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CHAPTER124

ReliabilitybasedoptimaldesignofverticalbreakwatersmodelledasaseriessystemoffailureE.ChristianilH.F.Burcharth2J.DalsgaardS0rensen3

AbstractReliabilitybaseddesignofmonolithicverticalbreakwatersisconsidered.Probabilisticmodelsofimportantailuremodessuchsslidinganduptureailurenheubblemoundandthesubsoilaredescribed.Characterisationoftherelevantstochasticpara-metersarepresented,andrelevantdesignvariablesareidentifiedandanoptimal systemreliabilityformulationispresented.Anillustrativeexampleisgiven.Keywords:erticalwallbreakwaters,slidingfailure,upturefailure,designoptimisa-tion,reliabilityIntroductionAnumberofbreakwaterfailureshavebeenreportedduringthelast2 0yearsfo rrubblemoundbreakwatersswellsorverticalbreakwaters.g.inesPortugal),Arzew(Algier),Mutsu-OgawaraPortJapan),GelaItaly)andAlgecirasPortSpain).Thishasesultednnewwaysofapproachingthedesignproblemsrelatedtobreakwaters.Probabilisticmethodshavebeenintroducedtosolvebreakwaterdesignproblemsintheearlystageofplanning.ReliabilitybaseddesignofbreakwatershasbeendiscussedbyNielsenet.l.1983),Burcharth(1991),1992a) ,1992b),Burcharthetal.1994)and(1995)andbyTakayama1994) .Inthefollowingacaissonverticalbreakwaterisanalysedwithrespecttoprobabilityoffailurefo rsinglefailuremodes.Emphasisisputonthefoundationfailuremodesandtheirrelativeimportance.IdentificationofthefailuremodesSlidinganduptureailurenheubblemoundoundationandnhesubsoilareusuallythemostcriticalfailuremodesorverticalbreakwaters.therailuremodesexist.g.ettlementofthecaisson,seawardsliding,scouratheoesubsoiland/orrubblemound),instabilityofthearmourstonesinthefoundation,and structuralfailureofthecaisson.

'Ph.D.-student,HydraulicsandCoastalEng.Lab.,Dept.fCivilEng.,AalborgUniversity2Prof.dr.techn.,HydraulicsandCoastalEng.Lab.,Dept.fCivilEng.,AalborgUniversity3Assoc.Prof.Ph.D.,Dept.fBuildingTechnologyandStructuralEngineering,AalborgUniversity1589

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1590 COASTALENGINEERING996

Ninefoundationfailuremechanismsincludingslidingareidentifiedcf.igure.Over-turningisarelevantfailuremodeonlyinthecasesofmonolithicstructuresplacedonverystrongfoundationsoilsorrock.Thedesignsconsideredcoververticalbreakwatersplacedonloworhighrubblemoundsandsandorclaysubsoils.

(T) Slidingfailure2)Failureintherubble (3)ailureintherubblemoundmoundlidinginclayorsandsubsoilSlipfailureplane v-~1 - ^ - ^ --^ ^5*t s ^~**C\Rubblemouri

T\Failumou reintid heubble

Claysandorrock

/eNFailureintherubble /*?\^^ moundandsand ^subsoil

Clayorsand

Failureintherubblemoundandsandsubsoil

?* ?+*,S| M>^ V WAXClayorsa ndorn>< Slipfen < an d i Sand ta-f -y

(7)Failureintheubbleg)ailureintherubble ftsailureintherubblemoundandsandoundandclaymoundandclaysubsoilsubsoil(rotation)ubsoil(rotation)55(x,z)iTO(x,z)' 'randtlzoneFigure:NinefoundationrupturefailuremechanismsSlidingSliding,.e.orizontaldisplacementofthecaisson,canoccursaslipeitheratheinterfacebetweenhecaissonconcretebaseplateandherubblematerial,orentirelyintherubblematerial.Corresponding tothefirstmentionedcasestabilityagainstslidingexistswhentheratiooftheresultanthorizontalforce,Fg,otheresultantverticalforceisequaltoorlessthantan/i,i.e.

FH

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VERTICALBREAKWATERS 591

whereFQsheweightofthecaissoneducedorbuoyancy,Fushewavenducedupliftforceonthebaseplate,and/istheangleoffrictionbetweentheconcretebaseplateandheubblemoundorhebeddingayer.eferencesgivenoTakayama(1992)orvaluesoftanfi .Iftheslidingailureakesplaceentirelynheubblematerial,.g .nhebeddinglayer,henj ,mustbesubstitutedbytheeffectiveangleoffrictionip 'ofthematerial.Themostcriticalofthetw ocasesshouldbeconsideredinthedesign.FHandFuareinthispapercalculatedbyusingthewaveloadformulationbyGodaetal.1972)and1974)extendedtoncludeimpulsivepressure,Takahashi1994) .ThedesignwaveheightisadjustedinthesurfzoneasdescribedbyGoda1975).TheresultantoftheforcesFH,QandFusindicatedinFigure2asFR.Rupturefailurenrubble,sandandclaysubsoil cases1 9Toevaluatethestabilityofthefoundation,consistingoftherubblemound,sandorclaysubsoil,heupperboundheoremofgeneralplasticityheorysused.hisheoremcanalsobeappliedinaprobabilisticapproachofdesign.Applicationoftheupperboundtheoremrequiresthathenormalityconditionisul- filled.Experienceshowsthatgoodestimatesofthebearingcapacitycanbeobtainedbyintroductionofareducedeffectiveangleoffriction ,Hansen1979)definedby

sinip 'co sib .tan

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Figure2 :Rupturenheu b b lemoundandclaysubsoil.Thegeometricallengthsandtheradiusfo rthekinematicallyadmissiblerupturefigurearecf.igure2 .

he=Bz+Brm+2/i/j hnta,n(ipdl+62) (3 )Asnotedabove,hesliplineABsapproximatedbyastraightine.TheadiusR2thenbecomes

#2=1BCIsin02FurtherI ADbecomes

IAD +xhcta,n(ipdl+02) 2 Thecentreofgravityfo rzoneand2isdefinedbythelengthIQ,cf .Figure2

(IAD-\IBC){\IAD+\IBC hn+\{lBc~2hnlBC+\h2u

(4 )

(5 )

Externalwor done

MAD+jhc hn(6 )

TheexternalworkWEonebyhewaveoads,heporepressurealongheruptureboundarylineandtheweightoftheverticalbreakwaterisfo raninfinitesimalrotation5aroundpointD

WE=SM0 (7 )whereM0isthemomentaroundDofthewaveloads,theporepressureandtheweightofthecaisson.Thewor donedu eoheweightofzones and2sarotationaroundD

wh2=y sw ) i Di) a 1 n2) 8

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wherefliandS ^ 2aretheareasofzoneand2 andI AD 'Gstheperpendicularlengthbetweenhepointofrotationandthecenterofgravityorzoneand.Theworkdonebyheweightofzone szeroasheresultantdisplacementofthecentreofgravityshorizontal.Thenternalworkdonenzone3alongherimoftheruptureboundaryBCs

z-202W3 = 6B% Cu(0)d0 9)Jowherec^sheundrainedshearstrengthoftheclay.TheimitstateequationforherupturemechanisminFigure shen:

W Wh2 WE=0 (10)02sheunknownangleobedeterminedbyminimisingtheratiobetweenhestabil-isingworkanddrivingwork.SystemmodeloffailuremodesforrupturefailureandoverturningIndesignofverticalbreakwaters,themain concernissliding,overturning,andrupturefailuresnherubblemoundandnhesubsoil.Theseailuremodescanbemodelledbyaseriessystemsf.igure3.

9Rubblemoundandsandsubsoil|Slidingfailure^Overturning[|ftf [[fupturefailureofIIRupturefailureofthe rabble Jqthe rabble,Ruptureof the rabbleandsand subsoil 3.u uptureofthe rabblean dsandsubsoil 5. I Iuptureof therabblean dsandsubsoil g,uptureof therabbleandsandsubsoil(rotation).

b)ubblemoundandclaysubsoil-siidingfailureLfZZTJTRupturefailureofthe rabble 4 ,I 3 Ruptureoftherubblean dclaysubsoil 3, u uptureoftherubblean dclaysubsoil(rotation).n uptureof therubblean dclaysubsoil 9.

Figure3:eriessystemoffailuremodesforrupturefailureofthesandandclaysubsoil.CharacterisationofthestochasticvariablesAllvariablesarenprinciplestochasticvariablesnaimitstateormulation.omeparameterse.g.eometricalparametershavesmallcoefficientofvariationandmightberegardedasdeterministicvariables.Theparameterswhichhaveasignificantdegreeofuncertaintynbreakwaterdesignwillbediscussednheollowing.tsassumedthatallstochasticvariablesarendependent,unlessotherwisestatednheext.

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ThewaveinducedhorizontalforceandtheupliftforcecanbecalculatedfromequationsgivenbyGodaet.l.1974)andTakahashi1994).Themodeluncertaintyrelatedtotheod awaveoa dismodelledbyanormaldistribution.ThishasbeenclarifiedbyBruining(1994)whocomparedanumberoflaboratorytestsresultswiththeGodafor-mulainordertoevaluatetheuncertaintyrelatedtothehorizontalwaveload,thewaveinducedupliftforce,thehorizontalmomentandthewaveinducedupliftmoment.ThemodeluncertaintiesarerepresentedbyvariablesU p H,UFU,UMHndUMVxpectedvaluesbias)andstandarddeviationsaregiveninTable2 .ThedeepwaterwaveclimatecharacterisedbythesignificantwaveheightHsoisassumedtofollowaWeibulldistribution.ThedistributionfunctionofthemaximumsignificantwaveheightwithinTyearsisgivenby

FBTH)= exp IHsoBw \\ A )XT

(11)where sheaveragenumberofHsodatavaluesperyear.w .6 9susuallyregardedasadeterministicparameter.DuetothelimitednumberofdataA andkaresubjecttostatisticaluncertainty.Aandkvaluesaremodelledasnormaldistributedstochasticvariableswithavariancebasedonthemaximumlikelihoodestimates.TheexpectedvalueandthestandarddeviationofAandkarepresentedinTable1.

standarddeviation,a

(approximation)

pA* T l+2/pk)NVr2(i+i/w0 PA( )'

Table:MeanandstandarddeviationofAandkvaluesinheWeibulldistribution.InTableNshenumberofavailableffso-valuesandTshegammaunction.PA=0-58andp^=1.14andN=30willbeusedintheillustrativeexample,cf.Table2.Asthewaterdepthdecreasesfromdeepwatertoshallowwater,waveransformationwillresultinrefraction(whenwavesarenotheadon),shoalingandfinallywavebreak-ing.Thereforetheuncertaintyofthebreakerheightsshouldbeconsideredindesignindepthlimitedcases.ThedesignwaveheightHdesign=#1/250oDeappliedintheGodaformulaisincaseofnosurf zoneinfrontofthestructuretakenas.8 Hs0.

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i f -Si^* d ? 1 1 - ^ 9- ts i = 1 I I CN O w CO t- 00 OJ O I-H CN O ^c" ID

Table2:tochasticmodelofthestochasticvariables

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1596 OASTALENGINEERING99 6

InaseofasurfzonenrontofthestructurehebreakerheightsakensGoda,1994)Hb=L00.17 l-exp(-1-5 (l+15tan4/36>))) 1 2 )

where/,shewaterdepthatadistance5HSseawardofthestructure,andL o=HS O/0.035sthedeepwaterwavelength.Consequently,Hdesign=min[HS0,Hi,].Tidalelevation sassumedoollowaosinedistributionunction,eeTakayama(1992).

*C(0=--arccosf-l)-1 1 3) where(variesbetweenQ=0.75m.Stormurge%shouldbeconsideredwhenhestructuresnshallowwater,dueopossiblechangeinbreakerwaveheightsandbuoyancyofthestructure.orsimplicitythestormsurgeisnottakenintoaccount.Thetotalwaterdepthinfrontthestructureishtot hs+ ,wherehssthemeanseawaterlevelatthefootofthestructure,withoutinfluencefromthestormsurgeortidallevel.Theaveragemassdensityofaconventionalverticalbreakwaterincludingsandballast,reinforcedconcretewallsandconcretecapcanbeassumedobenormaldistributedwithameanvalueintherangepc=2 .15-2 .3t/m3andacoefficientofvariationof5% ,Burcharth1992) .Itisgenerallyacceptedthatthevariabilityoftheeffectivefrictionangleofawellknownsoilsamplessmall,butauthorssuchsNadimetl.1994)andCherubini1992)haveencounteredvariationcoefficientsntherange3%o5% .tsassumedinthispaperthatthengleofdilationandeffectivefrictionanglehaveavariationcoefficientcorrespondingto0%.Thefrictioncoefficientbetweenthe baseplateand rubble isassumednormaldistributedwithameanvaluetan i=0.636andacoefficientofvariation of15%,Takayama(1992) .Evenhomogeneousoilayersexhibitchangenstrengthrompointopoint.heundrainedhearstrengthofclaysanexamplewherespatialvariabilityexists.tsassumedobemodelledbyaog-Gaussianstochasticield{cu(x,z)}wherexshehorizontalcoordinateandzheverticalcoordinate,ee.g.Andersenetl.1992) .Themeanvaluefunctionandcovariancefunctionareinthispaperassumedtobe

E[cu(x,z)] = J , Cu+az1 4 )Cov[cuxi,zi),cu(x2,z2)] = 1 5)

ex(-| )exp(-(-1))

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whereC usinkPaandx,zaretakeninmetreswithorigoa; ,z)=(0,0)equaltopointB,eef.igure,iC usheexpectedvalueoftheundrainedshearstrength,aC usthestandarddeviationoftheundrainedshearstrengthanda=3kPa/misaconstantsignifyingthelinearincreaseoftheundrainedshearstrengthoftheclaysubsoil.MethodofreliabilityanalysisTheuncertaintiesphysical,statisticalandmodel)elatedtoheaboveailuremodescf .igureremodelledsstochasticvariables,andimitstateunctionsorhefailuremodesareformulatedasdescribedabove.ThewaveoadsareestimatedusingtheGodaformula,includingimpulsivepressuremodification)withmodeluncertaintyincluded.Forthefoundationfailuremodesthestrengthoftheclaysubsoilismodelledasastochasticieldandaprobabilisticlimitstateunctionshenormulatedusingkinematicallyadmissiblefailuremechanisms,Christiani(1996).TheprobabilityoffailureofthefailuremodesareestimatedusingFirstOrderReliabil-ityMethods.ystemfailureismodelledbyaseriessystemandthesystemprobabilityoffailurecanbeevaluatedonhebasisoftheFORManalysisofthesingleailuremodes,seeMadsenetal .1986)andBurcharth1992) .LimitstatefunctionsSlidingfailure.Failurecorrespondingtoslidingcanbemodelledbythelimitstatefunction,cf eq.1 ):

0 failure= 0 limitstate1 6)>0 nofailure

FoundationfailurenherubbleandclaysubsoilAsmentionedaboveheundrainedshearstrengthofthelaysmodelledsaog- Gaussianstochasticield{cu{x,z)}.hecorrelationengthsor{cu(x,z)}resmallcomparedtotheintegrationintervalsanditfollowsfromthecentrallimittheoremthatthetotalinternalworkin theclaysubsoilcanbeapproximatedbyanormaldistributedstochasticvariableW 3withmeanvalue(j,w 3andstandarddeviationaw 3 -heimitstatefunctioniswritten,seeeq .1 0)

Qday=E[W]+uwaW3-Wh2 WE 1 7)whereu\ysarealizationofanormaldistributedstochasticvariableUwwithmean0andunitstandarddeviation.TheexpectedvalueofW 3s

E[W]=8Rl[cu(e)]d0 1 8)Jo

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1598 OASTALENGINEERING996

whereE[cu(6)]istheexpectedvalueofcatthepositiondescribedbytheangle9alongthecircularrupturelineBC.ThevarianceofW zs

Var[W ]=< 7 J L=< 2/ / CouMflJ.c^OldBide19)7o JowhereCov[cu(6),cu(0i)\sthecovariancefunctionofcuatthepositionscorrespondingto0and0\.SystemsreliabilityoftheseriessystemsnvolvingsandandclaysubsoilsThesingleailuremodesnFigurereegardedsmailurecomponents.tsclearromadeterministicdesignofamonolithicstructurehatfoneoftheailurecomponentsfailthenthesystemfails,i.e.hebreakwaterhasnoloadcarryingcapacityafterthefailureofonecomponent.ThesystemprobabilityoffailurePfcanbewrittenasaprobabilityofunions.

/Pf=P\\J*(*) =.*(*)

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S.WX

bo=Rubblemoundblocksize

Harbourside

Figure4 :Definitionofth edesignariablesforoptimalreliabilitydesign.Iftheobjectivefunction ischosenasthetotalexpectedcostsCTofthestructureduringthelifetime,theoptimaldesigncanbefoundasthesolutiontotheoptimisationproblem

min CT(b)=C/(b)+CFPf{b)b\

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Cj{B,hu)=ChWcaisson(B)+C/2Wrubble{hll) (28)whereWcai s s < mstheweightofthecaissonandWrub b i esthecorrespondingweightoftheubble.ThedifferencenpriceperunitweightbetweentheweightoftheubbleandcaissonschosensTP-=.heostofconstructionsminimisedwithupperboundsontheprobabilityoffailureofthesignificantfailuremodes.Optimaldesignsfo rdifferentlevelsoftheacceptableprobabilityoffailurePfwithoutconsideringwaveheightreductioninthesurfzoneispresentedinTable3: ApplyingJapanesedesignguidelines Pi B(sand&clay)hn Bsandkclay),hnH=5.56m, B=17.8m,hn=6m waterdepthatthetoeh,=1 4 m waterdepth,footofthecaissond=8m

0.1 0.2 0.324.7m),.0m )20.6m),.0m )18.2m),.0m )

26.5m),.0m )21.6m),.0m )19.0m),.0m )Table3:Optimaldesignfo rdifferentlevelsofacceptableprobabilitywithoutconsider-

ingshoalingandbreakerheightsinthesurfzone.ConclusionsFoundationfailuremodesforverticalbreakwatersareformulatedsuchthateliabilityanalysescanbeperformed.Thefailuremodesincludeslidingfailureandfailuremodesinvolvingsandandlaysubsoils.tochasticmodelsfo runcertainparametersarede- scribedusingtheinformationfromexperimentaltestsandfromtheliterature.Furtheritismentionedthatfoundation failuremodescanbemodelledascomponentsin aseriessystem.Reliabilitybasedoptimisationformulationsfo rrationaldesignofverticalwallbreak-watersaregiven.Finallyanillustrativeexampleispresented,wherethereliabilityofabreakwateronahighrubblemoundisinvestigatedandoptimaldesignsaredetermined.AcknowledgementsThisworkissupportedbytheDanishresearchprogramMarinTeknik sponsoredbytheDanishTechnicalResearchCouncil.ReferencesAndersen,E.Y.&B.S.Andreasen&P.Ostenfeld-Rosenthat1992) .FoundationRe-liabilityofAnchorBlockforSuspensionBridge.roc.FIPWG7.5,LecturenotesnEng.Vol.6,SpringerVerlag,992pp.31-140 .Bruining,J.W.1994) .aveforcesonverticalreakwaters.eliabilityofdesignfor-mula.elftHydraulicsReportH1903,MASTIIcontractMAS2-CT92-0047.Burcharth,H.F.1991) .ntroductionfpartialoefficientsnheesignfu b b l emoundreakwaters.onf.-nCoastalStructuresandBreakwaters,nst.fCivilEngineering,London.Burcharth,H.F.1992a) .Developmentofapartialcoefficientsystemforth edesignofrubblemoundreakwaters.IANCPTCIIWorkingGroup2 ,SubgroupFreport.Burcharth,H.F.1992b).Reliabilityevaluationofastructureatea .roceedingsoftheShortCourseonDesignandReliabilityofCoastalStructures.Venice,ScuoladiS.GiovanniEvangelista,992.3rdInternationalConferenceonCoastalEngineering.

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1602 OASTALENGINEERING996

Burcharth,H.F.&J.D.S0rensen&E.Christiani(1994).nth eEvaluationof FailureProbabilityofMonolithicVerticalWallBreakwaters.roc.WaveBarriersinDeepwa-ter ,PortandHarbourResearchInstitute,Yokosuka,Japan,994,pp.58-46.BurcharthH.F.,J.D.S0rensen& ;E.Christiani(1995):Applicationofreliabilityanalysisforoptimaldesignofmonolithicverticalwallbreakwaters,Proc.COPEDEC995,RiodeJaneiro,Brazil.CherubiniC.&GiasiC.I.1993).hecoefficientofvariationsofsomegeotechnicalparameters.robabilisticmethodsingeotechnical,A.ABalkema1993).Christiani,E.1996).tatisticallyaseddesignmethodsforreakwaters.h.D.,Aal-borgUniversityinprint).Goda,Y.andT.Fukumori1972) .aboratorynvestigationofwavepressuresexertedup onverticalandcompositewalls.oastalEngineeringinJapan,Vol.5.p81-90,1972.Goda,Y.(1974).Anewmethodofwavepressurecalculationforth edesignofcompositebreakwater.roc.4thInt.onf.CoastalEng.,Copenhagen,Denmark.Goda,Y.1975).rregularwavedeformationnhesurfzoneCoastalEngineeringinJapan,JSCE,Vol.18pp.3-16.Hansen,.1979).efinitionnduseffrictionngles.roc.nt.onf.IIECSMFE,Brighton,UK ,979.Nadim,F .&S.Lacasse,andT.R.Cuttormsen1994) .robabilisticfoundationtabil-ityanalysis:Mobilisedfrictionanglesavailablehearstrengthapproach.tructuralSafety&Reliability,994,Balkema,Rotterdam.NielsenS.R.K.andBurcharthH.F.1983).tochasticdesignofrubb lemoundbreakwa-ters.roc.1thIFIPConferenceonsystemmodellingandoptimization,Copenhagen.Madsen,H.O.,S.Krenk&N.C.Lind1986).MethodsofStructuralSafety.rentice-Hall,986.S0rensen,J.D.,H.F.BurcharthandE.Christiani1994) .Reliabilityanalysisandop -timaldesignofmonolithicverticalwallreakwaters.roc.thFIPWG7.5Assissi,Italy.Chapman&Hall.S0rensen,C.S.,Clausen,C.J.F.,Andersen,.,1993).earingapacityAnalysesforhereatBeltEastBridgeAnchorBlocks.imitStateDesignneotechnicalEngineering.SLAD93 ,pp.305-312 Takahashi,S.,Tanimoto,K.,andK .Shimosako(1994).DynamicResponseandSlidingofBreakwaterCaissonagainstImpulsiveBreakingWaveforces.roc.WaveBarriersinDeepwater ,PortandHarbourResearchInstitute,Yokosuka,Japan,994,pp.62 -399.Takayama,T.1992) .EstimationofslidingFailureProbabili tyofpresentBreakwatersforProbabilisticDesign.eportofPortandHarbourResearchInst.,Vol.1 ,No.,1992.

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