International Symposium on Improvement of Nuclear Safety Using Probabilistic Fracture Mechanics
Yinsheng Li Nuclear Safety Research Center
Japan Atomic Energy Agency (JAEA) October 24, 2014
PFM Applications to Seismic Safety Evaluation - Seismic Fragility Evaluation Using PFM -
Contents 1
1. Background and Objective
2. Application to Seismic Safety Evaluation for Aged
Piping Using PFM
- Analysis code
- Benchmark analysis
- Application to seismic fragility evaluation
3. Evaluation Methodology of Crack Growth and
Fragility for Piping Subjected to Severe Earthquake
4. Summary 【本資料は公開文献に基づくものである】
Background 2
On March 11, 2011, Fukushima Dai-ichi Nuclear Power Station (NPS) experienced an extremely massive earthquake (the Tohoku District - off the Pacific Ocean Earthquake) and a severe accident followed at an unprecedented scale and over a lengthy period.
Many lessons have been learned from the Fukushima NPS accident. One of them is “Effective use of probabilistic safety assessment (PSA) in risk management” (Report of Japanese Government to the IAEA Ministerial Conference on Nuclear Safety, June, 2011)
On the other hand, about one-third of the nuclear power plants (NPPs) in Japan have been operating for more than 30 years, and cracks due to age-related degradation mechanisms have been detected within some components including pipes.
Therefore, seismic PRA and seismic safety evaluation considering aging mechanisms for aged components have become increasingly important.
PFM has been recognized as a rational methodology for risk assessment of aged components, because it can evaluate the failure probabilities considering the age-related degradation mechanisms and the influence parameters with their inherent probabilistic distributions.
In order to conduct seismic PRA considering aging mechanisms for aged components, a PFM analysis code for piping has been improved, considering typical aging mechanisms of pipes and fracture mechanics analysis models provided in Japan.
Based on the analysis results of PFM code, failure probabilities, fragility curves of aged pipes were investigated. These data of failure probabilities and fragility curves are useful for seismic safety evaluation for aged components.
Objective 3
Applications to Seismic Safety Evaluation
Application to safety advancement evaluation 「実用発電用原子炉の安全性向上評価に関する運用ガイド」では経年事象や設計上の想定を超える事象を考慮したリスク評価や安全裕度の評価を求めている。 Application to seismic PRA evaluation 改訂中の原子力学会標準 地震PRA実施基準では経年事象を考慮したフラジリティ評価及びリスク評価を明記している。 Application to risk evaluation and RI-ISI 地震荷重の影響、検査の効果を考慮したリスク評価。リスク情報の活用、RI-ISIの評価。 Application to seismic safety margin evaluation 偶然的不確実さ及び認識論的不確実さを考慮した信頼度評価及び裕度評価。 Application to evaluation of combination effect of main-shock and after-shock,
etc. 本震と余震の重畳を考慮した経年配管のフラジリティ評価
4
Applications to Seismic Safety Evaluation Building & Component
Fragility Evaluation Accident Sequence Evaluation Seismic Hazard Evaluation
Seismic ground motion strength
Occ
urre
nce
frequ
ency
Seismic hazard curve
Cor
e da
mag
e fre
quen
cy
Core damage probability
Core damage frequency
Accident sequence occurrence frequency
Seismic hazard curve
Component A Fragility curve
Seismic ground motion strength
Component B Fa
ilure
pr
obab
ility
Process of seismic PRA and failure probability considering age-related degradation
Aged component such as piping
Fragility curve
Failu
re
prob
abili
ty
Prog
ress
of
agi
ng
Seismic ground motion strength
Applications of PFM for aged piping
5
Seismic ground motion strength
Example analysis flow of PFM code for piping
In order to conduct seismic PRA or seismic safety evaluation for existing NPPs, a PFM analysis code for aged piping has been developed to evaluate failure probabilities and fragility curves of cracked pipes considering aging mechanisms and seismic loads.
To evaluate domestic aged piping at NPPs, this code has been improved based on the fracture mechanics analysis models and experimental data provided in Japan,
such as: • Crack initiation and distribution due to
SCC and PWSCC • Crack growth rates of SCC, PWSCC and
fatigue • Solutions of stress intensity factor • Detection probability of in-service
inspection • Failure evaluation of cracked pipe
PFM Analysis Code for Piping
6
Start
Select a sample weld joint of pipe
Sampling cracks based on the crack size distribution
Determine the crack detection probability in pre-service inspection
Calculate the crack growth considering SCC, PWSCC or/and
fatigue
Determine the crack detection probability in in-service inspection
Evaluate the failure of cracked pipe
Evaluate the leak detection
Calculate the failure and leak probabilities of the pipe
End
Nex
t tim
e st
ep
Nex
t sam
ple
(M
onte
Car
lo si
mul
atio
n)
Circumferential semi-elliptical surface cracks in weld joints
x
y
a1
R
t
Inner surface
c1
Outer surface
Crack examples: Circumferential semi-elliptical surface cracks occurrence in weld joints.
SCC in a weld joint
Base metal
Inner surface of pipe
Weld metal
SCC
Outer surface of pipe
Aging mechanisms: Cracks caused by SCC or initial cracks; Crack growth due to SCC and fatigue.
PFM Analysis Code for Piping
7
Analysis function for BWR piping
Crack initiation or distribution: Models based on Japanese measurement data.
PLR line
：Weld joints where crack has been detected
：Weld joints
Examples of SCC detected in BWR plants
Probabilistic models of crack growth rates for both SCC and fatigue: Models based on the data provided in Japanese Fitness-For-Service Code (JSME NA1-2012)
Crack growth model
溶接金属 Base metal
Inner surface of pipe
Weld metal
SCC
Outer surface of pipe Given in JSME NA1-2012
Cra
ck g
row
th ra
te o
f SC
C (m
/s)
mCKdtda
=
K (MPa√m)
Cra
ck g
row
th ra
te o
f SC
C (m
/s)
Given in JSME NA1-2012
K (MPa√m)
mCKdtda
=
Crack growth rate (CGR) for sensitized type 304 stainless steel
1.0E-12
1.0E-11
1.0E-10
1.0E-09
1.0E-08
10 100
応力拡大係数 [MPa √m]
SC
Cき
裂進
展速
度 [
m/se
c]
実験値平均値±1σ±2σ維持規格
K (MPa√m)
Cra
ck g
row
th ra
te o
f SC
C (m
/s)
10-12
10-11
10-10
10-9
10-8
C: Log-normal distribution Mean ln(C) ：-30.02 S. D. of ln(C) ：0.3091
1σ 2σ
1.0E-13
1.0E-12
1.0E-11
1.0E-10
1.0E-09
1.0E-08
10 100
応力拡大係数 [MPa √m]
SC
Cき
裂進
展速
度 [
m/s
ec]
実験値平均値±1σ±2σ維持規格
Cra
ck g
row
th ra
te o
f SC
C (m
/s)
10-12
10-11
10-13
C: Log-normal distribution Mean ln(C) ：-32.22 S. D. of ln(C) ：11.80
10-10
10-9
10-8
K (MPa√m)
1σ 2σ
CGR for low carbon austenitic stainless steel
PFM Analysis Code for Piping
8
Analysis function for BWR piping
Ni-based alloy weld joint
x
y
R
t
2θ
Nozzle (Low alloy)
Safe end (Stainless steel)
Weld metal (Ni-based alloy)
W
t Depth direction
Width direction 2 With direction 1
Inner surface
Outer surface
PFM Analysis Code for Piping
9
Analysis function for PWR piping
Examples of PWSCC detected in PWR plants
Crack examples: Circumferential or axial semi-elliptical surface cracks occurrence in weld joints.
Aging mechanisms: Cracks caused by PWSCC or initial cracks; Crack growth due to PWSCC and fatigue.
CGR models: CGRs for different materials.
SG nozzle
WRS: WRS in dissimilar materials.
Circumferential semi-elliptical surface cracks in weld joints
Axial semi-elliptical surface cracks in weld joints
PWSCC
Inner surface
Outer surface
PWSCC
Weld metal (Ni-based alloy)
( )βα thref
g KKTTR
Qdtda
−
−−=
11exp ( ) ( )
∆<∆∆≥∆−∆
=th
thrcN
KKKKRKtTc
dNda
01/ 34.125.324.077.0
R:stress ratio, tr: loading increasing time, Tc:temperature, cN ：CGR coefficient
PFM Analysis Code for Piping
10
Analysis function for PWR piping CGR for PWSCC : Models based on the data provided in JSME
CGR for fatigue: Models based on the data provided in JSME
Q: thermal activation energy, R: universal gas constant T: temperature, Tref: reference temperature α: CGR coefficient, β: CGR exponent
Stress intensity factor K (MPa√m)
Cra
ck g
row
th ra
te d
a/dt
(m/s
)
Mean curve
2σ curve
Exp.
fatig
ue C
GR
(m/c
ycle
)
Predicted fatigue CGR (m/cycle)
Mean curve
2σ curve
PWSCC
Outlet nozzle
SIF solutions for semi-elliptical surface cracks with large aspect ratio in plates and cylinders
• a/ = 0.5, 1.0, 2.0, 4.0 • a/t = 0.0, 0.1, 0.2, 0.4, 0.6, 0.8 • t/Ri = 0 (plate), 1/80, 1/40, 1/20, 1/10, 1/5, 1/2 (cylinder)
PWSCC
PFM Analysis Code for Piping Example PWSCC with large aspect ratio detected in NPPs
11
• Cracks in both circumferential and axial directions
PFM Analysis Code for Piping
0
10
20
30
40
50
60
70
0.0 0.2 0.4 0.6 0.8 1.0Crack depth, a/t (-)
Solu
tion
of K
, (M
Pa √
m)
31%
き裂深さ a/t, （-）
応力
拡大係数の解
K, (
MPa
√m
)
本提案手法による解
FEAによる解従来法による解
-300
-200
-100
0
100
200
300
400
500
600
0.0 0.2 0.4 0.6 0.8 1.0Distance from inner surface, x /t (-)
Res
idua
l stre
ss,
σres
(MPa
)
1 区間2 3 区間4 区間5
溶接残留応力
4次多項式による補間
応力分布
（M
Pa）
表面までの距離 x/t, （-）
区間別4次多項式による補間
・Semi-elliptical surface crack ・a/： 1/8
Curve fitting of stress distribution by segment division SIF solution using proposed method
SIF calculation method for complicated stress distributions
∑∑= =
=
m
j
n
i
i
ijijI
j
Qa
taAFK
1 0
π
12
a: crack depth, Q : flaw shape parameter m: number of divided segments nj: order of polynomial at j-th segment Aij : coefficients of stress polynomial distribution at segment j Fij : coefficients obtained from weight function
Stre
ss d
istri
butio
n, (M
Pa)
Distance from inner surface x/t, (-)
Solu
tion
of S
IF, (
MPa√a
)
Crack depth a/t, (-)
Present method
Conventional method
FEA solution
y
βn
R
t
x'
y' γj
ξ
x
∑
∑∑
=
==−
= n
iiii
n
jjjj
n
iiii
a
aaji
1
11
sincos
sinsinsinsin
tanθγ
θγθγ
ξ
−= ∑
=
n
iii
in
fcbn t
a
1sincossin2
2θγβ
πσ
σ
−−= ∑
= f
mn
ii
in t
aσσ
πθπβ12
1
The failure bending stress :
The neutral angle :
The failure evaluation method for a pipe containing multiple cracks in the same cross section of pipe [1]:
The direction of coordinates that provides the minimum failure strength:
[1] Y. Li. et al, PVP2009-77061
ni: number of cracks on the right side of y-axis nj: number of cracks on the left side of y-axis n: total number of cracks
corresponding to (x', y') coordinates
Pipe section with multiple cracks
PFM Analysis Code for Piping
13
PFM Analysis Code for Piping
14
• Consideration of equivalent cyclic stress • Consideration of uncertainty of seismic
response stress • Consideration of seismic waves • Consideration of response stress from
severe earthquake
ln(Stress) -250
-200
-150
-100
-50
0
50
100
150
200
32 33 34 35 36 37 38 39 40 41 42
時間 [sec]
曲げ
応力
[M
Pa]
レンジ計数法による応力レンジの計数
Analysis function for seismic response stress
Realistic response
Prob
abili
stic
den
sity
Failure Probability
Ultimate capacity
Uncertainty of seismic response stress Consideration of seismic response waves
Cra
ck d
epth
Crack length
time 1 time 2
Crack growth due to age-related degradation
Crack growth due to postulated earthquake
Crack growth due to postulated earthquake
Benchmark Analysis Results 15
1.0E-10
1.0E-08
1.0E-06
1.0E-04
1.0E-02
1.0E+00
0 5 10 15 20Operation time (year)
Cum
ulat
ive
failu
re p
roba
bilit
y
10-10
10-8
10-6
10-4
10-2
100
Cum
ulat
ive
failu
re p
roba
bilit
y (-
)
1100 gal
750 gal
450 gal
Without earthquake
PASCAL-SPPRAISE-JNES
1100 gal
750 gal
450 gal
Without earthquake
PASCAL-SPPRAISE-JNES
1.0E-10
1.0E-08
1.0E-06
1.0E-04
1.0E-02
1.0E+00
0 5 10 15 20Operation time (year)
Cum
ulat
ive
failu
re p
roba
bilit
y
10-10
10-8
10-6
10-4
10-2
100
Cum
ulat
ive
failu
re p
roba
bilit
y ( -
)
1100 gal
750 gal
450 gal
Without earthquake
PASCAL-SPPRAISE-JNES
1100 gal
750 gal
450 gal
Without earthquake
PASCAL-SPPRAISE-JNES
Example results of failure probability
• Objective：confirm the reliability of PFM codes developed by different organizations • PFM analysis codes：PASCAL-SP (Developed by JAEA), PRAISE-JNES (Improved based
on pc-PRAISE by JNES) • Pipe：PLR pipe in BWR plant • Crack: SCC; Circumferential inner surface semi-elliptical crack in weld joints • Crack growth: crack growth due to SCC and fatigue • Failure probability: with and without seismic loads
Results considering crack initiated by SCC
300A pipe 400A pipe
Cum
ulat
ive
cond
ition
al b
reak
pro
babi
lity
Cum
ulat
ive
cond
ition
al b
reak
pro
babi
lity
Operation year after inspection (Year) Operation year after inspection (Year)
1.0E-08
1.0E-06
1.0E-04
1.0E-02
1.0E+00
0 5 10 15 20Operation time (year)
Cum
ulat
ive
failu
re p
roba
bilit
y (-
)
10-8
10-6
10-4
10-2
100
Cum
ulat
ive
failu
re p
roba
bilit
y (-
)
750 gal
450 gal
Without earthquake
PASCAL-SPPRAISE-JNES
750 gal
450 gal
Without earthquake
PASCAL-SPPRAISE-JNES
1.0E-08
1.0E-06
1.0E-04
1.0E-02
1.0E+00
0 5 10 15 20Operetion time (year)
Cum
ulat
ive
failu
re p
roba
bilit
y
10-8
10-6
10-4
10-2
100
Cum
ulat
ive
failu
re p
roba
bilit
y (-
)
750 gal
450 gal
Without earthquake
PASCAL-SPPRAISE-JNES
750 gal
450 gal
Without earthquake
PASCAL-SPPRAISE-JNES
16 Benchmark Analysis Results
Example results of failure probability
• PFM analysis codes：PASCAL-SP (JAEA), PRAISE-JNES (JNES) • Pipe：stainless pipe in BWR plant • Crack: initial crack in weld joints caused by welding or etc. • Crack growth: crack growth due to fatigue • Failure probability: with and without seismic stresses
Results considering crack growth due to fatigue
300A pipe 400A pipe
Cum
ulat
ive
cond
ition
al b
reak
pro
babi
lity
Cum
ulat
ive
cond
ition
al b
reak
pro
babi
lity
Example Analysis Results for Seismic Safety
1.0E-05
1.0E-04
1.0E-03
1.0E-02
1.0E-01
1.0E+00
0 5 10 15 20
Cum
ulat
ive
(con
ditio
nal)
failu
re p
roba
bilit
y
Operation time [year]Operation time (year)
Cum
ulat
ive
failu
re p
roba
bilit
y
: PRAISE-JNES
: PASCAL-SPW/o earthquake
Seismic acc.:450 gal
750 gal
1150 gal
In-service Inspection
• Analysis model of detection probability: Proposed by Khaleel [1].
• Parameters: Corresponding to the “outstanding level” of the detection performance and crack due to SCC.
[1] M. A. Khaleel et al, ASME PVP 1995. Operation time(year)
Cum
ulat
ive
cond
ition
al b
reak
pro
babi
lity
1100 gal
10-5
10-4
10-3
10-2
10-1
100
w/o earthquake
• PFM analysis codes：PASCAL-SP (JAEA), PRAISE-JNES (JNES) • Pipe：PLR pipe in BWR plant ・Cracks: SCCs • Crack growth: crack growth due to SCC and fatigue • Failure probability: with and without seismic stresses
Results considering seismic stress and in-service inspection
Operation year after inspection (Year)
17
• Analysis for two initial cracks • Located at 0 deg. and 60 deg.
x
y
a1
c2
R t
18
1.0E-05
1.0E-04
1.0E-03
1.0E-02
1.0E-01
1.0E+00
0 5 10 15 20
Cum
ulat
ive
(con
ditio
nal)
failu
re p
roba
bilit
y
Operation time [year]Operation time (year)
Cum
ulat
ive
failu
re p
roba
bilit
y
: PRAISE-JNES
: PASCAL-SP
W/o earthquake
Seismic acc. : 1150 gal
750 gal
450 gal
• Analysis for two initiated cracks
• Initiated at 0 deg. and 60 deg.
x
y
a1
c1
R t
Cum
ulat
ive
cond
ition
al b
reak
pro
babi
lity
1100 gal 10-5
10-4
10-3
10-2
10-1
100
w/o earthquake
Benchmark Analysis Results
• PFM analysis codes：PASCAL-SP (JAEA), PRAISE-JNES (JNES) • Pipe：PLR pipe in BWR plant • Cracks: SCCs; Multiple cracks; Circumferential inner surface semi-elliptical crack in weld joints • Crack growth: crack growth due to SCC and fatigue • Failure probability: with and without seismic stresses
Results considering crack initiation and growth due to SCC
Operation year (Year)
0.00
0.01
0.02
0.03
0.04
0 100 200 300 400
応力（MPa)
確率
密度
0 year
4th year
Progress of aging
2nd year
Stress (MPa)
Prob
abili
stic
den
sity
Realistic response
Ultimate capacity
300A, ばらつきあり（β=0.15)
1.0E-12
1.0E-10
1.0E-08
1.0E-06
1.0E-04
1.0E-02
1.0E+00
0 1 2 3 4 5
検査後運転年数 (年）
破損
確率
地震無し
Ratio=1.5
Ratio=3.0
Ratio=4.0
Ratio=4.5
Ratio=5.0
Ratio=5.5
Ratio=6.0
Operation year after inspection (Year)
10-12
10-6
10-4
10-2
10-0
10-10
10-8
Cum
ulat
ive
failu
re p
roba
bilit
y
300A pipe, β=0.15
Failure probability considering seismic stress and probabilistic distribution of ultimate capacity
w/o seismic stress
19
Failure probabilities and probabilistic distribution of ultimate capacity
Example Analysis Results of Seismic Fragility
• Through PFM analyses, the failure probabilities considering the effects of age-related degradation and seismic stresses can be obtained.
• Corresponding to failure probabilities obtained from PFM, the probabilistic distribution of the ultimate capacity and its decrease due to progress of aging degradation can be evaluated.
Cum
ulat
ive
cond
ition
al fa
ilure
pro
babi
lity
300A, ばらつきあり（β=0.15)
1.0E-12
1.0E-10
1.0E-08
1.0E-06
1.0E-04
1.0E-02
1.0E+00
0 1 2 3 4 5
検査後運転年数 (年）
破損
確率
地震無し
Ratio=1.5
Ratio=3.0
Ratio=4.0
Ratio=4.5
Ratio=5.0
Ratio=5.5
Ratio=6.00
0.01
0.02
0.03
0.04
0.05
0 1 2 3 4 5 6 7
地震動比率
破損
確率
1year (β=0.15)
3year (β=0.15)
5year (β=0.15)
• Through PFM analyses, the failure probabilities considering the effects of age-related degradation and seismic stresses can be obtained.
• Based on the failure probabilities obtained from PFM, the fragility curves which are useful in risk evaluation can be obtained for different operation years.
• Comparing the general fragility curve, the fragility curve considering age-related degradation is a function of operation year. The fragility curve goes up with the increasing operation year, due to the progress of the aging mechanisms.
Operation year after inspection (Year)
10-12
10-6
10-4
10-2
10-0
10-10
10-8
Cum
ulat
ive
failu
re p
roba
bilit
y
300A pipe; β=0.15 Incr
easi
ng o
f ope
ratio
n ye
ars
Ratio of seismic motion
Cum
ulat
ive
failu
re p
roba
bilit
y
300A pipe; β=0.15
Failure probabilities and fragility curves considering age related degradation
w/o seismic stress
20 Example Analysis Results for Seismic Fragility Seismic fragility curves considering age-related degradation
Cum
ulat
ive
cond
ition
al fa
ilure
pro
babi
lity
Cum
ulat
ive
cond
ition
al fa
ilure
pro
babi
lity
0
0.01
0.02
0.03
0.04
0.05
0 1 2 3 4 5 6 7
地震動比率
破損
確率
1year (β=0.15)
3year (β=0.15)
5year (β=0.15)
Failure probability basis
50
60
70
80
90
100
0 1 2 3 4 5
検査後の運転年数 （年）
耐震
裕度
の変
化率
(%
)
Incr
easi
ng o
f ope
ratio
n ye
ars
Ratio of seismic motion
Cum
ulat
ive
failu
re p
roba
bilit
y
Operation year after inspection (Year)
Red
uctio
n of
pro
babi
lity-
base
d SM
(%)
• Based on the fragility curves and the assumed failure probability basis, the probability-based seismic margin and its reduction rate considering the age-related degradation can be obtained.
• Probability-based seismic margin decreases with the increasing of operation years. The reduction rate of seismic margin depends on the progress rate of age-related degradation, such as crack growth rate, the distribution of residual stress and so on.
Mcp
Tentatively assumed to be 1%
300A pipe
300A pipe
21 Example Analysis Results for Seismic Fragility Seismic fragility curves and seismic safety margin
Failure probabilities and fragility curves considering age related degradation
Cum
ulat
ive
cond
ition
al fa
ilure
pro
babi
lity
22
Seismic fragility curves considering the effect of after shock
Example Analysis Results for Seismic Fragility
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0 5 10 15 20
条件
付漏
洩確
率
経過時間 [年]
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0 0.5 1 1.5 2 2.5 3
条件
付漏
洩確
率
基準地震動レベル比率
Cum
ulat
ive
cond
ition
al fa
ilure
pro
babi
lity
Cum
ulat
ive
cond
ition
al fa
ilure
pro
babi
lity
Operation year after inspection (Year) Ratio of seismic motion
Main-shock
Main-shock + after-shock
w/o seismic stress
Ratio of seismic motion: 1.5
Ratio of seismic motion: 1.0
Main-shock
Main-shock + after-shock
5yr
0yr
• Because PFM can consider the age-related degradation in mechanism, it can evaluate the combination effect of main-shock and after-shock.
• In the following case, the after-shock occurred soon after the main-shock with a same magnitude is evaluated.
Failure probability and seismic fragility curve with the effect of after-shock
CGR and Fragility for Severe Earthquake In order to contribute to the seismic PRA and seismic safety evaluation for aged
components, it is necessary to consider the earthquake beyond design basis ground motion.
Therefore, we are developing the evaluation methodology of crack growth rate (CGR) beyond the small scale yielding condition, considering large seismic stresses corresponding to response of severe earthquakes.
Building & Component Fragility Evaluation Accident Sequence Evaluation Seismic Hazard Evaluation
Seismic ground motion strength
Occ
urre
nce
frequ
ency
Seismic hazard curve
Cor
e da
mag
e fre
quen
cy
Core damage probability
Core damage frequency
Accident sequence occurrence frequency
Seismic hazard curve
Component A Fragility curve
Seismic ground motion strength
Component B
Failu
re
prob
abili
ty
Process of seismic PRA
DB
GM
Seismic ground motion strength
1.E-07
1.E-06
1.E-05
1.E-04
1.E+01 1.E+02 1.E+03
da/d
N(m
/cyc
le)
∆K (MPa√m)
系…系…da/dN = 1.60 x 10-13 ∆K3.58
Crack growth rate
CGR for cyclic stress under SSY:
( )m
fatigue
KCdNda
∆=
( )m
fatigue
JCdNda
∆=
It can not represent the CGR for cyclic stresses beyond SSY.
10-4
10-5
10-6
10-7
10 102 103 Cra
ck g
row
th ra
te (m
/cyc
le)
∆K (MPa√m)
1.E-07
1.E-06
1.E-05
1.E-04
1.E+01 1.E+02 1.E+03
da/d
N(m
/cyc
le)
∆J (kJ/m2)
da/dN = 2.42 x 10-9 ∆J1.79
Crack growth rate (Case 1)
10-4
10-5
10-6
10-7 Cra
ck g
row
th ra
te (m
/cyc
le)
10 102 103
∆J (kJ/m2)
Experimental data
( )m
fatigue
KCdNda
∆=
( )m
fatigue
JCdNda
∆=
24 CGR and Fragility for Severe Earthquake CGR for large seismic stresses • For constant cyclic stress beyond small scale yielding condition (SSY)
Experimental data
CGR of ∆J basis for cyclic stress beyond SSY:
It can represent the CGR for cyclic stresses beyond SSY.
𝑑𝑑𝑑𝑑
= 𝐶′𝛽𝑎𝛽𝑏∆𝐽 𝑑𝑖
𝑟𝑝𝑖′
𝑟𝑝𝑝𝑝′ + 𝑑𝑝𝑝 − 𝑑𝑖
𝛾𝑅′
− ∆𝐽𝑖=1𝐽𝑚𝑎𝑚,𝑝𝑝 − 𝐽𝑚𝑎𝑚,𝑖
𝐽𝑚𝑎𝑚,𝑝𝑝 − 𝐽𝑚𝑎𝑚,1
𝑚′
1.E-07
1.E-06
1.E-05
1.E+00 1.E+01 1.E+02
Cra
ck g
row
th ra
te (m
/cyc
le)
∆J (kJ/m2)
Experimental dataPrediction with excessive loadingPrediction without excessive loading
Excessive compressive load > excessive tensile load
Evaluation method of CGR for large seismic stresses beyond SSY
Cra
ck g
row
th ra
te (m
/cyc
le)
Cra
ck g
row
th ra
te (m
/cyc
le) 10-5
10-6
10-7
10 100 1 ∆J (kJ/m2)
10 100 1 ∆J (kJ/m2)
10-5
10-6
10-7
The case that CGR is accelerated
( )mJCdNda
∆=
( )mJCdNda
∆=
25 CGR and Fragility for Severe Earthquake CGR for large seismic stresses • For random cyclic stress beyond SSY
The case that CGR is retarded
Excessive tensile load > excessive compressive load
0.0
0.2
0.4
0.6
0.8
1.0
0 50 100 150 200 250
き裂進展量
(mm
)
サイクル数 (-)
試験体
荷重負荷
Crack Loading points
Loa
d
Cyclic No. Loa
d
Cyclic No.
Example of severe seismic wave
Cra
ck g
row
th v
alue
(mm
)
Cyclic No. (-)
Present method
Previous method under SSY
Experimental result Pipe
specimen
Experiment using seismic load and pipe specimen Comparison of experimental and predicted results
(Case of stainless steel pipe)
26
• Stainless steel pipe（SUS316） • Carbon steel pipe（STPT410）
• Do: 114.3 mm x t: 8.6 mm • Circumferential through wall crack
Pipe/Materials Dimension of pipe/crack
CGR and Fragility for Severe Earthquake Confirmation of CGR for large seismic stresses
0 10 20 30 40
Con
ditio
nal c
umul
ativ
e br
eak
prob
abili
ty (-
) 10-7
10-8
10-9
10-10
Operating time (year)
Case 3: 2/3 σ0
Case 1: 1.5 σ0
Case 2: σ0
• When the maximum amplitude of the seismic stress is larger than the yield stress σ0 , break probability based on the elastic fracture mechanics is not conservative.
Present method Previous method under SSY
27 CGR and Fragility for Severe Earthquake Failure probability evaluation considering different levels
of seismic stresses
Analysis condition • Pipe：stainless pipe in
BWR plant • Crack: initial crack in
weld joints caused by welding or etc.
• Crack growth: fatigue
28 Summary
Seismic PRA and seismic safety evaluation considering aging mechanisms for aged components are important and the ongoing issues.
Example applications to seismic fragility evaluation using PFM considering IGSCC and fatigue are introduced in this presentation.
We are making efforts considering other important aged components and age-related degradation mechanisms such as
• PWSCC
• NiSCC
• Flow Accelerated Corrosion
• Thermal aging embrittlement
We are also making efforts to link with seismic hazard and accident sequence evaluation, and to utilize failure probabilities considering age-related degradation mechanisms and seismic stresses.
