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Organization (S): EDF/AMA
Handbook of Référence
R7.04 booklet: Evaluation of the damage
Document: R7.04.03
Postprocessing according to RCC-M'S

Summary:

Operator POST_RCCM allows to check the criteria of level 0 and certain criteria of level A of
RCC-M-B3200, for modelings of continuous mediums 2D or 3D.

It also allows the calculation of the criteria of fatigue of the §B3600 in postprocessing of calculations of pipings.

The criteria defined in the B3200 chapter of the RCC-M utilize significant sizes that one
compare with limiting values.

The criteria of level 0 aim at securing the hardware against the damage of excessive deformation,
of plastic instability and elastic and elastoplastic instability. These criteria require the calculation of
equivalent constraints of Pm membrane, local membrane Pl and membrane plus Pm+Pb inflection.
order POST_RCCM calculates Pm or Pl and Pm+Pb.

The criteria of level A aim at securing the hardware against the damage of progressive deformation and of
tire. Except fatigue, they require the calculation of the amplitude of variation of linearized, noted constraint Sn, and
possibly of the quantity Sn *. For fatigue, they require in more calculation of the amplitude of variation of
constraint in a point, noted Sp.

Command POST_RCCM [U4.67.04] carries out calculations of Sn, Sn *, Sp and of the number of acceptable cycles
in fatigue. In postprocessing of analyzes of pipings, option FATIGUE_B3600 allows calculation factor
of use in fatigue by taking into account all the calculated situations.
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Count

matters

1 Criteria of the RCC-M B3200. Adaptation to Code_Aster ........................................................................ 4
1.1 Geometrical data ................................................................................................................... 4
1.2 Data of loading ................................................................................................................. 4
1.3 Data material ........................................................................................................................... 4
1.4 Simplifying assumptions ............................................................................................................. 4
1.5 Calculations carried out by POST_RCCM ................................................................................................... 5
2 Criteria of level 0 (key word PM_PB) ................................................................................................... 6
2.1 Criteria of level 0 specified by the RCC-M .................................................................................. 6
2.1.1 General primary equivalent constraint of membrane Pm ................................................ 6
2.1.2 Primary equivalent constraint of local membrane Pl ....................................................... 6
2.1.3 Primary equivalent constraint of membrane+flection Pmb (or Plb) ................................... 6
2.2 Calculations carried out by Aster .............................................................................................................. 7
3 Criteria of level A (except fatigue) (key word SN) .................................................................................. 8
3.1 Criteria of level A specified by the RCC-M .................................................................................. 8
3.1.1 Sn calculation ........................................................................................................................... 8
3.1.2 Sn calculation * .......................................................................................................................... 8
3.2 Calculations carried out by Aster .............................................................................................................. 9
3.2.1 Sn calculation ........................................................................................................................... 9
3.2.2 Sn calculation * ........................................................................................................................... 9
4 Criteria of fatigue (of level A) (key word FATIGUE) ......................................................................... 10
4.1 First method: maximum amplitude in a transient ....................................................... 10
4.1.1 Calculation of Sp ......................................................................................................................... 11
4.1.2 Sn calculation by the algorithm describes ............................................................ 11 previously
4.2 Second method: combination of several transients and under-cycles, method ZH210 ..... 12
4.2.1 Calculation of the elementary factors of use ........................................................................... 12
4.2.2 Algorithm of office plurality ............................................................................................................. 13
5 Criteria of fatigue for the analysis simplified of pipings according to RCC-M'S B3600 ........................ 14
5.1 Calculations of all the states of loading ...................................................................................... 14
5.1.1 Calculations of the static states of loading ......................................................................... 14
5.1.2 Calculation of the seismic loadings ...................................................................................... 14
5.1.3 Calculation of the thermal transients ....................................................................................... 15
5.2 Calculations of the amplitudes of constraints ......................................................................................... 16
5.2.1 Calculation of the combinations of loading (I, J) inside each group of situations16
5.2.1.1 Case of the under-cycles ................................................................................................ 19
5.2.2 Calculation of the combinations of loading (I, J) for the situations of passage between group
situations ......................................................................................................................... 19
5.3 Calculation of the factor of use .............................................................................................................. 19
6 Course of the analysis of the behavior to fatigue according to RCC-M B3200 .................................. 22
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6.1 Calculations of all the states of loading ...................................................................................... 22
6.1.1 Linear combination of the tensors of constraints .............................................................. 22
6.1.2 Calculation of the thermal transients ....................................................................................... 22
6.1.3 Case of the seismic loadings ......................................................................................... 23
6.1.4 Calculations of the amplitudes of constraints inside each group of situations ......... 23
6.1.4.1 Calculation of the alt (I, J) without taking into account of the seism ............................................... 23
6.1.4.2 Calculation of the alt (I, J) with taking into account of the seism ............................................... 26
6.1.4.3 Calculation of Sp (I, J) with taking into account of the seism .................................................. 27
6.1.5 Calculation of the amplitudes of constraints for the situations of passage between group of
situations .............................................................................................................................. 28
6.1.6 Storage of the amplitudes of constraints for all the combinations ............................. 28
6.2 Calculation of the factor of use .............................................................................................................. 29
7 Bibliography ........................................................................................................................................ 31
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1
Criteria of the RCC-M B3200. Adaptation to Code_Aster

The B3200 chapter of the RCC-M [bib1] described the general rules of analysis of the behavior of
hardware of level 1 of Centrales Nucléaires. These rules make it possible to ensure the safeties
necessary with respect to the damage to which this hardware is subjected. For that one defines
various levels of criteria in each category of situation which make it possible to compare
significant sizes with limiting values. The adaptations necessary to Code_Aster are
described here, and justified in [bib2].

1.1 Data
geometrical

The user of the RCC-M must distinguish in his structure the zones of major discontinuity, the zones
of minor discontinuity and zones comprising of the geometrical singularities. The RCC-M defines
“segments of support” which are used to linearize the constraints. These segments are, out of the zones of
discontinuity, of the generally normal segments on the median surface of the wall, and in
zones of discontinuity, shortest segments allowing to join the 2 faces of the wall.

The user of ASTER must thus define the whole of the sections of the structure where calculations of post-
processing will be made (it is him which knows if these sections pass by current zones, or zones
of geometrical discontinuity). In practice, one works on a segment provided by INTE_MAIL_2D or
INTE_MAIL_3D. One systematically calculates all the criteria at the two ends of the segment, or
more precisely with the two intersections of the segment with the edges of the structure.

1.2
Data of loading

The user of the RCC-M must give the number of occurrences of each situation of operation
(for example: heating of the boiler, hot stop, etc.). A situation of operation can
to be broken up into transients, i.e. of the evolutions of the parameters of operation
total (pressure, temperature) according to time.

In ASTER, one treats mechanical results (produced by MECA_STATIQUE or
STAT_NON_LINE), therefore transients. For each transient, the stress fields are
provided to the moments of discretization of calculation.

1.3 Data
material

The data material required are as follows:

· Sm: acceptable value (tabulée in the RCC-M Annexe Z1).
· m, N: constant material for the calculation of Ke (defined in the RCC-M B3234.6)
· EC., E: moduli of elasticity (for the correction of the curve of fatigue, annexes Z1).
· Curves of fatigue of material: according to RCC-M'S annexes Z1.

1.4 Assumptions
simplifying

In the RCC-M, the user must be able to say, after analysis of the results of calculation, if them
principal directions in a given point are fixed or if they turn in the course of time.

On the other hand, in command POST_RCCM, one can not make an assumption. One will not consider
that the case where the principal directions are unspecified.
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Moreover, the user must be able to classify the constraints in the following categories:

· General primary education of membrane: Pm
· Primary education of local membrane: Pl
· Primary education of inflection: Pb
· Thermal expansion: EP
· Secondary: Q
· Of point: F

This choice cannot be made by POST_RCCM. Only the user can qualify a stress field
(“Primary education”, “secondary”, or summon it of both). The criteria which are to be checked are calculated to leave
stress fields (constant or function of time) provided by the user. It is him which ensures
coherence enters the calculation of these fields and the criteria applied.

However, to fix the ideas, classification is simpler in the following cases:

· a constant or variable loading with imposed force or pressure is primary, except for
certain very particular structures,
· a constant or variable loading with imposed displacement is in theory, secondary (except
in the case of “the effect spring”),
· a permanent or transitory thermal loading is in theory secondary.

On the other hand, the combination of these types of loadings leads to a result which cannot be any more
qualified of primary education or secondary. According to the criteria, the user could thus be brought to
to break up its loadings.

1.5
Calculations carried out by POST_RCCM

One describes here the operation of command POST_RCCM allowing to carry out the calculation of
certain criteria RCC-M B3200 of levels 0 and A. the realization described here does not take into account
touts the criteria of B3200 and could be supplemented (for example for other levels of criteria,
or for criteria of the RCC-MR,…).

The principal data is the segment (of support) where calculations will be carried out. It is the user who
the segment chooses and which with the responsibility to find that for which quantities intervening in
the criteria are maximum. The automatic search for this segment carrying out the maximum is one
difficult problem, and is not programmed.

After having calculated one or more results by MECA_STATIQUE or STAT_NON_LINE, and having defined it
segment by INTE_MAIL_2D or _3D, the user requires the calculation of the criteria by the operator
POST_RCCM.

Three types of criteria are accessible each one by a key word factor:

· criteria of level 0 by key word PM_PB,
· criteria of level A (except fatigue) by the key word SN,
· criteria of fatigue (also of level A) by key word FATIGUE.
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2
Criteria of level 0 (key word PM_PB)

2.1
Criteria of level 0 specified by the RCC-M

The criteria of level 0 aim at securing the hardware against the damage of excessive deformation,
of plastic instability and elastic and elastoplastic instability. They must be checked by the situation
of reference (see B3121 and B3151). These criteria require the calculation of the constraints
equivalent P, P, P
m
I
B which is below defined.

2.1.1 General primary equivalent constraint of Pm membrane

Being given the primary constraint of the situation of reference (1e category) and a segment located out
of a zone of major discontinuity. In each point end of this segment length L, one
calculate:

L
1
P = my (moy
moy
X ij)

=
ds
m
ij
ij,
T
Eq T
. resca
L 0

where (ij)
= max -

are the principal constraints
Eq Tresc
.
has
I
J
(I I=, 13
)
I, J

The criterion is written:

P S
m
m

2.1.2 Primary equivalent constraint of local membrane Pl

Being given the primary constraint of the situation of reference (1e category) and a segment located
in a zone of major discontinuity, the definition of pi is identical to that of Pm on it
segment.

The criterion is written:

P 15
. S
I
m

2.1.3 Primary equivalent constraint of membrane+flection Pmb (or Plb)

Being given the primary constraint of the situation of reference (1e category) and a segment (directed).
In each point end of this segment length L, (ends corresponding to the skins
external and intern), one calculates:

P
moy
fle
flax
m = max ij
B = max ij
mb = max ij
T (
)
P
Eq T
. resca
T (
)
P
()
Eq T
. resca
T
Eq Tr
. ESCA
L
L
moy
1
fle
6
L

=

ds


=
S
2
-
ds
flax
moy
fle
ij



L
ij
ij
L
ij
ij
ij
ij


2
=
±
0
0

flax
moy
fle
ij (S =)
0 =ij
- ij
flax
moy
fle
ij (S = L) =ij +ij

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The criteria are written:

P
15
. S
mb
m
P 15
. S
Ib
m

2.2
Calculations carried out by Aster

It is with the user to know if one calculates Pm (forced general of membrane: out of the zones of
geometrical singularity) or Pl (forced local of membrane: in the singularities). To leave
stress fields provided (in the result), a membrane stress is thus calculated.

The concept result comprises either only one stress field, or of the fields resulting from one
evolution. In this last case, one will seek the maximum compared to the list of the sequence numbers
terms intervening in the criteria.

The algorithm is as follows:

Impression of the segment (cf POST_RELEVE)

· On the whole of the nbmax, sequence numbers n=1
- extraction of the moment T
- on each end of the segment
- calculation of Pm and Pmb by integrations on the segment

P
moy
m = max ij
T (
) EqTres
.
Ca
L

moy

1
=
ds
ij

,
L
ij
0
L
fle
fle
6
L
Pb = max ij

ij = 2 - ij,
T (
)
S
ds
Eq Tresca
.
L


2
0
P (S
moy
fle
moy
fle
mb
=)
0 = max ij
- ij
mb (=) = max ij
+ ij
T (
)
P
S L
Eq Tres
.
Ca
T (
) EqT.resca

- Search of the maximum of Pm, Pmb (s=0), Pmb (s=l).
- Output and storage in the table of the result.

The values limit are Sm and 1.5 Sm, Sm being working stress function of material and of
temperature, given by mot_clé the SM_KE_RCCM of behavior FATIGUE in DEFI_MATERIAU.
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3
Criteria of level A (except fatigue) (key word SN)

3.1
Criteria of level A specified by the RCC-M

This option makes it possible to calculate the criteria of level A (except fatigue) which aim at securing the hardware
against the damage of progressive deformation. They require the calculation of the amplitude of variation
linearized more secondary primary constraints, noted Sn.

3.1.1 Sn calculation

One takes into account the more secondary primary constraints and the constraints resulting from
opposed thermal dilations: Pl + Pb + EP + Q which thus represents the linearized constraints
associated all the loading (mechanical and thermal).

The points of calculation are the two ends of the segment (given by key word CHEMIN). In each
not end of this segment length L, one calculates:



S
flax
flax
N = max my (
X ij (T1) - ij (t2)

T1 T
Eq Tre
.
sca
2

L
L
moy
1
fle
6
L

=

ds


=
S




2
-
ds
flax
moy
fle
ij
L
ij
ij
L
ij
ij
ij
ij


2
=
±

0
0
flax
moy
fle

ij (S =)
0 =ij
- ij
flax
moy
fle
ij (S = L) =ij +ij

The criterion (of total adaptation) is written:

S 3S
N
m

Sm being the working stress function of material and the temperature, given by mot_clé
SM_KE_RCCM of behavior FATIGUE in DEFI_MATERIAU.

If this criterion is not checked, one can practice the simplified elastoplastic analysis of B3234.3. It is necessary
to carry out the three following operations:

· to check the criterion:

S * 3S
N
m

· to make an elastoplastic correction (Ke > 1) in the analysis with fatigue,
· to check the rule of Bree (B3234.8) in the current parts of the cylindrical hulls (and
pipes) subjected to a pressure and a variation in cyclic temperature. This relates to one
very particular situation and will thus not be described here.

3.1.2 Sn calculation *

S is noted *
N the amplitude Sn calculated without taking into account bending stresses of origin
thermics. One calculates for each end:



S *
flax
fleth
flax
fleth
= max max
1 -
1
-
2 -

N *
ij
ij
ij
ij
2
T1 T (
(T)
(T) ((T)
(T))
2
Eq Tres
.
Ca
L

fleth
6
L

=
S
HT

2
-
ds
ij
L
ij


2
0
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thij coming from a calculation carried out with the thermal loading only (i.e. which one removes from
complete calculation, having led to the Sn value, all the loadings other than the loading
thermics).

3.2
Calculations carried out by Aster

3.2.1 Sn calculation

From the moments of calculation selected in the result, one thus calculates Sn at each end of
segment. If it y does not have calculation at the moment T = 0, one creates a stress field identically no one with
T = 0.

The algorithm is as follows:

Impression of the characteristics of the segment (cf POST_RELEVE)

· on the whole of the n1=1, sequence numbers, nbmax
- Extraction of moment T1
- calculation of flax
flax
ij (T, S =)
1
0 and ij (T, S L)
1
=
- For varying N2 of n1+1 with nbmax
- Extraction of the moment t2
- calculation of flax
flax
ij (T, S =)
2
0 and ij (T, S L)
2
= and of
flax
flax
flax
flax
ij (T, S =)
0 - ij (T, S =)
2
1
0 and ij (T, S = L) - ij (T, S = L)
2
1

- calculation of the principal directions and the criterion of Tresca:
(flax
flax
(flax
flax
ij
T, S = L - ij T, S = L
2
1
)
ij (T, S =)
0 - ij (T, S
2
1
=)
0
and
(
)
(

Eq Tre
.
sca
Eq T
. resca
- search of the maximum thus of Sn

Output and storage in the table of the result.

3.2.2 Sn calculation *

This calculation is carried out if operand RESU_SIGM_THER is present. Only the user ensures
coherence, i.e. this result must be produced by a thermomechanical calculation under
thermal loading only, knowing that the result given by RESULTAT can be due to one
combination of this thermal loading with other loadings. It is necessary thus that the moments of
this result correspond to those of the result associated with key word RESULTAT.

The algorithm is identical to the precedent but relates to two stress fields.
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4
Criteria of fatigue (of level A) (key word FATIGUE)

In the case of fatigue it should be made sure that the factor of total use - which integrates the effect of
combinations of situations 2 to 2 - is lower than 1. For this analysis one needs:

· on the one hand, of Sn previously defined, which requires the linearization of the constraints, this
to calculate the elastoplastic concentration factor Ke,
· in addition, of Sp which is the amplitude of variation of total constraints (Pl + Pb + EP + Q + F),
who does not require linearization, and who whose definition follows. This Sp is used to calculate
alternate equivalent constraint Salt which, via the curves of fatigue, makes it possible to determine it
factor of use.

We present two methods:

· the first makes it possible to calculate the factor of use for only one transient (according to the RCC-M
B3234.5). It is supposed here that the transient does not comprise secondary fluctuations
(each quantity varies between an always identical minimum and a maximum),
· the second method makes it possible to combine several transients and to take account of
factors of use specific to the secondary fluctuations (additional RCC-M ZH210).

It is also supposed that one is not located in zones comprising of the singularities
geometrical (if not, it is necessary to apply the methods of calculation to the starting of the singular zones which
are the subject of appendix ZD of the RCC-M).

4.1
First method: maximum amplitude in a transient

It is a first approach of the calculation of the damage of fatigue of the RCC-M B3200, limited to
processing of only one transient (not of combination of transients) and without consideration of
under-cycles. This method is the only available one in version 3 of Code_Aster (it is activated by
key word “FATIGUE_SPMAX” in version 4). One calculates the amplitude of variation of constraint in one
, not noted Sp, and the amplitude of variation of linearized constraint Sn for the calculation of the factor of
elastoplastic correction Ke (according to the RCC-M B3200).

In each point end of the segment length L, one calculates:





S
flax
flax
p = max max ij 1 - ij
2
N = maxmax ij 1 - ij 2

T1 T (
(T)
(T))
S
((T)
(T)
Eq T
. resca
2

T1 T
Eq T
. resca
2

L
L
moy
1
fle
6
L

=

ds


=
S




2
-
ds
flax
moy
fle
ij
L
ij
ij
L
ij
ij
ij
ij


2
=
±
0
0

then

1 E
S
C
=
Ke (Sn) S
alt
and by the curve of fatigue of Wöhler: NR
= F (S
adm
alt).
2nd
p

An acceptable value of Ke can be given according to the RCC-M B3200 as follows:

Ke (Sn) = 1
if S < 3S
N
m
1 - N S

K
N
E (Sn) = 1 +

-
1
if
3S < S < 3M S
N (M-1) S
m
N
m
3 m

1
Ke (Sn) =
if
S > 3M S
N
N
m
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The values of m and N are given in B3234.6 of the RCC-M. Ces parameters and the curve of
fatigue are introduced into command DEFI_MATERIAU [U4.23.01] under the key word factor
FATIGUE.

·
N corresponds to N_KE_RCCM
·
m corresponds to M_KE_RCCM
·
Sm corresponds to SM_KE_RCCM
· The Young modulus of reference of the curve of fatigue EC. corresponds to E_REFE.
Young modulus corresponding to calculation carried out is defined classically under the key word
factor ELAS.
· The curve of fatigue NR
= F (S
adm
alt) is a function defined by DEFI_FONCTION, and
introduced into DEFI_MATERIAU by key word WOHLER of the key word factor FATIGUE.

This algorithm is directly deduced from calculation from Sn and maximum Sp for only one transient and
does not take into account the under-cycles. It thus does not correspond to that of POST_FATIGUE.

If the user wishes at the same time the Sn and fatigue analysis for the transient, it can occur to use
the key word Sn, because the calculation of Salt implies two calculations:

· that of Sp
· that of Sn carried out previously.

As for the criterion Sn < 3Sm, as from every moment of calculation, one calculates Sp with each
end of the segment. If only one moment ago, one creates a fictitious transient between this moment and the state
identically no one.

The algorithm is almost identical to that used for the Sn calculation, without linearization. It is written:

· Impression of the segment (cf POST_RELEVE)
· on the whole of the n1=1, sequence numbers, nbmax

4.1.1 Calculation of Sp

· Extraction of moment T1
· For varying N2 of n1+1 with nbmax
- Extraction of the moment t2
- calculation of ij (T, S =)
0 - ij (T, S
2
1
=)
0 and ij (T, S = L) - ij (T, S = L
2
1
)
- Calculation of the principal directions and calculation of
-
(
(ij T, s=l - ij T, s=l
1
2
)
ij (T, S =)
0 - ij (T, S
1
2
=)
0)
and
(
)
(

Eq Tres
.
Ca
Eq Tre
.
sca
- search of the maximum to obtain:



S (S =)
0 = max max ij,
1
= 0 - ij,
2
= 0

T1 T (
(T S)
(T S
p
))EqTres
.
Ca
2



S (S = L) = max max ij,
1
= - ij,
2
=

T1 T (
(T S L)
(T S L
p
))EqTres
.
Ca
2


4.1.2 Sn calculation by the algorithm describes previously


S (S =)
0 = max my (flax
flax
X ij (T, S
1
=)
0 - ij (T, S
N
2
=)
0

T1 T
Eq Tres
.
Ca
2



S (S = L) = max my (flax
flax
X ij (T, S
1
= L) - ij (T, S
2
= L
N
)

T1 T
Eq Tr
. ESCA
2

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Note:

The moments which maximize Sp are not inevitably identical to those which maximize Sn.

· Output and storage in the table of the result.

After the calculation of Sp and Sn, one obtains the number of acceptable cycles by:

- The calculation of Ke for each end, as by the preceding formula.
- The calculation of Salt starting from Sp, EC., E, Ke:

1 E
S
C
=
Ke (Sn) S
alt

2nd
p

- One deduces then Nadm de Salt and from the curve of fatigue.

4.2 Second method: combination of several transients and
under-cycles, method ZH210

The first method does not take into account the possible under-cycles, and does not combine them
transients between them. One describes here another possibility, available only in version 4 of
Code_Aster (key word “FATIGUE_ZH210”).

The algorithm is similar to that of POST_FATIGUE. More precisely, the algorithm used in
POST_FATIGUE is a restriction on the uniaxial case of method ZH210. Indeed, the data of
order POST-FATIGUE is a scalar function of time (whereas POST_RCCM treats
tensors of constraints functions of time).

This method resulting from appendix ZH210 of the RCC-M was preferentially selected with others
methods described in the RCC-M [bib1].

The principal advantage of this method is to consider all the under-cycles automatically
possible. Its disadvantage is the number of calculations to be carried out if one does not restrict the whole of
moments used in calculation.

Indeed, one defines for each transient a whole “of states of loading”, which are the moments
significant where the constraints pass by a local extremum. By defect, in Code_Aster, all them
moments of calculation are used. One associates each one of them the number of occurrences of the transient.
definition is thus:

State of loading = {urgent, tensor of constraints, numbers occurrences}.

Then, one builds the whole of all the states of loading by sweeping all the transients. With
boils of the account, the concept of transient is forgotten: one does not work any more but on one whole of states
of loading. One then calculates all the elementary factors of use associated all them
combinations taken two to two. One uses then a method of office plurality of the factors of use
elementary, based on the assumption of the linear office plurality of the damage, to obtain the factor of use
total.

4.2.1 Calculation of the elementary factors of use

At each end of the segment, for any couple of states of loading K and L, one calculates the quantities
Sp (K, L) and Sn (K, L) by:





S (K, L) =



-

((K) - (L))
S
flax
flax
N (K, L)


= (ij (K) ij (L
p
ij
ij
))

Eq Tres
.
Ca
Eq T
. resca

From Sn (K, L), one calculates Ke (K, L) like previously, then:

1 E
S (K, L)
C
=
Ke (K, L) S p (K, L
alt
) and by the curve of fatigue of Wöhler: NR (K, L) = F (S
adm
alt).
2nd
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The method to determine Ke is identical to the preceding one.

From the number of acceptable cycles Nadm (K, L), one calculates the factor of use of the combination:

U (K, L) = Nkl/Nadm (K, L), with Nkl = min (Nocc (K), Nocc (L)).

This calculation is carried out for each combination of two states of loading. One thus obtains
(always for each end of the segment) a symmetrical matrix U (K, L), of command the number of states
of loading.

4.2.2 Algorithm of office plurality

For each end:

Data: Numbers total states of loading NR
stamp U (K, L), K, l=1, NR
vector of entireties Nocc (I), i=1, NR

U (M1) =0 (factor of total use)

· seek maximum of U (M1, K, L) on all the combinations (K, L) such as Nocc (K) >0 and
Nocc (L) >0. That is to say U (M1, m, N).
· U (M1) = U (M1) + U (M1, m, N).
· If Nocc (m) < Nocc (N) then
Nocc (N) =Nocc (N) - Nocc (m)
Nocc (m) =0
and conversely
If Nocc (N) < Nocc (m) then
Nocc (m) =Nocc (m) - Nocc (N)
Nocc (N) =0
· If there are still states of loading I such as Nocc (I) >0, return in 1.

Two remarks can be made:

If the number of moments defined for each transient is large, calculation can be prohibitory.
It is thus necessary to be able to restrict it. It is what is made in POST_FATIGUE, by a sorting
preliminary of the moments. One eliminates the moments such as the scalar function is linear for
to keep only the ends of the segments of straight line. One eliminates also the very small ones
variations. Here, in multiaxial situation, the sorting is more delicate. Concept of constraints
proportional could be used, but it is necessary to envisage in more one possibility for
the user to define itself the list of the moments (key word NUME_ORDRE)

By this method, one is sure not to forget no under-cycle. On the other hand, it is desirable
to eliminate the moments which do not correspond to local extrema, because they could
to generate factitious under-cycles, augmentatnt the factor of use (these moments are
only used for the numerical discretization of the mechanical or thermal problem).
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5
Criteria of fatigue for the simplified analysis of pipings
according to the RCC-M B3600

Vocabulary used: compared to the preceding options, which treat complete transients
(mechanical modeling of the structure subjected to stories of temperatures and loading
mechanics), it is of use in B3600 of defines each situation as the passage of a stabilized state
With (correspondent with a pressure interns given in the line of piping, a uniform temperature
data, and of the fixed mechanical loadings) in a state stabilized B (with constant loadings
different from the precedents). One associates this situation a thermal transient.

The processing which is described here is carried out for each node of each mesh of the line of
piping considered. The result obtained will be thus a factor of use (total or partial) for each
node of each mesh requested by the user.

5.1
Calculations of all the states of loading

For each node of each mesh, the present stage consists in calculating, for all the situations,
the moments relating to each stabilized state (by cumulating the various loadings which intervene).

5.1.1 Calculations of the static states of loading

One treats the results of static calculations (field EFGE_ELNO_DEPL or SIEF_ELNO_ELGA) for
stabilized states of the list of the situations undergone by the line. Torques for each stabilized state
are obtained by algebraic summation of the torques corresponding to the various loading cases of
situation (signed).

M
M
M

I =
I CHAR +

I CHAR
+ I




_
1
_
2
{
X;
}
Z

y;

(the loadings are for example opposed thermal dilation, the displacement of anchoring).

5.1.2 Calculation of the seismic loadings

The seismic loading breaks up into 2 parts:

· An inertial part

It is calculated by imposing on the whole of anchorings the same movement characterized by
spectrum envelope of the various spectra of floor, in the horizontal directions X and Y
on the one hand, and vertical Z on the other hand (in the total reference mark). With this intention, the command is used
COMB_SISM_MODAL, which produces generalized efforts which correspond to each direction of
seism as well as the quadratic office plurality of these efforts:
The inertial contribution of the seism to component I of the moment is written:
M
=
2 I, J; ; ; ; ;

_ _
I _S _ DYN
J
J (M
(spectrum) () {X y} Z {X Y Z
I S DYN
}}
with: Mi_S_DYN (spectrej) moment in direction I resulting from the dynamic loading in
direction J. This office plurality is already made by COMB_SISM_MODAL.
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· A quasi-static part

It is estimated by imposing static differential displacements corresponding to the maximum one
differences of the seismic movements of the points of anchoring in the course of time. Calculations
are thus realized for each unit loading (a calculation by displacement in one
direction given for an end of the line).
The loadings must then be combined by quadratic average by
POST_RCCM_B3600 (this calculation is not carried out as a preliminary).

The quasi-static contribution of differential displacements of anchoring to component I of
moment is written:

NR _ ANC
M
=
(M K
) 2
I _S _ ANC
I _S _ ANC
K =1
with: M K
the component iéme of the moment corresponding to the kéme displacement of anchoring.
I _S _ ANC

Combination of the inertial components and differentials due to the seism:

The iéme component resultant is obtained by quadratic average of the inertial component iéme
and differentials:

M
=
2 +
2 I; ; what amounts carrying out it average quadratic
_
(M _ _) (M _ _)
{X y} Z
I S
I S ANC
I S DYN
from every inertial and differential moment,

M
2
2
=

+
I
; ;
I _
(Mk
S
I _ S _ ANC)
(Mi_S_DYN)
{X y Z}
k=,
1 NR _ ANC
The result of this office plurality is to be stored in the table above.

Each one as of the these loadings (inertial answer, displacement of anchoring) is defined by one
occurrence of key word RESU_MECA.

5.1.3 Calculation of the thermal transients

In the § B3653 of the RCC-M which describes the method of analysis to fatigue for a line of piping,
the loadings of the type “heat gradient in the thickness” are taken into account by the intermediary
of four variables:

T1: amplitude of the variation enters the two stabilized states of the difference in temperature enters
walls internal and external, for an equivalent linear distribution of the temperature.

T2: nonlinear part of the distribution in the thickness of wall of the amplitude of variation of
the temperature enters the two stabilized states.

Your and Tb: amplitude of variation between two stabilized states of the average temperatures in
respective zones has and B located on both sides discontinuity of material or structure.

Methodology: For each one of the transients and each section of piping of the line (and each
junction), one realizes as a preliminary, according to the geometrical complexity of the problem studied a calculation
thermics 2D or 3D.

Each calculation is stripped using two calls to POST_RELEVE_T (OPERATION =EXTRACTION
and OPERATION=MOYENNE) in order to extract, for each moment I, variation in the temperature on
the selected section and average values (moments of command 0 and 1).
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From these values, one calculates the quantities T1, T2, Your and Tb in the following way:

For each moment of the transient, one calculates (by the same routines as in POST_RELEVE_T):
T

T y Dy: average value of T () on the ligament
moy ()
T
= 1 2 (,).
- T
T
2
Possibly (material discontinuity or junction):
B
T

T y Dy: average value of T () on the ligament corresponding to the node B located
moy ()
T
= 1 2 B (,).
- T
T
2
on the other side of the junction
V

y T y Dy: variation of a linear distribution equivalent to T ().
moy () = 12
T
. (,).
2 2
- T
T
2
then:
T1 () = Vmoy ()
T
-
-

ext. ()
Tmoy () 1

T
1 ()
2
T
=
T - T
-


moy
T
2 ()
max
int ()
() 1

1 ()
2
0

In the case of a discontinuity of material or a junction, one calculates:

T
= T
has ()
moy ()
T
= T

B ()
B
moy ()
T
- T
has has ()
B B ()

by using the possibly different dilation coefficients for the two convergent meshs
with the treated node.
In practice, the zones has and B will correspond to segments chosen by the user in
POST_RELEVE, and the produced tables will be associated the two adjacent meshs having jointly
the node which corresponds to the junction.

5.2
Calculations of the amplitudes of constraints

5.2.1 Calculation of the combinations of loading (I, J) inside each group of
situations

The first phase consists in calculating the amplitudes of constraints which correspond to
combinations of all the stabilized states pertaining to the situations of a given group, in
choosing the moments of the thermal transients which maximize these amplitudes of constraints. In
effect, the thermal transients defined in Dossier d' Analyze of Comportement are associated
situations. During the analysis of the behavior to fatigue, we are brought to define
cycles of fictitious loadings by associating stabilized states pertaining to different situations.
In this case, the thermal transient associated the fictitious cycle corresponding to the stabilized states I and J
will be selected in order to maximize the amplitude of constraints.

The whole of the combinations (I, J) is thus considered with (I, J) ({1, 2. , NR}, {1, 2. , NR}) (NR being it
a number of states stabilized except seism, i.e. 2 times the number of situations of the group), and one
built a matrix [NR;NR] of the values alt (I, J).
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For each combination, alt (I, J) was obtained in the following way:
Are two stabilized states, I and J belonging respectively to the situations p and Q:

One calculates
Sp (I, J): amplitude of variation of the total constraints (eq. (11) of the §B3653 of the RCC-M) by:

P I, J .D
0
0
D
1
S I, J, T = K C
.
+ K C
. 0 M
.
I, J +
.K .E. T

T, T
p (
p)
()
1
1
2
2
I (
)
3
1 (p
p
K
L)
2 T.
2.I
2 (.1 -)

1
+ K C
. .E. T
. T, T - T
. T, T
+
.E. T
T, T
3
3
ab
has
(p has
p
K
L)
B
B (p
p
K
L)
2 (p
p
K
L)
1
(p p
T, T indicates two unspecified moments of the transient associated with the situation p.
K
L)

One calculates also the same type of term, with the thermal transient associated the situation Q:
P I, J .D
0
0
D
1
S I, J, T = K C
.
+ K C
. 0 M
.
I, J +
.K .E. T

T, T
p (
Q)
()
1
1
2
2
I (
)
3
1 (Q
Q
K
L)
2 T.
2.I
2 (.1 -)

1
+ K C
. .E. T
. T, T - T
. T, T +
.E. T
T, T
3
3
ab
has
(Q has
Q
K
L)
B
B (Q
Q
K
L)
2 (Q
Q
K
L)
1
then:
S I, J
max max S I, J, T, max S I, J, T
p (
)

=


Q
Q (
p (
p)
Q
Q (
p (
Q)
(T T,
,
K
L)
(T tkl)



with:

· C1, C2, C3, K1, K2, K3 indices of constraints provided to the §B3680 of the RCC-M
· E: modulus of elasticity of piping at ambient temperature
· : Poisson's ratio
·
: dilation coefficient of piping at ambient temperature (with T_REF)
· Eab: average modulus of elasticity enters the two zones separated by a discontinuity to
ambient temperature (TEMP_REF).
· D0: diameter external of piping
· T: nominal thickness of the wall

I =
2
2 2
.
.
64 (D0 - (D0 - T))
· I: moment of inertia of piping

· Semi (I, J): variation of moment resulting from the various loadings of the situations to which
belong the stabilized states I and J:
M =
-
+
-
+
-
I
(MR. X (I) MR. X (J) 2 (MY (I) MY (J) 2 (MZ (I) MZ (J) 2
·

· P0 (I, J): difference in pressure between states I and J.

The terms utilizing the temperature are:

1
1
1
T
T, T =
2 T T, T (y Dy
).
=
2 T T (y Dy
2
).
T T (y Dy
).
T
=
T - T
T
moy (Q
Q
-
-

K
L)
T
Q
Q
T
(K L)
T
Q
T
(K)
T
Q
T
-
(L)
(Q
moy
K)
(Q
moy
L)
T
2
T
2
T
2
12
12
12
T

,
.
,
().
.
().
.
().
1 (Q
Q
T T =
2 y T T T
y Dy =
2 y T T
y Dy -
2 y T T
y Dy V
=
T - V
T
2 -
2 -
2
K
L)
T
Q
Q
T
(K L)
T
Q
T
(K)
T
Q
T
-
(L)
(Q
moy
K)
(Q
moy
L)
T
2
T
2
T
2

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Q
Q
Q
Q
Q
Q
T
T T - T
T T -
T
T T
max (,
K
L)
moy (K
L)
1
,
1 (,
K
L)
2
and
Q
Q
Q
Q
Q
Q
Q
Q
T
T T =
T
T T - T
T T -
T
T T
2 (,
K
L)
max min (, K L) moy (K L) 1
,
1 (,
K
L)
2
0

One also calculates:

Sn (I, J): amplitude of variation of the linearized constraints (eq. (10) of the §B3653 of the RCC-M)

P I J D
D
S I J T = C
+ C
M I J +
E T
T T + C E T T T - T T T
N (, p)
0 (,
).
.
0
. 0.
p
p
p
p
p
p
1
2
I (
)
1
,
….
1 (
,
K
L)
.
.
3
ab
has
has (
,
K
L)
.
B
B (
,
K
L)
.
2 T
.
2 I
(.
2 1 -)
P I J D
D
S I J T = C
+ C
M I J +
E T
T T + C E T T T - T T T
N (, Q)
0 (,
).
.
0
. 0.
Q
Q
Q
Q
Q
Q
1
2
I (
)
1
,
….
1 (,
K
L)
.
.
3
ab
has
has (,
K
L)
.
B
B (,
K
L)
.
2 T
.
2 I
(.
2 1 -)
S I J =
S I J T
S I J T
N (,
) max (max N
Q
Q
(, p), max N
Q
Q
(, Q)
(T, tkl)
(T, tkl)
)

One calculates then S p, Q = max S I, J, for I and J sweeping the whole of the stabilized states of both
N (
)
N (
)
I, J
situations p and Q (in general, 6 possible combinations).

One obtains finally the amplitude of constraint between states I and J, by:

1
,
=.
.
,
.
,
alt (I
)
E
J
C Ke (Sn (p Q) S p (I J)
2nd

with:
EC.: Young modulus of reference for the construction of the curve of Wöhler, provided by the user
in DEFI_MATERIAU, under key word E_REFE, of the key word factor FATIGUE.
E: Smaller of the Young moduli used for the calculation of states I and J, i.e. evaluated with
temperatures of these stabilized states.

Ke the elastoplastic concentration factor defined in the §B3234.6 of the RCC-M.

1


if
S

N (p, Q)
3.Sm
-


K
=
+
-
<
<

E (S N (
N
S p Q
p, Q)
1
N (,
)
1

N (
S p Q
m
. m -).
1 3.S

if

,
3. .S




m
N (
)
m
1
3.S

m

1


if
S

N (p, Q)
3. .S
m m
N

The values of m and N depend on material, and are provided by the user in DEFI_MATERIAU,
under key words M_KE and N_KE, key word factor FATIGUE. The value of S is smallest of
m
values corresponding to the stabilized states I and J. If key words TEMP_REF_A and TEMP_REF_B are
present, S is interpolated for this temperature (which must correspond to the average temperature
N
transient). If not, S is taken at ambient temperature.
N
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Note:

In the case of mixed loadings mechanics and thermics, the RCC-M (from
modifying of June 1994) the decomposition of the concentration factor authorizes élasto-
plastic in a mechanical component (Ke_meca) and a thermal component (Ke_ther).
This method of calculation is generally (but not in all the cases) a little less
penalizing that method above. We chose here not to use this
possibility for two reasons. On the one hand the decomposition of Ke is profitable only if the share
of thermal loading of origin is important (and it complicates the analysis with fatigue).
In addition the expression of Ke_ther proposed in the RCC-M is valid only for steels
austenitic. In the case of ferritic steels, coefficients of the expression of Ke_ther
must be the subject of a validation on a case-by-case basis, which seems not very compatible with
objectives of our schedule of conditions.

One builds thus, for each group of situation, a symmetrical square matrix containing
the whole of the alt (I, J) thus obtained. In this unit, one identifies the combination (K, L)
corresponding to the greatest value of alt. One associates this matrix a vector containing it
a number of occurrences of each stabilized state

5.2.1.1 Case of the under-cycles

The under-cycles correspond either to the taking into account of the under-cycles related to the seism, or with
situations for which key word COMBINABLE=' NON' was indicated. In both case, one calculates
the amplitude of constraints while utilizing only constraints related to these under-cycles (not
of combination of states of loading apart from this situation). For the calculation of alt, it is necessary
to use the Ke factor which corresponds to the principal situation from which the under-cycle results.

5.2.2 Calculation of the combinations of loading (I, J) for the situations of passage
between group of situations

Two states of loading are combinable only if they belong to the same situation or if it
exist a situation of passage between the groups to which they belong. In this last case, one
will associate combination I, J the number of occurrences of the situation of passage. If
situation of passage belongs to the one of the two groups (what is not excluded a priori), it is
naturally combined with the other situations of this group, then is used for the combination of the situations
of its group with the situations of the group in relation.

For each situation of passage of a group with another, one thus considers the whole of
combinations (I, J) with I pertaining to the first group (of dimension NR) and J pertaining to
second group (of dimension M). For each combination, alt (I, J) was obtained in the same way
that previously and one associates to him the number of occurrences of the situation of passage. One builds
still a matrix (rectangular) containing all the alt (I, J),

5.3
Calculation of the factor of use

One notes:

nk the number of cycles associated with the situation p to which belongs the state stabilized K;
nl the number of cycles associated with the situation Q to which belongs the state stabilized L;
NS the number of occurrences of the seism (only SNA is considered in second category)
NS numbers under-cycles associated with each occurrence with the seism.
npass a number of cycles associated with a possible situation with passage between p and Q if these
situations do not belong to the same group, but if there exists a situation of passage enters
both.
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For the whole of the combinations of states of loading (inside a group of
situations or associated a situation of passage):

If NS 0, one selects the NS/2 combinations of stabilized states K and L more penalizing,
i.e. NS/2 combinations (K, L) driving with the greatest values of alt (K, L).

For each of these NS/2 combinations:

· one superimposes the loadings of seism to the variation of moment resulting from different
loadings of the stabilized states K and L:

2
2
2
M = (M




1 (K) - M1 (L) +
M 1) + (m2 (K) - m2 (L) + m2) + (m3 (K) - m3 (L) + M
I
S
S
S 3)
with:

MX (K) and MX (L): components in the direction X (X {1 2
; ; }
3) of the moments associated with
states K and L
MSx: Total amplitude of variation in direction X of the moments due to the seism.

· One calculates then Sp and Sn with the new value the Semi one and one calculates:

1
, =.
.
,
.
,
alt _ S (K
)
E
L
C Ke (Sn_ S (m N) Sp_ S (K L)
2nd

· one calculates the number of acceptable cycles NR (K, L) for the amplitude of constraint alt_S (K, L).
NR (K, L) corresponds to the X-coordinate of the point of ordinate alt_S (K, L) in the curve of Wöhler
associated material.
1
· one calculates finally U
=

1 (K, L)
NR (K, L)
· one takes into account the under-cycles due to the seism while calculating:

E
D
K,
C
L =
.K S
K, L .K .C.
.
M

+ M

+ M


alt _ SC (
)
E (N _ S (
)
0
2
2
2
2
2
S1
S 2
S 3
E
4.I
2 N
. - 1
· one calculates with this value: U
=
with:
NR
2 (K, L)
(S)
NR
,
SC (K, L) a number of cycles
SC (K L)
acceptable for the amplitude of constraint alt_SC (K, L). It should be noted that one uses
value K
,
previously calculated for the principal cycle.
E (S N _ S (K L)

· One then cumulates these factors of use partial in the factor of total use: U = U + u1 (K, L) +
u2 (K, L)

One starts again this calculation until exhaustion of the NS/2 combinations more penalizing.

The calculation of the factor of use is then continued without taking into account the seism:

If NS = 0, or after having taken into account the seism for the NS/2 combinations more
unfavourable:

· One selects the combination (K, L) leading to the maximum value of alt (K, L), on
the whole of the combinations, such that the number of occurrences n0 is nonnull.
With n0 = min {nk, nl, npass,} if npass is nonnull, or n0 = min {nk, nl,} if npass is null.
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· one calculates the number of acceptable cycles NR (K, L) for the amplitude of constraint alt (K, L).
NR (K, L) corresponds to the X-coordinate of the point of ordinate alt_S (K, L) in the curve of Wöhler
associated material.

1
, =.
.
,
.
,
alt (K
)
E
L
C Ke (Sn (p Q) S p (K L)
2nd
N
· one calculates U (K, L)
0
=

NR (K, L)
· one replaces then
nk by (nk - n0)
nl by (nl - n0)
if it is about a situation of passage, npass by (npass - n0)

then:

if nk = 0, the column and the line corresponding at the state stabilized K of the matrix alt (I, J) are
settings with 0.
if nl = 0, the column and the line corresponding at the state stabilized L of the matrix alt (I, J) are
settings with 0.

The loop is repeated until exhaustion of the number of cycles. The factor of use U of the line
G
for the group considered is then defined by:

U
. It is cumulated with the factor of total use: U = U +U
G = U (K L)
early
early
G

Note:

Appendix ZI of code RCC-M defines the curves of Wöhler until an amplitude of
constraint minimum corresponding to one lifespan of 106 cycles. If the value alt calculated
for a combination (I, J) of stabilized state is lower than this amplitude minimum, the factor
of use is equal to 0 for the combination (I, J) considered. This returns implicitly to
to consider the existence of a limit of endurance to 106 cycles.

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6
Course of the analysis of the behavior to fatigue
according to RCC-M B3200

The processing which is described here (cf [bib2]) is to be carried out for the segment considered. The result obtained
will be thus a factor of use (total or partial) in each end of this segment.

6.1
Calculations of all the states of loading

6.1.1 Linear combination of the tensors of constraints

The present stage consists in reconstituting, for all the situations, (including the situations of
passage) tensors of constraints in each node of the segment relating to each state stabilized (in
cumulating the various tensors of the constraints which intervene).

For each calculation of unit loading, one extracts the tensor from the constraints along the segment
of analysis. The tensors all of the constraints must be expressed in the same reference mark (the reference mark
total related to modeling 2D or 3D).

This reference mark must be coherent with that in which the total efforts resulting from calculation are expressed
beam.

One notes _ () with
U

{XX, YY, ZZ, XY, XZ,
}
YZ components of the tensor of
constraints associated with the unit loading. The calculation of the tensor of constraints corresponding to
mechanical loading pertaining in a stabilized state is then obtained in the following way:

that is to say F
,
,
,
,
,
the torque of effort associated with the loading (I).
X (I) Fy (I) Fz (I) M X (I) M y (I) M Z (I)

one has then, by linear combination:


=

+

+

(I)
F I.
F
F I.
F
F I.
F
+
X ()
_U (X _U)
y ()
_U (Y _U)
Z ()
_U (Z _U)

M I.
M
+ M I.
M
+ M I.
M
X ()
_U (
X _ U)
y ()
_U (
Y _ U)
Z ()
_U (
Z _ U)

One then linearly cumulates the tensors of constraints for all the loadings of the state
stabilized considered.

One stores, on this level, the tensors of constraints for each node of the segment.

6.1.2 Calculation of the thermal transients

As the thermal transients lead to statements of constraints (on a section
corresponding to the node studied) variable according to time, it is necessary to store all these
values in order to maximize the amplitudes of constraints correctly.
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6.1.3 Case of the seismic loadings

The seismic loading breaks up into 2 parts:

has) Inertial left Une

It is calculated by imposing on the whole of anchorings the same movement characterized by
spectrum envelope of the various spectra of floor, in the horizontal directions X and Y
on the one hand, and vertical Z on the other hand (in the total reference mark). With the exit of this calculation beam, one
obtains generalized efforts which with the quadratic office plurality of these efforts for each direction of
seism, therefore not signed efforts. One stores these constraints in table 1 above, for
to seek thereafter the combination of sign which maximizes the amplitude of constraints.


b) Quasi-static left Une

It is estimated by imposing static differential displacements corresponding to
maximum of the differences of the seismic movements of the points of anchoring in the course of time. Of
even, the efforts are combined by quadratic average, therefore not signed. The result is with
to store in table 1 above.

6.1.4 Calculations of the amplitudes of constraints inside each group of
situations

The first phase consists in calculating the amplitudes of constraints which correspond to
combinations of all the stabilized states pertaining to the situations of a given group, in
choosing the moments of the thermal transients which maximize these amplitudes of constraints.

Indeed, the thermal transients defined in Dossier d' Analyze of Comportement are
associated situations. During the analysis of the behavior to fatigue, we are brought to
to define cycles of fictitious loadings by associating stabilized states pertaining to situations
different. In this case, the thermal transient associated the fictitious cycle corresponding to the states
stabilized I and J will be selected in order to maximize the amplitude of constraints.

The whole of the combinations (I, J) is thus considered with (I, J) ({1, 2. , NR}, {1, 2. , NR}) (NR being it
a number of states stabilized except seism, i.e. 2 times the number of situations of the group), and one
built a matrix [NR;NR] of the values alt (I, J).

6.1.4.1 Calculation of the alt (I, J) without taking into account of the seism

The calculation of alt (I, J) was carried out, for each couple of stabilized states (I, J), and each end of
segment, starting from the tensor of the constraints S
, and of the tensor of the linearized constraints
p (I J)
S
:
N (p Q)

1
,
=.
.
,
.
,
alt (I
)
E
J
C Ke (Sn (p Q) S p (I J)
2nd

with:

EC.: Young modulus of reference for the construction of the curve of Wöhler, provided by the user
in DEFI_MATERIAU, under key word E_REFE, of the key word factor FATIGUE.
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Ke the elastoplastic concentration factor defined in the §B3234.6 of the RCC-M.


1


if
S

N (p, Q)
3.Sm
-


K
=
+
-
<
<

E (S N (
N
S p Q
p, Q)
1
N (,
)
1

N (
S p Q
m
. m -).
1 3.S

if

,
3. .S




m
N (
)
m
1
3.S

m

1


if
S

N (p, Q)
3. .S
m m
N

The values of m and N depend on material, and are provided by the user in DEFI_MATERIAU,
under key words M_KE_RCCM and N_KE_RCCM, key word factor FATIGUE.

Note:

In the case of mixed loadings mechanics and thermics, the RCC-M (from
modifying of June 1994) the decomposition of the concentration factor authorizes élasto-
plastic in a mechanical component (Ke_meca) and a thermal component (Ke_ther).
This method of calculation is generally (but not in all the cases) a little less
penalizing that method above. Mechanical decomposition - thermics leads in
effect with more important values of Ke for certain Sn values.
We chose here not to use this possibility for two reasons. On the one hand
decomposition of Ke is profitable only if the share of thermal loading of origin is
important (and it complicates the analysis with fatigue). In addition the expression of Ke_ther
proposed in the RCC-M is valid only for the austenitic steels. In the case of them
ferritic steels, the coefficients of the expression of Ke_ther must be the subject of a validation
on a case-by-case basis, which seems not very compatible with the objectives of our schedule of conditions.

One builds thus, for each group of situation, a symmetrical square matrix containing
the whole of the alt (I, J) thus obtained. One associates this matrix a vector containing the number
occurrences of each stabilized state. In fact, the process this calculation of the factor of use requires
to calculate two matrices alt (I, J): one without taking into account the seism, the other with taking into account
seism. One is interested here in the matrix without seism.

For each combination, alt (I, J) was obtained in two stages: Sn (p, Q) should initially be calculated, then
Sp (I, J).

It is necessary to seek the maximum of alt (I, J) for each combination of states of loading I, J. But it
is necessary to obtain these values to take account of the variation of the constraints due to the transients
thermics, which are variable according to time. The step presented here thus consists with
to seek the moment of the thermal stresses which make maximum alt (I, J).

The following paragraphs thus correspond to calculations to carry out for any couple of state of
loading I, J pertaining to the situations p and Q of the same group of situations, and for any couple
moments T p and tq transitory respectively associated with the situations p and Q.

Sn calculation (p, Q) without seism

One calculates the equivalent amplitude of the linearized constraints Sn for any couple of stabilized states
belonging to the situations p and Q of couple I and J current (in practice there are two states stabilized by
situation, there are thus 4 combinations to calculate).
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All along the ligament of analysis, one calculates the tensor of amplitude of the constraints (I, J)

ij (I, J) = ij (I) - ij (J)

with () the tensor of the constraints associated with the state with loading.

Moreover, it is necessary to tarnish account of the thermal transients associated with the situations p and Q. One superimposes
thus with (I, J) the tensors of constraints corresponding to the thermal transients associated
situations p and Q:

I, J, T = I, J + T and I, J, T = I, J + T
ij (
Q)
ij (
) ij (Q)
ij (
p)
ij (
) ij (p)

T and T represent two moments of the transients respectively associated with the situations p and Q.
p
Q
One then linearizes (I, J, T and (I, J, T, component by component, along the ligament:
Q)
p)
T
T
(
2
moy


fle
6
y Dy and (
y
y Dy
ij)
=.
.
.
2
(ij
mn
) ()
ij
)
2
= 1. (ij
mn
) ().
mn
T
mn
-
T
T
- T
2
2
(flax

=
+

ij)
(moy
ij
mn
) (fleij
mn
) mn

with: T thickness of the wall

y radial position of the point considered (is null there with mid thickness and positive on the surface interns)

For each combination (I, J), one obtains two values
: S I, J, T, S I, J, T which are
N (
Q)
N (
p)
respectively equal to the equivalent constraint (within the meaning of Tresca) of the two tensors
flax

I, J, T,
flax

I, J, T
ij
(Q)
ij
(p)
One retains finally largest of these two values, for each 4 combinations I, J and
then the greatest value among the 4 combinations:

S p, Q
max max S I, J, T, max S I, J, T

N (
)

=

N (
p)
N (
Q)

I, J T
T
p
Q


One stores, for the couple of situation of the group considered, the values of S
,
. Indeed,
N (p Q)
maximization over the moments could not be made on this level, but only on the level of
value of alt (I, J). In fact one chooses here to return maximum Sn (p, Q) so that the factor of correction
elastoplastic Ke is maximum (what is in conformity with the RCC-M, to see ZH210 for example).

Calculation of Sp (I, J) without seism

The calculation of Sp (equivalent amplitude of the total constraints) is to be realized for each of both
ends of the segment, in the following way:
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For the couple (I, J), at each end of the segment, one calculates the tensor of amplitude of the constraints
(I, J)

ij (I, J) = ij (I) - ij (J)

with () the tensor of the constraints associated with the state with loading.

Moreover, it is necessary to tarnish account of the thermal transients associated with the situations p and Q. One superimposes
thus with (I, J) the tensors of constraints corresponding to the thermal transients associated
situations p and Q:

I, J, T = I, J + T and I, J, T = I, J + T
ij (
Q)
ij (
) ij (Q)
ij (
p)
ij (
) ij (p)

S (I J
I, J
p,) is equal to the constraint equivalent (within the meaning of Tresca) to the tensor (
):
That is to say 1, 2 and 3 components of (I, J) in the principal reference mark,

S (I J
1 2 2 3 3
p,) = my {
X
-
,
-
,
- 1}

For each combination (I, J), one obtains two values
: S I, J, T, S I, J, T which are
p (
Q)
p (
p)
respectively equal to the equivalent constraint (within the meaning of Tresca) of the two tensors
I, J, T, I, J, T
ij (
Q)
ij (
p)

Final expression of alt (I, J)

The value retained for alt (I, J) is that which makes maximum the expression:

1 E
I, J, T =.
.K S p, Q S
.
I, J, T
alt (
)
C
p
E (N (
) p (
p)
2nd



That is to say: S I, J
max max S I, J, T, max S I, J, T
alt (
) =

(alt (p)
(alt (Q)
T
T
p
Q

One then stores this value in a symmetrical square matrix containing the whole of the alt (I, J)
without seism. One stores also the symmetrical matrix square containing the values of K
,
, which
E (Sn (p Q)
could be used for the under-cycles.

Case of the under-cycles

The under-cycles correspond either to the taking into account of the under-cycles related to the seism, or with
situations for which key word COMBINABLE=' NON' was indicated. In both case, one calculates
the amplitude of constraints while utilizing only constraints related to these under-cycles (not
of combination of states of loading apart from this situation). For the calculation of alt, it is necessary
to use the Ke factor which corresponds to the principal situation from which the under-cycle results.

6.1.4.2 Calculation of the alt (I, J) with taking into account of the seism

The second phase consists in calculating the amplitudes of constraints which correspond to
combinations of all the stabilized states pertaining to the situations of a given group, in
choosing on the one hand, moments of the thermal transients which maximize these amplitudes of
constraints, and in addition, combination of signs for the not signed loadings which provide
also the maximum.
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Sn calculation (p, Q) with taking into account of the seism

The seismic loadings are not signed. Each component of the tensor of the constraints can
thus to take two values (positive and negative). At the time of the superposition of a loading not signed
with a signed loading, the RCC-M forces to retain on each component a sign such as
calculated constraint (makes alt of it) is raised. The tensor of the constraints cash six components, it
thus exist 64 combinations of signs to be examined.
After having reconstituted the stress fields corresponding to the stabilized states, on the one hand and with
seism, in addition, one proceeds in the following way:
one considers the combination of two stabilized states I and J, and one calculates (I, J, T and (I, J, T,
Q)
p)
as above. One defines, for each node of the segment, and each of the two transients
thermics, 64 Sk tensors of seismic loadings corresponding to the 64 combinations of sign
possible, for the 6 components of the tensor of the constraints corresponding to the loading
seismic, and one numbers them from 1 to 64.
One calculates then the equivalent amplitude of constraint linearized on the segment: S I, J, T, S
N (
p
K)
corresponding to each of the 64 combinations:
S I, J, T, S = forced equivalent of Tresca (
flax

I, J, T, S).
ij
(p K)
N (
p
K)
S

S p, Q
max max max S I, J, T, S, max max S I, J, T, S

N
=

I, J
T
[N p K
K
] T [N Q K
K
]
One retains finally
()
(
)
(
)
p
Q

One stores, for the whole of the couples of situations of the group considered, the values of S S
,
N (m N)

6.1.4.3 Calculation of Sp (I, J) with taking into account of the seism

Just as previously, the seismic loadings not being signed, it is necessary to retain on each
component a sign such as the calculated constraint is raised. The tensor of the constraints cash
six components, there are thus 64 combinations of signs to be examined. For each combination of
sign and each moment, two values are obtained: S I, J, T, S, S I, J, T, S which are
p (
Q
K)
p (
p
K)
respectively equal to the equivalent constraint (within the meaning of Tresca) of the two tensors
I, J, T, I, J, T. These values are stored.
ij (
Q)
ij (
p)

Final expression of alt (I, J)

The value retained for alt (I, J) is that which makes maximum the expression:

1 E
I, J, T, S =.
K
.
S p, Q S
.
I, J, T, S
alt (
p
)
C
K
E (S
N (
) p (p K)
2nd



That is to say: S I, J, S
max max max S I, J, T, S
, max max S I, J, T, S
alt (
) =


T
([alt (p K)]
K
) T ([alt (Q K)]
K
)
p
Q

One then stores this value in a symmetrical square matrix containing the whole of the alt (I, J)
with seism.

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6.1.5 Calculation of the amplitudes of constraints for the situations of passage enters
group situations

For each situation of passage of a group with another, one considers the whole of the combinations
(I, J) with I pertaining to the first group (of dimension NR) and J pertaining to the second group (of
dimension M). For each combination, alt (I, J) was obtained in the same way that previously.
One still builds a matrix (rectangular) containing all the alt (I, J), which one associates it
a number of occurrences of the situation of passage.

Indeed, two states of loading are combinable only if they belong to the same situation
or if there is a situation of passage between the groups to which they belong. In it
last case, one will associate combination I, J the number of occurrences of the situation of passage.
If the situation of passage belongs to the one of both groups (what is not excluded has
priori), it is naturally combined with the other situations of this group, then is used for the combination
situations of its group with the situations of the group in relation.

6.1.6 Storage of the amplitudes of constraints for all the combinations

To carry out the calculation of the factor of use, amplitudes of the constraints calculated previously and
the associated numbers of occurrences are stored in a square matrix containing all them
amplitudes of constraints alt except seism, for all the possible combinations of situations (with
interior of each group of situations, and between two groups if there is a situation of passage).
The matrix has as a dimension the sum of the number of situations of all the groups:

Alt

Group 1
Group 2
Group 3


Etat1

State J



State L


.
State N
State NR
Group 1
State 1
Alt
… …… 0 0 0
0………

State
I


Alt (I, J)

0 0 0
0…… alt (I, N)



0 0 0
0………



0 0 0
0………

Group 2
….








0 0 0
0
State
K




Alt (K, L)

0 0 0
0









0 0 0
0
….


SYM





0 0 0
0
Group 3
….












State
m





Alt (m, N)














State
NR








In the table above, one associates value 0 the combinations of states between groups 1 and 2 and
groups 1 and 3, because there is not situation of passage between these groups. On the other hand it in
exist between groups 1 and 3, one thus associates herring barrel combination states of groups 1 and 3
the value of alt.
The number of the states of loading will have to be built starting from the number of situation (single) and of
relative number (1 or 2) of the stabilized state of the situation, to obtain a univocal classification of
states of loading.

In the case of the under-cycles, only the diagonal term is filled.

This matrix is also to build for the values of S
,
who take into account
alt (I J S)
seism. There are thus two matrices giving all the possible values of alt.
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In correspondence of this matrix, one can associate a table giving the number of occurrences, it
number of situation and the number of group of each stabilized state, and a table giving the number
occurrences of the situations of passage:

State
Situation groups
Under-cycle
nocc

group 1 group 2
Npass
State
1
1 1 0 N1 1 1 0
… 1 1 0 N1 1 2 0
State I
2
1
0
Ni

1
3
npass
… 2 1 0 Ni………
State
K
3 2 0 Nk 2 1 0
… ………… 2 2 0
State
m
… 3
nsous
0 2 3 0
… ………… 3 1
npass
State NR
NR
3
0
NR





6.2
Calculation of the factor of use

For the whole of the combinations of states of loading (inside a group of situations or
associated a situation of passage), and for each end of the segment:

The general step described by the §ZH210 of code RCC-M is as follows:

One considers the whole of the NR stabilized states, and one builds:

· the matrix [NR, NR] of the equivalent amplitudes of constraint alt (I, J, T, S) corresponding to
superposition of the transient of passage of the state stabilized I in a state stabilized J, of the transient
thermics associated and with the seismic loading.
· the matrix [NR, NR] of the equivalent amplitudes of constraint alt (I, J, T) corresponding to
superposition of the transient of passage of the state stabilized I in a state stabilized J and of the transient
associated thermics (without seism).
One notes:
nk
the number of cycles associated with the situation p to which belongs the state stabilized K;
nl
the number of cycles associated with the situation Q to which belongs the state stabilized L;
NS
the number of occurrences of the seism (in general only of SNA is considered in second
category)
NS
a number of under-cycles associated with each seism.
npass a number of cycles associated with a possible situation with passage between p and Q if these
situations do not belong to the same group, but if there is a situation of
passage enters both.

If NS 0, one selects the NS/2 combinations more penalizing, i.e. the NS/2
combinations (K, L) driving with the greatest values of alt (K, L), without taking into account it
seism.

For each of these NS/2 combinations:

1
· One calculates the factor of use u1 (K, L) associated alt (I, J, T, S): U
=
, with NR (K, L) it
1 (K, L)
NR (K, L)
a number of acceptable cycles associated alt (I, J, T, S).
· One takes into account the NS under-cycles of the seism by:
, K and L being both
alt _ SC (K L)
extreme states of the under-cycle. Then:
2. - 1
U
=
with NR
2 (K, L)
(NS)
NR
,
SC (K, L) a number of acceptable cycles for the amplitude of
SC (K L)
constraint alt_SC (K, L).
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· One obtains then: U (K, L) = u1 (K, L) + u2 (K, L)

· One starts again this calculation until exhaustion of the NS/2 combinations more penalizing.

The calculation of the factor of use is then continued without taking into account the seism:

If NS = 0, or after having taken into account the seism for the NS/2 combinations more
unfavourable:

· One selects the combination (K, L) leading to the maximum value of alt (K, L), on
the whole of the combinations, such that the number of occurrences n0 is nonnull.
With n0 = min {nk, nl, npass,} if npass is nonnull, or n0 = min {nk, nl,} if npass is null.
· one calculates the number of acceptable cycles NR (K, L) for the amplitude of constraint alt (K, L).
NR (K, L) corresponds to the X-coordinate of the point of ordinate alt (K, L) in the curve of Wöhler
associated material.
· One calculates the factor of use U (K, L) associated the amplitude of constraint alt (K, L):
U (K,)
N
L
0
=

NR (K, L)
· One increments the factor of total use: U = U + U (K, L)
· One replaces:
nk by (nk - n0)
nl by (nl - n0)
if it is about a situation of passage, npass by (npass - n0)
if nk = 0, the column and the line K of the matrix of the alt (I, J) were put at 0.
if nl = 0, the column and the line L of the matrix of the alt (I, J) were put at 0.

· The loop is repeated until complete exhaustion of the number of cycles.

Note:

Appendix ZI of code RCC-M defines the curve of Wöhler until an amplitude of
constraint minimum corresponding to one lifespan of 106 cycles. If the value alt
calculated for a combination (I, J) of stabilized state is lower than this amplitude
minimum, the factor of use is equal to 0 for the combination (I, J) considered.

The loop is repeated until exhaustion of the number of cycles.

The factor of use U of each end of the segment for the group considered is then defined by:
G

U
. It is cumulated with the factor of total use: U = U +U
G = U (K L)
early
early
G
Handbook of Référence
R7.04 booklet: Evaluation of the damage
HT-66/02/004/A

Code_Aster ®
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7 Bibliography

[1]
“RCC-M: Rules of Conception and Construction of the mechanical hardware of the small islands
nuclear PWR. Edition 1991 “Edité by the AFCEN: French association for the rules of
design and of construction of the hardware of the nuclear boilers.
[2]
Y. WADIER, J.M. PROIX, “Spécifications for an ordering of Aster allowing of
analyzes according to rules' of the RCC-M B3200 “. Note EDF/DER/HI-70/95/022/0 “RCC-M:
Rules of Conception and Construction of the mechanical hardware of the nuclear small islands
PWR. Edition 1993 “Edité by the AFCEN: French association for the rules of design
and of construction of the hardware of the nuclear boilers.
[3]
I. BAKER, K. AABADI, A.M. DONORE: “Project OAR: Description of “file OAR”,
filing system of feeding of the data base “Note EDF/R & D/HI-75/01/008/C
[4]
F. CURTIT “Réalisation of a software tool for analysis to fatigue for a line of piping -
schedule of conditions “Note EDF/R & D/HT-26/02/010/A
[5]
F. CURTIT “Analyze with the fatigue of an interior line VVP Br with under-thickness” Note
EDF/R & D/HT-26/00/057/A
Handbook of Référence
R7.04 booklet: Evaluation of the damage
HT-66/02/004/A

Code_Aster ®
Version
6.4
Titrate:
Postprocessing according to RCC-M'S


Date:
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Author (S):
Key J.M. PROIX
:
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: 32/32

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