E-Journal of Advanced Maintenance Vol. 11 No. 4 (2020) 172-178
Japan Society of Maintenology

Verification of Safety Margin of Reactor Pressure Vessel Exposed toVarious Thermal Transients Based on Probabilistic Approach
Masahide SUZUKI1, Kenta MURAKAMI1, Takashi SUZUKI1*, Ryuta OKAYAMA1**,
Jinya KATSUYAMA2 and Yinsheng LI2
1Nagaoka University of Technology, Kamitomioka 1603-1, Nagaoka, Niigata, 940-2188, Japan
Formerly graduate student, currently Japan Nuclear Fuel Limited, 1** Formerly graduate student,
currently Tokyo Electric Power Company, 2 Japan Atomic Energy Agency, Tokai-mura, Nakagun,Ibaraki, 319-1195, Japan

1*ABSTRACT
For the reactor pressure vessel (RPV) integrity assessment, deterministic fracture mechanics (DFM) is appliedconsidering factors such as neutron irradiation embrittlement, postulated crack and thermals transients etc. On
the other hand, probabilistic fracture mechanics (PFM) can evaluate the failure probability considering theuncertainties of several kinds of factors affecting the RPV integrity. The purpose of present study is to obtaininsight concerning the further improvement of RPV structural integrity from the probabilistic approachconsideration through careful comparisons with DFM approach. It was found that the transients which give riseto the re-pressurization have a possibility to deteriorate the deterministic – probabilistic correlation.
KEYWORDS
Reactor Pressure Vessel, Probabilistic Fracture Mechanics, Pressurized Thermal Shock, Structural Integrity,
Irradiation Embrittlement, Fracture Toughness, Thermal Transients

1.

Introduction

Kic,KI [MPa m^0.5]

More than 15 light water reactors (LWRs) in Japan have been operated for more than 30 years.
Regarding the structural integrity of aging reactor pressure vessels (RPVs), pressurized thermal shock(PTS), as shown in Fig.1, is considered the most important event for pressurized water reactor (PWR).
In the current Japanese Electric Association Code [7], deterministic fracture mechanics (DFM) is
applied considering the safety margins of influencing factors such as the degree of neutron irradiation
embrittlement, postulated crack and thermals transient conditions during PTS. The structural integrityof RPVs can be assessed together with the safety margin in terms of temperature margin (ΔTm) shownin Fig. 2. The influencing factors, however, generally contain the uncertainties. On the other hand,probabilistic fracture mechanics (PFM) has a feature which can take into account such uncertainties
pertaining to these factors and can calculate the conditional probability of crack initiation (CPI) andfailure (CPF) values.
stress intensity factor : KI

150
Safety margin(ΔTm)

100fracture toughness : KIc50(Tkmax,Kmax)

00Fig. 1. Schematic view of the PTS events.
The inner surface of the vessel wall is cooled
as the ECCS coolant water is injected at theLOCA event, and the tensile stress is
produced at near surface region.

Failure region

Safety region

20050100150200Temperature [℃]250300Fig. 2. RPV integrity evaluation scheme. When KIC<KI ,fracture occurs. When the warm pre-stress (WPS) effect isconsidered, safety margin is expressed as the temperature
difference between the safety and failure regions, ΔTm.
(Tkmax,Kmax) shows the maximum KI point.

172
ISSN-1883-9894/10 © 2010 – JSM and the authors. All rights reserved.

M.Suzuki, et al./ Verification of Safety Margin of Reactor Pressure Vessel Exposedto Various Thermal Transients based on Probabilistic Approach

Probabilistic

Conditional Fracture Probability

Then, the question arises as to whether there exists any certain tendency on thecorrelation between the deterministic and probabilistic approaches relating to safety assessment(see Fig.3). To expand the use of probabilistic approach in the integrity assessment, detailed
comparative analyses on these two approaches are needed. In the present study, the deterministic –probabilistic correlation in the RPV integrity assessment has been investigated from thepoint of ΔTm – CPI or ΔTm – CPF relationships. Although generally good relationship wasrecognized [1,2], more systematic analysis is needed to clarify the realistic RPV structural orthermal transient conditions, which may deteriorate the deterministic – probabilistic correlation. For
this purpose, effects of cladding, crack types (surface crack or under-clad crack) and various transients
including ones with re-pressurization are investigated.

Fig. 3. Schematic illustration of the deterministic –probabilistic correlation. The variation is expected as
to the correlation in accordance with the effects ofcladding, crack types and transient conditions. If thecorrelation is with larger scattering, then, the safetymargin determined by the deterministic approach
does not necessary mean the same safety level.

How scattering may bechanged with the calculation
condition from the relatively
small scattering (goodrelationship)

Safety Margin, ΔTm

Deterministic

2. Analytical Method
2.1. Parameters for Calculation
A model plant adopted for the analysis is 2-loop PWR. The inner radius of RPV is
1669mm, with wall thickness of 165mm with cladding of 6mm thickness. The steel
contains 0.12wt%Cu and 0.56wt%Ni, and the RTNDT shifts due to neutron irradiation arecalculated according to JEAC 4201 -2007 (2013). Assumed defect types are surface andunder-clad cracks. On crack data including the distribution formula, we followed theguideline and the code manual [3,4,8,9] .
Table 1. Parameters and assumptions for the probabilistic analysis. (a) Model RPV , (b) Assumed crack(a)(b)

Inner radiusof RPV

1669 [mm]

RPV steelthickness

165 [mm]

Cladthickness

6 or 0 [mm]Operatingtemperature288 [℃]

Chemicalcompositionof steel,Standard
deviation

Cu 0.12 [wt.%]
Ni 0.56 [wt.%]

Embrittlement
equation

JEAC 4201-2007（2013 addendum)

Initial crack(deterministic)

Semi-elliptical
10 (length) × 6(depth)[mm]

Initial crack
distribution(probabilistic)

Length: lognormal
distribution ＊Depth: exponential
distribution ＊＊Plasticity
correction forunder-clad
crack

RSE-M
（β-correction)

Cu 0.01 [wt.%]
Ni 0.02 [wt.%]＊𝑝 ｂ ＝＊＊𝑝 𝑎𝐶
𝜎ｂ 2𝜋

173

𝜆 exp

𝑒𝑥𝑝
𝜆𝑎 ,

1 𝑙𝑛 ｂ⁄𝜇𝜎2,

𝑎: 𝑐𝑟𝑎𝑐𝑘 𝑑𝑒𝑝𝑡ℎ𝑏: 𝑎𝑠𝑝𝑒𝑐𝑡 𝑟𝑎𝑡𝑖𝑜

M.Suzuki, et al./ Verification of Safety Margin of Reactor Pressure Vessel Exposed
to Various Thermal Transients based on Probabilistic Approach

Thermal transients used for the calculation were those which correspond to large
break loss of coolant accident (LBLOCA), small break LOCA (SBLOCA), main steamline break (MSLB) and stuck open valve events for pressurizer (SO) as shown inFig.4. In the SBLOCA and MSLB, two different transient conditions (case1 and 2)
were used. As for the SO, thermal transient behavior is very complex, and standard
pattern has not been established. So, in the present analysis, we used the data describedin reference [6]. Five different transients were used for the calculation. It is to be notedthat re-pressurization occurs for this type of transients. (The SO numbers in the figurescorrespond to reference literature [6].)In the structural or fracture analysis, WPS (warm pre-stress) effect, welding
residual stress, plasticity correction, and so on were considered.

Fig.4.

Temperature and pressure change at various transients used in the calculation

2.2 PFM Analysis
The PFM code used to calculate CPI and CPF in this study is mostly PASCAL4.0A, developedby Japan Atomic Energy Agency. The detailed analysis procedure is described in reference [3-5]. As
the essential concept of the code is shown in Fig.5, the failure probability which corresponds to theinterception of probability curves of fracture toughness and stress intensity factor can be calculated
based on the fracture criteria for the aging pressure vessel at the time of the thermal transient. Monte
Carlo method is used for the probability calculation.

174M.Suzuki, et al./ Verification of Safety Margin of Reactor Pressure Vessel Exposed
to Various Thermal Transients based on Probabilistic Approach

Criteria on crack extension orcrack arrest
Fracture toughness
Initiation: KIC
Arrest:KIa

Fig.5.

>=<Stress intensity
factor: KI

The concept of failure probability and relating fracture criteria in the PASCAL Code.

3. Calculation Results and Discussion
The relationship between failure probability in terms of CPI and ΔTm or CPF and ΔTm as adeterministic safety margin is to be discussed with focus on the possible influencing factors in thefollowing sections. Basically, previous study [1,2] as well as this study show a very good relationshipbetween CPI and ΔTm (see Fig.6 (a)), when the cladding is not considered and when the classicaltransients like LBLOCA, SBLOCA and MSLB are considered. This correlation would be a starting or
reference relation for considering the deterministic–probabilistic correlation.
3.1.

The effect of cladding

Fig. 6 compares the calculation results for CPI when the RPV is with and without cladding. WhileCPI has a very good relationship with ΔTm for the RPV without cladding (Fig.6 (a)), the same plotting
exhibits broader correlation when the RPV is with cladding (Fig.6 (b)). In this case, additional stressis produced during PTS events because thermal expansion coefficients are much different between
base metal and stainless overlay cladding. KI at the crack tip is affected by the thermal stress (i.e.
residual stress), which is different depending on the transient types which have different cooling rates.
Hence, this broadening is thought to be caused by the cladding effect.

(a)1.E‐01(b)1.E‐01
1.E‐03

1.E‐05SBLOCA 1
SBLOCA 2
MSLB 1MSLB 2
LBLOCA

1.E‐07
1.E‐091.E‐110Fig.6.

50100
Safety margin (ΔTm) [℃]

CPICPI

1.E‐03

1.E‐05SBLOCA 1
SBLOCA 2
MSLB 1MSLB 2
LBLOCA

1.E‐07
1.E‐091.E‐11150

050100
Safety margin (ΔTm)[℃.]

The relationship between CPI and ΔTm, for RPV (a) without and (b) with stainless overlay
cladding .

175150M.Suzuki, et al./ Verification of Safety Margin of Reactor Pressure Vessel Exposed
to Various Thermal Transients based on Probabilistic Approach

3.2.

On the difference between CPI and CPF

When CPF (conditional probability of failure) is considered, ΔTm – CPF relationship broadened
even more than that of CPI (conditional probability of fracture initiation). LBLOCA plotting is, inparticular, much deviated (see Fig.7). The difference of CPF and CPI is considered to be related with
the increase of crack arrests for LBLOCA. Larger temperature gradients through wall thickness are
created during the LBLOCA as shown in Table 2. Hence, in the high probability, crack is arrested inthe events for LBLOCA as is clear from Fig. 8 which indicates the crack arrest ratio by CPF/CPIvalue.

1.E‐011.21

1.E‐05SBLOCA 1
SBLOCA 2
MSLB 1MSLB 2
LBLOCA

1.E‐07
1.E‐091.E‐110Fig.7.

SBLOCA 1
SBLOCA 2
MSLB 1MSLB 2
LBLOCA

0.8

CPF/CPICPF

1.E‐03

LBLOCA

0.6
0.4
0.2
0050
100
Safety margin (ΔTm) [℃]150
Fig.8.

The relationship between CPF and ΔTm .

24681012fluence [n/cm^2, E>1MeV]

The crack arrest ratio (CPF/CPI).

Table.2. Temperature gap between the inner and outer surface.

Max gap[℃]

3.3

InnerOutersurface [℃] surface [℃]

Time[min]SBLCOA1

113.9

98.7

231.2

32

SBLOCA2

147.6

81.5

229.1

25

MSLB1
MSLB2
LBLOCA

134.2
135.4
196.3

147.6
129.6
83.2

281.8
265
279.4

6
105

Effect of transients with re-pressurization

The effect of transients with re-pressurization is considered very important for integrity evaluation.
As shown in Fig.9, in the SO events, KI curves show multiple maximum values caused by not only
thermal stress due to rapid cooling, but also the state of the pressurizer including re-closing of safety
relief valve (SRV). Therefore, KI curves become complicated, and it is also true for its relation withKIC curve.

176

M.Suzuki, et al./ Verification of Safety Margin of Reactor Pressure Vessel Exposed
to Various Thermal Transients based on Probabilistic Approach

Temperature
Temperature (℃)

KI change in some SO events

Pressure (Mpa)

Pressure

Re-pressurization

Temperature （℃）
Time, minFig.9.

Temperature, pressure and KI changes in SO events [6] .

CPF

Fig.10 shows the ΔTm – CPF relationship for several transients including the SO events. The
correlation curves were roughly diverged to 3 lines. It was found that ΔTm – CPF relationship variedmuch for SO transients depending on the characteristic feature of the individual transients, mostpossibly reflecting the re-pressurization effects. Namely, fracture point on the KI line is mostimportant. When KI curve crosses with the KIC curves at the higher temperature region, it resembles
the normal PTS events. But when it crosses at lower temperature at the time of re-pressurization,correlation line goes to right (or upper).

ΔTm (℃）
Fig.10.

3.4

The relationship between CPF and ΔTm for SO events .

Summarizing of the analysis

Other factors were also investigated whether they affect the relationship. For example, the effectof crack types was investigated, because in the latest Japanese code, the crack used for the PTS
evaluation is changed from the surface crack to under-clad crack [7]. As a result, however, significant

177

M.Suzuki, et al./ Verification of Safety Margin of Reactor Pressure Vessel Exposed
to Various Thermal Transients based on Probabilistic Approach

influence has not been found from the point of the ΔTm – CPF relationship. Hence, most distinct
results could be summarized as the following three.
(1) For the RPV without cladding, CPI has a very good relationship with ΔTm. While, in the case of
the RPV with cladding, the broader correlation is revealed. The broadening is thought to be caused by
the residual stress due to overlay cladding.
(2) For CPF, the relationship even more broadens than that of CPI. LBLOCA plotting is, in particular,much deviated. The deviation is related with the increase of crack arrests for LBLOCA because oflarger temperature gradients through wall thickness.
(3) In the SO events, KI curves show multiple maximum values caused by not only thermal stressdue to rapid cooling, but also full water state of the pressurizer or re-closing of safety relief valve(SRV). As a result, the SO events exhibit broader ΔTm – CPF relationship.

4.

Conclusion

From the intensive set of PFM calculations, it was found that the SO transients which give rise to
the re-pressurization have a possibility to deteriorate the deterministic – probabilistic correlation.
Implication was also obtained that useful safety consideration could be possible by probabilistic safetyassessment, which cannot be obtained by deterministic assessment alone. To improve further thereliability or structural integrity of aged RPV, important factors can be extracted. Even if the safetymargin is the same from the deterministic evaluation, actual reliability is not the same. This difference
is made clear from the present correlation analysis, and hence PFM evaluation could be used for thecontinuous improvement of the safety of aged plants.

Acknowledgement
Thanks are due to JAEA technical staffs for their supports.

References
[1]
[2]
[3]
[4]
[5]
[6]

[7]
[8]
[9]

K. Onizawa, K. Masaki, J. Katsuyama,“Probabilistic Structural Integrity Analysis of Reactor Pressure
Vessels during PTS Events”, Proceedings of the ASME 2012 Pressure Vessels & Piping
R. Okayama, M. Suzuki1, Y. Li, J. Katsuyama, “Structural Integrity Sensitivity Analysis of Reactor
Pressure Vessel Based on Probabilistic Fracture Mechanics During Thermal Transient Condition”,AESJ Domestic Meeting, September 2016 (in Japanese).
J.Katsuyama, K.Osakabe, S.Uno, Y.Li, "Guideline on Probabilistic Fracture Mechanics Analysis forJapanese Reactor Pressure vessels", PVP2017, 16-20(2017)
J.Katsuyama, K.Osakabe, S.Uno, Y.Li, " Guideline on a Structural Integrity Assessment for Reactor
Pressure Vessel Based on Probabilistic Fracture Mechanics (Contract Research)JAEA-Research
2016-022 (2016),
Y. Li, G. Katsumata, K. Masaki, S.Hayashi, Y.Itabashi, M.Nagai, M. Suzuki, Y. Kanto, "Verification of
Probabilistic Fracture Mechanics Code PASCAL", ICONE25-66468, July 2-6, 2017, Shanghai, China
M. EricksonKirk, M. Junge, W. Arcieri, B.R. Bass, R. Beaton, D. Bessette, T.H.J. Chang, T.
Dickson, C.D. Fletcher, A. Kolaczkowski, S. Malik, T. Mintz,
C. Pugh, F. Simonen, N. Siu, D.
Whitehead, P. Williams, R. Woods, S. Yin, “Technical Basis for Revision of the Pressurized
Thermal Shock (PTS) Screening Limit in the PTS rule (10CFR50.61)”, NUREG-1806, U.S. Nuclear
Regulatory Commission, (2006).
Japanese Electric Association Code, JEAC4206 (2016)
K.Masaki, H.Nishikawa, K. Osakabe, K. Onizawa, "User's Manual and Analysis Methodology ofProbabilistic Fracture Mechanics Analysis Code PASCAL3 for Reactr Pressure Vessel"
JAEA-Data/Code 2010-033, March 2011
Japanese Electric Association Guide JEAG4640 (2018)

178