K
C
=
, =
=. 1 - 2
and
m
.
2 m
D
.
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3
Seismic response of a system to several degrees of
freedom
3.1
Equations of the movement in the absolute reference mark
The balance of a mechanical system consists in writing, some is the moment of calculation considered, that
summon internal forces, inertias and of damping is equal to the imposed external forces
on this known as system: F
+ F
+ F
= F
iner
amo
int
ext.
In the case of a linear behavior, known the system is represented by a model of finite elements
or of discrete elements, one has (after discretization):
F
iner = MR. X!has
F
int = K Teststemxà
·
Teststemxà is the vector of nodal displacements of the discretized structure, in the reference mark
absolute;
·
M is the matrix masses structure;
·
K is the matrix stiffness of the structure;
·
F
= F - F
ext.
E
C is the vector of the forces imposed on the studied structure, FC that of
possible forces of shock (cf [R5.06.03]).
To simplify the presentation, it is considered that the structure is only requested by
displacements imposed on the level of its various supports. Thus, Fe = 0.
With an aim of simplifying the presentation, one generally separates the degrees of freedom into two, in
function of their type:
· degrees of freedom of structure not subjected to an imposed movement - also called
active degrees of freedom - they are the unknown factors of the problem;
· degrees of freedom of structure subjected to an imposed movement - also called
ddl_impo - they are the boundary conditions in displacement of the problem (limiting conditions
of Dirichlet).
On the edges of the structure where Xs displacements are imposed, one a: B X = X
has
S.B is
stamp passage of all the degrees of freedom of the structure to the degrees of freedom of structure
subjected to an imposed movement.
The balance of the system is written then, some is v pertaining to the space of displacements
kinematically acceptable i.e., some such as B v = 0 are v:
MR. X!+ Famo has + K Teststemxà - Fext, v = 0
B X

= Xs has
That is to say:
M
X!
T
+ Famo has + K Teststemxà - Fext = - B.

éq 3.1-1
B X

= Xs has
F
B
= - T has. is the vector of the forces of reactions exerted by the supports on the structure.
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By taking account of the partition of the degrees of freedom, the vector of displacements in the reference mark
X
has
m
m xs
absolute is written: Teststemxà =. The operators describing the structure become: M =
,
X


S
m
m
sx
S
K
K xs
K =
with m
MT
=
and K
kT
=
and the vector of the external forces applied to
K
K

sx
xs
sx
xs
sx
S
- FC
the structure is written: Fext =
.
0
The fundamental equation of dynamics in the absolute reference frame is written then, by taking account of
the partition of the degrees of freedom:
m

m X
!
K
K X
- F
xs
has

. + F
xs
has
C
+
. =

m

m
X
amo

!
K
K
X
F
sx
S S
sx
S S
has
Maybe, by considering only the active degrees of freedom:
m X! + F
+ K X = - F - m X
has
amo
has
C
xs!
- K X
S
xs
S
This approach requires the knowledge of displacements and absolute velocities associated
the accélérogramme (T) but the recorders measure either of accelerations or speeds. One can
to go up with displacements by simple or double integration with command CALC_FONCTION
[U4.62.04]. However, uncertainties of measurement give drifts which it is advisable to correct:
displacements are thus well-known than speeds and accelerations. One will keep in memory
orders of magnitude of the maximum amplitudes following:
· some tenth of “G” for accelerations;
· a few tens of cm/S for speeds;
· a few tens of cm for displacements.
One will also make sure that at the end of the seism speed and displacement are realistic i.e. with
more few tens of cm for displacement, null for speed.
3.2
Equations of the movement in the relative reference mark
3.2.1 Decomposition of the absolute movement
The stresses undergone by a structure at the time of a seism are classified in two types in the rules
of construction (ASME, RCC-M):
· constraints induced by the relative movement of the structure compared to its deformation
primary statics or constraints. These stresses are due to the effects inertial of the seism;
· constraints induced by differential displacements of anchorings or constraints
secondaries.
Generally, one thus breaks up the study of the structures into the study of the static deformation due to
movements of the supports (it is the movement of drive) and in the study of the vibrations induced by
accelerations of the supports around this deformation (it is the relative movement).
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X

Teststemxà
S displacement imposed of the supports
Xe
absolute movement Teststemxà
Xr
movement of drive Xe
relative movement Xr
(Ra)
The absolute displacement of any point M of the structure, not subjected to an imposed displacement, is equal
with the sum of relative displacement and displacement of drive of this point:
X (M) = X (M) + X (M
has
R
E
)
éq 3.2.1-1
That is to say:
·
Teststemxà, the vector of displacements in the absolute reference frame;
·
Xr, the vector of definite relative displacements like the vector of displacements of
structure compared to the deformation which it would have under the static action of displacements
imposed on the level of the supports. Xr is thus null at the points of anchoring: B Xr = 0;
·
Xe, the vector of the displacements of drive defined as displacements of
structure requested statically by imposed displacements of the supports
B
X = X
E
S

= R +
X
.X.
K X = - B

T
E
E =
S
E
. with

E
is the matrix of the static modes. The static modes represent, in the absence of
external forces, the response of the structure to a unit displacement imposed on each
degree of freedom of connection (others being blocked).
3.2.2 Simple or multiple excitation
To clarify more in detail the approach moving relative, and more particularly the calculation of
components of drive, it is necessary to introduce concept of the simple or multiple excitation.
3.2.2.1 Excitation
simple
It is considered that the imposed seismic movement is a solid movement of body. One says
generally that the structure mono - is supported.
Teststemxà (T)
Xr (T)
Xs (T)
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The absolute displacement of any point M of the structure, not subjected to an imposed displacement
thus break up into a relative displacement compared to a pointer related to the support where is
imposed the seismic movement and in a rigid displacement of drive.
In this case, the static modes correspond to the six modes of rigid body. Like the structure
is linear rubber band, one separately studies the effects of the six components of the seismic movement.
For each seismic direction, one writes simply the inertias induced by the seism under
following form:
P (T) = - Mr. X
! S = - (T) .M
·
(T) seismic movement in a direction is the accélérogramme;
·
is mode of the solid and unit body in this direction;
· The seismographs measure only signals of translation. To consider that the structure
studied mono is supported amounts supposing that all its points of supports undergo
even translation. In this case, the components of [] are worth 1 for the degrees of freedom
who correspond to displacement in the seismic direction considered and 0 for
degrees of freedom which correspond to displacement in seismic directions
perpendiculars with that considered or rotations.

However, considering the size of the models, the complete seismic analysis of equipment is carried out
generally in several stages. Detailed seismic analysis of the equipment considered
use then as excitations, the accelerations calculated at the time of the first stage. They
compose of the six accélérogrammes of translation and rotation. The three are thus calculated
modes corresponding to imposed displacements of translation and the three modes
corresponding to imposed displacements of rotation. If the seismic movement is one


imposed rotation, in a point M, M = OM for the degrees of freedom which
correspond to displacement of translation and for the degrees of freedom which
correspond to rotations.
3.2.2.2 Excitation
multiple
One cannot always only consider:
· the accelerations undergone by the whole of the points of anchoring of the studied structure are
identical and in phase;
· the supports indeformable and are actuated by the same movement of rigid body.

In this case, one says that the structure is multi - supported. Static modes =
Id
correspond then to the 6.nb_supports static modes (or 3.nb_supports modes) where nb_supports is
the number of accélérogrammes different undergone simultaneously by the structure. They are calculated by
operator MODE_STATIQUE [U4.52.04] with option DDL_IMPO. They are solution of the equation
following:
X = X
E
S
K

K 0

xs
that is to say
. =

éq 3.2.2.2-1
K X = - B



T
E
. E
K
K
Id
F
sx
S

has
E
Maybe, by considering only the active degrees of freedom: K. + K .Id = 0.
xs
The inertias induced by the seism are written then simply:
nb_supports
P (T) = -
Mr. .X
m! S (T)
m
m=1
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3.2.3 Modeling of damping
It is considered that the damping dissipated by the structure is of viscous type i.e. the force
of damping is proportional to the relative speed of the structure:
F
= C X
amo
! R where C is the matrix of damping of the structure.
That amounts neglecting the effect imposed speed. Indeed, one can more generally write:
F
= C X! = C X! + C .X
amo
has
R
! S.
In the case of a uniform excitation at the base (case of the mono-support), damping only intervenes
on relative displacements (the forces of damping are null for a rigid displacement).
In the case of a multiple excitation where the static solution is not any more one rigid displacement,
to consider that the force of damping is proportional to the relative speed of the structure is one
simplifying assumption.
3.2.4 Fundamental equation of dynamics
The fundamental equation of dynamics [éq 3.1-1], in the relative reference mark, is written then, taking into account
equations [éq 3.2.1-1] and [éq 3.2.2.2-1]:
MR. X
! + C X! + K X = - M.X! + F - BT.
R
R
R
S
ext.
R
éq 3.2.4-1
Maybe, by partitionnant the degrees of freedom:
m

m X
! C
C X
! K

K X
- F
.
.

xs
R
xs
R
xs
R
C
(m + m I
xs
) dx!S
. +
. +
. =
-

m

m

0
C
C
0
K

K
0
F
sx
S
sx
S
sx
S

has
R
(Mr. + Mr. I

sx
S
) dx!S
with C
= cT
sx
xs
Maybe, by considering only the active degrees of freedom:
Mr. X! + c.x
R
! + k.x = - F
R
R
C - (Mr. + Mr. I
xs
) dx!S
Principal advantages of the approach in relative displacement compared to that in displacement
absolute are as follows:
· it is not necessary to integrate the accélérogramme (T);
· relative displacements obtained make it possible to determine the primary constraints directly
induced by the seism.
3.3
Calculation of the seismic loading
The seismic loading (cf [§3.2]) - Mr. is - (Mr. + Mr. I
xs
) dx!S on the degrees of freedom
credits is built by operator CALC_CHAR_SEISME [U4.43.01]. It is usable directly at the time
of a direct transitory analysis with DYNA_LINE_TRAN [U4.54.01] or of a transitory analysis by
modal synthesis with DYNA_TRAN_MODAL [U4.54.03]. On the other hand, during a transitory analysis
direct nonlinear with DYNA_NON_LINE [U4.32.02], it should be transformed into a concept of the type
charge. This is carried out starting from operator AFFE_CHAR_MECA [U4.25.01] in the following way:
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char_sei = CALC_CHAR_SEISME (...);
charge = AFFE_CHAR_MECA (MODELE:…, VECT_ASSE: char_sei);
dyna_nlin = DYNA_NON_LINE (
excit (CHARGE: con_lim, CHARGE: cham_no, FONC_MULT: acceler)
…) ;
In the case of a supported mono structure, it is enough to indicate the direction of the seism:
mono_x = CALC_CHAR_SEISME (MATR_ASSE: mass,
DIRECTORATE (...), MONO_APPUI:“OUI”);
In the case of a structure multi supported, should as a preliminary have been calculated the static modes. One
calculate as many seismic loadings of supports which undergo a different acceleration.
multi_xi = CALC_CHAR_SEISME (MATR_ASSE: mass, DIRECTION (...),
NOEUD: NOI, MODE_STAT: mode_stat,);
3.4
Loading of incidental the wave type
It is also possible to impose a seismic loading by wave planes via
order AFFE_CHAR_MECA and the key word factor ONDE_PLANE. That corresponds to the loadings
classically met during calculations of interaction ground-structure by the integral equations.
In harmonic, a wave planes elastic is characterized by its direction, its pulsation and its type
(wave P for the waves of compression, waves SV or HS for the waves of shearing). In
transient, the data of the pulsation, corresponding to a standing wave in time, must be
replaced by the data of a profile of displacement which one will take into account the propagation with
run from time in the direction of the wave.
More precisely, one characterizes:
· a wave P by the function U (X, T) = F (k.x - C T) K
p
· a wave S by the function U (X, T) = F (k.x - C T) K
S
With:
·
K, unit vector of direction
·
F then represents the profile of the wave given according to the direction K.
O
“Principal” face of wave
K
corresponding at the origin
profile
H
Function F
H0 is the distance from the principal face of wave in the beginning O, carried by the directing vector of the wave with
the initial moment of calculation, H the distance from the principal face of wave in the beginning O, one unspecified moment.
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Note:
This type of load is available in a direct transitory calculation linear DYNA_LINE_TRAN or
not DYNA_NON_LINE.
The use of this type of loading will be detailed in a specific note.
4
Transitory seismic response by modal synthesis
4.1
Description of the method
The method of modal recombination consists in breaking up the relative movement of the structure on
the base of the clean modes. As this one is null on the level of the supports, one projects the equation of
dynamics on the basis of blocked clean mode (clean modes obtained by blocking all them
degrees of freedom of connection).
X
Q
R =.
·
is the matrix of the blocked clean modes;
·
Q the vector of the unknown factors generalized on the basis of blocked clean mode.
The blocked clean modes are solution of:
(
0
K -- 2i.M) I = where F
F

I are the modal reactions at the points of supports.
I
The equation of the movement projected on the basis of dynamic mode is written then:
MR. Q
! (T) + C! (
Q T) + K Q (T)
T
= - .M..X
T
! S + .F
T
ext. - .B T
G
G
G
.r
where MR. G, CG and K G are the matrices of mass, of generalized damping and stiffness. For
to simplify, one considers that they are diagonal. The matrix of damping generalized CG too
because it is supposed that the assumption of Basile is checked (the matrix of damping is a combination
linear of the matrices masses and stiffness).
Maybe, by considering only the active degrees of freedom:
Mr.! (
Q T) + C. !(
Q T) + K. (
Q T)
T
= - .f
T
C -.(Mr. + Mr. I
xs
) D X
G
G
G
! S
In the absence of shock, one is thus led to solve a whole of uncoupled equations (there is
as much as clean modes).
Note:
It is possible to calculate a modal base with nondiagonal matrices. It is enough to
to specify during the construction of the classification generalized by the key word
STOCKAGE: “PLEIN” of command NUME_DDL_GENE [U4.55.07].
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4.2
Choice of the modal base
For the seismic analysis of a linear structure, it would be necessary in theory to retain all the modes of which them
Eigen frequencies are lower than the cut-off frequency (generally about 33 Hz).
In practice, one is often satisfied to preserve in the modal base only the modes which
contribute to a significant degree to the answer. One then preserves only the modes of which mass
effective unit in a direction is higher than 1 and one also makes sure that, for
the whole of these modes selected, the unit effective mass cumulated in each direction is little
different from the total mass of the structure (higher than 90%). The criterion of office plurality of the masses
modal effective is reached by connecting the following operators:
· Calculation of the total mass of the structure: POST_ELEM [U4.61.04]
masse_in = POST_ELEM (MASS_INER: (TOUT: “OUI”))
· Calculation of the blocked dynamic clean modes
: they are calculated in the operator
MODE_ITER_SIMULT [U4.52.02] or in MODE_ITER_INV [U4.52.01] according to the selected method.
mode = MODE_ITER_SIMULT (); or mode = MODE_ITER_INV ();
· Standardization of the modes compared to the generalized mass: NORM_MODE [U4.64.02]
NORM_MODE (MODE: mode, NORME:“MASSE_GENE”, MASSE_INER: masse_in);
· Extraction of the modal base of the modes whose unit effective mass exceeds a certain threshold
(1 for example) and checking which the extracted modes represent at least 90% of the mass
total: EXTR_MODE [U4.64.03]
EXTR_MODE (
FILTRE_MODE (MODE: mode, CRIT_EXTRE: “MASSE_EFFE_UN”, THRESHOLD:1.e-3)
IMPRESSION (OFFICE PLURALITY:“OUI”);
Note:
Macro command MACRO_MODE_MECA [U4.52.05] makes it possible to connect the unit directly
of the three last preceding commands.
Attention, certain local answers (in the particular case of nonlocalized linearities) can be
strongly influenced by modes of a higher nature whose frequency is beyond the frequency
of cut and whose effective modal mass is low (lower than 1). Key word VERI_CHOC of
order DYNA_TRAN_MODAL [U4.54.03] allows to check a posteriori that the selected modal base
is sufficient. If it is not the case, one highly advises to supplement it.
4.3 Calculation of the dynamic response of the structure studied by
modal synthesis
After having calculated the base of the dynamic clean modes and having built a generalized classification
by NUME_DDL_GENE [U4.55.07], one projects then the matrices of mass, damping and of
stiffness, on this same basis with the operator PROJ_MATR_BASE [U4.55.01], vectors second
member with PROJ_VECT_BASE [U4.55.02].
Note:
Macro command MACRO_PROJ_BASE [U4.55.11] makes it possible to connect the unit directly
of the three operations.
The matrices and vectors thus projected, one calculates the generalized response of the mono system or
multi-excited using operator DYNA_TRAN_MODAL [U4.54.03].
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Seismic response by transitory analysis
Date:
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Author (S):
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Key:
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4.4
Taking into account of the modes neglected by static correction
During the calculation of the generalized response of an excited mono structure, it is possible to take in
count, a posteriori, the static effect of the neglected modes. In this case, once reconsidered the base
physique one corrects the value of relative displacement calculated (respectively relative speed and
relative acceleration) by the contribution of a pseudo-mode. The pseudo-mode is defined by
difference between the static mode associated the unit loading of constant acceleration type
imposed and projection on the calculated dynamic modes of displacement (respectively
relative speed and relative acceleration).
One has then:


p

X
= X +
F (T).
I
I
-
.
r_corrigé
R


J
J
I


j=1




p


!X
F T

r_corrigé =!
X +! ().
R
I
I -! .
J
J

I


j=1




p

!X
F T

r_corrigé =!X +! ().
R
I
I -! .

J
J
I


j=1



Multiplicative functions of time F (T
I
) correspond to the accélérogramme imposed I (T) in
each direction I considered.
The step to be followed is as follows:
· Calculation of the unit loading of type forces imposed (constant acceleration) in the direction of
seism: AFFE_CHAR_MECA [U4.25.01]. One will pay attention to permute the sign of the direction
since the seismic inertia is form (
P T) = - Mr.
. X!S
cham_no = AFFE_CHAR_MECA (MODELE:model, PESANTEUR:(VALE, DIRECTION)) ;
· Calculation of the linear static response of the structure to the preceding loading case:
MACRO_ELAS_MULT [U4.31.03].
mode_cor = MACRO_ELAS_MULT (CHAR_MECA_GLOBAL: con_lim,…
CAS_CHARGE: (NOM_CAS:“xx”, CHAR_MECA: cham_no)) ;
It will be noted that there is as many loading case of direction of seism
· Calculation of the derived first and second of the accélérogramme: CALC_FONCTION [U4.62.04].
deri_pre and deri_sec = CALC_FONCTION (OPTION: DERIVE);
· Calculation of the answer generalized by taking of to account the modes neglected by correction
statics:
dyna_mod = DYNA_TRAN_MODAL (MASS_GENE: , RIGI_GENE:
MODE_CORR: mode_cor
EXCIT (CORR_STAT: “OUI”
D_FONC_DT: deri_pre, D_FONC_DT2: deri_sec.)
…) ;
Note:
In the case of an excited structure multi, the taking into account of the modes neglected by correction
statics is not developed. One postraite absolute displacement in this case.
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4.5
Taking into account of the character multi - supported of a structure
It was seen previously (cf [§3.3]) that to calculate the seismic loading in the case of one
structure multi supported, should as a preliminary have been calculated the static modes. If one wants to be able
to restore the sizes calculated in the absolute reference mark or if one wants to be able to take into account
not located linearities, it also should be specified in DYNA_TRAN_MODAL that the studied structure
is multi excited. Indeed, in this last case, one compares at every moment, the vector of
absolute displacements of each point of shock considered, in order to determine if there is shock and of
to calculate the corresponding forces of shock.
The step to be followed is as follows:
· Calculation of the static modes: MODE_STATIQUE [U4.52.04].
mode_stat = MODE_STATIQUE (DDL_IMPO: (...));
· Calculation of the answer generalized by taking of account the component of drive:
dyna_mod = DYNA_TRAN_MODAL (MASS_GENE: , RIGI_GENE:
MODE_STAT: mode_stat
EXCIT (MULT_APPUI: “OUI”
ACCE: accelero, VITE: speed, DEPL: move
DIRECTION: (...), NOEUD:NO1
…)
…) ;
4.6 Post
processing
Operators REST_BASE_PHYS [U4.64.01] or RECU_FONCTION [U4.62.03] can then restore
in physical space calculated evolutions:
· the operator
REST_BASE_PHYS restores overall (the complete field) displacements,
speeds and accelerations;
· the operator
RECU_FONCTION restores locally (temporal evolution of a degree of freedom)
displacements, speeds and accelerations.
One can restore the relative sizes by specifying (MULT_APPUI: “NON”) or sizes
absolute by (MULT_APPUI: “OUI”).
One obtains then displacements of drive necessary to the calculation of the secondary sizes in
withdrawing from absolute displacements relative displacements. This is carried out by the command
CALC_FONCTION [U4.62.04] option COMB.
From the preceding evolutions, one can also extract the max. values and RMS and calculate it
spectrum of response of associated oscillator. This is carried out by command CALC_FONCTION options
MAX, RMS and SRO.
Handbook of Référence
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Seismic response by transitory analysis
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5
Direct transitory seismic answer
Direct integration is realizable is with assumptions of linear behavior: operator
DYNA_LINE_TRAN [U4.54.01] is with assumptions of nonlinear behavior: operator
DYNA_NON_LINE [U4.32.02]. Setting with share the way of taking into account the seismic loading
(cf [§3.3]), syntaxes of DYNA_NON_LINE and DYNA_LINE_TRAN are identical.
5.1 Taking into account of a damping are equivalent to damping
modal
Generally, the most precise information that one has on damping comes from the tests from
vibration which makes it possible to determine, for a frequency of resonance given fi, the width of
corresponding resonance and thus damping reduces I to this resonance. It is thus
necessary to be able to take into account, in a direct transitory calculation, a damping
equivalent with modal damping.
From the spectral development of the matrix identity:
n_mod be X XT K n_modes X XT K
Id =
I
I
=
I
I
T

2
I 1
X K X
I 1
M
=
I
I
=
G_i .i
one shows:
· that one can develop the matrix of damping of the structure C in series of modes
clean:
n_mod be
T
C = have.(K.I
) (K.I
)
i=1
· and that, account held of the definition of the critical percentage of damping:

T
I
I.
C I =.
2 MG_i .i. I .ai =.
2 KG_i.i
It is thus advised with the user to specify (syntaxes of DYNA_NON_LINE and DYNA_LINE_TRAN
are identical), the values of modal depreciation for each Eigen frequency by
the intermediary of the key word factor AMOR_MODAL.
That amounts imposing a force of damping proportional to the relative speed of the structure:
n_mod be

F
= C X
I
T
amo
! R with C = 2.
.(K.I
) (K.I
)
K
i=
.
G I
1
_
I
5.2 Taking into account of a stress multi supports with restitutions
relative and absolute fields
By defect, the sizes are calculated in the relative reference mark. In DYNA_NON_LINE and
DYNA_LINE_TRAN, one uses a syntax identical to that of DYNA_TRAN_MODAL (presence of the words
keys MODE_STAT and MULT_APPUI: “OUI”) to calculate them in the absolute reference mark.
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6 Interaction
ground-structure
The seismic behavior of a building depends on the characteristics of the ground on which it is posed
since it depends on the seismic movement imposed on the ground and the dynamic behavior of the building and
of its foundations. The interaction ground-structure most frequently contributes to decrease the answer of
studied structure.
6.1
Impedance of a foundation
That is to say a surface rigid foundation without mass, subjected to a harmonic force of pulsation:
P T
it
() = P.E
0
It is thus actuated by a movement X (T) of the same frequency. One calls
impedance of the foundation, the complex number (
K), function of the frequency such as:
(
P (T)
K) =
.
X (T)
Several analytical or numerical methods make it possible to calculate the impedance of a foundation
according to the complexity of the foundation and ground on which it is posed or partially hidden. Among
most frequently used, one quotes:
· analytical methods within the competences of WOLF or DELEUZE where it is supposed that it
to erase circular, rigid and is posed on a homogeneous ground. The foundation must be surface;
· numerical method of code CLASSI where it is supposed that the foundation raft is of form
unspecified, rigid and posed on a possibly laminated ground. The foundation must be
surface;
· numerical method of the code MISS 3D where the foundation raft can be of an unspecified form,
possibly deformable and posed on a possibly laminated ground.
It is possible to treat the interaction ground-foundation by the frequential method of coupling (taken in
count frequency response of the matrix of impedance) by carrying out a coupled calculation
MISS 3D/Code_Aster. This type of calculations is not detailed in this reference material. One
present here only the case more the current where the interaction ground-foundation is treated by the method
springs of ground (it is considered that the terms of the matrix of impedance are independent of
frequency).
In the case of a surface rigid foundation, the impedance is calculated in the center of gravity of
surface in contact in a reference mark related to the principal axes of inertia of this surface. For each
frequency, it is expressed in the shape of a matrix of dimension (6, 6). One adjusts then the value
of each term according to a particular clean mode of the building studied in blocked base:
· frequency of the first mode of swinging 0 for the horizontal stiffnesses
Kx (0), Ky (0) and of rotation Krx (0), Kry (0);
· frequency of the first mode of pumping

1 for the vertical stiffness Kz (1) and of torsion
Krz (1).
As the Eigen frequencies of the building depend on the stiffnesses of ground, the calculation of the values
total within the six competences of ground results from an illustrated iterative process appears [Figure 6.1-a].
first stiffnesses of ground Kx (0), Ky (0), Kz (1), Krx (0), Kry (0) and Krz (1) are selected
according to the first Eigen frequencies of swinging (0) and pumping (1) of
structure in blocked base. The stiffnesses of grounds are then adjusted at the first frequencies
clean significant of the structure on spring until correspondence of the frequencies to which
functions of impedance are calculated with the values of the Eigen frequencies of the coupled system
ground-building.
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Selection of the stiffnesses of ground
Kv (J) and K (I) in the matrix
of impedance
Code_Aster
Modal calculation with structure on
comes out from ground.
Not
Eigen frequencies
models on spring
correspond to
frequencies I and J
Yes
End
Appear 6.1-a: Processus of adjustment of the stiffnesses of ground
6.2 Taking into account of a modal damping calculated according to the rule
RCC-G
One breaks up damping due on the ground into part of material origin and a part
geometrical: damping due to the reflection of the elastic waves in the ground.
The rule of the RCC_G consists in summoning, for each mode, depreciation of each under
structure constitutive of the building considered and depreciation structural and geometrical of the ground
balanced by their respective rate of potential energy compared to total potential energy:
Eki.k + Esi.si

K
S
I =
Eki + Esi
K
S
with:
·
I, damping reduces average mode I;
·
K, the reduced damping of the kème element of the structure;
·
if, the reduced damping within the competence of ground S for mode I;
·
Eki, potential energy of the kème element of the structure for mode I;
· and
Esi, potential energy within the competence of ground S for mode I.
In the payment, modal damping is limited to a maximum value of 0,3.
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The part of material origin of the damping of the ground is calculated by balancing damping of
each under structure by the report/ratio: rate of potential energy on total potential energy. As for
the geometrical part of damping, it is calculated by distributing the values of damping
for each direction (three translations and three rotations) balanced by the rate of potential energy
in the ground of the direction. The directional values of damping are obtained while interpolating,
for each calculated Eigen frequency, directional functions of damping exit of a code
of interaction ground-structure (PARASOL, CLASSI or MISS 3D). The report/ratio of the imaginary part on two
Im (K ())
time the real part of the matrix of impedance:
, provides the values of this damping
.R
2nd (K ())
radiative.
The step to be followed is as follows:
· Calculation of the potential energy dissipated in the studied structure: POST_ELEM [U4.61.04]
Ek = POST_ELEM (ENER_POT: (TOUT: “OUI”)) ;
· Calculation of modal damping by the rule of the RCC_G: CALC_AMOR_MODAL [U4.64.04]
l_amor = CALC_AMOR_MODAL (
ENER_SOL: (MODE_MECA: base_modale, GROUP_NO_RADIER: ,
Kx (
Ky (
Kz (1)
0)
0)
KX:
, KY:
, KZ:
,
Krx (
Kry (
Krz (1)
0)
0)
KRX:
, KRY:
, KRZ:
)) ;
AMOR_INTERN: (GROUP_MA:, ENER_POT: E K, AMOR_REDUIT: K)
Im (K ())
AMOR_SOL: (FONC_AMOR_GEO:
.R
2nd (K ()) )
);
The calculation of the contribution of the ground to the potential energy Es (key word factor ENER_SOL) is calculated with
to leave the values of impedance of ground determined previously (cf [§6.1]). It can be calculated
according to two different methods according to whether one average modal efforts (key word RIGI_PARASOL)
or modal displacements with the node of the foundation raft.
The reduced damping within the competence of ground S S (key word factor AMOR_SOL) is calculated from
values of radiative damping.
6.3
Distribution of the stiffnesses and damping of ground
If one wants to study the effect of a seism on the possible separation of the foundation raft for example, one can be
brought to model the ground either by a single spring in the center of gravity of the interface ground-building
but by a carpet of springs. This is possible thanks to command AFFE_CARA_ELEM [U4.24.01]
option RIGI_PARASOL.
The step consists in calculating in each node of the grid of the foundation raft the elementary stiffnesses
(K, K, K, Kr, Kr, Kr
X
y
Z
X
y
Z) to apply starting from the total values within the three competences of translations:
kx, ky, kz and within the three competences of rotations: krx, kry, krz resulting from a code of interaction ground-structure
(or calculated analytically).
It is supposed that the elementary stiffnesses of translation are proportional to surface S (P)
represented by the node P and with a function of distribution F (R) depend on the distance R of the node
P in the center of gravity of the foundation raft O:
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K = K
.
X
X (P) = K
S (P). F (P
X
O)

P
P

K = K
y
y (P) = K. S (P). F (COp
y
)

P
P
K = K
Z
Z (P) = K. S (P). F (COp
Z
)


P
P
One deducts K X then from them then K (P
X
) starting from calculation:
S (P). F (COp)
kx (P) = K .S (P). F (P
O) = K
X
X.
.
S (P). F (COp)
P
One deducts of the same K (P from them
y
) and K (P
Z
).
For the elementary stiffnesses of rotation, one distributes what remains after having removed the contributions
had with the translations in the same way that translations:

K = K
2
2
2
2
X-ray
X-ray (P) + [K y (P) .zO + K
P
Z (P). yOP] = K. S (P). F (COp
X-ray
) + [ky (P) .zO + K
P
Z (P). y P
O]

P
P
P
P

K = K
2
2
2
2
ry
ry (P) + [kx (P) .zO + K
P
Z (P). xOP] = K. S (P). F (COp
ry
) + [kx (P) .zO + K
P
Z (P). X P
O]

P
P
P
P
K = K
2
2

=
. (). (O) +
2
2

K
S P F
P
[kx (P) .y PO +ky (P) .x
rz
COp]
rz
rz (P) +
[kx (P) .yO +k
P
y (P). X P
O]

P
P
P
P
One deduces krx then from it then K (P
X-ray
) starting from calculation:
krx (P) = K .S (P). F (COp
X-ray
)

S (P). F (COp)
= K
. 2
. 2
X-ray - [K y (P) zO + K
P
Z (P) y P
O].

S (P). F (COp
P
)
P
One deducts of the same K (P from them
ry
) and K (P
rz
).
Note:
By defect, one considers that the function of distribution is constant and unit i.e.
each surface is affected same weight.
One can distribute in the same way six total values of damping, analytical or calculated by one
code interaction ground-structure.
6.4
Taking into account of an absorbing border
If one wants to calculate the seismic response of a stopping, it is necessary, amongst other things, capacity to take into account
not reflection of the waves in the valley. This is possible thanks to elements at absorbing border:
option IMPE_ABSO in DYNA_NON_LINE and DYNA_LINE_TRAN. This functionality is not
detailed in this document. It will be the subject of a specific note.
Handbook of Référence
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7 Bibliography
[1]
R.W. CLOUGH, J. PENZIEN - Mc GRAW-HILL: “Dynamics off structures” - (1975).
[2]
“Paraseismic Engineering” - collective work under the direction of V. DAVIDOVICI - Presses of
the E.N.P.C. (1995).
[3]
P. LABBE, A. PECKER, J.P.
TOURET
: “
Seismic behavior of the structures
industrial “- course IPSI from the 20 to September 22, 1994.
[4]
Fe WAECKEL: “Method for calculation by modal superposition of the seismic answer
of a multimedia structure “- Note HP62/95.017B (09/95).
[5]
V. GUYONVARH, G. DEVESA: “Method for calculation of the seismic excitations with
anchorings of the CPP N4 “- Note HP52/99.006 (09/99).
[6]
“Seismic Calculation of the buildings” - Règles RCC-G, additional A
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