Nuclear Engineering and Design 236 (2006) 350–358

Failure probability assessment of wall-thinned nuclear pipesusing probabilistic fracture mechanics

Sang-Min Lee, Yoon-Suk Chang, Jae-Boong Choi, Young-Jin Kim∗School of Mechanical Engineering, Sungkyunkwan University, 300 Chunchun-dong, Jangan-gu,

Suwon, Kyonggi-do 440-746, Korea

Received 14 April 2005; received in revised form 15 September 2005; accepted 16 September 2005

Abstract

The integrity of nuclear piping system has to be maintained during operation. In order to maintain the integrity, reliable assessment proceduresincluding fracture mechanics analysis, etc., are required. Up to now, this has been performed using conventional deterministic approaches eventhough there are many uncertainties to hinder a rational evaluation. In this respect, probabilistic approaches are considered as an appropriatemethod for piping system evaluation. The objectives of this paper are to estimate the failure probabilities of wall-thinned pipes in nuclear secondarys gram usingr to internp sults showeda dition ando©

1

btptmstrycmctaie

inties,

hasnsid-n and

sincesev-egrityle, ausingtress

APilitiessting

oderate

wclear

0d

ystems and to propose limited operating conditions under different types of loadings. To do this, a probabilistic assessment proeliability index and simulation techniques was developed and applied to evaluate failure probabilities of wall-thinned pipes subjectedalressure, bending moment and combined loading of them. The sensitivity analysis results as well as prototypal integrity assessment repromising applicability of the probabilistic assessment program, necessity of practical evaluation reflecting combined loading con

peration considering limited condition.2005 Elsevier B.V. All rights reserved.

. Introduction

The wall thinning phenomenon due to erosion/corrosion haseen considered as an important issue for integrity evalua-

ion of nuclear piping. Recent failure accident of feedwateripe at Mihama Unit 3 nuclear power plant in Japan was a

ypical case caused by it (Korea, 2004). The integrity of pri-ary and secondary piping systems in nuclear power plants

hould be maintained during operation. In order to maintainhe integrity of nuclear piping, usually complicated procedu-al assessment is required including fracture mechanics anal-sis, etc. Until now, it has been mainly carried out by usingonventional deterministic approaches even though there areany uncertainties to hinder a rational evaluation of structural

omponents. All kinds of uncertainties related to loading his-ory, material property and failure mechanism are taken intoccount in engineering safety factors. However, from a real-

stic assessment point of view of structural components, it isasily expected that the deterministic evaluation results are gen-

erally too conservative because all the relevant uncertaare accumulated into the unique safety factors (Yagawa et al.2001).

In this respect, probabilistic fracture mechanics (PFM)risen as a useful alternative. The probabilistic approach is coered as an appropriate methodology in reasonable evaluatiorisk-based decision making for major nuclear componentsit can deal with various uncertainties quantitatively. To date,eral PFM programs have been developed as means of intevaluation tools to resolve industrial issues. For exampmethod to establish steam generator plugging strategyMonte Carlo simulation was proposed in case of axial scorrosion cracking at tube expansion transition zone (Cizelj andMavko, 1995). And a computer program based on the SINTprocedure has been developed to calculate failure probabwhen a defect size is either given by non-destructive te(NDT)/non-destructive examination (NDE) or not (Dillstrom,2000). Also, for leak-before-break applications, a computer ctitled PSQUIRT was developed to evaluate probabilistic leakin nuclear piping (Rahman et al., 1996). However, there are feefficient programs for field use to assess wall-thinned nu

∗ Corresponding author. Tel.: +82 31 290 5274; fax: +82 31 290 5276.E-mail address: yjkim50@skku.edu (Y.-J. Kim).

pipes under combined loading of internal pressure and bendingmoment.

029-5493/$ – see front matter © 2005 Elsevier B.V. All rights reserved.oi:10.1016/j.nucengdes.2005.09.008

S.-M. Lee et al. / Nuclear Engineering and Design 236 (2006) 350–358 351

Nomenclature

d defect depth (mm)Do outer diameter of pipe (mm)l defect length (mm)Mc plastic collapse moment (kN m)Mcrit. critical bending moment (kN m)ML plastic limit moment (kN m)Mop bending moment (kN m)Mref reference normalizing moment subjected to bend-

ing moment (kN m)MC

ref reference normalizing moment subjected to com-bined loading (kN m)

Mβ=3.0 limited bending moment (kN m)Nfailure number of simulation cycles when the failure

occurredNtarget total number of simulation cyclespcrit. critical internal pressure (MPa)Pf failure probabilitypL plastic limit pressure (MPa)pop internal pressure (MPa)pβ=3.0 limited internal pressure (MPa)Ri inner radius of pipe (mm)Rm mean radius of pipe (mm)t thickness of pipe (mm)

Greek lettersβ reliability indexθ half of total defect angle (rad)κ curvatureµ mean value of probabilistic variableσ standard deviation of probabilistic variableσf flow stress (MPa)σ local equivalent stress averaged in the minimum liga-

ment of the defect (MPa)σy yield strength (MPa)σu ultimate tensile strength (MPa)Φ cumulative standard normal distribution function

The purposes of this paper are to estimate the failure probabilities of wall-thinned pipes in the nuclear secondary systemand to propose limited operating conditions under differenttypes of loadings. To do this, an integrity assessment programbased on PFM was developed, which adopts both of reliabilityindex and simulation techniques. Then, it was applied to evaluate the failure probabilities of wall-thinned pipes subjectedto internal pressure, bending moment and combined loadingof them. During the estimation, effects of several limit statefunctions (LSFs) are investigated and extent of contributionby each variable on failure probability is examined throughsensitivity analyses as well. Based on the prototypal integrityassessment results, finally, limited operating conditions of wall-thinned nuclear pipes are suggested at reference reliabilitindex.

2. Development of probabilistic assessment program

2.1. Fundamentals of probabilistic approach

The PFM can be used to determine failure probabilities (Pf )of components by treating the scatter of applied loads, struc-tural geometries and material properties adequately. The fail-ure behavior of component is described by limit state function(LSF),g(x), depending on basic random variablesx = (x1, x2, . . .,xn) which denote the several parameters. By definition,g(x) < 0implies failure condition whereas no failure occurs forg(x) > 0and 0 defines the limit state. Then the failure probability isobtained by integrating the probability density function (PDF) ofrespective basic variablesxi over the region ofg(x) < 0 (Haldarand Mahadevan, 2000).

In order to estimate the failure probability, two techniquesare generally used: the one is reliability index technique suchas first order reliability method (FORM) and second order reli-ability method (SORM), and the other is simulation technique.In FORM, through a linearization of the LSF at design point, anapproximate failure probability can be determined as followingwell-known expression

Pf = Φ(−β) = 1 − Φ(β) (1)

whereΦ is the cumulative standard normal distribution functiona nceb esignp

ratich limits forms rdera

P

w aleo hems

usedt domv f theb ilurep

P

w ureo

2

learp undert sure,

-

-

y

ndβ the reliability index that represents the minimum distaetween the origin of the space of basic variables and the doint on failure surface.

In SORM, the failure surface is approximated by a quadyper-surface associated with the curvature of non-lineartate around the minimum distance point. A simple closed-olution for the probability computation using second-opproximation is given as follows:

f ≈ Φ(−β)n−1∏i=1

(1 + βκi)−1/2 (2)

hereκi is theith main curvatures of the limit state and the vf∏n−1

i=1 (1 + βκi)−1/2 is a specific term of SORM called as tultiplication factor, even though the definitions ofΦ andβ are

ame with those in FORM.The Monte Carlo simulation (MCS) method, also, can be

o estimate the failure probability. It generates sets of ranariables according to the given probabilistic distributions oasic variables and puts them into the LSF. Thereby, the farobability can be determined by Eq.(3)

f = limNtarget→∞

[Nfailure

Ntarget

]≈ Nfailure

Ntarget(3)

hereNfailure is the number of simulation cycles when the failccurred andNtarget is the total number of simulation cycles.

.2. Specifics of probabilistic integrity assessment

Probabilistic integrity assessment of wall-thinned nucipes is conducted by using the three evaluation methods

he pre-described loading conditions that are internal pres

352 S.-M. Lee et al. / Nuclear Engineering and Design 236 (2006) 350–358

bending moment and combined loading. The correspondingLSFs are constructed based on deterministic criteria proposedby Kanninen and Kim.

The LSF based on Kanninen’s estimation equation (Kanninenet al., 1982) that was derived from a net-section collapse stressunder pure bending condition can be expressed as follows:

LSF : g(xi) = Mc − Mop (4)

whereMop is the bending moment andMc is the plastic collapsemoment of pipes with constant depth flaws represented as below

LSF : Mc = 2σfR2mt

(2 sin α − d

tsin θ

);

α = 1

2

(π − d

tθ

)(5)

in the above equation,σf is the flow stress given by the averageof yield strength (σy) and ultimate tensile strength (σu), Rm isthe mean radius of pipe andt is the pipe wall thickness. Notethat Eq.(5) is just for pure bending condition and thus the effectof axial tension due to internal pressure is not considered.

The LSF based on Kim’s estimation equation (Kim et al.,2004) that uses a local stress at the deepest point in semi-elliptically wall-thinned area is as follows:

L

w ts neda binel

In case of internal pressure, since only hoop and axial stressesare components of principal stress under thin-wall approxima-tion, the equivalent stress in the minimum ligament can be givenby

LSF : σplocal =

pop

(pref/σy)(7)

where

pref

σy= t

Rm

1√A2 − AB + B2

(8)

A = 1

(1 − (d/t)) + (d/t)(1/φ);

B = Ri

2Rm; φ =

√1 + 1.61l2

Rit(9)

In case of bending moment, the limit-load solution derived fromequilibrium stress fields is represented as

LSF : σMlocal =

Mop

(Mref/σy)(10)

where

M = ML ; M = 4σ R2 t

{cos (

π dθ) − d f (θ)

}

f

SF : g(xi) = σu − σlocal (6)

hereσ local indicated in Eqs.(7), (10) and(13) is equivalentress averaged over the minimum ligament in locally thinrea under internal pressure, bending moment and com

oading (Shim et al., 2005).

Fig. 1. The flowchart of wall-thinn

d

ref 1.333L y m 8 t t 2θ

(11)

(θ) = 0.7854θ2 − 0.09817θ4 + 0.0040906θ6 − 0.000085θ8

(12)

ed piping assessment program.

S.-M. Lee et al. / Nuclear Engineering and Design 236 (2006) 350–358 353

Fig. 2. The main window of wall-thinned pipe assessment program.

In case of combined loading, from Eqs.(7) and(10), the equiv-alent stress (σC

local) in the minimum ligament is expressed as

LSF : σp+Mlocal =

√√√√(pRi

2t− M

MCref/σy

)2

−(

pRi

2t− M

MCref/σy

)(p

pL/σy

)+(

p

pL/σy

)2

(13)

where

pL = σyt

Rm

(1 − d

t+ d

t

1

ϕ

); ϕ =

√1 + 1.61l2

Rit(14) MC

ref = 3R2mtσy

{cos

(πdθ

8t+ πpRm

4tσy

)− d

t

f (θ)

2θ

}(15)

uatio

Fig. 3. The sample screens of piping eval n: (a) FORM/SORM modules; (b) MCS module.

354 S.-M. Lee et al. / Nuclear Engineering and Design 236 (2006) 350–358

Table 1Input variables of wall-thinned nuclear pipes (Caleyo et al., 2002; Miyazaki etal., 1999)

Variable µ CV

Defect depth,d (mm) 4.3, 6.9 0.1Defect length,l (mm) 100 0.1Defect angle,θ/π 0.5 0.1Outer diameter,Do (mm) 114.3 0.02Thickness,t (mm) 8.6 0.02Yield strength,σy (MPa) 326 0.07Ultimate tensile strength,σu (MPa) 490 0.07Internal pressure,pop (MPa) 30–65 –Bending moment,Mop (kN m) 5–40 –

2.3. Probabilistic assessment program

The wall-thinned piping integrity assessment program basedon PFM was developed using Microsoft Visual C++ 6.0 andMicrosoft Access 2000.Fig. 1 shows the flowchart of devel-oped program that consists of database part and evaluation partincorporating the reliability index and MCS techniques. In thedatabase part, basic information such as operating conditions,pipe/defect geometries and material properties is contained,which has been collected during each overhaul period. On the

Fn

other hand, in the evaluation part, the failure probability of wall-thinned nuclear pipes can be assessed and its results are fed backinto the database part. Specifically, input variable transforma-tion, iteration and numerical analysis functions are incorporatedin the FORM and SORM modules while random number genera-tion, probability distribution generation and evaluation functionsare included in the MCS module.

Fig. 2 depicts the main window of the probabilistic assess-ment program, in which the user can select one of integrityevaluation methods such as FORM/SORM or Monte Carlo sim-ulation. Fig. 3 shows the sample screens of piping evaluationusing FORM/SORM and MCS modules, respectively. In gen-eral, for the integrity evaluation, a set of input data includingthe material properties of yield and ultimate tensile strengths,defect and pipe geometries associated with the correspondingPDF type, mean (µ) and standard deviation (σ) are requiredunder deterministic loading conditions. From the input data,corresponding coefficient of variation (CV) that is a function ofµ andσ is determined automatically. For MCS, additionally, anappropriate replication number and sufficient simulation numberare required to get reliable results by controlling the simulation.Thereby, as the evaluation results, the calculated failure prob-ability is returned with reliability index in FORM module andmultiplication factor in SORM module.

ig. 4. The prototypal failure probability evaluation results for wall-thinneduclear pipes: (a) internal pressure; (b) bending moment.

Fwv

ig. 5. The comparison of prototypal failure probability evaluation results forall-thinned nuclear pipes with a corroded depth ofd/t = 0.5: (a) internal pressures. combined loading; (b) bending moment vs. combined loading.

S.-M. Lee et al. / Nuclear Engineering and Design 236 (2006) 350–358 355

3. Probabilistic integrity assessment of wall-thinnednuclear pipes

3.1. Determination of PDF

The PDFs related to the defect shape, pipe geometryand material property should be determined for probabilis-tic integrity assessment of wall-thinned nuclear pipes. In thisresearch, due to lack of actual field data, theµ and CV of prob-abilistic variables were quoted from references with respect toSTS410 carbon steel (Caleyo et al., 2002; Miyazaki et al., 1999).Table 1indicates the probabilistic variables as well as determin-istic variables for applied loading condition. With respect to the

mean of defect depth, especially, two values were chosen forfurther investigation since it is regarded as the most importantvariable.

3.2. Failure probability assessment

The failure probabilities of wall-thinned nuclear pipes havebeen calculated using the aforementioned LSFs and relatedprobabilistic variables.Fig. 4 shows the prototypal failureprobability assessment results for nuclear pipes subjected tointernal pressure (pop) and bending moment (Mop), respec-tively. In the figure, to represent the failure probability asgeneralized form, governing parameters were normalized with

Fo

ig. 6. The effect of probabilistic variables on failure probability for wall-thinneuter diameterDo; (d) thickness,t; (e) ultimate tensile strength,σu.

d nuclear pipes under internal pressure: (a) defect depth,d; (b) defect lengthl; (c)

356 S.-M. Lee et al. / Nuclear Engineering and Design 236 (2006) 350–358

corresponding material properties such asσ local/σu and Mop/Mc.

In case of wall-thinned pipe withd/t = 0.5 (d = 4.3 mmandt = 8.6 mm) under internal pressure, the failure probabilitywas estimated as below 10−8 at internal pressure of 30 MPaand increased monotonically up to about 1 at 65 MPa. Also,the critical pressure (pcrit.) was determined as 60 MPa at themoment when the estimated local equivalent stress (σ local) inthe minimum ligament of pipe equals to the ultimate tensilestrength (σu). Besides, for pipe withd/t = 0.8 (d = 6.9 mm andt = 8.6 mm), the failure probability reached about 1 at 45 MPaand the critical pressure was 40 MPa although the relevantfigure was not included in this paper. The design pressureof secondary piping system in Korean nuclear power plantis usually less than 10 MPa. Thus, in the conventional deter-

ministic approach, it can be easily perceived that the wall-thinned nuclear pipes withd/t = 0.5 or 0.8 under the threefoldvalue of the typical design pressure (30 MPa) have sufficientmargin.

For wall-thinned pipes under bending moment, the criticalvalue (Mcrit.) can be defined as the estimated equivalent stressof pipe equals to the ultimate tensile strength in Kim’s equa-tion or the applied moment (Mop) equals to the plastic collapsemoment (Mc) in Kanninen’s equation. By adopting these two cri-teria, the critical moment of wall-thinned pipe withd/t = 0.5 wasdetermined as 25.8 kN m from Kim’s equation and 26.4 kN mfrom Kanninen’s equation. So, in this case, the Kim’s equationestimated slightly conservative results than Kanninen’s equa-tion. However, in case of nuclear pipe withd/t = 0.8, the criticalmoment was calculated as about 16 kN m from Kanninen’s equa-

Fl

ig. 7. The optimum operating conditions of internal pressure and bending mooading; (b) bending moment vs. combined loading.

ment for wall-thinned nuclear pipes withd/t = 0.5: (a) internal pressure vs. combined

S.-M. Lee et al. / Nuclear Engineering and Design 236 (2006) 350–358 357

tion which was 20% lower than the one estimated by using Kim’sequation.

Fig. 5 represents the comparison of prototypal failure prob-ability assessment results for wall-thinned nuclear pipe withd/t = 0.5 under combined loading. In accordance with the addi-tional bending moment or internal pressure, from 0 to 30% ofpre-defined critical values calculated by Kim’s equation, thecorresponding failure probabilities were varied significantly. Asdepicted inFig. 5(a), by adding the bending moment from 0 to30% of critical value to original pure internal pressure condition,the failure probabilities at 35 MPa level were increased about1070–11,880% per each increment. Also, as shown inFig. 5(b),by adding the internal pressure from 0 to 30% of critical valueto original pure bending condition, the failure probabilities at15 kN m level were increased about 2160–13,770% per eachincrement. Thereby, it is believed that, under combined loadingcondition, the effect of internal pressure was slightly higher thanthat of the bending moment.

3.3. Sensitivity analyses

Sensitivity analyses are necessary to confirm the reliabilitiesand to measure the uncertainties of probabilistic input variables,which affect on failure probability. In this paper, as representa-tive case, the sensitivity analyses under pure internal pressurecondition have been conducted using Kim’s estimation equational es ev le.

ith– incet vel oi as thr gt bil-i h CVv ectl thef CV.T ail-u Vf seda rentud litiesw heref havt andu

3

bilityi nts( ighc minor

consequence of failure. In this paper, the limited operating con-ditions of wall-thinned nuclear pipes under internal pressureand bending moment were proposed using LSF based on Kim’sequation (Kim et al., 2004) at reference reliability index (β = 3.0,Pf = 0.00135).

Fig. 7(a) represents the proposed limited operating conditionsof wall-thinned nuclear pipes withd/t = 0.5 under pure internalpressure and combined loading. According to the variation ofadditional bending moment, cross the board, the proposed lim-ited internal pressure (pβ=3.0) acting on nuclear pipe varied from27.1 to 42.7 MPa. When the loading condition was changed from0 to 30% of critical moment at reference reliability index level,the proposed limited pressure was decreased about 12–16% pereach increment of it due to combined loading effect.Fig. 7(b)shows the proposed limited operating conditions under purebending moment and combined loading at the reference relia-bility index level. For the case under pure bending condition, theproposed limited bending moment (Mβ=3.0) was about 19 kN mthat corresponds to the three-quarters of the critical moment.By increasing the values from 0 to 30% of critical moment, theones were decreased about 10–13% per each increment. Thus itmay be possible to propose limited operating conditions basedon the assignedβ = 3.0. For example, when considering pureloading conditions, the wall-thinned nuclear pipes withd/t = 0.5

Fig. 8. The normalized limited operating conditions for wall-thinned nuclearpipes withd/t = 0.5 and 0.8: (a) internal pressure vs. combined loading; (b)bending moment vs. combined loading.

s LSF. The five probabilistic variables (defect depth,d; defectength, l; outer diameter,Do; thickness,t and ultimate tensiltrength,σu) represented in Eqs.(6)–(9) were selected and thariational effects of CV (=σ/µ) were analyzed for each variab

Fig. 6depicts the effects of CVs on failure probabilities w25%, 25% and 50% variation from the original values. S

he estimated failure probabilities were dependent on the lenternal pressure, in this sub-section, 35 MPa was chosenepresentative pressure for comparison. InFig. 6(a), by changinhe CV of defect depth from 0.15 to 0.075, the failure probaties were significantly decreased about 90–98% per eacariation.Fig. 6(b) and (c) shows the effects of different defength and outer diameter distributions, however, in whichailure probabilities were rarely affected by variations ofhe effect of different pipe wall thickness distribution on fre probability is shown inFig. 6(d). By decreasing the C

rom 0.03 to 0.015, the failure probabilities were decreabout 60% per each variation. Finally, the effect of diffeltimate tensile strength distribution is shown inFig. 6(e). Byecreasing the CV from 0.105 to 0.0525, the failure probabiere decreased approximately 85% per each variation. T

ore, when the same mean value is considered, cautionso be devoted for setting the distributions of defect depthltimate tensile strength.

.4. Limited operating condition

The factor of three can be assigned as a target reliandex for limit state of structure in various nuclear power plaFaber and Sorensen, 2002). It was suggested considering host of measure for defects at structural components and

fe

-e

358 S.-M. Lee et al. / Nuclear Engineering and Design 236 (2006) 350–358

should be operated within 71% (=42.7/60.0 MPa) of the criticalpressure or 74% (=19.0/25.8 kN m) of the critical moment.

The limited operating conditions for wall-thinned nuclearpipes withd/t = 0.8 was also calculated. From pure internal pres-sure condition to internal pressure +30% of critical moment,the proposed limited internal pressures were varied from 7.8to 20.4 MPa. By increasing the values from 0 to 30% of crit-ical moment, the ones were decreased approximately 23–35%per each increment owing to combined loading effect. For thecase under pure bending moment, the proposed limited bendingmoment was about 9.7 kN m that was a half of critical momentof the pipe. By increasing the values from 0 to 30% of criticalpressure, the limited ones decreased about 21–31%. Thus, inthis case, it may also be possible to recommend limited operat-ing conditions based on the assignedβ = 3.0. For instance, thewall-thinned nuclear pipes withd/t = 0.8 subjected to pure load-ing conditions should be operated within 51% (=20.4/40.0 MPa)of the critical pressure or 51% (=9.7/19.2 kN m) of the criti-cal moment.Fig. 8represents the normalized limited operatingconditions for wall-thinned nuclear pipes withd/t = 0.5 and 0.8under internal pressure, bending moment and combined loadingconditions.

4. Conclusions

In this paper, the structural integrity assessment programb d ana pes.T

( bilityarlogrity

( d foreadribed

en

( lureulti-idesrel-

atively higher than those of defect length and outer diameterof pipe.

(4) The limited operating conditions for wall-thinned pipes withdifferent d/t values were proposed at reliability index = 3.Those were decreased from about 75% level of critical valuewhend/t= 0.5–50% level of critical value whend/t = 0.8.

Acknowledgements

The authors are grateful for the support provided by agrant from Safety and Structural Integrity Research Center atSungkyunkwan University.

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