Reliability-based service life assessment of concrete structures in nuclear power plants: optimum inspection and repair

Nuclear Engineering and Design 175 (1997) 247–258

Reliability-based service life assessment of concrete structures innuclear power plants: optimum inspection and repair

Bruce R. Ellingwood a,*, Yasuhiro Mori b

a Johns Hopkins Uni6ersity, Baltimore, MD 21218, USAb Nagoya Uni6ersity, Nagoya, Japan

Received 1 December 1996; accepted 20 December 1996

Abstract

Aging effects in reinforced concrete structures brought on by severe service conditions may cause their structuralcapacities to deteriorate gradually over their service life. Research is being conducted to address aging managementof safety-related reinforced concrete structures in nuclear power plants (NPPs). Documentation is being prepared toidentify potential structural safety issues and to recommend criteria for use in evaluating reinforced concretestructures for continued service. Time-dependent reliability analysis provides the framework and quantitative tools forthe condition assessment. The role of in-service inspection and repair in ensuring continued reliability in service isexamined. Optimum strategies can be determined on the basis of minimum life-cycle cost. © 1997 Elsevier ScienceS.A.

1. Introduction

Structural aging of reinforced concrete struc-tures due to severe service conditions, aggressiveenvironments, or accidents may cause theirstrength and stiffness to deteriorate over time.Among the most pervasive aging mechanisms arecorrosion of reinforcement and deterioration ofconcrete due to chemical attack, expansive aggre-gate reactions or freeze–thaw damage, and fa-tigue from repetitive loads. The random fashionin which degradation occurs must be taken intoaccount in devising policies for in-service moni-toring and risk management of engineered facili-ties.

During the next 15 years, the operating licensesfor a number of nuclear power plants (NPPs) inthe USA will expire. Utilities may decide to seekrenewals of these licenses rather than to face thesubstantial costs associated with replacing lostgenerating capacity, plant shutdown and decom-missioning. Evaluation of a NPP facility withregard to its suitability for continued service en-compasses structural, mechanical and electricalcomponents (Vora et al., 1991; Shah andHookham, 1994). While structural componentsgenerally play a passive role in mitigating acci-dents from internal events, they play a significantrole in maintaining plant safety when accidentsare initiated by extreme environmental events. Incontrast to many mechanical and electrical com-ponents, structural replacement is not practical* Corresponding author.

0029-5493/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved.

PII S0029 -5493 (97 )00042 -3

B.R. Ellingwood, Y. Mori / Nuclear Engineering and Design 175 (1997) 247–258248

and structural failure may lead to malfunction ofother appurtenant systems. The evaluation of ag-ing structures for continued service must providequantitative evidence that their condition has notdeteriorated to the point where their capacity tomitigate future extreme operating or environmen-tal events within the proposed future service pe-riod is impaired. To achieve the desiredperformance goals, structures may have to beinspected and maintained periodically to ensurethe necessary level of reliability during a servicelife extension.

The Structural Aging Program, recently com-pleted for the US Nuclear Regulatory Commis-sion by Oak Ridge National Laboratory, wasconducted to: (1) identify and evaluate potentialstructural degradation processes in safety-relatedconcrete structures; (2) define issues to be ad-dressed during reviews for continued service andtechnical bases and criteria for resolution of theseissues; (3) identify and evaluate relevant in-serviceinspection, assessment and remedial measures;and (4) develop reliability-based methodologiesand engineering decision tools to perform suchassessments and evaluations (Naus et al., 1993,1996). Significant progress was made in identify-ing material property databases, critical structuralcomponents in NPPs (Hookham, 1991), nonde-structive strength evaluation methods for concrete(Snyder et al., 1992), and repair strategies fordifferent damage mechanisms (Krauss, 1994).

Reliability-based methodologies integrate infor-mation on design requirements, material andstructural degradation and damage accumulation,environmental factors and non-destructive evalua-tion (NDE) technology into a decision tool thatprovides a quantitative measure of structural reli-ability under projected future service conditions(Ellingwood and Mori, 1993; Mori and Elling-wood, 1993, 1994a,b). Optimum in-service inspec-tion–maintenance strategies can be determinedwithin this structural reliability framework tomeet performance goals stated in probabilisticterms. Research to develop reliability-based con-dition assessment methods has highlighted theneed for quantitative modeling of strength degra-dation and the impact of NDE on in-servicecondition assessment.

2. Mechanisms of deterioration in concretestructures

2.1. Ser6ice history of concrete structures

The performance of concrete containments andother safety-related structures in NPPs generallyhas been very good. Where deterioration has beenobserved, it generally has occurred early in the lifeof the plant and has been corrected (Naus et al.,1993; Shah and Hookham, 1994). Mechanisms ofpotential aging and deterioration in reinforcedconcrete structures in NPPs include corrosion ofreinforcement, detensioning of prestressing ten-dons, and deterioration of concrete due to attackby sulfates (particularly magnesium sulfates) orother chemicals, frost, expansive aggregate (pri-marily alkali–silica) reactions, and leaching orsalt recrystallization. One concern in structuralaging studies is the inaccessibility of certain rein-forced concrete components that have beenjudged to be important for maintaining plantsafety (Hookham, 1991) particularly componentsbelow grade that are exposed to soil and ground-water. Another is that most concrete contain-ments and other structures in NPPs have seen lessthan 25–30 years of service. The question arisesas to whether degradation mechanisms such asthose identified above could cause damage toreinforced concrete structures after years of ap-parently satisfactory service.

2.2. Mathematical models of degradation

Time-dependent reliability analysis and servicelife predictions for reinforced concrete structuresrequire time-dependent stochastic models of struc-tural strength. In concept, these strength modelscan be derived from: (1) mathematical modelsdescribing the effects of service and environmentalfactors on aging steel and concrete materials; (2)accelerated life testing; or (3) or combination ofthe two. At the current state of the art, agingeffects usually are known qualitatively; however,quantitative models that describe material degra-dation processes often are empirical in nature(Clifton and Knab, 1989).

B.R. Ellingwood, Y. Mori / Nuclear Engineering and Design 175 (1997) 247–258 249

The depth of deteriorated concrete or steel of-ten can be modeled to an acceptable approxima-tion by

X(t)=C(t−Ti)a (1)

in which t= time, Ti= induction or initiation timerequired to activate the deterioration process,C=rate parameter and a= time–order parame-ter. The parameters C and a must be determinedfrom experimental data. One must be cautious inusing the results of accelerated aging tests todetermine C and a as the mechanisms may notscale properly from the laboratory to the proto-type. For processes that are essentially diffusion-controlled (carbonation leading to initiation ofcorrosion of steel reinforcement), a=1/2; forother degradation mechanisms (sulfate attack), a

may be greater than 1.As noted above, corrosion is a pervasive mech-

anism of deterioration in structures. A review ofdata on uniform corrosion for carbon steels, someof which are similar in composition to the steelsused for reinforcement or concrete liners, indi-cates that the rate parameter, C, typically rangesfrom about 50–125 (when X(t) is measured in mmand t is in years); parameter a typically is between1/2 and 1. The variability in C reported in theliterature is quite large, coefficients of variation(COV) being as high as 0.7 but typically closer toabout 0.5. In comparison a evidently can be as-sumed to be deterministic. Because the uniformcorrosion rate generally decreases with time, onemust be cautious about drawing interferences re-garding future corrosion from a single in-serviceobservation. The induction time, Ti, clearly de-pends on the amount of concrete cover andwhether the concrete surface has cracked. In pre-vious studies, Ti has been assumed to be a log-normal random variable with a median typicallyof approximately 10 years (corresponding to rein-forcement cover of 28 mm) and a COV of 0.20(Vesikari, 1988).

Structural resistance, R(t), can be related toX(t) for a given behavioral limit state of interest,such as flexure, shear or compression. It can beassumed that the degraded material is ineffectivein resisting force. The relation between R(t) andX(t) may be nonlinear, depending on the nature

of the attack and the mechanics of the limit state.To place the above relation in perspective forcorrosion, if C=80 and a=3/4, X(t) would beabout 1 mm after 40 years. If such corrosion wereto occur in No. 8 reinforcing bars (nominally 25mm diameter) in a beam or slab, the section losswould cause a decrease in flexural strength toabout 85% of the initial strength after 40 years.

3. Reliability-based methods for conditionassessment

3.1. Time-dependent reliability analysis

Structural loads, engineering material proper-ties and strength degradation mechanisms are ran-dom in nature. Time-dependent reliability analysismethods provide a framework for the analysis ofuncertainty in loads and residual strength of agingreinforced concrete structures. Sources of uncer-tainty that complicate the evaluation of agingeffects on the residual strength arise from: (1)differences in design codes and standards for com-ponents of different ages; (2) lack of in-servicemeasurements and records; (3) variations in ser-vice loads; (4) limitations in available models forquantifying time-dependent material changes andtheir contribution to structural capacity; (5) limi-tations in nondestructive evaluation (NDE) tech-nologies applied in difficult field circumstances;and (6) shortcomings in existing methods for re-habilitation and repair.

The failure probability of a structural compo-nent can be evaluated as a function of (an intervalof) time if stochastic processes defining the resid-ual strength and the probabilistic characteristicsof the loads at any time are known. The strength,R(t), of the structure and applied loads, S(t),both are random functions of time. At any time, t,the margin of safety, M(t), is,

M(t)=R(t)−S(t) (2)

Making the customary assumption that R and Sare statistically independent, the (instantaneous)probability of failure is,


Pf(t)=P [M(t)B0]=&�

0

FR(x)fS(x) dx (3)

in which FR(x) and fS(x) are the cumulativedistribution function (CDF) of R and probabilitydensity function (PDF) of S. Eq. (3) provides aninstantaneous quantitative measure of structuralreliability, provided that Pf(t) can be estimatedand/or validated (Ellingwood, 1992).

For service life prediction and reliability assess-ment, one if more interested in the probability ofsatisfactory performance over some period oftime, say (0, T), than in the snapshot of thereliability at a particular time provided by Eq. (3).Indeed, it is difficult to use reliability-based engi-neering decision analysis without having sometime period (expected service life or in-serviceinspection interval) in mind. The probability thata structure survives during interval (0, T) isdefined by a reliability function, L(0, T). If, forexample, n discrete loads S1, S2,…, Sn occur attimes (t1, t2,…, tn) during (0, T), L(0, T) becomes,

L(0, T)=P [R(t1)\S1,…, R(tn)\Sn ] (4)

If the load process is continuous rather thandiscrete, a conservative approximation to L(0, T)can be obtained from a down-crossing analysis ofthe process, M(t), in Eq. (2) (Veneziano et al.,1977). We observe from Eq. (2) that failure in theinterval (0, T) is associated with the first excursionby M(T) of level zero from above. It can beshown that,

L(0, T)\1−E [NM(0, T)] (5)

in which E [NM(0, T)] is the expected number ofdown-crossings by M(t) of level zero during(0, T), determined as,

E [NM(0, T)]=& T

0

nM(x) dx (6)

in which nM(x)=mean rate of downcrossings ofM through zero. The down-crossing functionnM(x), being an average, is easier to determinethan the complete probability law of randomprocess, M(t).

The reliability function can be related to thehazard function, h(t), defined as the conditionalprobability of failure within time interval (t, t+dt), given that the component has survived during(0, t). One has (Ellingwood and Mori, 1993),

L(0, T)=exp�

−& T

0

h(x) dxn

(7)

The probability that the random time at struc-tural failure, Tf, occurs prior to future mainte-nance scheduled at t+Dt, given that the structurehas survived to t, then can be evaluated as,

P [Tf5 t+Dt �Tf\ t ]=1−exp�

−& t+Dt

t

h(x) dxn

(8)

In turn, the structural reliability for a successionof inspection periods is,

L(0, T)=5i

L(ti−1, ti) exp�

−& T

ti

h(x) dxn

(9)

in which ti−1=0 when i=1. A comparison ofEq. (5) and Eq. (6) with Eq. (7) indicates that forsmall limit state probabilities and failure rates,h(x):nM(x).

The hazard function for purely chance failuresis constant in time. When structural aging occursand strength deteriorates, h(t) characteristicallyincreases nonlinearly with time. This is unlike thesituation with mechanical or electrical compo-nents, where aging often has been assumed tocause the failure rate to increase linearly in time(Vesely, 1987).

It will be assumed in the sequel that significantstructural loads can be modeled as a sequence ofload pulses, the occurrence of which is describedby a Poisson process with mean rate of occurrencel, random intensity Sj, and duration t. Such asimple load process has been shown to be aneffective model for extreme service and environ-mental loads on NPP structures (Hwang et al.,1987), since normal service loads challenge thestructure to only a small fraction of its strength.With this assumption, the reliability function inEq. (4) becomes

L(0, T)

=&�

0

exp�

−lT(1−T−1 & T

0

FS(rg(x)) dx)n

fR(r)

× dr (10)

in which fR(r)=PDF of initial strength, R(0),and g(t)/R(0) is a function that describes thedegradation of strength in time. The limit state


Table 1Statistics of strength and load

Mean COVParameter Rate Duration PDF

1.12Mn Log-normal0.14—Initial flexure strength —60 years 1.0D 0.07Dead load Normal—0.25 years Type I0.4LLive load 0.500.5 years

probability or probability of failure during (0, T)is F(0, T)=1−L(0, T); note that F(0, T) is notthe same as Pf(t) in Eq. (3).

3.2. Illustration of ser6ice life prediction

Concepts of time-dependent reliability analysisare illustrated with an example of a reinforcedconcrete slab drawn from the recently completedStructural Aging Program research (Mori andEllingwood, 1993). The slab is 9.1m×9.1 m×305mm with 25.4 mm concrete cover on the rein-forcement. The specified compression strengthof the concrete is 28 MPa while the specifiedyield strength of the reinforcement is 414 MPa.The slab is uniformly loaded by dead and liveload, and is designed using the requirements forflexural strength in ACI Standard 349 (ACI,1990):

0.9Mn=1.4D+1.7L (11)

in which Mn is nominal or code (characteristic)flexural strength and D and L are moments (struc-tural actions) due to dead and live loads specifiedin the code. The reinforcement ratio for the slab is0.0110, and its nominal flexural strength is 2423kNm. It is assumed in this illustration that theslab is exposed to an environment that gives riseto corrosion of the reinforcement following car-bonation of the concrete cover.

The strength of the slab changes in time, ini-tially increasing a modest amount as the concretematures and then decreasing as active corrosiontakes place in the reinforcement. As the concretematures, the mean compressive strength increasesaccording to

fc(t)=15.5+3.95 ln tB47.9 MPa (12)

( fc measured in MPa, t is measured in days), from28.7 MPa at 28 days to 47.9 MPa at 10 years. Atthe same time, depth of corrosion of reinforce-ment is modeled by Eq. (1), in which the meanand COV in C are 30 and 0.50, respectively, anda=1.0 (Vesikari, 1988). The induction period isassumed to be deterministic and equal to 10 years;sensitivity studies reported elsewhere (Mori andEllingwood, 1993) indicated that variability in Ti

had a negligible effect in this particular problem.The dead load is modeled as a random variablethat is constant in time while the live load ismodeled as a Poisson pulse process. The load andinitial strength statistics used in this illustrationare summarized in Table 1; their basis is presentedelsewhere (Mori and Ellingwood, 1993).

The time-dependent change in the flexuralstrength of the slab and the live load process areillustrated conceptually by the sample functions,r(t) and s(t) (Fig. 1). The time-dependent strengthparameters (Eq. (1) and Eq. (12)) lead, on aver-age, to a decrease in flexural strength to 95% ofthe initial strength after 40 years. Two otherscenarios are illustrated in Fig. 1 for comparison,one in which strength degrades linearly to 90% ofinitial strength at 40 years and a second in whichstrength is constant in time.

Fig. 2 compares the limit state probabilitiesF(0, T)=1−L(0, T) for the three degradationmodels illustrated in Fig. 1 and for intervals (0, T)ranging up to 60 years. When R(t)=R(0) and nodegradation in strength occurs, one obtains aresult that is analogous to what has been done inprobability-based code work to date (Ellingwood,1992). Neglecting strength degradation entirely ina time-dependent reliability assessment can bequite unconservative, depending on the time-de-pendent characteristics of strength.


Fig. 1. Sample functions of load and strength of concrete slab.

4. In-service inspection and repair

4.1. Condition assessment

Changing load configurations, deteriorationand environmental degradation continually alterthe condition of a reinforced concrete structure inservice. Forecasts of time-dependent reliability(e.g. Fig. 2) enable the analyst to determine thetime period beyond which the desired reliability ofthe structure cannot be ensured. At such a time,the structure should be inspected and its conditionshould be evaluated. Intervals of inspection andmaintenance that may be required as a conditionfor continued operation can be determined fromthe time-dependent reliability analysis. ISI/M is aroutine part of managing aging and deteriorationin many engineered facilities (ASME, 1992); workalready is under way to develop reliability-basedpolicies for offshore platforms (Madsen et al.,1989) and aircraft (Yang, 1994). Reportedly, how-ever, utilities generally do not perform regular

inspections of NPP structures other than visualobservations or the monitoring of ungrouted pre-stressing tendons required by RG 1.35 (Regula-tory Guide, 1990). Many aging mechanisms maynot be easily detectable visually and can lead tocracking and strength degradation.

Structurally significant flaws must be locatedwith high probability during an inspection. Theprobability of detecting a flaw depends on thenature of the flaw and the NDE technology em-ployed. An ideal NDE technology would have ahigh probability of locating defective structureand, at the same time, a low probability of falseindications that might lead to unnecessary andcostly repair. No NDE method is perfect. Flawdetection with current methods is particularlydifficult in reinforced concrete structures becauseof the nonheterogeneity of the material, hostilefield conditions and the lack of accessibility tosome structural components of interest.

Intuition and experience suggests that larger ormore severe flaws should be easier to detect than


Fig. 2. Failure probability of concrete slab.

small flaws. Flaw detectability can be described bya function in the form of a cumulative probabilitydistribution, d(x), which simply expresses theprobability that an NDE device will be able todetect a flaw with size x. Several such functionsare illustrated in Fig. 3. Each NDE method has athreshold of detection or sensitivity, xth, belowwhich it is virtually certain that the flaw will notbe detected. Attempts to reduce xth by increasingthe resolution of NDE often lead to an increase inthe ‘false call’ probability. Conversely, there is anupper limit above which no improvement in de-tectability is noted, and d(x) may approach avalue dmaxB1.0 if flaw detection is not certain.Conceptually, there is such a function associatedwith each NDE method. Practically, d(x) oftenhas a large dispersion for field inspection, andsufficient data seldom exist to define d(x) formethods used to inspect concrete structures insitu. Sensitivity studies incorporating differentd(x) in time-dependent reliability analyses of con-crete structures indicate that the threshold of de-tection, xth, is more important than the shape ofd(x) above xth (Mori and Ellingwood, 1993).Thus, in the absence of further information, xth

can be taken as a ‘first-order’ indication of flawdetectability in the reliability analysis of concretestructures and d(x) becomes a step function asillustrated in Fig. 3. Coupled with uncertainty indetection is uncertainty in measurement. Deci-sions regarding repair are based on the flaw thatis observed while the actual flaw that is presentmay be different.

Inspecting a structure reveals something aboutits in-service condition that enables the PDF ofstrength to be updated. The information gainedfrom inspection usually involves several structuralvariables, including dimensions, defects and per-haps an indirect measure of strength or stiffnesssuch as ultrasonic pulse velocity (Snyder et al.,1992). If the results of the inspection can becollected in event B, the update PDF of resistancecan be obtained from Bayes theorem:

fR(r �B)=CK(B �r)fR(r) (13)

in which fR(r) is the PDF prior to inspection,K(B �r) is denoted the likelihood function, and Cis a normalizing constant to make fR(r �B) a legiti-mate PDF. If the structure subsequently is re-paired, the PDF of strength is again updated


Fig. 3. Probability of detecting flaw of size s or larger

using Eq. (13), in which the conditioning event Bnow depends on the effectiveness of the repairoperation. Various repair strategies have beenidentified as part of the Structural Aging Program(Krauss, 1994).

Although in theory one can associate a likeli-hood function K(B �r) with each inspection tech-nology and repair method, in practice this hasbeen found to be very difficult to do for rein-forced concrete structures. The effects of nonde-structive evaluation and repair on structuralcapacity are difficult to determine because of theheterogenous nature of reinforced concrete as aconstruction material, inaccessibility of certaincritical components, and difficult measurementconditions in situ. Uncertainties in the ISI/Mprocess stem from the imperfect nature of the flawdetection process, from unpredictable gain instrength following repair (depending on repairmethods selected) and other factors. These uncer-tainties must be reflected properly in the condi-tional PDF, fR(r �B).

The time-dependent reliability analysis is re-ini-tialized following ISI/M using the updated fR(r)in Eq. (13) in place of fR(r). This updating causesthe hazard function, h(t), to be discontinuous andL(0, T) to be discontinuous in its derivative.Beneficial maintenance causes the hazard functionto decrease and leads to improvement in theoverall reliability estimated with respect to somefuture service or inspection interval.

4.2. Illustration of reliability updating

As an illustration of the updating process, weconsider the same reinforced concrete slab as be-fore in Figs. 1 and 2. The mechanism causingdegradation is assumed to be corrosion of thereinforcement. We consider a 60-year service lifeand envision three alternative strategies: (1) Theslab is fully inspected at 30 years with a NDEtechnique capable of detecting defects causing a1% (or more) reduction in strength (denoted xth=0.01), and is fully repaired; (2) The slab is in-


Fig. 4. Mean degradation functions for a slab under alternate ISI/M policies.

spected at 20, 30, 40 and 50 years with a NDEtechnique capable of detecting only defects caus-ing an 8% (or more) reduction in strength (de-noted xth=0.08), and is fully repaired; or (3) Theslab is not inspected or repaired. Policy (1) in-volves a single thorough inspection at midlife,while policy (2) involves several more superficialinspections but at more frequent intervals. Themean degradation functions and time-dependentfailure probabilities, F(0, T)=1−L(0, T), are il-lustrated in Figs. 4 and 5. Both strategies 1 and 2lead to approximately the same failure probabili-ties. Among several alternate ISI/M policies, theselection of an appropriate policy depends on thetotal cost associated with each.

4.3. Optimum inspection–repair strategies

Costs of inspection and maintenance can be asignificant part of the overall life-cycle cost of areinforced concrete structure. Tradeoffs betweenthe extent of inspection, cost, and required levelof reliability can be performed systematically byreliability-based optimization methods using anobjective function that takes into account costs ofinspection, repair and failure. One might considerthe following optimization problem:

Minimize: Ct=Cins+Crep+CfF(0, T) (14a)

Subject to: F(0, T)5Ptarget (14b)

in which Cins= inspection cost, Crep=repair cost,Cf= loss due to failure, and Ptarget=establishedtarget failure probability during service life T. Thevariables in the optimization are the inspectiontimes, the extent of inspection and the thresholdof detection of the NDE device, xth, and theeffectiveness or quality of repair.Cost Cins dependson the quality and extent of inspection;

Cins=ainsAins(1−xth)bins (15a)

in which Ains is the area or volume of structureinspected, xth is the threshold of detection, andains, bins are constants. bins is greater than unity,reflecting the fact that NDE with low xth is rela-tively more costly. Crep is a linear function of thearea to be repaired and a nonlinear function ofthe damage to be repaired (Mori and Ellingwood,1994a,b);

Crep=arepArep(xmax)brep (15b)

in which Arep is the area or volume of structurerepaired, xmax is the maximum damage intensityto be repaired and arep, brep are constants. Allcosts should be discounted to present worth.


Fig. 5. Failure probability of a slab for alternate ISI/M policies.

The socioeconomic impact of failure (unservice-ability) of concrete components in NPPs isdifficult to assess in absolute terms at present.However, sensitivity studies can be conducted byvarying the relative costs in Eq. (14a) in fixedratios Cins:Crep:Cf (Mori and Ellingwood, 1994b).Fig. 6 illustrates an analysis of costs from Eqs.(14a), (14b), (15) and (15b) for the slab analyzedearlier, with ains:arep:Cf in the ratios 1:5000:10 000,xth=0.005 and F (0,60)51.5×10−4 (Mori andEllingwood, 1993). The optimal number of inspec-tions in this case is 2. Similar studies show thatwhen failure costs dominate over other costs andthe aging mechanism causes essentially lineardegradation in strength over time, the optimalpolicy is to perform in-service inspection–mainte-nance at essentially uniform intervals. Recent re-search has shown that computational complexities(if not physical insights) of time-dependent reli-ability and decision analysis can be managed us-ing the theory of Markov process (Rahman andGrigoriu, 1993; Tao et al., 1995). This approach isworthy of further consideration in developing fa-cility risk management policies.

5. Conclusions

When there are limited resources for ISI/Mavailable, it often is most effective to select a fewsafety-critical components and focus the monitor-ing efforts on these components (Hookham, 1991;Ellingwood and Mori, 1993). The reliability anal-ysis can be used to identify those components thatare most significant from a risk viewpoint, thusavoiding the need for costly monitoring. It ap-pears that if the cost of failure is orders of magni-tude larger than inspection and maintenancecosts, the optimal ISI/M policy is to inspect atnearly uniform intervals over the projected servicelife or service life extension.

Acknowledgements

This research was supported through GrantNo. 19X-SD684V from the Oak Ridge NationalLaboratory with Dr D.J. Naus as program man-ager. This support is gratefully acknowledged.


Fig. 6. Illustration of minimum expected cost analysis.

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Reliability-based service life assessment of concrete structures in nuclear power plants: optimum inspection and repair