RISKBASEDSHIPHULLSTRUCTURALDESIGNANDMAINTENANCE

PLANNING

Federico Sisci

Thesis to obtain the Master of Science Degree in

Naval Architecture and Marine Engineering

Supervisor: Prof. Yordan Garbatov

Jury (Examination Committee)

Chairperson: Prof. Carlos Guedes Soares Supervisor: Prof. Yordan Garbatov

Member: Prof. Manuel Ventura

July 2017

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AbstractObjectiveofthisworkistodevelopatoolforamulti-purposevesselthatestimatesoptimalshipspecifications,

defines the optimal scantlings of themidship section and finds out the optimalmaintenance strategy. The

measure parameters for the optimization tasks are the cost, aiming to a consistent model that, based on

structuralsafety,isthemostconvenientfortheshipowner.Thefirsttask,throughanoptimizationalgorithm,

findstheminimumoftheRequiredFreightRatebychangingthedesignvariables(Lpp,B,D,T,Cb,Vs),takinginto

a consideration of constraints related to the ship’s operational profile and stability. Employing MARS2000

software, a typical midship section of a multi-purpose vessel has been designed and the ultimate bending

momentisevaluated.

ThesecondtaskaimstheoptimizationofthescantlingsthroughtheFirstOrderReliabilityMethod,inwhichthe

probabilisticvariablesofthewave-induced,stillwaterandultimatebendingmomentsandtherespectivemodel

uncertaintyfactorsareaccountedforinthelimitstatefunction,thatdefinestheboundsforthefailureofthe

ship.Theestimatedreliabilityindexesareusedtocalculatetherisk-associatedcostsbeingpossibletofindout

itsminimum.

The third task consists of analyzing six differentmaintenance policies,which, considering different levels of

corrosiontolerancesandrepairconsequences,definetheoptimalstrategybasedonthecosts,downtimeand

availability.

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Contents

1 Introduction.............................................................................................................8

2 Objectives and methodology..............................................................................10

3 Theoretical background.......................................................................................113.1 Conceptualdesign.............................................................................................113.2 Risk-basedmodel..............................................................................................123.3 Maintenancebasedmodel................................................................................14

4 Conceptual design................................................................................................164.1 Power calculation..........................................................................................174.2 Weight estimate.............................................................................................234.3 Cost estimate.................................................................................................294.4 Stability...........................................................................................................334.5 Freeboard.......................................................................................................344.6 Rolling period................................................................................................354.7 Single-objective constraint optimization...................................................35

5 ScantlingsthroughMARS2000..................................................................................40

6 Risk based model.................................................................................................436.1 Limitstatefunction...........................................................................................436.2 Reliabilityofthestructure.................................................................................516.3 Reliabilitycostmodel........................................................................................53

7 Maintenance based model..................................................................................571.1 Probabilityandseverityofstructuralfailureduetocorrosion...........................581.2 Optimalreplacementintervalwithminimizationoftotalcosts.........................621.3 Optimalreplacementagewithminimizationoftotalcosts................................631.4 Optimalreplacementageaccountingfortimerequiredtoperformreplacement 651.5 Optimalreplacementintervalwithminimizationofdowntime..........................661.6 Optimalreplacementagewithminimizationofdowntime................................671.7 Optimalinspectionintervalwithmaximizationofavailability...........................68

8 Conclusions...............................................................................................................71

Bibliography..................................................................................................................73

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ListofFiguresFigure1-1:Flowchartofthetasksinthecode..........................................................................9Figure3-1:Stagesoftheshipdesign.......................................................................................11Figure3-2:FSAMethodology..................................................................................................14Figure3-3:Thicknesswastageduetocorrosionasfunctionoftime[19]...............................15Figure4-1:Conceptualdesigntool..........................................................................................16Figure5-1:Materialsintheplatesofthemidshipsection(greenforHS,redformarinesteel)

.........................................................................................................................................40Figure5-2:DMFvsultimatebendingmoment.......................................................................42Figure6-1:Risk-basedmodelflowchart.................................................................................43Figure6-2:Sensitivityanalysisforvariablesofthelimitstatefunction..................................51Figure6-3:Beta-indexasfunctionoftheDMF.......................................................................51Figure6-4:ProbabilityoffailureasfunctionoftheDMF........................................................53Figure6-5:Totalcostsasfunctionofthebeta-index..............................................................55Figure7-1:Hazardrateasfunctionoftime,theso-calledbathtubfunction..........................58Figure7-2:Reliabilityofcorrodedplates................................................................................60Figure7-3:Costsasfunctionofthepreventivereplacementintervals..................................63Figure7-4:Totalcostperunittimeasfunctionofthereplacementages..............................64Figure7-5:Totalcostperunittimeasfunctionofthereplacementageaccountingfortime

requiredtoperformreplacement...................................................................................66Figure7-6:Downtimeperunittimeasfunctionofthereplacementinterval........................67Figure7-7:Downtimeperunittimeasfunctionofthereplacementage...............................68Figure7-8:Availabilityperunittimeasfunctionoftheinspectioninterval...........................70

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ListofTablesTable1:Boundsfordesignvariables.......................................................................................36Table2:Boundsforproportionsbetweenmaindimensions..................................................36Table3:Inputvariable.............................................................................................................37Table4:Fourmodelspresentedasexample...........................................................................38Table5:Dimensionsusedforthescantlings,risk-basedandmaintenance-basedmodels....39Table6:RelationbetweenthicknessmodificationandDMF..................................................41Table7:UltimatebendingmomentobtainedfromMARS2000..............................................41Table8:GrossandnetmodulusatdeckofthemidshipsectionobtainedfromMARS2000.42Table9:Modeluncertaintyfactors.........................................................................................44Table10:valuesforthecoefficients!0, !1, !0....................................................................46Table11:Beta-indexandprobabilityoffailureforthedifferentDMFforintactandcorroded

condition.........................................................................................................................52Table12:Valuesforthecostsforeachbeta-index.................................................................56Table13:Corrosiontolerancelimits.......................................................................................59Table14:Tolerance/consequence..........................................................................................61Table15:Optimalpreventivereplacementintervalinyears..................................................63Table16:Optimaforreplacementages..................................................................................64Table17:Optimalpreventivereplacementtimeageaccountingforthetimerequiredto

performreplacement......................................................................................................65Table18:Minimizationofthedowntimebyoptimizingthereplacementinterval................66Table19:Minimizationofthedowntimebyoptimizingthereplacementage.......................67Table20:Optimalinspectionintervalswithmaximizedavailability.......................................69

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1 Introduction

Sinceengineeringandarchitectureareappliedtobuildanykindofvessel,thedesignprocesshasbeenkeptfor

years always more or less the same, with a lot of estimations, complex analyses, iterations and data

extrapolations,leadingtoabignumberofcompromises,wheretheobjectivewasasatisfactorygoodmodel,and

nottheoptimalmodel.Forthisreason,ithasbeendecidedtoapplyacomputer-aidedoptimizationonthesame

methodologies,withtheresultofanincreaseinspeedandquantityofoutputsforthedesignprocess,i.e.the

timeinvestedfortheconceptualdesignhasbeenreducedfromfewmonthstofewdays[1],givingasoutputan

optimumdesignfortheshipmodel.

Besidesthereducedtimeandhigherreliability,thesoftware-basedprocesshasbecomeessentialtoengineers

andnavalarchitectsasitallowstoanalyzeabignumberofoptionsandsolutions,achievinganoptimizedmodel

thattakesintoaccounttechnicalandeconomicissues.

Forthispurpose,standardengineeringprincipleshavebeenappliedwiththeobjectivetoformulateatoolthat

can achieve an optimized design based on the costs, structural reliability and maintenance strategy, by

consideringconstraintsregardingtheship’soperationandship’sintegrity.Astodesignthemodeltherearemany

challengesbecauseoflackofdata,modeluncertaintiesandinterdependabilityofdifferentfactors,inlinewith

thetraditionalengineeringdesign,assumptionsandapproximateestimationshadtobecarriedout,inorderto

achieveagoodbalancebetweentoolspeedandreliability.

TheoptimizationhasbeenperformeddevelopingatoolthroughtheuseofMATLAB,amulti-paradigmnumerical

computingenvironmentthatwithaniterativemethodfindstheoptimumbychangingtheobjectivevariables.

Twooptimizationalgorithmshavebeenusedforthedifferenttasks,whichusewillbeexplainedfurtherinthe

chapter“Objectivesandmethodology”.

Thetoolconsistsofthreeconsecutivetasks,whicharerespectivelytheconceptualdesign,structuralreliability

assessmentbasedonscantlingmodificationandmaintenancestrategypolicies(Figure1-1).

Conceptualdesignisthefirststageoftheship’sdesignanditsfunctionistodefinetheinitialdataofthemodel

basedonrelationsbetweenship’scharacteristics,weights,transportablecargoandcosts,subdividedintocapital

costs (CAPEX) and operative costs (OPEX). In the tool, it is implemented the possibility to personalize the

optimization problem, by considering owner’s specification requirements, such as shipyard capacities, total

distanceandrestrictionsofthevoyage,beingtakenasmeasureofmerittheRequiredFreightRate(RFR),acost

measureforthecargofreight[2].Inthisway,itcanbeshapedthemidshipsectionusingthesoftwareMARS2000

developedbyBureauVeritas inordertoobtainmidshipsectionmodulusandtheultimatebendingmoment,

whicharecalculatedbasedonstipulatedmethodsbasedontheClassificationSociety.

Moredetailed shipdesign is achieved ina secondstepbyemploying the risk-basedandmaintenance-based

models,aswiththesewillbecalculatedtheoptimalscantlingsandtheoptimalmaintenancestrategy.

Inthesecondtask,thebendingmomentsdescribedbyprobabilisticdistributionsarecalculatedinthelimitstate

functionandthereliabilityindexiscomputedthroughtheFirstOrderReliabilityMethod(FORM).Thereliability

indexiscalculatedfordifferentscantlingsandhavingasmeasureofmeritthetotalcost,themostconvenient

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scantlingsischosen[3].Atlastsixdifferentmaintenancepolicieshavebeenadopted,whichhaveasanobjective

thecosts,downtimeandavailability,inordertoestimatethemaintenancestrategythatallowstooptimizeship’s

operation.Forthistask,inordertotakeintoconsiderationdifferentprobabilitiesrelatedtothefailurestateof

corrodedplatesandtotheconsequencesofarepair,havebeencalculatedtheobjectivevaluescomputingthe

combinationmatrixbetweenthedifferentcorrosiontoleranceandrepairconsequencelevels[4].

Inordertobeabletomakemoredetailedassumptionsandtoanalyzemoreinthespecificthemodelproblem,

ithasbeenchosentofocusthecodeononeshipcategory,whichisthemultipurposeship.Thistypeofvessel

allowsthecarriageofmanydifferentcargoes,fromwoodtobulk,althoughinthistoolthecargotankisdesigned

calculatedbasedonTEUblocks.

Asthischosenshiptypewouldpresentawiderangeofdifferentsolutions,themodelhadtobeconstrainedeven

moretoarangeofdimensions,whichwillbediscussedmoreindetailinthechapter“Conceptualdesign”.

Figure1-1:Flowchartofthetasksinthecode

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2 Objectives and methodology

ThetoolforthethreetasksiscodedinMATLAB,whichisamulti-paradigmnumericalcomputing,allowingeasy

implementationoftheoptimizationprocess.

Two optimization algorithms are used for this code, the non-linear optimization algorithm, that performs a

sensitivityanalysisforthevariablesandanalyzesthegradientoftheobjectivefunctiontofindtheoptimum,and

theoptimizationwiththeuseofageneticalgorithm,whichcategorizesthefirstsolutionsregardingtheobjective

functionandafterkeepschangingthevariablesuntiltheobjectivevaluedoesnotvaryanymoreconvergingto

theoptimum[5].

Fortheconceptualdesign,ageneticalgorithmisadopted,which,althoughslower,ismorereliableinthequality

of thesolution.Theobjectiveof this task isminimizing theRequiredFreightRate (RFR),bymodifying length

betweenperpendiculars(%&&),breadth('),depth((),draft()),blockcoefficient(*+)andservicespeed(,-),

definedasdesignvariables.Moreover,constraintsregardingship’soperationandtechnicalspecificationsare

applied,inordertoreducethenumberofoutputsthathavetobeanalyzed.Torelatetheobjectivevaluewith

thedesignvariables,inlinewiththetraditionalengineering,empiricalformulasbasedontheregressionfrom

existing ships, formulas from classification society and estimations to reduce complexity of theproblemare

applied.

Themidship sectionof the resulting ship from the firstoptimizationprocess isdesignedbyMARS2000. This

program calculatesmidship sectionmodulus and ultimate bendingmoment by shaping the typical midship

sectionofamultipurposevessel.Differentscantlingsarecalculatedtoobtaintherespectiveultimatebending

moment,usedinthereliabilitymodel,andthemidshipsectionmodulus.

TheFirstOrderReliabilityMethod(FORM)isusedtocalculatethereliabilityindexfortherespectiveultimate

bending moment with the non-linear optimization algorithm, as it delivers the same values as the genetic

algorithm,beingatthesametimefaster.Asthedifferentreliability indexesrelativetothescantlings implya

modificationinthecostsregardingtheriskoffailureandinvestmentinbuildingthestructure,thecostsforthe

differentscantlingsarecalculatedandthesolutionwiththelowesttotalcostsischosen.

Asimilarapproachisalsoadoptedforthemaintenancemodel,whereafunctionbetweenthedifferentstrategies

andthetotalfinalcostsiscalculatedandtheminimumchosenasanoptimalmodel.Inordertotakeintoaccount

materialbehaviorandthedifferencesintheacceptance,fourdifferentlevelsofcorrosiontoleranceandrepair

consequenceareapplied.Forthistask,differentmaintenancepoliciesareemployed,basedonminimizingcosts,

downtimeoronmaximizingavailabilityofthevessel.

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3 Theoretical background The present tool, coded in MATLAB, is based on three consecutive tasks, which focus respectively on the

conceptualdesign,risk-basedandmaintenance-basedanalyses,andforthisreasonhaveadifferenttheoretical

background,whichwillbeindividuallypresentedinthefollowingchapters.

3.1 ConceptualdesignTheshipdesignconsistsinasequenceoftasksthataimtodefinealldimensionsandspecificationsneededto

buildaship.Itisgenerallydividedintofivemainstages:conceptualdesign,preliminarydesign,contractdesign,

detaildesignandproductiondesign(Figure3-1).

Theconceptualdesignistheinitialstepofthevesseldesignanditsobjectiveistodefinethemaindimensions,

cargocapacity,speedandtoperformaneconomicfeasibilityanalysis,takingintoaccountrequirementssetby

theshipowneroveroperabilityandfunctionalityofthevessel[6].

Duringthelastcentury,theconceptualdesignhaschangedhisroleintheshipbuilding,asfromanestimative

tool ithasbeentransformedintoamoreconsistentprocessuptothepointwheretheestimationsaremore

realistic and it is possible to create a series ofmodels betweenwhich can be chosen themost convenient

solution.

Figure3-1:Stagesoftheshipdesign

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HarryBenford[7]appliedtothedecision-makingprocessinbuildingashipthecostsassessment,withwhichitis

possibletoestimateoptionsforthevesselascostdependent,astheshipneedstobeaprofitableinvestment.

For this purpose, it has been introduced asmeasure ofmerit the Required Freight Rate (RFR),which is the

minimumpriceofaunitofcargo(TEUinthistool)tobedeliveredfromoneporttoanother.

AsinitiallyengineersweremanuallycalculatingtherelationbetweenthedesignvariablesandtheRFR,itwas

possible to define just fewmodels for the same problem, as a higher number of iterations would be time

consuming,leadingtoacceptablesolutions.Lyon[8]adaptedthisprocesstoacomputeraidedoptimization,so

that the quality of models is higher, by increasing the number of output models and choosing the most

convenientbetweenthese,loweringdrasticallythetimeneeded.Inthisway,it ispossibletoconsiderawide

rangeofinterconnectedmodelsbasedontechnicalefficiency,costs,operationandpriorities,withtheobjective

tominimizetheRFR.

Michalski [9] proposed amodel where the design objective is the RFR based on the optimization of ship’s

deadweight,whichisfasterandanalyzestheproblemundertheperspectiveoftheshipowner.Tothishasbeen

preferredtosettheshipmaindimensionsasdesignvariables,asthesearedynamicallyusedtodescribealsothe

riskandthemaintenancebased-models.

As in reality the ship design problem can’t be described just by a single-objective optimization, but more

objectives have to be taken into account, Ray and Sha [10] proposed amulti-objectivemodel optimization.

Indeed, as a ship can be also optimized based on e.g. maneuverability, sea keeping or specific ship type

characteristics, the different optimizing objectives can be associated to aweightage factor definedwith the

AnalyticHierarchyProcess,usedtoformulatemathematicallytheshipowner’spriorities.Theproposedmulti-

objectivemodeloptimizationisestimatedtobemoredetailedthanrequestedbythisproject,astheproblem

restrictedtoasingle-objectiveoptimizationgivesasatisfactoryandmoresimplesolution.

Chen[2]proposedasingle-objectiveoptimizationmodelappliedoncontainership,where,besidesthedescribed

standardprocedure,adoptedasmoothfunctiontocorrelatethenumberofTEUavailableonthevesselandthe

shipmain dimensions using the least square fittingmethod, to solve the problem of having a non-smooth

functionduetotheintegernumberofTEU.

3.2 Risk-basedmodelAstheloads,thatashipexperiencesduringitslifetimeandthestrengthofthestructuralmaterial,presentmany

uncertainties,itisnotpossibletopredictifthevesselwillfail.Forthisreason,Mansour[11]proposedamodel

basedonstatisticallydistributedrandomvalues,asalsothestructureresponsefollowsasuchpattern.Inthe

model have been considered the extreme values of the wave-load bendingmoment, as its variance is not

negligibleand,thus,alsostatisticallydistributedmaximahavetobeconsidered,whilethestillwaterbending

momentisassumedtohaveadeterministicvalue.

FaulknerandSadden[12]applyonsuchproblemthelimitstatefunctionandproposeashipdesignpolicybased

onpartialsafetyfactorsapproachandonthesafetyindex(beta-index),analyzedwiththeFirstOrderReliability

Model(FORM).Thelimitstatefunctionisdesigncriteriathat,appliedontheship’sbendingmoments,defines

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thelimitbetweenfailureandsaferegion,wherebyanincreaseofwave-inducedandstillwaterbendingmoments

increasesalso theprobabilityof failure, thus theprobability toovercome the limit state representedby the

ultimatebendingmomentinadefinedperiod.

Moreinvestigationhasbeendoneonthemodelsthatdescribethestructuralbehaviorofavesselsubjectedto

loads, leading to an efficient solution by allowing to consider in the limit state function different types of

distributionforthevariables[13]. Inthisway, it ispossibletoconsidertheextremevaluesforthewave-load

bendingmomentandtoconsiderstillwaterbendingmomentnormaldistributed.

Zayed et al [14] consider the probability of failure being time-dependent, as the reliability of the structure

decreaseswiththetime.Alsointhismodel,usingrandomvariablesandstochasticrandomprocesses,thefailure

isdescribedbytheupcrossingofthelimitstaterepresentedbytheultimatebendingmoment.

GarbatovandSoares[3]analyzedamodelappliedonacontainership,analyzingitstypicalmidshipsectionand

takingintoaccountloadsinfull,partialandharborconditions.Moreover,thereliabilityindexhasbeenexamined

asdependentfromtheheelingangle.

BesidestheFORM,havebeenanalyzedalsoSecondOrderReliabilityModels(SORM),buttheirapplicationto

thisproblemisratherinconvenient,asitdefinesamorecomplexlimitstatefunctionasitisactuallyneededby

theproblem[15].

The FORM is a probabilistic reliabilitymethod based on the first-order expansion aboutmean value (Taylor

expansion)andcalculatesthereliabilityindexthroughanalyticalequations.Ifitisnotpossibletoapproximate

thelimitstatefunctiontoalinearcurvebutrathertoaquadraticfunction,itisneededtoadopttheSORM,which

ismorecomplexbutallowsabetterestimationoflimitstatefunctionsofhigherorderthan1.

The Monte Carlo method is a simulation method that consists in generating sample data, considered as

observations, calculates the solutions based on such samples and produces statistical distributions for the

results,withhigherprecision if thenumberofdata sampled ishigher.Thismethod isusuallyadopted if the

probabilitydistributionsarenotknownandhavetobeassumedortheproblempresentsrandomvariables.

ThecurrentstateofinvestigationhasbeenconsideredinparalleltotheFormalSafetyAssessment(FSA)[16],a

systematicmodelusedtoanalyzetherisksandtheassociatedcostsandevaluatethebenefitsofchangesinthe

structure.

The FSAmethodology (Figure 3-2) identifies five stepswhere the hazard, the risk and the cost benefits are

calculatedandevaluatedbasedondecisionmakingrecommendations.

Thehazardisdefinedas“apotentialtothreatenhumanlife,health,propertyortheenvironment”andtherisk

isthe“combinationofthefrequencyandtheseverityoftheconsequence”[16],andbasedontheseconcepts

theaimoftheproblemistominimizethecostsreducingtherisks.

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3.3 MaintenancebasedmodelTheagingstructuresexposedtomarineenvironmentspresentadecreaseinthereliability,beinginthiswaymore

likely that the system failsafteraperiod.For this reason, ithas tobeplannedan intervalafterwhich some

componentsofthestructurehavetobereplaced(maintenanceinterval), inordertosubstituteitbeforelose

strengthandtoguaranteetheminimumdesignreliabilityofthesystem.

Themain reason to performmaintenance is the corrosion, described as gradual loss ofmaterial caused by

chemicalorelectrochemicalreactionwiththeenvironment.Itisofamajorissueforthestructuralreliabilityas,

besidesthelossofweightandmidshipsectionmodulus,itcausesalsolossofmaterialstrengthproperties.

Giventhatthecorrosionpresentsahighnumberofuncertainties,Melchers[17]proposesaphenomenological

modelwheretherate,extent, localizationandvariabilityofexpectedcorrosionareanalyzed,consideringthe

lossofmaterialduetocorrosionastime-dependent.

Yamamotoand Ikagaki [18]define the corrosionprocessbasedon three stages: degradationof the coating,

generationofcorrosionpitsandpropagationofthecorrosionpitsuntilthestructuralfailure.Themodelthat

describesthecorrosionprogressionisbasedonprobabilisticdataobtainedbyanalyzingthethicknessdecreases

inthemeasuredplates. Inthisway,thecorrosionmodeldesignedresultstobeanon-lineartimedependent

process,wheretheprogressofthedepthofthecorrosionisnon-linearaswell.

Amoredetailedmodelwithconsiderationofnon-linearbehaviorand randomprotectioncoatingduration is

presentedbyGuedesSoaresandGarbatov[19],whichdefinethefailurewhenthedepthofcorrosionwastage

reachesthecorrosiontoleranceofthecomponent.Also,thismodeldefinesthreestages(Figure3-3),wherethe

first,definedas./,isthetimewherethecoatinghaseffect,andforthisreasonthereisnomaterialwastagedue

Figure3-2:FSAMethodology

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tocorrosion,theseconddefinestheperiod.0whencorrosionpitsaregeneratedwithanon-linearincreaseof

thethicknesswastageandthethirdstagepresentsanapproximatelinearwastageofmaterial.Thus,inthismodel

thereliabilityisassessedtobetime-dependentandforthisreasonitiseasytoimplementareplacementpolicy

forthecomponentsthatsurpasstheacceptablecorrosiondepth.

Consideringthemaintenancestrategy,Singh[20]proposedthreepoliciesbasedontheminimizationofcosts

and maximization of availability: replacement age, minimal repair with replacement age and minimal

repair/failurereplacement.Inthismodelthefailurerate,definedasthefrequencyatwhichonecomponentor

asystemfailsinadeterminedunitoftime,issupposedtofollowtheWeibulldistribution.

Garbatov [4]proposessixdifferentmaintenancepolicies:optimal replacement interval,optimal replacement

ageandoptimalreplacementageaccountingfortimerequiredtoperformreplacementbasedonminimization

of the total costs, optimal replacement interval and optimal replacement age based on minimization of

downtime,andoptimalreplacementintervalbasedontheavailabilityofthesystem.Moreover,inthismodelare

calculatedalsofourdifferentlevelsofcorrosiontoleranceandfourofrepairconsequence,inordertoconsider

thevariationsinthematerialstrengthandrepaireffectiveness.

Thedowntimeistheperiodwhenthecomponentorsystemisexpectedtobeunavailable,whiletheavailability

istheprobabilitythatacomponentorsystematadefinedunittimeisabletooperate.

Figure3-3:Thicknesswastageduetocorrosionasfunctionoftime[19]

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4 Conceptual design Theconceptualdesignisthefirstphaseofashipdesignwheretheinitialdimensionsanddataofthemodelare

estimated. In this work, the conceptual design has been aimed coding the single-objective constrained

optimization withMATLAB, which consists in changing a number of variables within constrains in order to

minimizeormaximizeanobjectivevaluethroughageneticalgorithmandanon-linearsolver.

Initially,physicalandempiricalformulashavetobedefined,inordertoallowtoestimatethefirstdataofthe

shipstartingfromsomeassumptionsexplainedalongthechapterandfewmaindimensions,whichforthiscase

are:

• %&&,lengthbetweenperpendicularsinm

• ',breadthinm

• (,depthinm

• ),draftinm

• *+,blockcoefficient

• ,-,vessel’sspeedinkn

Afterthatthestructureofformulasthatrelatemaindimensionstomoredetaileddatahasbeencreated,the

maindimensionslistedabovearesetasvariablessothattheycanchangewithinconstraints,whicharechosen

basedonrequirementsontheship,onphysicallimitationsandondatafromshipsofsimilartype(moredetailed

descriptioninChapter“Single-objectiveconstraintoptimization”).

Finally,thevariableshavebeensetinordertominimizetheRequiredFreightRate121,whichhasbeenchosen

becauseitresumesallthemaindataoftheship(dimensions,power,cargocapacity,weightsandcosts)andat

thesametimeitisaneasywaytoevaluatetheresults,asitistheminimumvalueforacontainer(TEU)tobe

shipped.

Figure4-1:Conceptualdesigntool

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Underthepointofviewofthecodeexecution(Figure4-1),theprogram,startingfromthedefaultinitialdata,

computesifthemodelpresentstheminimumobjectivevalue,ifthisshallnotbethecase,thecodegenerates

newrandomdesignvariables.Withthesevaluesestimationsaboutpower,weightsandcostsaredoneandthe

resultsarecheckedfortheconstrainsimposedtothemodels.Oncetheprogramhasfoundtheminimumforthe

objectivevalue,thecodeendsanddisplaysthefinaloptimizedvalues.

4.1 Power calculation

Inordertoestimatethepowerassociatedwiththemaindimensionsoftheship,awiderangeofmethodshas

beendevelopedintheyearsadoptingapproachesmoreorlessaccurate,wheretoincreasinglevelofaccuracy

oftheresults followhigher levelofdetailingandestimationofsomedata.Anexampleofaquickandrather

imprecisemethodtoestimatethepoweristheso-calledAdmiraltyformula,whichdependsfromthespeed,the

displacementandthetypeofthesship.

Tothismethod,itwaspreferredamoreaccurateanddetailedapproach,whichconsistsfirstinestimatingthe

total hull resistance in order to obtain the necessary effective power required for the given speed. Also, to

estimatethetotalresistanceactingonthehullhavebeendevelopedseveralmethods,amongwhichhasbeen

chosentheHoltropandMennen’smethod[21].Thismethodwaschosenbecausehasbeenprovedtobemore

reliableconcerningmerchantships,andhavingbeenupdatedfewtimes,itisstillnowadaysawideusedmethod.

TheHoltropandMennenmethodsubdividesthetotalresistancethatactsonthehullin:

1343 = 16 1 + 89 + 1:;; + 1< + 1= + 13> + 1: (1)

Where:

1343–TotalresistanceoftheshipinN

16 –FrictionalresistanceinN

1 + 89–Formfactorforviscousresistance

1:;;–AppendagesresistanceinN

1<–WaveresistanceinN

1=–Pressureresistanceonbulbousbow

13>–TransomresistanceinN

1:–CorrelationresistancefrommodeltoshipinN

Thefrictionalresistanceresultsfrom:

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16 = 0.5B,-C*DE (2)

where*Disthefrictionalresistancecoefficientcalculatedas:

*D =

0.0075GHI 1J − 2 C

(3)

andSistheprojectedwettedsurfaceofthebarehull,calculatedasfollows:

E = %&& 2) + ' *M 0.453 + 0.4425*+ − 0.2862*M − 0.003467

')

+ 0.3696*S + 2.38T=3*+

(4)

*M = 1 − 0.0622JU.VWC

(5)

Theterm 1 + 89 representstheformfactorofthehullandisisdefinedby:

1 + 89 = 0.93 +

0.487118X9Y '%&&

9.UZ[UZ

)%&&

U.YZ9UZ

%&&%\

U.U9C9]Z^

%&&^

_

U.^ZY[Z

1

− *&`U.ZUCYV

(6)

%\ = %&&(1 − *& + 0.06*&%Xb/(4*& − 1)) (7)

%Xb = 8.80 − 38.92J (8)

*& =

*+*M

(9)

Theresistancederivedbytheappendages1:;;hasbeenignoredandconsideredequalto0N,asassumedin

[2].

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Inordertocalculatethewaveresistance,havebeendevelopedthreeformulas,whichcorrespondtothreeranges

ofFroudenumber[2].If2e > 0.55:

1<`= = gBIX9VXCX]hij[l^2em + lYXHn(o>2e`C)] (10)

With:

X9V = 691.3*M`9.^^YZ

g%^

C.UUWVV

%'− 2

9.YUZWC

(11)

XC = hij(−1.89 X^) (12)

X^ = 0.56

T=39.]

') 0.31 T=3 + )6 − ℎ=

(13)

X] = 1 − 0.8

T3')*M

(14)

l^ = −7.2035

'%&&

U.^CZ[ZW

)'

U.ZU]^V]

(15)

r = −0.9 (16)

lY = X9]0.4hij(−0.0342e`^.CW) (17)

X9] =

−1.69385,%&&^

g< 512

−1.69385 +

%&&

g9^

− 8

2.36,512 <

%&&^

g< 1726.91

0,%&&^

g> 1726.91

(18)

20

o> =1.446*& − 0.03

%&&',%&&'< 12

1.446*& − 0.036,%&&'> 12

(19)

If2e < 0.4:

1<`: = gBIX9XCX]hij[l92em + lYXHn(o>2e`C)]

(20)

With:

X9 = 2223105XV

^.V[Z9^ )'

9.UVWZ9

90 − tu `9.^V]Z](21)

XV =

0.229577'%&&

U.^^^^^

,'%&&< 0.11

'%&&,0.11 <

'%&&< 0.25

0.5 − 0.0625%&&','%&&> 0.25

(22)

tu = 1 − 89hij[−

%&&'

U.[U[]Z

1 − *< U.^UY[Y 1 − *&

− 0.0225%/+U.Z^ZV %>

'

U.^Y]VY

100g

%&&^

U.9Z^UC

]

(23)

l9 = 0.0140407%&&)− 1.75254

g9^

%&&− 4.79323

'%&&− X9Z

(24)

X9Z =

8.07981*& − 13.8673*&C + 6.984388*&^,*& < 0.81.73014 − 0.7067*&,*& > 0.8

(25)

If0.4 < 2e < 0.55:

21

1< = 1<`:U.Y + 102e − 4 (1<`=U.]] − 1<`:U.Y)/1.5 (26)

Where:

)6 –draftattheforeperpendicular,assumedtobeequaltoT,asinafirstestimationthereisnotrim

%Xb–positionoflongitudinalcenterofbuoyancy,estimatedwiththeSchneekluthformula[22].This

valueispositiveforwardthecenteroftheshipandisexpressedaspercentageof%&&.

*M–midshipcoefficient,calculatedwiththeformulafromAlvarino,AzpírozandMeizoso[23]

*&–prismaticcoefficient

T0–transomarea,assumedtobe4m2

%v–parameterforthelengthoftherun

B–saltwaterdensityequalto1.027kg/m3

1J–Reynoldsnumber

Thepressureresistanceonthebow1=iscalculatedwiththefollowingformula:

1= = 0.11 hij −3w=`C 2ex

^ T=39.]BI/(1 + 2ex

C ) (27)

T=3 = 0.08TM (28)

TM = *M') (29)

Where:

TM–transversemidshipsection’sarea

T=3–transversesectionalareaofthebulb,approximatedtobe8%ofthemidshipsectionarea

Thecoefficientfortheemergenceoftheboww=,theverticalpositionofthecenterofthebowℎ=andtheFroude

numberbasedontheimmersion2exaredeterminedby:

w= = 0.56T=3U.]/()6 − 1.5ℎ=) (30)

2ex = ,/ I )6 − ℎ= − 0.25T=3U.] + 0.15,C U.] (31)

ℎ= = 0.9

)2

(32)

22

Ifpresent,theimmersedtransomhasinfluenceontheresistanceonthehullandisderivedbytheformula:

13> = 0.5,CT3XZ (33)

With:

XZ =

0.2 1 − 0.22e3 ,2e3 < 50,2e3 ≥ 5

(34)

2e3 = ,/ 2I

T3' + '*<

U.]

(35)

*S = 0.95*& + 0.17 1 − *&

9^

(36)

Where:

2e3–Froudenumberrelatedwiththetransom

*S–waterlinecoefficient

As the formulas presented above are calculated based on a model, an additive resistance, the model-ship

correlationresistance1:,hastobeaddedtoobtainthetotalresistance.Thisiscalculatedasfollows:

1: = 0.5B,CE*: (37)

*: = 0.006 %;; + 100 `U.9Z − 0.00205 + 0.003

%&&7.5

U.]

*+YXC(0.04

− XY)

(38)

XY =

)6%&&,)6%&&≤ 0.04

0.04,)6%&&> 0.04

(39)

Havingobtainedthetotalresistancethatactsonthehullandknowingthevelocity,itispossibletocalculatethe

requiredEffectiveHorsepower(EHP)andShaftHorsepower(SHP):

23

{|w =

1343,746.0

(40)

E|w =

{|w0.65

(41)

Where:

EHP,SHP–inhp

1343–inN

,–velocityinm/sec

4.2 Weight estimate Thetotalweightofashipcanbesplitintothepropership’sweight,thelightweight}~x�Ä0,andtheconsumables

andcargoweight,thedeadweight}mÅÇm,forthisreasonwhilethelightweightwillbecalculatedinordertoget

an estimation of the building costs, the deadweight is the determinant for operative costs of the ship. The

lightweightisthesumofweightsregardingthehull}ÄÉ~~,whichincludethemainhullstructure,superstructure

andbulkheadsoftheship, theoutfitandhullengineering}ÑÄ, thattakes intoaccounthull insulation, joiner

bulkheads, pipes, deck fittings, cargo booms, anchors, rudder, galley equipment and hatch covers, and the

machinery}M,whichisthesumoftheweightsregardingthewholepropulsionsystem,frompropellertofunnel

[7].

Thedeadweightiscomposedbytheweightofthecargo}/Ñe(containersinthiscase),thefuelweight}DÉÅ~and

theweightoffreshwater,lubricatingoil,storesandcrewandotherweightrelatedtothemachinerybeingidle

}Mx-/.

Thesamesubdivisionhasbeendonetoestimatetheweights[2]:

}343 = }~x�Ä0 +}mÅÇm (42)

}~x�Ä0 = }Ä +}ÑÄ +}M (43)

}mÅÇm = }/Ñe +}DÉÅ~ +}Mx-/ (44)

Thecalculationstoestimatethelightshipweightareasfollows:

}Ä = *-*%9*%C*%^

*Ö1000

U.W

(45)

24

*- = 8550 (46)

*%9 = 0.675 + 0.5*+ (47)

*%C = 1 + 0.36

%-%&&

(48)

*%^ = 0.00585

%ÑÇ(− 8.3

9.[

+ 0.939(49)

*Ö =

%ÑÇ'(100

(50)

ÜáÄ = [48 + 0.15 0.85 − *+

%ÑÇ(

C

](-(

(51)

}ÑÄ = }Ñ +}ÄÅ (52)

}Ñ = 2412

*Ö1000

U.[C]

(53)

}ÄÅ = 1196

*Ö1000

U.[C]

(54)

ÜáÑÄ = 1.005 − 0.0000689%ÑÇ ( (55)

}M = 215*D

E|w1000

U.VC

(56)

ÜáM = 0.47( (57)

Üá~x�Ä0 =

}ÄÜáÄ +}ÑÄÜáÑÄ +}MÜáM}Ä +}ÑÄ +}M

+ 0.3(58)

25

Where:

}Ä–weightofhullstructureinton

}ÑÄ–weightofoutfitandhullengineeringinton

}Ñ–weightofoutfitinton

}ÄÅ–weightofhullengineeringinton

}M–machineryweightinton

%-–lengthofsuperstructureinm

ÜáÄ–verticalcenterofgravityofthehullweightàâà–increaseddepthbecauseofshearandhatchwayapproximatedas1.008[2]

Beforecalculating thedeadweightof theship, it isnecessary tocalculate themaximalnumberofcontainers

(TEU)thattheshipcancarrywiththefollowingformulas[2]:

){ä = ){ä+ + ){äm (59)

){ä+ = E+%+'+(+ (60)

){äm = Em%m'm)Öm (61)

%+ = %m = tJã(

%/Ñe%U)

(62)

'+ = tJã(

' − 2}''U

)(63)

(+ = tJã(

( − ('|(U

)(64)

'm = tJã(

''U)

(65)

)Öm =

4,' < 32.2l

4 +' − 32.27.8

,32.2l ≤ ' ≤ 40l

5 +' − 403

,40l ≤ ' ≤ 43l

6,' > 43l

(66)

26

E+ = 0.8479*+ − 0.0918 (67)

Em = 0.7534 (68)

Where:

){ä+,){äm–numbersofcontainersrespectivelybelowandonthedeck

%+, '+, (+, %m, 'm–containernumberalonglength,breadthandheightdirectionrespectivelybelow and

onthedeck

)Öm–numberofrowsofcontainersabovedeck

E+, Em–stowagefactorsforcontainersrespectivelybelowandonthedeck

%/Ñe–effectivelengthtocarrycontainers

}'–wingtankbreadth,assumedtobe1.83m[2]

DBH–doublebottomheight,assumedtobe1.83m[2]

%U, 'U, (U–length,widthandheightofacontainer,respectivelyequalto6.10m,2.44mand2.59m

Theusageof the integer function in the formulas for the calculationof thenumberofmaximumcontainers

possibleinthethreedimensionsdeliversastepwisefunction,whichleadstheoptimizationcodetodeliverwrong

results, as thegradientof such function is always0. For this reason, it hasbeenusedanestimation for the

calculationsabove,basedonregressionsofexistingships[2]:

){ä+_D~ÑÇ0 = E+(0.0196%ÑÇ'( − 148.6129) (69)

){äm_D~ÑÇ0 = 0.050117%ÑÇ')Öm − 82.6702 (70)

){ä = ){ä+_D~ÑÇ0 + ){äm_D~ÑÇ0 (71)

The lengthoverall%ÑÇhasbeenestimatedtobe2%bigger thanthe lengthbetweenperpendiculars%&&.The

lengthatthewaterline%S~hasbeenassumedtobeequalto%&&.

Theround-tripnumberNTiscalculatedwiththeformulas[2]:

Ö) =

çã(vã

(72)

27

çã = 365 − Hℎ (73)

(vã = %éã + wèã + Eã (74)

%éã = 4

){ä)E%äÖ/\ÇeÅ_D~ÑÇ0

(75)

Ö/\ÇeÅ_D~ÑÇ0 = 0.0187%ÑÇ + 0.3572 (76)

Eã =

(nã,

(77)

Where:

çã–ship’sannualoperationtime

Hℎ–off-hiretimeoftheship,estimatedequalto15days[2]

wèã–Portwaitingtime,estimatedtobe2days[2]

)E%ä–numberofcontainershandledinonedaybyonecrane,equalto1440

Üá+ = ('| + *8Ib(+(U (78)

*ê�+ =

12,*+ = 1

23, *+ = 0.5

hGnh, GtJh!vtJãhvjHG!ãtHJ

(79)

Üám = ( + |*| + )Öm

(U2

(80)

}DÉÅ~ = 1.1

E|w(nãE2*,

(81)

ÜáDÉÅ~ =

6.13385

}DÉÅ~ (82)

28

}Mx-/ = }/& +}DS +}~Ñ +}Ä/ (83)

}/& = 6(vã + 50 *D (84)

}DS = 280*D (85)

}~Ñ = 50*D (86)

}Ä/ = 5wèã*D (87)

ÜáMx-/ = 0.5( (88)

ÜámÅÇm

=}w*){ä+_D~ÑÇ0Üá+ +}w*){äm_D~ÑÇ0Üám +}DÉÅ~ÜáDÉÅ~ +}Mx-/ÜáMx-/

}w*C){ä+_D~ÑÇ0){äm_D~ÑÇ0}DÉÅ~}Mx-/

(89)

Where:

Üá+–verticalcenterofgravityofthecontainersbelowdeck

Üám–verticalcenterofgravityofthecontainersabovedeck

|*|–hatchcoverheight

}DÉÅ~–fuelweightinton

(nã–servicerangeofshipinnm

ÜáDÉÅ~–verticalcenterofgravityofthefuel

}/&–crewandprovisionsweightinton

}DS–freshwaterweightinton

}~Ñ–lubricateoilweightinton

}Ä/–machinerybeingidlerelatedweightinton

*D–conversionfactorlongtonstometrictonsequalto1.016[2]

}w*–weightpercontainer,estimatedtobeequalto12ton[2]

29

4.3 Cost estimate Oncetheweightsoftheshiparedefined,itispossibletoestimatethecoststobuildandoperatethevesselwith

theadditionofmorevalues,suchastheworkingtimetobuildthevessel,laborcostandothers.Assuggestedby

Chen[2]thelaborcost%/hasbeenassumedtobe20$perman-hour.Inaccordancetothisinitialvalue,allthe

formulasusedtoestimatethecostofashipareinUSDandtheweightsinton.

ThetotalannualcostoftheshipissubdividedintocapitalcostTXX(CAPEX)andoperativecostTHX(OPEX).The

firstderivesfromthebuildingoftheshiptakingintoaccountofmateriallaborcostsandprofits,whilethesecond

takes intoaccounttheexpensesrelativetotheoperationoftheship,whichmeansboththeyearlyexpenses

(insurance,crewsalary,administrationandmaintenance)andtheexpensesrelatedtotheactualvoyage(fuel

andportfees).BothCAPEXandOPEXhavebeenfinallyaddedinordertoobtaintheminimumsellingcostof

eachcontainer121inordertorecovertheexpensesoftheship.

Firsthasbeencalculatedthecapitalcostswiththefollowingformulas[2]:

%ℎn = ëℎn%X (90)

ëℎn = *lℎn

}Ä1000

U.[]

(91)

ë!ãn = *nℎ}Ä (92)

%H = ëℎH%X (93)

ëℎH = *H

}Ñ100

U.W

(94)

ë!ãH = *Hí}Ñ (95)

%ℎh = ëℎℎh%X (96)

ëℎℎh = *ℎℎh

}ÄÅ100

U.V]

(97)

30

ë!ãℎh = *ℎéGG}ÄÅ (98)

%l = ëℎl%X (99)

ëℎl = *ℎl

E|w1000

U.Z

(100)

ë!ãl = *ll

E|w1000

U.Z

(101)

ëtnXX = 0.10(ë!ãn + ë!ãH +ë!ãℎh +ë!ãl) (102)

TXXHX = 180000ÖXvhèU.]Z (103)

çìℎX = 0.70(%ℎn + %H + %ℎh + %l) (104)

îãX = ë!ãn + ë!ãH +ë!ãℎh +ë!ãl +ëtnXX + TXXHX + çìℎX (105)

wv = 0.05îãX (106)

îbX = îãX + wv (107)

çèh = 0.05îbX (108)

çèX = îbX + çèh (109)

TXX = çèX*v (110)

31

*v =

1 + ïv ñ~ïv1 + ïv ñ~ − 1

(111)

Where:

%ℎn–laborcosthull

ëℎn–man-hoursforhull

*lℎn–coefficientfortheeffectivenessoftheyard,assumedtobe3160[2]

ë!ãn–materialcostforhull

*nℎ–costofhullmaterialperton,assumedtobe400USD/ton[2]

%H–laborcostforoutfit

ëℎH–man-hoursforoutfit

*H–outfitcoefficient,assumedtobe8000[2]

ë!ãH–materialcostforoutfit

*Hí–costofoutfitperton,assumedtobe1500USD/ton[2]

%ℎh–laborcostforhullengineering

ëℎl–man-hoursforhullengineering

*ℎℎh–hullengineeringcoefficient,assumedtobe20400[2]

ë!ãℎh–materialcostforhullengineering

*ℎéGG–costofhullengineeringperton,assumedtobe3500USD/ton[2]

%l–laborcostformachinery

ëℎl–man-hoursformachinery

*ℎl–machinerycoefficient,assumedtobe6773[2]

ë!ãl–materialcostformachinery

*ll–coefficientformachinerycost,assumedtobe38867[2]

ëtnXX–miscellaneouscost

TXXHX–accommodationcost

çìℎX–overheadcost

îãX–yard’stotalcost

wv–yard’sprofit

îbX–yard’sbuildingprice

çèh–ownerexpense

çèX–ownercost

TXX–annualcapitalcost(CapEx)

*v–capitalrecoveryfactor

ïv–interestrate

EG–ship’slife

32

TheoperatingcostsarecalculatedbasedmostlyonthenumberofcrewmembersintheshipÖ/\ÅS,onthecubic

number*ÖandontheShaftHorsepowerSHP,asfollows[2]:

Ö/\ÅS = *-0[*r8

*Ö1000

9Z+ *Åe�

E|w1000

9/]

(112)

}!Ih = 37800Ö/\ÅS

Y]

(113)

En = 112

Ö/\ÅS10

Y

,Ö/\ÅS < 50

70000 + 5600(Ö/\ÅS − 50),Ö/\ÅS ≥ 50

(114)

ïJn9 = 1351Ö/\ÅS (115)

ïJnC = 14000 + 0.0098(ë!ãℎh + ë!ãn + ë!ãH +ë!ãl + %l + %ℎn +

%H + %ℎh)

(116)

ëvℎ = 151200

*Ö1000

C^

(117)

ëvl = 14000

E|w1000

C^

(118)

wHvã = 20 + 290

*Ö1000

wèãÖ)(119)

TíX = }D2XHnãÖ) (120)

THX = }!Ih + En + ïJn9 + ïJnC + wHvã + TíX (121)

T!X = TXX + THX (122)

33

121 =

T!XÖ)){ä0.8

(123)

Where:

*-0–stewarddepartmentcoefficientassumedtobe1.25[2]

*mê–deckdepartmentcoefficientassumedtobe15.4[2]

*Åe�–enginedepartmentcoefficientassumedtobe10[2]

En–costsrelatedtostoresandsupplies

ïJn9–protectionandindemnityinsurance

ïJnC–hullandmachineryinsurance

ëvℎ–maintenanceandrepaircostsforthehull

ëvl–maintenanceandrepaircostsforthemachinery

wHvã–portexpenses

TíX–annualfuelcost

2XHnã–averagefuelcost,assumedtobe80USD/ton[2]

THX–annualoperatingcost(OpEx)

T!X–annualaveragecost

121–requiredfreightrate,objectivetominimizedintheoptimization

4.4 Stability Inordertomakesomeassessmentsregardingtheinitialstabilityoftheship,hasbeenusedasparameterthe

minimumheelingarmáëMxecalculatedfollowingtheformulaprovidedby[24]:

áëMxe =

wT|_ã!Jó

(124)

w = 0.055 +

%&&1309

C

(125)

T = %ÑÇ ( − ) + )Öm(U%ÑÇ (126)

| =

0.5%ÑÇ ( − ) C + 0.5%ÑÇ )Öm(U C

T+ 0.5)

(127)

_ = B%&&')*+ (128)

Where:

34

w–Pressureontransverseprojectedareasubjecttoheelingangleóinton/m2

T–projectedlateralareainm2

|–verticaldistancebetweencenterofAandcenterofunderwaterlateralareainm

_–displacementoftheshipinton

ó–angleofheel,notsmallerthan14degrees[2]

Thisvalueiscomparedtotheactualheelingarm,calculatedinthestandardway[2]:

áë = Üë − Üá (129)

Üë = Ü' + 'ë (130)

'ë = 0.085*+ − 0.002

'C

)*+

(131)

Ü' = )(0.9 − 0.3*ò − 0.1*=) (132)

Üá =

(Üá~x�Ä0}~x�Ä0 + ÜámÅÇm}mÅÇm)}~x�Ä0 +}mÅÇm

(133)

Where:

áë–transversalmetacentricheight

Üë–transversalmetacentricradius,estimatedwiththeXuebinformula[25]

Üá–heightofcenterofgravity

Ü'–heightofcenterofbuoyancy,estimatedwiththeSchneekluthformula[22]

4.5 Freeboard

Also,theminimumvalueforthefreeboardhastobecalculatedwiththefollowingapproximation[2]:

2'Mxe = X! + b (134)

X = 0.025633%ÑÇU.W9YZ (135)

35

b =0.25 ( −

%ÑÇ15

,%ÑÇ(< 15

0,%ÑÇ(≥ 15

(136)

! =

*+ + 0.681.36

,*+ > 0.68

1,*+ ≤ 0.68

(137)

Theactualfreeboard2'oftheshipiscalculatedas:

2' = ( − ) (138)

Where:

(–depthoftheshipinm

)–draftoftheshipinm

4.6 Rolling period Beingtherollingperiodameasurefortheaccelerationsontheship, ithastobecheckedagainstaminimum

value(1HGGtJIwhvtHrMxe),inthiscaseassumedtobeequalto15seconds,inordertoavoiddiscomfortofthe

crewcausedbyhighaccelerations.Therollingperiodoftheshipisapproximatedwiththefollowingformula[2]:

1HGGtJIwhvtHr = 0.58'C + 4ÜáC

áë

(139)

4.7 Single-objective constraint optimization

Theformulaspresentedabovearemostlyestimationsthatrelateship’smaindimensionstootherparameters

usefulfortheinitialshipdesign,sothateithertheinitialship’smaindimensionsaredecidedandfixedbefore

thecalculationsareperformed,orthesearekeptasvariableinordertodynamicallyoptimizetheproblemand

find an optimal solution. The second procedure has been chosen, as it offers amore flexible design, being

possiblealsotosetconstraintsandlimitstothedataanalyzed.

Forthisreason,havebeenchosensixvariablesasdesignvariables,identifiedastheship’smaindimensions,from

whichallthepreviouscalculationsdepend.Theseare:

%&&–lengthbetweenperpendiculars

36

'–breadth

(–depth

)–draft

*+–blockcoefficient

,-–servicespeed

Asthedesignvariablesduringtheoptimizationcouldassumewhatevervalue,itwasnecessarytoconstrainthese

incertainintervalsthathavebeendefinedbasedondataofexistingships.Thechosenconstraintsareshownin

thefollowingtable:

Table1:Boundsfordesignvariables

Lowerbound Upperboundôöö[m] 100.00 300.00õ[m] 15.00 40.00ú[m] 8.00 30.00ù[m] 6.00 15.00ûü 0.68 0.80†°[kn] 14.0 20.0

Moreover,typicalproportionsbetweenmaindimensionsofmulti-purposevesselshavebeenanalyzedandset

asconstraintsinordertobuildamodelwhichisclosestaspossibletoreality.TheconstraintsareshowninTable

2.

Table2:Boundsforproportionsbetweenmaindimensions

Lowerbound Upperboundôöö/õ 4 6.5õ/ú 1.5 2.5õ/ù 2.25 3.75

As the shipduringonevoyage trip couldhave topass through somecanalsor shallowwaters, alsophysical

constraints had to be taken into consideration. For this purpose, has been estimated the Panama Canal to

presentthehighestrestrictions,sothatitsmaximaldraft() < 12.6l)andmaximalbreadth(' ≤ 32.3l)have

beentakenaswellasconstraintsforthemodel.

Besidesthese,havebeenusedotherconstraintsthatneedtobefulfilledfortheshiptocomplywithRulesand

Regulations, regarding metacentric height (GM) and freeboard (FB), and to avoid discomfort of the crew,

regardingtherollingperiod.Theconstraintsare:

1HGGtJIwhvtHr > 1HGGtJIwhvtHrMxe (140)

37

áë > áëMxe (141)

2' > 2'Mxe (142)

As the ship’s displacement_ and the total ship’s weight}343 have been calculated with two different

procedures(respectivelythroughtheblockcoefficient*+andwiththeformulaspresentedabove)andasthey

havetobesetasequalbecauseoftheArchimedes’principle,ithasbeenintroducedanotherconstraint,which

isfulfillediftheequalitypresentedbelowisverified:

_ = }343 (143)

Thetoolasdesignedallowstoimplementtheinsertionofdifferentvariablesthatarespecifictotheshipyard

(e.g.availablescantlingsorlaborcosts),tothevoyage(e.g.voyagedistance,off-timeperiodorminimumservice

speed)ortotheshipowner’srequirements(e.g.amountofshippedTEUs),sothatamorespecificmodelcanbe

achieved.Inthepresentedcodehasbeenconsideredjustoneinputvariable,showninTable3below.

Table3:Inputvariable

Variable Description DefaultvaluetotalDist distanceofonevoyage 5000nm

Asthemodelisorientatedtotheshipowner,arepresentedfourexamplesforanoptimizedmodelbasedonthe

requirementsforthevoyage(

38

).ThesemodelshavebeenobtainedsettingamaximumnumberofTEUthathavetobeshipped,respectively

650,700,750and800TEU.

Thefourmodelsarepresentedasexampleinordertoshowwhicharethemostconvenientmodelsthat it is

possibletodesignwiththeuseoftheproposedoptimizationtoolbyconstrainingthenumberofTEU.Although

theseareoptimalsolutionsregardingtheRFR,toperformthescantlings,riskandmaintenanceanalysis,ithas

beendecidedtoadoptarealshipmodel.Theusedmodelhasbeentakenfromadesignofamulti-purposeship

bytheBulyardShipbuildingIndustry,beingavailablethemidshipsectionandmaindata.Amoredetailedanalysis

hasbeenperformedforsuchship,applyingtheformulasfortheconceptualdesignovertheprovidedinitialdata.

TheresultsarelistedinTable5.

Definition Variable Model1 Model2 Model3 Model4 UnitLengthbetweenperpendiculars Lpp 137.20 129.02 128.73 131.94 m

Lengthoverall Loa 139.94 131.60 131.3 134.57 mBreadth B 21.11 22.80 23.86 23.685 mDepth D 11.13 11.69 13.09 13.53 mDraft T 8.13 8.594 9.10 8.915 m

Servicespeed Vs 14.39 14.794 15.271 14.41 knBlockcoefficient Cb 0.795 0.749 0.713 0.729 -Midshipsection

coefficient Cm 0.983 0.982 0.981 0.982 -

Displacement Displ 19216 19452 20448 20862 tonShafthorsepower SHP 2358 2496 2678 1975 hp

Power Pd 1143 1210 1298 957 kWNumberofcrew Ncrew 30 31 32 31 -Numberofcranes Ncrane 3 3 3 3 -Metacentricheight GMt 1.90 2.03 1.65 1.40 m

Freeboard FB 3.00 3.09 3.99 4.61 mLightshipweight Wlight 10711 10810 11048 11033 tonDeadweight Wdead 8505 8643 9400 9829 ton

CAPEX Aac 2.29*106 2.35*106 2.48*106 2.39*106 USDOPEX Aoc 1.18*106 1.21*106 1.26*106 1.16*106 USD

Numberofcontainers TEU 650 662 724 763 -

RequiredFreightRate RFR 214.73 212.21 199.739 191.95 USD/TEU

39

Table4:Fourmodelspresentedasexample

Definition Variable Model1 Model2 Model3 Model4 UnitLengthbetweenperpendiculars Lpp 137.20 129.02 128.73 131.94 m

Lengthoverall Loa 139.94 131.60 131.3 134.57 mBreadth B 21.11 22.80 23.86 23.685 mDepth D 11.13 11.69 13.09 13.53 mDraft T 8.13 8.594 9.10 8.915 m

Servicespeed Vs 14.39 14.794 15.271 14.41 knBlockcoefficient Cb 0.795 0.749 0.713 0.729 -Midshipsection

coefficient Cm 0.983 0.982 0.981 0.982 -

Displacement Displ 19216 19452 20448 20862 tonShafthorsepower SHP 2358 2496 2678 1975 hp

Power Pd 1143 1210 1298 957 kWNumberofcrew Ncrew 30 31 32 31 -Numberofcranes Ncrane 3 3 3 3 -Metacentricheight GMt 1.90 2.03 1.65 1.40 m

Freeboard FB 3.00 3.09 3.99 4.61 mLightshipweight Wlight 10711 10810 11048 11033 tonDeadweight Wdead 8505 8643 9400 9829 ton

CAPEX Aac 2.29*106 2.35*106 2.48*106 2.39*106 USDOPEX Aoc 1.18*106 1.21*106 1.26*106 1.16*106 USD

Numberofcontainers TEU 650 662 724 763 -

RequiredFreightRate RFR 214.73 212.21 199.739 191.95 USD/TEU

40

Table5:Dimensionsusedforthescantlings,risk-basedandmaintenance-basedmodels

Definition Variable Value UnitLengthbetweenperpendiculars Lpp 115.07 m

Lengthoverall Loa 117.37 mBreadth B 20.00 mDepth D 10.40 mDraft T 8.30 m

Servicespeed Vs 15.98 knBlockcoefficient Cb 0.719 -Midshipsection

coefficient Cm 0.980 -

Displacement Displ 16591 tonShafthorsepower SHP 3557 hp

Power Pd 1724 kWNumberofcrew Ncrew 31 -Numberofcranes Ncrane 3 -Metacentricheight GMt 1.78 m

Freeboard FB 2.10 mLightshipweight Wlight 10531 tonDeadweight Wdead 6060 ton

CAPEX Aac 2.34*106 USDOPEX Aoc 1.34*106 USD

Numberofcontainers TEU 441 -

RequiredFreightRate RFR 294.35 USD/TEU

41

5 ScantlingsthroughMARS2000Astherisk-basedmodelestimatesthefailurestateofthestructurethroughthelimitstatefunction,theultimate

bendingmomentisneededandiscalculatedusingtheprogramMARS2000developedbyBureauVeritas.

Tocalculatetheultimatebendingmomentinitiallythemidshipsectionhadtobegeometricallydrawn(Figure

5-1),withrelativepanels,stiffeners,compartmentsandmaterials,whichdatahavebeentakenfromthemidship

sectionofthemultipurposeship,initiallydesignedbyBulyardShipbuildingIndustry.Theestimationoftheship

specificationsiscarriedinchapter“Conceptualdesign”andthefinalvaluesusedareshowninTable5.

Asthemidshipsectionshallbecheckedwithdifferentscantlingsinordertoobservetheinfluenceofincreased

ordecreasedmaterial,ithasbeendecidedtomodifyjustthethicknessofthehightensilesteelplates(shownin

greeninFigure5-1)over2mminarange±10llfromtheoriginalscantling.Ithasbeendecidedtojustmodify

theupperplates’thicknessasthisisthepartofthemidshipsectionthatexperiencesthehigheststresses,sothat

itisagoodassumptionthatiftheupperstructurewillnotfail,norwilltherestofthemidshipsection.

Inordertohaveadimensionalfactorthattakesintoaccountthechangeofthicknessithasbeenintroduceda

DesignModificationFactor(DMF)[26],thatistherelationbetweenthemodifiedsectionalareaandtheoriginal

one,asshowninTable6.

Figure5-1:Materialsintheplatesofthemidshipsection(greenforHS,redformarinesteel)

42

Table6:RelationbetweenthicknessmodificationandDMF

Scantlings Sectionalarea DMF-6 1.528 0.94-4 1.560 0.96-2 1.593 0.980 1.626 1.002 1.658 1.024 1.691 1.046 1.724 1.068 1.756 1.0810 1.789 1.10

Ithastobenotedthatalthoughthethicknessmodificationshallhavea±10llrangefromtheoriginalvalue,

in this section has been calculated the scantlingmodification just from−6ll to+10ll, aswith a lower

thicknesstheusedprogramMARS2000doesnotdeliveranyvalue.

With these inputs, it was finally possible to obtain the ultimate bendingmoments in hogging condition (as

standardforcontainervessels)forbothintactëÉ,xe0Ç/0andcorrodedëÉ,/Ñ\\ÑmÅmsectionsforeachoftheDMF,

asshowninTable7anddisplayedinFigure5-2.InTable8aredisplayedthegrossandnetsectionmoduliatdeck

foreachoftheDMF.

Table7:UltimatebendingmomentobtainedfromMARS2000

DMF £§,•¶ß®©ß(MPa) £§,©™´´™¨≠¨(MPa)0.94 772.79 592.980.96 877.37 716.420.98 989.39 797.291.00 1102.18 906.701.02 1214.78 1018.151.04 1324.68 1129.201.06 1431.80 1237.401.08 1515.86 1276.691.10 1540.82 1289.12

43

Table8:GrossandnetmodulusatdeckofthemidshipsectionobtainedfromMARS2000

DMF Grosssectionmodulus(m3) Netsectionmodulus(m3)0.94 3.098 2.560.96 3.396 2.8590.98 3.695 3.1581.00 3.995 3.4581.02 4.294 3.7581.04 4.595 4.0841.06 4.895 4.3591.08 5.196 4.6611.10 5.497 4.963

Figure5-2:DMFvsultimatebendingmoment

44

6 Risk based model Having estimated in the previous chapter themain dimensions of themodel, it is possible tomake further

calculations about the reliability of the structure and find the optimal thickness basedon the probability of

failure. The reliability of the structure has been estimated taking into account the asymmetrical bending

momentsthatactonthehull(stillwaterandwave-loadinducedbendingmoment)andcomparingitwiththe

ultimatebendingmomentofthestructureitself,obtainedbytheprogramMARS2000.Thishasbeenaimedby

calculatingthelimitstatefunctionoftheproblemwiththeFirstOrderReliabilityMethod(FORM),whichprovided

theprobabilityoffailure.Inthisway,itwaspossibletocalculatethecostassociatedwiththeriskoffailureand,

summingitwiththecostresultingbychangingthethickness,itresultsinthetotalexpectedcostasfunctionof

thechangeinthickness,sothatthelowestvalueofthisfunctionpresentstheoptimalthicknessbasedoncosts

relatedtotheprobabilityoffailure.Thetoolfortherisk-basedmodelisshowninFigure6-1.

6.1 Limitstatefunction

Inordertodescribethereliabilityofthestructureundertheasymmetricalbendingmomentstowhichtheship

issubjectduringitslifetimeandtopredicttheprobabilitiesofthistocollapse,ithasbeenusedtheFirstOrder

ReliabilityMethod(FORM)[3].Thismethodhasbeenadoptedtoevaluatethevariablesthathaveinfluenceon

Figure6-1:Risk-basedmodelflowchart

45

thefailureofthestructureandinthiswaytodefinealimitstatefunctionforthereliabilityá Æ ,thatdescribes

whethertheshipisinthesaferegion(á Æ ≥ 0).

IthasbeenpreferredtheFORMtoevaluatethereliabilityoftheship,becauseittakesintoaccountdifferent

typologiesofdistributionofthedifferentvariables,asitisneededintheultimatestrengthanalysis,andithas

beenpreferredthistotheSecondOrderReliabilityMethod(SORM)asthelinearmethodofFORMprovidesa

simplerandsatisfyinglyaccurateapproach[15].

Inthisway,thelimitstatefunctionperformedwithFORMovertheultimatestrengthanalysishasasvariables

theultimate,stillwaterandwave-inducedbendingmoments,andtherelateduncertaintyfactors,summedup

intheequation

á Æ = ØÉëÉ − (Ø-Së-S + ØS∞Øe~ëS∞)

(144)

Where:

ëÉ–ultimatebendingmomentofthestructureinMN.m

ë-S–stillwaterbendingmomentinMN.m

ëS∞–wave-loadinducedbendingmomentinMN.m

ØÉ, Ø-S, ØS∞–modeluncertaintyfactorsforultimate,stillwaterandwave-loadbending moment, see

Table9

χ≤≥–modeluncertaintyfactorfornon-lineareffects,takenas1

Table9:Modeluncertaintyfactors

Mean Deviation COV¥§ 1.05 0.1 0.1¥°µ 1.00 0.1 0.1¥µ∂ 1.00 0.1 0.1

The limit state functionpresentedabove takes intoaccountdifferent typesofdistributions for thedifferent

variables,sothatitisnecessarytodescribethevariablesindependentlyandshowforeachofthemwhichisthe

procedureadoptedrelativetothetypicaldistributiontoobtainthemeanvalueëh!Jandstandarddeviation

Eãr(hì.

Inordertogetthelimitstatefunction,itwasnecessarytoproceedthroughtwosteps:thedefinitionofthemean

valueandstandarddeviationforthedifferentvariables(definedbytheindex0),andtheoptimizationofthem

inordertogetthelimitstatefunctioná(Æ) = 0,sothatthecombinationofthevariablesistheoptimaloneand

obtaininthiswaytheminimumprobabilityoffailurepossible(definedbytheindex1).

ThestillwaterbendingmomentisassumedtobedescribedbyaNormaldistribution[3,42],whichstatistical

descriptorsdependfromtheratiobetweendeadweightanddisplacementoftheship}andthelengthofthe

ship%&&[27,42],whiletypicalvaluesforstillwaterandwaveinducedbendingmomentshavebeencalculated

followingtherulesofClassificationSocieties(DetNorskeVeritasrules[28]).Meanandstandarddeviationofthe

46

stillwaterbendingmomentarecalculatedbasedontheformulas(145)and(146)developedbyGuedesSoares

andMoan[42].

Eãr(hì-S,U = Eãr(hì-S,MÇ∑ë-S,∏ñ10`C

(145)

ëh!J-S,U = ëh!J-S,MÇ∑ë-S,∏ñ10`C

(146)

Eãr(hì-S,MÇ∑ = !U,mÅ∞ + !9,mÅ∞} + !C,mÅ∞%&&

(147)

ëh!J-S,MÇ∑ = !U,MÅÇe + !9,MÅÇe} + !C,MÅÇe%&&

(148)

} =

}mÅÇm}343

(149)

ë-S,/- = 175J*%&&C ' *+ + 0.7 10`Z − ëS∞,/-

(150)

* = 10.75 −

300 − %&&100

9.]

(151)

Where:

ë-S,/-–stillwaterbendingmomentasgivenbytheClassificationSociety[28]

ëS∞,/-–verticalwave-inducedbendingmomentasgivenbytheClassificationSociety[28]

!U, !9, !U–coefficientstakenasshowinTable10

}–relationbetweendeadweightandtotalship’sweight

J – coefficient for the probability level, taken as 1.0 for the strength assessment

correspondingtotheprobabilityof10-8

47

*–waveparameterfor90 ≤ %&& < 300[28]

Table10:valuesforthecoefficients!U, !9, !U

!U,mÅ∞ 17.4!9,mÅ∞ -7.0!C,mÅ∞ 0.035!U,MÅÇe 114.7!9,MÅÇe -105.6!C,MÅÇe -0.154

Inasimilarwayhavebeenobtainedmeanvalueandstandarddeviationfortheverticalwave-inducedbending

momentëS∞,/-,forwhichhavebeentakenintoconsiderationthestochasticextremevaluesatarandomtime

pointoverareferencetimeperiod)\forameanwaveperiod)S.Forthisreason,ithasbeenfittedtoaGumbel

distribution[3]andiscalculatedasfollows:

ëh!JS∞,U = πM,S∞ + ∫M,S∞ªS∞

(152)

πM,S∞ = èS∞ GHI J/ ê

(153)

∫M,S∞ =

èS∞8GHI J/

9`êê

(154)

èS∞ =

ëS∞,/-GHI 10[ 9/ê

(155)

ëS∞,/- = 1902lJ*%&&C '*+10`Z

(156)

8 = 2.26 − 0.54GHI(%&&) (157)

48

J/ = 0.35

)\)S

(158)

Where:

πM,S∞–parameteroftheGumbeldistribution

∫M,S∞–parameteroftheGumbeldistribution

ªS∞–takenequalto0.577[26]

èS∞–scalefactoroftheWeibulldistribution[28]

ëS∞,/-–verticalwave-inducedbendingmomentasgivenbytheClassificationSociety[28]

2l–distributionfactor,equalto1forbendingmomentatºΩΩC

8–shapefactoroftheWeibulldistribution[28]

J/–meannumberofloadcyclesinareferencetimeperiod)\forameanwave

period)S

)\–referencetimeperiod,takenequalto1year[26]

)S–meanwaveperiod,takenequalto8sec[26]

Theultimatebendingmomentdistributionhasbeencalculated,asshowninchapter“Reliabilityofthestructure”,

computingforeachofthescantlingsdescribedinchapter“ScantlingsthroughMARS2000”theultimatebending

moment for the hogging condition resulting from geometry of themidship section and still water bending

moments.With thesevalues ispossible tocalculatemeanvalueandstandarddeviationofultimatebending

moment(intactandcorroded)forthedifferentDesignModificationFactors(DMF).

Standard deviationEãr(hì andmean valueëh!J for ultimate bendingmoment are fitted to a Log-normal

distribution[3]andarecalculatedintwostepsmarkedwiththeindexes0and1(wherethesecondisthefinal

valueoftheequivalentnormalmeanandstandarddeviationusedintheFORMoptimization)andisvalidforboth

intactandcorrodedconditionwiththerelativeultimatebendingmomentshowninTable7asfollows:

Eãr(hìÉ,U = hij æòÉ

C − 1 (hij 2øòÉ + æòÉC )

(159)

ëh!JÉ,U = hij(øòÉ +

æòÉC

2)

(160)

øòÉ = 2æòÉC GHI 0.05ëÉæòÉ 2¿ + GHI(ëÉ) (161)

49

æòÉ = GHI(*ç,ÉC + 1)

(162)

WhereCOVisthecoefficientofvariation,takenequalto0.08[3].

Secondsteptocalculatethelimitstatefunctionconsistsintheoptimizationoftheobjectivevariablewiththe

FORMthroughthechangeofmodificationfactorsƒ∗toobtainthedesignpointsi∗inthe limitstatefunction,

which represent themost likely combinationofvaluesof theparameters that lead the structure to fail. The

designpointsarecalculatedformeanvaluesofultimate,stillwaterandwave-inducedbendingmomentsand

modeluncertaintyfactorsasfollows:

i∗ = ëh!JUƒ∗

(163)

Initiallythesemodificationfactorshaveallbeensetequalto1,sothatthecalculatedparameterandthedesign

pointwere the same, and just in a secondmoment, during theoptimization, theywere changed to get the

G X = 0and the lowest beta-index possible, as this gives the combination of variables that represent the

thresholdbetweenfailure(negativeá Æ )andsaferegion(positiveá Æ ).Inputvaluesforthisoptimizationare

meanvaluesandstandarddeviationoftheultimate,wave-inducedandstillwaterbendingmomentsandthe

respectivemodeluncertainty factors,asshownandcalculated inthefirststep. Asthese inputvalues follow

different distributions, different formulas were used to get the equivalent normal meanëh!J9 and the

equivalentnormalstandarddeviationEãr(hì9.

As the still water bendingmoment and all the uncertainty factors are fitted to a Normal distribution, their

equivalentnormalmeanandstandarddeviationcorrespond to thepreviouslycalculatedmeanandstandard

deviation:

ëh!J-S,9 = ëh!J-S,U (164)

Eãr(hì-S,9 = Eãr(hì-S,U

(165)

Thewave-induced load is fitted to aGumbel distribution, so that its equivalent normalmean and standard

deviationareasfollows:

50

ëh!JS∞,9 = ëh!JS∞,U − Eãr(hìS∞,92`9 2S∞ (166)

Eãr(hìS∞,9 = í(2`9(2S∞, 0,1)/íS∞ (167)

2S∞ = hij(−hij(−

ëh!JS∞,U − πM,S∞∫M,S∞

))

(168)

íS∞ =

1∫M,S∞

hij(−( −ëh!JS∞,U − πM,S∞

∫M,S∞+ hij −

ëh!JS∞,U − πM,S∞∫M,S∞

)

(169)

Whereí(i)representsthenormalprobabilitydensityfunctionandthe2(i)isthenormalcumulativedensity

function.

Themean and standard deviation of the ultimate bendingmoment are calculated following the Log-normal

distributionasshownbelow:

ëh!JÉ,9 = iÉ∗ (1 − GHI iÉ∗ + oÉ) (170)

Eãr(hìÉ,9 = iÉ∗ GHI(1 +Eãr(hìÉ,Uëh!JÉ,U

C

)

(171)

oÉ = GHI ëh!JÉ,U − 0.5GHI(1 +

Eãr(hìÉ,Uëh!JÉ,U

C

)(172)

Inordertobeabletocalculatethereliabilityindex(β-index)havebeenobtainedthedesignpointsinthereduced

coordinatesystemforalltheparametersofthelimitstatefunctionasfollows:

Ji =

i∗ − ëh!J9Eãr(hì9

(173)

51

Beingthereliabilityindextheminimumdistancefromtheoriginpointofthereducedcoordinatesystemtothe

limitstatesurface,itiscalculatedasfollows:

∫ = Ji 3(Ji)

(174)

TooptimizetheFORMproblempresentedabovehasbeenusedanon-linearoptimizationalgorithmcodedon

MATLAB,wheretheobjectivewasthelimitstatefunctionequaltoG X = 0,obtainedbychangingthedesign

pointsof thevariables inordertoget the lowestreliability indexpossibleandtotake intoconsiderationthe

constraintsandboundariesset.

Inordertokeepthebeta-indexaslowaspossible,thishasbeenconstrainedinthecodeandachievedwithan

iterativeprocess.

Thereliabilityindexhasbeenconstrainedinthecodeand,withaniterativeprocessloweringtheconstraintof

themaximalbeta-indexallowedbutstillkeepingtheá Æ = 0,itwasaimedtogetthelowestreliabilityindex

possible.Itwasdecidedtominimizethisparameterastoalowerreliabilityindexcorrespondsahigherprobability

offailure,sothatbeingalsothiscaseinthe“saferegion”ofthelimitstatefunction,itisassuredthatalsoallthe

othercaseswillhavenohigherprobabilityoffailure.

Moreover,inordernottoallowthemodificationfactorsƒ∗tohaveeverypossiblevalue,theywereconstrained

tobebetween-2and2.

Anassessmentabouttheinfluencethatthesevenvariableshaveoverthelimitstatefunctioncanbedonewith

asensitivityanalysis,wherethesensitivitiesarecalculatedforeachvariableasfollows:

πx = −

1

…I i…ix

C xÀ9

…I(i)…ix

(175)

InFigure6-2 isshownthesensitivityanalysis,where itcanbeseenthatthevariablesregardingtheultimate

bendingmomenthavehighpositivevalues,whichmeansthatbyanincreaseofthem,thelimitstatefunctionis

positivelyaffected,whileitistheoppositeforwave-inducedandstillwaterbendingmoments.

52

6.2 ReliabilityofthestructureThe reliability index has been calculated for the scantlings presented above for both intact and corroded

conditions(Table11andFigure6-3).

Figure6-2:Sensitivityanalysisforvariablesofthelimitstatefunction

Figure6-3:Beta-indexasfunctionoftheDMF

53

The∫-indexesforintactandcorrodedconditioncanbeunifiedunderonecommon∫-indexifalsothetimeis

takenintoaccountassumingthecorrosionisalinearprocessfortheship’slifetime(assumedtobe25years[26]):

∫ = −∫�\Ñ-- + ã − 1

∫�\Ñ-- − ∫eÅ024

(176)

Where:

ã–lifeoftheshipinyears,withamaximumof25years

Inthisway,itwaspossibletoobtaintheprobabilityoffailurewDbycalculatingthenormalcumulativedensity

functionofthebetaindexasfunctionofthetime(Table11andFigure6-4):

wD(ã) = 2(−∫(ã)) (177)

Table11:Beta-indexandprobabilityoffailureforthedifferentDMFforintactandcorrodedcondition

DMF ∫�\Ñ-- wD,�\Ñ-- DMF ∫eÅ0 wD,eÅ0

0.94 1.435 7.56E-02 0.94 0.201 4.20E-010.96 2.269 1.16E-02 0.96 1.005 1.58E-010.98 3.176 7.46E-04 0.98 1.768 3.85E-021.00 3.484 2.47E-04 1.00 2.437 7.40E-031.02 3.928 4.28E-05 1.02 2.938 1.65E-031.04 4.345 6.96E-06 1.04 3.481 2.50E-041.06 4.826 6.98E-07 1.06 3.984 3.39E-051.08 5.285 6.30E-08 1.08 4.244 1.10E-051.10 5.558 1.37E-08 1.10 4.454 4.21E-06

54

6.3 ReliabilitycostmodelToevaluatethecostassociatedwiththemodificationofthestructurehasbeenperformedacostbenefitanalysis

[26]:

*0Ã = *3D

à + *MÅÃ

(178)

Where:

*0Ötotalexpectedcost,USD

*3DÃ –costassociatedwiththefailureoftheship,USD

*MÅÃ –costofimplementingastructuralsafetymeasure,USD

Thecostassociatedwiththefailureoftheship*3DÃ isthesumofthecostoflostoftheshipcalculatedintheyear

oflifeã*e(ã),thelossofcargo*/,theaccidentalspilloffueloil*mandthelossofhumanlife*∞,assumingthat

theoperationallifeoftheship)is25years[26]:

*3Dà (ã) = wD ∫ [*e ã */ + *m + *∞ ] hij(−ªã)

3

0À9

(179)

Figure6-4:ProbabilityoffailureasfunctionoftheDMF

55

*e ã = (çèX − çèX − *-/\Ç& )(

ã25)

(180)

*-/\Ç& = *MÇ0Å\xÇ~/0Ñe}~x�Ä0

(181)

*/ = */Ñe0ÇxeÅ\){ä

(182)

*m = }DÉÅ~w-&x~~w-~*T)E

(183)

*∞ = Ö/\ÅSw/\ÅSï*T2

(184)

Where:

*e(ã)–costassociatedwiththelostoftheshipintheyearoflifet,inUSD

*/–costassociatedwiththelossofcargo,inUSD

*m–costassociatedwiththedamagescausedbytheaccidentalspilloffueloil,in USD

*∞–costassociatedwiththelossofhumanlife,inUSD

ª–discountrate,assumedtobe5%

çèX–ownerinitialcostoftheship,calculatedinChapter4.3,inUSD

*-/\Ç&–scrappingvalueoftheship,inUSD

*MÇ0Å\xÇ~/0Ñe–costofstructuralmaterialperton,assumedtobe277USD

*/Ñe0ÇxeÅ\–valueofonesinglecontainer,estimatedtobe2500USD

}DÉÅ~–weightoffuelestimatedinChapter4.2,inton

w-&x~~–percentageoffueloilspilledafterthestructuralfailure,takenas25%

w-~–probabilitythatfueloilspilledreachestheshore,takenas10%

*T)E–CostofAvertingaTonofSplit,takenequalto60000USD

w/\ÅS–probabilityoflossofhumanlifeafterstructuralfailure,assumedas25%

ï*T2–ImpliedCostofAvertingaFatality,valuegiventohumanlifeassumedtobe3.85106USD

56

Thecostofimplementingastructuralsafetymeasure*MÅtakesintoaccountthedecreasedorincreasedcostof

thedeckstructure,iftheDMFistakenrespectivelynegativeorpositiveanditiscalculatedwiththefollowing

formula:

*MÅÃ = (ë2Ã − 1 }-0\É/0É\Å*SÑ\ê/0Ñe

(185)

Where:

}-0\É/0É\Å–weightofthecylindricalbodyoftheship,estimatedtobe70%ofthetotal lightship

weight

*SÑ\ê/0Ñe–costtoworkontonofstructuralmaterial,takenequalto2500USD

Ithasbeenobservedthatwhilethecostofimplementingastructuralsafetymeasureisroughlyproportionalto

theaddedmaterialtothetoppartofthesection,thecostassociatedwiththestructuralfailureoftheshiphighly

dependsontheprobabilityoffailure,whichisalmostnegligiblewhen(ë2 ≥ 1.02,loweringthistypeofcostto

apointwhereithasnoinfluencemore,asshowninFigure6-5,wherethepointsmarkthevaluesforeveryDMF

from0.94to1.10.

Figure6-5andTable12showthatwhileforanegativeDMF,thetotalexpectedcostisinfluencedbothbythe

costassociatedwiththefailureoftheshipandthecostofimplementingasafetymeasure,forpositiveDMFthe

total cost ismostly determined by the cost of implementing a structural safetymeasure, being in thisway

inconvenienttohaveabiggerDesignModificationFactor(DMF)than1.00.

Figure6-5:Totalcostsasfunctionofthebeta-index

57

Thescatterdiagramaboveshowsthatthetotalcostasthefunctionofthebeta-indexhastheminimumbetween

theDMFof0.98and1.00,beingthesecondjustthousandUSDcheaperthanthefirst.Forthisreason,ithasbeen

estimatedthattheinitialscantlingoftheproposedship((ë2 = 1.00)isalreadyintheoptimumconditionfor

thereliability,althoughwithacontinuousfunctionofthecostasfunctionofthebeta-indextheminimumwould

bebetweentheDMF0.98and1.00.ThisrangeofDMFofminimumtotalexpectedcostisverifiedbetweenbeta-

indexof1.70and2.5,asalowerindexwouldincreasedrasticallythecostofafailure,beingtheprobabilityof

collapse too high, and a higher increase would lead to high cost of implementing a safetymeasure for no

advantagesinthereliability,asfora(ë2 ≥ 1.00thecostrelatedtothefailureoftheshiparealmostnegligible.

Table12:Valuesforthecostsforeachbeta-index

Õ-index ûŒ≠ ûùœ ûß0.201 -897960 4053475 31555151.005 -598640 1519504 9208641.768 -299320 371461 721412.437 0 71419 714192.938 299320 15941 3152613.481 598640 2411 6010513.984 897960 327 8982874.244 1197280 106 11973864.454 1496600 41 1496641

58

7 Maintenance based model Duringthelifeofaship,whichisgenerallydesignedtobe25years[4],manystructuralcomponentshavetobe

replacedtoguaranteefullfunctionalityandsafetyforthewholestructure,aswiththeusage,thesecomponents

maynotbeanymorefunctionalfortheirpurposeandleadthesysteminwhichtheyareintegratedtofailure.As

inthischaptertheshiphasbeenanalyzedunderthestructuralaspect,thethicknessofthestructuralcomponents

hasbeencheckedwithrespecttoplannedmaintenance.

Thebiggest issueforthemetalcomponentsduringtheship’s lifetimeiscorrosion,whichisthecauseforthe

materialdegradationandlossofstrength[31].Corrosionisdefinedasatimeandenvironmentdependentloss

ofmaterialcausedbyachemicalorelectrochemicalreactionwiththeenvironment[32],forwhichreasonits

behaviorhastobeanalyzedinordertopreventfailuresofthestructure.

Theprocessofcorrosionleadstoafailureofthecomponentwhentheplatethicknessislessthanaminimum

safetyleveldefinedtakingintoconsiderationstructuralsafetylevels.Indeed,thelossofmaterialdoesnotjust

causelossofthethicknessinthecomponentandsurfacedestruction,butitisalsoresponsibleforthelossof

tensile strength and for non-linear reduction in yield strength. For these reasons, the most important

consequencesofcorrosiononstructuralcomponentsarereductionofstructuralstrength,unfavorablestress

distribution(causedbycorrosionpits)andreducednetsectionalarea[31,33,34,4].

Consequently,themaintenancehasbeensetasacentraloptimizationobjective,byminimizingthecoststhat

derivefrompreventiveandcorrectivemaintenance,asthesecanbeestimatedwiththehelpofstochasticmodels

andlistedbetweentheOPEX.Inthismodelhavebeenconsideredbothpreventiveandcorrectivemaintenance,

being the first the substitutionof a component before it fails and the second the substitutionof it after its

structuralfailure.Asitisnotpossibletoinspectsteadilythestructuretopredictthestructuralfailure,statistical

calculations based onWeibull distribution have been used and implemented in the cost calculations.With

estimationsabouttheconsequencesonthecostsderivingfromafailureandrelatingthesewitharisklevel,it

was possible to find themost convenientmaintenance strategy regarding costs, downtime and availability,

analyzedusingdifferentmaintenancepolicies.

TheWeibullfamilyprobabilitymodelshavebeenadoptedtodescribethelikelihoodofacomponenttofail[35,

36]inordertodefineoptimalreplacementintervalsandoptimalreplacementage(alsotakingintoconsideration

thetimerequiredtoperformthereplacementitself),byminimizingthetotalcost,thetotaldowntimeofthe

systemperunittimeorbymaximizingtheavailabilityofit.

Thecorrosionprogressionhasbeendefinedbythenon-linearmodelproposedbyYamamotoandIkagaki[18],

although this isnot sufficient todescribe thebehavior.GuedesSoaresandGarbatov [19]andPaiketal [37]

proposeda corrosionmodel formarine structuresbasedon three steps: afteraperiodwhen theprotective

coatinghas itseffect, it starts totakeplaceonthesurfaceof themetalliccomponentanon-linearcorrosion

processthatleadstoincreasinglyhighercorrosionratesuptothepointwheretheprocessgetsstableandthe

rateisstabilizedtoasteady-state.Exceedingthelimitonstructuralstrengthofonecomponentcausesthefailure

ofit,whichisalsofittedtoaWeibulldistribution.

59

ThismodelisbasedonpreviousstudiesdonebyGarbatov[4],analyzesandimplementsinthemainmodelthe

probabilityandseverityofstructuralfailurescausedbycorrosion,calculatescostsderivingfrompreventiveand

correctivemaintenance,withtheobjectivetominimizecosts,downtimeormaximizetheavailability,optimizing

the replacement interval, replacement age and replacement age with consideration of the time needed to

replacethecomponent,factorthatcanleadtoabettermodeldesign.

1.1 ProbabilityandseverityofstructuralfailureduetocorrosionInordertoestimatetheprobabilityofacomponenttofailasaconsequenceofcorrosion,thefailurerate(or

hazardrate)ℎ(ã)hasbeenanalysed.Thisfunctiongivesthenumberofcomponentsthatoutofthetotalityfails

inadeterminedunitoftime,or,saidinotherwords,itistheinstantaneousprobabilityofacomponentnotto

surviveataget[4,20].Followingthisdefinition,thefailurerateiscalculatedas:

ℎ ã =í(ã)rã

0–—00

í(ã)rã 0

=2 ã + “ã − 2(ã)1 − 2(ã)

=í(ã)1 − 2(ã)

(186)

Thehazardratefunctioncanbesubdividedintothreeregionswithdifferentbehaviorregardingthenumberof

failuresperunittime(Figure7-1).

Indeed,whileinthefirstregion(RegionA)thenumberoffailuresstartingfromahighratedecreasequickly,as

thenewbuiltcomponentsneedtimetogetstable,mostlyregardingfailuresintheproductionandmounting

(burn-in failures), theRegionB is characterizedbya constanthazard rateandby random failures,being the

periodwhenthecomponentsareinnormaloperationconditions.Thethirdregion,characterizedbywear-out

failures,representsthelastperiodoflifeofacomponent,aswithanincreasinglyhigherhazardrate,itwillfail

orwillbesubstituted.Consequently,theaimofthismodelistoavoidthatcomponentsenterintothewear-out

Figure7-1:Hazardrateasfunctionoftime,theso-calledbathtubfunction.

60

region,donebyidentifyingthemomentinthelife-cycleofastructuralcomponentwhenitstatisticallygoesfrom

RegionBtoRegionC,asjustinthewear-outregionitisconvenientandfavorabletoreplaceacomponent.

As statedabove, thedistributionof stochastic failureshasbeen fitted toaWeibulldistribution [4],with the

probabilitydensityfunction,í ã expressedas:

í ã =∫S”Sã − ªS”S

hij −ã − ªS”S

Ñ ,íHvã > ªS

0,íHvã ≤ ªS

(187)

wheretheshape,∫S,scale,”Sandlocation,ªSparametersoftheWeibulldistributionarebiggerthan0.

Inorder toestimatenumerically thestructural failureofacomponent,havebeenconsidered fourgradesof

corrosiontolerance(low,moderate,highandextreme),whichcorrespondtofourpermissibledepthsofcorroded

material,afterwhichthecomponentisconsideredtobecompletelyfailed.Thesecorrespondtofourvaluesfor

thethreedifferentWeibulldistributionparameters,listedinTable13:

Table13:Corrosiontolerancelimits

Low Moderate High Extreme∫S 4.43 4.75 4.13 4.18

”S[years] 11.13 12.19 17.22 22.06ªS[years] 5 5 5 5

Consequently,thehazardratemodelledbyaWeibulldistributionisequalto:

ℎ ã =∫S”Sã − ªS”S

Ñ`9

,íHvã > ªS

0,íHvã ≤ ªS

(188)

Basedontheshapeparameter∫Scanbedefinedthebathtubfunctiononthehazardrate,sothatRegionAis

describedby∫S < 1,RegionBhasafailurerate,nottime-dependentwitha∫S = 1andthewear-outregion

characterizedbya∫S > 1.

Theintegraloftheprobabilitydensityfunction,í ã resultstobethecumulativedensityfunction,2 ã fitted

alsotoaWeibulldistribution:

2 ã = í ã rã

0

U= 1 − hij −

ã − ªS”S

Ñ`9

(189)

Asalsofortheprobabilitydensityfunction(pdf),ifã ≤ ªS,thevalueforthecumulativedensityfunction(cdf)is

0.

61

The reliability function (or survival function)1 ã is strictly related to the cdf, being the probability that a

componentwillfulfilitsfunctionintheunitoftime(forthismodelithasbeenchosenoneyear).Thereliability

functionisdefinedas:

1 ã = í ã rã

0= 1 − 2(ã)

(190)

ForthedifferentcorrosiontoleranceshasbeenplottedthereliabilityofcorrodedplatesasshowninFigure7-2.

Ithasbeenestimatedthatthepreventiveandcorrectivemaintenancehasabiginfluenceoverthetotalcostsof

a ship, responsible for 25%up to35%of theoperating costs (OPEX) [38], dependingon factors suchas the

shipyard,geographicallocationoftheship,operationalprofileandmore[4].

Inthecalculationofthetotalcostsofthemodelithasbeenincludedthetimeofstoppageduetopreventiveand

correctivemaintenance,beingpossibleinthiswaytomakeassessmentsabouttheavailabilityofthecomponent

orofthewholesystem,inthiscasethevessel.

InaccordancetoDod[39],Kececioglu [40]andLangford [41],havebeenappliedprocedurestoestimatethe

probabilityandseverityofafailure,beingpossibletomakeassessmentsaboutdifferentconfigurations.Methods

thathavebeenappliedarebasedontheriskprioritynumbers,occurrence/severitymatrix,riskrankingtables

andcriticalityanalysis[4].

Thesemethodshavebeenappliedoverthefollowingparameterstobeabletoestimateandclassifythemon

fourlevels(valuesinTable14):

- Corrosiontolerance,CT

- Consequenceofpreventivereplacement,CPR

- Consequenceoffailure/correctivereplacement,CFR

Figure7-2:Reliabilityofcorrodedplates

62

- Consequenceofreplacementsaccountingfortheinspectioninterval,CII

- Timerequiredtomakearepairorreplacement,CRTable14:Tolerance/consequence

Lowi=1

Moderatei=2

Highi=3

Extremei=4

CTi 0.1 0.15 0.2 0.25CPRi 1 2 4 8CFRi 2 3 4 5CIIi 4 3 2 1CRi 1 2 4 8

Basedonthepresentedvalues,costsandtimeneededhavebeencalculatedforthemodelwiththefollowing

formulas:

*jx = *Hnã*)x*w1xEãvéXãév!G}htIℎã

(191)

*íx = *jx*21x

(192)

)jx =

)H*w1x1 − *)x

(193)

)íx = )jx*21x

(194)

)tx =

)Hï*ïï1 − *)x

(195)

)vx =

)H1*1x1 − *)x

(196)

where*jxisthetotalcostofthereplacementbeforefailure,*Hnãisthetotalrepaircostpertonofstructural

material,assumedtobe1000USD/ton[4],EãvéXãév!G}htIℎãistheweightofthematerialofthestructure,

takenascalculatedinchapter“Conceptual design”,*íxisthetotalcostofareplacementafterfailure,)jxis

thepreventivereplacementinterval,)íxisthecorrectivereplacementinterval,)txtheinspectioninterval,)vx

theintervalofrepair/replacement,andtisthetoleranceorconsequencelevel,with1 ≤ t ≤ 4.

63

1.2 Optimalreplacementintervalwithminimizationoftotalcosts

Maintenancepolicyandplanningforcommercialvesselsiscriticalfortheoperationalcosts,becauseitaimsto

findtheoptimumformaintenance,avoidingunnecessarycosts,andreducesdrasticallytheprobabilityoffailure

ofavessel.Forthisreason,theobjectiveistofindthebestmaintenancepolicytomaximizetheeffectivityofthe

system(analyzedunderthepointofviewofcosts,downtimeoravailability)overoperationandmaintenance

cycleofthevessel[20].

Ithastobetaken intoaccountthatanoptimalmaintenance isbasedonpreventivemaintenanceperformed

whenthelife-cycleofacomponentgoesfromRegionBtoRegionC,sothatintervalsshallnotbedesignedtoo

short,asthiswouldreducethetimewhenthecomponentisinthenormaloperationconditions,andatthesame

timetheyshallnotbeplannedtoolong,astheprobabilityoffailureandthusthecostsofcorrectivemaintenance

wouldbehigher[4].

Afirstmaintenancepolicythathasbeenanalyzedisbasedonfindingtheoptimalintervalbetweenpreventive

maintenanceoperations,whichonly target is theoptimummomentwhen thecomponentshallbe replaced,

withouttakingintoconsiderationthecorrectivemaintenance,performedwheneverneeded.Thisoptimization

iscalculatedbyminimizingthecosttoreplaceonecomponentsummingthecostofpreventivemaintenance*&

andthecostofcorrectivemaintenance*Dtimestheprobabilityoffailurebeforetimetp,dividedbythelength

ofthechosenintervaltp:

* ã& =

*& + *D|(ã&)ã&

(197)

| ) = 1 + | ) − t − 1 í ã rãx–9

x

3`9

xÀU

,) ≥ 1(198)

Renewal theoryapproachhasbeenadoptedtocalculate| ) that is theexpectednumberof failures inthe

timeinterval(0,T),with| 0 = 0,whichmeansthatwhenacomponenthasbeenimplementedinasystem,it

isnewandthustherearenopre-existingfailuresattime0.

Ithasbeencalculatedthetotalcostasafunctionoftheyearsofthecomponent’slifeforthefourdifferentlevels

of corrosion tolerances and repair consequences (Figure 7-3) and it has been found theminimum for each

combinationbetweenthecorrosiontoleranceandrepairconsequencelevels,asshowninTable15.

64

Table15:Optimalpreventivereplacementintervalinyears

RepairconsequenceLow Moderate High Extreme

Corrosiontolerance

Low 8 7 6 6Moderate 8 8 7 7

High 11 10 10 9Extreme 11 11 11 11

1.3 OptimalreplacementagewithminimizationoftotalcostsAlso for thismaintenance policy it has been aimed theminimization of the total costs resulting from both

preventiveandcorrectivereplacement,butinsteadoftheinterval,ithasbeentakenintoconsiderationtheage

of theanalyzedcomponent [4]. In thisway, the intervalbetween twomaintenance tasksdoesnot take into

consideration just the scheduled preventive maintenance, but it takes account as well of the corrective

maintenance, toconsider theactualageofacomponentandtoavoidtoreplaceacomponentthat isstill in

RegionAorB.

Hence,theobjectiveofthispolicyistooptimizetheageofacomponentafterwhichitshallbereplaced,keeping

thetotalcostperunittimederivingfrommaintenanceaslowaspossibleandtakingintoaccountatthesame

timethebenefitsofsuchmaintenance.Thisisobtainedbysummingthecostofpreventivemaintenancetimes

Figure7-3:Costsasfunctionofthepreventivereplacementintervals

65

the reliability, summed with the cost of corrective maintenance times the probability of having a failure,

everythingdividedbytheunittime,whichisthemaintenanceintervaltpplusthetimeafterwhichacorrective

replacementwouldhavetobeperformed,asfollows:

* ã& =

*&1 ã& + *D 1 − 1 ã&

ã&1 ã& + ãí ã rã0Ω`

(199)

Thetotalcostsderiving frommaintenanceareshownas functionof replacementages inFigure7-4,andthe

optimaforeverycombinationbetweencorrosiontolerancesandrepairconsequencesaredisplayedinTable16.

Table16:Optimaforreplacementages

RepairconsequenceLow Moderate High Extreme

Corrosiontolerance

Low 11 11 10 10Moderate 12 11 11 11

High 15 14 13 13Extreme 18 17 16 15

Figure7-4:Totalcostperunittimeasfunctionofthereplacementages

66

1.4 OptimalreplacementageaccountingfortimerequiredtoperformreplacementAsapreventiveandacorrectivemaintenancerequiretimeforthereplacement,thiscouldhaveeffectsonthe

costsandleadtoadifferentdistributionofthese.Forthisreason,ithasbeenconsideredapolicythatcalculates

theoptimalreplacementagetaking intoaccountthetimerequiredtoperformthereplacement[4].Thishas

beenachievedbyminimizingthetotalcostsperunittimederivingfrommaintenance,as,ifitwouldhavebeen

minimizedthedowntime,itwouldhaveoptimizednotonlythereplacementage,butalsothetimerequiredfor

themaintenance,leadingtoincorrectresults.Thetotalcostsresultbyimplementingintheformulathetimeto

performthepreventiveandcorrectivemaintenance,asshowninthefollowingformula:

* ã& =

*&1 ã& + *D 1 − 1 ã&ã& + )& 1 ã& + ë ã& + )D 1 − 1 ã&

(200)

where)& is time required to perform preventive maintenance,)D the time required to perform corrective

maintenanceandë(ã&)themeantimetofailureafterareplacementdoneatmomenttp.

To be able to compute such optimization, themean time to failure (MTTF),which is the expected age of a

componentbeforeitfails,hasbeendefinedas:

ë ã =

ãí ã rã1 − 1(ã&)

0Ω

`

(201)

Inthisway,itwaspossibletodisplaythefunctionoftotalcostperunittimerelatedtothereplacementagein

Figure7-5,andtoshowtheoptimaforeveryfunctioninTable17.

Table17:Optimalpreventivereplacementtimeageaccountingforthetimerequiredtoperformreplacement

RepairconsequenceLow Moderate High Extreme

Corrosiontolerance

Low 11 11 10 10Moderate 12 11 11 11

High 15 14 13 13Extreme 18 17 16 15

67

1.5 OptimalreplacementintervalwithminimizationofdowntimeAnotherwaytoconsidertheproblemisanalyzingandminimizing,insteadofthetotalcosts,thedowntimeper

unittime,asthiscangiveameasureofthetimeneededtoreplaceacomponent[4].Ithasbeenminimizedthe

downtimeD(tp)bysummingthetimerequiredtoperformacorrectivereplacementtimesthehazardrateand

thetimerequiredforthepreventivemaintenance,dividedbytheunittimeã& + )&,asfollows:

( ã& =

| ã& )D + )&ã& + )&

(202)

It has been analyzed the relation between replacement intervals and downtime for the different levels of

corrosion tolerance and repair consequence as done in the previous chapters, in order to understand this

behavior(Figure7-6)andfindtheminimaforeachofthecombinationsbetweencorrosiontoleranceandrepair

consequenceTable18.

Table18:Minimizationofthedowntimebyoptimizingthereplacementinterval

RepairconsequenceLow Moderate High Extreme

Corrosiontolerance

Low 7 6 5 4Moderate 8 7 6 5

High 11 10 8 7Extreme 11 11 11 9

Figure7-5:Totalcostperunittimeasfunctionofthereplacementageaccountingfortimerequiredtoperformreplacement

68

1.6 OptimalreplacementagewithminimizationofdowntimeAsimilarapproachhasbeenappliedtofindouttheoptimalreplacementagetpbyminimizingthedowntimeper

unittime[4].Thisresultsbydividingthedowntimeinacyclebythecyclelength,wherethedowntimeinacycle

isthemultiplicationofthetimeneededforapreventivereplacementtimethereliabilityofthecomponent,plus

themultiplicationofthetimeneededforacorrectivereplacementtimestheprobabilityoffailureofthesame

component,asshowninthefollowingformula:

( ã& =

)&1 ã& + )D 1 − 1 ã&ã& + )& 1 ã& + ë ã& + )D 1 − 1 ã&

(203)

Thedowntimeasafunctionofthereplacementageforallthecombinationsbetweenthefourlevelsofcorrosion

toleranceandrepairconsequenceisshowninFigure7-7anditsoptimaaredisplayedinTable19.

Table19:Minimizationofthedowntimebyoptimizingthereplacementage

RepairconsequenceLow Moderate High Extreme

Low 11 11 10 10

Figure7-6:Downtimeperunittimeasfunctionofthereplacementinterval

69

Corrosiontolerance

Moderate 12 11 11 11High 15 14 13 13

Extreme 18 17 16 15

1.7 OptimalinspectionintervalwithmaximizationofavailabilityAnotheroptimizationpolicyistomaximizetheavailabilityofthestructure,byconsideringtheintervalsbetween

inspections.Asthecorrosiondevelopmentwithacomponentdeliversjustestimationsbasedonstochasticdata

and does not provide the actual state of it, it is necessary to plan inspections tomeasure the state of the

structure.Onthisconceptisbasedtheoptimalpolicydescribedinthischapter,consideringthecomponentafter

the inspection in as-new condition, as either the inspection reveals a failure, and the component has to be

replaced,orshowsthattherearenomajorfailures,hencenoreplacementisneeded[4].

Inthisway,theinspectionincreasesthereliabilityofthestructure,sothatthemaingoalistoidentifytheoptimal

so-called failure-finding interval ti,which is theoptimal interval between two inspectionsbymaximizing the

availability.

AvailabilityT ãx isdefinedasthemostimportantmeasureforasystemeffectivenessandgivestheprobability

thatthecomponentisreadytobeusedatanytimeintheinterval(0,ti).Itdependsontheprobabilityoffailure,

Figure7-7:Downtimeperunittimeasfunctionofthereplacementage

70

theanalyzedcomponenthaswhileitisusedbythesystemandbythetimeandresourcesneededtobringthe

componentbacktoanoperationalstate[20],andthelong-runavailabilityisdefinedas:

T =

{(ä){ ä + {(()

(204)

Thismeansthatthetotalavailabilityofasystem(orcomponent)isequaltotheexpecteduptime{(ä)divided

bythetotaltimeconsidered{ ä + {((),wherethesecondtermistheexpecteddowntimeofthesystem.

Following this definition, it has been applied over the analyzed system, where ti is the interval between

inspections(i.e.themomentwhentheinspectionisperformed),)xisthetimeneededfortheinspectionand)\

isthetimetorepairorreplacethecomponent,asshowninthefollowingformula:

T ãx =ãx1 ãx + ãí ã rã

0’`

ãx + )x 1 ãx + ãx + )x + )\ 1 − 1 ãx

(205)

Theobjectivewastoobtaintheoptimalintervalbetweeninspectionssothattheavailabilitycouldhavebeen

maximized,bytakingintoaccountthecostsderivingfromtheinspectionandrelatingthesetothebenefitsthat

theinspectiongives.

Availabilityperunittimeasafunctionoftheinspectionintervalsforeverycombinationofcorrosiontolerance

andrepairconsequenceisdisplayedinFigure7-8andtheoptima(maxima)foreachofthefunctionsarelisted

inTable20.

Table20:Optimalinspectionintervalswithmaximizedavailability

RepairconsequenceLow Moderate High Extreme

Corrosiontolerance

Low 11 10 10 9Moderate 11 11 10 9

High 14 13 12 11Extreme 16 14 13 12

71

Figure7-8:Availabilityperunittimeasfunctionoftheinspectioninterval

72

8 ConclusionsThedevelopedtoolhasfulfilledtheobjectivesofthework,deliveringafunctionalitythatinaneasyandfastway

canestimateoptimalmaindimensions,scantlingsandmaintenancestrategyforamulti-purposevessel.

ThefirsttaskconsistsincalculatingshipmaindimensionsbasedonrulesbyClassificationSocieties,estimations,

regression analysis from existing vessels and approximations, obtained by previously developedmodels for

conceptualdesignandimplementedtodeliveraconsistentandreliablefirstestimation.Themaindimensions

calculatedareobtainedbyoptimizing throughan iterativealgorithm theobjective function,whichhasbeen

chosentobetheRequiredFreightRate(RFR),asthisvariablesummostoftheship’sspecificationsandtakesinto

accountCAPEX,OPEXandnumberofTEUtransported,allowingtheshipownertodecidetheamountofTEUto

beshippedandtoanalyzeiftheproposedmodelisprofitable.Onthetoolareimposedconstraintsregarding

shipstability,physicalrestrictions(e.g.thePanamaCanal)andvoyagerequirements.

Further,theseconstraintsandresultsfromformulaslinkedtothedesignvaluesshallbecomparedtoexisting

multi-purposevessels,tovalidatethemodelandincasetocorrectsomeestimationsdone.Also,thecodecan

beimplementedwiththeinsertionofmoredatabytheuser,inordertoconsidermorerequirementsimposed

by the ship owner regarding operational life of the ship. Another improvement to the suggested tool is to

considerdifferentcomputational formulastoestimateandanalyzetheproposedmaindimensions,ase.g.to

estimate the structural weight there have been developed a wide number of models, based on regression

analysis,midshipsectionmodulusandmore.BytheresultsithasbeennotedthenumberofTEUthatcanbe

loadedontheshipmodelisbiggerthantheamountthatamulti-purposeshipnormallycarries,asthistypeof

vesselisloadedalsowithothertypeofcargo.

Thesecondtaskistheoptimizationofthescantlingsbasedonariskanalysis,afterthattheultimatebending

momenthasbeencalculatedthroughthesoftwareMARS2000implementingthemaindimensionsofthemodel

inatypicalmidshipsectionofamultipurposeship.Thereliabilityofthemodelhasbeencalculatedapplyingthe

FirstOrderReliabilityMethod (FORM)on the limit state function,being theboundarybetweensafeand fail

conditionestimatedtobelinear[15,3].Thevariablesthatwereconsideredinthelimitstatefunctionarethe

wave-induced,stillwaterandultimatebendingmomentsandtherespectivemodeluncertaintyfactors,andeach

hasbeendescribedbythemostsuitablestatisticaldistribution.Thisprocesshasbeencalculatedoverscantlings

inarangefrom−6llto+10llfromtheoriginalwithanintervalof2ll,obtainingtherespectivereliability

index (∫ -index) and thus the probability of failure. The different probabilities of failure have been used to

calculate the risk-associated costs, which, summed with the cost of modifying the structure by the Design

ModificationFactor(DMF),resultedinthetotalcostsassociatedtoeachDMF,beinginthiswaypossibletofind

outthesolutionthatpresentsthelowesteconomicalrisk.

Withthistoolcanbecheckedifanotherreliability-basedmethodinsubstitutiontoFORMcanbeadopted,in

ordertolink,makedynamicallyinteractiveandfastertheprocessestocalculatethe∫-indexandtherisk-related

costs.Inadditiontothis,itisinterestingtocheckifmodifyingthescantlingsofthewholemidshipsectionwould

leadtomoreaccuratevalues.

73

Thethirdtaskisbasedondefiningtheoptimalmaintenancestrategybasedonminimizingcosts,downtimeor

maximizing availability, calculating the optimal replacement interval, the optimal replacement age and the

optimal replacement age accounting for time required to perform replacement. In this model have been

consideredfourlevelsofcorrosiontolerancecombinedwithfourlevelsofrepairconsequence,inordertotake

intoaccountmaterialbehaviorandvariablefactors.

74

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