Nuclear Engineering and Design 195 (2000) 239–260

Probabilistic elastic-plastic fracture analysis ofcircumferentially cracked pipes with finite-length surface

flaws

Sharif Rahman *Department of Mechanical Engineering, College of Engineering, The Uni6ersity of Iowa, 2140 Seamans Center, Iowa City,

IA 52242 1527, USA

Received 26 January 1999; received in revised form 23 August 1999; accepted 31 August 1999

Abstract

A probabilistic model was developed for predicting elastic-plastic fracture response and reliability of circumferen-tially cracked pipes with finite-length, constant-depth, internal surface cracks subject to remote bending loads. Itinvolves engineering estimation of energy release rate, J-tearing theory for characterizing ductile fracture, andstandard methods of structural reliability theory. The underlying J-estimation model is based on the deformationtheory of plasticity, a constitutive law characterized by power law model for stress-strain curve, and an equivalencecriterion incorporating reduced thickness analogy for simulating system compliance due to the presence of a crack.New analytical equations were developed to predict J-integral and are shown to be fairly accurate when comparedwith generally more accurate elastic-plastic finite-element results. Using this J-estimation method, fast probabilityintegrators and simulation methods were formulated to determine the probabilistic characteristics of J. The samemethods were used later to predict the probability of crack initiation and net-section collapse as a function of theapplied load. Numerical examples are provided to illustrate the proposed methodology. The results show thatprobabilistic analysis based on net-section collapse (without any margin) may significantly overpredict the reliabilityof surface-cracked pipes. © 2000 Elsevier Science S.A. All rights reserved.

www.elsevier.com/locate/nucengdes

1. Introduction

Degraded structural systems comprised ofcracked piping systems are often found in nuclearpower plants, off-shore drilling platforms, gastransmission lines, fossil power generation plants

and others. Of critical importance is the predic-tion of fracture response and reliability of crackedpipes under various operating conditions, so thatappropriate flaw acceptance criteria can be devel-oped. The fracture analysis can be based on lin-ear-elastic or more complex elastic-plastic(nonlinear) models. It is now well established thatthe nonlinear fracture-mechanics methods providemore realistic measures of fracture behavior ofcracked pipes with high toughness and low

* Tel.: +1-319-335-5679; fax: +1-319-335-5669; http://www.icaen.uiowa.edu/�rahman.

E-mail address: rahman@icaen.uiowa.edu (S. Rahman)

0029-5493/00/$ - see front matter © 2000 Elsevier Science S.A. All rights reserved.

PII: S0029 -5493 (99 )00214 -9

S. Rahman / Nuclear Engineering and Design 195 (2000) 239–260240

strength materials compared with the elasticmethods. In much or all of the working tempera-ture regime of these piping components, the mate-rial is being typically stressed above thebrittle-to-ductile transition temperature where thefracture response is essentially ductile and thematerial is capable of considerable inelastic defor-mation. As such, elastic-plastic theories should beused for fracture analyses of these structural com-ponents. While the development is still ongoing,significant progress has been made in the deter-ministic modeling of both linear-elastic fracturemechanics (LEFM) and elastic-plastic fracturemechanics (EPFM). Probabilistic models havealso been developed to estimate various responsestatistics and reliability (Provan, 1987). Currently,there are many applications of probabilistic frac-ture mechanics in the field of oil and gas, nuclear,naval, aerospace, and other industries. Nearly allof these methods have been developed strictlybased on LEFM models. On the other hand, theprobabilistic analysis based on EPFM models hasreceived only a limited attention to date. Proba-bilistic analyses based on EPFM models are justbeginning to appear, particularly, for applicationsin pressure boundary components (Rahman,1995; Rahman and Brust, 1995; Rahman et al.,1995; Rahman, 1996, 1997; Rahman and Kim,1997a; Rahman and Kim 1997b; Rahman,1998a,b; Rahman and Kim 1998).

In EPFM, the crack-driving force is frequentlydescribed in terms of J-integral. The J-integral isan appropriate fracture parameter that describesthe crack-tip stress and strain fields adequatelywhen there are no constraint effects. Similar toany deterministic EPFM problem, the evaluationof J-integral for probabilistic analysis can be per-formed by (1) numerical method and (2) engineer-ing estimation method. Traditionally, a numericalstudy has been based on elastic-plastic finite ele-ment method (FEM). Using FEM, one can calcu-late J for any crack geometry and loadconditions. However, it is also useful to havesimplified estimation methods for routine engi-neering calculations. Accordingly, the probabilis-tic EPFM analyses based on both methods havebeen reported (Rahman, 1995; Rahman andBrust, 1995; Rahman et al., 1995; Rahman, 1996,

1997; Rahman and Kim, 1997a; Rahman andKim 1997b; Rahman, 1998a,b; Rahman and Kim1998). For example, in the U.S. Nuclear Regula-tory Commission’s Short Cracks in Piping andPiping Welds Program (Wilkowski et al., 1991), aprobabilistic model was developed by the authorfor elastic-plastic analysis of circumferentialthrough-wall cracks in pipes for leak-before-breakapplications (Rahman et al., 1995). This modelinvolves a J-estimation method, statistical repre-sentation of uncertainties in loads, crack size, andmaterial properties, and first- and second-orderreliability methods (FORM/SORM). Shortlythereafter, similar probabilistic models based onother J-estimation formulas have also been re-ported (Rahman, 1995; Rahman and Brust, 1995;Rahman, 1996, 1997). In these models, the estima-tion formulas typically consist of closed-formequations of J as a function of load, crack size,and material properties of a structure and hence,do not require any expensive finite element calcu-lations. Actually, this is a major reason why theFORM/SORM algorithms have been successfullydeveloped for probabilistic analysis of elastic-plas-tic structures. These methods predict reasonablyaccurate failure probability and have been re-cently verified by the generally more accuratenonlinear FEM-based probabilistic analysis ofthrough-wall cracks in pipes (Rahman and Kim,1997a; Rahman and Kim 1997b; Rahman,1998a,b; Rahman and Kim 1998).

For analyzing circumferentially part-throughcracks in pipes by nonlinear FEM, the three-di-mensional nature of the surface crack requiresimmense computational effort to ensure adequatemesh design and enormous computer storage anddata reduction. This is quite a formidable taskeven for a deterministic study that typically re-quires a large number of analyses to developrobust criteria for pipe flaw evaluations (Krish-naswamy et al., 1995). To alleviate this problem,the use of line-spring/shell model, which essen-tially reduces the three-dimensional surface-crackproblem into a more tractable two-dimensionalshell problem, has been suggested by various re-searchers (Rice, 1972; Rice and Levy, 1972; Shira-tori and Miyoshi, 1980; Parks and White, 1982;Lee and Parks, 1995). The line-spring/shell model

S. Rahman / Nuclear Engineering and Design 195 (2000) 239–260 241

has been recently claimed to provide fairly accu-rate predictions of fracture response characteris-tics with much less computational effort than thethree-dimensional finite element analysis (Krish-naswamy et al., 1995; Mohan, 1998). Neverthe-less, the need to use FEM for surface cracks raisesthe degree of complexity in reliability analysis,because the performance function (limit statefunction) for FORM/SORM is not available asan explicit, closed-form function of input randomvariables. Although calculating J and other rele-vant fracture parameters by nonlinear FEM is notunduly difficult, the evaluation of response deriva-tives or sensitivities (e.g. gradient vector and Hes-sian matrix of J), required for FORM/SORManalysis, is a formidable task. Using finite-differ-ence method to calculate these derivatives is oftencomputationally expensive, since many repetitionsof nonlinear deterministic analysis are required,particularly when there is a large number of ran-dom parameters. An attractive alternative, whichis used in this study, is to develop approximateanalytical methods, which sidestep the need toperform full-scale three-dimensional finite elementanalysis from the very beginning. If these analyti-cal methods are accurate enough for deterministiccalculations, then a corresponding reliabilitymodel can be easily developed for probabilisticpipe fracture evaluations.

The objective of this study was to develop astochastic model for predicting elastic-plastic frac-ture response and reliability of circumferentiallycracked pipes with finite-length, constant-depth,internal surface flaws subject to pure bendingloads. It is based on (1) engineering estimation ofJ-integral, (2) J-tearing theory for characterizingductile fracture, and (3) standard computationalmethods of structural reliability theory. The J-es-timation method in the underlying deterministicmodel involves the deformation theory of plastic-ity, a constitutive law characterized by power lawmodel for stress-strain curve, and an equivalencecriterion incorporating reduced thickness analogyfor simulating system compliance due to the pres-ence of a crack. The equations for J-integral in acracked pipe were derived in closed form in termsof elementary functions. Both analytical and sim-ulation methods were formulated to determine the

probabilistic characteristics of J. The same meth-ods were used later to predict the probability ofcrack initiation and net-section collapse as a func-tion of the applied bending load. Numerical ex-amples are presented to illustrate both thedeterministic and probabilistic models of the pro-posed methodology. In the deterministic analysis,the validity of proposed J-estimation formula isevaluated by comparisons with generally moreaccurate finite-element results. In the probabilisticanalysis, the failure probabilities due to initiationof crack growth and net-section collapse are eval-uated for a cracked pipe with random materialproperties and crack size.

2. Elastic-plastic fracture analysis

Consider Fig. 1, which illustrates a pipe with aconstant-depth, internal, surface crack symmetri-cally placed in its bending plane. The pipe hasmean radius, Rm, wall thickness, t, and is subjectto pure bending moment, M applied at remoteends. The depth and total angle of the surfacecrack are denoted by a and 2u, respectively. It isassumed that the well-known Ramberg–Osgoodmodel can represent the material’s stress-strainresponse and is given by

o

o0

=s

s0

+a� s

s0

�n

(1)

in which s0 is a reference stress, which can bearbitrary, but usually assumed to be the yieldstress, E is the modulus of elasticity, o0=s0/E isthe associated reference strain, and a and n aremodel parameters usually chosen from best fit ofactual laboratory data. Also, the J-resistancecurve from the compact-tension specimen isdeemed to be adequately characterized by apower-law equation

JR(Da)=JIc+C�Da

k�m

(2)

in which Da is the extension of crack depth duringcrack growth, JIc is the plane strain mode-I frac-ture toughness at crack initiation, and C and mare model parameters also obtained from best fitof experimental data. In Eq. (2), k is a dummy

S. Rahman / Nuclear Engineering and Design 195 (2000) 239–260242

Fig. 1. A pipe with a circumferential surface crack subjected to pure bending.

parameter with a value of 1 introduced here onlyto dimensionalize C. Note that ‘Da ’ here is thephysical crack extension, i.e. without blunting.This is because blunting is automatically ac-counted for in the pipe estimation schemes as wellas finite element analysis.

2.1. The J-integral

Under elastic-plastic conditions and deforma-tion theory of plasticity when the stress-straincurve is modeled by Eq. (1), the total crack driv-ing force, J, can be obtained by adding the elasticcomponent, Je and the plastic component, Jp, i.e.

J=Je+Jp (3)

For a surface-cracked pipe under pure bending,closed-form expressions can be developed forboth Je and Jp. They are discussed in the follow-ing subsections.

2.1.1. Elastic solutionThe elastic component, Je, at the point of max-

imum depth, is given by (Krishnaswamy et al.,1995; Kumar and German, 1988)

Je=KI

2

E %(4)

where E %=E/(1−n2) for plane strain conditionwith E and n representing elastic modulus andPoisson’s ratio of the material, respectively, andKI is the mode-I stress intensity factor. From theLEFM theory, KI at the deepest point of thecrack is given by

KI=M

pRm2 t

FB(a/t, u/p)pa (5)

in which FB(a/t, u/p) is a geometry functionrelating KI of a cracked pipe to that for the samesize (depth) of a through-wall crack in an infiniteplate. In general, FB should be a function of a/t,u/p, and Rm/t. But, according to Article IWB-3650 in Section XI of the ASME Code (Evalua-tion of Flaws in Austenitic Steel Piping, 1986),

FB(a/t, u/p)

=1.1+at�

−0.09967+5.0057�a

tu

p

�0.565

−2.8329�a

tu

p

�n. (6)

S. Rahman / Nuclear Engineering and Design 195 (2000) 239–260 243

Fig. 2. Reduced section analogy by the SC.ENG1 and SC.ENG2 methods.

Hence, the elastic J is

Je=aFB(a/t, u/p)2 M2

pRm4 t2E %

. (7)

2.1.2. Plastic solutionThe plastic component, Jp, also at the point of

maximum depth, can be defined as

Jp=&M

0

(fpc

(AdM=

(

(A&M

0

fpc dM, (8)

the evaluation of which requires determination ofM−fp

c relationship. In a parallel study, the au-thor used an equivalence approach to determinethis relationship by replacing the actual crackedpipe with an uncracked pipe with reduced thick-ness, te, as shown in Fig. 2. Far from the crackplane, the rotation of the pipe is not greatlyinfluenced by whether a crack exists or some otherdiscontinuity is present as long as the discontinu-ity can approximate the effects of crack. Thereduced thickness section, which actually resultsin material discontinuity, is an attempt to simu-late the reduced system compliance due to thepresence of crack. Using this approach, the M−fp

c relationship can be estimated to be (Rahmanand Brust, 1997)

fpc =

� tte

�n−1� p

4K.�n

a�M

M0

�n−1

2MpRm

4 t2E %IB(a/t, u/p) (9)

where

K. =p

2

G�

1+12n�

G�3

2+

12n�, (10)

IB(a/t, u/p)

=2u�

Rm−t2�&

aFB2 (a/t, u/p)da

+2u&

a2FB2 (a/t, u/p)da, (11)

G(u)=&�

0

ju−1 exp(−j)dj, (12)

and M0=pRm2 ts0 is a suitable reference moment.

Placing this M−fpc relationship in Eq. (8) yields

Jp=a(M/M0)n+1(p/4K. )n

2tu(Rm− t/2+a)(n+1)�

H(a/t)n−1 dG(a/t)d(a/t)

+ (n−1)G(a/t)H(a/t)n−2 dH(a/t)d(a/t)

n(13)

where

H(a/t)=tte

(14)

and

G(a/t)=2

pRm4 t2E %

IB(a/t, u/p). (15)

S. Rahman / Nuclear Engineering and Design 195 (2000) 239–260244

Eqs. (7) and (13) provide closed-form solutionsof Je and Jp, respectively. This makes the pro-posed estimation method computationally feasibleand attractive for the development of probabilisticfracture-mechanics models.

2.2. The SC.ENG1 and SC.ENG2 methods

The evaluation of Jp in Eq. (13) requires deter-mination of H(a/t) and dH(a/t)/d(a/t). Accord-ing to the definition of H(a/t) (see Eq. (14)), thisalso requires an estimation of the equivalentthickness, te for the uncracked pipe. In the equiva-lence method proposed here te can be determinedby forcing the net-section-collapse moment of theequivalent uncracked pipe to be equal to thenet-section-collapse moment of the actual crackedpipe. For an uncracked pipe with reduced thick-ness, te, the net-section-collapse moment, ML

d , is

MLd =4sfRm

2 te (16)

where sf is the flow or collapse stress of thematerial1. However, in determining the net-sec-tion-collapse moment, ML

c , for circumferentialsurface-cracked pipe, there are several solutionsavailable in the current literature. In this study,two such equations, based on original and Kuri-hara modifications, were used to determine H(a/t) and its derivative for the evaluation of Jp.Accordingly, the expressions of Jp based on H(a/t) and dH(a/t)/d(a/t) obtained from the originalnet-section-collapse equations (Kanninen et al.,1976) and Kurihara modification to the net-sec-tion-collapse equations (Kurihara et al., 1988) aredefined as the SC.ENG1 and the SC.ENG2 meth-ods, respectively. The explicit details for the eval-uations of H(a/t) and dH(a/t)/d(a/t) by thesetwo methods are given in the next twosubsections.

2.2.1. The SC.ENG1 methodThe following are the original equations for the

net-section-collapse moment, MLc (Kanninen et

al., 1976) of a surface-cracked pipe under pure

bending and the resulting expressions for H(a/t)and dH(a/t)/d(a/t) used by the SC.ENG1method. For b5p−u,

MLc =2Rm

2 tsf

�2 sin b−

at

sin un

(17)

where

b=p−u(a/t)

2. (18)

When the limit loads from Eqs. (16) and (17) aremade equal,

H(a/t)=2

2 sinp−u(a/t)

2−

at

sin u

(19)

and

dH(a/t)d(a/t)

=2�

u cosp−u(a/t)

2+sin u

n�

2 sinp−u(a/t)

2−

at

sin un2. (20)

For b]p−u,

MLc =2Rm

2 tsf�

2−atn

sin b (21)

where

b=p(1−a/t)

2−a/t. (22)

When the limit loads from Eqs. (16) and (21) aremade equal,

H(a/t)=2

(2−a/t)sinp(1−a/t)

2−a/t

(23)

and

dH(a/t)d(a/t)

=

2p

2−a/tcos

p(1−a/t)2−a/t

+2 sinp(1−a/t)

2−a/t�(2−a/t) sin

p(1−a/t)2−a/t

n2 . (24)1 The flow stress is a stress value between the material’s yieldstrength and ultimate strength. Typically, it is assumed to bethe average of yield and ultimate strengths of a material.

S. Rahman / Nuclear Engineering and Design 195 (2000) 239–260 245

2.2.2. The SC.ENG2 methodThe following are the Kurihara modifications

(Kurihara et al., 1988) to the equations for thenet-section-collapse moment, ML

c of a surface-cracked pipe under pure bending and the resultingexpressions for H(a/t) and dH(a/t)/d(a/t) usedby the SC.ENG2 method. For b5p−u,

MLc =2Rm

2 tsf

�2m sin b+

�1−

at−m

�sin u

n(25)

where

b=p

2+

u(1−a/t−m)2m

. (26)

When the limit loads from Eqs. (16) and (25) aremade equal,

H(a/t)=2

K1(a/t)(27)

and

dH(a/t)d(a/t)

= −2

K1(a/t)2

dK1(a/t)d(a/t)

(28)

where

K1(a/t)

=2m sin�p

2+

u(1−a/t−m)2m

n+ (1−a/t−m) sin u (29)

and

dK1(a/t)d(a/t)

= −1m

cos�p

2+

u(1−a/t−m)2m

n�

mu+u(1−a/t)(m((a/t)

n+

2(m((a/t)

sin�p

2+

u(1−a/t−m)2m

n−�

1+(m((a/t)

nsin u. (30)

For b]p−u

MLc =2Rm

2 tsf

�1−

at+m

nsin b (31)

where

b=p(1−a/t)

1−a/t+m. (32)

When the limit loads from Eqs. (16) and (31) aremade equal,

H(a/t)=2

K2(a/t)(33)

and

dH(a/t)d(a/t)

= −2

K2(a/t)2

dK2(a/t)d(a/t)

(34)

where

K2(a/t)=Q1(a/t) sin Q2(a/t) (35)

dK2(a/t)d(a/t)

=Q1 cos Q2

dQ2

d(a/t)+sin Q2

dQ1

d(a/t)(36)

Q1(a/t)=1−at+m (37)

Q2(a/t)=p(1−a/t)

Q1(a/t)(38)

dQ1

d(a/t)= −1+

(m((a/t)

(39)

and

dQ2

d(a/t)= −

pQ1+p(1−a/t)dQ1

d(a/t)Q1

2 . (40)

In Eq. (25) through to Eq. (40), the functionsm(a/t, u/p) and #m(a/t, u/p)/#(a/t) are defined as

m(a/t, u/p)=1− (a/t)2(u/p)0.2 (41)

(m(a/t, u/p)((a/t)

= −2(a/t)(u/p)0.2 (42)

which are empirical functions developed by Kuri-hara et al. (1988). When the exponents in theequation for m(a/t, u/p) are chosen to be 2 and0.2 (see Eq. (41)), the predicted net-section-col-lapse moments of pipes with both short and longdeep flaws compare reasonably well with the ex-perimental data (Kurihara et al., 1988). Whenthese exponents are assigned large positive values,m approaches 1 and the resulting Kurihara mod-ifications to the net-section-collapse equations de-generate to the original equations. In that case,the difference between the SC.ENG1 and

S. Rahman / Nuclear Engineering and Design 195 (2000) 239–260246

SC.ENG2 methods also vanishes. In this study,however, the exponents suggested by Kurihara etal. (i.e. 2 and 0.2) were used for the developmentof SC.ENG2 method and subsequent numericalcalculations presented in this paper.

2.3. Deterministic e6aluation of SC.ENG1 andSC.ENG2 methods

To evaluate the accuracy of the SC.ENG1 andSC.ENG2 methods, the J-integral solutions forseveral surface-cracked pipes under pure bendingloads were compared with finite-element resultsfrom author’s previous work (Rahman and Brust,1997). Four pipe cases were considered. In allcases, the outer pipe diameter, Do=404.2 mm(15.91 inches) and pipe wall thickness, t=26.42mm (1.04 inches). The stress-strain curve wasidealized with the Ramberg–Osgood model (i.e.Eq. (1)). Two different crack lengths, such asu/p=1/4 (long crack) and u/p=1/16 (shortcrack), and two different hardening exponents,such as n=3 and n=10, were selected. In allcases, the crack depth ratio, a/t was fixed to be 50percent. Also, the following values were assumed:s0=241 MPa (35 ksi), a=1, E=207 GPa(30,000 ksi), and n=0.29.

Figs. 3–6 show the plots of J versus M ob-

tained by the proposed SC.ENG1 and SC.ENG2methods and FEM. The FEM solutions wereobtained by using the ABAQUS computer code(Version 5.3) with the line-spring/shell elements.Finite-element results from the line-spring/shellmodel were previously validated against thosefrom the full-scale three-dimensional model (Kr-ishnaswamy et al., 1995). The comparisons withthe FEM results suggest that the SC.ENG2method provides reasonably accurate estimates ofJ-integral for various applied loads for variouscombinations of crack length (u/p) and materialconstant (n). The agreement between theSC.ENG2 and FEM solutions is excellent whenu/p or n is larger. Figs. 3–6 also show the corre-sponding results from the SC.ENG1 methodwhich predicted smaller J compared with bothSC.ENG2 and finite-element solutions for anygiven moment. Since, the Kurihara modificationin SC.ENG2 method lowers the Net-Section-Col-lapse load from the original equations, the equiva-lent thickness, te is larger in SC.ENG1 methodthan that in SC.ENG2 method. Therefore, in theSC.ENG1 method, the values of H(a/t) [recall,H(a/t)= t/te ] would be smaller resulting insmaller Jp by the SC.ENG1 method as comparedwith that by the SC.ENG2 method (see Eq. (13)).

Fig. 3. Predicted J by the proposed methods and FEM (u/p=1/4, n=3).

S. Rahman / Nuclear Engineering and Design 195 (2000) 239–260 247

Fig. 4. Predicted J by the proposed methods and FEM (u/p=1/4, n=10).

Fig. 5. Predicted J by the proposed methods and FEM (u/p=1/16, n=3).

by the proposed SC.ENG1 and SC.ENG2 meth-ods, which are presented in this paper, are limitedto the specific pipe geometry, crack size, andmaterial constants defined above. More numericalresults from FEM are needed to systematicallyverify the accuracy of the proposed methods. Theyare subjects of current research by the author.

Hence, the trend shown in these figures is ex-pected. Since, the SC.ENG2 method providedbetter J predictions than the SC.ENG1 method, itwas decided to use the SC.ENG2 method forconducting subsequent probabilistic analyses to bediscussed in the forthcoming sections.

Note that the evaluations of J-integral predicted

S. Rahman / Nuclear Engineering and Design 195 (2000) 239–260248

3. Failure load

In order to evaluate structural integrity, it isrequired to know the load-carrying capacity of apiping system. There are several means by whichit can be estimated. They are based on variousdefinitions of failure criteria such as, initiation ofcrack growth and unstable crack growth in elas-tic-plastic fracture mechanics, and the Net-Sec-tion-Collapse in limit-load analysis. They arebriefly described in the following subsections.

3.1. Initiation load

The initiation load, Mi can be defined as thebending moment which corresponds to initiationof crack growth in a pipe. If J is a relevantcrack-driving force, it can be estimated by solvingthe following nonlinear equation

J(Mi, a0)−JIc=0 (43)

in which J(Mi, a0) is the energy release rate (ie.J-integral) for load Mi and initial crack depth a0,which can be obtained from Eqs. (3), (7) and (13),and JIc is the plane strain mode-I fracture tough-ness at crack initiation. Standard numerical meth-ods, such as the bisection method, Newton–Raphson method, and others, can be applied tosolve Eq. (43) (Press et al., 1990).

3.2. Maximum or instability load

In applications of nonlinear fracture mechanics,the J-tearing theory is a very prominent conceptfor calculating maximum load-carrying capacityof a pipe. It is based on the fact that fractureinstability can occur after some amount of stablecrack growth in tough and ductile materials withan attendant higher applied load level at fracture.Let J and JR (a−a0) denote the crack-drivingforce and fracture toughness of a ductile pipingmaterial as a function of load and crack-depthextension, respectively. The limit state characteriz-ing fracture instability based on J-tearing theoryis given by

J(Mmax, a*)−JR(a*−a0)=0 (44)

(J(a

(Mmax, a*)−(JR

(a(a*−a)=0 (45)

where Mmax and a* are load and crack depthwhen crack growth becomes unstable. Eqs. (44)and (45) are two nonlinear simultaneous equa-tions with the independent variables Mmax and a*.Again, they can be solved by standard methods,such as the Newton–Raphson method (Press etal., 1990).

Fig. 6. Predicted J by the proposed methods and FEM (u/p=1/16, n=10).

S. Rahman / Nuclear Engineering and Design 195 (2000) 239–260 249

Note, the J-resistance curve, which is typicallygenerated from small-scale laboratory specimens,must be prescribed in such a way that similarconstraint conditions exist in both cracked struc-ture and laboratory specimens. Otherwise, con-straint effects should be accounted for in thefracture-mechanics analysis.

3.3. Net-section-collapse load

The Net-Section-Collapse analysis is a simpleand straightforward method for predicting failureload of a cracked pipe. In this analysis, it isassumed that (1) the failure load occurs when thepipe section containing the crack becomes fullyplastic, (2) there is insignificant crack growth fromcrack initiation to failure, and (3) the toughnessof the material is sufficiently high so that failure isgoverned by the strength of materials (i.e. the flowstress). The flow stress is a value between the yieldand ultimate strengths of a material and repre-sents an average critical net-section stressthroughout the flawed ligament of the structure.Based on this assumptions, the original Net-Sec-tion-Collapse load, Mnsc, is given by (Rahman,1998; Rahman and Wilkowski, 1998)

Mnsc=

2sfRm2 t�

2 sin b−at

sin un

, b5p−u

2sfRm2 �2−

atn

sin b, b]p−u

(46)

where,

b=

p−uat

2, b5p−u

p�

1−at�

2−at

, b]p−u

(47)

which is a repetition of Eqs. (17) and (21). Previ-ously, the flow stress sf was not needed to bequantified to calculate H(a/t) for the SC.ENG1and SC.ENG2 methods. For calculating Net-Sec-tion-Collapse moment of a pipe, however, it must

be specified explicitly. In this paper, sf was as-sumed to be the average of yield and ultimatestrengths of the pipe material. Note that the alter-native limit-load equations based on Kuriharamodifications (see Eqs. (25) and (31)) could havebeen used, but they were not used here for calcu-lating the Net-Section-Collapse moment explicitly.This is consistent with the load calculations in theASME Section XI Code (Rahman and Brust,1997).

4. Probabilistic fracture mechanics and reliability

4.1. Random parameters and fracture response

Consider a cracked pipe with uncertain me-chanical and geometric characteristics that is sub-ject to random loads. Denote by X anN-dimensional random vector with componentsX1, X2,…, XN characterizing all uncertainty in thesystem and load parameters. For example, when asurface-cracked pipe is considered, the possiblerandom components are: crack length ratio, u/p,initial crack depth ratio, a0/t, pipe radius-to-thick-ness ratio, Rm/t, elastic modulus, E, basic tensilestrength parameters, sy and su, Ramberg-Osgoodconstitutive parameters, a and n, fracture tough-ness parameters, JIc, C, and m, and applied bend-ing moment, M. All or some of these variablescan be modeled as random variables. Hence, anyrelevant response, such as the J-integral, shouldbe evaluated by the probability

FJ( j0)=def

Pr[J(X)B j0]=def&

J(x)B j 0

fX(x)dx (48)

or the probability density fJ( j0)=dFJ( j0)/dj0,where FJ( j0) is the cumulative probability distri-bution function of J and fX(x) is the known jointprobability density function of X.

The above fracture parameter J can also beapplied to determine load-carrying capacity ofsurface-cracked pipes. Several fracture criteriabased on this J-integral parameter and Net-Sec-tion-Collapse are discussed in this paper. In ageneric sense, let Mf(X) denote the failure mo-ment for a given surface-cracked pipe under purebending. Note that Mf(X) is always random be-

>

S. Rahman / Nuclear Engineering and Design 195 (2000) 239–260250

cause it depends on input vector X which israndom. It can be evaluated when a relevantcrack driving force from deterministic fracture(e.g. J-integral from finite element analysis orEqs. (7) and (13)) and an appropriate fracturecriteria (e.g. Eq. (43) or Eqs. (44) and (45)) areknown. Suppose that the design requires Mf(X) tobe always greater than the applied load M (M canbe random as well). This requirement cannot besatisfied with certainty because both the systemand the load parameters are uncertain. Hence, theperformance of the cracked pipe should be evalu-ated by the reliability PS or its complement, theprobability of failure, PF(PS=1−PF) defined as

PF=def

Pr[g(X)B0]=def&

g(x)B0

fX(x)dx (49)

where g(X) is the performance function given by

g(X)=Mf−M

=h(sy, su, a, n, JIc, C, m, u/p, a0/t, Rm/t, E)

−M (50)

in which h is a function (implicit) of randomparameters characterizing pipe’s structural resis-tance (only the random arguments are shown inEq. (50)). The failure probability, defined by Eq.(49), can be evaluated if the appropriate fracturecriterion is known. For example, when Mf(X) isequal to the initiation load Mi(X), PF in Eq. (49)corresponds to the probability of initiation ofcrack growth which may provide a conservativeestimate of pipe’s structural performance. A morerealistic evaluation of pipe’s reliability can beevaluated if Mf(X) is equal to maximum loadMmax(X) (which allows the crack to grow until itbecomes unstable or through-wall) in which casePF represents failure probability due to the ex-ceedance of pipe’s maximum load-carrying capac-ity. When EPFM-based failure criteria are notnecessary, simple performance function based onlimit-load analysis [i.e., Mf(X)=Mnsc(X)] can al-so be used to determine failure probability of pipes.

Numerical efforts are often required to com-pute Mf(X), particularly when Mf(X)=Mi(X) orMmax(X). This is true in spite of analytical repre-sentation of Je- and Jp-integrals. In this paper, theNewton-Raphson method was used to conduct

the numerical analysis.

4.2. Structural reliability analysis

The generic expression for both probabilities inEqs. (48) and (49) involves a multidimensionalprobability integration for their evaluation. In thisstudy, standard reliability methods, such as first-and second-order reliability methods (Hasoferand Lind, 1974; Rackwitz and Fiessler, 1978;Ditlevsen, 1979a,b; Fiessler et al., 1979; Chen andLind, 1982; Breitung, 1984; Der Kiureghian andLiu, 1986; Madsen et al., 1986; Der Kiureghian etal., 1987; Hohenbichler et al., 1987; Breitung,1989; Tvedt Tvedt, 1990; Haldar and Mahadevan,1995), Monte Carlo with importance sampling(MCIS) (Harbitz, 1986; Hohenbichler, 1988;Ibrahim and Rahman, 1991), and direct MonteCarlo simulation (MCS) (Rubinstein, 1981), wereused to compute these probabilities. They arebriefly described here to compute the probabilityof failure PF in Eq. (49) assuming a genericN-dimensional random vector X and the perfor-mance function g(x) defined by Eq. (50). Thesame methods can be applied to determine theprobability FJ( j0) defined by Eq. (48).

4.2.1. First- and second-order reliability methodsThe first- and second-order reliability methods

are based on linear (first-order) and quadratic(second-order) approximations of the limit statesurface g(x)=0 tangent to the closest point of thesurface to the origin of the space. The determina-tion of this point involves nonlinear constrainedoptimization and is usually performed in the stan-dard Gaussian image of the original space. TheFORM/SORM algorithms involve several steps.First, the space x of uncertain parameters X istransformed into a new N-dimensional space uconsisting of independent standard Gaussian vari-ables U. The original limit state g(x)=0 thenbecomes mapped into the new limit state gU(u)=0 in the u space. Second, the point on the limitstate gU(u)=0 having the shortest distance to theorigin of the u space is determined by using anappropriate nonlinear optimization algorithm.This point is referred to as the design point orbeta point, and has a distance bHL (known as

S. Rahman / Nuclear Engineering and Design 195 (2000) 239–260 251

reliability index) to the origin of the u space. Third,the limit state gU(u)=0 is approximated by asurface tangent to it at the design point. Let suchlimit states be gL(u)=0 and gQ(u)=0, whichcorrespond to approximating surfaces as hyper-plane (linear or first-order) and hyperparaboloid(quadratic or second-order), respectively. Theprobability of failure PF (Eq. (49)) is thus approx-imated by Pr[gL(U)B0] in FORM andPrÈgQ(U)B0É in FORM. These first-order andsecond-order estimates PF,1 and PF,2 are given by(Hasofer and Lind, 1974; Rackwitz and Fiessler,1978; Ditlevsen, 1979a,b; Fiessler et al., 1979; Chenand Lind, 1982; Breitung, 1984; Der Kiureghianand Liu, 1986; Madsen et al., 1986; Der Ki-ureghian et al., 1987; Hohenbichler et al., 1987;Breitung, 1989; Tvedt; Tvedt, 1990; Haldar andMahadevan, 1995)

PF,1=F(−bHL) (51)

PF,2$F(−bHL) 5N−1

i=1

�1−ki

f(−bHL)F(−bHL)

�−1/2

(52)

where

f(u)=1

2pexp

�−

12

u2� (53)

and

F(u)=1

2p

& u

−�

exp�

−12

j2�dj (54)

are the probability density and cumulative distri-bution functions, respectively, of a standard Gaus-sian random variable, and ki is the ith principalcurvatures of the limit state surface at the designpoint. Further details of FORM/SORM equationsare available in the following references, Hasoferand Lind (1974), Rackwitz and Fiessler (1978),Ditlevsen (1979a,b), Fiessler et al. (1979), Chenand Lind (1982), Breitung (1984), Der Kiureghianand Liu (1986), Madsen et al. (1986), Der Ki-ureghian et al. (1987), Hohenbichler et al. (1987),Breitung (1989), Tvedt; Tvedt (1990), Haldar andMahadevan (1995).

FORM/SORM are standard computationalmethods of structural reliability theory. In this

study, a modified HL-RF algorithm, described inAppendix A, was used to solve the associatedoptimization problem. The first- and second-ordersensitivities were calculated numerically by thefinite difference method.

4.2.2. Monte Carlo simulationConsider a generic N-dimensional random vec-

tor X which characterizes uncertainty in all loadand system parameters with the known joint distri-bution function, FX(x). Suppose that x (1), x (2),…,x (L) are L realizations of input random vector, X,which can be generated independently. Rubinstein(1981) provides a simple method to generate Xfrom its known probability distribution. Let g (1),g (2),…, g (L) be the output samples of g(X) corre-sponding to the input x (1), x (2),…, x (L) that can beobtained by conducting repeated deterministicevaluation of the performance function in Eq. (50).Define Lf as the number of trials, which areassociated with negative values of the performancefunction. Then, the estimate PF,MCS by simulationis

PF,MCS=Lf

L, (55)

which approaches the exact failure probability PF

when L approaches infinity. When L is finite, astatistical estimate on the probability estimatormay be needed. In general, the required sample sizemust be at least 10/Min(PF, 1−PF) for a 30%coefficient of variation of the estimator

4.2.3. Monte Carlo with importance samplingIn Monte Carlo with importance sampling (Har-

bitz, 1986; Hohenbichler, 1988; Ibrahim and Rah-man, 1991), the random variables are sampledfrom a different probability density, known as thesampling density. The purpose is to generate moreoutcomes from the region of interest, e.g. thefailure set F={x : g(x)B0}. Using informationfrom FORM/SORM analyses, good sampling den-sities can be constructed. According to Hohenbich-ler (1988), the failure probability estimate PF,IS byimportance sampling based on SORM improve-ment is given by

S. Rahman / Nuclear Engineering and Design 195 (2000) 239–260252

and J-resistance curves of a specific pipe material(e.g. Type 304 stainless steel) at 288°C (550°F)were obtained from the following references,Landes et al. (1984), Wilkowski et al. (1985),Hiser and Callahan (1987), Van Der Sluys (1988),Schmidt et al. (1991), Chopra (1993). These datawere then fitted with the Eqs. (1) and (2) todetermine the constitutive model parameters a

and n, and fracture toughness parameters, JIc, C,and m. (This is a standard practice in the deter-ministic pipe fracture evaluations.) The basicstrength parameters, such as yield strength, sy

(0.2% offset) and ultimate strength, su, were de-termined as well. These provided the independentmeasurements of the random vectors{sy, su}T, {a,n}T, and {JIc, C, m}T representing pipe materialproperties. Following standard statistical analy-ses, conducted in Rahman et al. (1995), Table 1shows the mean and covariance for each of theserandom vectors. It was assumed that the jointprobability distribution of each vector was log-normal. This was justified via comparisons withactual data, which indicated that the marginalprobability of each component of the above vec-tors would follow lognormal distribution reason-ably well (Rahman et al., 1995). A Gaussiandistribution also seems to be good choice, butthere are some concerns on the possible negativerealizations of some of these positive randomvariables, which have large coefficients of varia-tion. Hence, X was modeled with lognormal prob-ability although no rigorous proof is providedhere to validate this assumption by comparing themultivariate joint probability distributions. Also,no correlations were permitted between thestrength and toughness properties because eachset of laboratory data did not always includesimultaneous measurement of all properties.However, the components within each vector arecorrelated and their correlation characteristics aredefined in the covariance matrices provided inTable 1.

The methods to generate samples of genericrandom vector, X, which are needed in FORM/SORM and simulation analyses, require Rosen-blatt transformation to obtain standard Gaussianvector, U (Rosenblatt, 1952). For special cases,

PF,IS$F(−bHL) 5N−1

i=1

[1−kiC(−bHL)]−1/2

×1

NIS

%NIS

j=1

F[hQ(wj)]F(−bHL)

exp�

−12

C(bHL) %N−1

k=1

kkwk, j2 n

(56)

where

C(−bHL)=f(−bHL)F(−bHL)

, (57)

wj={w1,j, w2,j,…, wN−1,j}T is the jth realization ofan N−1 dimensional independent Gaussian ran-dom vector W with the mean and variance of itsith component being zero and 1/[1−C+ (−bHL)], respectively, hQ(wj) is the quadratic ap-proximant in the form of rotationalhyperparaboloid, and NIS is the sample size forimportance sampling. Further details are availableelsewhere (Hohenbichler, 1988).

5. Numerical applications

5.1. Description of the problem

Consider a surface-cracked side riser pipe madeof Type 304 Stainless Steel from a Boiling WaterReactor (BWR) plant. The pipe has outer diame-ter, Do=709.2 mm (27.92 inches), wall thickness,t=33.77 mm (1.33 inches) [i.e. Rm/t=10], elasticmodulus, E=182 700 MPa (26 500 ksi), Poisson’sratio, n=0.3, and reference stress, s0=152 MPa(22 ksi). These parameters were treated as deter-ministic variables since no significant statisticalvariability was found from their actual measure-ments. The operating temperature at BWR condi-tion was assumed to be 288°C (550°F). Therandom parameters included crack size parame-ters, u/p and a0/t, yield strength, sy, ultimatestrength, su, Ramberg-Osgood constitutiveparameters, a and n, and mode-I fracture tough-ness parameters, JIc, C, and m. The statisticalproperties of these variables are described below.

5.1.1. Statistical characteristics of materialproperties

The samples of raw data for the stress-strain

S. Rahman / Nuclear Engineering and Design 195 (2000) 239–260 253

Table 1Mean and covariance of material properties for Type 304 stainless steel pipe at 288°C (550°F)

Random vector Covanance matrixMean vector

!sy

su

"a !151.526

450.632

" �220.881

118.615

118.615

652.654

n!a

n

"b !8.942

3.615

" � 10.920

−1.202

−1.202

0.208

nÍÁ

Ä

JIc

C

m

ÌÂ

Åc

ÃÆ

È

2.204×105

−58.937

−25.530

−58.937

1.006×104

6.842

−25.530

6.842

0.0242

ÃÇ

ÉÍÁ

Ä

1059.56

345.087

0.652

ÌÂ

Å

a Both sy and su are in MPa unit.b a and n are dimensionless; s0=152 MPa, E=182 700 MPa (see Eq. (1)).c Both JIc and C are in kJ/m2 unit with k=1 mm (see Eq. (2)); m is dimensionless; Da is to be expressed in mm unit.

5.2. Probabilistic characteristics of J-integral

The second-order reliability method was ap-plied to determine the probabilistic characteristicsof J-integral for the side riser pipe for severalapplied moments. To understand better the rela-tive magnitude of these moments, the Net-Sec-tion-Collapse moment (M0 nsc) of this pipe when allrandom input variables are assigned to their meanvalues was calculated using Eq. (46), and wasfound to be: M0 nsc=3.73 MN-m. Fig. 7 shows thecomputed probability densities [ fJ( j0)] of J-inte-gral for values of the applied moment, M=1.49MN-m (M/M0 nsc=0.4), 1.87 MN-m (M/M0 nsc=0.5), and 2.24 MN-m (M/M0 nsc=0.6). They wereobtained by repeated SORM analysis for variousthresholds of J, i.e. by calculating the probabilityin Eq. (48) as a function of j0 and then takingnumerical derivative of this probability with re-spect to j0. As expected, the probability massshifts to the right when the applied moments arehigher. Also presented in the same figures are thecorresponding histograms of J-integral developedby conducting direct MCS for the same values ofapplied moments. The sample size for each MonteCarlo analysis was 10 000. The results indicatethat SORM can predict probabilistic characteris-

when X is either correlated normal or correlatedlognormal, the above transformation can besidestepped by invoking Cholesky decompositionof the covariance matrix (Rahman et al., 1995).Further details on the sample generation of X,either generic or normal/lognormal, are availablein Rahman et al. (1995).

5.1.2. Statistical properties of initial flaw sizeIn order to perform probabilistic analysis, the

probability distributions of initial crack lengthratio, u/p and crack depth ratio, a0/t need to bespecified as well. Ideally, this information shouldcome from in-service inspection data so that anyuncertainty in detected crack size can be charac-terized by their probability. In this example prob-lem, it was assumed that both u/p and a/t wouldfollow Gaussian probability distribution withmeans 0.25 and 0.5, respectively, and standarddeviations 0.025 and 0.05, respectively. Thesestatistical parameters and their probability distri-butions were chosen quite arbitrarily only to illus-trate the proposed methodology. Actualdetermination of their statistical properties willdepend on inspection quality (e.g. measurementerror in crack size), material type, and other fac-tors. They were not considered in this study.

S. Rahman / Nuclear Engineering and Design 195 (2000) 239–260254

(M/M0 nsc=0.22) in Fig. 9, and M=1.49 MN-m(M/M0 nsc=0.4) in Fig. 10, representing small, in-termediate, and large magnitudes of applied mo-ment, respectively. For each moment, theprobability densities of Je, Jp, and J (J=Je+Jp)were computed by SORM. From Fig. 8, it appears that when the applied moment is small(M=0.37 MN-m), the elastic component of J is

tics of J with very good accuracy when comparedwith the Monte Carlo method for all values ofapplied moment considered here.

Figs. 8–10 show the componential probabilitydensities (as well as total) of J-integral computedfor several values of applied moment. Three loadcases were considered and they were: M=0.37MN-m (M/M0 nsc=0.1) in Fig. 8, M=0.82 MN-m

Fig. 7. Accuracy of SORM in predicting probabilistic characteristics of J-integral.

Fig. 8. Probabilistic characteristics of Je-, Jp- and J-integrals for M=0.37 MN-m.

S. Rahman / Nuclear Engineering and Design 195 (2000) 239–260 255

Fig. 9. Probabilistic characteristics of Je-, Jp- and J-integrals for M=0.82 MN-m.

Fig. 10. Probabilistic characteristics of Je-, Jp- and J-integrals for M=1.49 MN-m.

is more pronounced and thus fully plastic fracture-mechanics analysis becomes necessary to evaluatepiping integrity. Finally, in Fig. 9, when the loadmagnitude is somewhat intermediate (M=0.82MN-m), both components of J and hence, EPFMare needed to predict the fracture behavior of pipes.

much greater than the plastic component of J andhence, the fracture behavior is significantly gov-erned by the elastic properties of pipe. This isequivalent to an LEFM analysis. On the otherhand, when the applied load is large (M=1.49MN-m) as in Fig. 10, the plastic component of J

S. Rahman / Nuclear Engineering and Design 195 (2000) 239–260256

MCS, the sample size was varied according to thelevel of probability being estimated. In all cases,the sample size was targeted to be 10/Min(PF,1−PF) for obtaining a 30% coefficient of varia-tion of the probability estimator.

The computer program used to illustrate thisexample problem did not have the calculation ofmaximum load coded. Hence, comparisons offailure probabilities due to exceedance of maxi-mum load and crack initiation load could not bemade in this study. However, for a pipe with asurface crack, the difference between initiationand maximum loads is not typically large sincethe amount of crack growth is limited to theuncracked ligament of the pipe thickness. Hence,the failure probabilities of pipes based on maxi-mum and initiation loads are expected to be veryclose. This situation is, however, very different fora through-wall-cracked pipe in which case a sig-nificant amount of circumferential crack growthcan occur for a large-diameter pipe if the essentialfracture conditions exist. In that case, probabilityof failure based on maximum and initiation loadscan be very different. For further details, seeauthor’s previous work on through-wall-crackedpipes (Rahman, 1995; Rahman and Brust, 1995;Rahman et al., 1995; Rahman, 1996, 1997; Rah-man and Kim, 1997a; Rahman and Kim 1997b;Rahman, 1998a,b; Rahman and Kim 1998).

5.3. Piping reliability assessment

Fig. 11 shows the plots of exceedance probabil-ities of crack initiation moment, Mi and Net-Sec-tion-Collapse moment, Mnsc versus appliedmoment M for the side riser pipe described ear-lier. Various reliability methods, such as FORMand SORM, and simulation methods, such asMCIS and MCS, were used to determine thefailure probability. They all consistently indicatethat PF increases as M increases, and it ap-proaches unity when M becomes very large. Com-pared with the failure probability due to initiationof crack growth, the failure probability based onNet-Section-Collapse was found to be muchlower. Unless there is sufficient evidence that thepipe will fail with a limit-load criterion, an analy-sis based on the Net-Section-Collapse load (with-out any safety margin) may overpredict thereliability of pipes.

Fig. 11 also shows that the results obtained bythe approximate methods, e.g., FORM andSORM, provide satisfactory probability estimateswhen compared with the results from ImportanceSampling and MCS methods. No meaningful dif-ferences were found between the results of FORMand SORM and their probability estimates arevirtually identical. During the performance of

Fig. 11. Probability of failure by various reliability methods.

S. Rahman / Nuclear Engineering and Design 195 (2000) 239–260 257

Fig. 12. Computational efficiency of FORM/SORM and MCIS.

cient than the direct MCS and are far superiorparticularly when the failure probabilities are inthe lower range.

6. Summary and conclusions

A probabilistic model was developed for pre-dicting elastic-plastic fracture response of circum-ferentially cracked pipes with finite-length,constant-depth, internal surface flaws subject toremote bending loads. It involves engineering esti-mation of energy release rate, J-tearing theory forcharacterizing ductile fracture, and standardmethods of structural reliability theory. The un-derlying J-estimation model is based on the defor-mation theory of plasticity, a constitutive lawcharacterized by power law model for stress-straincurve, and an equivalence criterion incorporatingreduced thickness analogy for simulating systemcompliance due to the presence of a crack. Newanalytical equations were developed to predict theJ-integral for a surface-cracked pipe under purebending. Both analytical and simulation methodswere formulated to determine the probabilisticcharacteristics of J. The same methods were usedlater to predict the probability of crack initiationand net-section collapse as a function of the ap-

Finally, Fig. 12 exhibits the computational ef-fort needed to produce the results of Fig. 11 byFORM, SORM, Importance Sampling, and MCSmethods. In particular, it was measured in termsof total number of deterministic analyses requiredby each of these methods to predict the probabil-ity of crack initiation. The plots in Fig. 12 showhow the required number of analyses by thesemethods varies with the range of probability esti-mates made in this study. It appears that thenumber of analyses and hence, the computationaleffort required by FORM, SORM, and Impor-tance Sampling methods are almost constant re-gardless of the probability estimates that variedfrom 10−9 to 1. On the other hand, the number ofanalyses required by MCS method was inverselyproportional to PF (PF in this case is the probabil-ity of failure due to crack initiation) which sug-gests that for a probability estimate of about2.5×10−2 or lower, a significant amount of com-puter time can be saved by using FORM, SORM,and Importance Sampling methods instead of us-ing MCS. Also, the differences in number ofanalyses required by FORM, SORM, and Impor-tance Sampling method are quite negligible whencompared with the number of analyses requiredby MCS. Clearly, the FORM/SORM algorithmsand Importance Sampling method are more effi-

S. Rahman / Nuclear Engineering and Design 195 (2000) 239–260258

plied load. Several numerical examples are pre-sented. The results showed that:� The predicted J-integral by the proposed ana-

lytical methods, in particular the SC.ENG2method, agreed reasonably well with the resultsof generally more accurate elastic-plastic finiteelement analyses.

� Current reliability methods, such as FORMand SORM, provided accurate probabilisticcharacteristics of J-integral and failure loadsfor surface-cracked pipes under bending withmuch less computational effort when comparedwith those obtained by direct MCS. Similaraccuracy and computational efficiency werealso demonstrated by the Importance Samplingmethod.

� The failure probabilities due to the exceedanceof initiation load were much lower than thatdue to the exceedance of Net-Section-Collapseload. Large differences may exist in the resultsproduced by each of these two failure criteria,especially when the applied load levels aresmaller for which failure probabilities are alsosmaller. Unless there is adequate evidence thatthe pipe will fail with a limit-load criterion, ananalysis based on Net-Section-Collapse load(without any margin) may overpredict the reli-ability of a piping system significantly.

Acknowledgements

The author would like to thank Dr. F.W. Brustof Battelle for his support during the developmentof SC.ENG1 and SC.ENG2 methods. Part of thiswork was also supported by the Faculty EarlyCareer Development Program of the U.S. Na-tional Science Foundation (Grant No. CMS-9833058). The program directors were Drs. KenChong and Sunil Saigal.

Appendix A. A modified HL-RF method

In FORM/SORM, the main effort is calculat-ing the reliability index, bHL= u* by finding thedesign point, u*, which can be formulated as aconstrained optimization problem defined by

minu�RN

u subject to gU(u)=0

(58)

where RN is an N-dimensional real vector space,u�RN is the space of standard Gaussian vector,U�RN, and gU(u) :RN � R is the transformedperformance function in u-space, and

u =def' %

N

i= i

u i2 (59)

is the Euclidean L2-norm of the N-dimensionalvector, u. A modified HL-RF method, originallyproposed by Hasofer and Lind (1974) and laterextended by Rackwitz and Fiessler (1978) andmodified by Liu and Kiureghian (1991), is one ofthe most widely used and robust optimizationmethods to solve the reliability problem in Eq.(58) (Rackwitz and Fiessler, 1978; Liu and Ki-ureghian, 1991). The original ML-RF methodinvolves an iterative algorithm given by the fol-lowing recursive formula

uk+1=1

9gU(uk) 2 [9gU(uk)Tuk−gU(uk)]9gU(u)

(60)

where, uk is the vector at kth iteration, 9={#/#u1, #/#u2,…, #/#uN}T is a vector of gradientoperators, and 9gU(uk) is the gradient of scalarfield, gU(uk). The algorithm proceeds iterativelyuntil convergence is achieved, i.e. when

�uik+1−ui

k�5ocon, for all i (61)

and

�gU(u*)�$ �gU(uk+1)�5ocon (62)

where ocon is a small control parameter assignedby the user. From the past experience of authors,a value of ocon=10−4 to 10−3 usually yieldssatisfactory estimates of bHL.

To improve the robustness of Eq. (60), Liu andKiureghian proposed a non-negative merit func-tion, m(uk), which is defined as (Liu and Ki-ureghian, 1991)

m(uk)

=12*

uk−9gU(uk)Tuk

9gU(uk) 2 9gU(uk)*2

+12

cgU(uk)2

(63)

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where, c is some scalar positive constant. Themerit function in Eq. (63) is a convenient guidefor selecting step size, since it is a function ofquantities already known at the current iterationpoint, uk. This modification greatly improves theconvergence (although not strictly guaranteed) ofthe original HL-RF method (Liu and Kiureghian,1991).

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