r7.04.03 Post-treatment according to RCC -M#

Summary:

The operator POST_RCCM [U4.83.11] allows you to check the level 0 criteria and some level A criteria of chapter B3200 of the RCC ‑M, for 2D or 3D continuous media models. Level 0 criteria aim to protect the material against damage from excessive deformation, plastic, elastic, and elastoplastic instability. Level A criteria, on the other hand, aim to protect the equipment against damage caused by progressive deformation and fatigue.

It also allows the calculation of the level A criteria of chapters B3600 and ZE200 in post-processing pipe calculations.

Finally, it makes it possible to assess environmental fatigue resistance for chapters B3200 and ZE200.

Moreover, the operator POST_RCCM makes it possible to calculate the priming factor at the level of a singular zone, in the sense of appendix ZD of RCC -M.

Introduction

The RCC -M [65] describes the general rules for analyzing the behavior of nuclear power plant equipment. These rules aim to ensure that REP power plant equipment has sufficient safety margins with respect to the various types of damage to which they could be exposed as a result of the loads applied to them: excessive deformation and plastic instability, elastic or elastoplastic instability, gradual deformation, gradual deformation under the effect of repeated stresses, fatigue (progressive cracking), sudden rupture, etc.

In c*ode_aster*, it is possible to perform four types of calculations:

calculation of level 0 and A criteria of chapter B3200 in post-processing calculations on 2D or 3D structures;
calculation of the fatigue criteria of chapter B3600 by post-processing pipe calculations;
calculation of the fatigue criteria in annex ZE200 in post-processing pipe calculations. The method according to appendix ZE200 is a mixed method that combines simplified equations from the B3600 and detailed analysis from the B3200.
calculation of environmental fatigue compatible with chapters B3200 and annex ZE200 of RCC -M. Environmental fatigue resistance was integrated into the RCC -M 2016 edition in the form of a Probationary Phase Rule (RPP).

Chapter 2 provides general information on chapter B3200 and annex ZE200 of RCC -M.

The B3200 criteria correspond in operator POST_RCCM to the methods (TYPE_RESU_MECA) “B3200” and “EVOLUTION” (). Their calculation is detailed in chapters 3 and 6. “EVOLUTION” makes it possible to calculate the priming factor for geometric singularities and is well suited to cases where there are few situations. In contrast, “B3200” is well suited to calculations for numerous situations, which can be distributed over several operating groups. Earthquake, transit situations and environmental fatigue can be taken into account with this TYPE_RESU_MECA.

In the POST_RCCM operator, the criteria for ZE200 correspond to the “ZE200a” and “ZE200b” methods. Their calculation is detailed in Chapter 4.

The calculation of the resistance to environmental fatigue is described in chapter 5.

In operator POST_RCCM, the fatigue criterion of the B3600 corresponds to the “B3600” method. Their calculation is detailed in Chapter 7.

General

This chapter aims to recall some basic definitions associated with Chapter B3200 and Appendix ZE200 of RCC -M. It does not apply to chapter B3600** . **

This chapter also makes it possible to describe the necessary adaptations made in c*ode_aster*, which are explained in [65] and [66]. In operator POST_RCCM, the criteria in chapter B3200 correspond to the methods TYPE_RESU_MECA =” EVOLUTION “and TYPE_RESU_MECA =”B3200”. In the POST_RCCM operator, the criteria in appendix ZE200 correspond to the TYPE_RESU_MECA =” ZE200a “and TYPE_RESU_MECA =” ZE200b” methods.

Geometric data

The user of the RCC -M must distinguish in its structure between areas of major discontinuity, areas of minor discontinuity and areas with geometric singularities. These require specific treatment (described in section 6.5).

Current area (excluding geometric singularity) :

The RCC -M defines « support segments » that are used to linearize constraints. These segments are, outside the areas of discontinuity, segments that are generally normal to the median surface of the wall, and in the areas of discontinuity, the shortest segments making it possible to join the 2 faces of the wall.

The user of c*ode_aster* must therefore define all the sections of the structure where the post-processing calculations will be made (it is he who knows if these sections pass through current areas, or areas of geometric discontinuity). In practice, we work:

or on an existing segment in the mesh;
or on a segment defined in MACR_LIGN_COUPE.

All the criteria are calculated systematically at both ends of the segment (calculation of the Pm, the use factor…).

Geometric singularity:

Areas of local discontinuities whose geometric outline varies abruptly are the site of acute stress concentrations. In this case, the classical methods associated with current areas are unsuitable and the concept of priming factor is introduced. This parameter must be calculated on a circle (with an imposed radius, depending on the material) around the singularity. The priming factor can only be calculated with the “EVOLUTION” type.

The user must therefore define this cut line. In practice, we work:

or on a circular cut line existing in the mesh;
or on a circular cut line defined in MACR_LIGN_COUPE.

Material data

The material data required for the calculation are as follows:

\(>\): allowable value (tabulated in RCC -M Appendix ZI).
\(>\): conventional elastic limit (tabulated in RCC -M Annex ZI 2.1).
\(>`*, * :math:\)>`: material constants for the calculation of \(>\) (defined in RCC -M B3234.6)
\(>`*, * :math:\)>`: elasticity modules (for the correction of the fatigue curve, annex ZI).
Material fatigue curves: according to RCC -M annex ZI.
Distance*d* to the geometric singularity and the priming law of the material (as defined and tabulated in appendix ZD2200 of RCC -M) for calculating the priming factor.

Simplifying hypotheses

In the RCC -M, the user should be able to say, after analyzing the results of the calculation, whether the main directions at a given point are fixed or if they rotate over time. In the POST_RCCM command, you can make no assumptions. Only the case where the main directions are any will be considered.

In addition, the user should be able to classify constraints into the following categories:

General membrane primer: \(>\)
Local membrane primer: \(>\)
Bending primer: \(>\)
Thermal expansion: \(>\)
Secondary: \(>\)
Cutting-edge: \(>\)

This choice cannot be made by POST_RCCM. Only the user can qualify a field as constraints (« primary », « secondary », or the sum of both). The criteria to be verified are calculated from constraint fields (constants or a function of time) provided by the user. It is he who ensures the consistency between the calculation of these fields and the criteria applied.

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

a loading that is constant or variable under imposed force or pressure is primary, except for certain very specific structures,
a constant or variable loading with imposed displacement is in principle secondary (except in the case of « spring effect »),
a permanent or transitory thermal loading is in principle secondary.

On the other hand, the combination of these types of loads leads to a result that can no longer be described as primary or secondary. Depending on the criteria, the user may therefore be required to break down his loads.

Calculations made by POST_RCCM

The operation of the command POST_RCCM making it possible to perform the calculation of certain criteria of levels 0 and A of the RCC -M B3200 and of the criteria of level A of the annex ZE200 is described below. The implementation described here does not take into account all the criteria of the B3200 and may be completed.

The main data is the (support) segment where the calculations will be carried out. It is the user who chooses the segment and who is responsible for finding the one for which the quantities involved in the criteria are maximum. The automatic search for this segment is a difficult problem, and is not scheduled.

After calculating one or more results by MECA_STATIQUE or STAT_NON_LINE, the user must extract the constraints on the analysis segment by POST_RELEVE_T or MACR_LIGN_COUPE. Finally, the user requests the calculation of the criterion (s) by the operator POST_RCCM.

Four types of criteria are each accessible by the keyword factor “OPTION”:

level A criteria (excluding fatigue) using the SN keyword,

level A fatigue criteria using the keywords FATIGUE (for the “B3200”, “ZE200a” and “ZE200b” types) or FATIGUE_ZH210 (for the “EVOLUTION” type),

an environmental fatigue criterion using the keyword EFAT. This criterion is compatible with the options” B3200 “,” ZE200a “and” ZE200b “.

Moreover, with the “EVOLUTION” method only, it is possible to check the priming criterion (level A criterion) in a singular zone (keyword AMORCAGE).

Level 0 criteria specified by the RCC -M (keyword PM_PB )

Level 0 criteria aim to protect the material from damage caused by excessive deformation, plastic instability, and elastic and elastoplastic instability. They must be verified by the reference situation (see B3121 and B3151). These criteria require the calculation of the equivalent stresses \(>\), \(>\), \(>\) which are defined below and are available for the “B3200” and “EVOLUTION” types.

General primary equivalent membrane stress

Given the primary constraint of the reference situation (1st category) and a segment located outside a zone of major discontinuity. At each end point of this segment of length l, we calculate \(>\) and the criterion is written (B3233.1):

\(>\)

\(>\) is the acceptable equivalent stress, tabulated in Annex ZI of RCC -M. \(>\) is defined in the calculation by the operand SM of the key word factor RCCM (or RCCM_FO) of DEFI_MATERIAU. It may be a function of temperature.

Primary equivalent local membrane stress

Given the primary constraint of the reference situation (1st category) and a segment located in a zone of major discontinuity, the definition of \(>\) is identical to that of \(>\) on this segment.

The criterion is written (B3233.2):

\(>\)

Primary equivalent membrane stress+flexion

Given the primary constraint of the reference situation (1st category) and a segment (oriented). At each end point of this segment of length \(>\), (ends corresponding to the external and internal skins), \(>\) and \(>\) are calculated and the criteria are written (B3233.3):

\(>\)

A-level criteria specified by the RCC -M (keywords SN and FATIGUE/FATIGUE_ZH210 )

Level A criteria aim to protect the equipment against damage caused by progressive deformation and progressive cracking. With the methods” EVOLUTION “,” B3200 “,” ZE200a “, and” ZE200b “, four types of criteria can be verified:

Range of variation of \(>\) (“SN” option);

Thermal ratchet calculation (“SN” option);

Calculation of the fatigue use factor (options” FATIGUE “/” FATIGUE_ZH210 “)

These various parameters and associated criteria are described below as defined in RCC -M. In part 7, we introduce a simplifying hypothesis before detailing their calculation in c*ode_aster* in chapters 3, 4 and 6.

Snet Sn calculation*

We take into account the more secondary primary stresses and the stresses resulting from thwarted thermal expansions: \(>\) which therefore represents the linearized stresses associated with all the loading (mechanical and thermal).

The calculation points are the two ends of the segment. At each end point of this segment of length \(>\), \(>\) is calculated according to paragraph B3232.6 and the global adaptation criterion is written (B3234.2):

\(>\)

\(>\) being the allowable stress as a function of the material and the temperature, given by the operand SM of the keyword factor RCCM (or RCCM_FO) of DEFI_MATERIAU.

If this criterion is not met, simplified elastoplastic analysis of B3234.3 can be performed. The following three operations must be performed:

We calculate \(>\) the amplitude \(>\) calculated without taking into account thermal flexural stresses and we must check the criterion:

\(>\)

make an elastoplastic correction (\(>\)) in the fatigue analysis,
check the thermal ratchet criterion (B3234.8) in the common parts of cylindrical shells (and pipes) subjected to pressure and a cyclic temperature gradient.

Thermal ratchet calculation

The wall of an apparatus subjected simultaneously to constant pressure and to cyclical variations in temperature can undergo large deformations under a thermal ratchet. It is a particular mechanism of progressive deformation in which the deformation increases by approximately the same amount with each cycle.

The condition to be met is written below and relates to the maximum allowable value of the amplitude of variation of the thermal stress, see B3234.8. It relates to the case of a shell with symmetry of revolution loaded by a constant internal pressure. We note:

\(>\), maximum allowable value of the amplitude of variation of the thermal stress,

\(>\), maximum of the general membrane stress (or average) due to pressure,

\(>\), conventional elastic limit read from tables Z I 2.1, for the maximum temperature reached during the cycle.

The criterion is of the form: \(>\). By asking \(>\) and \(>\), we have

if the temperature variation is linear across the wall:

For \(>\)

\(>\) for \(>\)

if the temperature variation is parabolic in the wall:

\(>\) for \(>\)

and for \(>\): linear interpolation between points: \(>\) and

\(>\).

In summary, the membrane stress \(>\) is therefore calculated by linearization of the pressure stress, then two quantities \(>\) and \(>\) are deduced from it using the equations above. These two quantities are the respective maximum allowable values of \(>\) and \(>\).

Fatigue use factor calculation

The general principle of fatigue calculation is to combine each of the situations two by two and to ensure that the total use factor defined in this paragraph is less than 1.

\(>\)

Algorithm for calculating the total use factor

Schematically, the algorithm for calculating \(>\) defined in chapter B3200 of RCC -M is as follows:

The elementary use factor for each combination of situations is calculated. The combination of situations p and q is based on the definition of two fictional transients 1 and 2. The basic use factor is the sum of the use factors due to each of these transitions. At the end of this step, a symmetric matrix (nxn) of elementary use factors is available (n being the number of situations),

\(>\)

The total use factor is initialized to 0,

\(>\)

We identify the most penalizing combination of situations k and l (maximum elementary use factor) and we multiply it by the minimum number of occurrences of these two situations,

\(>\)

We update the number of occurrences of the situations k and l,

\(>\) \(>\)

Go back to step 3 until all occurrences are exhausted.

Defining the two fictional transients is a delicate step in this algorithm. The rule is different depending on whether the main directions are fixed or variable.

\(>\) the elementary use factor for a combination of situations p and q is calculated by introducing into the material fatigue curve (Wöhler curve) the amplitudes of variation of the alternating stresses of the two fictional transients \(>\) and \(>\) into the material fatigue curve (Wöhler curve).

Salt calculation

\(>\) and \(>\) are defined from the amplitude of variation of the linearized stresses \(>\) and the amplitude of variation of the total stresses of the two fictional transients \(>\) and \(>\).

Two formulas are proposed (cf. §B3234.6):

KE_MECA:

\(>\)

The Young reference module (\(>\)) is provided by the user in DEFI_MATERIAU, under the keyword E_REFE, of the keyword factor FATIGUE. \(>\) is the elasto-plastic concentration factor that is a function of \(>\):

\(>\)

Parameters \(>\) and \(>\) are provided in DEFI_MATERIAU, under the keywords M_KE and N_KE, under the key word factor RCCM. If the keywords TEMP_REF_A and TEMP_REF_B are present, \(>\) is interpolated for this temperature (which must correspond to the mean temperature of the transient). Otherwise \(>\) is taken at room temperature.

KE_MIXTE: since the 1997 modification of RCC -M [1], we can choose another formula, based on a decomposition of \(>\):

\(>\)

with:

\(>\) is equal to the \(>\) defined above,
\(>\),
\(>\) and \(>\) represent the amplitudes of variation in the mechanical part of the total stresses of the fictional transients 1 and 2
\(>\) and \(>\) are calculated from the total stresses \(>\) and \(>\) from which the mechanical stresses \(>\) and \(>\) are subtracted respectively.

Finally, we calculate \(>\) the elementary use factor associated with the combination of p and q situations, defined from the fatigue curve of the material \(>\):

\(>\).

Note:

1) To calculate the elementary use factor of a situation p combined with itself, in KE_MECA, \(>\) and \(>\) .

En KE_MIXTE, \(>\) and \(>\)

2) In RCC -M, the contribution of sub-cycles is also taken into account in the elementary use factor. In code_aster, this quantity is not taken into account.

Type “B3200”

Charging data

The “B3200” type is well suited to calculations on a component subject to numerous situations. Several operating groups can also be defined, with possible situations of transition between these groups. Sharing groups and an earthquake can also be taken into account.

A situation is defined by its thermal, pressure and mechanical load (forces and moments). In code_aster the loads can be entered in various forms.

the thermal can only be entered as a transient \(>\) (RESU_THER)
pressure loads can be entered in two ways:
- as a transient \(>\) (RESU_PRES)
- in the form of a unit load \(>\) with two pressures \(>\) and \(>\) for stabilized states (RESU_MECA_UNIT, PRES_A and PRES_B),
the constraints related to mechanical loads (forces and moments):
- in the form of a transient \(>\) (RESU_MECA)
- with unit loads (global unit forces and moments applied to the limits of the model) with two torsors for the stabilized states (CHAR_ETAT_A and CHAR_ETAT_B). These efforts can either be calculated with c*ode_aster*, or taken from the OAR database.
- with unit loads to which a torsor is applied, this torsor is calculated by the interpolation between two torsors (CHAR_ETAT_A and CHAR_ETAT_B) which correspond to the temperatures TEMP_A, and TEMP_B and thanks to the temperature during the situation TABL_TEMP.

All types of situations can be combined in code_aster. For example, the user can enter a first situation where all the loads are in the form of transient then a second situation where the thermal and the pressure are in the form of transient and the mechanical with unit loads.

For the sake of clarity, only the equations in the case where all situations are described by transient loads are presented in this chapter.

Appendix 1 gives the equations in the case where all situations are described in unit form. Appendix 2 gives the equations in the case where all situations are described in unit form with interpolation on the temperature.

In this chapter, all operating situations can be broken down into transients, i.e. changes in constraints due to different loads as a function of time:

a transitory sum of the transients due to efforts and moments (defined under “RESU_MECA”),
a transient due to pressure (defined under “RESU_PRES”),
a thermal transient (defined under “RESU_THER”).

Notes:

For stitches, the user must then enter under RESU_MECA a transient mechanical tensor which is the sum of the two force tensors associated with the body and the tube. Appendix 3 summarizes the equations in this case.
The thermal transient, the pressure transient and the mechanical transient corresponding to a situation must be defined at the same instants

Calculations performed with the option “PM_PB”

For now, this option is available if all the mechanical and pressure loads in the situations are in unitary form.

Given the primary constraint of the reference situation (1st category) and a segment located outside a zone of major discontinuity. At each end point of this segment of length*l*, we calculate for a situation:

With \(>\)

\(>\)

Calculations performed with the option “SN”

The calculation points are the two ends of the segment. For a given situation, at each end point of this segment of length \(>\), \(>\) is calculated according to paragraph B3232.6:

\(>\) where \(>\) and \(>\)

In the case where the situation is described instantaneously (see part 12), this definition is applicable directly.

In code_aster, when the situation is defined in a unitary manner with two stabilized states and one thermal transient, the formula was adapted by introducing a method for selecting the moments. This method is described in part 14.

Method for selecting instants

In the “B3200” method developed in c*ode_aster*, a method for selecting moments has been implemented (“TRESCA” under the “METHODE” keyword)

It will thus be assumed that the times corresponding to the extremes of the amplitude of variation of the constraints (linearized or total) of the combination of two situations are also the moments corresponding to the extremes of the constraints of each of the situations taken alone.

Four moments \(>\), \(>\), \(>\) and \(>\) are therefore identified beforehand for each situation. We note \(>\) the stress tensor sum of the tensors in the form of a transient.

\(>\) and \(>\) correspond to the extremes of the linearized transient stress of the situation, in the sense of an equivalent Tresca stress signed by the stress trace:

\(>\)

\(>\) and \(>\) correspond to the extremes of the total transitory stress of the situation, in the sense of an equivalent Tresca stress signed by the stress trace:

\(>\)

Note:

There are actually 2x4 moments that are identified beforehand for each situation: at the origin and at the end of the analysis segment.

This method of selecting moments saves a significant amount of time. It is available by choosing the value “TRESCA” under the keyword “METHODE “.The method for selecting moments by the signed tresca is taken by default if the user does not specify anything. But it may lack robustness in particular when the coordinate system of the main constraints is rotating.

It is then possible to test all the moments of the transients, by choosing the value “TOUT_INST” under the keyword “METHODE”. It is not possible to differentiate the method for selecting the times for Sn and Sp.

Sn calculation

Note \(>\) the transient tensor associated with the situation; and \(>\) and \(>\) the extreme moments as defined at 14.

With “METHODE” = “TRESCA”, the parameter \(>\) for the situation is set by:

\(>\).

With “METHODE” = “TOUT_INST”, the parameter \(>\) for the situation is set by:

\(>\).

Note:

In this case, \(>\) .

Sn calculation*

Note \(>\) the amplitude \(>\) calculated without taking into account thermal flexural stresses. This definition is translated as:

At the origin of the segment:

\(>\)

At the end of the segment:

\(>\)

with \(>\).

Thermal ratchet calculation

It is necessary to have previously defined the conventional elastic limit for the maximum temperature reached during the cycle either by the operand SY_MAX of POST_RCCM; or by the operand SY_02 of the keyword RCCM in DEFI_MATERIAU [U4.43.01]. If no elastic limit is defined, thermal ratchet calculation is impossible.

In the table generated by the command, for each end of each analysis segment, for the situations and for the combinations of situations appear:

the maximum of the general membrane stress due to pressure \(>\) (“SIG_PRES_MOY”)
the linearized amplitude of variation of the thermal stress \(>\) and its maximum allowable value \(>\) (“SN_THER” and “CRIT_LINE”)
the amplitude of variation of the thermal stress \(>\) and its maximum allowable value \(>\) (“SP_THER” and “CRIT_PARAB”)

Calculations performed with the option “FATIGUE”

It is recalled that the calculation of the elementary use factor requires beforehand the calculation of the amplitude of variation of the linearized \(>\) and total \(>\) stresses for each of the combinations of situations (part 10). This calculation is carried out successively for the situations within each group with or without an earthquake, then for the combinations of transition situations between groups of situations.

A method for the accumulation of elementary use factors, based on the hypothesis of the linear accumulation of damage, is then used to obtain the global use factor.

Combining situations within each situation group

Sn calculation

We must not forget the case where the most penalizing combination of linearized constraints corresponds to the two extremes of the same situation. For the combination of p and q situations:

\(>\)

The quantities \(>\) and \(>\) are calculated according to part 13 and the calculation of the quantity \(>\) is described in the rest of this part.

Calculation of Sn (p, q)

We note \(>\) the transient tensor associated with the situation p and \(>\) the transient tensor associated with the situation q. \(>\) and \(>\) the extreme moments of the transition of the situation p and \(>\) and \(>\) the extreme moments of the transitory of the situation q as defined in 14.

With “METHODE” = “TRESCA”, the parameter \(>\) for the combination of p and q situations is defined by:

\(>\).

With “METHODE” = “TOUT_INST”, the parameter \(>\) for the combination of p and q situations is defined by:

\(>\).

Note:

In this case, \(>\) .

Calculating S p

In fact, we calculate the amplitude of variation of the total stresses of the two fictional transients \(>\) and \(>\). We must not forget the case where the most penalizing combination of total constraints corresponds to the two extremes of the same situation. So we change the definition of \(>\) and \(>\) as follows:

\(>\)

If \(>\), then \(>\);

If \(>\), then \(>\).

The quantities \(>\) and \(>\) are calculated according to Annex 4 and the calculation of the quantities \(>\) and \(>\) is described in the rest of this part.

Calculating S p1 (p, q) and S p2 (p, q)

We note \(>\) the transient tensor associated with situation \(>\) and \(>\) the transient tensor associated with situation \(>\). \(>\) and \(>\) the extreme moments of the transition of the situation \(>\) and \(>\) and \(>\) the extreme moments of the transitory of the situation \(>\) as defined in 14.

With “METHODE” = “TRESCA”, the parameters \(>\) and \(>\) for combining situations p and q are defined by:

\(>\)

\(>\).

With “METHODE” = “TOUT_INST”, the parameter \(>\) for the combination of p and q situations is defined by:

\(>\).

If \(>\) and \(>\) are the moments of the fictional transient 1, then the times of the fictional transient 2 \(>\) and \(>\) are determined according to the method described in Annex 5 and the quantity \(>\) is equal to:

\(>\)

Note:

The user has the possibility to enter stress indices under the keyword INDI_SIGM in order to compare the results obtained with the method “ZE200a “ or “ ZE200b “.The corresponding equations are described in Appendix 6.
In this part, \(>\) .

Calculation Spméca and Spther

In the case where method KE_MIXTE is used, it is necessary to decompose the amplitude of variation of the stresses into a mechanical part and a thermal part. For the definition of \(>\), the RCC -M (§B3234.6) leaves freedom between (cf. §2.6.3.1):

take the mechanical part of the amplitude of the maximum stresses between the two transients;
take the maximum value of the amplitude of the mechanical stresses during these transients.

It is this last method, which is more conservative but easier to implement, that was adopted.

We must not forget the case where the most penalizing combination of total constraints corresponds to the two extremes of the same situation. So we change the definition of \(>\) and \(>\) as follows:

If \(>\), then \(>\) and \(>\).

The quantities \(>\) and \(>\) are calculated according to Annex 4 and the calculation of the quantities \(>\) and \(>\) is described in the rest of this part. The thermal stress amplitudes are \(>\) and \(>\).

We note \(>\) and \(>\) the extreme moments of the transition of the situation p and \(>\) and \(>\) the extreme moments of the transient of the situation q as defined in 14.

With “METHODE” = “TRESCA”, the parameters \(>\) and \(>\) for combining situations p and q are defined by:

If \(>\), then

\(>\)

and \(>\).

If \(>\), then

\(>\)

and \(>\).

With “METHODE” = “TOUT_INST”, the moments \(>\), \(>\),, \(>\), \(>\), defining the fictional transients 1 and 2 (cf. part 16) also come into play in the parameters \(>\) and \(>\)

\(>\)

\(>\).

Calculating S ALT and FU ELEM

The stress amplitudes \(>\), \(>\) and \(>\) (\(>\), \(>\),, \(>\) and and \(>\) si KE_MIXTE) make it possible to arrive at the stress amplitudes \(>\) and \(>\) according to the equations in the part. 10

We deduce from this via the Wöhler curve f the numbers of admissible cycles \(>\) and \(>\) such as \(>\) and \(>\).

The elementary use factor of the situation combination is then equal to

\(>\)

Transitional situations

Two situations p and q can only be combined if they belong to the same group or if there is a transition situation between the groups to which they belong. In the latter case, the number of occurrences of the transition situation will be associated with the combination of situations p and q. Once this number of occurrences \(>\) is exhausted then these situations are no longer combinable. An example is given in part 19.

A transit situation can connect a maximum of 20 operating groups. The transition situation must belong to all the groups it connects.

Several transit situations can be declared at the same time. If several transition situations link the same groups, we will take the number of occurrences of the one that gives the least penalizing elementary use factor.

Sharing group

Situations that are part of the same sharing group share their number of occurrences. This sharing group is numbered under the keyword “NUME_PARTAGE” and has nothing to do with the operating group under “NUME_GROUPE”. A given situation can only belong to one sharing group.

Sub-cycle management

The method for calculating the stress amplitude with fictional transients selects a pair of moments. It is also necessary to take into account the sub-cycles that each situation involved. The user has this option by specifying SOUS_CYCL = “OUI” but only when METHODE =” TOUT_INST “

At the stage of calculating the quantities for the situations alone, once the pair of extreme moments is found, the sub-cycles of each situation are also extracted. For example, for a situation p, np subcycles Spi, p are extracted and for a situation q, nq subcycles Spj, q are extracted.

To the elementary use factor calculated previously, the contribution of the sub-cycles is then added,

\(>\). This contribution is a function of the sub-cycles extracted previously and of the equivalent Ke that was used to combine the situations.

\(>\) with \(>\) and \(>\),

\(>\) and \(>\).

Storage

To perform the calculation of the total use factor, the elementary use factors calculated previously and the associated numbers of occurrences are stored in a square matrix containing all the elementary use factors \(>\) excluding the earthquake, for all possible combinations of situations, i.e. within each group of situations, i.e. within each group of situations, and between two groups if there is a transition situation. The dimension of the matrix is the sum of the number of situations in all the groups and being symmetric, it is filled only above the diagonal.

Example 1

In the table below, an example of a calculation with three operating groups is given.

Group 1 contains the situations numbered 1, 2, and 3
Group 2 contains situations numbered 4, 5, and 6
Group 3 contains situations numbered 7, 8, and 9

We calculate the \(>\) of the possible combinations, otherwise we put a zero in the table.

\(>\)	Group 1			Group 2			Group 3
	Situ 1	Situ 2	Situ 3	Situ 4	Situ 5	Situ 6	Situ 7	Situ 8	Situ 9
Situ 1	FIRE (1.1)	FIRE (1.2)	FIRE (1.3)	0000	0000	0000	0000	0000	0000
Situ 2		FIRE (2.2)	FIRE (2.3)	0000	0000	0000	0000	0000	0000
Situ 3			FIRE (3.3)	0000	0000	0000	0000	0000	0000
Situ 4				FIRE (4.4)	FIRE (4,5)	FIRE (4.6)	0000	0000	0000
Situ 5					FIRE (5.5)	FIRE (5.6)	0000	0000	0000
Situ 6						FIRE (6.6)	0000	0000	0000
Situ 7							FIRE (7.7)	FIRE (7.8)	FIRE (7.9)
Situ 8								FIRE (8.8)	FIRE (8.9)
Situ 9									FIRE (9.9)

Example 2

In the table below, an example of a calculation with three operating groups is given. Situation 7 is a transition situation between groups 1 and 3.

Group 1 contains situations numbered 1, 2, 3, and 7
Group 2 contains situations numbered 4, 5, and 6
Group 3 contains situations numbered 7, 8, and 9

We calculate the \(>\) of the possible combinations, otherwise we put a zero in the table. Situation 7 is not put back into the table twice even if it belongs to two groups.

FU (1.7), FU (2.7), and FU (3.7) are now calculated because situations 1, 2, 3, and 7 are part of the same group.

The passage situation number 7 created the passage between groups 1 and 3 and we therefore calculate the terms **FU (1.8), FU (1.9), FU (2.8), FU (2.9), FU (2.9), FU (2.9), FU (2.9), FU (2.9), FU (2.9), FU (2.9), FU (2.9), FU (2.9), FU (2.9), FU (2.9), FU (2.9), FU (

	Table of \(>\)
	Situ 1	Situ 2	Situ 3	Situ 4	Situ 5	Situ 6	Situ 7	Situ 8	Situ 9
Situ 1	FIRE (1.1)	FIRE (1.2)	FIRE (1.3)	0000	0000	0000	FIRE (1.7)	FU (1.8)	FU (1.9)
Situ 2		FIRE (2.2)	FIRE (2.3)	0000	0000	0000	FIRE (2.7)	FU (2.8)	FU (2.9)
Situ 3			FIRE (3.3)	0000	0000	0000	FIRE (3.7)	FU (3.8)	FU (3.9)
Situ 4				FIRE (4.4)	FIRE (4,5)	FIRE (4.6)	0000	0000	0000
Situ 5					FIRE (5.5)	FIRE (5.6)	0000	0000	0000
Situ 6						FIRE (6.6)	0000	0000	0000
Situ 7							FIRE (7.7)	FIRE (7.8)	FIRE (7.9)
Situ 8								FIRE (8.8)	FIRE (8.9)
Situ 9									FIRE (9.9)

Earthquake factor

When an earthquake is taken into account, a second matrix of elementary use factors with earthquake is constructed. This matrix is filled in the same way as the matrix without an earthquake, taking into account only possible combinations (groups and situations of passage).

Finally, the elementary use factors with earthquake are the sum of the use factor of the combination of the situation p and q with earthquake, of the contribution of the sub-cycles of the situations (cf 19) and of the contribution of the seismic sub-cycles \(>\).

\(>\)

with \(>\) where \(>\) is the use factor due to the earthquake alone and \(>\) is the number of seismic sub-cycles (“NB_CYCL_SEISME”).

If the user has chosen “SOUS_CYCL” = “OUI”, for the calculation of \(>\), the method applied in part 19 is repeated using the Ke depending on the earthquake only.

To obtain \(>\), the second phase consists in calculating the stress amplitudes that correspond to the combinations of situations in a given group, taking into account the seismic loads.

Depending on the definition of the situations, an earthquake can also be defined in two different ways:

unitary: it is described by a mechanical state (S) and the corresponding torsor \(>\) under CHAR_ETAT, the keyword “RESU_MECA_UNIT” must be entered.
with six tensors corresponding to efforts and moments \(>\), \(>\),, \(>\), \(>\), \(>\), \(>\)

Note:

For stitching, the force twister goes from 6 to 12 components in units and the number of tensors increases from 6 to 12 instantly

The seismic loads are not signed. Each component of the stress tensor can therefore take two values (positive and negative). When superimposing an unsigned load with a signed load, the RCC -M requires a sign to be retained on each component such that the calculated constraint (in fact \(>\)) is increased.

The parameters \(>\), \(>\), \(>\) and \(>\) with earthquake are calculated in the same way as without an earthquake, but maximizing the amplitude of the constraints in relation to all sign possibilities. In order to better understand, an example is given in the rest of this paragraph, more equations with the earthquake are summarized in Appendix 7.

For example, for the moment selection method “TOUT_INST”,

\(>\).

Total use factor calculation

If we have N situations, at the end of the previous steps, we therefore have:

a \(>\) matrix of elementary use factors \(>\) with earthquake
a \(>\) matrix of \(>\) elementary use factors without an earthquake.

We note:

\(>\)	the number of occurrences associated with situation \(>\)
\(>\)	the number of occurrences associated with situation \(>\)
\(>\)	the number of occurrences of the earthquake
\(>\)	number of cycles associated with a possible transition situation between p and q if these situations do not belong to the same group

1. We initialize the total use factor \(>\)
- If \(>\), we look for the largest elementary use factor with earthquake in table \(>\)
- If \(>\), we go to step 8
1. We multiply this elementary use factor \(>\) by its number of occurrences
- \(>\) in general
- \(>\) if the p and q situations are only linked by a passing situation
1. We get the partial use factor due to this combination \(>\).
1. We increment the total use factor with the partial use factor found in the previous step \(>\)

1. We update the number of occurrences
- In general, \(>\), \(>\), and \(>\).
- If a situation has made the transition, we also update \(>\)
1. If the situation p or the situation q belongs to a sharing group, the numbers of occurrences of the situations d and the same sharing group are updated. Then the loop is repeated in step 2.
1. We look for the greatest elementary use factor in table \(>\)
1. We multiply this elementary use factor \(>\) by its number of occurrences
- \(>\) in general
- \(>\) if the p and q situations are only linked by a passing situation
1. We get the partial use factor due to this combination \(>\)
1. We increment the total use factor with the partial use factor found in the previous step \(>\)

1. We update the number of occurrences
- In general, \(>\), \(>\).
- If a situation has made the transition, we also update \(>\)

1. If the situation p or the situation q belongs to a sharing group, the numbers of occurrences of the situations d and the same sharing group are updated. Then the loop is repeated at step 8 until all the numbers of occurrences of all the situations are exhausted.

Note:

Annex ZI of code RCC -M defines Wöhler curves up to a minimum stress amplitude corresponding to a lifespan of 106 cycles. If the value \(>\) calculated for a combination \(>\) of stabilized state is less than this minimum amplitude, the use factor is equal to 0 for the combination \(>\) considered.

Types” ZE200a “and” ZE200b “

Charging data

The types” ZE200a “and” ZE200b “are well suited to calculations on pipes or joints subject to numerous situations. Several operating groups can also be defined, with possible situations of transition between these groups. Sharing groups and an earthquake can also be taken into account.

Each situation is described by two stabilized states and one thermal transient. Stabilized states describe the loadings due to moments via a \(>\) twister.

Pressure can be described in two different ways:

“ZE200a”: it is associated with stabilized states which are then defined by a \(>\) pressure
“ZE200b”: it is in the form of a transitory under the keyword “RESU_PRES”.

The use of these options requires the prior calculation of the stress fields for each of the thermal transients; these fields are to be provided on the analysis segment at the moments of discretization of the calculation using tables.

Additional data is required for the calculation according to Appendix ZE200 of RCC -M. These data are used in the simplified equations from chapter B3600. They are as follows:

Geometric characteristics of the pipe: EP thickness, radius R and moment of inertia I under the keyword TUYAU.
Constraint indexes from paragraph B3680 of RCC -M: C1, C2, C3, K1, K2, K3 under the keyword INDI_SIGM

Notes:

For stitches, it is also possible to use the types ZE200a and ZE200b by defining two torsors of moments associated respectively with the body and the tubing. It is also necessary to define the radii of the body and of the tubing under “TUYAU “ and their stress indices under” INDI_SIGM “. Appendix 3 summarizes the equations in this case.
Efforts are not taken into account in ZE200
En “ZE200b” , the user must also provide the constraint fields for each of the pressure transients and they must be defined at the same times as the thermal transients
The two methods for selecting instants “ TRESCA “ “*”* and* “ TOUT_INST “ are available (see part 14 )

Calculations performed with the option “SN”

Type “ZE200a”

We note \(>\) the thermal transient stress tensor associated with the situation and \(>\) and \(>\) the extreme moments of this transient as defined at 14. We index A and B the magnitudes of the stabilized states of the situation (pressure and twisting on the moments). R, e, and I are the geometric characteristics of the pipe, C1 and C2 are the stress indices of RCC -M.

With “METHODE” = “TRESCA”, the parameter \(>\) for the situation is set by:

\(>\)

With “METHODE” = “TOUT_INST”, the parameter \(>\) for the situation is set by:

\(>\)

Type “ZE200b”

We note \(>\) the sum of the transients tensor associated with the situation and \(>\) and \(>\) the extreme moments of this transient as defined at 14. We index A and B the magnitudes of the stabilized states of the situation (twisting over the moments). R, e, and I are the geometric characteristics of the pipe, C2 is the stress index of RCC -M.

With “METHODE” = “TRESCA”, the parameter \(>\) for the situation is set by:

\(>\)

With “METHODE” = “TOUT_INST”, the parameter \(>\) for the situation is set by:

\(>\)

Note:

In this paragraph, \(>\) .

Calculations performed with the option “FATIGUE”

Combination of situations within each situation group

Calculating S n

We must not forget the case where the most penalizing combination of linearized constraints corresponds to the two extremes of the same situation. For the combination of p and q situations:

\(>\)

The quantities \(>\) and \(>\) are calculated according to part 24 and the calculation of the quantity \(>\) is described in the rest of this paragraph.

We note \(>\) the sum of the transients tensor associated with the situation p and \(>\) the sum of the transients tensor associated with the situation q. \(>\), \(>\), \(>\), \(>\), the extreme moments of these transients as defined in 14. We index p and q the magnitudes of the stabilized states of the two situations (pressures and torsors on the moments). R, e, and I are the geometric characteristics of the pipe, C1 and C2 are the stress indices of RCC -M.

For the two types” ZE200a “and” ZE200b “, we first maximize the quantity \(>\) out of the four possibilities of combining stabilized states.

For “ZE200a”

\(>\)

For “ZE200b”

\(>\)

Then, for both types, the rest of the calculation is identical.

With “METHODE” = “TRESCA”, the parameter \(>\) for the situation is set by:

\(>\) with \(>\)

\(>\) and \(>\).

If \(>\), then \(>\) and if “ZE200b”, \(>\).

Or, with “METHODE” = “TOUT_INST”, the parameter \(>\) for the situation is set by:

\(>\) with \(>\).

If \(>\), then \(>\) and if “ZE200b”, \(>\).

Calculating S p

\(>\)

If \(>\), then \(>\);

If \(>\), then \(>\).

The quantities \(>\) and \(>\) are calculated according to Annex 4 and the calculation of the quantities \(>\) and \(>\) is described in the rest of this paragraph.

For the two types” ZE200a “and” ZE200b “, we first maximize the quantity \(>\) out of the four possibilities of combining stabilized states, \(>\) and \(>\) having been determined in part 25.

For “ZE200a”

\(>\)

\(>\) is the complement of \(>\) on stabilized states.

For example, by indicating A and B the stabilized states of situations p and q, if \(>\) then

\(>\).

\(>\)

For “ZE200 b”

\(>\)

\(>\)

\(>\) is the complement of \(>\) on stabilized states.

For example, by indicating A and B the stabilized states of situations p and q, if \(>\) then

\(>\).

\(>\)

For both types, the rest of the calculation is identical.

\(>\)

With “METHODE” = “TRESCA”, the parameters \(>\) and \(>\) for combining situations p and q are defined by:

\(>\)

With “METHODE” = “TOUT_INST”, the parameter \(>\) for the combination of p and q situations is defined by:

\(>\).

If \(>\) and \(>\) are the moments of the fictional transient 1 \(>\), then the times of the fictional transient 2 \(>\) and \(>\) are determined according to the method described in Appendix 5 and the quantity \(>\) is equal to:

\(>\)

Calculation Spméca and Spther

take the mechanical part of the amplitude of the maximum stresses between the two transients;
take the maximum value of the amplitude of the mechanical stresses during these transients.

The latter method, which is more conservative but easier to implement, was adopted.

If \(>\), then \(>\) and \(>\).

The quantities \(>\) and \(>\) are calculated according to Annex 4 and the calculation of the quantities \(>\) and \(>\) is described in the rest of this paragraph.

The thermal stress amplitude \(>\) (resp. \(>\)) is defined by taking the total stress amplitude \(>\) (resp. \(>\)) from which we subtract \(>\) (resp. \(>\)).

We note \(>\) the tensor due to the pressure associated with the situation p and \(>\) the tensor due to the pressure associated with the situation q. \(>\), \(>\), \(>\), \(>\), the extreme moments of these transients as defined in 14. We index p and q the magnitudes of the stabilized states of the two situations (pressures and torsors on the moments). R, e, and I are the geometric characteristics of the pipe, K1, K2, K2, K3, C1, C2 and C3 are the stress indices of RCC -M.

For the two types” ZE200a “and” ZE200b “, we first maximize the quantity \(>\) out of the four possibilities of combining stabilized states, \(>\) having been determined in part 25.

For “ZE200a”

The states p and q are the states that maximized the magnitude \(>\) (part 26).

\(>\)

\(>\) is the complement of \(>\) on stabilized states.

For “ZE200b”

\(>\)

The states p and q are the states that maximized the magnitude \(>\) (part 26).

\(>\), \(>\) is the complement of \(>\) on stabilized states and \(>\).

With “METHODE” = “TRESCA”, the parameters \(>\) and \(>\) for the combination of situations p and q are defined using the quantities \(>\) and \(>\) of the part 26:

If \(>\), then \(>\)

and \(>\).

If \(>\), then \(>\)

and \(>\).

With “METHODE” = “TOUT_INST”, we do not perform a new time search with respect to the KE_MECA method (part 26). Let \(>\) and \(>\) be the moments of the fictional transitory 1 \(>\), and \(>\) and \(>\) the moments of the fictional transitory 2 \(>\). So, the parameters \(>\) and \(>\) for the combination of situations p and q are defined by:

\(>\),

\(>\).

Total Use Factor Calculation

Taking into account the earthquake, sharing groups, sub-cycles, managing transit situations, storing elementary use factors and calculating the total use factor are the same as for the method “B3200” * ** (see parts*:ref:`18 <RefHeading__42536_1832330888>`*to*:ref:`21 <RefHeading__42538_1832330888>`*) (see parts to ) .

Environmental fatigue

Taking into account the effects of the environment on fatigue resistance in code_aster is available for types B3200, ZE200a and ZE200b. This consideration is taken into account after combining the situations and calculating the usual use factor described in part 21.

Calculation of FEN

The environmental factor for the combination of situations \(>\) and \(>\) is expressed as a function of the \(>\) deformation increment, the \(>\) partial environmental factor and the moments \(>\) and \(>\). Clue \(>\) scans the transient moments of \(>\) and clue \(>\) scans the transient moments of \(>\).

\(>\).

Regardless of the material studied, the general form of the partial environmental factor \(>\) is as follows:

\(>\).

A, B and C are constants that depend on the nature of the material: ferritic, austenitic, nickel-base (keywords A_ENV, B_ENV and C_ENV),
\(>\) is the sulfur content of the metal analyzed, therefore common to all situations (keyword S_ETOILE),
\(>\) is the degree of oxygen dissolved in the water in contact with the analyzed section. This quantity may be different for each situation (keyword O_ETOILE),
\(>\) is a function that depends on the average temperature T.*The user must therefore provide a table that contains the evolution of the temperature during the transition* (keyword TABL_TEMP).

\(>\).

The \(>\) function is described below and depends on thresholds \(>\) (keywords SEUIL_T_SUP and SEUIL_T_INF) and threshold values \(>\) (keywords VALE_T_SUP, VALE_T_INF, VALE_T_MOY_NUM and VALE_T_MOY_DEN):

\(>\).

\(>\) is a function that depends on the deformation rate \(>\). This function is described below and depends on thresholds \(>\) (keywords SEUIL_EPSI_SUP and SEUIL_EPSI_INF)

\(>\).

The deformation rate \(>\) is equal to:

\(>\).

The deformation increment \(>\) intervenes both in the expression of the environmental factor of the situation \(>\) and in the partial environmental factor \(>\) via \(>\). \(>\) is calculated from the stress tensor in the form of transient \(>\) in the following way: we calculate the tensor \(>\) such that

\(>\).

After diagonalization,

\(>\) with \(>\).

We can then calculate \(>\) which is a function of the main constraints, of the \(>\) of the situation combination and of the Young’s modulus \(>\) taken at the mean temperature of the time step \(>\). \(>\) was stored when calculating the usual use factor (part 10 10) (part) and \(>\) is calculated by linear interpolation on the \(>\) curve given under the keyword TABL_YOUNG.

\(>\).

Note:

The temperature table entered under the keyword TABL_TEMP must be defined at the same times as the tables that contain the stresses in the form of transient (thermal, pressure, mechanical depending on the calculation method)

Calculation of the use factor with environmental effect

To calculate the use factor with environmental effect, two quantities are still involved at this stage: a deformation criterion (keyword CRIT_EPSI) and the integrated FEN (keyword FEN_INTEGRE). These two quantities do not depend on the combination of situations, so we only enter one value for all combinations of situations.

Minimum deformation criterion

When calculating \(>\), if the sum of the deformation increments due to the situations p and q is less than CRIT_EPSI then the environment effect is not taken into account for this combination.

More specifically,

\(>\).

Finally, to obtain the elementary use factor with an environmental effect, we multiply the elementary use factor \(>\) (part 10) by the FEN of the combination of the situations p and q:

\(>\).

The total use factor with environmental effect \(>\) is calculated by taking into account \(>\) instead of \(>\) but without going through the algorithm in part 10.

FEN global and FEN integrated

A final check is then made: the global FEN is defined as:

\(>\)

If \({FEN}_{global}>{FEN}_{integre}\) then the total use factor with environment effect \({U}_{env}^{TOT}\) is updated by dividing them by FEN_INTEGRE.

Type “ EVOLUTION “

Charging data

“EVOLUTION” is well suited to calculations on a component subject to few loading situations and no earthquake. The RCC -M user must give the number of occurrences of each operating situation (for example: heating the boiler, hot shutdown, etc.). An operating situation can be broken down into transients, i.e. changes in global operating parameters (pressure, temperature) as a function of time.

In c*ode_aster*, we treat mechanical results (produced by MECA_STATIQUE or STAT_NON_LINE), and therefore transients. For each transient, the constraint fields are to be provided on the analysis segment at the moments of discretization of the calculation via tables created by calling POST_RELEVE_T or MACR_LIGN_COUPE.

Several types of results may be necessary for each transient: constraints for thermomechanical loads (TABL_RESU_MECA), constraints for thermal loading alone (TABL_SIGM_THER), constraints for pressure loading alone (TABL_RESU_PRES) and constraints for singular areas (TABL_SIGM_THETA).

This TYPE_RESU_MECA is the one that leads to the most accurate results. In fact, it does not require the introduction of any simplifying hypothesis either on the definition of loads, or on the calculation of the various level 0 or level A criteria.

Moreover, it makes it possible to calculate the priming factor at the level of a singular zone, in the sense of appendix ZD of RCC -M.

Calculations made with the option “AXIS” and “RAYON_MOYEN”

The treatment of stress linearization using the 3D method may lead to an erroneous result due to the inhomogeneous radial mass distribution under the axisymmetric hypothesis. For an axisymmetric model, the center axis must be offset outside by a distance so that the resulting bending moment is cancelled out on the offset axis [9].

Figure 6.2-1 Axisymmetric model and variables involved in the stress linearization method. Left: model without curvature; Right: model with curvature in plane XOY. (Ref. ANSYS APDL Theory Reference )

By convention, the model is axisymmetric around the Y axis. Since the stress tensor has been converted into the local 2D coordinate system, depending on the physical design, we can quantify the offset distance to linearize the normal stress:

\({x}_{f}=\frac{{M}_{z}}{{F}_{y}}=\frac{{t}^{2}cos(\varphi )}{12{R}_{c}}\text{ }(6.2-1)\)

Where \({M}_{z}\text{ }\) is the total bending moment in the peripheral direction induced by the normal stress and \({F}_{y}\text{ }\) is the total normal force. The other variables are shown in Figure 6.2-1.

Likewise, the offset distance to linearize the circumferential stress is expressed as: the offset distance to linearize the circumferential stress is expressed as:

\({x}_{h}=\frac{{M}_{y}}{{F}_{z}}=\frac{{t}^{2}}{12\rho }\text{ }(6.2-2)\)

The main idea of calculating membrane stress \({\sigma }_{ij}^{m}\) is to average the total force by the elementary area in the plane that is orthogonal to the stress component. The flexural stress calculation uses the equation in section 3.2 when in 3D cases. The only difference with 3D models is the consideration of the offset axis. Here, the detailed calculation is presented as an example for the circumferential stress component whose formulation is the most complicated.

The simplified expression for the other linearized stress components can be found in ANSYS APDL Theory Reference [10].

The newly developed code within the code_aster POST_RCCM command allows users to switch to the optimized calculation routine if an axisymmetric model is declared, without interfering with the original functionality of the command. In addition, it was possible to indicate the curvature of the \(\rho\) axis in the XOY plane if necessary, with reference to Figure 6.2-1. The value \(\rho\) by default has been set to -1 representing a perfectly straight axis.

Calculations performed with the option “PM_PB”

The stress table includes either a single time step or a complete transient (nb_inst time step). In the latter case, we will search for the maximum, in relation to the list of order numbers, of the various terms involved in the criteria.

It is up to the user to know whether to calculate \(Pm\) (general membrane stress: outside the zones of geometric singularity) or \(Pl\) (local membrane stress: in the singularities). From the stress readings provided, a membrane stress is therefore calculated.

The algorithm is as follows. On all order numbers \(n=1,\text{nb\_inst}\):

extraction of constraints at time \(t\)
On each end of the segment:
- calculation of \({P}_{m}(t)\), \({P}_{b}(t)\), \({P}_{\text{mb}}(t,s=0)\) and \({P}_{\text{mb}}(t,s=l)\) by integration on the segment

\(\begin{array}{c}{\text{\%}sigma}_{\text{ij}}^{\text{moy}}(t)=\frac{1}{l}\underset{0}{\overset{l}{\int }}{\text{\%}sigma}_{\text{ij}}(t)\text{ds},{P}_{m}(t)={\text{\%}sigma}_{\text{ij}}^{\text{moy}}(t)\parallel \\ \end{array}\)

\(>\)

Search for as many as \(>\), \(>\), and \(>\)
Output and storage in the result table.

Note:

Thermal stresses are of a secondary type and should therefore not be taken into account in the calculation of level 0 criteria. In POST_RCCM, if TABL_RESU_MECA and and TABL_SIGM_THER are present simultaneously, it is assumed that the result TABL_RESU_MECA corresponds to the complete thermomechanical load, and the thermal stresses are therefore subtracted from it.

Calculations performed with the option “SN”

Sn calculation

We note \(>\) the number of moments selected in the transient in question.

The algorithm for calculating \(>\) is as follows:

on all order numbers, \(>\)
Extraction of moment \(>\)
Calculation of \(>\) and \(>\)
For \(>\) varying from \(>`*to* :math:\)>`
Extraction of the moment t2
calculation of \(>\) and \(>\) and of

\(>\) and \(>\)

calculation of the main directions and the Tresca criterion:

\(>\) and \(>\)

search for the maximum so \(>\) at each end
Output and storage in the result table.

Note:

The quantity \(>\) calculated here corresponds to an amplitude. It is therefore essential that all states of the system be considered, including states with zero stress (e.g. cold stop: zero pressure and applied moments and ambient temperature) .

Calculating S n *

This calculation is performed if the TABL_SIGM_THER operand is present. Only the user ensures the consistency of the data, that is to say that this result must be produced by a thermo-mechanical calculation under thermal load alone, knowing that the result given by TABL_RESU_MECA may be due to a combination of this thermal load with other loads. In particular, it is therefore necessary that the times in tables TABL_RESU_MECA and TABL_SIGM_THER correspond.

The algorithm is identical to the previous one but focuses on two constraint fields.

Thermal ratchet calculation

The calculation is performed if the operands TABL_SIGM_THER and TABL_RESU_PRES are present. It is also necessary to have previously defined the conventional elastic limit for the maximum temperature reached during the cycle either by the operand SY_MAX of POST_RCCM; or by the operand SY_02 of the keyword RCCM in DEFI_MATERIAU [U4.43.01]. If no elastic limit is defined, thermal ratchet calculation is impossible.

In the result table, for each end of each analysis segment, the elastic limit SY, the amplitude of variation of the thermal stress SP_THER, the maximum of the general membrane stress due to pressure SIGM_M_PRES and two maximum admissible values of the amplitude of variation of the thermal stress, calculated either by assuming a linear temperature variation in the wall (VALE_MAXI_LINE), or by assuming a variation in temperature in the wall (), and by assuming a variation in temperature. parabolic temperature in the wall (VALE_MAXI_PARAB).

Fatigue calculations with the option “FATIGUE_ZH210”

The requirements for calculating the use factor are defined in §2.6.3.

The « EVOLUTION » method corresponds to appendix ZH210 of the RCC -M. It consists in « forgetting » the concept of situation and in directly combining loading states, which are the significant moments of all the transients where the constraints pass through a local extremum. By default, in c*ode_aster*, all calculation times are used. Each of them is associated with the number of occurrences \(>\) of the transitory. So the definition is:

Load state = {instant, stress tensor, number of occurrences} .

Then, the set of all the loading states is constructed by scanning all the transients. In the end, the concept of transient is forgotten: we are only working on a set of loading states. The elementary use factors associated with all the combinations taken two by two are then calculated. A method for the accumulation of elementary use factors, based on the hypothesis of the linear accumulation of damage, is then used to obtain the global use factor.

The main advantage of this method is that it automatically considers all possible sub-cycles: there is no need to identify fictional transients combining situations together. Its disadvantage is the number of calculations to be performed if the set of times used in the calculation is not restricted.

Note:

The algorithm described here is similar to POST_FATIGUE. More specifically, the algorithm used in POST_FATIGUE is a restriction to the uniaxial case of the ZH210 method. In fact, the data in the command POST - FATIGUEest is a scalar time function, while POST_RCCM treats constraint tensors/time functions.

Calculation of elementary use factors

At each end of the segment, for any pair of loading states \(>\) and \(>\), the quantities \(>\) and \(>\) defined by:

\(>\)

For the calculation of \(>\), two formulas are proposed (cf. part 10):

the original method (KE_MECA) which does not make a distinction between the mechanical part and the thermal part:

the method* KE_MIXTEi * introduced in the 1997 amendment of RCC -M [1] which is based on a decomposition of \(>\) between the mechanical part and the thermal part.

The fatigue curve \(>\) is a function defined by DEFI_FONCTION, and introduced in DEFI_MATERIAU by the keyword WOHLER of the keyword factor FATIGUE. It makes it possible to calculate the allowable number of cycles \(>\) associated with \(>\), then the elementary use factor:

\(>\).

This calculation is performed for each combination of two loading states. We therefore obtain (always for each end of the segment) a symmetric matrix \(>\), in order of the number of loading states \(>\).

Total use factor calculation

The algorithm for calculating the total use factor, for each end of the cut line, is as follows:

\(>\)
\(>\)
Loop \(>\) (find the maximum in the table) Si \(>\): Loop \(>\) If \(>\) and \(>\): \(>\)
\(>\)
\(>\)
Refreshing the number of occurrences: \(>\)
\(>\)
Return to the beginning of the procedure until all occurrences have been eliminated

Notes:

If the number of times defined for each transient is large, the calculation time can be prohibitive. So you have to be able to restrict it. This is what is done in POST_FATIGUE, by a preliminary sorting of the moments. Instants such that the scalar function is linear are eliminated to keep only the extremities of the line segments. Very small variations are also eliminated. Here, in a multi-axial situation, sorting is more delicate. The concept of proportional constraints could be used, but in practice the user can define the list of moments himself (keyword NUME_ORDRE) .
By this method, we are sure not to forget any sub-cycle. On the other hand, it is desirable to eliminate the moments that do not correspond to local extremes, because they could generate artificial sub-cycles, increasing the use factor (these moments are only used for the numerical discretization of the mechanical or thermal problem) .
With the option “ FATIGUE_ZH210 “, the combinations of transients are taken into account in the calculation of \(>\) and \(>\) .

example

This paragraph aims to illustrate the algorithm for calculating the use factor on a simple example, taken from the basic validation test case rccm01a [V1.01.107]. It is assumed that there are three situations of two time steps each, the number of occurrences being 1, 5 and 10 respectively.

The usage factor matrix as calculated by the first part of the algorithm is given below. To simplify the presentation, except the upper part of the symmetric matrix is written.

	j	1	1	2	3	4	5	6
i	Nocc	1	1	5	5	10	10	10
1	1	0	0	1.10 -4	0	3.10 -4	2.10 -4	1.10 -4	1.10 -4
2	1		0	0	1.10 4	2.10 -4	1.10 4	0
3	5				0	3.10 -4	2.10 -4	1.10 -4	1.10 -4
4	5					0	1.10 -4	2.10 -4	2.10 -4
5	10						0	1.10 -4
6	10							0

Table 6.4.2-1: Initial matrix of use factors

The most penalizing combination is \(>\), whose number of occurrences is 1:

\(>\)

The numbers of occurrences are updated: \(>\), \(>\). The usage factor matrix is updated; if row \(>\) or column \(>\) has zero occurrences, it is set to zero.

	j	1	1	2	3	4	5	6
i	Nocc	0	0	1	4	10	10	10
1	0
2	1		0	0	1.10 -4	2.10 -4	1.10 -4	0
3	5				0	3.10 -4	2.10 -4	1.10 -4	1.10 -4
4	4					0	1.10 -4	2.10 -4	2.10 -4
5	10						0	1.10 -4
6	10							0

Table 6.4.2-2: Matrix of use factors — iteration 1 of the calculation

The calculation continues in the same way: the most penalizing combination is now \(>\), whose number of occurrences is 4:

\(>\)

The penalizing combinations are then successively \(>\) of the number of occurences 1; \(>\) of the number of occurences 9.

The total use factor is then:

\(>\)

Priming factor calculations with the option “AMORCAGE”

Principle of calculating the priming factor

Areas of local discontinuities whose contours vary abruptly are the site of acute stress concentrations. In this case, the concept of use factor defined above is no longer appropriate and it must be replaced by the concept of priming factor (B3234.7).

The priming factor is calculated from the magnitude of variation of the stress in the structure at a distance \(>\) from the singularity, and from a priming law. The analysis procedure is defined in appendix ZD2200. The distance \(>\) and the priming laws are material characteristics and are tabulated in table ZD2300.

The priming law defined in RCC -M is of the form:

\(>\)

with \(>\) the number of acceptable cycles and \(>\) the amplitude of variation of the tangential stresses, in the local coordinate system, at the distance \(>\) from the singularity.

The priming law developed in operator POST_RCCM takes into account the load ratio \(>\) of the load, as recommended in the RSE -M (ref 15):

\(>\)

with the following relationship between the magnitude of variation of the real stresses \(>\) and the effective \(>\):

\(>\).

Note:

To use a priming law as defined in RCC -M, i.e. without taking into account the load ratio, simply define a R_AMORCgrand (1000 for example) .

Calculation in code_aster

The parameters of the priming law (A_AMORC, B_AMORC, R_AMORC) and the distance to the singularity D_AMORC are to be defined under the keyword factor RCCM of DEFI_MATERIAU.

The expected input table, under the keyword TABL_SIGM_THETA, corresponds to the statement of constraints on a circular cut line (of radius D_AMORC) around the singularity. The constraints must be expressed in the local reference.

images/1013EFC4000069D500004D690EB23ABDC7EC016C.svg

Figure 6.5.2-a: definition of the local coordinate system

Such a table can be created using the MACR_LIGN_COUPE command. As in fatigue calculation (cf. §6.4), it is considered that all the moments provided correspond to extremes of the transient. Moreover, the concept of transient is forgotten and we are only working on a set of loading states. Note \(>\) the number of occurrences associated with the load state \(>\) and \(>\) the total number of load states.

The calculation algorithm is then as follows:

Loop on the points of the cut line

Verify that the point is at the distance*d* from the singularity
Loop \(>\) Extraction of \(>\) Loop \(>\) Extraction of \(>\) Calculation of Calculation of \(>\) Calculation of Calculation of \(>\) Calculation of the number of admissible cycles and the elementary priming factor \(>\) \(>\) \(>\)

At the end of this first part, we therefore have a matrix of the factors for initiating all the combinations of loading states. The size of the matrix is \(>\) but only the part above the diagonal is entered.

The algorithm for calculating the total priming factor, for a given point on the cut line, is then as follows:

\(>\)
\(>\)
Loop \(>\) (find the maximum in the table) Si \(>\): Loop \(>\) If \(>\) and \(>\): \(>\)
\(>\)
\(>\)
Update the number of occurrences: \(>\) \(>\)
Return to the beginning of the procedure until all occurrences have been eliminated

At the end of this algorithm, we thus have the priming factor for each of the points (i.e. for each of the angles) of the cut line.

Type “B3600”

In code_aster, it is possible to evaluate level A criteria (fatigue) according to chapter B3600 of RCC -M. It is customary in B3600 to define each situation as the transition from a stabilized state A (corresponding to a given internal pressure in the pipe line, a given uniform temperature, and fixed mechanical loads) to a stabilized state B (with constant loads different from the previous ones). This situation is associated with a thermal transient. Thermal stresses can also be taken into account in the calculation.

The treatment that is described here is carried out for each node of each mesh of the pipe line in question. The result obtained will therefore be a use factor (total or partial) for each node of each mesh requested by the user.

Prior calculation of all loading status

For each node in each mesh, the present step consists in calculating, for all situations, the moments relating to each stabilized state (by accumulating the various loads that occur).

Static load state calculations

The results of static calculations (field EFGE_ELNO or SIEF_ELNO) are treated for the stabilized states in the list of situations experienced by the line.

A stabilized state can be defined by a list of load cases, each load being signed. In this case, the torsors of the stabilized state are obtained by algebraic summation of the torsors of each of the load cases:

\(>\)

Loads are for example thwarted thermal expansion, anchor displacement.

Calculation of seismic loads

The seismic load is divided into two parts:

An inertial part

It is calculated by imposing on all the anchors the same movement characterized by the envelope spectrum of the various floor spectra, in the horizontal \(>\) and \(>\) directions on the one hand, and vertical \(>\) on the other hand (in the global coordinate system). To do this, we use the COMB_SISM_MODAL command, which produces generalized efforts that correspond to each earthquake direction as well as the quadratic accumulation of these efforts.

The inertial contribution of the earthquake to the \(>\) component of the moment is written:

\(>\)

with Mi_ S_DYN (spectrej) the moment in the i direction resulting from dynamic loading in the j direction. This accumulation is done directly by COMB_SISM_MODAL.

A semi-static part

It is estimated by imposing static differential displacements corresponding to the maxima of the differences in the seismic movements of the anchor points over time. The calculations are therefore carried out for each unit load (a calculation by displacement in a given direction for one end of the line).

Note \(>\) the number of anchor points in the structure. The quasistatic contribution of the differential anchor movements to the i component of the moment is written as:

\(>\)

with \(>\) the \(>\) th component of the moment corresponding to the \(>\) th anchor displacement.

Combination of inertial and differential components due to the seism

The resulting \(>\) th component is obtained by root mean square of the \(>\) th inertial and differential components:

\(>\)

which in fact amounts to performing the root mean square of all inertial and differential moments,

\(>\)

For the user, the earthquake situation is defined by the list of results corresponding to the inertial response and to the responses to the movement of the \(>\) successive anchor points. The recombination by mean squared is done directly by the operator POST_RCCM.

Calculation of thermal transients

Loads of the « thermal gradient in thickness » type are broken down into three parts, see Figure 7.1.3-a:

a constant value which is the average temperature value:

\(>\), where \(>\) is the nominal wall thickness.

a linear distribution with zero mean (moment of order 1):

\(>\)

a non-linear distribution with zero mean and zero moment with respect to the mean fiber.

For each of the transients and each pipe section of the line (and each junction), a 2D or 3D thermal calculation is therefore carried out beforehand, depending on the geometric complexity of the problem studied.

Each calculation is then analyzed in order to extract, for each moment of the transient, the temperature on the selected section and the average values (moments of order 0 and 1). This operation can be done for example using two calls to POST_RELEVE_T (OPERATION =” EXTRACTION “and OPERATION =” MOYENNE “).

In the case of a material discontinuity or a junction, the average temperature (noted \(>\) and \(>\)) on both sides of the junction is calculated. In practice, the areas \(>\) and \(>\) will correspond to segments chosen by the user in POST_RELEVE_T, and the tables produced will be associated with the two adjacent cells having in common the node that corresponds to the junction.

images/101ADC3A0000485C00001450594B9C77257FF513.svg

Figure 7.1.3-a: Decomposition of the temperature distribution in wall thickness (figure taken from RCC -M, §B3653.4)

Calculations of the amplitudes of stress variation

Principle of the method

The stress variation amplitudes are defined in paragraph B3653 of RCC -M for combinations between two moments or two loading states. By noting \(>\) and \(>\) these two moments, we have schematically for the amplitude of variation of a quantity \(>\):

\(>\)

In the method as developed in c*ode_aster*, the situations are defined in a simplified manner by two stabilized states and a thermal transient: it is then not possible to work directly on each of the moments of the situations and hypotheses must be introduced.

We therefore consider all the combinations \(>\) with \(>\), \(>\) being the number of stabilized states excluding the earthquake (i.e. 2 times the number of situations in the group). Let’s say two stabilized states, \(>\) and \(>\), belonging to situations \(>\) and \(>\) respectively. The amplitude of variation \(>\) will then be calculated in the following way:

\(>\)

by noting \(>\) the amplitude of variation of the thermal stress of the transient \(>\).

Notes:

It is important to note that the amplitude of stress variation is achieved by maximizing the amplitude of thermal stresses for each thermal transient independently of each other. The calculation method for the case “B3600” * is therefore different from that adopted for the case “B320 0_UNIT “ .
As stated in paragraph B3653.2, all system states should be considered, including states with zero stress (e.g. cold shutdown: zero pressure and applied moments and ambient temperature).

Calculating load combinations within each group

The objective is to build, for each group of situations, a symmetric square matrix containing all the amplitudes of variation of the alternating stress \(>\), with \(>\) and \(>\) two stabilized states associated respectively with situations \(>\) and \(>\). This calculation requires the prior calculation of the quantities \(>\) (amplitude of the total stress) and \(>\) (amplitude of the linearized stress).

Notations and definitions

We note:

\(>\), \(>\), \(>\), \(>\), \(>\), \(>\)	=	Stress indices provided in §B3680 of RCC -M
\(>\)	=	Modulus of elasticity of pipes at room temperature
\(>\)	=	Poisson’s ratio
\(>\)	=	Coefficient of pipe expansion at room temperature
\(>\)	=	Average elastic modulus between the two zones separated by a discontinuity at room temperature
\(>\)	=	Outer pipe diameter
\(>\)	=	Nominal wall thickness
\(>\)	=	Piping moment of inertia: \(>\)
\(>\)	=	Variation in moment resulting from the different loadings of the situations to which the stabilized states \(>\) and \(>\) belong: \(>\)
\(>\)	=	Pressure difference between states \(>\) and \(>\)
\(>\), \(>\)	=	Amplitude of variation of the average temperatures in zones \(>\) and \(>\) between the times \(>\) and \(>\)
\(>\), \(>\)	=	Amplitude of variation in temperatures at the level of the external/internal wall between the moments \(>\) and \(>\)
\(>\)	=	Amplitude of the variation between the two moments of the temperature difference between the two moments of the temperature difference between the inner and outer walls, for an equivalent linear temperature distribution: \(>\)
\(>\)	=	Nonlinear part of the distribution in wall thickness of the amplitude of temperature variation between the moments \(>\) and \(>\): \(>\)

Calculation Sp

The amplitude of variation in total stresses \(>\) for pipes is defined in equation 11 of §B3653 of RCC -M. We calculate:

\(>\)

\(>\) refer to any two transient moments associated with the \(>\) situation. In the event of a material discontinuity or a junction, the terms \(>\) and \(>\) to remember are those associated with the most penalizing section.

In the same way, \(>\) is calculated with the thermal transient associated with the situation \(>\). The amplitude \(>\) for the \(>\) combination is then:

\(>\)

Sn calculation

The amplitude of variation of linearized stresses \(>\) for pipes is defined in equation 10 of §B3653 of RCC -M. We calculate:

\(>\)

We then calculate \(>\), for \(>\) and \(>\), scanning all the stabilized states of the two situations \(>\) and \(>\) (4 possible combinations).

Salt calculation

Two formulas are proposed to define the amplitude of variation \(>\) between states \(>\) and \(>\)

KE_MECA: this is the original method, the only one available in versions prior to version 7.2:

\(>\)

with:

\(>\): Young’s reference module for constructing the Wöhler curve, provided by the user in DEFI_MATERIAU, under the keyword E_REFE, from the keyword factor FATIGUE.
\(>\): Smallest of the Young’s modules used to calculate states \(>\) and \(>\), i.e. evaluated at the temperatures of these stabilized states.

\(>\)

with \(>\) and \(>\) depending on the material, and provided by the user in DEFI_MATERIAU, under the keywords M_KE and N_KE, of the keyword factor RCCM. If the keywords TEMP_REF_A and TEMP_REF_B are present, \(>\) is interpolated for this temperature (which must correspond to the mean temperature of the transient). If not, \(>\) is taken at room temperature.

KE_MIXTE: since the 1997 modification of RCC -M, we can choose another formula, based on a decomposition of \(>\):

\(>\)

with:

\(>\) is equal to the \(>\) defined above
\(>\)
\(>\) represents the amplitude of variation in the mechanical part of the quantity \(>\), between the states \(>\) and \(>\). It is calculated on the basis of mechanical stresses: pressure, self-weight, earthquake (inertial and anchor movements), thermal expansion.
\(>\) the amplitude of variation in the thermal part of the quantity \(>\), between the states \(>\) and \(>\) (terms dependent on \(>\), \(>\), \(>\) and \(>\) in the definition of §7.2.2.2).

Case of sub-cycles

The sub-cycles correspond either to the consideration of sub-cycles linked to the earthquake, or to situations for which the keyword COMBINABLE =” NON “has been entered. In both cases, the amplitude of stresses is calculated by involving only the stresses associated with these sub-cycles (no combination of loading states outside this situation). To calculate \(>\), you must use the factor \(>\) which corresponds to the main situation from which the sub-cycle comes.

Calculation of load combinations for transit situations

Two loading states can only be combined if they belong to the same situation or if there is a transition situation between the groups to which they belong. In the latter case, the number of occurrences of the transit situation will be associated with the combination \(>\). In the case where the transition situation belongs to one of the two groups (which is not excluded a priory step), it is naturally combined with the other situations in this group, then used to combine the situations of its group with the situations of the group in relation.

For each situation of passage from one group to another, we therefore consider all the combinations \(>\) with \(>\) belonging to the first group (of dimension \(>\)) and \(>\) belonging to the second group (of dimension \(>\)). For each combination, \(>\) is obtained in the same way as before and the number of occurrences of the transition situation is associated with it. We build a (rectangular) matrix containing all \(>\).

Use factor calculation

We note:

\(>\)	:	number of cycles associated with the situation \(>\) to which the stabilized state belongs \(>\);
\(>\)	:	number of cycles associated with the situation \(>\) to which the stabilized state belongs \(>\);
\(>\)	:	number of occurrences of the earthquake;
\(>\)	:	number of sub-cycles associated with each occurrence of the earthquake;
\(>\)	:	number of cycles associated with a possible transition situation between \(>\) and \(>\) if these situations do not belong to the same group, but if there is a transition situation between the two.

For all combinations of loading states (within a group of situations or associated with a transit situation) :

If \(>\) , we select the combinations of stabilized states \(>\) and \(>\) the most penalizing, i.e. the combinations \(>\) leading to the largest values of \(>\) .

For each of these combinations:

A) Superposition of the moments of seismic origin and the combination \(>\) :
The earthquake loads are superimposed on the variation in moment resulting from the different loadings of the stabilized states \(>\) and \(>\):

\(>\)

with:

\(>\) and \(>\): components in the \(>\) (\(>\)) direction of the moments associated with the states \(>\) and \(>\);

\(>\): total amplitude of variation in the \(>\) direction of the moments due to the earthquake (\(>\) where \(>\) is the total resulting moment (inertia and anchor movements) as defined in §7.1.2).

We then calculate \(>\) and \(>\) as defined above with the new value of \(>\) (noted respectively \(>\) and \(>\)) and we calculate:

\(>\)

The number of allowable cycles \(>\) for the stress amplitude \(>\) is calculated using the Wöhler curve associated with the material.
We finally calculate \(>\)
B) Consideration of \(>\) seismic cycles considered as sub-cycles:
Amplitude of variation of the seismic stress alone:

\(>\)

The number of allowable cycles \(>\) is calculated for the stress amplitude \(>\). Note that the value \(>\) previously calculated for the main cycle is used.
We finally calculate \(>\)
C) Cumulation
\(>\)

This calculation is repeated until the most penalizing combinations are exhausted.

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

If \(>\) , or after taking into account the earthquake for the most unfavorable combinations:

We select the combination \(>\) leading to the maximum value of \(>\), out of all combinations, such that the number of occurrences \(>\) is non-zero, with:

\(>\) if \(>\) is non-zero

\(>\) if \(>\) sucks

The number of allowable cycles \(>\) is calculated for the stress amplitude \(>\), using the Wöhler curve associated with the material.
The elementary use factor is then calculated: \(>\).
We finally replace:

\(>\) by \(>\)

if it is a temporary situation, \(>\) by \(>\)

so:

if \(>\), the column and the row corresponding to the stabilized state \(>\) of the matrix \(>\) are set to 0.

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

Note:

Appendix ZI of code RCC -M defines Wöhler curves up to a minimum stress amplitude corresponding to a lifespan of 106 cycles. If the value \(>\) calculated for a combination \(>\) of stabilized state is less than this minimum amplitude, the usage factor is equal to 0 for the combination (i, j) in question. This is implicitly equivalent to considering the existence of an endurance limit of 106 cycles.

Appendix 1: B3200 equations for situations in unit form

Each stabilized mechanical state is described using a pressure \(>\) and a force twister \(>\) defined under the keyword “CHAR_MECA”. The stress tensors are reconstituted by linear combination from the stress tensors associated with each of the unit loads. For example, we note \(>\) the stress tensor associated with unit force loading in the x direction. The calculation of the stress tensor corresponding to a mechanical loading belonging to a stabilized state is then obtained in the following way:

\(>\)

The use of this option requires the prior calculation of the stress fields for the 7 unit loads and the stress fields for each of the thermal transients; the unit fields are to be provided on the analysis segment via tables under the keyword “RESU_MECA_UNIT”.

Notes:

For stitches, it is also possible to define two force tensors associated respectively with the body and the tubing. The force twister goes from 6 to 12 components and we go from 7 to 13 unit loads. Appendix 5 summarizes the equations in this case.

Calculations performed with the option “PM_PB”

For now, this option is available if situation data is in unit form only, not in instant form.

\(>\)

By noting \(>\) and \(>\) the two stabilized mechanical states of the situation, we have:

\(>\) and \(>\)

\(>\).

Calculations performed with the option “SN”

Sn calculation

The calculation points are the two ends of the segment. For a given situation, at each end point of this segment of length \(>\), \(>\) is calculated according to paragraph B3232.6:

\(>\)

We note \(>\) and \(>\) the mechanical stresses associated with the two stabilized states of the situation and \(>\) the thermal transient associated with this situation. We then have,

\(>\)

According to the method for selecting the times chosen (see 14), the amplitude Sn is obtained.

With “METHODE” = “TRESCA”, with \(>\) and \(>\) being the extreme moments of this transient as defined in 14., the \(>\) parameter for the situation is defined by:

\(>\).

With “METHODE” = “TOUT_INST”, the parameter \(>\) for the situation is set by:

\(>\),

with \(>\)

and \(>\).

Sn calculation*

Note \(>\) the amplitude \(>\) calculated without taking into account thermal flexural stresses.

If a thermal transient is defined (i.e. the keyword NUME_RESU_THER is entered), the calculation of \(>\) for a situation is done in a manner similar to that of \(>\):

\(>\)

Thermal ratches

See part 15.

Calculations performed with the option “FATIGUE”

Sn calculation

The mechanical stresses associated with a stabilized state of the situation p (respectively of the situation q) are denoted \(>\) (respectively \(>\)).

We note \(>\) the transient tensor associated with the \(>\) situation and \(>\) the transient tensor associated with the q situation. \(>\) and \(>\) the extreme moments of the transition of the situation \(>\) and \(>\) and \(>\) the extreme moments of the transitory of the situation \(>\) as defined in 14.

With “METHODE” = “TRESCA”, the parameter \(>\) for the combination of p and q situations is defined by:

\(>\)

with \(>\),

\(>\),

\(>\).

With “METHODE” = “TOUT_INST”, the parameter \(>\) for the combination of p and q situations is defined by:

\(>\)

with \(>\)

and \(>\).

Note:

In this case, \(>\) .

SP calculation

The mechanical stresses associated with a stabilized state of the situation p (respectively of the situation q) are denoted \(>\) (respectively \(>\)).

We note \(>\) the transient tensor associated with situation p and \(>\) the transient tensor associated with situation q. \(>\) and \(>\) the extreme moments of the transition of the situation p and \(>\) and \(>\) the extreme moments of the transitory of the situation q as defined in 14.

With “METHODE” = “TRESCA”, the parameters \(>\) and \(>\) for the combination of situations p and q are defined by:

\(>\)

By maximizing on the four possible combinations of stabilized states \(>\),

\(>\),

i.e. 8 possibilities.

\(>\),

i.e. 8 possibilities.

With “METHODE” = “TOUT_INST”, the parameter \(>\) for the combination of p and q situations is defined by:

\(>\)

with \(>\) and \(>\).

If \(>\) and \(>\) are the moments of the fictional transient 1 \(>\), then the moments of the fictional transient 2 \(>\) and \(>\) are determined according to the method described in Appendix 2 and the quantity \(>\) applies by maximizing over the four possible combinations of stabilized states \(>\),:

\(>\)

Notes:

In this part, \(>\) .

SP calculation meca

We note \(>\) (respectively \(>\)) the mechanical stresses associated with the stabilized states of the situation p and the situation q that maximized the magnitude \(>\). We note \(>\) (respectively \(>\)) the mechanical stresses associated with the stabilized states that have maximized the magnitude \(>\).

With “METHODE” = “TRESCA” and “METHODE” = “TOUT_INST”:

\(>\) and \(>\)

Appendix 2: B3200 equations for situations in unit form with temperature interpolation

Each situation is defined by two stabilized mechanical states A and B. Each state is described using a pressure \(>\) and an effort twister \(>\) defined under the keyword “CHAR_MECA” and corresponds to a temperature (TEMP_A or TEMP_B). In this example, the situations do not have pressure loading.

The user must also enter the temperature profile as a function of the weather during the situation (keyword “TABL_TEMP” under the keyword factor “SITUATION”)

The stress tensors are then reconstituted by linear interpolation using this temperature as a function of time and of the two torsors. The use of this option requires the prior calculation of the stress fields for the 6 unit loads (“RESU_MECA_UNIT”) and the stress fields for each of the thermal transients (“RESU_THER”).

For the moment, the calculation of PM_PB is not available if the situation data is in unit form with interpolation on the temperature.

Calculating Sn for a situation

We note \(>\) and \(>\) the linearized mechanical stresses associated with the moments of the two stabilized states of the situation and \(>\) the thermal transient associated with this situation. We then have,

\(>\)

At times t1 and t2, we have \(>\) and \(>\). If \(>\), by interpolation we have

\(>\) \(>\)

The equations are similar for the other five components. From this we deduce the expression of the linearized mechanical stresses due to the moments at the times t 1 and t 2

\(>\)

\(>\).

With “METHODE” = “TOUT_INST”, the parameter \(>\) for the situation is set by:

\(>\)

With “METHODE” = “TRESCA”, the parameter \(>\) for the situation is set by:

\(>\)

Calculations performed with the option “FATIGUE”

Calculating Sn for a combination of p and q situations

We note \(>\) and \(>\) the linearized mechanical stresses associated with the moments of the two stabilized states of the situation p and \(>\) the thermal transient associated with this situation.

For the moment tp belonging to \(>\), we have \(>\) and if \(>\), by interpolation we have

\(>\)

For now tq belonging to \(>\), we have \(>\) and if \(>\), by interpolation we have

\(>\).

The equations are similar for the other five components. From this we deduce the expression of the linearized mechanical stresses due to the moments at the times tp and tq.

\(>\)

\(>\).

With “METHODE” = “TOUT_INST”, the parameter \(>\) for combining p and q situations is:

\(>\)

Calculating Sp for a combination of p and q situations

We note \(>\) and \(>\) the mechanical stresses associated with the moments of the two stabilized states of the situation p and \(>\) the thermal transient associated with this situation.

For the moment tp belonging to \(>\), we have \(>\) and if \(>\), by interpolation we have

\(>\)

For now tq belonging to \(>\), we have \(>\) and if \(>\), by interpolation we have

\(>\).

The equations are similar for the other five components. From this we deduce the expression of the mechanical stresses due to the moments at the times tp and tq.

\(>\)

\(>\).

With “METHODE” = “TOUT_INST”, the parameter \(>\) for combining p and q situations is:

\(>\)

The moments of \(>\) are determined according to Appendix 5.

Appendix 3: Equations for a pipe junction (pitting)

Type “B3200”

Unitary type of situation

We note \(>\) and \(>\) the mechanical stresses associated with the two stabilized states of the situation. Only the definitions of \(>\) and \(>\) change compared to the case of a single set of external torsors.

For a component, each stabilized mechanical state is described using a pressure \(>\) and a force twister \(>\) defined under the keyword “CHAR_MECA”. For a pipe junction, two force torsors are provided:

\(>\)

\(>\).

The stress tensors are reconstituted by linear combination from the stress tensors associated with each of the unit loads. For example, we note \(>\) the stress tensor associated with the unit load in force in the x direction for the tubing. The calculation of the stress tensor corresponding to a mechanical loading belonging to a stabilized state is then obtained in the following way:

\(>\)

The calculation of the quantities \(>\) and \(>\) is then identical to a single component.

Instantaneous situation

Note \(>\) the transient tensor associated with the situation. For a component, we enter the thermal transient under RESU_THER, the pressure transient under RESU_PRES and the transient due to forces and moments under RESU_MECA and \(>\).

For a pipe junction, you must enter under RESU_MECA the tensor sum of the tensors of the body and the tubing.

\(>\)

The calculation of the quantities \(>\) and \(>\) is then identical to a single component.

Type “ZE200a”

We note \(>\) the sum of the transients tensor associated with the situation and \(>\) and and \(>\), \(>\) and \(>\) the extreme moments of this transient as defined in 14. We index A and B the magnitudes of the stabilized states of the situation (pressure and twisting on the moments). R, Rtubu, Rcorp, and I, Itubu, Icorp are the geometric characteristics of the pipe, K1, K2, tubu, K2, corp, K3, C1, C2, Tubu, Tubu, C2U, Tubu, C2, Icorp, and C3 are the stress indices of the RCC -M.

Parameter \(>\) for a situation is defined by:

\(>\) with

\(>\)

With “METHODE” = “TRESCA”,

\(>\)

With “METHODE” = “TOUT_INST”,

\(>\)

For both methods,

\(>\) and \(>\) with

\(>\)

and \(>\)

With “METHODE” = “TRESCA”,

\(>\) and

\(>\).

With “METHODE” = “TOUT_INST”,

we go back to the moments \(>\) and \(>\) which maximize the magnitude \(>\) and \(>\),
we calculate the quantity \(>\) such as \(>\).

Note:

In this paragraph, \(>\) .

Type “ZE200b”

Parameter \(>\) for a situation is defined by:

\(>\)

With “METHODE” = “TRESCA”, \(>\) and with “METHODE” = “TOUT_INST”, \(>\).

For both methods,

\(>\) and \(>\) with

\(>\)

and \(>\) , \(>\) .

With “METHODE” = “TRESCA”,

\(>\) and \(>\)

With “METHODE” = “TOUT_INST”, we use the moments \(>\) and \(>\) that maximize the magnitude \(>\): \(>\) and \(>\).

\(>\) and \(>\) (we take the moments \(>\) and \(>\) which maximize the magnitude \(>\)).

Note:

In this part, \(>\) .

Appendix 4: Calculation of SP and SP meca of a single situation ( “B3200” , “ZE200a” and “ZE200b” )) **

Type “B3200”

Unitary type of situation

We note \(>\) and the mechanical stresses associated with the stabilized states of the situation p.

\(>\)

With “METHODE” = “TRESCA”, the parameter \(>\) for situation p is defined by:

\(>\)

with \(>\) and \(>\),

With “METHODE” = “TOUT_INST”, the parameter \(>\) for situation p is defined by:

\(>\)

with \(>\) and \(>\).

Instantaneous situation

With “METHODE” = “TRESCA”, the parameter \(>\) for situation p is defined by:

\(>\),

\(>\)

With “METHODE” = “TOUT_INST”, the parameter \(>\) for situation p is defined by:

\(>\).

If \(>\) and \(>\) are the moments that maximize \(>\), then

\(>\)

Type “ZE200a”

\(>\) and \(>\) with

\(>\) and \(>\)

With “METHODE” = “TRESCA”,

\(>\).

\(>\)

With “METHODE” = “TOUT_INST”,

\(>\),

\(>\)

Type “ZE200b”

\(>\)

with \(>\) and \(>\) and \(>\)

We note \(>\)

With “METHODE” = “TRESCA”,

\(>\)

\(>\) and \(>\).

With “METHODE” = “TOUT_INST”,

\(>\)

If \(>\) and \(>\) are the moments that maximize \(>\), then

\(>\)

If \(>\) and \(>\) are the moments that maximize \(>\), then

\(>\) and \(>\).

Appendix 5: Method for calculating the fictional transient 2 with “TOUT_INST”

We note \(>\) the sum of the transients tensor associated with the situation p and \(>\) the sum of the transients tensor associated with the situation q.

We note If \(>\) and \(>\) are the moments of the fictional transitory 1 \(>\), then we determine the moments of the fictional transitory 2 \(>\) and \(>\).

We are looking for \(>\) that maximizes the magnitude \(>\) such as

\(>\).

We are looking for \(>\) that maximizes the magnitude \(>\) such as

\(>\).

The moments \(>\) and \(>\). are then used to calculate the quantity \(>\) regardless of the chosen TYPE_RESU_MECA (ZE200a, ZE200b, B3200).

Appendix 6: “B3200” method with stress indices

The user can enter stress indices under the keyword INDI_SIGM in order to compare the results obtained with “ZE200a” or “ZE200b” or to integrate the effects of a non-modelled weld. The stress index equations for “B3200 “are shown below. This is only possible with instantaneous situations.

Parameters \(>\) and \(>\) were determined in part 16. We add the size \(>\) to them.

\(>\)

With “METHODE” = “TRESCA”, the parameter \(>\) for the combination of p and q situations is defined by:

\(>\) with

\(>\) and \(>\).

If \(>\), then \(>\), \(>\), and \(>\).

With “METHODE” = “TOUT_INST”, we use the times that occur in the quantity \(>\) to calculate \(>\), \(>\) and \(>\).

If \(>\), then \(>\),

\(>\)

\(>\).

Appendix 7: Equations taking into account the seism

Calculating quantities for a situation

Type “B3200”

Unitary type of situation

We note \(>\) and \(>\) the mechanical stresses associated with the two stabilized states of the situation and \(>\) the transient tensor associated with this situation. \(>\), \(>\), \(>\), and \(>\) the extreme moments of this transitory as defined in 14.

The earthquake is described by a stabilized mechanical state (S) and the corresponding torsor \(>\) under CHAR_ETAT, the keyword “RESU_MECA_UNIT” must be entered.

With “METHODE” = “TRESCA”, we test all the sign possibilities on the components of the earthquake and the parameters \(>\) and \(>\) for a situation are defined by:

\(>\)

\(>\) (64 possibilities)

\(>\)

\(>\) (64 possibilities)

With “METHODE” = “TOUT_INST”, we test all the sign possibilities on the components of the earthquake and the parameters \(>\) and \(>\) for a situation are defined by:

\(>\),

\(>\)

\(>\) (64 possibilities)

\(>\)

\(>\) (64 possibilities)

For both methods, the \(>\) parameter for a situation is defined by:

\(>\)

Note:

In this part, \(>\) .

Instantaneous situation

Note \(>\) the transient tensor associated with the situation. \(>\), \(>\), \(>\) and \(>\) the extreme moments of the transitory situation as defined in 14.

The earthquake is described by six tensors corresponding to the efforts and moments \(>\), \(>\),, \(>\), \(>\), \(>\), \(>\).

With “METHODE” = “TRESCA”, we test all the sign possibilities on the components of the earthquake and the parameters \(>\) and \(>\) for a situation are defined by:

\(>\) (64 possibilities)

\(>\)

With “METHODE” = “TOUT_INST”, we test all the sign possibilities on the components of the earthquake and the parameters \(>\) and \(>\) for a situation are defined by:

\(>\) (64 possibilities)

We take the moments \(>\) and \(>\) that maximize \(>\)

\(>\)

Note:

In this part, \(>\) .

Type “ZE200a”

We note \(>\) the thermal transient tensor associated with the situation and \(>\) and \(>\) the extreme moments of this transient as defined at 14. We index A and B the magnitudes of the stabilized states of the situation (pressure and twisting on the moments). R, e, and I are the geometric characteristics of the pipe, K1, K2, K2, K3, C1, C2 and C3 are the stress indices of RCC -M.

The earthquake is described by a stabilized mechanical state (S) and the corresponding torsor \(>\) under CHAR_ETAT. We test all the possible signs on the components of the earthquake and the parameter \(>\) for a situation is defined by:

\(>\)

With “METHODE” = “TRESCA”, \(>\).

With “METHODE” = “TOUT_INST”, \(>\).

For both methods,

\(>\) and \(>\) with

\(>\)

and \(>\)

With “METHODE” = “TRESCA”,

\(>\). and \(>\)

With “METHODE” = “TOUT_INST”,

We take the moments \(>\) and \(>\) that maximize \(>\)

\(>\) and \(>\)

Type “ZE200b”

We note \(>\) the sum of the transients tensor associated with the situation and \(>\) and \(>\) the extreme moments of this transient as defined at 14. We index A and B the magnitudes of the stabilized states of the situation (pressure and twisting on the moments). R, e, and I are the geometric characteristics of the pipe, K1, K2, K2, K3, C1, C2 and C3 are the stress indices of RCC -M.

The earthquake is described by a stabilized mechanical state (S) and the corresponding torsor \(>\) under CHAR_ETAT.

We test all the possible signs on the components of the earthquake and the parameter \(>\) for a situation is defined by:

\(>\) with

\(>\)

With “METHODE” = “TRESCA”, \(>\).

With “METHODE” = “TOUT_INST”, \(>\).

For both methods,

\(>\) and \(>\) with

\(>\)

and \(>\) , \(>\) .

With “METHODE” = “TRESCA”,

\(>\) and \(>\)

With “METHODE” = “TOUT_INST”, we use the moments \(>\) and \(>\) which maximize the magnitude \(>\) and

\(>\) and \(>\).

\(>\).

We go back to the moments \(>\) and \(>\) which maximize the magnitude \(>\) and

\(>\)

Note:

In this part, \(>\) .

Bibliography

« RCC -M: Rules for the Design and Construction of Mechanical Equipment for Nuclear Islands PWR. June 2000 edition, amended June 2007 » Edited by AFCEN: French Association for the rules of design and construction of electro-nuclear boiler equipment.
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1. CURTIT « Fatigue analysis of a VVP line inside BR with underthickness » Note EDF/R&D/HT-26/00/057/A
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« RSE -M: Operational Supervision Rules for Mechanical Equipment on Nuclear Islands REP. Edition 1997, modified 2005 » Edited by AFCEN: French Association for the rules of design and construction of electro-nuclear boiler equipment.

1. METAIS: « Technical specification book (CPT) for modifying fatigue calculations for code_aster operator POST_RCCM » Note EDF/SEPTEN/D305914013267
1. 1. L, “Outcur: An automated evaluation of two-dimensional finite element stresses according to ASME Section III stress requirements. “, Pap Am Soc Mech Eng, no. 76-WA/ PVP -16, p. 8, 1976.
“ANSYS Mechanical APDL Theory Reference”, p. 960, 2020.

Description of document versions

Version Aster	Author (s) Organization (s)	Description of changes
5	J.M. Proix, EDF -R&D/ AMA	Initial text
7.4	J.M. Proix, EDF -R&D/ AMA	Unit method added
8.4	Galenne, EDF -R&D/ AMA	Adding B320 0_UNIT
9.4	Galenne, EDF -R&D/ AMA	Addition of the calculation of the priming factor (sheet 10429) Modification of the fatigue formulation for B320 0_UNIT (sheet 12297)
10.4	Galenne, Chau T.H., EDF -R&D/ AMA	Adding the KE_MIXTEpour EVOLUTION method (sheet 12818)
13.1	Plessis, EDF -R&D/ AMA	Added ZE200a and ZE200b Addition of environmental fatigue EFAT
13.3	Plessis, EDF -R&D/ AMA	Creation of B3200 which absorbs B320 0_UNIT and allows other forms of loading to be entered at times