E-Journal of Advanced Maintenance Vol. 11 No. 1 (2019) 8-15
Japan Society of Maintenology

Development of Reliability Assessment Model for Seismic Loading thatconsiders Fatigue Damage
Masayuki KAMAYA1,* and Takao NAKAMURA2
1
2

Institute of Nuclear Safety System, Inc., 64 Sata, Mihama-cho, Fukui 919-1205, Japan
Osaka University, 2-1 Yamadaoka, Suita, Osaka 565-0871, Japan

ABSTRACT
Failure probability of static components is used for assessing plant safety or resilience index of nuclear power
plants. In this study, a failure probability assessment model was developed that considers the effect of material
degradation caused by low-cycle fatigue. Crack initiation due to fatigue damage was assumed and its depth was
determined from the magnitude of fatigue damage. Then, crack growth was predicted for seismic loads
consisting of a main earthquake and aftershocks. Finally, the failure probability was calculated for the seismic
load of various magnitudes. The model was successfully applied to a pipe of the residual heat removal system ofa pressurized water reactor power plant. It was shown that the failure probability was hardly affected by the
degree of fatigue damage. Although the crack depth had little influence on the fracture strength of the cracked
pipe, an increase in the number of aftershocks could increase the probability of leakage.
KEYWORDS

ARTICLE INFORMATION

low-cycle fatigue, crack growth, usage factor (UF), reliability analysis,
static components

Article history:
Received 22 Octorber 2018
Accepted 9 May 20191.

Introduction

Failure probability of static components is used for assessing plant safety or resilience index of
nuclear power plants (NPPs) [1]. For aged NPPs, it is important to consider material degradation in
the calculation of the failure (fracture or leakage) probability. It has been shown that low-cycle fatigue
is the primary degradation mode which can affect the failure probability [2]. In design of NPP
components, the fatigue damage is assumed to be accumulated during plant operation and the degreeof accumulated damage is controlled so as not to exceed a critical value during plant operation.
Namely, the usage factor (UF) is confirmed not to exceed 1. Even if the UF is less than 1, theaccumulated damage may affect the failure probability. Moreover, the UF may exceed 1 due to large
earthquakes which could lead to a severe accident occurrence.
This paper aimed at developing a reliability assessment model to obtain failure probability ofstatic components considering the fatigue damage accumulated during plant operation. The UF wastransformed to crack depth in order to quantify the impact of fatigue damage on failure probability.
The developed model was applied to the residual heat removal system of a pressurized water reactor(PWR) NPP.

2.

Development of Reliability Assessment Model

2.1. Overview of model

Input data to perform the assessment were the UF value of the component of interest, themagnitude of the seismic load and the number of aftershocks. As schematically shown in Fig.
1, first, crack depth distribution was determined from the given UF. The relationship between
the crack depth and UF was derived by a Monte Carlo simulation using the probabilistic
fracture mechanics code p-CRESTA [3] considering variation of initial crack depth, crack*

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

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

8

M. Kamaya and T. Nakamura/ Development of Reliability Assessment Modelfor Seismic Loading Considering Fatigue Damage

MaximumLoad

UF of component
Hazard level– Seismic load
– Number of aftershocks

Failure probability

Crack depth vs. Failure probability
Input

・Magnitude of seismic load
・Variation in flow stress
Crack depth
vs.
Failureprobability
Crack depth

UF0

Crack depth

Crack depth

Crack growth due to earthquake

UF

Crack depth
distribution

Seismic load + no.of aftershocks

1

Crack depth distribution from UF

Failureprobability

Crack depth distribution after earthquake

Fig. 1 Schematic diagram for the reliability assessment model considering degradation of components.

growth rate, material properties and so on. Then, the crack depth distribution was derived foreach UF. The crack for each size grew due to an assumed main quake and aftershocks, thenthe crack depth distribution after the earthquake was obtained. Meanwhile, the relationshipbetween the fracture probability and crack depth (not the crack depth distribution) was
obtained [4]. By integrating the fracture probability for the crack depth distribution, failure(fracture or leakage) probability was obtained.
2.2. Relationship between degree of fatigue damage (UF) and crack depth
2.2.1. Fatigue life prediction by crack growth analyses
Fatigue life can be divided into two parts: initiation life and propagation life. The fatigue tests
using Type 316 stainless steel [5] showed that, if the crack initiation was defined as the emergence ofa crack of several tens of micrometers, the initiation life was relatively short, and the fatigue life could
be estimated by crack growth prediction without considering the incubation period before the crack
initiation. A later study showed that the fatigue crack growth rate (da/dN) of Type 316 stainless steel
was predicted by Eq. (1) [6]:2.85
da  m  3.33 1012 Keq ｍ0.5 dN  cycle 





(1)

where

Keq  f  E  a(2)

and Δε and a are the strain range and crack depth, respectively. f is the geometrical constant for stressintensity factor [7] and E is Young’s modulus. The value of E = 174 GPa for 325C is used in thispaper.
It was assumed that initial crack depth followed a log-normal distribution, of which the mean

9

M. Kamaya and T. Nakamura/ Development of Reliability Assessment Modelfor Seismic Loading Considering Fatigue Damage
2.0

0.06
Δε = 0.8%

Simulation result
Regression

1.5

Probability

Strain range Δε , %

0.05

1.0

0.5

0.0
1.E+02

pCRESTA: ai=0.1 mm+2σ‐2σ
Fatigue test

0.04
0.03
0.02
0.01
0.00

1.E+03

1.E+04

1.E+05

1.E+06

0

50

100

150

200

Nf  103 , cycles

Number of cycles to failure Nf

Fig. 2 Fatigue life predicted by probability
fracture mechanics analysis and fatigue test.

Fig. 3 Fatigue life distribution obtained byprobability fracture mechanics analysis.

value was 0.1 and the coefficient of variation (COV: standard deviation/mean value) was 0.5. The
crack aspect ratio (depth/surface length) was assumed to be 0.5 [8] and the growth rate had a standard
variation of 2.7 in log-normal distribution (10± σ) [9]. The fatigue life, which was defined as thenumber of cycles for the initial crack to grow to 3 mm deep, for a round-bar specimen of 10 mmdiameter was calculated by p-CRESTA. Fig. 2 shows the calculated fatigue life and its variation. The
mean value of the predicted fatigue life corresponded well with that obtained experimentally [10]. The
fatigue life distribution followed the log-normal distribution as shown in Fig. 3.
2.2.2. Considering safety factor to design fatigue curve

The UF was derived using the design fatigue curve, for which the 20 safety margin wasconsidered in the allowable number of cycles from the best-fit regression of fatigue test results [11].
The 20 safety margin included the variation of test data (2), surface roughness (4), size effect(2.5) and load history effect (1) [12]. Then, these four factors were considered in the crack growth
analyses. First, the variation of test data was considered for the variation of initiation size and crack
growth rate as assumed for Fig. 2. The effect of surface roughness was reflected by increasing themean value of the initial crack depth to 0.3 mm, which was determined considering the surface
roughness of 0.2 mm as a machined condition [13]. The COV of the initial depth was set to 0.5. The
size effect can be considered by increasing the risk for crack initiation. A large component has morechance for crack initiation. Since the incubation period was not considered in the crack growth
analysis, the reduction in the incubation period was fully considered. Then, in the assessment model,
the size effect was not considered. Finally, the load-history effect was taken into account byconsidering the change in crack closure [14]. The variable loading alters the crack mouth opening
point. A large compressive load decreases the crack opening point (enhances the crack mouth
opening) and accelerates the crack growth. When the applied strain range  was constant, theeffective strain range eff, which corresponds to the strain range while the crack mouth opened, waspredicted by Eq. (3) [14]: eff   100  200
.
E

(3)

The variable loading alters the eff, although the magnitude of eff depends on various factors and isdifficult to predict. However, the maximum eff is always less than . Then, in this model, the Keq
was calculated using statistical his instead of  using the uniform random number Runiform.

 his    Runiform

100  200
.
E

10

(4)

M. Kamaya and T. Nakamura/ Development of Reliability Assessment Modelfor Seismic Loading Considering Fatigue Damage

0.5UF

0.05

1.5

Probability

Strain range Δε , %0.06

pCRESTA: ai=0.1 mm (test)
pCRESTA: ai=0.3 mm (actual)‐3σ
Fatigue test
DFC

2.0

1.0

Simulation result
Regression

0.04
0.03
0.02
0.01

0.5

0.00
0.0
1.E+020
1.E+03

1.E+04

1.E+051

2

3

Crack depth, mm

1.E+06

Fig. 6 Crack depth distribution for UF = 0.5,Δε = 1.2%.

Number of cycles to failure Nf

Fig. 4 Fatigue life prediction for actual components.
Mean crack depth μc, mm

0.5

Stress-3σ(for UF = 1) Best-fit regression of fatigue life

0.4
0.3

c  0.305exp  0.178UF 

0.2

Δε = 0.4%
Δε = 0.8%Δε = 1.2%
Δε = 1.6%

0.1
0.0
0.0

0.5

1.0

1.5

2.0

UF

(a)Mean value
7

Crack depth

Number of cycles

6

3 mm

COVc

5

COVc  6.319 exp  0.097UF 

43

Δε = 0.4%
Δε = 0.8%Δε = 1.2%
Δε = 1.6%2
10

UF = 0.5 UF = 1

Number of cycles

Fig. 5 Schematic drawing representing the meaningof safety margin in the design fatigue curveand correlation with crack depth.

0.0

0.5

1.0

UF

1.5

2.0

(b) COV
Fig. 7 Change in crack depth distribution
parameters.

Fig. 4 shows the predicted fatigue life considering the above conditions. The variation of − 3from the mean value corresponded well with the design fatigue curve prescribed in the design andconstruction code of the Japan Society of Mechanical Engineers (JSME) [11].
2.2.3. Relationship between UF and crack depth

Since the design fatigue curve (fatigue life corresponds to UF =1) could be predicted as the −3 variation of the predicted fatigue life, by using the crack depth change during the simulation, it ispossible to derive the relationship between UF and crack depth as schematically shown in Fig. 5. The
previous calculations assumed the crack growth was not in actual components but in a round-bar
specimen. Then, the crack growth was simulated for a plate having infinite width by setting thegeometrical constant to be f = 0.896 [15]. Fig. 6 shows the crack depth distribution at UF = 0.5, where
UF = 1 corresponds to the predicted fatigue life obtained by the simulation (the “− 3” line shown in11

M. Kamaya and T. Nakamura/ Development of Reliability Assessment Modelfor Seismic Loading Considering Fatigue Damage
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-12

Before loading
Afer loading:(Pm+Pb)/Sy=1
Afer loading:(Pm+Pb)/Sy=2
Afer loading:(Pm+Pb)/Sy=2.5

1.E-01

Fracture probability

Probability density

1.E+00

1.E-02
1.E-03
1.E-04
1.E-05
0

2

4

6

8

10

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

3

12

4

5

6

7

8

9

10

Crack depth af , mm

Crack depth, mm

Fig. 9 Relationship between failure probability
and crack depth [4].

Fig. 8 Crack depth distribution before and after
seismic loading.

Fig. 5). The crack depth distribution could be approximated by a log-normal distribution. The crack
depth distribution was obtained for each UF and its mean value (denoted as μc) and COV (COVc) are
shown in Figs. 7a and 7b, respectively. μc and COVc exhibited almost the same value for various
applied strain ranges. The value of μc + 3 was about 3.25 mm, which corresponded to the criticaldepth for fatigue life determination. The best-fit regressions of μc and COVc were obtained as:c  0.305exp 0.178UF (5)COVc  6.319exp 0.097UF (6)Table 1 Geometrical and operational conditions
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 (200ºC)
Young’s modulus: E
183 GPa
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

3. Calculation of Failure Probability
3.1. Crack growth due to seismic load

As an example, the residual heat removal (RHR) piping system was assumed to be subjected to aseismic load. The geometrical and operational conditions are summarized in Table 1. The material
properties shown in Table 2 were obtained from the JSME code [16]. As shown in Fig. 1, the analysis
input values were UF and loading conditions. Once the UF was determined, the crack depth
distribution was obtained from Eqs. (5) and (6). Then, the crack growth analysis was performed foreach crack depth of the distribution. Here, the fatigue crack growth in air environment was assumed
using the growth rate prescribed in the JSME fitness-for-service code [17]. ΔKeq instead of the stressintensity factor was used to calculate the growth rate for a given Δε [18]. The elastic-plastic strain
range was obtained using Ke factor [4][11] and fixed f = 0.896 was applied. The magnitude of theseismic load at the component surface was represented by a sum of the membrane and bending
stresses, which is denoted by Pm + Pb. The membrane stress (axial stress) was assumed to be brought
about by the internal pressure of 5 MPa and the bending stress was caused by the seismic loading.
Fig. 8 shows the crack depth distribution after the crack growth analyses. The magnitude of the121.E+00

1.E+00

1.E-02

1.E-02

Failure probability

Failure probability

M. Kamaya and T. Nakamura/ Development of Reliability Assessment Modelfor Seismic Loading Considering Fatigue Damage

1.E-04
1.E-06
1.E-08
1.E-10UF=0.5UF=1
UF=2

1.E-12

no. of aftershocks: 0
no. of aftershocks: 1
no. of aftershocks: 2

1.E-04
1.E-06
1.E-08
1.E-101.E-12

1.E-14

1.E-14

0

1

2

0

3

1

2

3

Applied load (Pm+Pb)/Sy

Applied load (Pm+Pb)/Sy

Fig. 10 Failure probability for various amplitudesof seismic loading.

Fig. 11 Failure probability for various
numbers of aftershocks (UF = 1).

loading was normalized by the yield strength. The number of cycles was assumed to be 60 [19]. The
design criterion was Pm + Pb < 2Sy. The amount of crack growth was not significant even when Pm +Pb = 2Sy. The crack was judged to penetrate the wall thickness when crack depth reached 75% of thewall thickness. The crack penetrated the wall thickness when the applied load was Pm + Pb = 2.5Sy.

3.2. Failure probability
Change in the fracture probability of the cracked pipe with the crack depth has been obtained in
the authors’ previous study as shown in Fig. 9 [4]. The fracture mode assumed in the fracture analysis
was plastic collapse, for which the material parameter required was flow stress Sf. In the analysis, thevariation in flow stress was considered. Since the values of Sy and Su were determined by multiplying
the test results by 0.885 for a conservative design [11], the mean value of 308.5 MPa (=273/0.885)
and COV of 0.1 [4] were assumed for the flow stress variation.
By integrating the crack depth distribution shown in Fig. 8 and fracture probability for each crack
depth, the failure probability (probability of leakage or fracture) for a given crack distribution was
obtained as shown in Fig. 10. The failure probability was about 0.01% even if the design limit
condition was Pm + Pb = 2Sy and the failure probability hardly depended on UF. The fracture strength
due to plastic collapse greatly depends on the ligament length (net-section of uncracked part). Since
the ligament length was insensitive to the crack depth when the crack depth was small compared to
the wall thickness, the fracture probability was almost the same regardless of the crack depth and thefailure probability was almost the same for different UF values. The increase in the failure probability
above Pm + Pb = 1.6Sy of UF = 2 was brought about by the increase in the leakage probability.

4. Discussion
4.1. Effect of earthquake aftershocks

The influence of aftershocks was considered using the crack growth of 60 cycles at the sameloading magnitude. Fig. 11 shows the failure probability for UF = 1. The increase in the number ofaftershocks contributed to the increase in the failure probability. This was due to the increase in theleakage probability. Since the leakage due to the fatigue crack growth had a minor effect on the plantsafety, it could be concluded that the influence of the aftershocks was not significant.
4.2. Meaning of UF =1 in plant design

As shown in Fig. 10, the failure probability was almost the same regardless of UF. Since theactual loading condition considered in the plant design was deduced to be much less than Pm + Pb =13

M. Kamaya and T. Nakamura/ Development of Reliability Assessment Modelfor Seismic Loading Considering Fatigue Damage

2Sy, the failure probability was almost the same even when UF = 2. This implied the current criticalcondition for fatigue design, UF =1, could be relaxed to UF = 2. In other words, the ×20 safetymargin considered in the design fatigue curve could be reduced to ×10.

5. Conclusion
The failure probability assessment model was developed that considers fatigue damage
accumulation. The design fatigue curve was shown to be predicted statistically by the crack growth
analyses considering various factors. Then, the equations to obtain the crack depth distribution for agiven UF were derived. This model was applied to a PHR piping system subjected to seismic loads
consisting of a main earthquake and aftershocks. It was shown that the failure probability was hardly
affected by the degree of fatigue damage. Although the crack depth had little influence on the fracture
strength of the cracked pipe, an increase in the number of aftershocks could increase the probability ofleakage.

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