A neural network approach to estimating deflection of diaphragm walls caused by excavation in clays

2021-03-19 来源：步旅网

ComputersandGeotechnics34(2007)385–396

www.elsevier.com/locate/compgeo

Aneuralnetworkapproachtoestimatingdeﬂectionofdiaphragm

wallscausedbyexcavationinclays

GordonT.C.Kunga,EvanC.L.Hsiaob,MattSchusterb,C.HseinJuang

ab,*SustainableEnvironmentResearchCenter,NationalChengKungUniversity,Tainan701,Taiwan

bDepartmentofCivilEngineering,ClemsonUniversity,Clemson,SC29634,USAReceived17July2006;receivedinrevisedform20October2006;accepted31May2007

Availableonline15August2007

Abstract

Anartiﬁcialneuralnetwork(ANN)-basedapproachforpredictingdeﬂectionofdiaphragmwallscausedbybracedexcavationinsofttomediumclaysispresentedinthisstudy.Fiveinputvariables,includingexcavationdepth,systemstiﬀness,excavationwidth,shearstrengthnormalizedwithverticaleﬀectivestress,andYoung’smodulusnormalizedwithverticaleﬀectivestress,areadoptedasinputstotheANN.ThedatabasefortrainingandtestingtheANNisgeneratedfromhypotheticalcasesusingﬁniteelementmethod.Theper-formanceofthedevelopedANNrevealsthattheinﬂuenceofeachinputvariableonthewalldeﬂectionisconsistentwiththeexcavationbehaviorsgenerallyobservedintheﬁeld.Thevalidationusing12excavationcasehistoriescollectedinthisstudyshowsthatthewalldeﬂectioncausedbybracedexcavationcanbeaccuratelypredictedbythedevelopedANN.Ó2007ElsevierLtd.Allrightsreserved.

Keywords:Neuralnetwork;Bracedexcavation;Walldeﬂection;Finiteelementmethod;Clays;Casehistories

1.Introduction

Bracedexcavationforbasementsofhigh-risebuildings,stationsofundergroundtransportationsystems,andundergroundparkingspaceisfairlycommoninurbancon-struction.Inthedesignofabracedexcavation,itisessen-tialtoconsidernotonlythestabilityissuesuchasbasalheaveduringconstruction,butalsothepotentialservice-abilityproblemofadjacentbuildingscausedbyexcessivewalldeﬂectionsandgroundmovements.Fig.1showsthetypicalwalldeﬂectionandgroundmovementcausedbyexcavationinsofttomediumclays.Asillustrated,theadja-centbuildingmaybedamagedduetotheexcessivediﬀeren-tialsettlementcausedbytheinducedwalldeﬂection.Overthepasttwodecades,theissueofwalldeﬂectionand

*Correspondingauthor.Tel.:+18646563322;fax:+18646562670.E-mailaddress:hsein@clemson.edu(C.H.Juang).

groundmovementhasbecomemoreimportantwithincreasinglitigationassociatedwiththedamageofadjacentbuildings.Thefocusofthispaperislimitedtotheestima-tionofwalldeﬂection.

Tomitigatethepotentialdamagetothebuildingsadja-centtotheexcavation,itisessentialtobeabletopredictaccuratelytheexcavation-inducedwalldeﬂectionandgroundmovement.Resultsofﬁeldmonitoringandobser-vationsinthepastprojectscanprovidesomeguidanceforassessingtheperformanceofafuturebracedexcavation(e.g.,Long[1]).Severalempiricalandsemi-empiricalapproachesforpredictingwalldeﬂectionandgroundmovementhavebeendevelopedbasedonﬁeldobservationsand/orﬁniteelementmethod(FEM)analyses[2–7].Inrecentyears,useofartiﬁcialneuralnetwork(ANN)tech-niqueforpredictingwalldeﬂectionandgroundmovementhasalsobeenreported[8–11].

Generally,inasimpliﬁedempiricalchart,itisdesirabletoemployasmallnumberofinputvariables.Forexample,

0266-352X/$-seefrontmatterÓ2007ElsevierLtd.Allrightsreserved.doi:10.1016/j.compgeo.2007.05.007

386G.T.C.Kungetal./ComputersandGeotechnics34(2007)385–396

hmhmFig.1.Eﬀectscausedbyexcavationinclays:walldeﬂection,groundmovementandbuildingresponse.CloughandO’Rourke[5]relatedtwovariables,thefactorofsafetyagainstbasalheaveandthesystemstiﬀness,totheratioofthemaximumlateralwalldeﬂectionovertheexca-vationdepth.Whilesimpliﬁcationisdesirableandmaybeoutofnecessity,theaccuracyofpredictionsusingsuchsim-pliﬁedchartsmightnotbedesirable,dependingonthesite,soil,andconstructionconditions.Becausethemaximumlateralwalldeﬂectionisoftenusedasacriteriontoguardagainsttheexcavation-relatedproblems,itisdesirabletohaveasimpleandeasy-to-usemodelthatcanproduceanaccurateestimateoftheexcavation-inducedmaximumwalldeﬂection.

Theobjectiveofthispaperistodevelopasemi-empiri-calapproachtoestimatingmaximumlateralwalldeﬂec-tioninsofttomediumclaysusingartiﬁcialneuralnetworks(ANNs).Tothisend,alargenumberofhypo-theticalcasesbasedontwowell-documentedcasehistoriesaregeneratedthroughFEManalyses.ANNsarethentrainedandtestedusingthehypotheticalcasesandﬁnallyvalidatedthroughthecollectedcasehistories.TheresultsshowthatthedevelopedANNscanaccuratelypredictthelateralwalldeﬂectioncausedbybracedexcavationinsofttomediumclays.

2.ExistingapproachesforestimatingwalldeﬂectionExistingempiricalandsemi-empiricalapproachesforestimationofexcavation-inducedwalldeﬂectionarebrieﬂyreviewedherein.ManaandClough[3]studiedanumberofcasehistoriesandrecognizedthattherewasastrongcorre-lationbetweenthemaximumlateralwalldeﬂection(dhm)andthepotentialforbasalheave,intermsoffactorofsafetyagainstbasalheave.Anempiricalchartwasdevel-opedbasedonthecasehistorydataforpredictingthemax-imumwalldeﬂectioninsofttomediumclaysbymeansofthefactorofsafetyagainstbasalheave,deﬁnedbyTer-

zaghi[12].WongandBroms[4]proposedasimpleproce-duretoestimatethelateraldeﬂectionofstruttedoranchoredsheet-pilewalls.Theprocedurewasdevelopedbasedonanassumptionthatthewallsareﬂexibleandthelateraldeﬂectionsaregovernedbyplasticyieldingofthesoilbelowthebottomofexcavation.Theyieldingisinverselyproportionaltothefactorofsafetyofbasalheave.Theexcavationwidth,excavationdepth,andsecantortangentmoduliofthesoilareincludedintheanalysis.Thevalidationoftheirprocedurebasedonlimitedcasehis-toriesfromSingapore,OsloandSanFranciscoshowedthatsatisfactorypredictionsofthewalldeﬂectioncanbeobtained.

CloughandO’Rourke[5]proposedanempiricalchart(seeFig.2)forestimatingthemaximumlateralwalldeﬂectionforsofttomediumclays,whichsimultaneouslyconsidersthefactorofsafetyagainstbasalheaveandthesystemstiﬀness(EI=c4whavg;whereEIisthewallstiﬀness,cwistheunitweightofwaterandhavgistheaveragesupportspacing).HashashandWhittle[6]conductedaseriesofnumericalinvestigationsusingnonlinearFEMtostudytheeﬀectsofwallembedmentdepth,supportconditions,andstresshistoryontheundraineddeforma-tionsaroundabraceddiaphragmwallinaclaydeposit.Theiranalysesusedacomprehensiveeﬀectivestresssoilmodel,MIT-E3(WhittleandKavvadas[13]),todescribeimportantaspectsofclaybehavior,includingsmall-strainnonlinearityandanisotropicstress–strain-strengthbehav-iors.Itisgenerallyconsideredanecessitytoemployasoilmodelthatcanadequatelydescribethesmallstrainbehaviorofsoilinabracedexcavationanalysis[14–16].Severaldesignchartsforestimatingthemaximumlateralwalldeﬂectionasfunctionsoftheexcavationdepth,supportcondition,andtheeﬀectofwalllengthonbasestabilitywereestablishedbyHashashandWhit-tle[6].

Sheetpile Walls1 m Thick Slurry Walls h=3.5h=3.5He)%(eH/mhδ(EI)(γh4wavg)Fig.2.Designcurvesformaximumlateralwallmovementforexcavationsinsofttomediumclays(CloughandO’Rourke[5]).G.T.C.Kungetal./ComputersandGeotechnics34(2007)385–396

Table1

SummaryoftheexcavationcasehistoriesexaminedCaseno.123456789101112

Casename

Total

excavationstages747655555667

Excavationwidth(m)41.2031.2033.4070.0049.2854.1236.0035.0012.3015.0015.0017.60

Finaldepthofexcavation(m)19.7013.9018.4520.0012.2512.6013.7013.1514.5019.4019.4019.86

Wall

length(m)35.028.031.032.521.522.028.525.026.030.030.035.0

Wallthickness(m)0.90.70.80.70.60.60.70.80.81.11.11.2

WallstiﬀnessEI(kNm2/m)150538570829591562770829544604044604070829510572801057280274851527485153568320

dhm/He(%)0.540.400.330.620.370.480.350.440.240.310.320.26

Reference

387

TNECChichingFormosaFar-EasternSinyiTaikaiElectronicsTaiwanSugarSubway-1Subway-2Subway-3Subway-4

OuOuOuOuetetetetal.al.al.al.[21][19][19][19]

WuandOu[29]

Unpublishedﬁle

Tung[30]Unpublishedﬁleﬁle

Unpublishedﬁle

Note:CasesNo.1,2and4areconstructedusingtheTop-downmethod.AllothercasesareconstructedusingtheBottom-upmethod.dhmisthemaximumlateralwalldeﬂection,andHeistheexcavationdepth.

Addenbrookeetal.[7]conducted30nonlinearFEManalysesofundraineddeepexcavationinstiﬀclayandtheresultsareusedtosupporttheuseofanewdisplace-mentﬂexibilitynumber(D)inmulti-proppedretainingwalldesign.Theﬂexibilitynumberisdeﬁnedas:D=EI/h5,whichissimilartosystemstiﬀness(EI=cwh4avg)suggestedbyCloughandO’Rourke[5].TheresultsobtainedbyAddenbrookeetal.[7]showedthatthevariationofwalldeﬂectionsestimatedwithﬂexibilitynumberislessthanthatestimatedwithsystemstiﬀness.Thus,useoftheﬂexi-bilitynumbermightenhancetheengineer’sconﬁdenceindevelopingsupportstrategiesinagivenexcavation.3.Collectionofexcavationcasehistories

TwelvecasehistoriesofbracedexcavationinsofttomediumclaysinTaipei,whichwereconstructedbytheTop-downmethodorBottom-upmethod,arecollectedforthevalidationoftheANNsdevelopedandpresentedlater.ItisnotedthatwiththeTop-downmethod,slabs(orﬂoors)areusedassupportinlieuofstrutsandanchors;theyarecastfromthetopoftheexcavationandproceedingdownwardtothebottomoftheexcavation.WiththeBot-tom-upmethod,wheretheretainingwallissupportedbystrutsandanchorsduringtheexcavation,slabsarecastaftertheexcavationfromthebottomofexcavationandproceedingupwardtothetopoftheexcavation.Table1presentsasummaryoftheseexcavationcasehistories,includingexcavationwidth,ﬁnaldepthofexcavation,walllength,wallthickness,wallstiﬀness,andtheratioofmax-imumlateralwalldeﬂection(dhm)overtheﬁnaldepthofexcavation(He).Thedepthsofexcavationsanddepthswherethestrutsorconcreteﬂoorslabswereinstalledorconstructedforallstagesinthesecasehistoriesareshown

Table2

ProppingarrangementinexcavationcasehistoriesexaminedStageno.1234567

Case1He2.84.98.611.815.217.319.7

HpCase2HeHpCase3HeHpN/A1.03.76.29.512.515.5

Case4He4.958.5512.415.416.920N/A

HpN/A3.457.0510.913.916.4N/A

Case5He23.956.99.8212.25N/AN/A

HpN/A1.383.336.39.23N/AN/A

Case6He2.04.07.09.612.6N/AN/A

HpN/A1.133.056.189.28N/AN/A

Case7He2.13.87.011.113.7N/AN/A

HpN/A1.33.36.510.5N/AN/A

Case8He1.103.707.2510.7513.15N/AN/A

HpN/A0.553.056.5510.05N/AN/A

Case9He2.54.58.512.514.5N/AN/A

HpN/A2.04.08.012.0N/AN/A

Case10He2.06.210.313.616.819.4N/A

HpN/A1.255.359.312.716.0N/A

Case11He2.06.210.313.616.819.4N/A

HpN/A1.255.359.312.716.0N/A

Case12He2.45.869.0611.5614.1617.6619.86

HpN/A1.905.368.5611.0613.6617.16

N/A5.3N/A1.62.08.63.2&04.33.5&011.06.46.97.113.910.510.1510.3N/AN/A13.213.7N/AN/A16.216.5N/AN/A18.45

Note:Heisthedepthofexcavation(m);Hpisthedepthwherethestrutisinstalled(m).

388G.T.C.Kungetal./ComputersandGeotechnics34(2007)385–396

inTable2.ItisnotedthatinallthecasesshowninTables1and2,thediaphragmwallwasusedastheretainingstructure.

ThestratigraphyoftheTaipeibasinisgenerallyrepre-sentedbyathickalluviumformation(theSungshanFor-mation)overlainbytheChingmeigravelFormation.TheSungshanFormationwasformedapproximately5000–6000yearsagoanditsthicknessisabout40–50m.Typi-cally,theSungshanFormationhassixalternatingsiltysand(SM)andsiltyclay(CL)layers,butmainlyconsistingofsofttomedium,slightlyover-consolidatedclaywithlowplasticity.TypicalsoilpropertiesoftheSungshanForma-tionhavebeendocumentedinmanypreviousstudies[17,18].

Thewalldeﬂectionsobservedinthe12casehistoriesareshowninFig.3.Formostcasehistories,thecantileverwalldeﬂectionbehaviorcanbeobservedatﬁrstand/orsecondstages,atwhichtimethestruthadnotbeeninstalledorthestiﬀnessoftheinstalledstrutwasnothighenough.Thediaphragmwallthendisplaysthedeepinwardmove-mentsatsubsequentstages.Fig.4examinesthemaximumwalldeﬂectionsatgivenexcavationstageswiththecorre-spondingdepthsofexcavation.Mostdatapointsfallintotheregionboundedbythelineofdhm=0.2%Heandthelineofdhm=0.5%He,whichissimilartothosepresentedbyCloughandO’Rourke[5].Itisnotedthatthebest-ﬁtcurveormeantrendline,dhm=0.32%He,isalsoshowninFig.4.Thescatterofthedatareﬂectsthepossiblevari-ationofdhmatagivenexcavationdepth,sincethedatawerecollectedfromdiﬀerentcases.Thevariationalsopointstothenecessityofemployingmoreinputvariablesintheempiricalmodelforpredictingdhm.

Fig.5comparesthedepthwheredhmoccurredwiththedepthofexcavationforeachstageinallthesecase

Wall deflection (mm)Wall deflection (mm)Wall deflection (mm)Wall deflection (mm)005408012002040600204060800306090120150Depth (m)10152025303505020(a) Case 14060020(b) Case 2406080020(c) Case 34060020(d) Case 44060Depth (m)10152025303505010(e) Case 5203040020(f) Case 6406080020(g) Case 7406080020(h) Case 84060Depth (m)101520253035(i) Case 9(j) Case 10(k) Case 11(l) Case 12Fig.3.Observationsofwalldeﬂectionforthe12excavationcasehistories.G.T.C.Kungetal./ComputersandGeotechnics34(2007)385–396389

150Average of stages 3-7 of all cases(Mean trend line)120Stages 1-2 Stages 3-7%Hem=0.5m)m90δh ( mhm=0.32%Heδ60δh30δ.2%Hehm=000510152025Depth of excavation, He (m)Fig.4.Maximumwalldeﬂectionversusexcavationdepthofcasehistories.histories.Insomecases,thewallmovementswereofthecantileverpatternatearlystagesofexcavation(stages1and2),andthus,themaximumwalldeﬂectionoccurredatthetopofthewall(andthus,thedepthswheredhmoccurredarezero).Ingeneral,thedepthswheredhmoccurredatexcavationstages3–7areapproximatelyequaltothedepthsofexcavation.SimilarresultshavepreviouslybeenreportedbyOuetal.[19].Fig.6showstherelation-shipbetweennormalizedmaximumlateralwalldeﬂectionandfactorofsafetyagainstbasalheave,deﬁnedbyTer-zaghi[12],forallcasehistoriesatstages3–7.Thelimits

25)m20( derrucc15o mhδerehw10stnd htprdeDth5ththth00510152025Depth of excavation (m)Fig.5.Depthwherethemaximumwalldeﬂectionoccurredincasehistories.3.02.5Limits suggested by Mana and Clough [3]Case histories at stages 3-7 )2.0%( eH1.5/mhδ1.00.50.00.51.01.52.02.5Factor of safety against basal heaveFig.6.Relationshipbetweenfactorofsafetyagainstbasalheaveandnormalizedmaximumlateralwalldeﬂection.suggestedbyManaandClough[3]arealsoshowninthisﬁgure.ThecasehistoriesoftheTaipeiclaygenerallyfallintotheregionboundedbythetwolimitssuggestedbyManaandClough[3].Thedatapointsfromthecasehisto-riesexaminedareseentobeclosertothelowerlimit,whichcouldbeexplainedbythefactthatManaandClough’slimitswereestablishedmostlywithcasehistoriesofexcava-tionsconstructedwithlow-stiﬀnesssheet-pilewallsorsol-dierpilewalls,whereasthehigh-stiﬀnessdiaphragmwallswereusedinthecasehistoriesexaminedherein.4.Establishmentofhypotheticalcases

Becausethenumberofexcavationcasehistoriescol-lectedinthisstudyisnotsuﬃcientfortrainingandtestingoftheintendedANN,alargenumberofhypotheticalcasesaregeneratedandanalyzedusingFEM.Inthissection,fac-torsaﬀectingwalldeﬂectionsandgroundmovementsaroundbracedexcavationsareﬁrstreviewedandtheproce-durefornumericalexperimentationusingFEMtogenerateartiﬁcialdataisthendiscussed.4.1.Factorsaﬀectingwalldeﬂection

Behaviorofabracedexcavationmaybeaﬀectedbymanyfactorssuchaswallstiﬀness,strutspacingandstiﬀ-ness,excavationwidthanddepth,prestress,depthtoanunderlyingstiﬀlayer,soilstiﬀnessandstrengthdistribu-tion,dewatering,adjacentsurcharge,soilconsolidationandcreep,andworkmanship[2–6,20].Itisnotpracticaltoincorporateallthesefactorsinasimpliﬁedmethodforpredictingwalldeﬂection,andthusonlytheessentialfac-torsareusedtogeneratehypotheticalcasesfordevelopingtheintendedANNmodel.

ManaandClough[3]showedthatthewalldeﬂectionwassigniﬁcantlyaﬀectedbythefactorofsafetyagainstbasalheave,whichcomprisestheeﬀectofexcavationwidth,shearstrengthofclay,andexcavationdepth.

390G.T.C.Kungetal./ComputersandGeotechnics34(2007)385–396

CloughandO’Rourke[5]furtherincludedthesystemstiﬀ-nessasanessentialfactor,whichpointedtotheimportanceofthewallstiﬀnessandtheaverageverticalstrutspacing.Basedonanextensiveseriesofanalyses,HashashandWhittle[6]reachedaconclusionthatwalllengthhasamin-imaleﬀectonthepre-failuredeformationsforexcavationindeeplayersofclaywherethereisnoconstraintontoemovement,butdoeshaveamajorinﬂuenceonthelocationoffailuremechanismwithinthesoil.Theyalsodemon-stratedtheeﬀectsofstresshistory,includingtheundrainedshearstrength,soilstiﬀness,andthecoeﬃcientofearthpressureatrest,onthewalldeﬂection.

Basedmostlyonthesepreviousﬁndings,ﬁvefactorsareselectedfornumericalexperimentationinthisstudytogen-eratehypotheticalcases;theyareexcavationdepth(He),excavationwidth(B),systemstiﬀness(S),ratioofshearstrengthoververticaleﬀectivestress(su=r0oververticalv),andratioofinitialYoung’smoduluseﬀectivestress(Ei=r0v).

4.2.Numericalexperimentation

Twowell-documentedcasehistories,theTaipeiNationalEnterpriseCenter(TNEC)case(Ouetal.[21])andtheFormosacase(Ouetal.[19]),areusedastheref-erencemodeltoconductalargenumberofnumericalexperimentsusingFEManalysesforestablishingadatabaseforthedevelopmentoftheintendedANNmodel.Fig.7showsthecomparisonofwalldeﬂectionandgroundsurfacesettlementbetweenobservationsandpre-dictionsforthetwocasesusingFEMwithamodiﬁedpseudoplasticitysoilmodel(Kung[22]).Thissoilmodelcanaccuratelysimulatenonlinearstress–strainbehaviorofclaysatsmallstrain[23].Thepredictionsofthewalldeﬂectionandgroundsettlementinexcavationinclays,asthoseshowninFig.7,areconsideredsatisfactoryandformabasisforgeneratinghypotheticalcasesinanumericalexperimentation.

Asmentionedpreviously,theeﬀectofshearstrengthofclay,claystiﬀness,wallproperties,andexcavationwidthonbracedexcavationinsofttomediumclaysshouldbeconsideredinthenumericalexperiments.DetailsoftheparametersusedintheanalysesofhypotheticalcasesaregiveninTable3.Thedepthsofinstalledstrutsorcon-structedﬂoorslabs,theexcavationdepths,andthestrati-graphyconditionsfornumericalexperimentsareassumedtobethesameasthoseintheTNECandFormosacases.Atotalof720diﬀerentFEManalysesweremadetosimu-latebracedwallexcavationsinsofttomediumclaydepos-its.TheresultsofFEManalysesgeneratethousandsofinstances(note:aninstanceisadatapointthatconsistsofﬁveinputvariables,He,S,B,su=r0outputvariable,themaximumwallv,andEi=r0deﬂection,v,andonedhm).ItshouldbenotedthatthewalldeﬂectiondataattheﬁrsttwostagesofexcavationineachcasearenotincludedinthedatabaseforthedevelopmentoftheANNbecauseatleasttwo-levelstrutsmustbeinstalledpriortodetermining

Wall deflection (mm)Distance from the wall (m)1201008060402000102030405060001020)mm20)(40m t(n hetpm60Stage 1Stage 5eStage 2Stage 6l30eDttStage 3Stage 7eS80Stage 4Predictions40100Stage 1- Stage 750(a)TNEC case Wall deflection (mm)Distance from the wall (m)706050403020100010203040500010)m20)mm( ( ntheStage 1Stage 520tpmeeD40Stage 2Stage 6Stage 3Stage 7ettlSStage 430Predictions60Stage 1- Stage 740(b) Formosa case Fig.7.PredictionsofwalldeﬂectionandsurfacesettlementonTNECandFormosacasesusingﬁniteelementmethodwithasmall-strainsoilmodel.Table3

ParametersofhypotheticalcasesbasedontheTNECandFormosacasesVariables

Selectedvalues12345(a)BasedonTNECcasesu=r0v0.250.290.320.360.40Ei/su15001750203524003000Wallthickness(m)0.60.91.21.5N/AExcavationwidth(m)

2039.260100(b)BasedonFormosacasesu=r0v0.250.300.340.4N/AEi/su1500210025003000N/AWallthickness(m)0.60.81.01.21.4Excavationwidth(m)10

100

thesystemstiﬀness.Furthermore,forpracticalpurposes,thecaseswiththecalculatedmaximumlateralwalldeﬂec-tionofgreaterthan200mmarediscarded.Thus,atotalof3486instancesaregeneratedfordevelopingtheintendedANN.Theseinstancescollectivelyrepresentbehaviorofwalldeﬂectioninbracedexcavationsinsofttomediumclays.Intheseinstances,thevariableHeisintherangeof6.9to19.7(m);thevariableSisintherangeof286to10950;thevariableBisintherangeof10to100(m);thevariablesu=r0visintherangeof0.25to0.4;thevariableEi=r0visintherangeof375to1200.

G.T.C.Kungetal./ComputersandGeotechnics34(2007)385–396391

5.DevelopmentoftheANN5.1.Procedureandmethodology

Athree-layer,feed-forwardANNtopologyshowninFig.8isadoptedinthisstudy.Amongthe3486instances,abouttwothirds(2324instances)areselectedasthetrain-ingset,andtheotheronethird(1162cases)areusedasthetestingset.Thesamplingprocessisconsideredlargelyran-dom,sincenoeﬀortwasmadetokeeptrackofthecharacteristicsofinputandoutputvariables.Whileran-domnessinthedataselectionwaslargelymaintained,thetrainingdatasetisbelievedtoberepresentative,asdatafromallrangesweresampled.Thenetworkisﬁrsttrainedusingthetrainingdataset.Theobjectiveofthenetworktrainingistomaptheinputtotheoutputbydeterminingtheconnectionweightsandbiasesthroughaback-propagationprocedure.Forthethree-layernetwork,theoutputofthenetwork,dhm,iscalculatedasfollows(Juanget(al.[24]):

n\" m!#)

dhm¼f2B0þXWkf1BHkþX

WikPið1Þ

k¼1

i¼1

wheredhmisthemaximumwalldeﬂection(mm);B0isthebiasattheoutputlayer(justoneneuroninthislayer);Wkistheweightofconnectionbetweenneuronkofthehiddenlayerandthesingleoutputlayerneuron;BHkisthebiasatneuronofthehiddenlayer(k=1,n);Wikistheweightofconnectionbetweeninputvariablei(i=1,m)

Input layerHidden layerOutput layerNeuron 1Heinput 1Neuron 2Sinput 2Neuron 3Binput 3Neuron 4outputδhmsuσv'input 4Neuron 5Eσinput 5Neuron 6v'Neuron 7Fig.8.Aschematicdiagramofthree-layerartiﬁcialneuralnetwork.andneuronkofthehiddenlayer;Piistheinputvariable

i;f1(k)isthetransferfunctionofeachneuroninthehiddenlayer;andf2(k)isthetransferfunctionoftheneuronintheoutputlayer.Bothtransferfunctionsf1(k)andf2(k)adoptedinthisstudyaresigmoidfunctionsdeﬁnedby:fNðkÞ¼

11þeÀk

forN¼1;2

ð2Þ

InEq.(1),thenumberofinputvariables(m)is5;theinputvariablesareP1=He,P2=S,P3=B,P4¼su=r0Pv,and5¼Ei=r0v.Thenumberofhiddenneurons(n)isdeter-minedthroughatrialanderrorprocedure;normally,thesmallestnumberofneuronsthatyieldssatisfactoryresults(judgedbythenetworkperformance)shouldbeused.Inthisstudy,sevenhiddenneuronsareselected,asthisnumberisthesmallestnumberofneuronsthatarerequiredtoyieldsatisfactoryresults(judgedbythenetworkperformance).

Inthepresentstudy,theLevenberg–Marquardt(LM)algorithm(DemuthandBeale[25])isadoptedforitseﬃ-ciencyintrainingnetworks.TheLMalgorithmisimple-mentedwithMatlab(MathWork,Inc.[26])anditsneuralnetworktoolbox(DemuthandBeale[25]),althoughothersoftwareorin-housecomputercodesmayalsobeused.Itshouldbenotedthatallvariablesarescaledintovaluesintherange0.1to0.9beforetraining.Theﬁnalweightsandbiasesofthetrainednetwork,expressedasthecoeﬃ-cientsinEq.(1)areshowninTable4.

Fig.9showstheperformanceofthedevelopedANNveriﬁedwiththetrainingdatasetandthetestingdataset.TheresultsshowthatthedevelopedANNcanaccuratelypredictthemaximumwalldeﬂectionsofalargenumberofhypotheticalcasesasdeﬁnedbytheFEManalyses.5.2.AssessmentoftheANNapproach

TheperformanceofthedevelopedANNhasbeenshowntobesatisfactory.Additionalassessmentispre-sentedherein.First,theconsistencyofthedevelopedANNisassessedbyemployingthevarioussamplingschemessothatalargenumberofdiﬀerenttrainingandtestingdatasetsarecreatedandusedinthedevelopmentofANN.Theresultsshowthatregardlessofwhichsam-plingschemewasused,theANNasdescribedinEq.(1)withsevenhiddenneuronsyieldspracticallythesamemax-imumwalldeﬂectionforeachinstance.Thus,thedevel-opedANNisnotsensitivetothechangeofsamplingschemesandcanproduceconsistentpredictionsofthewalldeﬂection.

UseofagraphtoshowthevariationoftheANN-pre-dictedmaximumwalldeﬂectiondhmwitheachinputvari-ableisanotherwaytoexaminethebehaviorofthedevelopedANN.However,theANNishighlynonlinearandthus,theconventionalmeansofplottingtheoutputversusoneinputvariablewhilemaintainingallotherinputvariablesasaconstantmaynotalwaysbemeaningful.Inthisstudy,eachofthe12casesisusedforexaminingthe

392G.T.C.Kungetal./ComputersandGeotechnics34(2007)385–396

Table4

ConnectionweightsandbiasesoftheneuralnetworkdeﬁnedinEq.(1)

WeightWikInput1(i=1)

HiddenHiddenHiddenHiddenHiddenHiddenHidden

neuronneuronneuronneuronneuronneuronneuron

1234567(k=1)(k=2)(k=3)(k=4)(k=5)(k=6)(k=7)

0.5830.744À3.0184.8423.6987.254À3.413

Input2(i=2)À1.315À1.3873.272À36.397À1.979À48.6652.777

Input3(i=3)À5.501À0.03À0.869À0.0330.8661.773À1.326

Input4(i=4)À1.096À4.002À1.382À1.1280.295À7.058À2.635

Input5(i=5)À1.124À0.12416.5630.598À3.637À12.53637.619

WkOutputneuronÀ73.0557.240À4.4940.4161.1670.8913.925

BiasBkHiddenlayerÀ3.666À1.463À0.0422.615À0.0954.676À1.095

BoOutputlayer

À1.349

250ANN-estimated δhm (mm)200150100500(a) Training data: 2324 instances R2 = 0.95 RMSE = 8.36 COV = 0.13250200150100500(b) Testing data: 1162 instances R2 = 0.95 RMSE = 8.69 COV = 0.13050100150200FEM-based δhm(mm)250050100150200FEM-based δhm(mm)250Fig.9.PerformanceofthedevelopedANN:(a)training,and(b)testing.200160120804000.220016012080400δhm(mm)0.250.30.350.40.45δhm(mm)'su/σvFig.10.VariationoftheANN-predictedwalldeﬂectionwiththenormalizedshearstrength.400600800Ei/σ'v10001200Fig.11.VariationoftheANN-predictedwalldeﬂectionwiththenormalizedYoung’smodulus.inﬂuenceoftheundrainedshearstrengthoftheclay,repre-sentedbythevariable,su=r0v,andtheYoung’smodulusoftheclay,representedbythevariable,Ei=r0v.Foreachcase,allvariablesaremaintainedasaconstantexceptthatsu=r0visallowedtovary,andthevarianceoftheANN-predicteddhmwithsu=r0visthenplotted.AsshowninFig.10,the

G.T.C.Kungetal./ComputersandGeotechnics34(2007)385–396393

decreasingtrendofdhmvaryingwithsu=r0vforallcasesisconsistentandasexpected.

Fig.11showsaplotsimilartoFig.10,wheretheinﬂu-enceofEi=r0visshownandsimilarconclusioncanbedrawn.Finally,whilenotshownherein,thedecreasing

euvivFig.12.ImportanceindexesofinputvariablesinthedevelopedANN.trendofdhmversussystemstiﬀness(S),andtheincreasingtrendofdhmversuseachoftheothertwovariables,excava-tionwidth(B)andexcavationdepth(He),areobtained.Thus,thedevelopedANNisshowntobehaveasexpectedusingthe12casehistories.

Additionalsensitivityanalysiswasconductedtoesti-matetheimpactofinputvariablesontheoutput,dhm.YangandZhang[27]suggestedthattherelativestrengthoftheeﬀectofaninputvariableontheoutputcanbederivedbasedontheweightsstoredinthenetwork.Theydeﬁnedanimportanceindextoexpressthedegreeofsensi-tivityforeachinputvariableontheoutput.FollowingtheprocedurebyYangandZhang[27],theimportanceindexesfortheﬁveinputvariables,He,B,S,su=r0v,andEi=r0v,are0.89,0.21,0.23,0.89,and1.0,asshowninFig.12.How-ever,theseweightsshouldbeviewedonlyasaroughesti-mate,astheyaredeterminedbasedonthesameassumption,mentionedpreviously,thatonlyoneinputvar-iableatatimeisallowedtovaryalthoughthedevelopedANNishighlynonlinear.

Greaterimportanceindexvaluesforvariables,He,su=r0v,andEi=r0vshowninFig.12areexpected,sincedhm1501209060(c) Case 39009060300306090300906030(g) Case 790090603003060900906030(k) Case 11003060900030(l) Case 126090030(h) Case 860900(d) Case 43060901201501601208040906030(a) Case 1(b) Case 2306001201600906030(e) Case 5(f) Case 6900906030(i) Case 9(j) Case 10900030609003060906030Estimated wall deflection (mm)090603009060300040800306003060Observedwalldeflection(mm)Fig.13.ComparisonofwalldeﬂectionspredictedbytheANNandtheCloughandO’Rourkemethodatstages3–7ineachofthe12cases.394G.T.C.Kungetal./ComputersandGeotechnics34(2007)385–396

hasastrongcorrelationwitheachofthethreevariablesingeneral.ComparedwithHe,su=r0BisrelativelyunimportantvandEi=r0andreﬂectedv,theexcava-tionwidthinthebehaviorofthedevelopedANN.Thisbehaviorissup-portedbytheresultsofnumericalexperimentationusingFEM.Finally,theresultalsoindicatedthatthesystemstiﬀ-nessSisrelativelyunimportantinthedevelopedANN.Thisresultmaybeexplainedinthefollowing.Foragivenexcavationcase,thestiﬀnessofretainingwallitself(EI)isaconstant,andthus,thesystemstiﬀnessineachexcavationstagetendstobeapproximatelythesame,sincethesameverticalspacingofstrutsforeachstageofexcavationisoftenmaintained.Therefore,astheexcavationproceeds,thewalldeﬂectiontendstoincrease,althoughthesystemstiﬀnessremainsapproximatelythesame.Asaresult,theinﬂuenceofthesystemstiﬀnessSonthewalldeﬂectionisrelativelylow.ThesameconclusionmaybereachedwithacloseexaminationofFig.2.Thehypotheticalcasesweregeneratedtoemulatemoderndeepbracedexcavationswherethefactorofsafetyagainstbasalheaveissuﬃcientlyhigh(i.e.,greaterthan1.4asshowninFig.6)andthesys-temstiﬀnessisrelativelylarge(i.e.,greaterthan300inmostinstances).Undertheconditionsofadequatefactorofsafetyagainstbasalheaveandsuﬃcientlylargesystemstiﬀ-ness,thevariationofdhmwiththesystemstiﬀnessforagivenHeisrelativelyﬂataccordingtoFig.2.Thus,thebehaviorofthedevelopedANN,whichyieldsalowimpor-tanceindexvalueforthevariableS,issupportedbywell-establishedknowledge.

6.ValidationofthedevelopedANN

The12excavationcasehistoriesintheTaipeiclayareusedtovalidatethedevelopedANN.Threeinputvari-ables,He,BandS,aredeterminedbasedonthegiveninformationforeachcasehistory.Theothertwovariables,su=r0vandEi=r0v,areestimatedbasedontheresultsofsmallstraintriaxialtestsfortheTaipeiclay(Kung[22,28]);theyareestimatedtobeapproximatelyequalto0.31and650,respectively,forall12casehistories.UsingthedevelopedANN,dhmforeachofthe12casesisthencalculatedandcomparedwithﬁeldobservation.Fig.13showsthemea-sureddhmofthe12excavationcasehistoriesatstages3–7andthosepredictedbythedevelopedANN.Forcompari-son,theresultsobtainedwiththeCloughandO’Rourke[5]methodareshown.AlsoincludedinFig.13arethetwoboundingtrendlines,dhm=0.2%Heanddhm=0.5%He.Mostpredictions(datapoints)fallintotheregionboundedbythetwotrendlinesofdhm=0.2%Heanddhm=0.5%He.TheresultsshowninFig.13indicatethatingeneral,thewalldeﬂectionsforallcasehistoriesexaminedcanbesatis-factorilypredictedbytheCloughandO’RourkemethodandthedevelopedANN,althoughthelatterwasshowntobemoreaccurateoverall.

Fig.14furthercomparestheperformanceofthetwomethodsinpredictingthe12casehistoriesexamined.Thevaluesofroot-mean-square-error(RMSE)andcoeﬃcient

ofvariation(COV)ofthepredictionsarecalculatedforeachmethod.Here,theRMSEiscalculatedbasedonthepredictedandmeasuredwalldeﬂectionsatstages3–7inall12cases.TheCOViscalculatedastheratioofRMSE

150)m(a) ANN predictionsmRMSE = 9.63(120 noCOV = 0.20itcel90fed llaw60 detcide30rP00306090120150150)m(b) Clough & O'Rourke m(120RMSE = 20.81 noCOV = 0.43itcel90fed llaw60 detcid30erP00306090120150Observed wall deflection (mm) Fig.14.ComparisonofwalldeﬂectionspredictedbytheANNandtheCloughandO’Rourkemethodinallcases(stages3–7).)m90m(RMSE = 8.4 noCOV = 0.41itcelfe60d llaw detcid30erp-NNA00306090Observed wall deflection (mm)Fig.15.Comparisonofpredictedandobservedwalldeﬂectionsatstage2ineachofthe12excavationcases.G.T.C.Kungetal./ComputersandGeotechnics34(2007)385–396395

overthemeanofthe44measureddataofwalldeﬂections.Again,thedevelopedANNisshowntobemoreaccurateoverallinpredictingwalldeﬂections.

7.Discussion

Oneoftheﬁveinputvariablesrequiredinthedevel-opedANNissystemstiﬀness(S),whichdependsonthewallstiﬀness(EI)andtheaveragesupportspacing(havg),asnotedpreviously.Ingeneral,thesupportspacing(h)isbetweentwostrutlevels,andthus,theANNmodelwasdevelopedforpredictionofmaximumwalldeﬂectionsstartingfromstage3oftheexcavation.However,onemightconsiderthebottomoftheexcavationasalevelofsupportforthewall.Thus,thepredictionofmaximumwalldeﬂectionatthecompletionofstage-2excavationmaybepossible.ExtendingthedevelopedANNtopre-dictionofthemaximumwalldeﬂectionatstage2(with-outre-trainingANN)iscarriedoutforthe12casesexamined,andtheresultsareshowninFig.15.Theaccu-racyoftheANNpredictionofthemaximumwalldeﬂec-tionsatstage2isconsideredsatisfactory,althoughthepredictionis,asexpected,slightlylessaccuratethanthosepredictionsforthemaximumwalldeﬂectionsatlaterstages.

8.Summaryandconclusions

Withtheavailabilityofpersonalcomputersandsoft-waretoday,artiﬁcialneuralnetworkisincreasinglybecom-inganengineeringtoolforderivingdata-drivenpredictivemodels.ThedevelopedANNscaneasilybeimplementedinspreadsheetmodulesforpracticalapplications.Regard-lessofwhetherastraightforwardneuralnetworkoramoresophisticatedANNisemployed,thedevelopedANNmustconformtothephysicalprinciplesand/orbehaviorofasystemitisintendedtoemulate.ThedevelopedANNmustalsobevalidatedwithactualcases.ThispaperdocumentsthedevelopmentandvalidationofanANNintendedforpredictingthemaximumwalldeﬂectioninabracedexcava-tioninsofttomediumclays.

BasedontheanalysisofthecollectedcasehistoriesofbracedexcavationintheTaipeisofttomediumclaysandthedevelopment,experimentationandvalidationofthedevelopedANN,thefollowingconclusionsarereached:1.TwelvecasehistoriesofbracedexcavationintheTaipeisofttomediumclayconstructedusingthediaphragmwallmethodarecollectedandanalyzedinthisstudy.Theobservedwalldeﬂectionsgenerallyfallintotheregionboundedbythetrendlinesofdhm=0.2%Heanddhm=0.5%HesuggestedbyOuetal.[19].Thefac-torofsafetyagainstbasalheave(FS)forexcavationstages3–7inallofthesecasehistoriesisgenerallygreaterthan1.4.Whentheobserveddataareplottedfordhm/HeversusFS,theygenerallyfallintothelimitssuggestedbyManaandClough[3],althoughthese

observeddataareclosertothelowerlimitduetotheuseofstiﬀerdiaphragmwallsinthecollectedcasehistories.

Anartiﬁcialneuralnetwork(ANN)isdevelopedforestimatingtheexcavation-inducedwalldeﬂectionbasedonalargenumberofhypotheticalcasesgeneratedbyaﬁniteelementmodelthatcansimulateaccuratelythesmallstrainsoilbehavior.FiveinputvariablesareadoptedfortheintendedANN;theyareexcavationdepth,systemstiﬀness,excavationwidth,ratioofshearstrengthoververticaleﬀectivestress,andratiooftheYoung’smodulusoververticaleﬀectivestress.Theresultsoftheextensiveassessmentshowthatthedevel-opedANNconformstothetypicalbehaviorofbracedexcavationinsofttomediumclays.

TheimportanceindexesoftheﬁveinputvariableswerecalculatedusingthemethoddevelopedbyYangandZhang[27].Theresultsshowedthattheexcavationdepth,ratioofshearstrengthoververticaleﬀectivestress,andratiooftheYoung’smodulusoververticaleﬀectivestressarethemostimportantfactorsinthedevelopedANNmodelforpredictionofthelateralwalldeﬂection.Thesystemstiﬀnessandtheexcavationwidthwerefoundtoberelativelyunimportantcomparedtotheotherthreevariablesforatypicaldiaphragmwallinsofttomediumclayswithadequatefactorofsafetyagainstbasalheaveandsuﬃcientlylargesystemstiﬀ-ness.Foragivenexcavationprojectwheretheexcava-tiondepthisalreadydetermined,thevariationofshearstrengthandYoung’smodulusofclaysduetosoilvariabilityand/ormeasurementerrorduringsiteinvesti-gationcouldplayasigniﬁcantroleintheaccuracyofthewalldeﬂectionpredictedbytheANN.

ThedevelopedANNisshowntobeabletopredictsat-isfactorilytheexcavation-inducedwalldeﬂectionsasevi-dencedbytheresultsofvalidationusingthe12casehistoriesthatwerenotusedinthedevelopmentoftheANN.WhencomparedwiththeexistingmethodssuchastheCloughandO’Rourke[5]method,thedevelopedANNyieldsimprovedresultsbasedontheevaluationofthe12casehistories.

ThelimitationsofthedevelopedANNarenotedasfol-lows:(1)themodelassumesnormalworkmanshipandnobasalfailureinthebracedexcavation,(2)construc-tion-relatedissuessuchasdewateringandoverexcava-tionpriortosupportinstallationarenotconsidered,(3)thedevelopedmodelisapplicabletocaseswheretheinputvariablesareintherangesspeciﬁed,and(4)theinitialmodulusofelasticitymustbemeasuredwithsmall-straintriaxialtestsorbenderelementtests,insteadofconventionaltriaxialtests.

Acknowledgements

ThestudyonwhichthispaperisbasedwassupportedbytheNationalScienceFoundationthroughGrantNo.

396G.T.C.Kungetal./ComputersandGeotechnics34(2007)385–396

CMS-0300198.Thisﬁnancialsupportisgreatlyappreci-ated.TheresultsandopinionsexpressedinthispaperarethoseofthewritersanddonotnecessarilyreﬂecttheviewoftheNationalScienceFoundation.Dr.Chang-YuOuisthankedforprovidingpartoftheexcavationcasehistoriesexaminedinthispaperandthejournalreviewersarethankedfortheirconstructivecomments.References

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A neural network approach to estimating deflection of diaphragm walls caused by excavation in clays