The probabilistic structural integrity assessment of reactor pressure vessels under pressurized thermal shock loading

Nuclear Engineering and Design 294 (2015) 93–102

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The probabilistic structural integrity assessment of reactor pressure vessels under pressurized thermal shock loading Mingya Chen a,∗ , Feng Lu a , Rongshan Wang a , Weiwei Yu a , Donghui Wang b , Guodong Zhang a , Fei Xue a a b

Suzhou Nuclear Power Research Institute, 215004 Suzhou, Jiangsu Province, PR China State Nuclear Power Plant Service Company, 200237 Shanghai, PR China

h i g h l i g h t s • • • •

The methodology and the case study of the FAVOR software were shown. The over-conservative parameters in the DFM were shown. The differences between the PFM and the DFM were discussed. The limits in the current FAVOR were studied.

a r t i c l e

i n f o

Article history: Received 13 April 2015 Received in revised form 3 August 2015 Accepted 15 August 2015 Classification: L. Safety and risk analysis

a b s t r a c t The pressurized thermal shock (PTS) event poses a potentially significant challenge to the structural integrity of the reactor pressure vessel (RPV) during the long time operation (LTO). In the USA, the “screening criteria” for maximum allowable embrittlement of RPV material, which forms part of the USA regulations, is based on the probabilistic fracture mechanics (PFM). The FAVOR software developed by Oak Ridge National Laboratory (ORNL) is used to establish the regulation. As the technical basis of FAVOR is not the most widely-used and codified methodologies, such as the ASME and RCC-M codes, in most countries (with exception of the USA), proving RPV integrity under the PTS load is still based on the deterministic fracture mechanics (DFM). As the maximum nil-ductility-transition temperature (RTNDT ) of the beltline material for the 54 French RPVs after 40 years operation is higher than the critical values in the IAEA-TECDOC-1627 and European NEA/CSNI/R(99)3 reports (while still obviously lower than the “screening criteria” of the USA), it may conclude that the RPV will not be able to run in the LTO based on the DFM. In the FAVOR, the newest developments of fracture mechanics are applied, such as the warm pre-stress (WPS) effect, more accurate estimation of the flaw information and less conservation of the toughness (such as the three-parameter Weibull distribution of the fracture toughness). In this paper, the FAVOR software is first applied to show both the methodology and the results of the PFM, and then the limits in the current FAVOR software (Version 6.1, which represents the baseline for re-assessing the regulation of 10 CFR 50.61), lack of the impact of the constraint effect, ductile–tearing initiation failure analysis, elastic–plastic fracture mechanics analysis, safety assessment of the crack front interface points and interaction among multiple cracks, are also discussed. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Although the long time operation (LTO) is a main concern for nuclear power plants (NPPs), the pressurized thermal shock (PTS)

∗ Corresponding author. Tel.: +086 051268701169; fax: +086 051268602518. E-mail address: [email protected] (M. Chen). http://dx.doi.org/10.1016/j.nucengdes.2015.08.020 0029-5493/© 2015 Elsevier B.V. All rights reserved.

event poses a potentially significant challenge to the structural integrity of the reactor pressure vessel (RPV) (IAEA, 2010; Qian et al., 2015). The assessment of RPV integrity from the point of view of its resistance against fast fracture is based on comparison of crack driving forces (such as stress intensity factor KI ) calculated for assessed points along the crack front with its allowable value (fracture toughness KIC or KIa ) for PTS events (Chen et al., 2014, 2015a). As the crack driving force (related to loads and structures) and the material toughness (related to temperatures, fluence and

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M. Chen et al. / Nuclear Engineering and Design 294 (2015) 93–102

materials) are of stochastic nature (Vladislav et al., 2013), two main approaches can be used:

based on the PFM, and then the limits of the current FAVOR software (Version 6.1) are also discussed.

(1) Deterministic fracture mechanics (DFM), where certain conservative values are used, and the result of this assessment is prediction either “RPV fails” or “RPV does not fail” during a single PTS event. (2) Probabilistic fracture mechanics (PFM), where some input data are treated as statistical distributions. Usually Monte Carlo simulations are performed. The result of this assessment is expressed in terms of frequency of fast fracture initiation or frequency of RPV failure.

2. Methodology

Only in the USA, the PFM is used as a basis for developing the “screening criteria”, which forms part of the USA regulations. The 10 CFR 50.61 (U.S. NRC, 1984) establishes a screening criterion based on the reactor vessel nil–ductility–transition temperature (RTNDT ) for the PTS events. The screening criterion RTNDT (called RTPTS in the rule) was selected according to the studies that the risk due to PTS events is acceptable (U.S. NRC, 1987). The RTPTS is 132 ◦ C for plates, forgings and axial welds, and 149 ◦ C for circumferential welds in 10 CFR 50.61. As long as the “screening criteria” is reached, the risk caused by the PTS event is considered to be acceptable. As the technical basis of FAVOR is not the most widely-used and codified methodologies (such as the ASME and RCC-M codes), the DFM is still used as “basic” in most countries. In the IAEA-TECDOC1627 report (IAEA, 2010), the maximum allowable RTNDT is only about 70 ◦ C for the typical PTS event. And in European NEA report (NEA, 1999), the maximum allowable RTNDT is about 80 ◦ C for the typical PTS events based on the tangent method. It is easy to find some interesting questions: (1) The maximum allowable RTNDT of the DFM is obviously smaller than the “screening criteria” of PFM. Which parameters are over-conservative in the DFM? (2) In Bezdikian (2008), the maximum RTNDT is about 80 ◦ C for 54 French RPVs after 40 years operation. So it may conclude that the RPV will not be able to run in the LTO based on the DFM. Although the resistance of RPV against fast fracture has to be proven by comprehensive analyses, there are few published literatures to discuss the above questions. In this paper, the FAVOR software of Oak Ridge National Laboratory (ORNL) is first used to show both the methodology and the results of the typical PTS event

In U.S. NRC (2007) introduces the process to establish the PTS “screening criteria” based on the FAVOR software which has been approved by the USA NRC. The process of FAVOR is shown in Fig. 1. The probabilistic steps are marked with the background color in Fig. 1. The “screening criteria” was established according to the design, manufacture and service of the USA domestic RPVs, so it may be not suitable to other countries. The FAVOR software is divided into three models, FAVLOAD, FAVPFM and FAVPOST. 2.1. FAVLOAD model, generate deterministic load Before the assessment, the event trees shall be developed for all initiating events potentially occurring in NPPs that may lead to PTSs. The identified scenarios have to be divided into different groups of similar variations of thermal hydraulic parameters, and from each group a representative has to be selected for reducing the amount of the assessment. For each group, the frequency of occurrence of individual PTS scenarios within the group will be determined, and the occurrence frequency of the whole group, frj for the jth group, will be established, including uncertainties. In the FAVLOAD model, deterministic temperature and stress fields are calculated for the jth group using the 1D finite elements (FEs) to produce a load-definition input file for the FAVPFM model. 2.2. FAVPFM model, Monte Carlo simulation In the FAVPFM model, the Monte Carlo techniques are applied. For the jth representative group, the conditional probabilities of fast fracture initiation CPIj (by plane-strain cleavage initiation) are determined. For less conservative evaluation, conditional probabilities of RPV failure CPIj may be determined. The failure criteria is the prediction of sufficient flaw growth either to produce a net-section plastic collapse of the remaining ligament or to advance the crack tip through a user-specified fraction of the wall thickness in the FAVOR software (Oak Ridge National Laboratory (ORNL), 2006). The main stochastic parameters within the Monte Carlo simulation are (1) flaw information (flaw density, size, aspect ratio and flaw position in the RPV) and (2) material parameters, neutron fluence, initial value of the reference temperature and its shift

Fig. 1. The flow chart of FAVOR software. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

M. Chen et al. / Nuclear Engineering and Design 294 (2015) 93–102

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due to irradiation. For the jth representative group, basic parts of the assessment are Monte Carlo simulations in which the values of individual parameters are randomly sampled from their statistical distributions, “random RPVs” with “random flaws” are generated. In what follows, only the CPIj approach is discussed. In the FAVPFM model, based on time variation of flaw driving forces (KI in the FAVOR), the Monte Carlo simulations are used to calculate the CPI. For the jth representative group, vessel l, and flaw k, the conditional probabilities of fast fracture initiation CPIjlk (t) (varied with time) is determined:

  CPIjlk (t) = 1 − exp −

KI (t)jlk − aKIC

cK  IC

(1)

bKIC

where KI (t)jlk is the flaw driving forces for flaw k, vessel l, and jth transient. aKIC , bKIC and cKIC are the parameters of the Weibull distribution of the fracture toughness. CPIjlk is determined as the maximum CPIjlk (t) over the time interval of the PTS event:





CPIjlk = max CPIjlk (t)

Fig. 2. Pressure-temperature vs. time.

(2)

For vessel l, the probability of fracture initiation can be viewed as the probability of occurrence union of all the flaws, assuming that each flaw independently and marginally contributes to the vessel’s CPI. When there are n randomly generated flaws, the total conditional probability of fast fracture initiation CPIjl is determined:



n

CPIjl = 1 − ˘ 1 − CPIjlk



(3)

k=1

The outcome for each RPV in the Monte Carlo analysis is fracture initiation or no fracture initiation. When there are m RPVs, in which there are mfra RPVs fracture initiation, the CPIj for the jth representative group is determined: CPIj =

mfra m

(4)

Fig. 3. The core segment model of the RPV.

3. Example: Typical transient analysis

2.3. FAVPOST model, post-processor

3.1. Selection of the event

Combining the frequency of jth group frj with corresponding CPIj , the unconditional frequency of fast fracture initiation FIj is obtained for the particular jth scenario group. Summing up the frequencies corresponding to all scenario groups (p groups), the final frequency of fast fracture initiation FI is determined:

The event in the IAEA report (IAEA, 2010) was used to introduce the detailed methodology and result of the PFM using the FAVOR software. The focus of FAVOR is only the beltline region of the RPV wall. The time variations of pressure and fluid temperature around the beltline are shown in Fig. 2, and the heat transfer coefficient is shown in Table 1.

FI =

p

j=1

FIj =

p

frj · CPIj

(5)

3.2. Analysis model

j=1

The acceptance criterion f in 10 CFR 50.61 is as follow: FI < 5 × 10−6 / (reactor year)

(6)

As both frj and CPIj are statistical distributions, the final FI is statistical distribution as well, and its mean value should be used in the final evaluation.

In the IAEA-TECDOC-1627 report, as shown in Fig. 3, the inner radius of the beltline Ri is 1994 mm, the thickness of the base metal t is 200 mm and the thickness of cladding tcladding is 7.5 mm, respectively (IAEA, 2010). The inner side of a RPV is assumed to be subjected to thermal shocks, and the thermal load is assumed to be rotationally symmetric and homogeneous along the Z-axis in the analysis. The beltline region of RPV is sufficiently far away from the nozzle area to be treated as axisymmetric cylindrical shell.

Table 1 Heat transfer coefficient around the beltline. Temperature (◦ C)





Heat transfer coefficient W/(m2 ◦ C) Temperature (◦ C) Heat transfer coefficient W/(m2 ◦ C)

37

48

49

59

69

96

106

115

992

877

790

1147

602

710

1229

1057

206 1581

251 4834

261 1757

268 6232

276 3453

279 1054

287 24,696

295 24,125

152 1838 / /

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Table 2 Material properties of the base metal and cladding. Material

Temperature (◦ C)

Young modulus/E (GPa)

Poisson’s ration ()

Thermal conductivity (W/(m ◦ C))

Thermal diffusivity,  = /C (10−6 m2 /s)

Density (103 kg/m3 )

Base metal

20 300 20 300

204 185 197 176.5

0.3

54.6 45.8 14.7 18.6

14.7 10.6 4.1 4.3

7.6

Cladding 1

0.3

7.6

Coefficient of thermal expansion1 (10−6 /◦ C) 10.9 12.9 16.4 17.7

Yield stress (MPa) 588 517 380 270

Mean value between 20 ◦ C and the temperature.

3.3. Material properties The RPV was modeled with SA508 Class 3 steel and the cladding was made of austenitic stainless steel, respectively. Temperature dependent material properties used in the analyses are summarized in Table 2 (IAEA, 2010).

4. Results and discussion 4.1. DFM assessment In order to show the fundamental computational algorithms in the FAVOR software, the deterministic result of FAVOR was obtained and then compared to that in the IAEA report (the DFM is used in the IAEA report). In the IAEA report, as shown in Fig. 3, the depth of the axial surface crack is 12 mm with the flaw shape a/c = 1/3. The stress free temperature was assumed to be the initial temperature of the transient. The time variation KI result of FAVOR for the particular crack during the transient is shown in Fig. 4, and the KI result in the IAEA report is shown in Fig. 5. At the repressurization time of the transient (about 7200 s), the temperature of the crack deepest point is about 70 ◦ C for both the FAVOR result and that in IAEA report. The KI result of FAVOR is slightly

Fig. 4. The results of FAVOR software (the deepest point of the particular crack tip).

smaller than that in the IAEA report. As the range of variation √ of the KI value at the re-pressurization time is about 16 MPa m in the IAEA report (IAEA, 2010), the KI result of FAVOR is within the range of variation. Therefore, it can be concluded that the

Fig. 5. Results of the IAEA report (the deepest point of the particular crack tip).

M. Chen et al. / Nuclear Engineering and Design 294 (2015) 93–102

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Table 3 Result of the Monte Carlo simulations.



2

Fluence (E > 1 MeV) ×1019 n/cm 2 4 6 8 10



Years of operation

Flaw number

Initiation number

Cleavage number

Ductile number

Conditional initiation/ cleavage frequency

/ About 40

5814,870 5814,870 5814,870 5814,870 5814,870

24 40 46 47 49

24 40 46 47 49

0 0 0 0 0

3.967 × 10−9 3.132 × 10−8 7.001 × 10−8 1.092 × 10−7 1.451 × 10−7

About 60

DFM result of FAVOR is consistent with the results in the IAEA report. 4.2. PFM assessment 4.2.1. Conditional probability of failure In order to avoid larger initial flaws and higher embrittlement of the weld material, the beltline is usually without any weld (there is also no weld in the IAEA report). The height of the core segment is 4.27 m, and the mean weight-percent measured values of Cu, Ni and Mn for the base metal are 0.30%, 0.85% and 0.12%, respectively. The default flaw file of FAVOR and the default distributions of the random variables are used to do the study. The load on the crack face and ductile–tearing model are included. The stress free temperature is assumed to be the initial temperature of the transient, and the E2006 correlation for estimating radiation-induced shift in RTNDT is used. 10,000 times of the Monte Carlo simulation were carried out, and the result is shown in Table 3. In the Monte Carlo simulation, the RPV flaw-characterization data for the 1st stochastically generated RPV trial is taken from the first group of the default file records. The RPV flaw characterization for the 2nd stochastically generated RPV trial is determined from the second group of file records, etc. The RPV trials cycle through the flaw characterization file records sequentially up to 1000, and then restarts at the first file record (Oak Ridge National Laboratory (ORNL), 2006). As shown in Table 3, due to the sampling method, the flaw number for each time simulation (under different levels of fluence) is constant. It means that the sampling process of flaw is the deterministic process in the FAVOR software when the flaw file is determined. 5814,870 flaws are included in 10,000 times of the Monte Carlo simulation, and no more than 50 flaws initiate and propagate through the wall. Although the ductile–tearing model is considered, there is no ductile–tearing failure in the study. As the ductile–tearing model for the FAVOR (Version 6.1) only exists in the Initiation-Growth-Arrest model in which the flaw extension should be arrested first, the ductile–tearing analysis cannot reduce the CPI. The CPI or CPF is very low under the typical PTS event, even when the neutron fluence is increased to 10 × 1019 n/cm2 (about 60 years of operation). When the neutron fluence is increased to 10 × 1019 n/cm2 , the CPI or CPF is still in the level of 10−7 /(reactor year). After considering the event probability, the initiation or failure frequency will be far below than the allowable risk of the regulation. That is one of the main reasons why the maximum RTNDT of DFM is obviously lower than the “screening criteria” of the PFM. The relationship between the flaw initiation number and the neutron fluence is shown in Fig. 6. As shown in Fig. 6, the flaw initiation number increases with the increase of neutron fluence, and decreases with the increase of the mean weight-percent values of Cu. 4.2.2. Flaw distribution In the FAVOR, the flaws of concern are assumed to be present at the time of vessel fabrication but not detected and repaired before

Fig. 6. Crack initiation number vs. fluence and Cu content.

the vessel was placed into service (U.S. NRC, 2007). The evaluations assume that there are no credible mechanisms to cause servicerelated flawing of the RPV materials. It is also assumed that flaw growth mechanisms of fatigue and stress corrosion flawing can be neglected. Three categories of flaws are defined in the FAVOR: (1) Category I, surface-breaking flaws, (2) Category II, embedded flawsfully elliptic geometry with inner crack tip located between the clad/base interface and 1/8 thickness of the RPV wall from the inner surface, and (3) Category III, embedded flaws-fully elliptic geometry with inner crack tip located between 1/8 and 3/8 thickness of the RPV wall from the inner surface (Oak Ridge National Laboratory (ORNL), 2006). When the neutron fluence is 10 × 1019 n/cm2 , the statistics of three type flaws are shown in Table 4. As shown in Table 4, although the largest number of flaws is the type III flaw, only the type II flaws can initiate in the study (the type II flaws are closer to the inner surface where the high tensile stress occurs). Table 4 Flaw statistics in the result of FAVOR. Flaw type

Type I

Type II

Type III

Summary

Flaw number Initiation number

3538 0

1937,147 49

3874,185 0

5814,870 49

Table 5 Flaw size statistics in the result of FAVOR. Flaw size (mm)

2.07518 4.15036 6.22554 8.30072 Weight summary (%)

Weight (%) Type I

Type II

Type III

0.0000 0.0000 100.0000 0.0000 100.0000

75.7314 23.3408 0.8302 0.0976 100.0000

75.7314 23.3408 0.8302 0.0976 100.0000

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M. Chen et al. / Nuclear Engineering and Design 294 (2015) 93–102

The flaw size is shown in Table 5. As it is assumed that the distribution of flaw is uniform along the wall thickness, the weight of different size flaws is same for the type II and type III flaws. As shown in Table 5, the maximum depth of surface-breaking flaws in FAVOR (with default flaw file) is only about 6.23 mm, only equivalent to half of the postulated flaw in the IAEA report. In Chen et al. (2014, 2015a), IAEA (2010), the safety margin of the beltline decreases with the increasing of the crack size. So the larger crack in the DFM analysis leads to lower value of the critical RTNDT . The surface-breaking flaws only exist in the cladding in the FAVOR, which means that there is no embedded flaw propagating through the cladding and no surface-breaking flaw extending to the base metal. This is obviously different to the hypothesis in the ASME and RCC-M codes (based on the DPM). The model used in the DFM conservatively postulates that all fabrication flaws are inner-surface-breaking flaws. In Chen et al. (2014), the KI of the sub-clad crack is 50% less than that of the surface breaking crack. In the ASME and RCC-M codes, the flaw in the base metal is assumed to extend through the cladding before the PTS event. The flaw information of FAVOR is set up according to the actual NDT information. As shown in Table 5, if the coefficient of 2 is applied on the maximum crack in the DFM analysis (it is also recognized that the fabrication flaw data has the greatest level of uncertainty of the inputs required for the PTS evaluations), the depth of the maximum embedded flaw is 16.6 mm. After extend through the cladding (the thickness of the cladding is about 7 mm), the depth of the surface-breaking flaw is about 23.6 mm. In the ASME code Section XI Appendix E “Evaluation of unanticipated operating events”, which can be used to assess the safety under the level D condition, the maximum depth of the flaw is 25 mm. In the RCC-M code Version 2007, the size of postulated flaw is as follow:



a=

min(0.5t, 10 mm) t ≤ 40 mm min(0.25t, 20 mm) t > 40 mm

1 a = 6 2b

(7a)

(7b)

in which, a is the flaw depth, 2b is the flaw length and t is the thickness of the vessel. So the maximum depth of the postulated crack is 20 mm when the thickness of the vessel is more than 40 mm in the RCC-M code. 4.2.3. Fracture toughness The main stochastic material parameters within the PFM assessment are neutron fluence, initial value of the reference temperature and its shift due to irradiation. As shown in Fig. 7, the threeparameter Weibull distribution of fracture toughness KIC is used in the FAVOR, and the toughness used in the DPM (the envelope boundary) is obviously conservative. Experimental evidence for the warm pre-stressing (WPS) effect in ferritic steels was first reported almost 40 years ago (Schuster et al., 1999). Although the WPS effect does not increase the fracture toughness directly, the effect can change the stress field around the crack front. If the WPS happens, the applied KI of an arrested flaw must be greater than the previous maximum KI (of the arrested flaw geometry since the time of the arrest) for the flaw to reinitiate (Oak Ridge National Laboratory (ORNL), 2006). The WPS has been implemented in the FAVOR as a user-set option, thus allowing cases to be run with and without WPS effects. 4.2.4. Events frequencies In Westinghouse (2003), the frequency of the large LOCA transient is only about 10−7 /(reactor year) for the Westinghouse plant. The frequency of the event is already lower than the allowed risk in 10 CFR 50.61, so even the CPI or CPF is 100% for the large LOCA transient, it can be acceptable by the regulation.

Fig. 7. Three-parameter Weibull distribution of the fracture toughness in the FAVOR.

While the large LOCA transient may be one of the few events to be studied for the NPPs based on the DFM, the RTPTS of the PFM could obviously higher than the critical RTNDT of the DFM. 4.3. Improve of the analysis model in the FAVOR Although the FAVOR software has been approved by the USA NRC, there are still some shortcomings in the model of FAVOR, such as lack of the impact of the constraint effect, ductile–tearing initiation failure analysis, elastic–plastic fracture mechanics analysis, safety assessment of the crack front interface points and interaction among multiple cracks. 4.3.1. Study of the constraint effect Although the advanced fracture-mechanics methodologies can obviously increase the material toughness values for shallow surface-breaking or embedded flaws, the high-constraint fracturetoughness data is used in the FAVOR. And, the constraint effect on the RPV safety is one of the hottest unresolved issues. In the case of brittle fracture (linear-elastic stress analysis method is used in the FAVOR), essentially two different methods to quantify constraint have been proposed, the Q parameter and the Tstress parameter (Harlin et al., 1988). Although it is more sophisticated, the Q methods require the use of very detailed elastic–plastic FE analysis, whereas the Tstress only requires an elastic analysis. The Q parameter and the Tstress both give a quantitative value for the constraint (O’Dowd et al., 1992), but the connection to toughness is lacking (Wallin, 2001). As Wallin (2001), Qian et al. (2013) give a connection between Tstress and the master curve transition temperature T0 , it will be taken in this paper to do the study. In the DFM, an axial surfacebreaking flaw is often assumed. In general, most of the cracks found in beltline region are shallow cracks ranging from 10 to 15 mm deep (Marshall Committee, 1982; Chen et al., 2014), and the maximum depth of the postulated crack specified in the code is about 1/4 t (ASME, 2013; RCC-M, 2007). Therefore, three different cracks of a/t = 0.05, 0.15 and 0.25 were modeled with fixed crack aspect ratio of a/c = 1/3 to investigate the effect of crack depth on the safety of RPV. All cracks were placed on the surface of the inner wall considering the high tensile stress during the PTS transient. In order to evaluate the Tstress , elastic FE was carried out using the generalpurpose FE program ABAQUS, Version 6.12. One of the surface flaw models is illustrated in Fig. 8, and only one quarter of the beltline was modeled considering symmetric conditions. The FE model was

M. Chen et al. / Nuclear Engineering and Design 294 (2015) 93–102

99

-0.8

a/t=0.05 a/t=0.15 a/t=0.25

-1.0

T t /σs T-stress/

-1.2

-1.4

-1.6

x/a=1/6

-1.8

-2.0 0

20

40

60

80

100

x/a [%] Fig. 8. Three-dimensional mesh of the vessel containing a surface crack.

(a) 3600 s

T0 =

T0deep + Tstress



10 MPa/◦ C

 , Tstress < 0

a/t= =0.05 a/t= =0.15 a/t= =0.25

-1.75 -2.00 -2.25 -2.50 -2.75

x/a=1/6

-3.00 -3.25 0

20

40

60

80

100

x/a [%]

(b) 7200 s Fig. 9. The distribution of the Tstress along the crack front. (x is the distance from the crack tip to the interface and a is the crack depth as shown in Fig. 3.)

(8)

in which, T0deep means T0 without considering constraint effect. In the IAEA-TECDOC-429 report (IAEA, 2005), RTT0 can be used to instead of RTNDT to calculate the fracture toughness KIC : RTT0 = T0 + 19.44 ◦ C

-1.50

T stress/yield stress T-stress/yield

also used in Chen et al. (2014, 2015b), so the detail and validity of the model aren’t illustrated in this paper. The Tstress along the crack front at 3600 s and 7200 s of the PTS transient in the IAEA-TECDOC-1627 report is shown in Fig. 9 (the yield stress  s of the base metal is assumed to be 500 MPa). The negative value of Tstress means losing of the constraint effects and increasing of the fracture toughness. The differences of constraint effect between shallow and deep cracks are obvious when the dominate stress is the thermal stress (3600 s), while the differences are less obvious when the dominate stress is caused by the pressure load (7200 s). It means that there is obvious relationship between the constraint effect and the load type. The Mises stress around the crack front at 3600 s of the transient is shown in Fig. 10, and the red zones means the stress is larger than the yield stress. As shown in Fig. 10, the red zones increase with the increasing of the crack size (means higher constraint effect), and the red zones change from the twin fans shape to the approximate circular shape when the constraint effect increases (1/2 of the crack front is shown in Fig. 10). In Wallin (2001), Qian et al. (2013), the relationship between Tstress and T0 of Master Curve is as follow:

(9)

KIC = 22.783e0.036(T −RTNDT ) + 36.5 (10) √ ◦ where KIC is expressed in MPa m and T( C) is material temperature. At 3600 s of the transient, when the temperature of crack front is 60 ◦ C and T0 is 90 ◦ C, respectively (assume to be constant values along the wall thickness), the KIC along the thickness of the base metal considering the constraint effect is shown in Fig. 11. As shown in Fig. 11, the KIC increases obviously when the constraint effect is considered. In the region near the clad-base interface, the effect of cladding stress is dominant and the dominant distance of the cladding stress is about 1/6 crack depth a (Chen et al., 2014). Out of the dominant region, the fracture toughness is almost doubled when the constraint effect is considered, especially for the shallow crack in this study. The same result is also shown in Pennell et al. (1995), the fracture toughness of the larger specimen is almost half of the smaller

size specimen, so the constraint effect has obviously influence on the safety of RPV. As shown in O’Dowd et al. (1992), the Q parameter can show more accurate of the constraint effect than the Tstress parameter. Due to lack of the connection between Q parameters and fracture toughness, the Tstress parameter is used in this paper. So, in the next step, the Q parameter may be used to study the constraint effect on the safety of the RPV. 4.3.2. Tearing failure analysis and code differences Although the ductile–tearing model is considered in the FAVOR, it only exists in the Initiation–Growth–Arrest model in which the flaw extension should be arrested first. That means the ductile–tearing model has no effect on the value of CPI in the FAVOR. While in RCCM code, as shown in Fig. 12, the ductile–tearing toughness is obviously higher than the crack initiation toughness which can be transferred from JIC , the ductile–tearing initiation failure mode has not been considered in the FAVOR. In MAI (2014), the differences among the PFM results of FAVOR software of USA, PASCAL3 software of Japanese and TURNS software of French is about 15–20%. Research shows that these differences are caused by the codes of different countries. For example, parts

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Fig. 10. Stress distribution along the crack tip at 3600 s of the transient. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Table 6 The differences in the comparison of material between the RCC-M and ASME codes. Element

Weight (wt%)

Phosphorus Sulphur Silicon Nickel

RCC-M M.2111 beltline material

ASTM SA 503 CL 3

0.008 max 0.008 max 0.10–0.30 0.50–0.80

0.025 max 0.025 max 0.15–0.40 0.40–1.00

Table 7 The differences in the design rule between the RCC-M and ASME codes. RCC-M Appendix ZG

Fig. 11. KIC along the crack front considering the constraint effect. (x is the distance from the crack tip to the interface and a is the crack depth as shown in Fig. 3.)

Brittle and ductile Hydro test pressure 1.43 time of design value Ductile toughness Level D code1 , KCP ≤ + KIC /1.2, when T < RTNDT + 60 ◦ C KCP ≤ + KIC /1.0, when T > RTNDT + 40 ◦ C

ASME Only non-ductile Hydro test pressure 1.25 time of design value No Level D code2 , 1.4(KIM + KIt ) + KIr ≤ + KIC

1

KCP means the crack driving force considering the plastic correction. ASME Section XI Appendix E “Evaluation of unanticipated operating events” can be used to assess the safety under the level D condition. KIM means the stress intensity factor due to membrane stress, KIt means the stress intensity factor due to thermal stress, KIr means the stress intensity factor due to residual stress. 2

of the differences between the RCC-M code and ASME codes are shown in Tables 6 and 7.

Fig. 12. JR − a resistance curve in the RCC-M code.

4.3.3. Elastic–plastic fracture mechanics analysis Based on the assumption that plastic zone is small around the crack tip, the linear-elastic stress analysis method is used in the FAVOR. As shown in Fig. 13, Chen et al. (2014) show that pure elastic analysis may be non-conservative. In the typical PTS transient, as the elastic stress in the cladding region is higher than the yield stress, the stress within the cladding would be limited when the plasticity is incorporated, and the stress in the base metal near the interface region would increase slightly to compensate the lowered stress in the cladding zone. So the elastic–plastic stress intensity factor KJ is larger than the elastic KI for the interface points, and also for the deepest point of the most shallow crack as shown in Fig. 13. But out of the region where the

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Fig. 13. Compare the elastic and elastic–plastic results. (x is the distance from the crack tip to the interface and a is the crack depth as shown in Fig. 3.)

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4.3.5. Probabilistic distribution of the uncertainties In the PTS assessment, approximate distributions upon on the subjective judgment are often adopted (Jackson, 2001). For example, the probabilistic distribution of the crack depth is the Weibull distribution in the TURNS software, exponential distribution in the PASCAL3 software and deterministic values (in the default flaw file of plate) in the FAVOR, respectively. The end region of the distribution has significant effect on the failure probability, especially for the highly reliable components (Oak Ridge National Laboratory (ORNL), 2006). Although the Weibull distribution, normal distribution and log-normal distribution which have been usually used in the PFM may have similar shapes in the middle part, there are differences in the end region. So the choice of a statistical distribution to be used for a parameter has an important impact on the risk of the PTS events. There are other shortcomings in the FAVOR, such as lack of interaction among multiple flaws. Doel-3 plant of the Electrabel company found 931 flaws in the core segment of the RPV in 2012 (Tweer, 2013; FANC, 2013). There is no method in the ASME and RCC-M codes to assess the interaction among so many flaws, and the interaction among multiple flaws is also not considered in the FAVOR software according to Eq. (3). 5. Conclusions Based on the study, the following conclusions are obtained:

Fig. 14. Crack initiation safety margin under the thermal load. (x is the distance from the crack tip to the interface and a is the crack depth as shown in Fig. 3.)

effect of cladding stress is dominant, KJ tends to be smaller than KI . Since it is not a priori clear whether linear elastic or elastic–plastic calculation of KI is more conservative, the elastic–plastic calculation should be done. 4.3.4. Safety assessment of the crack front interface point Only the safety of deepest point along the crack front is assessed in the FAVOR, while both the deepest point and the interface point between cladding and base need to be calculated according to the ASME and RCC-M codes (ASME, 2013; RCC-M, 2007). The crack initiation safety margin along the different crack tips under the pure thermal load is shown in Fig. 14 (Chen et al., 2015a). For the deepest points, the crack initiation safety margin decreases obviously with the increase of crack size first, then increases obviously when the crack size a/t is bigger than 0.05. The crack initiation safety margin of the interface points always decreases with increasing crack size, and the crack initiation safety margin of the interface points is always smaller than that of the deepest points. Therefore, the critical part along the crack front is the interface points rather than the deepest point. The large-scale experiment results of ORNL (Pennell et al., 1997) also show that the interface point will initiate first during some PTS events, therefore the safety of the interface point also needs to be considered.

(1) In this study, for the typical PTS transient in which the maximum allowable RTNDT is only about 80 ◦ C, the CPI is already obviously lower than the allowable risk in the regulation even when the neutron fluence is increased to the value corresponding to 60 years of operation, so the “safety” of PRV based on DFM may be different to that based on PFM during the LTO. (2) As the coefficient on the crack size is applied and the crack type is also assumed to be surface-breaking in the DFM, the flaw distribution poses a significance influence on the result. (3) The large LOCA transient may be one of the few events to be studied for the NPPs based on the DFM. As the frequency of the large LOCA transient is already lower than the allowed risk in 10 CFR 50.61, the RTPTS in the PFM could obviously higher than the critical RTNDT in the DFM. (4) Although some newest developments of fracture mechanics are applied, there are still some shortcomings in the model of FAVOR. Acknowledgements The project is supported by National High Technology Research and Development Program of China (863 program) under Grant no. 2012AA050901, and the National Natural Science Foundation of China under Grant no. 51275338. The financial supports of both programs are greatly appreciated. Thanks for Professor Terry Dickson of the Oak Ridge National Laboratory for the right to use the FAVOR software. References ASME, 2013. ASME Boiler and Pressure Vessel Code, Section XI, Appendix E, Evaluation of Unanticipated Operating Events. ASME, New York, NY, pp. 23–40. Bezdikian, G., 2008. Nuclear Power Plants Life Management Reactor Pressure Vessel Strategic Evaluation for Fluence in Relation with Integrity Assessment. Americian Society Mechanical Engineers, pp. 579–787. Chen, M.Y., et al., 2015a. The deterministic structural integrity assessment of reactor pressure vessels under pressurized thermal shock loading. Nucl. Eng. Des. 288, 130–140. Chen, M.Y., et al., 2015b. Use of the failure assessment diagram to evaluate the safety of the reactor pressure vessel. J. Press. Vessel Technol. 137, 051203:1–8.

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