E-Journal of Advanced Maintenance Vol. 11 No. 4 (2020) 124-131
Japan Society of Maintenology

Verification Benchmark Analysis of Structural Reliability Evaluation
Codes for Fast Reactor Components
Shigeru TAKAYA1,*, Naoto SASAKI2, Toshiki OKAMOTO3 and Hideo MACHIDA3
1

Japan Atomic Energy Agency, 4002 Narita, Oarai, Ibaraki 311-1393, Japan
Ascend Co., Ltd., 4002 Narita, Oarai, Ibaraki 311-1313, Japan
3
TEPCO Systems Corporation, 2-37-28 Eitai, Koto-ku, Tokyo 135-0034, Japan
2

ABSTRACTThis paper describes benchmark analysis of independently programmed structural reliability evaluation codes,
REAL-P and GENPEP. An upper core structure of a prototype fast breeder reactor in Japan, MONJU, waschosen, and crack initiation time and crack propagation due to fatigue-creep interaction damage was evaluatedin deterministic and probabilistic manners. Evaluation procedures follow the new guidelines on reliability
evaluation of fast reactor components issued by JSME. The results obtained by two codes were compared, andthe effects of differences in treatments of which details are not prescribed in the guidelines on results werediscussed. As result, although slight difference was recognized in crack initiation evaluation especially due todifference in fairing treatment of fatigue life curves, the results estimated by two codes generally agreed very
well for both deterministic and probabilistic evaluations. It was shown that the effects of differences intreatments of which details are not prescribed in the guidelines on results are small for structural reliability
evaluation of fatigue-creep interaction damage, which is one of typical degradation mechanisms for fast reactor
passive components.
KEYWORDS

ARTICLE INFORMATION

System Based Code concept, Inservice inspection, Fatigue-creep interaction
damage, Crack initiation, Crack propagation, Deterministic evaluation,
Probabilistic evaluation, Plant safety

Article history:
Received 22 October 2018
Accepted 04 February 2020

1.

Introduction

A new code case on inservice inspection (ISI) requirements for sodium-cooled fast reactor
passive components was issued in the American Society of Mechanical Engineers (ASME) boiler andpressure vessel code [1]. The code case was developed by a joint task group between ASME and theJapan Society of Mechanical Engineers (JSME) based on the System Based Code concept that
pursues improvements in reliability and economy while meeting plant safety goals [2]. The code caseprovides a flow to determine ISI requirements which consists of two stages. The first stage focuses on
structural integrity of components. Potential active degradation mechanisms are considered based on
conditions including material and service environment, and then reliability of the specified component
that is a complement of failure probability is evaluated. If the reliability meets a target value, it ispossible to proceed to the next stage, where plant safety is focused on. An assessment is made of thedetectability of defects before they grow to an unacceptable size in consideration of plant safety. If
there is any feasible way to reliably detect degradation, it is adapted as ISI. Otherwise, a structural
integrity evaluation would be required while employing a sufficiently conservative hypothesis. If theresult meets the target value, ISI is not obligation. In this way, the proposed flow can determine ISI
suitable to fast reactors by taking key characteristics including material, environment, inspection
property and importance of components in light of plant safety into consideration.
Thus, structural reliability evaluation plays an important role to determine suitable ISI
requirements according to the code case. Therefore, a mandatory appendix for procedure for structuralreliability evaluation for passive components is included in the code case. Correspondingly, JSME has
published guidelines on reliability evaluation of fast reactor components [3]. In addition to the*Corresponding author, E-mail: takaya.shigeru@jaea.go.jp

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

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S. Takaya, et al./ Verification Benchmark Analysis of Structural Reliability
Evaluation Codes for Fast Reactor Components

guidelines, development of structural reliability evaluation codes which can conduct evaluations
following the guidelines is also needed. Furthermore, it is important to confirm the effects ofdifferences in treatments of which details are not prescribed in the guidelines on results.
Therefore, in this paper, benchmark analysis on crack initiation and propagation due tofatigue-creep interaction damage, which is one of typical degradation mechanisms for fast reactor
passive components, was conducted by using independently programmed codes, REAL-P and
GENPEP [4]. In sodium-cooled fast reactors, internal pressure is much lower than light-water-reactors
because of high boiling temperature of sodium while thermal stress is high due to relatively large
temperature changes during start-ups and shutdowns. In addition, evaluation of stress relaxation
behavior at elevated temperature is important. Both codes were developed to deal with these featuresof sodium-cooled fast reactors in the ways following the guidelines.

2.

Evaluation Procedures

Evaluation procedures of fatigue-creep interaction damage followed the guidelines on
reliability evaluation of fast reactor components [3]. As for crack initiation evaluation, fatigue damage,Df, and creep damage, Dc, were calculated respectively, and it was assumed that a crack initiated when
Df and Dc met the following equation;(1)
D = f D f , Dc()

where D is the critical value that connects points (Df, Dc) = (1, 0), (0.3, 0.3), and (0, 1). As for crack
propagation evaluation, the following equation was used;dam= C f ΔJ f f + ò Cc J c mc dt
dn
1cycle(2)where a is the crack depth, and Cf, Cc, mf and mc are material constants. DJf is J-integral range forfatigue crack growth evaluation and Jc is J-integral for creep crack growth evaluation. DJf wasevaluated by the procedures explained in Ref. [9] which is also employed in the guidelines. Moredetails in the procedures are explained somewhere [3, 5, 9].
An upper core structure (UCS) of a prototype fast breeder reactor in Japan, MONJU, waschosen as a target component for this benchmark analysis (Fig. 1). Initiation and propagation of acrack on the outer surface of the shroud at the sodium surface level during normal operations were
evaluated. As for the crack propagation evaluation, a full circumferential crack with the depth of 1
mm was assumed as an initial crack. Analysis conditions were determined as shown in Table 1, basedon a previous study conducting a hypothetical evaluation without a stress-relief structure of the shroud[6]. As mentioned above, the steady primary stress due to internal pressure and dead weight was small,and the thermal transient stress due to start-ups and shutdowns, which was corresponding to stressintensity range and secondary bending stress range in the vertical direction to the crack, was dominant.
In addition, probabilistic distributions of random variables were also set as shown in Table 2, basedon previous reports on statistical properties of material strength [7-9]. The thermal stress amplification
factor, c, was also employed as a random variable to consider potential scattering of temperature
range during start-ups and shutdowns. This factor was multiplied by cyclic stress range and axial
direction stress ranges in Table 1. aR, ac and af are random variables to consider variations of creep
rupture time, steady-state creep strain rate and fatigue strength. Creep rupture time and fatigue
strength used in probabilistic evaluations were obtained by dividing their mean values by aR and af,
respectively. Steady-state creep strain rate is a function of creep rupture time, and gets higher for
shorter creep rupture time. The creep rupture time used for calculating steady-state creep strain rate
was obtained by dividing the mean value by ac. As for material constants in Eq. (2), the coefficients,
Cf and Cc, were chosen as random variables while the exponents, mf and mc, were dealt
deterministically. This was because variations of the coefficients were much larger than those of
exponents in experiment results [9], and were considered to have dominant effect on variations ofcrack propagation rates. Values of mf and mc were 1.374 and 0.9069, respectively. In this study, both
deterministic evaluation and probabilistic evaluation were conducted. As for probabilistic evaluation,
not only a case where the whole set of random variables in Table 2 was considered but also cases
where only some of them were selected were prepared for detailed comparison of the codes as shown125

S. Takaya, et al./ Verification Benchmark Analysis of Structural Reliability
Evaluation Codes for Fast Reactor Components

in Table 3. Median values in Table 2 were used in the probabilistic evaluation if the parameters were
not chosen as random variables in the case, as well as the deterministic evaluation.
Both REAL-P and GENPEP employed direct sampling Monte-Carlo method which is the mostbasic sampling method. The total sampling numbers are 104 for both REAL-P and GENPEP.

Fig. 1 Reactor structure of Monju

Table 1 Evaluation Parameters

Parameter
Material
Outer diameter (mm)Wall thickness (mm)Operating temperature (°C)
Temperature in cold shutdown state (°C)
Total duration at elevated temperature (h)Temperature range for calculation of peakthermal strain range (°C)
Steady primary stress (MPa)Stress intensity range* (MPa)Primary membraneStress range inPrimary bending
the verticaldirection to theSecondary membrane
crack (MPa)Secondary bending
Shakedown range (MPa)Strain rate (mm/mm/s)
Number of cycles per 30 years*: The range of stress during a loading cycle

126

Value
304SS
1875
60
530
200
210000
5
1
350
0
0
0
175
215
1×10-8460

S. Takaya, et al./ Verification Benchmark Analysis of Structural Reliability
Evaluation Codes for Fast Reactor Components
Table 2 List of Random Variables

Random Variable

Distribution
type

MedianThermal stress amplification factor, c (-)
Creep rupture time reduction factor, aR (-)
Creep strain amplification factor, ac (-)
Fatigue life reduction factor, af (-)
1
Coefficient for creep crack growth, Cc ( mm m +・
h m -1 / N m )Coefficient for fatigue crack growth, Cf ( mm m +1 / N m )Log-normal
Log-normal
Log-normal
Log-normal
Log-normal
Log-normal

1.00
1.00
1.00
0.10
1.59×10-2
6.34×10-5

c

c

f

c

f

Logarithmic
standard
deviation
0.078
0.560
0.800
0.420
0.422
0.422

Table 3 Analysis Cases of Probabilistic Evaluation

Case 1Case 2
Case 3
Case 4

3.

Selected random variable(s)§c, aR, ac, af, Cc, Cf§
c§aR, ac, af§§§
Cc, CfRemarks
All of the random variables in Table 2 are considered.
Only a random variable related to stress is considered.
Only random variables related to material properties
except coefficients for crack propagation are considered.
aR and af are not used in crack propagation evaluation.
Only coefficients for crack propagation are considered.
Both parameters are not used in crack initiation
evaluation.

Results and Discussion

Tables 4 and 5 show comparison of deterministic evaluation results of crack initiation and crack
propagation, respectively. It should be noted again that they are results in the hypothetical condition
without a stress-relief structure. In both cases, REAL-P and GENPEP gave similar results, of which
difference is less than a few percentages.
A main reason of the slight difference of fatigue damage in Table 4 is whether fairing treatmentof fatigue life curves was conducted or not. The guidelines recommend to use best-fit curves offatigue life in the JSME code for design and construction of fast reactors (JSME FR code) [10]lowered by a factor of two for strain or 10 for cycles, whichever is more conservative, as averagefatigue life curves for reliability evaluation [3]. However, the guidelines do not mention the fairing
treatment. REAL-P conducted fairing treatment between these two curves as shown in Fig. 2 toprevent discontinuous change in fatigue life while GENPEP did not to avoid additional
conservativeness. Therefore, REAL-P tended to give slight conservative fatigue damage in a certain
strain range. Fatigue life curves are not used in crack propagation evaluation, so the difference of
fatigue crack propagation in Table 5 is smaller than that of fatigue damage in Table 4.
On the other hand, as for creep, GENPEP gave slightly conservative results for both creep
damage and creep crack propagation. These parameters are sensitive to stress relaxation behaviors.
The guidelines provide the evaluation procedure of stress relaxation, but do not prescribe specificconditions as for time steps. As a result of comparison of treatments in two codes, it was found that
REAL-P determined time steps in the same manner as the JSME FR code, while GENPEP employed
an original manner. Schematic explanation of the manner used by REAL-P is shown in Fig. 3. Timestep width was determined so that stress relaxation for one time step did not exceed the limit, Dsmax (=0.49 MPa). The default time step width for the first time step was 10-4 h, and those for time steps after
the second time step were determined by dividing holding time at elevated temperature equally on thelogarithmic axis: 20 parts for one order. If the calculated stress relaxation for one time step exceeded
the limit, recalculation would be conducted by shortening the time step width by half. On the other
hand, GENPEP determined a tentative time step width so that stress relaxation from initial stress for
one time step became smaller than 0.1 MPa, and then one hundredth of the tentative time step width127

S. Takaya, et al./ Verification Benchmark Analysis of Structural Reliability
Evaluation Codes for Fast Reactor Components

or 10-5 h, whichever was shorter was chosen a final time step width. The same time step width wereused for whole holding time at elevated temperature. The time step width used in GENPEP wasshorter than that used in REAL-P. Furthermore, there was also a minor difference in the way of
calculation of creep damage for each time step, dc, which is expressed analytically by Eq. (3).

dc = ò

1dt
tr (s )

(3)

where tr, s and t are creep rupture time, stress and time, respectively. REAL-P actually evaluated dc
discretely by using Eq. (4) while GENPEP used Eq. (5).

dc =

1
Dt
1
Dt
+
tr (s (t0 )) 2 tr (s (t0 + Dt )) 2
1
dc =
Dt
tr (s (t0 ))

(4)
(5)

where t0 and Dt are time at the start point of each time step and time step width, respectively. These
differences in treatments were considered to result in the difference of a few percentages as for creep
damage and creep crack propagation.

Table 4 Comparison of deterministic evaluation results of crack initiation

Parameter
Fatigue damage at the first cycle (-)
Creep damage at the first cycle (-)
Crack initiation time (years)

REAL-P: A
2.03×10-3
5.19×10-3
8.35

GENPEP: B
1.99×10-3
5.31×10-3
8.41(A-B)/A (%)
1.97
-2.31
-0.72

Table 5 Comparison of deterministic evaluation results of crack propagation

Parameter
Fatigue crack propagation at the first cycle (mm)Creep crack propagation at the first cycle (mm)Time for crack to reach half of wall thickness (years)

Fig.2 Fairing treatment of fatigue life curves(Schematic)

REAL-P: A
3.75×10-5
4.10×10-5
1.50×103

GENPEP: B
3.75×10-5
4.19×10-5
1.48×103(A-B)/A (%)
0.00
-2.20
1.33

Fig.3
Determination of time steps
evaluation of stress relaxation (REAL-P)

128

for

S. Takaya, et al./ Verification Benchmark Analysis of Structural Reliability
Evaluation Codes for Fast Reactor Components

Figure 4 shows comparison of accumulated crack initiation probability distributions evaluated by
REAL-P and GENPEP. Those of Case 1 (Fig. 4(a)) where the whole set of random variables in Table
2 was considered agree well although there is very slight difference in the range above theaccumulated probability of about 50%. This tendency is more apparent in Case 2 (Fig. 4(b)) while
such difference is not recognized in Case 3 (Fig. 4(c)).
Figure 5 shows comparison of accumulated probability distributions of fatigue and creep
damages at the first cycle evaluated by REAL-P and GENPEP in Case 2. Lower damage regions in
Fig. 5 correspond to longer life regions in Fig. 4. Slight difference is recognized in a low damage
region for the accumulated probability distributions of fatigue damage (Fig. 5(a)) while the
distributions agree very well for creep damage (Fig. 5(b)). Therefore, it can be seen that thisdifference in fatigue damage causes the difference shown in Fig. 4(b). As explained above, there wasa minor difference in fatigue damage evaluation between two codes. In order to examine the effect of
this difference regarding fairing treatment, analytically evaluated accumulated probabilities areplotted in Fig. 5(a). Case 2 treated only one random variable, which enables to analytical evaluation.
The plots coincide with corresponding curves, which means that the difference regarding fairing
treatment is a main reason of the differences recognized in Fig. 5(a) and also Fig. 4(b). In the contrast,
as for creep damage, very slight difference in the region where the accumulated probability is higher
than approximately 50% was recognized. It is considered to be due to the difference in detailed
evaluation procedures of stress relaxation behavior and creep damage discussed above. The influenceof this difference in creep damage to the accumulated crack initiation probability was too small to berecognized in Figure 4.
In Case 3 (Fig. 4(c)), the effect of difference regarding fairing treatment of fatigue life curves
was hardly recognized. It is because the fatigue life reduction factor, af, was chosen as a randomvariable in Case 3, and then an intersection point and a fairing region of two fatigue life curves
illustrated in Fig. 2 moved from a strain range concerned in the evaluation.
As results, it was found that the both codes generally gave very similar results on crack initiation
probability except for the very slight difference due to the fairing treatment of fatigue life curves.

(a) Case 1 (Random variables: c, aR, ac, af )(b) Case 2 (Random variables: c)(c) Case 3 (Random variables: aR, ac, af )Fig. 4 Comparison of crack initiation probabilities evaluated by REAL-P and GENPEP

129

S. Takaya, et al./ Verification Benchmark Analysis of Structural Reliability
Evaluation Codes for Fast Reactor Components(b) Creep damage at the first cycle(a) Fatigue damage at the first cycle

Fig. 5 Comparison of damage probabilities evaluated by REAL-P and GENPEP (Case 2)

Figure 6 shows comparison of accumulated crack propagation probability distributions at the firstcycle. In every case, distributions evaluated by REAL-P and GENPEP agreed very well. Fatigue life
curves are not used in the evaluation of crack propagation, so there is no influence of fairing treatment.
In Cases 2 and 3 where just one random variable is considered, analytically calculated accumulated
probabilities are also plotted, and the plots coincide curves evaluated by the two codes. It can beconsidered that both codes were programmed properly. As for the probabilistic evaluation of crack
propagation, difference in results which required detailed discussion on treatments in the codes was
not especially recognized.

(a) Case 1 (Random variables: c, ac, Cc, Cf)(b) Case 2 (Random variables: c)(c) Case 3 (Random variables: ac )(d) Case 4 (Random variables: Cc, Cf )Fig. 6 Comparison of crack propagation probabilities at the first cycle
evaluated by REAL-P and GENPEP

4.

Conclusion
Structural reliability evaluation plays one of key roles to determine suitable ISI requirements of130

S. Takaya, et al./ Verification Benchmark Analysis of Structural Reliability
Evaluation Codes for Fast Reactor Components

sodium-cooled fast reactor passive components in the code case N-875 newly issued by ASME. JSME
also has published new guidelines on reliability evaluation of fast reactor components. Thus, theimportance of structural reliability evaluation is increasing.
In this study, a set of benchmark analysis conditions was prepared by using information of UCS
of MONJU for verification of structural reliability evaluation codes. Deterministic and probabilistic
evaluations of crack initiation and propagation due to creep-fatigue interaction damage, which is oneof typical degradation mechanism for fast reactor components, were conducted by using
independently developed structural reliability evaluation codes, REAL-P and GENPEP. There wereseveral differences in treatments of which details are not mentioned in the guidelines such as fairing
treatment of fatigue life curves and determination of time step for evaluation of stress relaxation,
between two codes. Slight difference was recognized in crack initiation evaluation especially due tothe difference in fairing treatment of fatigue life curves. However, in general, the results agreed very
well. As a result, it is shown that the effects of differences in treatments of which details are notprescribed in the guidelines on results are small for structural reliability evaluation of fatigue-creep
interaction damage, which is one of typical degradation mechanisms for fast reactor passive
components.
It is also expected to use the set of input and output data presented in this paper for verification ofother codes. Verification benchmark analysis of these codes will be also continued by increasing
examples.

Acknowledgement
Present study includes the results of the “Technical development program on a commercialized
FBR plant” entrusted to Japan Atomic Energy Agency (JAEA) by the Ministry of Economy, Tradeand Industry of Japan (METI).

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