Directional methods for structural reliability analysis

www.elsevier.nl/locate/strusafe

Directionalmethodsforstructuralreliabilityanalysis

JinsuoNie,BruceR.Ellingwood*

DepartmentofCivilEngineering,TheJohnsHopkinsUniversity,Baltimore,MD21218,USA

Abstract

Directionalsimulationreducesthedimensionofthelimitstateprobabilityintegralbyidentifyingasetofdirectionsforintegration,integratingeitherinclosed-formorbyapproximationinthosedirections,andestimatingtheprobabilityasaweightedaverageofthedirectionalintegrals.Mostexistingmethodsidentifythesedirectionsbyasetofpointsdistributedontheunithypersphere.Theaccuracyofthedirectionalsimulationdependsonhowthepointsareidenti®ed.Whenthelimitstateishighlynonlinear,ortheinherentfailureprobabilityissmall,averylargenumberofpointsmayberequired,andthemethodcanbecomeine󰂁cient.ThispaperintroducesseveralnewapproachesforidentifyingdirectionsforevaluatingtheprobabilityintegralÐSphericalt-design,SpiralPoints,andFeketePointsÐandcomparesthefailureprobabilitieswiththosedeterminedinanumberofexamplesinpreviouslypublishedwork.Oncethesepointshavebeenidenti®edforaprobabilityintegralofgivendimension,theycanbeusedrepeatedlyforotherprobabilityintegralsofthesamedimensioninafashionanalogoustoGaussQuadrature.#2000ElsevierScienceLtd.Allrightsreserved.

Keywords:Directionalsampling;Engineeringmechanics;Limitstates;MonteCarlosimulation;Probability;Relia-bility;Statistics;Structuralengineering

1.Introduction

Practicalstructuralreliabilityanalysesoftenrequiretheevaluationofthefailureprobabilityforlimitstatesinvolvingavector,X,offrom5to20randomvariablesdescribedbyajointprob-abilitydensityfunctionfX󰂅x󰂆.GiventhelimitstatefunctionG󰂅x󰂆󰂈0,de®nedsuchthatsafedomain󰀊s󰂈fxjG󰂅x󰂆b0gandfailuredomain󰀊f󰂈fxjG󰂅x󰂆`0g,thefailureprobabilityisgivenby

󰂅Pf󰂈fX󰂅x󰂆dx󰂅I󰂆

󰀊f

Ingeneral,thisintegralishardtoevaluate,particularlyinhigh-dimensionalspace.Acommonapproachis®rsttotransformtherandomvectorX󰂈󰂉X1YX2YFFFYXd󰂊󰁔toanindependent

*Correspondingauthor.

0167-4730/00/$-seefrontmatter#2000ElsevierScienceLtd.Allrightsreserved.PII:S0167-4730(00)00014-X

234J.Nie,B.R.Ellingwood/StructuralSafety22(2000)233±249

standardnormalrandomvectorU󰂈󰂉U1YU2YFFFYUd󰂊󰁔bytheRosenblatttransformationU󰂈T󰂅X󰂆,andthenapply®rst-orderorsecond-orderreliabilitymethods(FORM,SORM),MonteCarlosimulation,orothermethods.WiththeRosenblatttransformation,thelimitstatefunctioninthespaceofindependentstandardnormalvariablesmaybecomeahighlynonlinearfunction.Forhighlynonlinearstructuralcomponentandsystemlimitstates,FORMandSORMmaynotbesu󰂁cientlyaccurate[1].

DirectionalsimulationandimportancedirectionalsimulationhavebeenstudiedbyDitlevsen,etal.[2],andMelchersetal.[3,4],amongothers.Thedirectionalsimulationmethodinvolvesgeneratinguniformlydistributeddirectionvectorsandperformingaone-dimensionalintegrationalongeachdirection.Theimportancedirectionalsimulationmethodusestheimportancesam-plingtechniquetoconcentratethedirectionvectorsintheregionsofinterest.Althoughdirec-tionalsimulationmethodsarerelativelye󰂁cientcomparedtootherMonteCarlosimulationapproaches,thesemethodsmaydiminishinaccuracywhenthelimitstateG󰂅u󰂆ishighlynonlinearunlessthenumberofsamplingdirectionsislarge.Moreover,forasystemreliabilityanalysisthatissupportedby®niteelementmodeling,thenumberofdirectionsrequiredtolimittheerrorinPfmustbeheldtoaminimumfortheanalysistobeperformede󰂁ciently.

k[6],KatsukiandFrangopol[7]DevelopinganideaproposedbyKendall[5]andDea

approximatedtheactuallimitstatesurfaceintheindependentstandardnormalspacebyaseriesofhypersphericalsegments,eachhavingaradiusdescribedbya12distribution.Thefailureprobabilitywasthenapproximatedbythesumofthefailureprobabilitiesassociatedwiththosesegmentsapproximatingthelimitstatehypersurface.ThisHyperspaceDivisionMethod(HDM)canachievefairaccuracyinapproximatingthefailureprobabilityforbothstructuralcomponentsandsystems,thelatterofwhichmayhavehighlynonlinearlimitstateswithmultiplelocalextrema.Improvementstoenhancetheconvergenceande󰂁ciencyoftheHDMforrealisticsys-temswerediscussedinamorerecentpaperbythesameauthors[8].

ThekeyideaindirectionalsimulationandtheHDMisthesame:(a)toseekasetofpoints(asdirections)uniformlydistributedontheunithypersphereeitherbysimulationorbyconstruction(inwhichallpointsareobtainedfromsomespeci®edformula),and(b)toperformthereliabilityanalysisasasequenceofone-dimensionalintegrationsinthedirectionsthusidenti®ed.Thisideaisdevelopedfurtherinthispaper,inwhichadirectionalmethodisproposedthatyieldsaccuratefailureprobabilitiesofcomponentsandsystemsdescribedbylimitstatesinvolvingnonlinearitiesandlocalextrema.Oncesuchpoints(directions)aredetermined,theycanbeusedrepeatedlyfordirectionalintegrationinananalogywithGaussianQuadrature,ratherthanhavingtoberegen-eratedforeachreliabilityanalysis.Numericalexamplesillustratethee󰂁ciencyofthemethodincomparisonwithothermethods.2.Fundamentalprocedure

Alldirectionalmethodsrequireidenti®cationofdirectionsalongwhichtheintegrationisper-formedinclosedform,bysimulationornumericalmeans.Intheindependentstandardnormalspace,theintegralalongeachdirectionisobtainedexactlybyutilizingthe12distribution.Tosummarize,givenahyperspace󰀊withdindependentstandardnormalvariablesU󰂈󰂉U1YU2YFFFYUd󰂊󰁔,thenewrandomvariableZ2,de®nedby

J.Nie,B.R.Ellingwood/StructuralSafety22(2000)233±249

22

Z2󰂈U21󰂇U2󰂇ÁÁÁ󰂇Ud

235

󰂅P󰂆

isachi-squarerandomvariablewithddegreesoffreedom.Ifthelimitstatefunctionisahyper-sphereofradiusRinthehyperspace󰀊,then

22222Gd󰂅u󰂆󰂈Àu21Àu2ÀÁÁÁÀud󰂇R󰂈Àz󰂇R󰂈0

󰂅Q󰂆

andthefailureprobabilityassociatedwiththishyperspherecanbeobtainedexactly[9]:

2Pf󰂈P󰂉Gd󰂅u󰂆40󰂊󰂈1À12d󰂅R󰂆

󰂅R󰂆

Fig.1showsatwo-dimensionalillustrationof󰀊withthelimitstatesimpli®edtoacircular

22

functionG2󰂅u󰂆󰂈Àu21Àu2󰂇R󰂈0XSubdomains󰀊fiand󰀊siareradiallysplitfrom󰀊fand󰀊srespectively.Letthearclengthonthelimitstateassociatedwith󰀊fibeAfi,andthetotallengthofthecircle(areainhigherdimensions)beA,whereAisthesurfaceareaofahypersphereind-dimensionspace.Aisgivenby

V

%da2dÀ1bbbifdiseven`d󰂅da2󰂆3r

A󰂈󰂅S󰂆

bd󰂅dÀ1󰂆a2b󰂅󰂅dÀ1󰂆a2󰂆3dÀ1bXd2%rifdisodd

d3Thenthefailureprobabilityassociatedwith󰀊fiis

2

Pfi󰂈P󰂉U&󰀊fi󰂊󰂈󰂉1À12d󰂅R󰂆󰂊AfiaA

󰂅T󰂆

AlthoughtheweightAfiaAallowsonetodividethehypersphereunevenly(itleadstoanadaptivedivisionscheme),itisdi󰂁cultto®ndAfiwhend54.Ifthehypersphereisradiallydividedevenly,thecontributionofeachsubdomain,Pfi,becomessimply[7]

22

Fig.1.Failuredomainandsegmentsforlimitstatesu21󰂇u2󰂈R(after[7]).

236J.Nie,B.R.Ellingwood/StructuralSafety22(2000)233±249

2

Pfi󰂈󰂉1À12d󰂅R󰂆󰂊am

󰂅U󰁡󰂆

wheremisthetotalnumberofsubdomains.Notethatmmayhavevariousinterpretationsindi󰂀erentpoint-generatingmethods,suchasthenumberofsamplingdirectionsindirectionalsimulation,thenumberofsubdomainsintheHDM,andthenumberofpointsintheSphericalt-design,SpiralpointsandFeketepointsmethodstobedescribedsubsequently.Inthemoreusualcasewherethelimitstateisahypersurfaceratherthanahypersphere(Fig.2),thehypersurfaceisapproximatedbyaseriesofhypersphericalsegments,eachwithitscentralpointQilyingontheactuallimitstate.Thefailureprobabilityofanysubdomain󰀊fiisapproximatedbyEq.(7b),

2

Pfi󰂈󰂉1À12d󰂅Ri󰂆󰂊am

󰂅U󰁢󰂆

whereRi󰂈radiusofthehyperspheresegmentofsubdomaini,Ri50.

ThetotalfailureprobabilityPfisthesummationoverallsubdomains,i.e.

Pf%

m󰁘i󰂈1

Pfi󰂈

m󰁘i󰂈1

2

󰂉1À12d󰂅Ri󰂆󰂊am

󰂅V󰂆

TheRi'scanbeobtainedthroughnumericalmethods(e.g.[10]).Thisapproacheliminatesthe

limitationsofFORMandSORMthatoccurwhennonlinearitiesinGd󰂅u󰂆ormultiplelocalextremumpointsoftheprobabilitydensityfunctionexistonthelimitstatehypersurface,becauseitapproximatesthefailuresurfaceoverawiderdomainofxthandoFORM/SORM.System

󰀐󰀑

Fig.2.LimitstateGu󰂈0anditsapproximationwithsphericalsegments(after[7]).

\"

J.Nie,B.R.Ellingwood/StructuralSafety22(2000)233±249237

reliabilityproblemsmaybesolvedinthesamewayascomponentproblemsexceptthatoneneedsaspecialwaytocalculatetheRi's.Givenklimitstatefunctionsdescribingcomponent(ormodal)failures,

@Ri󰂈

min󰂅Rik󰂆forseriessystemsm󰁡x󰂅Rik󰂆

kk

forp󰁡r󰁡llelsystems

󰂅W󰂆

Forsystemsthatarenotmodeledaseitherseriesorparallelsystems,onemightutilizetheresponsesurfacedeterminedfroma®niteelementanalysisasthelimitstatesurface.

Eq.(8)istantamounttoanequallyweightedaverageofprobabilitiesevaluatedinmdirections.Itsaccuracyiscontingentongettingevenlydistributedpointsonaunithypersphere.In2-dimensionalspace,thisistrivial;indimensionsof3andhigher,itisincreasinglydi󰂁cult.Ourinterestliesmainlyind53.

3.Implementationofdirectionalmethods3.1.Directionalsimulation

k[6]wasamongthe®rstwhostudiedthedirectionalsimulationmethodasatoolforDea

evaluatingmultidimensionalnormalprobabilityintegrals.Itisaverye󰂁cientmethodofMonteCarlosimulation,providedthatradiusvectorstothelimitstatesurfaceinanydirectioncanbeobtainede󰂁ciently[2].ThefailureprobabilityformulationissimilarasinEq.(8)exceptthatmnowisthesamplesize.Directionalsimulationmixessimulationin(dÀ1)dimensionswithnumericalintegrationinonedimension.Togetacceptable(withrespecttoevenness)directionalsamplesinthismethod,areasonablelargenumberofpointsontheunithypersphere(direc-tions)areneeded.Thisentailsextracostforcomputingradii,whichbecomescriticalwhenthelimitstatefunctionishighlynonlinearorthereliabilityanalysisinvolvesalargenumberofvariables.Indirectionalsimulation,asetofNpointsP󰂈fP1YP2YFFFYPNguniformlydistributedontheunithyperspherede®nethedirections.Twoapproachestogeneratingthesepointsarecommon.The®rstistogenerateNvectorsoftheform󰂉u1Yu2YFFFYud󰂊󰁔,wheredisthedimensionofthespace,andui'sarerealizationsofavectorofindependentstandardnormalrandomvariables,eachofwhichhasbeennormalizedtounitlength.ThesecondapproachistogenerateNvectorsoftheformx󰂈󰂉x1Yx2YFFFYxd󰂊󰁔bytherejectionmethod,wherexi'sareindependentsamplesfromaone-dimensionaluniformdistribution.Avectorisretainedifjxj41X0;otherwise,anotherisgenerated.Finally,allvectorsarenormalized.Thedirectionalsimulationexamplespresentedsubsequentlyforcomparisonstoothermethodsstudiedwerepreparedusingtherejectionmethod.3.2.Otherdirectionalmethods

Inrecentyears,anumberofalternatemethodstogenerate``evenlydistributed''points(direc-tions)ontheunithyperspherehavebeendevelopedinother®elds.Someofthesemayprovetobeusefulinstructuralreliabilityapplications,andaredescribedbelow.

238J.Nie,B.R.Ellingwood/StructuralSafety22(2000)233±249

3.2.1.Sphericalt-design

Asphericalt-designisde®nedasfollows,usingthenotationofHardinandSloane[11],De®nition.

AsetofNpointsP󰂈fP1YP2YFFFYPNgontheunithypersphereÈÉ

󰀊d󰂈SdÀ1󰂈x󰂈󰂉x1Yx2YFFFYxd󰂊PRdXxÁx󰂈1formsasphericalt-designiftheidentity

󰂅N

1󰁘

f󰂅x󰂆d\"󰂅x󰂆󰂈f󰂅Pi󰂆

N󰀊di󰂈1

󰂅IH󰂆

󰂅II󰂆

(where\"isaLebesguemeasureon󰀊dnormalizedtohavetotalmeasure1)holdsforallpoly-nomialsfofdegree4t.

Inotherwords,theintegralofapolynomialfunctionoverthehypersphere󰀊dcanbeapproximatedbyitsaveragevalueatthepointsP.IfPformsasphericalt-design,Eq.(11)isexactforanypolynomialsofdegree4t.Delsarteetal.[12]haveshownthat,givendandt,thesmallestnumberofpoints,M4N,ofasphericalt-designisobtainedfrom,

tÀ1

󰂇dÀ1M󰂈22dÀ1

2M󰂈

t

󰂇dÀ12dÀ1

3󰂇

2

3

iftisodd

2

t

󰂇dÀ22dÀ1

3

iftiseven

󰂅IP󰁢󰂆󰂅IP󰁡󰂆

inwhich󰂅󰂆isthebinomialcoe󰂁cient.Asphericalt-designwithMpointsissaidtobetight.Veryfewtightt-designsexist;however,Eq.(12)servesasabenchmarklaterinthispaperagainstwhichtocomparethenumberofpointsrequiredbyothermethods.Table1presentsthenumberofpoints,M,vsdandt.HardinandSloane[11]havedevelopedaseriesofsphericalt-designsuptot󰂈21inthree-dimensionalspaceusingapatternsearchalgorithm,butsofarhavebeenunabletoprovidet-designsinhyperspaceofdegree54.Someexamplesinthree-dimensionalspacewillbepresentedlaterinthispaper.Aswillbeseenintheseexamples,the240-pointspherical21-designleadstoaveryaccurateande󰂁cientestimateofPf.

3.2.2.Constructionmethods

Aconstructionmethodisoneinwhichallthe``evenly''distributedpoints(directions)aredescribedexplicitlybysomeformula.Thesemethods,ifavailable,arethemoste󰂁cientwaytogetthepoints.Twomethodsareintroducedbelow.

.Spiralpointsmethod

Rakhmanovetal.[13]introducedthespiralpointproceduretoconstructalargenumber,

J.Nie,B.R.Ellingwood/StructuralSafety22(2000)233±249239

N,ofpointsonasphereS2.Thismethodcanonlybeusedin3-dimensionalproblems.Forsphericalcoordinates󰂅󰀒Y0󰂆,04󰀒4%,04042%,󰀒k󰂈󰁡r󰁣os󰂅hk󰂆Yhk󰂈À1󰂇

H

2󰂅kÀ1󰂆

Y14k4N

NÀ1I

󰂅IQ󰂆

3X61gf

0k󰂈d0kÀ1󰂇p󰂁󰂁󰂁󰂁q󰂁󰂁󰂁󰂁󰂁󰂁󰂁󰂁󰂁󰂁󰂁󰂁󰂁e󰂅mod2%󰂆Y24k4NÀ1Y01󰂈0N󰂈0

N1Àh2

k

wheretheparameter3.6isbasedonHabichtandvanderWaerden's[14]bestpackingsuggestionandnumericalexperiments[13].

.Hyperspacedivisionmethod(HDM)

TheHDMprocedure[7]willbeexplainedfora3-dspaceforsimplicity.Itcanbeextendedtohyperspaces.Theunitspherex2󰂇y2󰂇z2󰂈1canbeexpressedinpolarcoordinates,byasystemofequa-tionswithparameters󰀒and9;

V

`x󰂈󰁣os9󰁣os󰀒y󰂈sin9󰁣os󰀒󰂅IR󰂆X

z󰂈sin󰀒

Table1

ThenumberofpointsMinatightt-designvsdimensiond[Eq.(12)]

Degreet

Dimensiond2345671011121314151617181920

t=2555105182294450660935128717292275294037404692581471258510,395

t=21221325722002600616,01638,687,516184,756369,512705,4321,293,2922,288,1323,922,5126,537,52010,623,47016,872,57026,246,22040,060,020

t=323321785872135,853128,877415,7011,225,7853,350,4798,580,49520,7,05547,805,615105,306,075222,981,435455,657,715300,329,863566,092,3601036,974,8725,915,6,470

240J.Nie,B.R.Ellingwood/StructuralSafety22(2000)233±249

where04942%,À%a24󰀒4%a2.Todeterminethepoints,®rst󰂉À%a2Y%a2󰂊isequallydividedintom󰀒À1intervals,inwhichm󰀒isspeci®edinadvance.Eachoftheseintervalsis%a󰂅m󰀒À1󰂆.Thisgivesaseriesoflatitudecircles,eachofwhichisde®nedbyafunction

x2󰂇y2󰂈1Àz2󰂈󰁣os2󰀒iwhere󰀒iisgivenby

󰀒i󰂈À%a2󰂇%󰂅iÀ1󰂆a󰂅m󰀒À1󰂆Yi󰂈1Y2YFFFYm󰀒

󰂅IT󰂆󰂅IS󰂆

Second,thecircleiisdividedequallyintom9iarcs,inwhichm9iisgivenbyanintegerfunctionINT(),i.e.

m9i󰂈INT󰂅2󰂅m󰀒À1󰂆󰁣os󰀒i󰂆

󰂅IU󰂆

Consequently,them9ipointsarealmostevenlydistributedonthelatitudecirclei.Theanglesofthesepointsare

9ij󰂈2%am9iÂjY

j󰂈1Y2YFFFYm9i

󰂅IV󰂆

andthetotalnumberofpointsare

m󰂈

m󰀒󰁘i󰂈1

m9i󰂅IW󰂆

Finally,therectangularCartesiancoordinatesofeachpointarecalculatedbyEq.(14).Gen-eratingthepointsisslowbecauseofthelargenumberofcosineoperations.Recentwork[8],has

enhancedthee󰂁ciencyoftheHDM.

3.2.3.PatternsearchforFeketePoints

Anumericalmethodcanbederivedtogeneratepointsevenlydistributedontheunitspherefromminimizingthepotentialenergy(PE)inasetofpointswithforcesofmutualrepulsion.SuchpointsaredenotedasFeketePointsin3-dspace[15];theyaregeneratedconceptuallyforhigherdimensionalspacesinthispaper.

De®nition.

ÃÃ

FeketePointsP󰂈fPÃ1YP2YFFFYPNgontheunitspherearepointsthatminimizeE󰂅1YP󰂆󰂈

󰁘󰀌󰀌

󰀌PjÀPk󰀌À1

14j`k4N

󰂅PH󰂆

inwhichPi󰂈󰂉xi1Yxi2Yxi3󰂊󰁔[15].Eq.(20)describesphysicallythepotentialenergyofNparticlesontheunitspherewithunitchargesthatrepeleachotheraccordingto󰀌Coulomb's󰀌Àslaw.Although󰁐

therearemoregeneralformsfortheenergy,e.g.E󰂅sYP󰂆󰂈14j`k4N󰀌PjÀPk󰀌,experiencehas

J.Nie,B.R.Ellingwood/StructuralSafety22(2000)233±249241

shownthattheiruseslowsdowntheconvergencetominimumPEormayleadtoconvergencetoalocalminimum.Therefore,thisapproachwillbebasedonCoulomb'slaw.ItisassumedtoremainvalidwhenEq.(20)isgeneralizedtodimensionshigherthan3.

WebeginbygeneratingNpointsontheunithyperspherebyanyappropriatemethod,asdescribedpreviously,assigningaunitchargetoeachpoint,andcalculatingtheinitialpotentialenergyofthesystem.Second,usingageneralizedpointrepulsionmethod(e.g.Leech'salgorithm,19961),inwhichallthepointsareconsideredtorepeleachotheraccordingtoa1ar2forcelaw(Coulomb'slaw),theforcesbetweenthepointsarecomputed.ThetangentcomponentFtiofthetotalforceFiactingoneachpointiisusedtodetermineapattern,whichde®nesthemovingdirectionandrelativemagnitudeofpointi;themaximumtangentforceFtm󰁡xamongthesetan-gentcomponentsoftheforcesactingonallpointsisrecorded.Third,setaninitialmaximummovingstep󰀎u󰂈󰂅AaN󰂆1a󰂅dÀ1󰂆a2,bywhichthepointwiththelargesttangentforcewillmove,and

Fti

󰀎u.TestthePEtoseeifitissearchforalowerPEbylettingeachpointimoveastepof

jFtm󰁡xjreducedbythisstep.Ifnot,reduce󰀎ubyhalf,andrepeatthisstepuntillalowerPEisobtained.Finally,whenthedi󰂀erenceofPEnÀ1andPEnbetweentwosteps(nÀ1)andnfallsbelowsomespeci®edtolerance(say,10À8PEn),thentheprocedurestopsandyieldsanapproximatesetofFeketepoints.AlthoughgeneratingFeketepoints(44d420)istime-consuming,oncetheyaregeneratedandstoredinthecomputer,theycanbeusedrepeatedlyforreliabilityanalyses.Examplesinspaceswithdimensionsfrom3to7arepresentedinthesequel.

4.Numericalexamples4.1.Linearlimitstatefunctions

Thefollowinglinearlimitstatefunctionswiththreeto®vevariables:

d󰁘p󰂁󰂁󰂁

gd󰂅x󰂆󰂈Àxi󰂇3dY󰂅34d45󰂆

i󰂈1

󰂅PI󰂆

wherex󰂈󰂉x1Yx2YFFFYxd󰂊󰁔,isavectorofindependentGaussianrandomvariables,wereevaluatedpreviouslybyKatsukiandFrangopol[7],andareanalyzedinordertocomparetheproposedmethodswithpreviousresults.ItisclearthattheFORMwill®ndtheexactsolutionssincethelimitstatefunctionsarelinear.Indeed,theprobabilityoffailureisPf󰂈1X34997Â10À3foreachoftheselimitstatefunctions.

Table2(a)showsthatforthethree-variablelinearlimitstate,theuseoft-designpointsorFeketepointsyieldsthemoste󰂁cient(withrespecttothecosttocomputeradiiandPfi)andaccuratesolutions.TheSpiralPointsMethodismoreaccuratethantheHDMforthesamenumberofpoints,fromwhichonecaninferthattheSpiralPointsMethoddistributespointsmore

Sourcecodeisavailableatftp://ftp.cs.unc.edu/pub/users/leech/points.tar.gz.Thecodedealswiththree-dimen-sionalspaceonly.

1242J.Nie,B.R.Ellingwood/StructuralSafety22(2000)233±249

Table2

Failureprobabilitiesforlinearlimitstates[Eq.(21)]computedbydi󰂀erentmethods

Methodtogeneratethepoints

a.g3󰂅X󰂆forthreevariablesHDMSpiralPointsSphericalt-design

t=8t=10t=11t=13t=14t=16t=21

32012923201292366072961081442403660729610814424030021701,303,124

1002403004000604229,79432048060408009602080

1.342751.351541.345601.348861.3861.346031.352081.3431.347571.349401.349991.357951.346981.354761.350301.349051.350341.349511.349791.352691.351801.347531.351651.350351.350001.350001.267671.361021.347091.353591.348011.3501.350971.353681.34866

À0.5350.116À0.324À0.101+1.103À0.292+0.156À0.262À0.178À0.042+0.0015+0.591À0.221+0.355+0.024À0.068+0.027À0.034À0.013+0.201+0.136À0.181+0.124+0.028+0.002+0.002À6.096+0.819À0.213+0.268À0.145+0.050+0.074+0.275À0.097

NumberofpointsN

Computedprobabilityoffailure(Â10À3)

ErrorwithrespecttotheexactPf(%)

FeketePoints

b.g4󰂅X󰂆forfourvariablesHDMFeketePoints

c.g5󰂅X󰂆for®vevariablesHDMFeketePoints

evenlythandoesHDM.Thesameconclusionscanbedrawnfromtheresultspresentedforthe4-and5-variablesproblemsinTables2(b)and(c);notethattheSphericalt-designandSpiralPointsmethodsareavailableonlyinthree-dimensionalspace.For6-and7-variablesproblems,thee󰂁-ciencyandaccuracyoftheFeketePointsmethodisshowninTable3.Notethatthemaximum

J.Nie,B.R.Ellingwood/StructuralSafety22(2000)233±249

Table3

FailureprobabilitiesforG6󰂅X󰂆andG7󰂅X󰂆computedbyFeketepointsmethodDimensionsofthespaced=6

NumberofpointsN012805120608012802560512020,000

Computedprobabilityoffailure(Â10À3)1.362591.353451.351911.350161.343921.336271.351371.34887

243

ErrorwithrespecttotheexactPf(%)+0.935+0.258+0.144+0.014À0.448À1.015+0.104À0.081

d=7

numberofpointsusedwhend󰂈6or7(6080or20,000)isonlyslightlylargerthantheminimumnumberrequiredbyEq.(12)foratight21-design(6006or16016inTable1).ThisnumberistheminimumnecessaryfortheapproximationofEq.(1)byEq.(8)tobeexactwhentheintegrandinEq.(1)isdescribedbyapolynomialofdegree21orless.4.2.Seriessystemoflinearlimitstatefunctions

Consideraseriessysteminwhichthefailureregionisboundedbythefollowingtwolimitstatefunctions:

p󰂁󰂁󰂁

󰂅PP󰁡󰂆gs1󰂈Àx1Àx2Àx3󰂇33

gs2󰂈Àx3󰂇3X0

󰂅PP󰁢󰂆

󰁓

Thefailureregionisspeci®edby󰂅gs1`0󰂆󰂅gs2`0󰂆,asshowninFig.3.Thesecond-orderbounds[16]onthefailureprobabilityofthisseriessystemare2.53734Â10À34Pf,series42.618Â10À3[7].The``exact''solution2.561Â10À3wasobtainedbydirectionalsimulationusing10,000directions.Thesamplingerroronthisestimateisapproximately7.58Â10À5.Table4showsthatmostofthefailureprobabilitiescalculatedfromthedi󰂀erentmethodsliewithinthebounds,excepttheSphericalt-designwhenN󰂈36or60.TheyallareconsistentwiththeresultfromtheMonteCarlosimulation.

4.3.Parallelsystemoflinearlimitstatefunctions

󰁔

Eqs.(22)nowareassumedtoboundthefailureregion󰂅gs1`0󰂆󰂅gs2`0󰂆(seeFig.3)ofaparallelsystem.Thesecond-orderboundsare8.12977Â10À54Pf41.62595Â10À4[6];thefailureprobabilitycalculatedbyHohenbichler'sapproximation[17]formultinormalintegrals,reportedinKatsukiandFrangopol[7],is1.24211Â10À4.The``exact''solution,1.54156Â10À4,wasobtainedfromdirectionalsimulationusing10,000directions;thesamplingerroronthisestimate

244J.Nie,B.R.Ellingwood/StructuralSafety22(2000)233±249

Fig.3.Failuredomainsforseriesandparallelsystems.

Table4

Failureprobabilitiesforseriessystem[Eq.(22)]MethodtogeneratethepointsHDM

NumberofpointsN1292518220,8001292518220,800

t=8t=10t=11t=13t=14t=16t=21

3660729610814424036607296108144240300

Computedprobabilityoffailure(Â10À3)2.566132.566252.570872.585022.577922.576202.653062.619882.551162.570212.567432.565442.573982.571772.610922.570392.584222.554612.580102.575302.57215

SpiralPoints

Sphericalt-design

FeketePoints

J.Nie,B.R.Ellingwood/StructuralSafety22(2000)233±249245

is6.20Â10À6.UsingthesamepointsetsasintheseriessystemyieldstheresultsshowninTable5.ItshouldbenotedthatwhentheFeketepointsortheSphericalt-designmethodsareappliedtothisparallelsystemproblem(withaconvexfailureregion,seeFig.3)withthesamepointsetsasintheseriessystem(Table4),toofewpointsareobtainedtodescribethenonlinearlimitstateadequately.However,whenNisincreasedto1200(stillrelativelysmall,cf.Table5),theresultisquiteaccurate.TheSpiralPointsmethodappearstobehighlyaccurate,andwasfoundtogen-eratethepointse󰂁cientlyforbothseriesandparallelsystemsde®nedbypiecewiselinearfunc-tions.

4.4.NonlinearlimitstatefunctionThelimitstatefunctiong󰁣on󰁣󰁡ve󰂈

À0X5󰂅x21

󰂇x22

󰂇x23

p󰂁󰂁󰂁

À2x1x2À2x2x3À2x3x1󰂆À󰂅x1󰂇x2󰂇x3󰂆a3󰂇3X0

󰂅PQ󰂆

hasafailuredomainthatisconcavewithrespecttotheorigin.The``exact''solution0.1979769wasobtainedbydirectionalsimulationusing10,000directions.Thesamplingerroronthis

Table5

Failureprobabilitiesforparallelsystem[Eq.(22)]MethodtogeneratethepointsHDM

NumberofpointsN1292518220,8001292518220,800

t=8t=10t=11t=13t=14t=16t=21

36607296108144240366072961081442403001200

Computedprobabilityoffailure(Â10À4)1.245521.244281.243081.477371.512061.551540.821691.266871.600221.454911.602331.660621.654861.553901.234571.4911.324221.475091.371521.559291.526211.55563

SpiralPoints

Sphericalt-design

FeketePoints

246J.Nie,B.R.Ellingwood/StructuralSafety22(2000)233±249

estimateis1.56Â10À3.Table6(a)showsthattheSphericalt-designandFeketepointsmethodsrequirefewerpointsthantheSpiralPointsmethodforcomparableaccuracy.Howeverallthreemethodsyieldsatisfactoryresultsfortheconcavefailuredomain.Ontheotherhand,ifthelimitstatefunctionhasafailureregionthatisconvex,

Table6

Failureprobabilities

Methodtogeneratethepoints

a.Concavelimitstatefunction[Eq.(23)]SpiralPoints

NumberofpointsN10824012925182

t=8t=10t=11t=13t=14t=16t=21

36607296108144240366072961081442403001082401292518236607296108144240366072961081442403001200

Computedprobabilityoffailure0.197300.197760.197980.198010.199390.197770.200550.198000.197270.198210.198010.198340.197030.198020.198010.197880.197930.198010.197921.971951.935281.938671.940072.2571.865941.813911.955931.931442.0471.929472.195941.942282.075871.2401.949161.907331.925101.954121.94081

Sphericalt-design

FeketePoints

b.Forconvexlimitstatefunction[Eq.(24)]SpiralPoints

Sphericalt-design

t=8t=10t=11t=13t=14t=16t=21

FeketePoints

J.Nie,B.R.Ellingwood/StructuralSafety22(2000)233±249247

p󰂁󰂁󰂁22

g󰁣onvex󰂈0X5󰂅x2󰂇x󰂇xÀ2xxÀ2xxÀ2xx󰂆À󰂅x󰂇x󰂇x󰂆a3󰂇3X0122331123123

󰂅PR󰂆

theSphericalt-designandFeketePointsmethodsneedmorepointstodescribethelimitstate,asinthepreviousparallelsystemexample.The``exact''solution1.93043Â10À2wasobtainedbydirectionalsimulationusing10,000directions.Thesamplingerroronthisestimateis8.04Â10À4.Table6(b)showsthattheFeketePointsmethod(with1200points)yieldsapproximatelythesameaccuracyastheSpiralpointsmethodwith5182points.Inbothcases,theresultsareclosetothe``exact''solution.4.5.Rigid-plasticframe

Ditlevsenetal.[2]examinedtherigid-plasticframestructure(illustratedinFig.4)bythedirectionalimportancesimulationmethod.Thisstructurecanbeanalyzedasaseriessystemofthreelinearlimitstatefunctions(collapsemechanisms),which,accordingtotheprincipleofvir-tualwork,arede®nedasfollows:

fe󰁡mX󰁓w󰁡yX

gom󰁢inedX

g󰁢e󰁡m󰂈X2󰂇2X3󰂇X4ÀGbgsw󰁡y󰂈X1󰂇X2󰂇X4󰂇X5ÀFa

󰂅PS󰂆󰂅PT󰂆󰂅PU󰂆

g󰁣om󰁢ined󰂈X1󰂇2X3󰂇2X4󰂇X5ÀFaÀGb

TheyieldmomentsXjYj󰂈1YFFFY5,atthehingepointsinFig.4,areindependentandidenti-callydistributedlognormalrandomvariables,withmean\"󰂈1andcoe󰂁cientofvariation󰀍󰂈0X25.ThelateralforceF,verticalforceGandthedistancesaandbareassumedconstant,withGb󰂈1X15andFa󰂈2X40.

Fig.4.Portalframemodeledasrigid±plasticsystem.

248J.Nie,B.R.Ellingwood/StructuralSafety22(2000)233±249

Table7

Failureprobabilitiesfortherigid±plasticframestructure[Eqs.28)±(30)]MethodtogeneratethepointsFeketePoints

NumberofpointsN48060408009602080

Computedprobabilityoffailure(Â10À5)5.427185.282535.382035.563525.494565.45102

Letlogarithmicstandarddeviation$󰂈ln󰂅1󰂇󰀍2󰂆󰂈0X2462andlogarithmicmeanl󰂈ln

lnXjÀl2

\"À1$󰂈À0X03031.WiththetransformationU󰂈,Ujareindependentstandardnor-j2$malvariables.Afterthetransformation,thelimitstatefunctionsbecomehighlynonlinearforms,

fe󰁡mX󰁓w󰁡yX

gom󰁢inedX

g󰁢e󰁡m󰂈e$u2󰂇l󰂇2e$u3󰂇l󰂇e$u4󰂇lÀ1X15gsw󰁡y󰂈e$u1󰂇l󰂇e$u2󰂇l󰂇e$u4󰂇l󰂇e$u5󰂇lÀ2X40

󰂅PV󰂆󰂅PW󰂆󰂅QH󰂆

g󰁣om󰁢ined󰂈e$u1󰂇l󰂇2e$u3󰂇l󰂇2e$u4󰂇l󰂇e$u5󰂇lÀ3X55

Á󰁓󰁓À

Thefailuredomainisthende®nedby󰂅g󰁢e󰁡m`0󰂆gsw󰁡y`0󰂅g󰁣om󰁢ined`0󰂆.The``exact''solution5.45191Â10À5wasobtainedbydirectionalsimulationusingÂ10À6.The®rst-orderboundsforthesystemfailureprobabilityare[3.12,5.68]Â10À5;thesecond-orderboundsreport-edlywerecoincidentat5.20Â10À5[16].Table7showsthattheFeketePointsmethodyieldsaveryaccuratesolutioncomparedtothe``exact''solutionwhenthenumberofpoints5960.5.Conclusion

Methodsthatapproximatethelimitstatesurfacebyaseriesofsphericalsegmentscanprovideaccurateestimatesoffailureprobabilitiesofcomponentsorsystems.Suchmethodscandealwiththeproblemsinvolvinghighnonlinearities,multipleextremaoftheprobabilitydensityalongthelimitstatefunction,andmultiplelimitstates.However,theiraccuracydependsontheabilitytogeneratee󰂁cientlyasetofdirectionsalongwhichtheprobabilityincrementsinEq.(8)areesti-mated.Thispaperhaspresentedsomeprocedurestogeneratethesepoints``evenlydistributed''ontheunithypersphere.Oncethepointshavebeendetermined,theycanbeusedrepeatedlyinthenumericalintegrationsinreliabilityanalysisinamannersomewhatanalogoustoGaussQuadrature.

J.Nie,B.R.Ellingwood/StructuralSafety22(2000)233±249249

ThreefactorsmaketheFeketePointsmethodattractiveforthisparticularmethodofreliabilityanalysis.First,advancesincomputationhavemadethecomputationsnecessarytoidentifythepointspossible.Second,storageofpoints,onceidenti®ed,isinexpensive.Third,manypracticalstructuralsystemreliabilityproblemsrequireonly®veto20randomvariables,makingthee󰂀orttoidentifythepointsfeasibleandpractical.

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