Wenxing Zhou1Assistant ProfessorDepartment of Civil and EnvironmentalEngineering,University of Western Ontario,London, ON, N6A 5B9, Canadae-mail: wzhou@eng.uwo.caMaher A. NessimC-FER Technologies,Edmonton, AB, T6N 1H2, Canadae-mail: m.nessim@cfertech.comOptimal Design of OnshoreNatural Gas PipelinesTheoptimaldesignlevelforonshorenaturalgaspipelineswasexploredthroughahy-potheticalexample,wherebythepipewallthicknesswasassumedtobethesoledesignparameter. Theprobabilitydistributionsofthelife-cyclecostsofvariouscandidatede-signsfortheexamplepipelinewereobtainedusingMonte-Carlosimulation. Thelife-cyclecostincludedthecostoffailureduetoequipmentimpactandexternalcorrosion,andthecostofperiodicmaintenanceactionsforexternalcorrosion.Thecostoffailureincluded both the cost of fatality and injury as well as the cost of property damage andvalue of lost product. The minimumexpectedlife-cycle cost criterionandstochasticdominanceruleswereemployedtodeterminetheoptimal designlevel. Theallowablesocietal risk level was considered as a constraint in the optimal design selection. It wasfoundthat theCanadianStandardAssociationdesignleadstotheminimumexpectedlife-cycle cost and satises the allowable societal risk constraint as well. A set of optimaldesignsforarisk-aversedecisionmakerwasidentiedusingthestochasticdominancerules. BoththeASMEandCSAdesignsbelongtotheoptimal designset andmeet theallowable societal risk constraint. DOI: 10.1115/1.4002496Keywords: optimal design, onshoregas pipeline, life-cyclecost, failuremode, failureconsequences, risk attitude1 IntroductionAlthoughtransmissionpipelinesarewidelyrecognizedasthesafest means for mass transportation of hydrocarbons, incidents ofpipeline failures, dened as loss of pressure containment, do occuralbeit infrequently duetovarious hazards suchas equipmentimpact due to third-party interference, corrosion, and groundmovement. Failures of pipelines can lead to severe consequences,including loss of human life, damage to both industry assets andpublic property, and damage to the environment. For example, thethermal radiationhazardresultingfromanignitedruptureof ahigh-pressuregastransmissionpipelinecancausefatalitiesandinjuries,aswellassignicantpropertydamagewithinanimpactradius as large as several hundred meters. Among many means toincrease the reliability of a new pipeline, increasing the pipe wallthicknessisconventionallythemostcommonlyusedmeasure. Athicker pipewall reduces theprobabilityof failureandconse-quently, the potential cost of failure and cost of maintenance overthe pipelines service life but at the same time it leads to a higherinitialconstructioncost.Therefore, thedesignofanewpipelinecan be approached from a cost-benet perspective by selecting anoptimal wall thickness to balance benet and cost.Optimal design of civil systems subjected to infrequent hazardshas attracted extensive research attention since the 1970s. Rosen-blueth and Mendoza 1 formulated the optimal design based on areliability-based cost-benet concept for structures that are eitherabandonedorsystematicallyrebuiltafterfailures.Morerecently,the formulation in Ref. 1 was extended by a number of studiesconsidering more sophisticated time-variant reliability analysis2, impact of maintenance3, riskattitudes andsocietal riskacceptancecriteria3,4, andmultiplefailuremodesandfailurecauses3,5. Theoptimal designframeworkdevelopedinthesestudies was demonstrated for building structures; however, reportsof its application to energy pipeline systems were limited in publicliterature. Zhou et al. 6 carried out reliability-based analyses ofthe life-cycle costs for two hypothetical examples of onshore gastransmission pipelines. The life-cycle cost in their study consistedof the initial construction cost and cost of corrosion maintenanceduring the design life of the pipeline. The cost of failure, however,wasnotcalculated, althoughfailureprobabilitiesoftheexamplepipelines due to corrosion and equipment impact were evaluated.A majorityoftheaforementionedworkisbasedonthemaxi-mum expected overall return or minimum expected life-cycle costcriterion. It waspointedout inRef. 4 that thisformulationisadequateifadecisionmakerisrisk-neutral but maynot bead-equatefor decisionmakerswhoarerisk-averseor risk-seeking.Althoughthemaximumexpectedutilitycriterioncanrigorouslyaddress a decision makers risk attitude, the selection of a widelyacceptableutilityfunctionoftenprovestobeverydifcult. Thestochastic dominance rules were therefore proposed in Ref. 4 toselect optimal designs. The stochastic dominance rules were origi-nallydevelopedforselectingasetofefcientoroptimalinvest-ment choices when only partial information on an investors pref-erences e.g., risk aversion is available. These rules areadvantageousinthattheycantakeintoaccountdifferentriskat-titudes without the need for establishing specic utility functions.On the other hand, they typically lead to a set of optimal or ef-cient designs as opposed to a single option.Theobjectiveoftheworkdescribedinthispaperwastoex-plore the application of the reliability-based optimal design frame-work to onshore gas transmission pipelines through one hypotheti-cal example. The optimal design was selected from theperspective of the investor of the example pipeline as opposed tothe general public. A set of candidate designs was created for thepipelinebyvaryingthepipewall thickness. Theanalysis thenfocused on evaluating the costs of failure and maintenance withinthe pipelines service life for eachdesignbymodelingfailureevents duetocommonintegrityhazards, failureconsequences,andtypicalpreventivemaintenanceactions. Keyuncertaintiesinthe pipe resistance, hazard occurrence, load effects associatedwithagivenhazardoccurrence, andfailureconsequencesweretakenintoconsideration. Boththeminimumexpectedlife-cyclecost criterionandstochasticdominanceruleswereemployedtoseek optimal or efcient designs. The work was intended to pro-vide some insights into the optimal design levels for onshore gas1Correspondence author.Contributed by the Pressure Vessel and Piping Division of ASME for publicationin the JOURNALOF PRESSURE VESSEL TECHNOLOGY. Manuscript received July 23, 2009;nalmanuscriptreceivedAugust17, 2010;publishedonlineApril6, 2011. Assoc.Editor: Shawn Kenny.Journal of Pressure Vessel Technology JUNE 2011, Vol. 133 / 031702-1 Copyright 2011 by ASMEDownloaded From: http://pressurevesseltech.asmedigitalcollection.asme.org/ on 03/30/2015 Terms of Use: http://asme.org/termspipelinesfromboththerisk-neutralandrisk-averseperspectives.Moreover,therelationshipsbetweentheoptimaldesignsandde-signs conforming to the current pipeline standards in Canada andthe United States were examined2 Criteria for Optimal Design Under UncertaintyThe overall return of a civil system over its design life H can beexpressed as follows 2:Hx = Bx Cx 1aCx = C0x + Cfx + Cmx 1bwherexisavectorofdesignparameters,Bxisthebenetde-rived from the existence of the system, Cx is the total life-cyclecostofthesystem,C0xistheinitialconstructioncost,Cfxisthecost of failure, andCmx isthecost of maintenance. It isassumedthat thebenet andcosts canall beexpressedinnetpresent monetary value.The return Hx is generally uncertain because of the uncertain-ties in the resistance of the structure, in the hazard occurrence, inthe load effects resulting from a given occurrence of a hazard, andin the failure consequences. An optimal design can be obtained bymaximizing the expected value of the return, i.e., EHx, whereE representstheexpectation. Theoptimal designcanalsobeobtainedusingthestochasticdominancerules. Threestochasticdominanceruleswereemployedinthiswork, namely, therst-degree stochastic dominance FSD, second-degree stochasticdominance SSD, and third-degree stochastic dominance TSD.Abrief descriptionof theserulesisprovidedinthefollowing.Details can be found in Ref. 7.Consider the problemof selectingthe more efcient designfrom two candidate designs, x1 and x2. Let F1h and F2h denotethe cumulative distribution functions of the return H for x1 and x2,respectively.Supposethattheonlyknowninformationaboutthedecision makers preference is that he/she prefers a higher return;that is, theutilityfunctionuh for thedecisionmaker satisesuh 0. The FSD rule then states that x1 is preferred to x2 or x1dominates x2 by FSD if and only if F1h F2h at all values ofh with a strong inequality for at least one value of h. An intuitiveinterpretation of the FSD rule is that x1 dominates x2 if the prob-ability of obtaining a return equal to or higher than h under F1his noless thanthat under F2h i.e., 1-F1h 1-F2h at allvalues of h 7. A graphic interpretation of the FSD rule is that x1dominates x2 if F1h is below or tangent F2h at all values of h.Two important necessary conditions for FSD are the mean and lefttail conditions. If x1dominatesx2byFSD, themeanconditiondictatesthattheexpectedreturnassociatedwithx1, EH1, mustbe greater than the expected return associated with x2, EH2; theleft tail condition dictates that the minimum return associated withx1 must be greater than or equal to the minimum return associatedwith x2.TheSSDruleaddressesriskaversion. If thedecisionmakerprefershigherreturnandisrisk-averse i.e., uh 0anduh0, thenx1dominates x2bySSDif andonlyif hF1sdshF2sds at all values of h with a strong inequality for at leastone value of h. The mean and left tail conditions as described forFSD are also the necessary conditions for SSD, except that only aweakinequalityinthemeanconditionisneededfor SSD, i.e.,EH1EH2.TheTSDrulewas developedfor decisionmakers whoseekhigher returns, are risk-averse and prefer positive skewness of thereturni.e., uh 0, uh 0, anduh 0. Thehypothesisthatdecisionmakersgenerallyfavorpositiveskewnesshasbeensupportedbystrongempirical evidence, suchaspeoplebuyinglottery tickets and insuring their homes, and by positive skewnessfor therateof returnonstocks7. Giventheseconditions, x1dominates x2by TSD if and only if htF1sdsdthtF2sdsdt at all valuesof handEH1EH2withastrong inequality for at least one of these two conditions: that is,either htF1sdsdt htF2sdsdt for some value of h orEH1EH2.Note that a stochastic dominance rule for risk-seeking behavioralso exists and was applied in Ref. 4 for optimal design of build-ingstructuressubjectedtoseismichazard. However, it washy-pothesizedthatmostpipelineoperatorsarerisk-aversegiventhepotentiallysevereconsequences of pipelinefailures. Therefore,the risk-seeking stochastic dominance rule was not considered inthis study.3 Optimal Design of Onshore Gas Pipelines Under Un-certainty3.1 Example Pipeline. A hypothetical pipeline, adapted froman example given in Ref. 8, was used to demonstrate the optimaldesign of onshore gas pipelines under uncertainty. The pipeline islocatedinCanadaanddesignatedasaclass2pipeline.Ithasanoutside diameter of 508 mm i.e., nominal pipe size NPS 20, adesign pressure of 9.653 MPa 1400 psi, and a design life of 50years. The specied minimum yield strength SMYS of the pipesteel is 483 MPa i.e., X70 steel. The pipeline will be constructedwithdoublejointsof24mlong. Theaveragepopulationdensityinthevicinityof thepipelineis1.54people/hectare1 hectare=10,000 m2.Thepipewallthicknessisthesoledesignparam-eter considered for the example.Accordingtothe UnitedStates andCanadianpipeline stan-dards, i.e., ASME B31.8 9 and CSA Z662-07 10, the nominalpipewallthicknesswtnisdeterminedfromthewell-knownBar-low equation as follows:wtn =PnD2 SMYS2wherePn is the design pressure, D is the pipe diameter, and isthe design factor, which primarily depends on the location class ofthepipeline. Foraclass2gaspipeline, thedesignfactorequals0.72perCSA Z662-07and0.6per ASMEB31.8. Therefore,theexamplepipelinewill haveawall thicknessof 7.05mmif de-signed per CSA Z662 and a wall thickness of 8.46 mm per ASMEB31.8. By varying the design factor, a suite of candidate designscan be developed, each corresponding to a unique pipe wall thick-ness.3.2 FailureCausesandFailureModes. Equipment impactduetothird-partyinterferenceandexternal corrosionwereas-sumed to be the only failure causes for the example pipeline. Thisassumptionpartlyresultedfromhistorical failurestatistics11,whichindicatethatequipmentimpactandexternalcorrosionarerelevant for all natural gas pipelines andaccount for approxi-mately6076%ofall signicant failureevents. Theassumptionwas also made to simplify the analysis, as deterministic andprobabilistic models with respect to these two failure causes havebeen well researched and are well documented in open literature.Failure of pipelines due to equipment impact is the result of twoevents, namely, pipeline being impacted by excavation equipmentandfailureofthepipelinegivenimpact10. Thefrequencyofimpact isafunctionofthefrequencyoftheexcavationactivityalong the pipeline right-of-way ROW and the equipment impactpreventionmeasures, suchasthepipeburial depth, above- andbelow-ground signage indicating the location of the pipeline, one-callsystemandROW patrolmethod aerialorground,andfre-quency 10. TheoccurrenceofimpacteventsisrandomintimeandcanbeassumedtobedescribedbyaPoissonprocess. Thisimplies that the time interval between subsequent impacts followsan exponential distribution.Twolimitstatesareinvolvedinagivenimpact 10. Therstone is the puncture of the pipe wall by an indentor that has a shapecorrespondingtothat ofanexcavatorbucket tooth. Thesecond031702-2 / Vol. 133, JUNE 2011 Transactions of the ASMEDownloaded From: http://pressurevesseltech.asmedigitalcollection.asme.org/ on 03/30/2015 Terms of Use: http://asme.org/termslimit state arises if the puncture force is not large enough to pen-etrate the pipe wall. In this case, the dent-gouge defect producedbytheindentor mayfail under thepipeinternal pressureuponremoval of the indentor. Details of the limit state functions asso-ciated with these two limit states are provided in Annex O of CSAZ662-0710. Notethat bothlimit statesaretimeindependent;thatis,theprobabilisticcharacteristicsoftheuncertainvariablesinvolved in these limit state functions do not vary with time.Three different failure modes are assumed to be associated witha given failure due to equipment impact, namely, rupture or full-bore rupture, large leak, and small leak 10. A full-bore ruptureresultsinadouble-endedgasrelease, withthediameter of therelease hole at eachendequal tothe pipe diameter 11. Theaverage hole diameters for large leak and small leak are typicallyassumed to be 50 mm and 10 mm, respectively 11. To differen-tiateleakandrupture,itisassumedthatpunctureorfailureofadent-gouge will result in a through-wall cracklike defect 10. Thelength of the through-wall defect is assumed to equal the excava-tor tooth length in the case of puncture and the gouge length in thecaseof dent-gougefailure. Thefailureisthencategorizedasarupture if the length of the through-wall defect is greater than thecritical lengthfor unstableaxial extensionof thedefect. Other-wise, the failure is categorized as a leak. The equation for calcu-latingthecritical defect lengthisgiveninRef. 10. Historicalfailure statistics suggest that two-thirds of the leaks are large leakswith the rest being small leaks 11. The consequences associatedwith rupture, large leak, and small leak are described in Sec. 3.3.Failure at an active corrosion defect can be either a small leakor a burst 10. A small leak results from the defect penetrating thepipe wall, whereas a burst occurs if the pipe wall undergoes plas-tic collapse under internal pressure prior to the defect penetratingthe pipe wall 10. A burst can be further categorized as a ruptureor a large leak. The former is denedas a failure where thethrough-walldefectresultingfromaburstislongenoughtoun-dergounstableaxialextension,whereasthelatterisdenedasaburst without unstableaxial extensionof theresultingthrough-wall defect 10. The limit state functions for burst and small leakaswellasthelimitstatefunctionfordistinguishingruptureandlarge leak are given in Ref. 10. Note that these limit state func-tions are time dependent because corrosiondefects growovertime. A linear defect growth model was adopted in this study; thatis, the defect depth and length were assumed to grow in constantdepth and length growth rates 11,12.3.3 Failure Consequences3.3.1 SafetyImplications. Twotypesoffailureconsequenceswere considered in this work, namely, the impact on human safetyand damage to the operators property including the pipeline andother nonoperator properties. Theimpact onhumansafetywasevaluated using a model known as the C-FER model 13, whichwas developed to predict the thermal radiation hazard zone asso-ciated with an ignited rupture of a lean gas pipeline. The C-FERmodel assumesdouble-endedgasreleasefor arupturewiththediameter of the release hole at each end equal to the pipe diameter,andaneffectivereleaserateofone-thirdthepeakinitial ratetoacknowledgetherapiddropinlinepressureassociatedwiththerupture. Using a circle to represent the hazard area given ignition,themodelcalculatestheheatintensitylevelasafunctionofgasreleaserateanddistancefromthefailuresite.Alternatively, theradius of the hazard area, rhr m, within which the heat intensitylevel exceeds a certain threshold, Ith, can be calculated as follows:rhr = 0.1547pD2Ith3wherepis pipe internal pressure at the time of failure Pa, Disthepipediameter m,andIthisagivenheatintensitythresholdkW/ m2. Note that the C-FER model is conditioned on ignition.Theprobabilityof ignitiongivenarupturePircanbeapproxi-mated by a linear function of the pipe diameter as follows 11:Pir = 0.000492D 4Thethermalradiationhazardradiusassociatedwithanignitedlargeleak, rhl, isestimatedbyslightlymodifyingEq. 3toac-count for the fact that a large leak is a single-ended release from ahole much smaller than the pipe diameter. Furthermore, the effec-tivereleaseratewasassumedtobethesameasthepeakinitialrelease rate given that the line pressure is likely to decrease slowlywith a small release hole. These modications resulted in the fol-lowing equation for calculating rhl m:rhl = 0.2321pDh2Ith5whereDhisthediameterofthereleaseholemwithatypicalvalue of 0.05 m. The probability of ignition given a large leak, Pil,was assumed to be a constant of 0.1 11.Asmall leakisassociatedwithalowprobabilityofignition.Furthermore, the thermal radiation radius corresponding to an ig-nited small leak is negligible because the size of the release hole isverysmall not exceeding10mmindiameter. Therefore, thesafety implication associated with a small leak is negligible.The harm to people caused by thermal radiation is related to thethermal dosage received. The thermal dosage is measured by thethermal dosage unit, which is a function of the heat intensity leveli.e., heat ux andtheexposuretime1315. For anignitedpipeline incident, it has been suggested in Ref. 15 that an expo-suretimeof30sisconsideredrepresentativebytakingintoac-counttypicalreactiontimeofanindividualexposedtotheinci-dent, escapespeedoftheindividual, andlikelihoodofndingashelter within a reasonable distance. Based on this exposure time,theheat intensitythresholdscorrespondingto100%and0%le-thalityfor outdoor andindoor exposureswererecommendedinRef. 15 and are summarized in Table 1. It was further suggestedinRef. 15 that thelethalitycorrespondingtoaheat intensitylevelbetweenthe100%and0%lethalitylevelsbe50%forout-door exposure and 25% for indoor exposure. The latter is based onassumingaprobabilityof ignitionof 50%for atypical wood-framed structure subjected to heat intensity levels between15.8 kW/ m2and31.6 kW/ m2andaprobabilityoflethalityof50%givenignition.Itthenfollowsthattheprobabilityofinjuryforheatintensitylevelsbetween15.8 kW/ m2and31.6 kW/ m2is also 25% for indoor exposure. For outdoor exposure, the prob-abilityof injuryfor heat intensitylevelsbetween12.6 kW/ m2and 31.6 kW/ m2is 50%.Basedona summaryof the effects of thermal radiationonpeople given in Ref. 13, a heat intensity threshold of6.3 kW/ m2with an exposure time of 30 s was selected to deneTable 1 Heat intensity thresholds for fatality and injuryExposure typeHeat intensity thresholdskW/ m2100% lethality 0% lethality Onset of injuryOutdoor 31.6 12.6 6.3Indoor 31.6 15.8 15.8Journal of Pressure Vessel Technology JUNE 2011, Vol. 133 / 031702-3Downloaded From: http://pressurevesseltech.asmedigitalcollection.asme.org/ on 03/30/2015 Terms of Use: http://asme.org/termsthe onset of burn injury for outdoor exposure. It was pointed outin Ref. 15 that a typical wood-framed structure is very unlikelytoignitewhensubjectedtoaradiationintensityof15.8 kW/ m2or lower and will therefore afford indenite protection to the oc-cupants. Giventhis, theheatintensitythresholdfortheonsetofinjurywasselectedtobeidentical tothat for0%lethalityi.e.,15.8 kW/ m2 for indoor exposure. Asummaryof theselectedheat intensitythresholdsandcorrespondingimplicationsforhu-man safety is depicted in Fig. 1.3.3.2 Failure Costs. Thecost of fatalityandinjurycanbedetermined from the values of a statistical life VSL and a statis-tical injuryVSI. Viscusi andAldy16 recentlycarriedout acomprehensive review of studies of mortality and injury risk pre-miumspublishedinthepast30years. ThereviewindicatedthatVSL in Canada is typically between $3$6 million in 2000 US$.VSI inCanadarangesfromapproximately$1300$190,000in2000US$, where the upper limit is associatedwitha severeinjury. Based on the data summarized in Ref. 16, VSL and VSIwere selected to be $4.5 million and $15,000, respectively.The cost of property damage including the value of lost prod-uct was estimated from the database of signicant pipeline inci-dents administered by the U.S. Department of TransportationDOT.2DOT requires pipeline operators to report an incident if itmeetsanyoneofasetofreportingthresholds e.g., resultinginfatality or injury, incurring a total cost of over $50,000 or more in1984$, etc..Thetotalpropertydamagecategoryinanincidentreport represents theestimatedtotal cost of thedamagetothepublicandprivatenonoperator properties, thedamagetotheoperator assets and the value of lost product.TheincidentsincludedintheDOTdatabasefor onshoregastransmissionpipelinesfrom2002to2008wereexaminedinthiswork.Althoughincidentsthat occurredbetween1986and2001were alsoavailable inthe database, the informationrelatedtotheseincidentswasnot reportedascompletelyasthat for inci-dents that occurred after 2002. For example, whether or not igni-tion occurred was not reported for the pre-2002 incidents. Further-more, it was unclear to the authors that whether the value of lostproductwasincludedinthetotalestimatedpropertydamagere-portedforthepre-2002incidents. Therefore, pre-2002incidentswere not considered in estimating the cost of property damage.Due to the limited number of incident data in the database, thecost of property damage CPD was estimated as a function of thefailuremoderuptureor largeleak andpresenceabsence ofignitiononly; theimpact of other parameterssuchaspressure,diameter, and wall thickness was ignored. Given that the exampleisaclass2pipeline, theimpact of locationclasswasapproxi-matelyaccountedfor byincludingtheincident datapoints forclass1andclass2pipelines thenumberofdatapointswill besignicantlylimitedif onlyincidents for class 2pipelines areincluded. To this end, 28 data points for ignited rupture incidentswerecollected, resultinginanaverageCPDof $1,550,000allestimatedCPDin2007US$. Notethat inderivingtheabovecost, an incident that occurred in 2005 and resulted in $94.5 mil-lion estimated property damage was excluded as it was consideredanextremecaseandnot representativeof typical ruptures. Theaverage CPD per nonignited rupture was estimated to be $445,900based on 47 data points.TheCPDforlargeleakwasestimatedbasedontheleakinci-dentsreportedinthedatabase.Thereareonlythreeignitedleakincidents on class 1 and class 2 pipelines between 2002 and 2008,withanaverageCPDof$241,000. TheaverageCPDpernonig-nited leak incident was estimated to be $402,400 based on 85 datapoints. The fact that the average CPD per ignited leak is less thanthe average CPD per nonignited leak was considered an anomalyand can be attributed to the small number of ignited leak incidentsincludedinthedatabase. Inlight ofthis, theignitedandnonig-nited leak incidents were combined together to result in an aver-age CPD of $396,900 per large leak.TheinformationintheDOTdatabaseisnot suitableforesti-mating the cost of property damage per small leak. This is becausemost small leaks arenot serious enoughtomeet thereportingthresholds andthereforearenot reported11. Notethat smallleaks are unlikely to result in casualty or property damage becauseof the low probability of ignition and insignicant thermal radia-tion hazard area given ignition. Therefore, the cost of a small leakwas assumed to be solely the cost of excavating and repairing thepipeline at the location of the leak.Ideally, indirect failure consequences should also be consideredin the analysis. Such consequences may include increased regula-tory pressure, lost public condence, and litigations. Due to a lackofinformation,itisnotpossibletoquantifytheseconsequences.Therefore, they were not considered in this study.3.4 AcceptableRiskLevels. AsdescribedinSec. 3.3, fail-ures of gas pipelines have signicant life safety implications. Themaximumrisklevelsacceptabletothegeneral publicshouldbetaken into account in the selection of optimal designs. Two typesoflifesafetyrisksareusuallyconsideredinrisk-orreliability-baseddesignandassessment of gas pipelines, namely, societalriskandindividual risk 8,11. Societal riskisameasureoftheoverall risk of fatality due to pipeline incidents and can be repre-sentedbytheexpectednumber of fatalities per unit lengthofpipelineperyear. Individual riskistypicallyrepresentedbytheannualprobabilityoffatalityduetopipelineincidentsforanin-2http://primis.phmsa.dot.gov/comm/reports/safety/CPI.htmlFig. 1 Heat intensitythresholdsandhumansafetyimplica-tions: a outdoor exposure and b indoor exposure031702-4 / Vol. 133, JUNE 2011 Transactions of the ASMEDownloaded From: http://pressurevesseltech.asmedigitalcollection.asme.org/ on 03/30/2015 Terms of Use: http://asme.org/termsdividual located in the vicinity of the pipeline.InarecentstudysponsoredbythePipelineResearchCouncilInternational PRCI, theaveragesocietal risklevel for theon-shoregastransmissionpipelinenetworkintheUnitedStates, asimplied by the current design and maintenance practice, was cali-bratedtobe1.6105fatality/ km year 8,11. This valuewasthen adopted as the allowable societal risk level for the develop-ment of a set of target reliability levels for onshore gas pipelines.ThePRCI-sponsoredstudyalsoconsideredindividual riskcrite-rioninthetarget reliabilitydevelopment andadopted104peryear as the allowable individual risk level for class 1 and class 2pipelines. However, it was found that the individual risk criteriontypically does not govern the target reliability selection except forsmall diameter lowpressureclass1pipelines. For thepipelineexample assumed in this study, an allowable societal risk level of1.6105fatality/ km year was considered as a constraint in theoptimal design selection. Given that the example is a high-pressurelargediameterclass2pipeline, theindividualriskcon-straint was not considered based on the ndings reported in Refs.8,11.3.5 Maintenance. The maintenance actions considered in thisstudy involved periodic inspection and preventive repair of exter-nal corrosiondefects.Althoughother maintenanceactivitiesarealsocommoninpractice e.g., regulargroundoraerialpatrolofthepipelineROWtoprevent equipment impact, theywerenotconsidered in the analysis because the implementation and impactof these measures are generally independent of pipe wallthickness.Inagivenmaintenanceevent, ahigh-resolutioninlineinspec-tiontool was assumedtobeusedtodetect andsizecorrosiondefects. Defectsthataredetectedbythetoolandexceedcertainrepair thresholds will be excavated and repaired immediately aftertheinspection. A detecteddefectwillberepairedifitmeetsanyone of the following two criteria 11,12:dmaxM wtn6arbM Pn6bwhere dmaxMis the maximum defect depth measured by the inspec-tion tool; rbMis the defect burst pressure predicted from the mea-sured defect depth and length as well as the nominal pipe geomet-ric and material properties;andare constants that dene therepair thresholds.The detection capability of the inspection tool is represented bythe probability of detection POD, which was assumed to followan exponential type of function as follows 12,17:POD= 1 eqdmax7where dmax is the maximum defect depth and qis a constant thatdenes the tool accuracy and can be quantied from tool speci-cations that give the POD value for a prescribed reference defectdepth.The measured defect depth and length, dmaxMand lM, are relatedtothe actual defect depthandlength, dmaxandl, throughthefollowing equations:dmaxM= dmax+ ed8alM= l + el8bwhereedandelarerandomvariablesrepresentingthedepthandlengthmeasurementerrorsassociatedwiththetool, respectively.They are typically characterized by normal distributions with thecorresponding mean and standard deviations quantied from toolspecicationse.g., speciederror bandfor depthmeasurementwith a certain probability level 12.3.6 Cost-Benet Formulationfor the Example Pipeline.GiventhedescriptionsinSecs. 3.23.4, thetotal returnontheexample pipeline can be calculated as follows:Hwtn = B Cwtn = B C0wtn Cfwtn Cmwtn 9aCfwtn = i=12j=1niCLrij + CPrijeij9bCmwtn = k=1vCI + CRkeij9cwhere B is the benet derived from the existence of the pipeline,considered to be independent of pipe wall thickness; ni is the totalnumber of failures duetothe ithfailurecausethroughout thedesignlifeofthepipeline;CLrijandCPrijarethecostoffatalityandinjuryandcostofpropertydamage, respectively, associatedwiththejthfailureduetotheithfailurecause, whererijisanindexrepresentingthefailuremodeassociatedwiththefailure,i.e., rij=1, 2, and 3 representing small leak, large leak and rupture,respectively; ij is the time of occurrence of thejth failure due totheithfailurecause; isthediscount rate; erepresentsthecontinuous discounting function1; v is the total number ofscheduledcorrosionmaintenanceevents; CIisthecostofinlineinspectionandassumedtobeaconstant; andCRkisthecost ofpossible defect excavation and repair at the kth maintenanceevent. Note that CRk may differ for different events. It is assumedthat the pipeline will be immediately restored to its original con-dition if a failure occurs.Since a pipeline is a linear system, the probability of concurrentfailureeventsduetoequipmentimpactandexternalcorrosionatthe same location is extremely unlikely. Therefore, the total failurecost in Eq. 7 is a summation of the contributions from individualfailure events due to equipment impact and corrosion applied in-dependently. Because the benet is independent of pipe wallthickness, thetotal returncanbesimpliedasHwtn =Cwtn=C0wtn Cfwtn Cmwtn.Itfollowsthattheoptimaldesigncan be determined by maximizing the expected return or equiva-lentlyminimizingthelife-cyclecost. Theoptimal designscanalso be determined from the probability distributions of the over-allreturn i.e.,Cassociatedwiththecandidatedesignsbyap-plying the stochastic dominance rules.3.7 Summary of Input Parameters and AnalysisProcedure. A total of ten candidate designs, summarized in Table2, was createdfor theexamplepipeline. Theinput parametersrequiredtoestimatetheexpectedlife-cyclecostsandprobabilitydistributionsoftheoverallreturnsforthesedesignsaresumma-rizedinTables48in Appendix. Allrandomvariableswereas-sumedtobeindependent of eachother. Theindependenceas-sumptionalsoextendstorandomvariablesat different locationsalong the pipeline; that is, failure events associated with differentcorrosion defects or different equipment impacts are independentTable 2 Summary of candidate designs for the examplepipelineCandidate designNominal wall thicknessmm Design factorD1 ASME design 8.46 0.60D2 8.19 0.62D3 7.93 0.64D4 7.69 0.66D5 7.47 0.68D6 7.25 0.70D7 CSA design 7.05 0.72D8 6.86 0.74D9 6.68 0.76D10 6.35 0.80Journal of Pressure Vessel Technology JUNE 2011, Vol. 133 / 031702-5Downloaded From: http://pressurevesseltech.asmedigitalcollection.asme.org/ on 03/30/2015 Terms of Use: http://asme.org/termsof each other.Foragivendesign, C0isadeterministicquantityandcanbecalculated from the corresponding nominal wall thickness and unitcosts given in Table 8. Note that cost components independent ofwall thickness e.g., excavation and coating costs were excludedfromC0. TheMonte-Carlotechniquewasemployedtoevaluatethe probability distribution of Cf+Cm associated with the design.Each simulation trial involved generating samples of input param-eters summarized in Tables 48 and using the samples to calculatethe cost of failure due to equipment impact and external corrosionas well as thecost of scheduledcorrosionmaintenanceeventsover the design life of the pipeline i.e., 50 years.Toanalyzeequipmentimpactinagivensimulationtrial,ase-quenceofimpact eventswithinaspanof50yearsisgeneratedalong a unit length of the pipeline i.e., 1 km based on the prop-erty of the Poisson process that describes the occurrence of impactevent. Thelimit statefunctionsassociatedwitheachimpact arethen checked to identify potential failure incidents and corre-spondingfailuremodes. Givenfailure, thecost of fatalityandinjurywillariseifignitionoccursandpeoplearepresentwithinthe hazard impact zone. The cost of property damage, on the otherhand, will always arise given failure.To analyze external corrosion in a given simulation trial, a num-ber of signicant corrosion defects is generated along 1 km of thepipelinebasedonthepropertyof thePoissonprocess that de-scribesthespatialdistributionofthedefects. Eachofthegener-ateddefectsisconservativelyassumedtoexistatthestartoftheservice life i.e., no corrosion initiation time and to grow in depthandlengthat constant growthratesthat aregeneratedfromthecorresponding growth rate distributions. Agiven defect is re-movedfromthetrialifitisdetectedandrepairedduringoneofthescheduledmaintenanceeventsgeneratedinthetrial. Failureand the corresponding failure mode at the defect are identied ifanyofthelimitstatefunctionsassociatedwiththedefectisvio-latedbeforethenextscheduledmaintenanceorbetweenthelastscheduledmaintenanceandendof 50years. Givenfailure, thecost of failureiscalculatedinthesamewayasfor equipmentimpact.Inestimatingthecostofmaintenance,itisassumedthateachdefecttoberepairedrequiresaseparateexcavationandre-pair.4 Results and DiscussionTheexpectedlife-cyclecostsEC andcorrespondingcostcomponents i.e., C0, ECf, and ECm for all the candidate de-signs are summarized in Table 3. In addition, the standard devia-tion of the life-cycle cost c and the average annual life safety riskin terms of fatality Rs are also summarized in the table. In Fig. 2,C0, ECf+Cm, and EC are plotted versus the design factor. ThegureshowsthatC0decreasesapproximatelylinearlyasthede-sign factor increases because a higher design factor leads to a nearproportional decrease inthe pipe wall thickness. Onthe otherhand, ECf+Cmincreases as the design factor increases becausea thinner pipe wall leads to increases in the probability of failureas well as theprobabilityof corrosiondefect repair, therefore,resulting in higher expected costs of failure and maintenance.For all the candidate designs, ECf accounts for less than 10%ofEC, whereasECm ismarkedlyhigherthanECf andac-countsforamoresignicantpercentage between8%and20%of EC. This suggests that the corrosion maintenance parameters,i.e.,inspectionscheduleandrepaircriteria,canbeconsideredasdesign parameters in addition to the wall thickness to improve thelocally optimal design obtained in the present analysis. This willbe studied in future investigations.Basedontheminimumexpectedlife-cyclecostcriterion i.e.,for a risk-neutral decision maker, the optimal design is D8, cor-responding to a design factor of 0.74 and a wall thickness of 6.86mm. Thedesignwiththelargestwallthickness,D1 i.e., ASMEdesign, hasthehighest EC, duetothehighvalueofC0. Thevalue of ECf due to corrosion for D10 is lower than that for D9seeTable3, whichappears counterintuitive. This canbeex-plainedbythefact that thedefect repairthresholds i.e., =0.5and=1.39, which are xed for all the designs, become signi-cantly more stringent for D10 than for D9 as the pipe wall thick-ness decreases from D9 to D10 and result in more defects beingTable 3 Summary of the calculated costs and life safety risk levelsDesignCostCAD$/kmRsfatality/ km yr C0ECfECm E Cf+Cm EC cEquipment impact CorrosionD1 ASME 269,682 2,423 415 22,140 24,977 294,660 251,135 3.3106D2 261,125 3,020 514 23,488 27,021 288,147 276,518 4.9106D3 253,095 3,445 632 24,944 29,020 282,115 297,439 6.5106D4 245,543 5,168 778 26,469 32,414 277,958 383,717 9.5106D5 238,429 6,675 925 28,104 35,704 274,133 400,168 1.2105D6 231,715 8,844 1024 30,161 40,028 271,744 465,775 1.4105D7 CSA 225,370 11,116 1084 32,636 44,836 270,206 528,434 1.5105D8 219,362 13,382 1143 35,905 50,430 269,792 592,300 1.7105D9 213,666 16,650 1130 40,476 58,255 271,922 644,470 1.8105D10 203,118 27,139 895 56,422 84,456 287,574 828,381 2.3105Fig. 2 Variationof theexpectedlife-cyclecost withdesignfactor031702-6 / Vol. 133, JUNE 2011 Transactions of the ASMEDownloaded From: http://pressurevesseltech.asmedigitalcollection.asme.org/ on 03/30/2015 Terms of Use: http://asme.org/termsrepaired as opposed to being left in the pipeline. As shown in Fig.2, theportionof theEC curveboundedbyD4andD9i.e.,designfactors0.66and0.76, respectivelyisrelativelyat.ThevaluesofEC associatedwithD4andD9arehigherthanthatassociated with the optimal design D8 by only 3.0% and 0.8%,respectively. Furthermore, EC for D7 CSA design is only 0.2%higherthanthatforD8. ThissuggeststhatECisinsensitivetowall thickness variation in the vicinity of the optimal design.The calculated societal risk levels Rs for D1D10 range from3.3106fatality/ km year to2.3105fatality/ km year. Notethat Rs increases approximately linearly from D1 to D10, which isincontrast tothemorerapidincreaseinECf+Cm. Thisisbe-cause the increase in the probability of large leak accounts for themajority of the increase in the overall failure probability from D1to D10. The safety consequences of large leak are relatively smallwhereasthenancialconsequencesoflargeleaksaresignicantsee Table 8; therefore, ECf increases substantially from D1 toD10 with a relatively small increase in the life safety risk. The Rsvalues associated with D8D10 exceed the allowable societal risklevel of 1.6105fatality/ km year. Therefore, theoptimal de-signbecomesD7, i.e., theCSA design, iftheallowablesocietalrisklevel is appliedas aconstraint. As showninTable2, cincreases markedly from D1 to D10. Although this is irrelevant inthecontextoftheminimumexpectedlife-cyclecostcriterion, ithas signicant implications for selecting the optimal design usingthe stochastic dominance rules as described in the following para-graphs.The cumulative distribution functions CDF of the returns i.e.,C per km associated with the designs are depicted in Fig. 3. Allthe returns are normalized by the initial construction cost associ-ated with the ASME design i.e., C0=$269, 682/ km. Log scalesare usedfor boththe normalizedreturnandCDFtofacilitateinterpreting the gure. Figure 3 shows that the distribution curvesintersect each other; therefore, there are no dominant designsbasedontheFSDrule.Inotherwords,alltencandidatedesignsare efcient or optimal choices based on the FSD rule. As showninthegure, theminimumreturndecreasesi.e., themaximumlife-cycle cost increases as the wall thickness decreases because asmallerwallthicknessincreasesthecostsoffailureandmainte-nance. Themaximumreturnincreases i.e., theminimumlife-cyclecost decreases asthewall thicknessdecreasesbecauseasmaller wall thickness reduces the initial construction cost.To identify optimal designs based on the SSD rule, the integra-tionsof theCDFsof thenormalizedreturnstheintegrationisreferred to as I1 for brevity for D1D10 are depicted in Figs. 4aand4b,wherethelatterisaclose-upviewoftheupperendoftheformer tofacilitatethedescriptionof theresults. Figure4indicatesthatD9andD10aresuboptimaldesignswhileD1D8are all optimal designs according to the SSD rule. In other words,for a risk-averse decisionmaker anydesigns involvingdesignfactorsgreater thanor equal to0.76aresuboptimal for theex-amplepipeline. D10is suboptimal becauseit is dominatedbyD3D8,3i.e., I1forD3D8isbelowI1forD10overtheentirerange of the return. D9 is suboptimal because it is dominated byD8.BecauseI1forD1D8intersectseachother,allofthesede-signs are optimal. The fact that D8, the design with the minimumexpected life-cycle cost, belongs to the optimal design set per theSSDruleisnot unexpectedbecausethemeanconditionfortheSSDrule seeSec.2requiresthattheoptimaldesignset,ifex-isting at all, must contain the design with the maximum expectedreturn orequivalentlytheminimumexpectedlife-cyclecostforthis example. Note that the pipe wall thickness corresponding toD8isthelowerboundofthewall thicknessescorrespondingtothe optimal design set based on the SSD rule. In other words, anydesign that utilizes a wall thickness greater than that correspond-ing to the minimum expected life-cycle cost is considered optimalbased on the SSD rule.The division of the optimal and suboptimal design sets can bequalitativelyexplainedthroughthestandarddeviationofthere-turn. A risk-averse decision maker dislikes uncertainty; this meansthattheexpectedutilityforagivenreturndecreasesasthestan-darddeviationofthereturnincreases otherstatisticsremainingthesame 7. Asshownin Table3, cincreasesmarkedlyfromD1 to D10, whereas the change in the expected life-cycle cost isrelatively small note that the return and life-cycle cost are iden-tical other than the sign for the example. It then follows that the3Accordingtothestochasticdominancetheory, aninefcientorsuboptimalde-sign need not be dominated by all the efcient designs. Instead, dominance by onlyone efcient design is sufcient to relegate a particular design to the inefcient set.Fig. 3 CDFs of the normalized returnsJournal of Pressure Vessel Technology JUNE 2011, Vol. 133 / 031702-7Downloaded From: http://pressurevesseltech.asmedigitalcollection.asme.org/ on 03/30/2015 Terms of Use: http://asme.org/termsuncertaintyofthereturnincreasessignicantlyfromD1toD10,eventuallymakingD9andD10suboptimal froma risk-averseperspective.DoubleintegrationsoftheCDFsofthereturns referredtoasI2 for D1D10 were carried out to identify optimal designs basedontheTSDrule. TheresultsaredepictedinFig. 5. Thegureindicates that I2 for D8 is below I2 for D9 and D10 over the entirerange of the return. Furthermore, EH for D8 is greater than EHfor D9 and D10 based on Table 3 EH =EC. Therefore, D9and D10 are dominated by D8 per the TSD rule and are subopti-mal. D1D8belongtotheoptimal designset becausenoneofthem dominates the others by satisfying the two conditions for theTSD rule. This means that the optimal design set identied usingthe SSD rule cannot be further narrowed down by including moreinformationabout thedecisionmakerspreference i.e., positiveskewness in the return in addition to the risk-averse attitude.If the allowable societal risk level of 1.6105fatality/ km year is appliedas a constraint, the optimaldesignset basedonthe three stochastic dominance rules thenconsists of D1D7. Among this optimal set, the wall thickness forthe design leading to the lowest expected life-cycle cost i.e., D7ortheCSA designisthelowerboundvalueofthewall thick-nesses for D1D7.Fig. 4 Integration of the CDFs of the normalized returns: a overall range and b close-up viewof the upper end031702-8 / Vol. 133, JUNE 2011 Transactions of the ASMEDownloaded From: http://pressurevesseltech.asmedigitalcollection.asme.org/ on 03/30/2015 Terms of Use: http://asme.org/terms5 ConclusionsTheoptimaldesignlevelforonshorenaturalgastransmissionpipelines was explored through a hypothetical pipeline example inthispaper. Theworkemployedthereliability-basedcost-benetanalysis framework established in literature and focused on evalu-atingthelife-cyclecostofthepipeline.Thepipewallthicknesswas selectedas thesoledesignparameter. Basedonhistoricalfailurestatistics, equipment impact andexternal corrosionwereassumedtobethetwofailurecausesforthepipeline. Thelife-cyclecostincludedthewallthickness-relatedinitialconstructioncostaswellasthecostsoffailureandmaintenance. Thecostoffailure included the cost of fatality and injury, costs of damage tothepublicandprivateproperties, andthevalueoflost product.Duetoalackofinformation, indirectfailureconsequencessuchasincreasedregulatorypressure, lost of publiccondence, andlitigationcost werenot considered. Thecost ofmaintenancein-cludedthecostofscheduledinlineinspectionandpreventivere-pair of external corrosion defects. The allowable societal risk wasconsidered as a constraint in the optimal design selection,wherebytheaveragesocietal riskassociatedwiththegastrans-missionpipelinenetworkintheUnitedStates,asimpliedbythecurrent design practice, was adopted as the allowable societal risklevel.Theoptimal designfor theexamplepipelinewas sought byapplying the minimum expected life-cycle cost criterion as well astherst-, second-, andthird-degreestochasticdominancerules.Theresultsshowthattheoptimaldesignbasedontheminimumexpected life-cycle cost criterion utilizes a design factor of 0.74. Iftheallowablesocietalriskconstraintisapplied,theCSA design,whichutilizesadesignfactorof0.72, becomestheoptimal de-sign.Itwasobservedthattheexpectedlife-cyclecostisinsensi-tivetowallthicknessvariationinthevicinityoftheoptimalde-sign. The expected life-cycle cost of the CSA design is practicallythesameasthat correspondingtoadesignfactor of 0.74. Theresults also show that the expected cost of maintenance accountsfor a signicant portion of the expected life-cycle cost. Therefore,theoptimal designachievedinthis analysis canbepotentiallyimprovedbyconsideringthemaintenanceparametersasdesignparameters in addition to the wall thickness.Basedonthesecond- andthird-degreestochasticdominancerules, any designs involving design factors of 0.76 or greater aresuboptimal for a risk-averse decision maker. Applying the allow-able societal risk constraint will further relegate the design involv-ing a design factor of 0.74 to the suboptimal design set. Both theCSAandASMEdesigns belongtotheoptimal designset andsatisfy the allowable societal risk constraint. Note that the ASMEdesignhasthehighest expectedlife-cyclecost of all candidatedesigns. Thewallthicknessofthedesignwiththeminimumex-pected life-cycle cost is the lower bound of those included in theoptimal set. Theresultsdemonstratetheimplicationofthevari-ability of the life-cycle cost for selecting the optimal design froma risk-averse perspective.Table 4 Basic pipeline attributesParameter Probability dist. Mean Std. dev. SourceWall thickness Normal 1.0wtn0.015wtnAssumedInternal pressure Deterministic 9.653 MPa NADiameter Deterministic 508 mm NA 10Yield strength Normal 530 MPa 18.6 MPaTensile strength Normal 615.9 MPa 18.5 MPaCharpy v-notch toughness Normal 108 J 17.7 JFig. 5 Double integration of the CDFs of the normalized returnsJournal of Pressure Vessel Technology JUNE 2011, Vol. 133 / 031702-9Downloaded From: http://pressurevesseltech.asmedigitalcollection.asme.org/ on 03/30/2015 Terms of Use: http://asme.org/termsTable 5 Input parameters for modeling equipment impact and external corrosionParameter Probability dist. Mean Std. dev. SourceNumber of impacts Poisson 0.004/ km yeara0.063/ km yr 10Excavator weight Gamma 15.2 tons 10.8 tons 10Excavator tooth length Uniform 90 mm 28.9 mmGouge depth Weibull 1.2 mm 1.1 mmGouge length Lognormal 201 mm 372 mm 8Number of signicant corrosion defects Poisson 4.0/km 2.0/kmInitial average defect depth Weibull 0.005 mm 0.0025 mmInitial defect length Lognormal 30 mm 15 mmAverage defect depth growth rate Weibull 0.06 mm/year 0.03 mm/yearDefect length growth rate Lognormal 1.0 mm/year 0.5 mm/yearMax. to avg. defect depth ratiobShifted lognormalc2.16 1.03aThe mean impact rate is suggested in Ref. 10 for a pipeline located in an undeveloped area i.e., class 1 and class 2 areaswith typical equipment impact prevention measures.bThe ratio was used to calculate the maximum defect depth from the average depth for the purpose of distinguishing small leakfrom burst.cLower bound=1.0.Table 6 Input parameters for modeling maintenanceParameter CharacterizationYear of rst maintenance event 10aSubsequent maintenance event interval 10 yearsa 0.5a 1.39aq 2.30/wall thicknessbedNormal distributioncMean=0; std. dev. =0.078wall thicknesselNormal distributioncMean=0; std. dev. =15.6 mmarbMwascalculatedusingtheB31Gmodiedcriterion 18. Theinspectionintervalandrepairthresholdsareconsistentwithtypical industry practice 11.bBased on assumed tool specications that give a POD of 90% for a reference defect depth of 10% wall thickness.cBased on assumed tool specications that give a depth measurement error band of 10% wall thickness with a probability of80%, and a length measurement error band of20 mm with a probability of 80%.Table 7 Input parameters for evaluating safety-related failure consequencesParameter CharacterizationPopulation in hazard zone Poisson distributionMean=1.54 people/hectare;Probability of occupancya0.4bIndoor-outdoor split given occupancy 90% indoor; 10% outdoorbaRepresent the probability of people being present within the hazard area given failure.bBased on the values given in Ref. 11.Table 8 Summary of unit costsCost item Unit CostaSourceBare pipe $/tonne 1400 6Transportation and double-jointing % bare pipe cost 66Welding 20Inline inspection $/km 4000Corrosion defect excavation $/defect 50,000Corrosion defect repair 5000Fatality $/fatality 5,962,000b16Injury $/injury 20,000bProperty damage due to rupture-ignition $/rupture 1,787,700 DOT databaseProperty damage due to rupture-no ignition $/rupture 514,300Property damage due to large leak $/large leak 457,800bDiscount rate % 5.0 5aAll absolute costs are in terms of 2006 Canadian dollars CAD$.bCoststhat wereoriginallygiveninUS$at yearsother than2006wereconvertedto2006CAD$byassuminganannualination rate of 2.0% and an exchange rate of 1.00 CAD$=0.85 US$.031702-10 / Vol. 133, JUNE 2011 Transactions of the ASMEDownloaded From: http://pressurevesseltech.asmedigitalcollection.asme.org/ on 03/30/2015 Terms of Use: http://asme.org/termsFinally,itmustbeemphasizedthattheconclusionsreachedinthisstudyareapplicabletotheparticularexampleonly. Furtherstudies will be required to investigate if any of these conclusionscan be generalized.AcknowledgmentThenancial support providedtotherst authorbytheUni-versity of Western Ontario and by the Natural Sciences and Engi-neeringResearchCouncil ofCanada NSERCunderGrant No.376295-2009isgratefullyacknowledged.TherstauthorwouldliketothankProfessorH.P. Hongforhelpfuldiscussionsduringthe preparation of the manuscript.AppendixTheinput parametersusedtoestimatetheexpectedlife-cyclecosts and probability distributions of the overall returns are sum-marized in Tables 48.References1 Rosenblueth, E., and Mendoza, E., 1971, Reliability Optimization in IsostaticStructures, J. Engrg. Mech. Div., 97, pp. 16251642.2 Streicher, H., and Rackwitz, R., 2004, Time-Variant Reliability-OrientedStructural Optimization and a Renewal Model for Life-Cycle Costing,Probab. Eng. 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