GENES4/ANP2003, Sep. 15-19, 2003, Kyoto, JAPAN
Paper 1089
An Integrated Approach in the Structural Design of Fast Breeder Reactors
Part 1: A Potential Concept
Tai Asayama1*, Nobuchika Kawasaki1 and Masaki Morishita1
Japan Nuclear Cycle Development Institute, 4002 Narita, Oarai, Higashi-Ibaraki, 311-1393, Japan
1
For the commercialization of fast breeder reactors, “System Based Code”, a completely new scheme of a
code on structural integrity, is being developed. One of the distinguished features of the System Based Code
is that it is able to determine a reasonable total margin on a structural system, by allowing the exchanges of
margins between various technical items. For achieving margin exchange, a methodology for quantitative
estimation of structural reliability is necessary. Moreover, a simple and practical method of margin exchange
evaluation based on the estimated reliability is necessary. This paper proposes a scheme of margin exchange
that consists of a flow of creep-fatigue failure probability assessment and a simple and practical method for
margin exchange evaluation.
KEYWORDS: Failure probability, Fatigue, Margin exchange
I. Introduction
Aiming at a leap of progress in the evaluation of structural
integrity of nuclear power plant components, a concept of
the System Based Code has been proposed by Asada et al [1,
2]. A key point of the concept is margin exchange, which
allows a designer to achieve given target reliability by
multiple of combinations of technical options that range
from loading conditions, material properties, fabrication
techniques, inspection methods to repair and replacement.
This leads to a more cost-effective design without sacrificing
reliability.
One of the issues that are critical to the realization of
margin exchange concept is the development of a
quantitative measure of reliability [3-5]. Introduction of
probabilistic methods to structural integrity evaluation is a
promising method. To achieve this, 1) the development of an
estimation method of failure probability and 2) the
development of a margin exchange evaluation method based
on calculated estimated reliability, are necessary.
This paper first examines a scheme of margin exchange
that consists of failure probability estimation, and margin
exchange evaluation based on the estimated failure
probability. Failure probability estimation methods proposed
so far are briefly examined and a new simplified method is
proposed. Then, creep-fatigue failure, one of the most
important failure modes to be prevented in fast breeder
reactors, is taken as an example and a general flow of failure
probability estimation is proposed. Moreover, a new margin
exchange evaluation method based on the estimated failure
probabilities, the “Vector Method” is proposed. Only
concept and an example of its application is presented in this
paper and details of the method is given in the part 2 of this
paper [6].
II. Frame of margin exchange
*
Corresponding author, Tel. +81-29-267-4141, Fax. +81-29-2663675, E-mail: [email protected]
1. Estimation of failure probability
There are a couple of methods, both theoretical and
numerical, for the estimation of failure probability. Methods
that have potential to be used in the System Based Code are
summarized below.
Monte Carlo Simulation
The most popular numerical method is Monte Carlo
simulation. From the viewpoint of structural integrity
assessment, this method has fallowing advantages:
(1) As many random basic variables as a designer needs to
deal with in structural integrity assessment can easily
be incorporated in the simulation.
(2) Any type of probabilistic distributions can be used.
Correlations among variables can also be taken into
account.
(3) Complex failure criteria can be easily incorporated.
On the other hand, following disadvantage exists:
(1) Computational load is not negligible and relatively time
consuming, even if a method such as layered sampling
is employed.
First Order Second Moment Method
Another method for failure probability assessment is the
First Order Second Moment (FOSM) method, which is a
theoretical one. This method is used in such areas like civil
engineering and architectural engineering. A limit state is
defined as a function of random basic variables, and
reliability indices are calculated using a mean value and a
standard deviation of random variables. The advantages are:
(1) Once a limit state function is formulated, reliability
indices can be calculated relatively easily.
Computational load is negligible.
(2) Failure probability can be calculated from reliability
indices when probabilistic distributions used are limited
to normal and lognormal distributions.
Disadvantages are:
1
(1) Reliability indices cannot be related to failure
probability when probabilistic distributions other than
normal and lognormal distributions are used.
(2) The effects of inspection and repair during a service life
cannot be explicitly incorporated in the formulation.
Simplified method
The third possible way to calculate failure probability is to
use a simplified method. This simplified method is proposed
by the authors in this paper in order to calculate failure
probabilities theoretically with taking account of the effects
of inspection and repair during a service life. Advantages are
summarized below:
(1) Failure probabilities are obtained very easily through a
theoretical formulation without any kind of numerical
calculations.
(2) The effects of inspection and repair during a service life
can be explicitly taken into account.
Disadvantage is as follows:
(1) Due to the limitation of theoretical formulation, it is not
practical for general applications.
2. Evaluation method of margin exchange
Margin exchange
Margin exchange evaluation consists of:
(1) Estimation of failure probability for more than two
combinations of technical options, which range from
material selection, design evaluation, fabrication and
inspection,
inservice
inspection,
repair
and
replacement.
(2) Comparison of the calculated failure probabilities and
identification of equivalent combinations in terms of
failure probability.
These can be done by trial and error, using one of the
three methods described above, that is, Monte Carlo
Simulation, FOSM method, or a simplified method for
failure probability calculation. However, for the convenience
of designers, a simplified method for margin exchange
evaluation is beneficial. For this purpose, Asada et al. [2] has
proposed “quality assurance index”, which should be
basically determined by expert panels. In this paper, a
method to determine the indices based on the results of
failure probability calculation, a Vector Method, is proposed.
The outline of the method is described in Section IV of this
paper. For details, see Part 2 of this paper.
III. Procedure of failure probability estimation
In this paper, one of the most important failure modes to
be prevented in fast breeder reactors (FBRs), creep-fatigue,
is considered. A basic flow of creep-fatigue failure
probability estimation is proposed below. A part of the flow
is used in the demonstration of margin exchange in Chapter
3. This basic flow does not give specific evaluation methods
such as evaluation methods for creep-fatigue damage or
creep-fatigue crack propagation. For those methods see
reference [4].
Definition of failure (Event F)
Failure is defined as the occurrence of one or both of the
following events:
1) Failure by the propagation of creep-fatigue crack
(Event C).
2) Unstable fracture (Event U)
Failure probability p(F) is expressed by Equation (1),
where p(C) is the probability by crack propagation and p(U)
is the probability of unstable fracture:
p( F ) = p(C ) + p(U ) − p(C ∩ U )
(1)
Definition of failure by creep-fatigue crack propagation
(Event C)
Failure by creep-fatigue failure crack propagation (Event
C) is defined by the occurrence of one of the following
events:
1) Initiation of a creep-fatigue crack and its propagation to
the critical depth. The critical depth that defines failure
is expressed either by the ratio to wall thickness, such
that 1/4, 1/2, and 3/4, or by the initiation of a crack
(Event CA).
2) Propagation of n initial defects to the critical depth
(Event CB).
In this paper, the two events CA and CB were considered as
independent and failure probability by creep-fatigue crack
propagation p(C) was defined by Equation (2):
p (C ) = p (CA) + p (CB ) − p (CA ∩ CB )
= p (CA) + p (CB ) − p (CA) P (CB )
(2)
Initiation of a creep-fatigue crack and its propagation to the
critical depth (Event CA)
The probability of the occurrence of the event CA is
expressed by Equation (3), where p(CAI1) is the probability
of initiation of a creep-fatigue crack and p(CA1) is the
probability of the propagation of an initiated crack to the
critical depth.
p(CA) = p(CAI 1 ) p(CA1 )
(3)
Cumulated failure probability at the n th cycle P(CA) n is a
function of failure probability at the i th cycle p(CA, i) (i<=n).
p(CA) i is the probability that a crack initiates at the j th
cycle (j=1,2,…,i-1) and propagates during the following i-j
cycles and that its depth becomes failure criteria at the i th
cycle. Therefore, Equation (3) becomes Equation (4):
N
i −1
P(CA) n = ∑∑ p(CAI 1 ) j p(CA1 ) i − j
i =1 j =1
(4)
Propagation of n initial defects to the critical depth (Event
CB)
The event CB is defined as an event that at least one of the
m initial defects propagates to reach the critical depth. The
2
probability of the occurrence of the event CB is expressed by
Equation (5), which is the weakest link model, using
P(CBIm) and P(CB1). P(CBIm) is the probability that m initial
defects exist, and P(CB1) is the probability that a crack
propagates from an initial defect and reach the critical depth.
{ (
P(CB ) = p (CBIm ) 1 − 1 − p m (CB1 )
)}
(5)
Failure probability at the n th cycle P(CB)n is a cumulated
value of failure probability at the i th cycle (i<=n) p(CB)i.
Therefore, P(CB)n is expressed by Equation (5).
{ (
n
P(CB ) n = ∑ p (CBIm ) 1 − 1 − p m (CB1 ) i
i =1
)}
(6)
Probability of creep-fatigue failure P(C)
From Equations (2) through (6), probability of
creep-fatigue failure at the n th cycle is represented by
Equation (7):
P(C ) n = P (CA) n + P (CB ) n − P (CA) n P (CB ) n
= P (CA) n + P(CB ) n (1 − P (CA) n )
i −1
n
n
{ (
= ∑∑ p(CAI 1 ) j p (CA1 ) i − j + ∑ p(CBIn ) 1 − 1 − p (CB1 ) i
i =1 j =1
i =1
m
)}


× 1 − ∑∑ p(CAI 1 ) j p(CA1 ) i − j 
i =1 j =1


n
i −1
(7)
Probability of unstable fracture P(U)
Probability of unstable fracture at the n th cycle is
expressed by Equation (8).
n
P (U ) n = ∑ p (U ) i
i =1
(8)
Probability of failure P(F)
From Equations (1), (7) and (8), probability of failure at
the n th cycle is obtained as Equation (9).
P ( F ) n = P(C ) n + P (U ) n − P(C ∩ U ) n
n
i −1
n
{ (
= ∑∑ p (CAI 1 ) j p (CA1 ) i − j + ∑ p (CBIn ) 1 − 1 − p (CB1 ) i
i =1 j =1
i =1
m
)}
n i −1

 n
× 1 − ∑∑ p (CAI 1 ) j p (CA1 ) i − j  + ∑ p (U ) i − P(C ∩ U ) n
i =1 j =1

 i =1
(9)
The determination of probability density as a function of n
is rest to a designer.
IV. Margin Exchange Evaluation
In this chapter, an example of margin exchange evaluation
is shown. Firstly, a problem for example calculation is
explained. Then, an example by trial and error is presented,
first by Monte Carlo simulation and then by the newly
proposed simplified method. Finally, in place of trial and
error, a newly proposed “Vector Method” is shown. In this
example, only the event CB in the flow descried in Chapter
III is considered.
1. Example problem for calculation
Outline of example
In this example, a vessel containing coolant subjected to
creep-fatigue loading is considered. Wall thickness is
12.7mm and failure is defined as propagation of a crack to
the depth of 3/4 of wall thickness. Internal pressure is
negligible. Crack propagation from an initial defect in a
welded joint was considered, and crack initiation was not
considered. For initial crack distribution, 2nd order
polynomial function with the maximum initial depth of
9.5mm was assumed. Periodic inservice inspection was
assumed to be performed and detected cracks were assumed
to be repaired and that original properties were restored by
repair. Crack propagation rate, which is a function of
geometry, material properties, and loading conditions, is
assumed to be given by Eq. (10), where c and m are
constants depending on geometry, material properties and
loading conditions, and a is crack depth.
da
= g (a) = ca m
dn
(10)
Technical options
Technical options are as follows:
a) Selection of material and geometry, and determination
of loading conditions. These are expressed in terms of
crack propagation rate. The coefficient c in Eq. (10)
was made variable.
b) Probability of detection (POD) of defects in inservice
inspections. For simplicity, this was assumed to be a
constant regardless of crack size.
c) Interval of inservice inspection.
Technical options selected for an original design were
assumed as follows:
a) Crack propagation rate: c=0.0005, m=0.5
b) POD=0.4
c) Interval=250 cycles
Problem
The problem for margin exchange is as follows. For the
reduction of construction cost, a designer wants to suggest
an alternative design. In the alternative design, the designer
is going to use cheaper material whose crack propagation
rate is faster than the original material by 5 times. For the
compensation of the inferior material, an advanced inservice
inspection technique (non destruction examination) whose
probability of detection is as high as 0.8, is available. In
order to maintain the reliability (failure probability) of the
original design, what should be the interval of inservice
inspection in the alternative design? Failure probability
should be evaluated at the 1000 th cycle, which is assumed
as the end of a service life. To summarize, for the alternative
design:
a') Crack propagation rate: c=0.0025, m=0.5
3
b') POD=0.8
c') Interval= x cycles, the unknown
2. Margin exchange by trial and error
In this section, an example of margin exchange evaluation
by trial and error is shown.
Monte Carlo simulation
Cumulated failure probability of the original design was
calculated by Monte Carlo simulation and the result is
shown in Fig.1, as a function of number of imposed
creep-fatigue cycles. From this figure, we can see that the
failure probability at the 1000 th cycle is approximately
1.2x10-3.
The next step is to find an interval of inservice inspection
that gives a failure probability equivalent to the original
design. This can be obtained by trial and error, and after a
number of cases of calculations, an answer is found to be
approximately 90 cycles. For this case, cumulated failure
probability at the 1000 th cycle is 1.2x10-3 and the
relationship between cumulated failure probability and the
number of cycles is shown in Fig.1.
Simplified method
For this example calculation, a simplified method
proposed by the authors can be used as an alternative for
Monte Carlo method. The simplified method calculates
failure probability by Equation (11) [6]:
(1 − POD )l (Pf (ln0 ) − Pf ((l − 1)n0 ))
PfISI (n ) = ∑
p
l =1 + (1 − POD ) (P (ln + q ) − P (ln ))
0
0
p
p
6
5
l
p
= A1 ∑ bs ∑ l s (1 − POD ) + ∑ cs q s (1 − POD ) 
s =1
 s =0 l =1

(11)
Where, A1, bs, cs are constants that depend on the maximum
depth of initial crack, the critical depth of crack, crack
propagation rate, the interval of inservice inspection, and the
number of imposed cycles. l and p are constants determined
by the interval of inservice inspections.
Using this method, an answer is obtained more easily than
using Monte Carlo simulation. Trial and error is not so
tiresome and more accurate answer of 89 cycles is quickly
obtained. The result obtained is over plotted in Fig.1.
Generally, the results of Monte Carlo simulation and the
simplified method agree very well.
3. Margin exchange by Vector Method
The “Vector Method” is a new method for margin
exchange evaluation that the authors propose. The details are
shown in the Part 2 of this paper [6]. In this section, the use
of the method is demonstrated without detailed descriptions.
The basic idea of the Vector Method is derived from the
approach proposed by Asada et al [2] that uses quality
assurance indices and expressed by Equation (12):
F = C (1)Q( I , J I ) + C (2)Q( I , J 2 ) + L + C ( K )Q( I , J K )
(12)
Where F is total quality index and Q(I) is quality assurance
index corresponding to an design option of technical field I.
Technical fields include loading conditions, material
properties, fabrication techniques, inspection methods to
repair and replacement. C(I) is an influence coefficient
corresponding to a technical field I.
In the Vector Method, a “margin exchange table” is
prepared. An example is shown in Table 1. Margin exchange
evaluation is performed using this table. Firstly, quality
assurance index is calculated from technical specifications.
For the original design, the first technical option (crack
propagation rate) is c=0.0005, and this corresponds to q1=5.
This can be obtained either by interpolation of Table 1 or by
calculation using equations shown in the part 2 of this paper.
The second technical option, probability of detection of
defects of 0.4, corresponds to q2=3.52. The third technical
option, the inservice inspection interval of 250 cycles
corresponds to q3=1.12. Using these values and the values of
influence coefficients ci’s, we obtain a total quality index F:
F = c1q1 + c2 q2 + c3 q3
= 0.488 × 5 + 0.393 × 3.52 + 0.779 × 1.12
= 4.69
(13)
The second step is to find a value of q3 given that
c=0.0025 and POD=0.8. By referring to Table 1 or to
equations shown in the part 2 of this paper, we can obtain
that q1=2.85 and q2=4.79. Therefore, the equation to be
solved is:
′
4.69 = 0.488 × 2.85 + 0.393 × 4.79 + 0.779 q3
(14)
q3’ is easily found to be 1.83. Again, by referring to Table 1
or to equations shown in the part 2 of this paper, we can
obtain that the interval of inservice inspection for the
alternative design should be 103 cycles.
The result obtained by the Vector Method (103 cycles) is
slightly different from the one obtained by simplified
method (89 cycles, which is essentially identical to the result
obtained by Monte Carlo simulation). This difference is due
to an error in the approximation in the course of establishing
the margin exchange table. However, the error is not
significant and is considered to be tolerable in practical use.
When a designer uses the Vector Method, trial and error is
not necessary and the design option that gives an equivalent
failure probability to the original design is easily obtained by
very simple calculations. The benefit obtained by the use of
the Vector Method is considered to be significant.
V. Discussions
For the achievement of practical use of margin exchange
concept that Asada et al proposed [1,2], this paper showed a
4
potential scheme of margin exchange evaluation method.
Several issues remain to be addressed.
As to the failure probability estimation methods, the
extension of the simplified method, which has the
advantages of Monte Carlo simulation and FOSM at the
same time, is necessary for the achievement of practical
estimation, which requires the incorporation of the effects of
inspection and repair, without computational burdens.
Another way is to extend FOSM method so that it can deal
with the effects of inspection and repair.
For reliable estimation of failure probabilities, the
determination of probabilistic distributions corresponding to
the identified events in Chapter III is critically important.
For events for which sufficient data are not available,
uncertainty should be taken into account for precise
estimation.
For the practical use of the Vector Method, the
applicability of a single table to various conditions has to be
addressed. The influence coefficients can be different in
conditions significantly different than those used in
establishing the table.
Moreover, margin exchange can be approached both by
empirical ways (expert panels, for example) and quantitative
ways, the latter of which this paper described. The
combinations of the two approaches can give more
beneficial than use of a single approach.
6)
VI. Conclusions
The conclusions obtained in this paper are as follows:
(1) A potential scheme for quantitative margin exchange
was proposed. This scheme consists of failure
probability estimation and margin exchange evaluation
based on the estimation.
(2) For failure probability estimation, a simplified method
was proposed. This method estimates the effects of
inspection and repair without numerical simulation.
(3) The outline of newly presented margin evaluation
method, the “Vector Method” was introduced. This
method allows a designer to perform margin exchange
evaluation without calculating failure probabilities.
References
1) Asada, Y., Tashimo, M. and Ueta, M., System Based
Code -Principal Concept-, Proceedings of ICONE10
(2002)
2) Asada, Y., Tashimo, M. and Ueta, M., System Based
Code -Basic Structure-, Proceedings of ICONE10
(2002)
3) Asayama, T., Morishita, M., Dozaki, K. and Higuchi,
M., Development of the System Based Code for Fast
Breeder Reactors and Light Water Reactors - Basic
scheme -, ICONE10
4) Asayama, T., Kawasaki, N., Morishita, M. and Dozaki,
K., Development of the System Based Code for Fast
Breeder Reactors - Probabilistic methods in
creep-fatigue evaluation -, ICONE10
5) Asayama, T., Morishita, M., Kawasaki, N. and Dozaki,
5
K., Development of the System Based Code for
Structural Integrity of FBRs, ASME PVP Vol.439
(2002) 265.
Asayama, T., Kato, T. and Morishita, M., A
Probabilistic Approach in the Structural Design of Fast
Breeder Reactors Part 2: A Simplified Procedure for
Margin Exchange Evaluation, GENES4/ANP2003, Sep.
15-19, 2003, Kyoto, JAPAN (In this proceeding)
Table 1
An example of a margin exchange table
F=c1*q1+c2*q2+c3*q3
q1=c11*q11+c12*q12+c13
・・・(1)
(c13=-3.447) ・・・(2)
Target failure probability Pf
Total quality assurance index F
1x10-3
4.9
-4
5.4
1x10-5
6.0
1x10
-6
6.6
1x10
-7
7.1
1x10
目標値
Main infulence coefficient
Partial code (i)
Technical
item
Coeff. (c)
Load (11)
c1=0.451
Load/Material (1)
Material (12)
ISI accuracy (2)
c1=0.451
c2=0.464
Inservice Inspection
（ISI）
ISI frequency (3
Detailed
Simplified
c3=0.760
Target failure
probability
Coeff. (c)
Sub influence
coefficient
(cij)
Options
Quality assurance index (q)
1x10
-3
0.488
Snom(※1)×0.2
1x10-4
0.455
Snom×0.4
1x10-5
0.440
1x10
-6
0.447
Snom×1.5
1x10-7
0.425
Snom×2.5
1
1x10-3
0.488
Snom(※2)×0.2
5
1x10
-4
0.455
1x10-5
0.440
1x10-6
0.447
Snom×1.5
1x10
-7
0.425
Snom×4
1
1x10-3
0.393
POD=0.9（※3）
5
1x10-4
0.446
1x10
-5
0.457
1x10-6
0.468
0.2
2
1x10
-7
0.555
0.1
1
1x10-3
0.779
Every 2 cycles
5
1x10-4
0.771
5
4
-5
0.773
-6
0.762
80
2
0.715
300
1
1x10
1x10
1x10-7
（※1）Ratio to nominal load level determined elsewhere.
（注2）Ratio to nominal crack propagation rate determined elsewhere.
（注3）Average value independent of crack depth.
6
c11=1.289
Snom×0.7
5
4
q11
2
4
Snom×0.25
c12=1.230
Snom×0.6
q12
-
0.3
20
3
2
0.5
-
3
4
q2
q3
3
3
1.E-01
Original design (Simlified Method)
Alternative design (Simplified Method)
Original design (Monte Carlo)
Alternative design (Monte Carlo)
Cumulated Failure Probability
1.E-02
1.E-03
1.E-04
1.E-05
1.E-06
1.E+00
Fig. 1
1.E+01
1.E+02
1.E+03
Number of cycles　(Cycles)
1.E+04
Margin exchange evaluation by Monte Carlo simulation and a simplified method
7