an analytical approach to probabilistic modeling of

ORE Open Research Exeter

TITLE

An analytical approach to probabilistic modeling of liquefaction based on shear wave velocity

AUTHORS

Johari, A; Khodaparast, AR; Javadi, AA

JOURNAL

Iranian Journal of Science and Technology Transactions of Civil Engineering

DEPOSITED IN ORE

06 September 2018

This version available at

http://hdl.handle.net/10871/33935

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http://hdl.handle.net/10871/33935

1

Ananalyticalapproachtoprobabilisticmodelingofliquefaction1

basedonshearwavevelocity2 A.Johari1,A.R.Khodaparast1,A.A.Javadi23

1-DepartmentofCivilandEnvironmentalEngineering,ShirazUniversityofTechnology,Shiraz,Iran4 2-ComputationalGeomechanicsGroup,DepartmentofEngineering,UniversityofExeter,Exeter,EX44QF,UK5

[email protected]

Abstract7

Evaluationof liquefactionpotentialof soils isan important stepinmanygeotechnicalinvestigationsin8 regionssusceptibletoearthquake.Forthispurpose, theuseofsiteshearwavevelocity(Vs)providesa9 promisingapproach.Thesafetyfactors in thedeterministicanalysisof liquefactionpotentialareoften10 difficult to interpret becauseof uncertaintiesin the soil and earthquakeparameters.To dealwith the11 uncertainties, probabilistic approaches have been employed. In this research, the Jointly Distributed12 Random Variables (JDRV) method is used as an analytical method for probabilistic assessment of13 liquefaction potential based on measurement of site shear wave velocity. The selected stochastic14 parametersarestress-correctedshear-wavevelocityandstressreductionfactor,whicharemodeledusing15 atruncatednormalprobabilitydensityfunctionandthepeakhorizontalearthquakeaccelerationratioand16 earthquakemagnitude,whichareconsideredtohaveatruncatedexponentialprobabilitydensityfunction.17 ComparisonoftheresultswiththoseofMonteCarloSimulation (MCS)indicatesverygoodperformanceof18 theproposedmethodinassessmentofreliability.Comparisonoftheresultsoftheproposedmodelanda19 StandardPenetrationTest(SPT)-basedmodeldevelopedusingJDRVshowsthatshearwavevelocity(Vs)-20 basedmodelprovidesamoreconservativepredictionofliquefactionpotentialthantheSPT-basemodel.21

Keywords:Reliability,Jointlydistributedrandomvariablesmethod,MonteCarlosimulation,Liquefaction,22 Shearwavevelocity23

1.Introduction24

Liquefactionresultsfromtendencyofsoildepositstodecreaseinvolumewhensubjected25 toshearingstresses.Inadeterministicanalysis,liquefactioncanbedeterminedusingtheCyclic26 ResistanceRatio(CRR)andCyclicStressRatio(CSR)duetoearthquake(Kramer1996).Thecyclic27 stressratioisobtainedusingsomesoilandearthquakecharacteristics(SeedandIdriss1971).28 ThecyclicresistanceratiocanbeobtainedbyseveralmethodsthatwereproposedbySeedand29 Idriss(1971),orasdemonstratedbyYoudandIdriss(2001),usingthestandardpenetrationtest30 (Seed,Tokimatsuetal.1984,BoltonSeed,Tokimatsuetal.1985,Arulanandan,Yogachandranet31 al.1986,SeedandDeAlba1986,SeedandHarder1990,Youd,Idrissetal.2001),conepenetration32 test(Arulanandan,Yogachandranetal.1986,SeedandDeAlba1986,MitchellandTseng1990,33 Robertson1990,JuangandChen1999,Youd,Idrissetal.2001,BaziarandNilipour2003,Juang,34 Yuan et al. 2003, Lees et al. 2015), triaxial test results (Silver 1977), or shearwave velocity35 (AndrusandStokoe1997,AndrusandStokoeII2000).36

Evaluationofsoil liquefactionusingsiteshearwavevelocityprovidesamoreapplicable37 methodthanothersitetestmethodssuchasstandardpenetrationandconepenetrationtests.38 Thismethodisparticularlyusefulforspecificsoilssuchasgravelwherepenetrationtestsmaybe39 unreliable,andatsiteswhereboringsmaynotbepermittedsuchasunderconstructedstructure40 and landfill (Dobry, Stokoe et al. 1981, Seed, Idriss et al. 1983, Stokoe, Roesset et al. 1988,41 TokimatsuandUchida1990).However,Andrusetal.(2004)pointedoutthatthismethodismore42 conservative than penetration-basedmethods for evaluation of liquefaction for the compiled43 Holocenedata.YoudandIdriss(2001)andAndrusetal.(2004)providefurtherdiscussiononthe44 advantagesanddisadvantagesofthismethodandpenetration-basedmethodsinevaluationof45 liquefactionpotential.46

However, theinherentuncertaintiesoftheparameters,whichaffectliquefaction,dictate47 that this problem is of a probabilisticnature rather thanbeing deterministic.Uncertainty in48 liquefactioncanbedividedintotwodistinctivecategories:uncertaintyinthecyclicstressratio49

2

due to earthquake characteristics and uncertainty in the cyclic resistance ratio due to soil1 properties. In the first category, the selection of appropriateearthquake parameters such as2 magnitude, location and recurrence to assess the liquefaction potentialof the site would be3 importantand in second category, the parameteruncertainty,model uncertaintyandhuman4 uncertaintywouldbeimportant(Morgenstern1995).Parameteruncertaintyistheuncertainty5 intheinputparametersforanalysis(Ishihara1996);modeluncertaintyisduetothelimitation6 of the theories and models used in performance prediction (Whitman 2000),while human7 uncertaintyisduetohumanerrorsandmistakes(Sowers1991).8

Reliabilityanalysisprovidesameansofevaluatingthecombinedeffectsofuncertainties9 andoffersalogicalframeworkforchoosingfactorsofsafetythatareappropriateforthedegree10 ofuncertaintyandtheconsequencesoffailure.Thus, asanalternativeorasupplementto the11 deterministicassessment,areliabilityassessmentof liquefactionpotentialwouldbeuseful in12 providingbetterengineeringdecisions.13

There aremany reliabilityapproaches thathave been developed to deal uncertainties in14 liquefactionpotentialbasedonshearwavevelocity.Theseapproachescanbegroupedintothree15 categories:approximatemethods,artificialintelligencemethodandregressionanalysis.16

Approximatemethods:Mostof theapproximatemethodsaremodifiedversionsof three17 methodsnamely,FirstOrderSecondMoment(FOSM)method(AlfredoandWilson1975),Point18 Estimate Method (PEM) (Rosenblueth 1975), and First Order Reliability Method (FORM)19 (Hasofer,Lindetal.1973).Theseapproachesrequireknowledgeofthemeanandvarianceofall20 inputvariablesaswellastheperformancefunctionthatdefinesliquefactionsafetyfactor.21

Juangetal.(2005)usedFORMtocharacterizetheuncertaintyofashearwavevelocity-based22 simplifiedmodelforliquefactionpotentialevaluationdevelopedbyAndrusandStokoe(1997,23 2000).Theyrepresentedtheuncertaintyofthissimplifiedmodelbyalognormalrandomvariable24 andperformeda trial-and-errorprocedure todetermine the twostatisticalparametersof the25 modeluncertainty(meanandtheCoefficientofVariation(COV))basedonaBayesianmapping26 functionthatwascalibratedwithadatabaseofcasehistories.27

Zouetal.(2017)usedFORMtocharacterizetheuncertaintyofaConepenetrationtestmodel28 for liquefactionpotentialevaluation. Itwasshown that the deterministicnatureof the CPTU29 observationscanbecapturedintheprobabilisticanalysisiftheproposedprocedureisapplied.30

Artificialintelligencemethod:Inrecentyears,bypervasivedevelopmentsincomputational31 softwareandhardware,severalalternativecomputer-aidedpatternrecognitionapproacheshave32 emerged.Themainideabehindpatternrecognitionsystemssuchasneuralnetwork,fuzzylogic33 or genetic programming is that they learn adaptively from experience and extract various34 discriminates,each appropriatefor itspurpose.ArtificialNeuralNetworks (ANNs)andMulti-35 LayerRegression(MLR)arethemostwidelyusedpatternrecognitionproceduresthathavebeen36 introducedfordeterminationofliquefactionpotential.Inthisapproachthereliabilityanalysisis37 donebasedonafunctionthatisdevelopedbyanappropriateartificialintelligencemethod(Chau38 andWu2010,Wu,Chauetal.2010,Taormina,Chauetal.2015).39

Juangetal.(2001)developedanewVs-basedempiricalequationforassessingtheliquefaction40 resistanceofsoilsusinganeuralnetwork.Adatabaseofcasehistorieswasusedtotrainandtest41 anartificialneuralnetworkmodel.Themodelcouldpredict theoccurrence/nonoccurrenceof42 liquefaction based on soil and seismic load parameters. Based on this deterministic model,43 probabilisticanalysisof thecasesinthedatabasewasconductedusingthe logisticregression44 approachandthemappingfunctionapproach.45

Goh (2002)used a ProbabilisticNeural Network (PNN) approachbased on the Bayesian46 classifiermethod to evaluate seismic liquefactionpotentialbasedon actual field records and47 performed twoseparateanalyses,onebasedonconepenetrationtestdataandonebasedon48 shearwavevelocitydata.ComparisonsoftheresultsshowedthatthePNNmodelsperformfar49 better than the conventional methods in predicting the occurrence or non-occurrence of50 liquefaction.51

3

Mudulietal.(2014)evaluatedliquefactionpotentialofsoilwithinaprobabilisticframework1 basedonthepost-liquefactionConePenetrationTest(CPT)datausinganevolutionaryartificial2 intelligencetechnique,multi-genegeneticprogramming(MGGP).3

Regressionanalysis:Therationalityofthereliabilityanalysisresultslargelydependsonthe4 amountandqualityofthecollecteddatausedtodeducethestatisticsofthecyclicsoilstrength.5 Thismethodrequirescollectingdataforliquefactionandnon-liquefactioncases.6

Juangetal.(2002)usedtwodifferentapproaches,logisticregressionandBayesianmapping7 functionsforcalculatingprobabilitiesofliquefactionbasedonthestandardpenetrationtest,cone8 penetrationtest,andshearwavevelocitymeasurementsandcompared the resultswith each9 other.TheyshowedthattheBayesianmappingapproachispreferredoverthelogisticregression10 approachforestimatingthesite-specificprobabilityofliquefaction,althoughbothmethodsyield11 comparableprobabilities.12

Nafday(2010)developedasoilliquefactionmodelsbasedonsurvivalanalysisparametric13 regressiontoevaluatethefactorofsafetyandprobabilityofliquefaction.14

Inadditiontotheabovementionedapproaches,analyticalmethodssuchasjointlydistributed15 randomvariablesmethodandnumericalmethodslikeMonteCarlosimulation(Metropolisand16 Ulam 1949, Metropolis and Ulam 1949) also can be employed for reliability analysis of17 liquefactionpotentialbasedoninsitushearwavevelocitymeasurement.18

In this research, the jointlydistributedrandomvariablesmethod is usedas ananalytical19 method to assess the reliabilityof the safety factor in the predictionof liquefactionpotential20 consideringtheuncertaintiesinparametersandMonteCarlosimulationisusedforverifyingthe21 resultsofJDRVmethod. 22

Inthisanalyticalmethod,thederivationisdoneonlyonceandafterthat, itcanbeusedin23 manyapplications.Itisalsoworthnotingthat,insomeproblemssuchasliquefactionpotential24 assessment,whenarelativelylargenumberofvariablesareinvolved,theMonteCarlosimulation25 may require hundreds of thousands of simulation runs that make the method excessively26 demanding in computational time and resources. Moreover, the jointly distributed random27 variablesmethod canbeused for stochasticparameterswith anydistribution curve (such as28 normal,exponential,gamma,uniform,...)whereassomeotheranalyticalmethodslikePEM,and29 FOSMrequirespecific(e.g.,normal)distributionfunctions.Thisabilityisveryimportantbecause30 thepeakhorizontalearthquakeaccelerationratio(α)andearthquakemagnitude(Mw),whichare31 presented in liquefactionpotential relationship,areconsidered tohave truncatedexponential32 probabilitydensityfunctions.Itisimportanttonotethatthemaindeferencebetweenthispaper33 andthepublishedpapersisthatthepreviouspaperswerefollowedthereliabilityassessmentof34 liquefactionbyJDRVmethodusingSPT,CPTandtriaxialdata(Johari, Javadietal.2012,Johari35 and Khodaparast 2013, Johari and Khodaparast 2014). However, this research assesses this36 reliabilitybyJDRVmethodusingshearwavevelocitydata. 37

2.FactorofSafetyagainstliquefactionbasedonsiteshearwavevelocity38

ThesoilliquefactionFactorofSafety(FS)isdefinedintermsofCyclicResistanceRatiofor39 earthquakeswithmagnitude of 7.5 (CRR7.5), Cyclic Stress Ratio (CSR), earthquakeMagnitude40 ScalingFactor(MSF),andoverburdenstresscorrectionfactor(Kσ)as:41

(1)42

NoliquefactionispredictedifFS>1,andontheotherhand,ifFS≤1,liquefactionispredicted.43 CyclicResistanceRatioforearthquakeswithmagnitudeof7.5(CRR7.5)canbeobtainedfromshear44 wavevelocitymeasurementas(Andrus,Stokoeetal.2004):45

(2)46

7.5σ

CRRFS= .MSF.KCSR

ì üï ïí ýï ïî þ

æ öæ öç ÷ ç ÷è ø è ø

2a1 S1cs

7.5 a2a1 S1cs

KK V 1 1CRR = 0.022 +2.8 -100 215-(K V ) 215

4

where:1

VS1cs:Theclean-sandequivalentoftheoverburdenstress-correctedshear-wavevelocity,defined2 as(Andrus,Stokoeetal.2004):3

(3)4

VS1:Thestress-correctedshear-wavevelocitynormalizedtotheeffectiveoverburdenstressof5 100kPacalculatedas(Youd,Idrissetal.2001): 6

(4)7

CVS:Afactortocorrectthemeasuredsitevelocityfor (amaximumCVSvalueof1.4isapplied8 atshallowdepths).9 Vs:Siteshearwavevelocity(m/s)10 Kcs:FinescontentcorrectiontoadjustVs1valuestoacleansoilequivalent.Itcanbeapproximated11 usingthefollowingequation(Juang,Jiangetal.2002):12

(5)13

where:14

(6)15

FC:Theaveragefinescontent.16

Ka1andKa2:Correctionfactorsforcementationandaging(Andrus,Stokoeetal.2004).17

ThefactorsKa1andKa2areincludedinEquation(2)toextendtheoriginalCRR-basedshearwave18 velocityequationbyAndrusandStokoe(2000)foruncementedHolocene-agesoilstooldersoils.19 Thetwocorrectionfactorsaresuggestedbecauseitisbelievedthattwomechanismsinfluence20 thepositionof theCRR-basedshearwavevelocity curve for oldersoils. The firstmechanism21 involvestheeffectofagingonVS1.ThesecondmechanisminvolvestheeffectofagingonCRR.22 BothKa1andKa2are1.0foruncementedsoilsofHoloceneage.Foroldersoils,the values of Ka1and23 Ka2wereproposedbyAndrusetal.(2004). 24

TheCyclicStressRatio(CSR)hasbeenproposedbySeedandIdriss(1971)as:25

(7)26

where:27

σ’v:Effectiveverticalstress28

σv:Totalverticalstress29

τav:Averageshearstresscausingliquefaction30

(amax/g)=α: Peakhorizontalgroundsurfaceaccelerationnormalizedwithrespecttoacceleration31 ofgravity 32

rd:Stressreductionfactor33

cs=KS1cs S1V V

æ öç ÷¢è ø

0.25a

s1 s VS sv

PV =V C =Vσ

¢vσ

cs

cs

cs

=1FC 5%=1+(FC-5)T5%<FC<35%=1+30TFC 35%

KKK

£ìïíï ³î

æ ö æ öç ÷ ç ÷è ø è ø

S1 S12V VT 0.009 0.0109 +0.0038

100 100= -

¢ ¢av v max

dv v

τ σ aCSR=( )=0.65( )( )(r )σ σ g

5

The stress reduction factor, rd, provides an approximate correction for flexibility in the soil1 profile.Thereareseveralempiricalrelationsfordeterminationofrd.Theearliestandmostwidely2 used recommendation for assessment of rd was proposed by Seed and Idriss (1971),3 approximatedbyLiaoandWhitman(1986),andexpressedinYoudandIdriss(2001)as:4

(8)5

where:6

h:depthbelowgroundsurface(m)7

Themagnitudescalingfactor,MSF,hasbeenusedtoadjusttheinducedCSRduringearthquake8 magnitudeMw to an equivalent CSR for an earthquakemagnitude,Mw=7.5. The MSF is thus9 expressedas(Youd,Idrissetal.2001):10

(9)11

Mw:earthquakemagnitude12

Theoverburdenpressurecorrectionfactor,Kσ isusedtoadjustthecyclicresistanceratiowhere13 the overburden stresses aremuch greater than100kPa.This factor is definedby Idrissand14 Boulanger(2006):15

(10)16

where:17

(11)18

Pa:Atmosphericairpressure19 DR:Fieldrelativedensity20 Foruncementedsoils,Equation(1)canberewrittenbasedonequations(2)to(11)asfollows:21

(12)

22

Equation(12)canbesimplifiedas:

23

(13)

24

where25

0.5 1.5

d 0.5 1.5 21.0-0.4113h +0.04052h+0.001753hr =

1.0-0.4177h +0.05729h-0.006205h +0.00121h

æ öç ÷è ø

-2.56wMMSF=

7.5

( )K =1-C ln σ /P 1.0σ σ v a¢ £

£σR

1C =18.9-17.3D

0.3

æ ö ¢æ ö æ ö æ æ ööç ÷ç ÷ ç ÷ ç ç ÷÷

è ø è ø è è øøè ø =

¢

æ öæ ö æ öç ÷ç ÷ ç ÷

è ø è øè ø

2S1cs v

S1cs R a7.5σ

2.56v maxd w

v2

cs S1

cs S1

V σ1 1 10.022 +2.8 - 1- ln100 215-V 215 18.9-17.3D PCRRFS= .MSF.K =CSR aσ0.65 × ×r ×(M /7.5)σ g

K V 1 1 10.022 +2.8 - 1-100 215-K V 215 18.( )

( )

æ æ ööç ç ÷÷è è øø

sat w

R a

2.56satd w

sat w

γ -γ hln9-17.3D Pγ0.65 ×α×r ×(M /7.5)γ -γ

( ) ( )a =´

-2.56S1 w

S1 d wd

L V MFS V ,r , ,Mr α

6

(14)1

Itshouldbenotedthatifσ'v≤100kPa,thenKσ=1.0andtheterm 2

isremovedfromequation(14).3

3.Developingrelationsbetweendependentvariables4

Thejointlydistributedrandomvariablesmethodthatisusedforreliabilityassessmentin5 this research assumes that the variables are independent. There are several stochastic6 parameters in equations (13) and (14). As the liquefaction classification problem is highly7 nonlinearinnature,itisdifficulttodevelopacomprehensivemodeltakingintoaccountallthe8 independent variables, such as the seismic and soil properties, using conventional modeling9 techniques.Hence, inmanyof the conventionalmethods thathavebeenproposed, simplified10 assumptionshavebeenmade.11

No direct correlation exists between α and earthquake magnitude. Several empirical12 relationships have been developed for estimating α as a function of earthquake magnitude,13 distance from the seismic energy source, and local site conditions. Preliminary attenuation14 relationshipshavealsobeendevelopedforalimitedrangeofsoftsoilsites(Idriss1991).Selection15 ofanattenuationrelationshipshouldbebasedonsuchfactorsastheregionofthecountry,type16 offaulting,andsitecondition(Youd,Idrissetal.2001).17

Ontheotherhand,inequation(14),theparametersDRand arerelatedtoVs1.Inthis18 section,therelationshipbetweenDRandVs1aswellas andVs1aredevelopedtoresolvethe19 dependencyproblemofvariablesinthisequation.Asaresultofthisderivation,equations(13)20 and(14)havefourstochasticparameters,peakgroundacceleration(α),earthquakemagnitude21 (Mw), corrected shear-wave velocity (Vs1), and stress reduction factor (rd) as well as the22 deterministicparametersmaximumpossibledryunitweight( ),minimumpossibledryunit23 weight( ),unitweightofwater( ),averageFinesContent(FC),andspecificgravity(Gs).24

3.1. RelationbetweenDRandVs1:25

Andrusetal.(2004) proposedthefollowingrelationbetweenVS1csandN1,60cs:26

(15)27

where:28 VS1cs:Theclean-sandequivalentoftheoverburdenstress-correctedshear-wavevelocity.Itcan29 becalculatedfromequation(3).30 N1,60cs:Theclean-sandequivalentoftheoverburdenstress-correctedSPTblowcountdefinedas31 (Youd,Idrissetal.2001):32

(16) 33

where:34 35 N1,60:ThecorrectedSPTblowcountnormalizedtotheeffectiveoverburdenstressof100kPa36

( )

( ) ( )

æ öæ ö æ öç ÷ ç ÷ç ÷è ø è øè ø´

æ æ ööç ç ÷÷è è øø

-2.56sat

sat wsat w

R a

S1L V =0.65×γ 7.5

γ -γ h1γ -γ 1- ln(18.9-17.3D ) P

2cs S1

cs S1

K V 1 10.022 +2.8 -100 215-K V 215

( )æ æ ööç ÷ç ÷è øè ø

sat w

R a

γ -γ h11- ln18.9-17.3D P

satγsatγ

maxdγ

mindγ wγ

( )S1cs 1,60cs0.253V =87.8 N

1,60cs 1,60N =a+b.N

7

aandbarecoefficientstoaccountfortheeffectofFinesContent(FC),definedas(Youd,Idrisset1 al.2001):2

(17)3

(18)4

SeveralrelationshipsbetweenrelativedensityandSPTblowcountshavebeenproposedinthe5 literature(TokimatsuandSeed1987,Terzaghi1996,IdrissandBoulanger2008).Cubrinovski6 andIshihara(1999)proposedarelationshipbetweenDRandcorrectedSPTblowcountas:7

(19)8

where:9

(20)10

emax:Maximumpossiblevoidratiofromlaboratoryexperiment11 emin:Minimumpossiblevoidratiofromlaboratoryexperiment12 Thevoidratiorange(emax-emin)canbecalculatedasfollows(Das2013):13

(21)14

where:15

:Maximumpossibledryunitweightfromlaboratoryexperiment16

:Minimumpossibledryunitweightfromlaboratoryexperiment17 18 Combiningequations(15)to(19),therelationshipbetweenDRandVs1canbedevelopedas:19

(22)20

3.2. RelationbetweenγsatandVs1:21

Therelationbetweenγsatandγdcanbederivedfromtheirbasicdefinitionsas(Das2013):22

£ìïíï ³î

2

a=0FC 5%

a=exp[1.76-(190/FC )]5%<FC<35%a=5.0FC 35%

£ìïíï ³î

1.5

b=1FC 5%

b=[0.99+(FC /1000)]5%<FC<35%b=1.2FC 35%

D

1,60ND =R C

( )-D

max min

9C =1.7e e

( )s wmax

max mins w dmin

s wdmax minmin

dmax

wmax min

G γe = -1 G γ γ - γs d dG γ γe= -1 e -e = G γγ γ .γd de = -1γ

ì üï ïï ï® ®í ýï ïï ïî þ

maxdγ

mindγ

æ öæ öç ÷-ç ÷ç ÷è øè ø

1000253

0.5

RD

1 5KcsD ab.C 439= S1V

8

(23)1

where:2

γsat:Saturatedunitweight3 γd:Dryunitweightinnaturalstateofsoil4 Gs:Specificgravityofsoilsolids5 γw:Unitweightofwater(9.81kN/m3)6 e:Voidratioinnaturalstateofsoil7 Therelationbetweenrelativedensity(DR)anddryunitweight(γd)is(Das2013):8

(24)9

Usingequation(23)and(24),therelationbetweenγsatandDRcanbedevelopedasfollows:10

(25)11

Withsubstitutingequation(22)inequation(25),therelationbetweenγsatandVs1canbe12 obtainedasbellow:13

(26)14

Bysubstitutingequation(22)and(26)intoequations(13)and(14),theseequationsconverttoa15 stochasticindependentvariablerelations.16

4. Jointlydistributedrandomvariablesmethod17

Jointly distributed random variables method is an analytical stochastic method. In this18 method,theprobabilitydensityfunction(pdf)ofinputvariablesareexpressedmathematically19 andjointedtogetherbystatisticalrelations.TheJDRVmethodisanexactmethodandcanbeused20 forstochasticparameterswithanydistributioncurve(suchasnormal,lognormal,exponential,21 gamma, uniform, …). This ability is very important because the peak horizontal earthquake22 accelerationratio(α)andearthquakemagnitude(Mw),whicharepresentedinFactorofSafety23 against liquefaction relationship, are considered to have truncated exponential probability24 densityfunctions.Theavailablestatisticalandprobabilisticrelationsbetweenrandomvariables25 aregivenintheliterature(Hoel,Portetal.1971,Tijms2012,RamachandranandTsokos2014).26

In recent years this method has been applied to a number of geotechnical engineering27 problems(JohariandJavadi2012,Johari,Javadietal.2012,Johari,Fazelietal.2013,Johariand28 Khodaparast2013, JohariandKhodaparast2014,Johariandkhodaparast2015).29

( ) æ öæ öüç ÷ç ÷ï ç ÷ï è øè ø®ý

ï®ïþ

ws ws dsat

dsat

s w s w sd

d

γG +e γ G 1+ -1 γγ = γ1+e γ =G ×γ G ×γ Gγ = e= -11+e γ

( )® =max min maxmin

max min max max min

d d dd dR d

d d d d R d d

γ γ .γγ - γD = × γ

γ - γ γ γ -D γ - γ

( )( )-

+ -d d smax min

s d d d Rmax min maxsatγ

γ .γ (G 1)=G γ γ γ D

( )

-

æ öæ öæ öç ÷ç ÷+ - -ç ÷ç ÷ç ÷è øç ÷è øè ø

d d smax min0.51000

253css d d dmax min max D

satγγ .γ (G 1)

=1 5K

G γ γ γ ab.C 439S1V

9

5.MonteCarlosimulation1

ThesimulationbyMonteCarlocansolveproblemsbygeneratingsuitablerandomnumbers2 (or pseudo-random numbers) and assessing the dependent variable for a large number of3 possibilities. The MCS involves the definition of the variables that generate uncertainty and4 probabilitydensityfunction(pdf);determinationofthevalueofthefunctionusingvariablevalues5 randomlyobtainedconsideringthepdf;andrepeatingthisprocedureuntilasufficientnumberof6 outputstobuildthepdfofthefunction.ThenumberofrequiredMonteCarlotrialsisdependent7 on the desired level of confidence in the solution as well as the number of variables being8 considered(Harr1987),andcanbeestimatedfrom:9

(27)10

where:11 N:NumberofMonteCarlotrials12 d:Thestandardnormaldeviatecorrespondingtothelevelofconfidence13 :Thedesiredlevelofconfidence(0to100%)expressedindecimalform14

n:Numberofvariables15 Iftheproblemhasnvariables,thenumberoftrialsincreasesgeometrically,accordingtopower16 n.17

6.Reliabilityassessmentbyjointlydistributedrandomvariablesmethod18

Forreliabilityassessmentofliquefactionpotentialandtoaccountfortheuncertainties,four19 independent input parameters have been defined as stochastic variables. The stochastic20 parametersarestresscorrectedshear-wavevelocity(Vs1)andstressreductionfactor(rd),which21 aremodeledusingtruncatednormalprobabilitydensityfunctions(pdf)andthepeakhorizontal22 earthquakeaccelerationratio(α)andearthquakemagnitude(Mw)whichareconsideredtohave23 truncated exponential probability density functions. The depth is regarded as a constant24 parameter.25

For reliabilityassessmentof liquefactionsafetyfactorusing the JDRVmethod, equation26 (13) is rewritten into terms of K1 to K7 as shown in equation (28).The terms K1 to K7, are27 introducedinequation(29).Theprobabilitydensityfunctionofeachtermisderivedseparately28 byequations(36)to(43).DerivationsoftheseequationsarepresentedintheAppendix1.29

(28)30

where:31

(29)

32

é ùê úê úë û

n2dN= 24(1-ε)

e

1 2 3 4 1 2 3 4 5 3 4 6 4 7K ,K ,K ,K K .K .K .K K .K .K K .K KFS( )= = = =

( )ìïïïïïïíïïïïïïî

1 S1

2d-2.56

3 w

4

5 1 2

6 5 3

7 6 4

K =L V1K = r

K =M1K = α

K =K ×K

K =K ×K

K =FS=K ×K

10

Using the abovemathematical functions for K1 to K7 and fK1(k1) to fK7(k7) a computer1 programwasdeveloped(codedinMATLAB)todeterminetheprobabilitydensityfunctioncurve2 ofliquefactionsafetyfactor.Inaddition,forcomparison,determinationof thesafetyfactorfor3 liquefactionusingtheMCS wasalsocodedinthesamecomputerprogram. 4

MATLABisamulti-paradigmnumericalcomputingenvironment.MATLABallowsmatrix5 manipulations,plottingof functionsanddata, implementationof algorithms,creation of user6 interfaces,andinterfacingwithprogramswritteninotherlanguages. 7

To show the abilityof the proposedmethod an example is presented in the following8 sections.9

7.Example10

Todemonstratethe efficiencyandaccuracyof theproposedmethod indeterminingthe11 probabilitydensityfunctioncurvefortheliquefactionsafetyfactorandreliabilityassessment,an12 exampleproblemwithselectedparametersvaluesfromliterature(Gabriels,Sniederetal.1987,13 Kramer1996,Duncan2000,Youd,Idrissetal.2001,MarosiandHiltunen2004)ispresented.The14 stochasticparameterswithtruncatednormalandtruncatedexponentialdistributionsareshown15 inTables(1)and(2)respectively,andthedeterministicparametersaregiveninTable(3).16

Table(1)_Stochasticparameterswithtruncatednormaldistribution 17 18

19 20

Table(2)_Stochasticparameterswithtruncatedexponentialdistribution 21 Mean Maximum Minimum λ Parameters 6.418 8.0 5.5 2/3 Mw 0.269 0.4 0.2 10 α

Table(3)_Deterministicparameters 22 Depthofwatertable(m) Depth(m) FC(%) (kN/m3) (kN/m3) Gs

0.0 12.0 10.0 14.0 19.0 2.65

InordertocomparetheresultsofthepresentedmethodwiththoseoftheMCS,thefinal23 probabilitydensityandcumulativedistributioncurvesforthefactorofsafetyagainstliquefaction24 aredeterminedusingthesamedataandbothmethods.Forthispurpose,10,000,000generation25 pointsareusedfortheMCS.TheresultsareshowninFigures(1)and(2).26

Figure(1)_Comparisonofprobabilitydensityfunctionof

safetyfactoragainstliquefactionbytwomethodsFigure(2)_Comparisonofcumulativedistributionfunctionof

safetyfactoragainstliquefactionbytwomethods

mindγ maxdγ

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Factorofsafety

ProbabilityDensityFunction

MCSJDRV

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

FactorofSafety

CumulativeDistributionFunction

JDRVMCS

Parameters Mean Standarddeviation Minimum Maximumrd 0.8565 0.0207 0.7737 0.9394Vs1 180 8 148 212

11

Asitcanbeseeninthesefigures,theresultsobtainedusingthedevelopedmethodarevery1 closetothoseoftheMCS.Theprobabilityofliquefaction,isshownbygreenregionforFS<1,in2 Figure(1).Figure(2)showsthecumulativedistributioncurveoftheliquefactionsafetyfactor.It3 canbeseentheprobabilityofliquefaction(FS≤1)forthissiteattheassesseddepth(12m)isabout4 76%.Table(4)indicatesthatatthisdepthliquefactionwouldmostlikelyoccur.5

Table(4)_Classesofliquefactionpotential(Juang,Jiangetal.2002)6 Probability Class Description(Likelihoodofliquefaction)

0.85<PL<1.00 5 Almostcertainthatitwillliquefy0.65<PL<0.85 4 Liquefactionverylikely0.35<PL<0.65 3 Liquefactionandnon-liquefactionequallylikelyo.15<PL<0.35 2 Liquefactionunlikely0.0<PL<0.15 1 Almostcertainthatitwillnotliquefy

On the other hand, a deterministic calculation using the mean value of the stochastic7 parametersshowsthat,thesafetyfactoragainstliquefactionisabout0.72.Thisdemonstrates8 thatatthisdepthliquefactionwouldoccur,but theprobabilityof liquefactionisnotspecified.9 Therefore,thedesignercannothaveanengineeringjudgment.Infact,reliabilityassessmentand10 engineeringjudgmentareemployedtogethertodevelopriskanddecisionanalyses. 11

8.Probabilisticmodeldevelopment12

8.1.Database13

For developing the model, a database consisting of 225 site case histories, collected by14 Andruset al. (1999),wasused. Thedatabaseiscomposedof 129non-liquefiedcasesand9615 liquefiedcases.Table(5)providesasummaryofthisdatabase,includingtherangesofparameter16 valuesofthecasehistoriesinthedatabase.17

Table(5)_Parametersrangesofcasehistoriesinthedatabase(Andrus,Stokoeetal.1999)18

Earthquake Mw amax(g)No.ofcases Depth

(m)G.W.L.(m) FC(%) Vs1

(m/s)Liq. Non-Liq.

1906SANFRANCISCO,CALIFORNIA 7.7 0.32-0.36 8 4 4.6-9.9 2.4-6.1 5-44 124-191 1957DALYCITY,CALIFORNIA 5.3 0.11 0 5 3.5-7.9 2.7-5.9 2-12 113-211 1964NIIGATA,JAPAN 7.5 0.16 3 1 3.2-6.2 1.2-5.0 5 136-164 1975HAICHENG,PRC 7.3 0.12 5 1 3.0-10.2 0.5-1.5 42-92 111-158 1979IMPERIALVALLEY 6.5 0.12-0.51 4 7 3.0-4.7 1.5-2.7 10-75 104-211 1980MID-CHIBA,JAPAN 5.9 0.08 0 2 6.1-14.8 1.3 20-35 173-185 1981WESTMORLAND,CALIFORNIA 5.9 0.02-0.36 6 5 3.0-4.7 1.5-2.7 10-75 104-211 1983BORAHPEAK,IDAHO 6.9 0.23-0.46 15 3 1.9-3.7 0.8-3.0 5-6 115-318 1985CHIBA-IBAARAGI,JAPAN 6.0 0.05 0 2 6.1-14.8 1.3 20-35 173-185 10/26/85TAIWAN(EVENTLSST2) 5.3 0.05 0 4 5.3-6.1 0.5 50 155-191 11/7/85TAIWAN(EVENTLSST3) 5.5 0.02 0 4 5.3-6.1 0.5 50 155-191 1/16/86TAIWAN(EVENTLSST4) 6.6 0.22 0 4 5.3-6.1 0.5 50 155-191 4/8/86TAIWAN(EVENTLSST6) 5.4 0.04 0 4 5.3-6.1 0.5 50 155-191 5/20/86TAIWAN(EVENTLSST7) 6.6 0.18 0 4 5.3-6.1 0.5 50 155-191 5/20/86TAIWAN(EVENTLSST8) 6.2 0.04 0 4 5.3-6.1 0.5 50 155-191 07/30/86TAIWAN(EVENTLSST12) 6.2 0.18 0 4 5.3-6.1 0.5 50 155-191 07/30/86TAIWAN(EVENTLSST13) 6.2 0.05 0 4 5.3-6.1 0.5 50 155-191 11/14/86TAIWAN(EVENTLSST16) 7.6 0.16 0 4 5.3-6.1 0.5 50 155-191 1987CHIBA-TOHO-OKI,JAPAN 6.5 0.03 0 1 9.0 1.8 15 141 1987ELMORORANCH 5.9 0.03-0.24 0 11 3.0-4.7 1.8 10-75 104-211 1987SUPERSTITIONHILLS,CALIFORNIA 6.5 0.18-0.21 3 8 3.0-4.7 1.5-2.7 10-75 104-211 1989LOMAPRIETA,CALIFORNIA 7.0 0.13-0.42 33 34 2.3-9.9 0.6-6.1 1-57 107-222 1993KUSHIRO-OKI,JAPAN 8.3 0.41 2 0 4.2-4.5 0.9-1.9 5-7 161-189 1993HOKKAIDO-NANSEI-OKI,JAPAN 8.3 0.15-0.19 3 1 2.0-7.0 1.0-1.4 5-54 99-166 1994NORTHRIDGE,CALIFORNIA 6.7 0.51 3 0 4.4-5.6 3.4 10 142-170 1995HYOGOKEN-NANBU,JAPAN 6.9 0.12-0.65 11 8 3.3-15 1.5-7.0 2-77 126-239 1906SANFRANCISCO,CALIFORNIA 7.7 0.32-0.36 8 4 4.6-9.9 2.4-6.1 5-44 124-191 1957DALYCITY,CALIFORNIA 5.3 0.11 0 5 3.5-7.9 2.7-5.9 2-12 113-211 1964NIIGATA,JAPAN 7.5 0.16 3 1 3.2-6.2 1.2-5.0 5 136-164

8.2.Modeldevelopment19

Todeveloptheprobabilisticliquefactionmodelthefollowingprocedurewasfollowed:20 • The uncertainty in the input parameters used in the calculation of safety factor of21

liquefaction was assessed for each series of database. The large majority of the22

12

liquefactioncasehistorieslacksufficientinformationtojustifyattemptingtodevelopsite-1 specificestimatesoftheseuncertaintiesforeachcasehistory.Forthisreason,the COVof2 Vs1wastakenasbeingthesameforallcasehistoriesandequalto0.05assuggestedby3 MarosiandHiltunen(2004).Thestandarddeviationofrdwasselectedbasedonthethree-4 sigmarule(Duncan2000)andthecurvesuggestedbySeedandIdriss(1971)foreach5 depth.To consider theuncertaintyof earthquakeparameters, reasonablevalueswere6 takenforthescaleparameterofearthquakeaccelerationratioandmomentmagnitude,7 asbeingthesameforallcasehistoriesandequalto0.05and0.8respectively(βa=0.058 andβMw=0.8).Furthermore the rangeof variationofa andMwwas taken0.2 and2.59 respectivelyforallcasehistories(MWmax-MWmin=2.5andamax-amin=0.2).10

• The cumulative distribution function of each data series from the database was11 determinedusingtheJDRVmethodasdescribedinsection5.12

• Theprobabilityofliquefactionwascomputedfromthecumulativedistributionfunction13 foreachdataseries.14

• The safety factorof each dataserieswascalculatedusing the deterministicapproach15 describedinsection2.16

• Theprobabilityofliquefactionandtherelatedfactorofsafetyfromtwoprevioussteps17 wereplottedwithrespecttoeachotherforalldataseries.TheresultsareshowninFigure18 (3).19

Figure(3)_PredictionsofthedevelopedprobabilityliquefactionmodelusingJDRV

• TheprobabilisticliquefactionmodelwasdevelopedusingMATLABcurvefittingtoolbox.20 Themodelhasthefollowingform:21

(30)22

Inequation(30),FSiscomputedusingthemethodrecommendedbyAndrusetal.(Andrus,Stokoe23 et al. 2004), as described in section2 andΦ is the standard normal cumulative distribution24 functiondefinedas:25

(31)26

Usingequation(31),equation(30)canberewrittenas:27

(32)28

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

FactorofSafety(FS)

ProbabilityofLiquefaction(PL)

Non-liquefiedCaseHistoryData(129cases))LiquefiedCaseHistoryData(96cases)FittedCurve

PL=1-F200[(ln(FS)+2.6)/0.93]

R=0.99902

( ) ( )é ù+= -F ê ú

ë û

200 ln 2.61

0.93LFS

P FS

( )-

21 1 1 1 zΦ z = exp - du = + erfu2 2 22π 2¥

æ öæ öç ÷ç ÷

è ø è øòz

( ) ( )1 1+ erf2 2 2

é ùæ ö+= - ê úç ÷ç ÷ê úè øë û

200ln 2.6

10.93

LFS

P FS

13

whereerfiserrorfunction,definedas:1

(33)2

8.3Comparisonofthemodel3

Inthispartthedevelopedmodelwascomparedtoanexistingmodelandempiricaldata.4 ForthispurposethemodelproposedbyJuangetal.(2002)wasselected.5

(34)6

whereFSmustbecomputedassuggestedbyAndrusandStokoe(1997,2000).7 Additionallythemodelwascomparedwithempiricaldata.ForthispurposetheFSswere8

calculatedforalldatausingthedeterministicapproachasdescribedinsection3.Theresultswere9 thenplacedindifferentbinwidths(classes)of0.1,0.2,0.3,0.4and0.5basedontheirFSvalues.10 Bycountingthenumberofliquefiedcases(n1)andnon-liquefiedcases(n2)inthesamebinthe11 empiricalprobabilityofliquefactionPLcorrespondingtothecenterofeachFSbinwasobtained12 asPL=n1/(n1+n2)(Juang,Chingetal.2012).13

ComparisonoftheprobabilisticmodelsproposedbyJDRVandJuangetal.(2012)andthe14 empiricaldataforbinwidths0.1,0.2,0.3,0.4and0.5ispresentedinFigure(4).Furthermore,15 liquefactionprobabilitypredictionsofthemodelsforsomesafetyfactorsaregiveninTable(6).16

Figure(4)_Comparisonofdifferentmodelsandempiricaldata

Table(6)_Liquefactionprobabilitypredictionsofthemodels 17 Probabilityofliquefaction(%)

JDRVModel Juangetal.(2002) DesignFS Model40.4625.541.0

Vs-based 24.24 15.58 1.2 11.58 7.95 1.5

18

It can be seen that the proposed model provides a more conservative prediction of19 liquefactionpotentialthantheJuangetal.(2002)modelandempiricaldatawhichisduetouse20 ofliquefactionboundarycurveproposedbyAndrusandStokoe(1997,2000).Thiscurvemissed21 theleastnumberofliquefiedcasesandthusisequivalenttothe26%probabilitycurve(Juang,22 Jiangetal.2002).Thiscurve’sconservationtransfertoJDRVmodeldirectly.However,theJuang23 et al. (2002)modelwasderivedusing logistic regressionandBayesianmapping functionson24 shearwavevelocitymeasurementdatabase.25

( ) ( )exp -ò0

22erf x = dttπ

x

( ) =æ ö+ ç ÷è ø

3.41

10.73

LP FSFS

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

FactorofSafety(FS)


ProposedModel(Eq.(49))Juangetal.(2002)(Eq.(53))Empericalaverage(bin=0.1)Empericalaverage(bin=0.2)Empericalaverage(bin=0.3)Empericalaverage(bin=0.4)Empericalaverage(bin=0.5)

14

8.4ComparisonoftheVsandSPTbasedprobabilisticmodels1

Acomparisonof theresultsof theproposedmodel(Eq. (30))andtheSPT-basedmodel2 developedusingJDRV(JohariandKhodaparast2013)(Eq.(35))ispresentedinFigure(5).3 4

(35)5

Inthisequation,FSiscomputedusingthemethodrecommendedbyYoudandIdriss.(2001).6 Itisshownthat,asexpected,theVs-basedmodelprovidesthemoreconservativepredictionof7 liquefactionpotentialthan the SPT-basedmodel for importantsafety factors(FS<1.3)(in the8 sameprobabilityofliquefactionoccurrence,theVs-basedmodelpredictssmallerfactorofsafety9 thanSPTmodel)althoughtheresultsofthemodelsareclosetoeachother.10

Figure(5)_ ComparisonofVsandSPTbasedJDRVmodels

9.Conclusion11

Thispaperhaspresentedthedevelopmentofaprobabilisticmodelforliquefactionbased12 onsiteshearwavevelocityusingthe JDRVmethod.TheMonteCarlosimulationwasused for13 verifyingtheresultsofJDRVmethod.Theselectedstochasticparameterswerestress-corrected14 shear-wavevelocityandstressreduction factor,whichweremodeledusingtruncatednormal15 probabilitydensity functions and the earthquake acceleration ratio and earthquakemoment16 magnitude,whichwereconsideredtohavetruncatedexponentialprobabilitydensityfunctions.17 Theresultsshowedthattheprobabilitydistributionoftheliquefactionsafetyfactorobtainedby18 the JDRVmethodisverycloseto thatpredictedby theMonteCarlosimulation.Moreover, the19 resultsindicatedthattheJDRVmethodwasabletocapturetheexpectedprobabilitydistribution20 ofthesafetyfactorofliquefactioncorrectly.Comparisonoftheresultsoftheproposedmodeland21 theSPT-basedmodel,bothdevelopedusingJDRV,showsthattheVs-basedmodelprovidesamore22 conservativepredictionofliquefactionpotentialthantheSPT-basemodel.23

10.Appendix124

DerivationofmathematicalfunctionsK1toK7andFSandtheirsdomainsispresentedinthis25 Appendix:26

(36)27

( ) ( )é ù+= -F ê ú

ë û

500 ln 2.711

0.89LFS

P FS

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

FactorofSafety(FS)


Vsmodel(Eq.(49))

SPTmodel(JohariandKhodaparast,2013)(Eq.(54))

f f f´ ´1

ddKk

1 S1 S1

-1-1 1

K 1 V 1 V S11

S1

L (k ) 1(k )= (L (k )) = (V )d

d V

15

1

2

where:3

4

VS1mean:Averagevalueofstress-correctedshear-wavevelocity5 σVs1:Standarddeviationofstress-correctedshear-wavevelocity6 VS1min:Minimumvalueofstress-correctedshear-wavevelocity7 VS1max:Maximumvalueofstress-correctedshear-wavevelocity8

(37)9

10

11

12

rdmean:Averagevalueofstressreductionfactor13 σrd:Standarddeviationofstressreductionfactor14 rdmin:Minimumvalueofstressreductionfactor15 rdmax:Maximumvalueofstressreductionfactor16

(38)17

18

19

where:20

MWmin:Minimumvalueofmomentmagnitude21 MWmax:Maximumvalueofmomentmagnitude22 λMw:Rateofchangeinmomentmagnitude(rateparameter)=1/βMw23 βMw:Scaleparameterofmomentmagnitude24

max minS1 1 S1L(V )<k <L(V )

ìíî

1

1

min max

max min

S1

S1

k =L(V )

k =L(V )

S1

S1

meanmin

max mean

S1 S1 V

S1 S1 V

V =V -4σ 0V =V +4σ

>ìïíïî

f fæ öæ öæ ö ç ÷ç ÷ç ÷ ç ÷ç ÷ç ÷è ø è øè ø

mean

2 d

dd

2d 2

K 2 r 22 2 2 r 2r 2

1-r .k1 d 1 1(k )= ( ) = exp -0.5k dk k σ .kσ 2π.k

£ £ ® £ £min max

max min

d d d 2d d

1 1r r r kr r

ìïïíïïî

min

max

max

min

2d

2d

1k =r

1k =r

meanmin d

max mean d

d d r

d d r

r = r -4σ 0r = r +4σ

>ìïíïî

( )f f w w

w min w max

25-25 25 64- (- ) M M 364 64K 3 M 3 3 893 w ( )3 64

3 M w M w

25×λ .exp(-λ k )d(k )= k × (k ) =dk

64×k exp(-λ M )-exp(-λ M )

( )

£ £ ® £ £min max max min

-2.56 -2.56w w w w 3 wM M M (M ) k (M )

ìïíïî

min max

max min

-2.563 w

-2.563 w

k =(M )

k =(M )

16

(39)1

2

3

where:

4

αmin:Minimumvalueofearthquakeaccelerationratio5 αmax:Maximumvalueofearthquakeaccelerationratio6 λα:Rateofchangeinearthquakeaccelerationratio(rateparameter)=1/βα 7 βα:Scaleparameterofearthquakeaccelerationratio8

(40)9

10

and 11

(41)12

13

and

14

(42)15

16

and

17

AndthecumulativedistributionfunctionofK7canbedeterminedasbellow:18

(43)19

f fæ öç ÷è ø4

αα

4K 4 α 2

4 4 4 4 α min α max

-λλ .exp( )k1 d 1(k )= ( ) =

k dk k k .exp(-λ .α )-exp(-λ .α )

£ £4max min

1 1kα α

ìïïíïïî

min

max

4max

4min

1k =α1k =

α

f f f fòβ

5K 5 K ×K 5 1 K 1 K 15 1 2 1 2

α 1

k(k )= k = k (k ) ( )dk

k( )

£ £min min max max1 2 5 1 2k k k k k

ìíî

min min min

max max max

5 1 2

5 1 2

k =k kk =k k

ìïïíïïî

min

max

max

min

51

2

51

2

kα =max[k & ]kkβ=min[k & ]k

f f f fòK K ×K K K6 5 3 3 5

β6

6 6 3 3 3α 3

k(k )= k = k (k ) ( )dk

k( )


ìíî

min min min

max max max

6 5 3

6 5 3

k =k kk =k k

ìïïíïïî

min

max

max

min

63

5

63

5


f f f fòK K ×K K K7 6 4 4 6

β7

7 7 4 4 4α 4

k(k )= k = k (k ) ( )dkk

( )


ìíî

min min min

max max max

7 6 4

7 6 4

k =k kk =k k

ìïïíïïî

min

max

max

min

74

6

74

6


{ }Î = ò[ , ]7

min7min

K K7 7

k

7 7 7 7k

(k )= K k k (t)dtPF f

17

1

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an analytical approach to probabilistic modeling of

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