E-Journal of Advanced Maintenance Vol.8-2 (2016) 32-42
Japan Society of Maintenology

Development of Resilience Evaluation Method for Nuclear Power Plant(Part 2: Development of Failure Probability Assessment Model for Static
Components)
Masayuki KAMAYA1,* and Takao NAKAMURA2
1

Institute of Nuclear Safety System, Inc., 64 Sata, Mihama-cho, Fukui 919-1205, Japan

2

Osaka University, 2-1 Yamadaoka, Suita, Osaka 565-0871, Japan

ABSTRACT
Resilience index is proposed to be applied to evaluate the capability to recover safety performance of systems
under severe accident of nuclear power plant. A failure probability assessment model was developed for the
resilience evaluation of static components in nuclear power plants. This model took into consideration the effectof material degradation caused by low-cycle fatigue. NPP components consist of dynamic and static components,
and this model focused on static components. Crack initiation due to fatigue damage was assumed and its depth
was determined by the magnitude of fatigue damage. Then, crack growth was predicted for seismic loads
consisting of a main quake and aftershocks. Finally, the failure probability was calculated for the seismic load.
The fracture strength of the cracked component was calculated according to the fitness-for-service code, which
took into account scatter in the flow stress. The developed model was successfully applied to a pipe of the residual
heat removal system of a pressurized water reactor power plant. It was shown that the failure probability was
hardly affected by the degree of fatigue damage, and that it was about 8.7  10-4 when the applied stress was
equivalent to the maximum allowable stress for the component design. Although the crack depth had little
influence on the fracture strength of the cracked pipe, an increase in the number of aftershocks could increase theprobability of leakage.
KEYWORDS
failure probability, low-cycle fatigue, seismic load, crack growth,
aftershock, residual heat removal system

1.

ARTICLE INFORMATION
Article history:
Received 3 February 2016
Accepted 24 May 2016

Introduction

In the design of nuclear power plant (NPP) components, it is confirmed that the stress acting on
the components does not exceed the material strength [1][2]. In order to secure further integrity of thecomponents, it is important to consider the structural integrity for a load magnitude greater than thatassumed in the design (hereafter, beyond design (BD) load). The NPP system has its own resilience forrecovering from damage due to BD events. In order to assess the robustness of the system for the BDload, the failure probability of the components must be quantified.
Although the failure strength for the BD load can be estimated in the same way as for thecomponent design, additional considerations are required for operating NPPs. Since various aging
phenomena may reduce the failure strength of NPP component subjected to long-term operation,
reduction in the failure probability due to the aging phenomena should be taken into account. In ourprevious study [3], it was pointed out that low-cycle fatigue is a degradation phenomenon that can affectthe failure probability of static components. Low-cycle fatigue damage may initiate a crack and reduce
the failure strength of components. It should be noted that changes in failure strength due to low-cycle
fatigue need not to be considered when the plant operation conditions are within those considered in thedesign. This is because the degree of accumulated low-cycle fatigue damage is monitored during plant
operation so that the accumulated damage is smaller than that predicted in the design. However, theinfluence of low-cycle fatigue damage must be taken into account for BD conditions.
In this study, an assessment model was developed for calculating the failure probability of static*

Corresponding author, E-mail: kamaya@inss.co.jp

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

M. Kamaya and T. Nakamura/ Title of the Paper

Inputs for the model

Output of the model
Crack depth vs. UF
obtained by P-curve
Failure probability vs. crack depth
obtained by JSME FFS code

Seismic loads

Crack depth after
degradation
0UF

Failure probability

Failure probability

Crack depth

UF

1

0

Crack depth

Crack depth

Crack growth by seismic loads
Final depth after
extension af

Initial depth determined by UF

Number of cycles for seismic loads(main and aftershocks)

Fig. 1 Assessment procedure for calculating the failure probability of components
subjected to beyond design loads.

components considering low-cycle fatigue damage during plant operation. This model is aimed at theapplication to resilience analyses [4] for DB conditions. The basic concept of this model has been shownin the previous study [4] and seismic loads were chosen as the BD conditions. Fatigue damage initiates
a crack and the initiated crack may bring about component failure (fracture or leakage). The first stepin assessing the failure probability was to calculate the depth of the crack caused by the low-cycle
fatigue damage. The degree of fatigue damage is quantified using the usage factor (UF) in the plantdesign and maintenance. The component is considered to be safe when the UF is less than unity.
However, a small crack may be initiated even when the UF is less than unity [5][6]. Therefore, in thismodel, a crack was assumed to exist regardless of the UF and the crack depth was determined using thepostulated fatigue crack growth curve developed in our previous study [7]. Second, crack growth due
to seismic loads was calculated. The equivalent stress intensity factor was used to predict the fatigue
crack growth for large cyclic load caused by the BD loads. Third, the fracture strength of thecomponents was calculated according to the fitness-for-service (FFS) assessment procedure prescribedin the FFS code for nuclear power plant components issued by the Japan Society of MechanicalEngineers (JSME) [8]. By considering the scatter in material strength, the failure probability of thecomponents was obtained. Finally, the model was applied to calculate the failure probability of a pipeof the residual heat removal system, which is a key system for recovering from core damage accidents
in various scenarios [3]. The influence of the material strength distribution and the effect of the cyclicload caused by aftershocks was also discussed.

2.

Failure Probability Assessment Model

2.1. Overview
The failure probability of the components was calculated according to the procedure schematicallyshown in Fig. 1. Inputs to the model were external hazard, represented by the seismic loads, and degreeof fatigue damage accumulated in the component, which was represented by the UF. A crack was
postulated in the assessment. The crack depth was determined from the UF using the postulated crack
growth curve (hereafter referred to as P-curve), which is described in detail in the next section. The
crack was extended by seismic loads, which consisted of a main quake and aftershocks. The fracture
strength of the cracked component was calculated according to the JSME FFS code. Then, the failureM. Kamaya and T. Nakamura/ Title of the Paper2.85
da 3.331012  Keq 
dN

2.0
Experiment (Ref.[18])Estimated fatigue life (Ni+Np)
Ni for 0.1mm crack (Eq. (3))

1.E-05
1.E-06

Strain range Δε , %Crack growth rate da/dN, m/cycle

1.E-04

1.E-07Δε = 1.2%
Δε = 2.0%
σa = 250 MPa(1)σa = 250 MPa(2)σa = 270 MPa
CT specimen

1.E-08
1.E-09
1.E-101.E-11
1

10

100

1.5

1.0

0.5

0.0
1.E+031000

Keq , MPa m1/2

1.E+04

1.E+05

1.E+06

Number of cycles to failure Nf

Fig. 2 Low- and high-cycle fatigue crack
growth rates obtained using Type 316
stainless steel.

Fig.3 Fatigue life estimated by crack growth
analysis together with the fatigue life
obtained by low-cycle fatigue tests.

probability was derived by considering the scatter in material strength in the fracture strength
calculation. An important characteristic of the model is that the failure probability can be calculatedusing an Excel sheet without any macros. Therefore, the model can easily be combined with theresilience evaluation model [9]. The details of each assessment are explained in the following
subsections.
2.2. Postulated crack growth curve (P-curve)
In order to determine the crack depth for a given UF, the P-curve has been developed by the present
authors [10][11]. The UF is determined using the fatigue life obtained by fatigue tests. It has been shownthat small cracks less than tens of micrometers in length are initiated at the early stage of fatigue tests.
Then, for a Type 316 stainless steel at room temperature, the growth rate for small cracks was obtainedas [12]:2.85
da 3.33  1012   K eq  .
dN(1)

The growth rate da/dN is given in m/cycle and ΔKeq in MPa m0.5. ΔKeq is the equivalent stress intensity
factor defined by:

Keq  f  E  a(2)

where Δε is the applied strain range, a is the crack depth and f is the geometrical constant [12]. E isYoung’s modulus and E = 195 GPa was applied for room temperature. The growth rates obtained using
compact tension (CT) specimens under small scale yielding conditions are also shown in Fig. 2. The
measured crack growth rates correlated well with ΔKeq regardless of the loading amplitude and yielding
conditions [13]-[15]. The growth rate predicted by Eq. (1) is applicable to both high- and low-cycle
regimes regardless of the stress and strain amplitude. The fatigue life of stainless steel correlated wellwith the strain range rather than the stress range [16]-[19]. The equivalent stress intensity factor together
with Eq. (1) enables the crack growth for a given strain range to be predicted without considering thestress amplitude.
Crack growth was predicted for a given strain range assuming an initial crack depth of 0.1 mm andan incubation period Ni determined by [10]:M. Kamaya and T. Nakamura/ Title of the Paper

3.5
Thermal loading
Uniform loading

Crack depth, mm

3.0
2.5
2.0
1.5
1.0
0.5
0.0

0.0

0.2

0.4

0.6

0.8

1.0

N/Nf
Fig.4 Change in crack depth with normalized number of cycles obtained foruniform and thermal loading conditions (P-curve).

N i  3948    % 

3.95

.

(3)

The fatigue life corresponds to the sum of Ni and the number of cycles for an initiated 0.1 mm deep
crack growing to 3 mm deep. Figure 3 shows the estimated fatigue life together with the fatigue life
obtained by low-cycle fatigue tests [20]. The fatigue life estimated using Eqs. (1) and (3) agree wellwith that obtained experimentally.
Figure 4 shows the change in crack depth with the number of cycles normalized by the estimatedfatigue life N/Nf obtained for a strain range of Δε = 0.8%. N/Nf defined for the fatigue tests corresponds
to the UF obtained for the actual component. Therefore, it is possible to determine the crack depth fora given UF by replacing N/Nf with UF in Fig. 4. Since the maximum crack depth for the fatigue test isapproximately 3 mm [12], the crack depth does not exceed 3 mm when UF < 1. The curve shown in
Fig. 4 corresponds to the P-curve.
Most of the fatigue damage considered in the component design of nuclear power plants is brought
about by fluid temperature fluctuations caused by changes in operating mode. The cyclic stress andstrain due to the fluid temperature fluctuation on a component surface forms a gradient in the depth
direction [21]. The P-curve considering the stress gradient in the crack growth prediction is also shownin Fig. 4 [10]. The thermal stress was calculated assuming ramp change in fluid temperature on thesurface. The incubation period of a 0.1 mm deep crack was determined by Eq. (3) and growth prediction
was made using Eq. (1). The crack depth obtained for the thermal load is larger than that obtained forthe uniform stress.

2.3. Crack growth by seismic loads
Cyclic loads caused by the seismic loads propagate the crack initiated by fatigue damage. In themodel, crack growth due to seismic loads is estimated using the equivalent stress intensity factor. As
mentioned, the equivalent stress intensity factor is applicable to the crack growth prediction for a large
cyclic load exceeding the small scale yielding conditions [15]. The cyclic loadings are caused not onlyby the main quake but also by aftershocks. The strain range for calculating the equivalent stress intensity
factor is derived from the stress range using the plastic strain correction factor Ke used for the component
design [2]. Crack depth after the growth analysis is denoted by af and is subjected to the failure
assessment. If af exceeds 75% of the component thickness t, the crack is judged to be a leakage failure.

2.4. Failure probability
The failure load of the cracked component is calculated according to the JSME FFS code. StainlessM. Kamaya and T. Nakamura/ Title of the Paper

steel is commonly used for nuclear power plants and the failure mode of the stainless steel assumed in
the JSME FFS code is plastic collapse. The maximum bending load (limit load) of the cracked pipe forplastic collapse failure is calculated by [8]:Pb 

2S f 
afc 
 2sin   sin 
t
Ri 
 

(4)

1
2

(5)

a c
P 
 m 
t Ri
Sf     fwhere c and Ri are the half crack length and the pipe inner radius, respectively. Pm is the axial load and
Sf is the flow stress, for which the average yield and ultimate strength are applied in the JSME FFS code
[8].
The applied load is assumed to be the same as the maximum peak load caused by the seismic load.
If the applied load is larger than the limit load Pb’, the pipe is judged to have failed. In the model, thevariations in flow stress are considered, and then the failure probability is calculated.

4.

Assessment for Residual Heat Removal System

The failure probability assessment model was applied to the residual heat removal (RHR) system.
The RHR system plays an important role in the recovery from reactor core damage accidents [3]. The
detailed assessment conditions are as follows.

4.1. Geometrical, operational and material conditions
The geometrical and operational conditions applied to the assessment are summarized in Table 1.
This data corresponds to a piping system of the RHR system. The material constants used in theassessment were obtained from the JSME codes for material [22] and are shown in Table 2. As
mentioned, the inputs required for deriving the failure probability are the UF and seismic loads. The
maximum peak stress of the seismic load is represented by:Pm  Pb   S y(6)

where Sy is the yield strength value defined in the design code [22] and  is the constant. In the designof Class 1 components of NPPs, the sum of the membrane stress Pm and bending stress Pb caused byseismic load is confirmed to be less than 2Sy [23]. Namely,  = 2 means the maximum stress condition
considered in the component design and  > 2 corresponds to BD conditions.
Table 1 Geometrical and operational conditions for RHR system
Operation temperature
200°C
Internal pressure: Pi
5 MPa
Pipe material
SUS304TB
Outer diameter: Do
267.4 mm (10B)
Thickness: t
15.1 mm (Sch 80)
Table 2 Material strength obtained from JSME material code [22] (200°C)
183 GPa
Young’s modulus: E
Design stress intensity value: Sm
129 MPa
Yield strength value: Sy
144 MPa
Tensile strength value: Su
402 MPa
Flow stress: Sf = (Sy + Su)/2
273 MPaM. Kamaya and T. Nakamura/ Title of the Paper

4.2. Crack growth analysis
The initial depth a of the postulated crack was determined from the given UF using the P-curve
shown in Fig. 4. The surface length of the crack 2c was assumed to be 2c = a. The initial crack was
extended by the seismic loads. To calculate the equivalent stress intensity factor, the strain range
corresponding to the applied load given by Eq. (6) was determined using the plastic strain correction
factor Ke. In the JSME design code [2], the value of Ke is obtained by: SSK e  1  0.7  n  1 when n  1
 2S

Sy
 y


(7)

 2S y SnKe  1  2.11 1
 when
Sn 
Sy


(8)

where Sn is the range of the membrane and bending stress fluctuations and it was assumed to be 2(Sm +Sb). Thus, the strain range Δε is calculated by:

 

Sn
 Ke .
E(9)

The geometrical constant f for the equivalent stress intensity factor was set to 0.82, whichcorresponds to the geometrical constant for a circumferential inner surface crack of a/t = 0.2 [7]. Then,the crack growth rate is calculated by:
3.3da 2.93  1012  Keq  .
dN

(10)

This equation corresponds to the fatigue crack growth rate in an air environment at 200°C given in the
JSME FFS code appendix E-2-10 [8]. The stress ratio R was assumed to be R = −1. The number ofcycles caused by the main quake was assumed to be 60 [24]. Also, the same number of cycles andmagnitude of cyclic loads were applied for a single aftershock. Namely, two aftershocks caused 120

Sy 1.5Sy 2Sy 2.5Sy

1.E+00
1.E-01
1.E-02
1.E-03
1.E-04
1.E-05
1.E-06
1.E-07
1.E-08
1.E-09
1.E-101.E-11
1.E-12Fracture probability

6

Crack depth af , mm

5
4
3
2

No. of aftershocks: 0
No. of aftershocks: 1
No. of aftershocks: 2

1
0
0

100

200

300

400

500

3

4

5

6

7

8

9

Crack depth af , mm

Applied load Pm + Pb, MPa

Fig. 5 Change in crack depth with applied load
(UF = 1).

Pb = 1Sy
Pb = 1.5Sy
Pb = 2Sy
Pb = 2.5Sy

Fig. 6 Fracture probability calculated by the
JSME FFS code.

10M. Kamaya and T. Nakamura/ Title of the Paper

Sy 1.5Sy 2Sy 2.5Sy

1.E+00

1.E+00

1.E-02

1.E-02

Failure probability

Failure probability

Sy 1.5Sy 2Sy 2.5Sy

1.E-04
1.E-06
1.E-08
1.E-10UF = 1
UF = 0.5

1.E-12

1.E-04
1.E-06
1.E-08
1.E-10

COV = 0.2
COV = 0.1
COV = 0.05

1.E-12

1.E-14

1.E-14

0

100

200

300

400

500

0

Applied load Pm + Pb , MPa

Fig. 7 Change in failure probability with applied
load obtained for UF = 1.0 and 0.5(No. of aftershocks = 0).

100

200

300

400

500

Applied load Pm + Pb , MPa

Fig. 8 Change in failure probability with applied
load obtained for various COVs
(UF = 1, No. of aftershocks = 0).

cycles of crack growth in addition to 60 cycles by the main quake. The crack was extended in the depth
direction while the surface length was kept at 2c = a.
Figure 5 shows the change in crack depth after growth by the seismic loads. The UF was assumed
to be UF = 1. Namely, the crack depth before the growth analysis was 3 mm. The crack depth af became
deeper as the applied load was increased. The depth was af = 3.68 mm when the applied load was 2Sy
without considering aftershocks. The depth became af = 6.12 mm when two aftershocks wereconsidered. If the crack depth af exceeded 75% of the pipe wall thickness, which was t = 15.1 mm, thecrack was judged to penetrate the wall thickness. This is an engineering treatment in the structural
integrity assessment of cracked pipes [8].

4.3. Fracture analyses
If the applied bending stress Pb exceeds the limit load Pb’, the pipe is judged to have failed due toplastic collapse. The material parameter determining the limit load is the flow stress Sf. In theassessment, the variation in flow stress was considered. The magnitude of the variation was represented
by COV (coefficient of variation: standard deviation/mean value). The values of Sy and Su were
determined by multiplying the test results by 0.885 for a conservative design [2]. If the variation
between Sy, Su and the test results corresponds to 3σ of the normal distribution, the COV for Sy and Su
is (1 − 0.885)/3 = 0.038. The FFS code of API (American Petroleum Institute) [25] states that theminimum yield strength used for the assessment should be 69 MPa lower than the yield strength
obtained by material tests. If 69 MPa corresponds to 3σ, the COV for flow stress is 69/(3  273) = 0.084.
Then, in this model, the COV and mean value of the flow stress were assumed to be 0.1 and 273/0.885= 308.5, respectively.
Figure 6 shows the change in fracture probability with crack depth. The minimum flow stressrequired for a given crack depth and applied bending stress can be derived using Eqs. (4) and (5). The
membrane stress was assumed to be caused by internal pressure, which was half the hoop stress 0.5Pi
 Rm/t. The fracture probability was calculated as the cumulative probability for Sf. As shown in Fig. 6,
the fracture probability greatly depended on the applied load and hardly depended on the crack depth.
The limit load largely depends on the net section area, which is the uncracked area on the cracked crosssection. The change in net section area due to crack growth was not significant and had little influenceon the change in limit load. The failure probability was about 0.1% regardless of the crack depth when
the bending stress of Pb = 2Sy was applied.

M. Kamaya and T. Nakamura/ Title of the Paper

Sy 1.5Sy 2Sy 2.5Sy
1.E+00

Failure probability

1.E-02
1.E-04
1.E-06
1.E-08
1.E-10No. of aftershocks: 0
No. of aftershocks: 1
No. of aftershocks: 2

1.E-12
1.E-14
0

100

200

300

400

500

Applied load Pm + Pb , MPa

Fig. 9 Change in failure probability with applied load obtained forvarious number of aftershocks (UF = 1).

4.4. Analysis results
The fracture probability was obtained for the crack depth as shown in Fig. 6. On the other hand,the crack depth af was obtained by the crack growth analysis for the seismic loads. Then, by using af
for the fracture probability calculation shown in Fig. 6, the failure probability (probability of leakage or
fracture) was obtained as shown in Fig. 7, for which growth by aftershocks was not considered. The
failure probability greatly depended on the magnitude of applied load and was about 8.7  10–4 whenPm + Pb = 2Sy. The failure probabilities obtained for UF = 1.0 and 0.5 were almost identical. The initial
depth used for UF = 0.5 was 0.158 mm, which was obtained for the uniform loading conditions shownin Fig. 4, whereas it was 3 mm for UF = 1. Although the initial depth of UF = 0.5 was much less thanthat of UF = 1, the failure probability was almost the same. Since the fracture probability hardly
depended on the crack depth as shown in Fig. 6, the failure probability had little dependency on the UF.
In the analyses shown in Fig. 7, the crack depth af did not reach 0.75t.
Figure 8 shows the change in failure probability for different variations in flow stress obtained for
UF = 1. A larger scatter in flow stress resulted in a higher failure probability. This implies that thevariation in material strength should be suppressed in order to reduce the failure probability.
The change in failure probability with a different number of aftershocks is shown in Fig. 9 for UF= 1. The number of aftershocks had little influence on the failure probability when Pm + Pb was lessthan 2Sy. The failure probability for larger applied stress increased abruptly when the number ofaftershocks was one or two. This was due to crack penetration of the wall thickness. The crack depth
after the growth analysis became larger than 0.75t when the number of cycles was large.

5. Discussion
As mentioned, the failure probability calculation by the developed model can be performed on an
Excel sheet without any macros. Figure 10 shows the Excel sheet for calculating the failure probability.
The red characters indicate values to be given by the user. In other words, these parameters can bechanged arbitrarily. The failure probability is derived for each applied load Pm + Pb. Crack growth byseismic load can be calculated using a single equation when the geometrical constant f for calculating
the equivalent stress intensity factor is a fixed value regardless of the crack depth [5]. Then, forsimplicity, the fixed value of f = 0.82 was used in this study.
It should be noted that stress corrosion cracking (SCC) is also a major degradation phenomenon
that occurs in NPP components. Therefore, cracking due to SCC may reduce the fracture strength of theM. Kamaya and T. Nakamura/ Title of the Paper

UF

1

Material Properties
E
183 Gpa
Sy
144 MPaSu
402 MPa
σf
273 MPa

Crack growth analysis
Initial size
3 mm
a/c
1
200 ℃
Temperatureda/dt
1.6E-10D
2.9E-12 -9.803
ｍ
3.3
f
0.82
No. of after quakes
1
No.of cycles
60

10
Pm+Pb
Sn
Sn/2Sy KeΔε(%)
af(mm) c(mm)144
288 1.0000 1.000
0.157
3.01
3.01
288
576 2.0000 2.050
0.645
3.68
3.68
360
720 2.5000 2.260
0.889
5.91
5.91
0.109
3.00
0.120
3.00
0.131
3.00
0.142
3.00
0.153
3.00
0.178
3.01
0.212
3.01
0.245
3.02
0.279
3.04
0.313
3.05
0.347
3.08
0.381
3.10
0.415
3.14
0.449
3.18
0.483
3.23
0.517
3.30
0.550
3.37
0.584
3.46
0.618
3.57
0.652
3.70
0.686
3.86
400 0.720500 4.05
0.754
4.28
0.788
4.56
0.821
4.91
0.855
5.35
0.889
5.91
0.923
6.64
0.957
7.61
0.991
8.97
1.025
10.95

Fracture assessment
SF
1
Flow stress
Mean
COV

308.5 MPa
0.1 30.85

2θ(de a/tβMlimitσf
Probability
2.90 0.199 1.462
1.3
97.5 3.99E-12
3.55 0.243 1.461
1.3
211.9 8.69E-04
5.71 0.391 1.455
1.3
270.8 1.11E-01

3.00
2.90 0.199
3.00
2.90 0.199
3.00
2.90 0.199
3.00
2.90 0.199
3.00
2.90 0.199
Fracture
probability
3.01
2.91 0.199
3.01
2.91 0.200
3.02 1.E+00
2.92 0.200
3.04
2.93 0.201
3.05 1.E-02
2.95 0.202
3.08
2.97 0.204
1.E-043.10
3.00 0.206
3.14
3.03 0.208
3.18 1.E-06
3.07 0.211
3.23
3.12 0.214
1.E-083.30
3.18 0.218
3.37
3.26 0.223
1.E-103.46
3.35 0.229
3.57
1.E-12 3.45 0.237
3.70
3.58 0.245
3.86 1.E-14
3.73 0.256
4.05
0.268
0 3.91 100
4.28
4.14 0.284
4.56
4.41 0.302
4.91
4.75 0.325
5.35
5.17 0.354
5.91
5.71 0.391
6.64
6.41 0.439
7.61
7.35 0.504
8.97
8.67 0.594
10.95
10.58 0.725

Fracture probability

Crack depth af , mm

100
200 0.6944 1.000
110
220 0.7639 1.000120
240 0.8333 1.000
130
260 0.9028 1.000
140
280 0.9722 1.000
Crack depth150
af 300 1.0417 1.084
160
320 1.1111 1.210
6170
340 1.1806 1.321
180
360 1.2500 1.420
5190
380 1.3194 1.508
200
400 1.3889 1.588
4210
420 1.4583 1.660
220
440 1.5278 1.725
230
460 1.5972 1.785
3
240
480 1.6667 1.840
250
500 1.7361 1.890
2260
520 1.8056 1.937
270
540 1.8750 1.980
1
280
560 1.9444 2.020
290
580 2.0139 2.057
0300
600 2.0833 2.092
0 310100 620 200
2.1528 300
2.125
320
640 2.2222 2.155
P+Pm
b 2.184330
660 2.2917
340
680 2.3611 2.211
350
700 2.4306 2.236
360
720 2.5000 2.260
370
740 2.5694 2.283
380
760 2.6389 2.304
390
780 2.7083 2.325
400
800 2.7778 2.344

Pipe geometryDo
267.4 mmt
15.1 mm
8.354
Rm/t
Pi
5 MPa
Pm
20.89 MPa

1.462
1.3
1.462
1.3
1.462
1.3
1.462
1.3
1.462
1.3
1.462
1.3
1.462
1.3
1.462
1.3
1.462
1.3
1.462
1.3
1.462
1.3
1.462
1.3
1.462
1.3
1.462
1.3
1.462
1.3
1.461
1.3
1.461
1.3
1.461
1.3
1.461
1.3
1.461
1.3
1.460
1.3
1.460
1.3
200
300
1.459
1.3
P
+
P
b
1.459m
1.3
1.458
1.3
1.456
1.3
1.455
1.3
1.452
1.2
1.448
1.2
1.442
1.2
1.431
1.2

62.7 8.03E-16
70.6 6.20E-15
78.5 4.49E-14
86.4 3.05E-13
94.3 1.94E-12
102.3 1.16E-11
110.2 6.47E-11
118.1 3.39E-10
126.0 1.67E-09
134.0 7.69E-09
141.9 3.33E-08
149.8 1.35E-07
157.8 5.15E-07
165.7 1.84E-06
173.6 6.18E-06
181.6 1.95E-05
189.5 5.77E-05
197.5 1.61E-04
205.5 4.21E-04
213.5 1.04E-03
221.5 2.40E-03
229.5
400 5.25E-03
500
237.6 1.08E-02
245.8 2.10E-02
254.0 3.86E-02
262.3 6.72E-02
270.8 1.11E-01
279.6 1.75E-01
288.9 2.63E-01
299.1 3.80E-01
311.2 5.35E-01

Fig. 10 Excel file for calculating failure probability.

component. The magnitude of damage induced by SCC can also be represented by the crack depth asin the case for fatigue damage, so it is relatively easy to take the effect of SCC into account in the failure
probability assessment model. Since SCC is not generally initiated in the stainless steel pipes ofpressurized water reactor NPPs, the effect of SCC was not considered in this study. It should be notedthat efforts to prevent SCC are taken in the component design, whereas accumulation of fatigue damage
is assumed. Therefore, if SCC is successfully prevented, it need not be considered in the failure
probability assessment model.

6. Conclusion
In order to calculate the failure probability for the resilience evaluation of NPP static components,
failure probability assessment model was developed. This model could consider material degradation
due to low-cycle fatigue damage accumulated during plant operation. The magnitude of the fatigue
damage was quantified by the crack depth using the P-curve together with the UF. After the analysis of
growth due to the seismic loads, the crack was subjected to fracture strength assessment according to
the JSME FFS code. Then, the failure probability of the component was quantified. This model was
successfully applied to the RHR system, which is the key system in various scenarios for recoveringfrom core damage accidents. The failure probability was about 8.7  10–4 when Pm + Pb = 2Sy, whichcorresponded to the maximum stress considered in the design. A larger scatter in the flow stress resulted
in a higher failure probability. It was revealed that the failure probability was hardly affected by themagnitude of UF. Although the initial crack depth and crack growth due to seismic loads had little
influence on the failure probability, the increase in the number of aftershocks increased the failure
probability because the crack penetrated the wall thickness during the aftershocks.

M. Kamaya and T. Nakamura/ Title of the Paper

Acknowledgement
This work has been performed as a part of the Japan Ageing Management Program on System
Safety (JAMPSS) sponsored by the Nuclear Regulation Authority (NRA).

References[1] ASME, “Rules for Construction of Nuclear Facility Components – ASME Boiler and Pressure Vessel Code
Section III”, ASME, New York (2013).
[2] Japan Society of Mechanical Engineers, “Codes for Nuclear Power Generation Facilities: Rules on Design andConstruction for Nuclear Power Plants”, JSME S NC1-2012 (2012).
[3] T. Nakamura, M. Kamaya, “Development of resilience evaluation method for nuclear power plant (part 2:
study on reliability analytical method for degradation of static components)”, Maintenology, Vol. 15, pp.7176 (2016).
[4] K. Demachi, M. Suzuki, T. Itoi, K. Murakami, N. Kasahara, H. Miyano, T. Nakamura, S. Arai, M. Kamaya,
A. Yamaguchi, M. Matsumoto, “Development of resilience evaluation method for nuclear power plant (part1: proposal of resilience index for assessment of safety of nuclear power plant under severe accident)”,
Maintenology, Vol. 15, pp.65-70 (2016).
[5] M. Kamaya, M. Kawakubo, “Strain-based modeling of fatigue crack growth – An experimental approach forstainless steel”, International Journal of Fatigue, Vol. 44 pp.131-140 (2012).
[6] S. Abe, T. Nakamura, M. Kamaya, “Statistical model of micro crack growth for the evaluation of accumulated
fatigue in NPPs”, E-Journal of Advanced Maintenance, Vol.7-1 pp.129-137 (2015).
[7] M. Kamaya, T. Nakamura, “A flaw tolerance concept for plant maintenance using virtual fatigue crack growth
curve”, Proceedings of the ASME 2013 Pressure Vessels and Piping Conference, Paris, France, July, paper
no. 97851 (2013).
[8] Japan Society of Mechanical Engineers, “Codes for Nuclear Power Generation Facilities: Rules of Fitnessfor-Service for Nuclear Power Plants”, JSME S NA1-2012 (2012).
[9] M. Suzuki, K. Demachi, K. Murakami, T. Itoi, N. Kasahara, H. Miyano, T. Nakamura, S. Arai, M. Kamaya,
A. Yamaguchi, M. Matsumoto, “Development of resilience evaluation method for nuclear power plant (part3: study on evaluation method and applicability of resilience index)”, Maintenology, in press.
[10] M. Kamaya, T. Nakamura, “Fatigue damage management based on postulated crack growth curve”, E-Journalof Advanced Maintenance, Vol.7-1 pp.43-49 (2015).
[11] M. Kamaya, “Damage assessment of low-cycle fatigue by crack growth prediction (fatigue life under cyclic
thermal stress)”, Transactions of the Japan Society of Mechanical Engineers A, Vol. 79 pp.1530-1544 (2013).
[12] M. Kamaya, M. Kawakubo, “Damage assessment of low-cycle fatigue by crack growth prediction(development of growth prediction model and its application)”, Transactions of the Japan Society of
Mechanical Engineers Ser. A, Vol. 78 pp.1518-1533 (2012).
[13] J. R. Haigh, R. P. Skelton, “A strain intensity approach to high temperature fatigue crack growth and failure”,
Materials Science and Engineering, Vol. 36, pp.133-137 (1978).
[14] H. Kitagawa, S. Takahashi, C.M. Suh, S. Miyashita, “Quantitative analysis of fatigue process – microcracks
and slip lines under cyclic strains”, ASTM STP 675, pp.420-449 (1979).
[15] M. Kamaya, “Low-cycle fatigue crack growth prediction by strain intensity factor”, International Journal of
Fatigue, Vol. 72 pp.80-89 (2015).
[16] C. E. Jaske, W. J. O’Donnell, “Fatigue design criteria for pressure vessel alloys”, ASME Journal of Pressure
Vessel Technology, Vol. 99, pp.584-592 (1977).
[17] J. Colin, A. Fatemi, “Variable amplitude cyclic deformation and fatigue behavior of stainless steel 304L
including step, periodic, and random loading”, Fatigue & Fracture of Engineering Materials & Structures,Vol. 33, pp.205-220 (2010).
[18] M. Kamaya, M. Kawakubo, “Mean stress effect on fatigue strength of stainless steel”, International Journalof Fatigue, Vol. 74 pp.20-29 (2015).
[19] M. Kamaya, M. Kawakubo, “Loading sequence effect on fatigue life of Type 316 stainless steel”,
International Journal of Fatigue, Vol. 81 pp.10-20 (2015).
[20] M. Kawakubo, M. Kamaya, “Fatigue life prediction of stainless steel under variable loading (damage factors
determining fatigue life and damage evaluation for two-step test)”, Journal of the Society of Materials
Science, Japan, Vol. 60, pp.871-878 (2011).
[21] M. Kamaya, “Assessment of thermal fatigue damage caused by local fluid temperature fluctuation (Part I:
Characteristics of constraint and stress caused by thermal striation and stratification)”, Nuclear Engineeringand Design, Vol. 268, pp.121-138 (2014).
[22] Japan Society of Mechanical Engineers, “Codes for Nuclear Power Generation Facilities: Rules on Materials

41

M. Kamaya and T. Nakamura/ Title of the Paper
for Nuclear Facilities”, JSME S NJ1-2012 (2012).
[23] Electric Association Nuclear Standards Board, “Seismic Design Technology Guidelines for Nuclear Power
Station”, JEAC 4601-2008 (2008), Japan Electric Association.
[24] Japan Nuclear Energy Safety, “Report of integrity of nuclear power plant components for seismic load:
ultimate strength of piping system (in Japanese)”, 04-KiKouHou-0002, p.4-8 (2004).
[25] American Petroleum Institute, “Fitness-for-Service, Second Edition API 579-1/ASME FFS-1”, American
Petroleum Institute (2007).

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