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9OptimizationofFoundationofBridgeonSoftGroundY.
Demura*andM.
Matsuo***DepartmentofCivilEngineering,IshikawaNationalCollegeofTechnology,Japan**DepartmentofGeotechnicalandEnvironmentalEngineering,NagoyaUniversity,JapanAbstractPresentedisaprocedureofoptimizingthedesignofbridge-pierfoundationsconstructedonsoftgroundwhichislikelytoexperiencethelong-timedeformationduetotheweightofthebridge.
Thewholestructureofabridgeconsistingofthesuperstructureandthefoundationsshouldbedesignedasawholeinsuchawaythatthetotalexpectedcostofthewholestructurebecomeminimum.
Thisprocedureistotallydifferentfromthecurrentdesignmethodinwhichthesuperstructureandthefoundationsaretreatedasasetofseparatesystemsratherthanasatotalsystemconsistingofsubsystems,i.
e.
,thesuperstructureandthefoundations.
Inevaluatingthetotalexpectedcostofabridge,theconstructioncostbothofthesuperstructureandthefoundationsaswellasthedamageoccurrenceprobabilityshouldbetakenintoaccount.
Keywords:FoundationofBridge,SoftGround,OptimumDesign,SystemReliability,Bayes'Theorem1.
INTRODUCTIONFigure1showsasketchofabridgeplacedonpile-supportedpiersrestingonthebearingstratumoverlainbythesoftclaylayer.
Thepierwillsettlebyamountofsduetotheconsolidationoftheground.
Thesettlementisinducedbythepenetrationofthepile-tipintothebearingstratum.
Thepilesaredrawndownbythenegativefrictioncausedbytheconsolidationoftheclaylayerloadedbytheweightoftheembankment.
Thepurposeofthisstudyistoproposethemethodologyofoptimizingthefoundationofstructureonsoftground.
SupposewehavetwobridgesAandB,oneofwhich,bridgeAisdesignedwithrelativelylowsafetyfactoroffoundationagainstthesettlement,whiletheother,bridgeBisdesignedwithrelativelyhighsafetyfactoroffoundation.
ThefoundationofbridgeR.
Rackwitzetal.
(eds.
),ReliabilityandOptimizationofStructuralSystemsSpringerScience+BusinessMediaDordrecht1995jpiles;Optimizationoffoundationofbridgeonsoftgroundbearingstratum1negativeskin11frictionFigure1BridgeConstructedonSoftGround113Aisinexpensive,butlikelytosufferfromtheunfavorablesettlementwithhighprobability.
Thesettlementoffoundationresultsintheadditionalstressesinthemaingirder,i.
e.
,themaingirderwillhavehighprobabilityoffailure.
Hence,themaintenancecostofmaingirderisexpensive.
Themaintenancecostincludestherepairworkstobeneededduetothefuturesettlement.
InthecaseoftheotherbridgeB,theconstructioncostoffoundationisexpensive,butthemaintenancecostofmaingirderisinexpensive.
ThecomparisonofthebridgesAandBindicatestheexistenceofthesafetyfactoragainstthesettlementwhichcorrespondstotheminimumsummationoftheconstructioncostandthemaintenancecost.
Figure2showstherelationshipbetweenthesafetyfactoroffoundationGsubandthecostsofthemaingirderandfoundation.
Itshouldberecognizedthatthemaintenancecostofmaingirdervariesasafunctionofthesafetyfactorofmaingirder.
Intheproceduredescribedinthispaper,(i)weconsiderthewholestructureasasystemconsistingoftwosubsystems,i.
e.
,thesuperstructure(maingirder)andsubstructure(foundation),and(ii)wechoosetheoptirnumdesignsoastorealizetheminimumofthetotalexpectedcost,i.
e.
,thesummationoftheconstructioncostandtheexpectedlossofthewholesystem.
Anaccuratepredictionofthesettlementofthepiersisunavoidablyneededinsuchatotalconstructioncost~offoundation8/maintenancecostofmaingirderFigure2RelationshipbetweenSafetyFactorandCost114PartTwoTechnicalContributionsprocedure.
Themodelproposedinthispaperincludestheprobabilisticsettlementpredictionmethoddevelopedbycollectinganumberofcaserecordsofthesettlementofbridgepiers.
2.
OPTIMIZATIONPROCEDURETheobjectivefunctionofthesystemtobeoptimizedisinprinciplegivenas(1}inwhichE[CT]denotesthetotalexpectedcost,Asubthedesignvariableofthesubstructure,Asupthedesignvariableofsuperstructure,Ce.
subtheconstructioncostofsubstructure,Ce.
suptheconstructioncostofthesuperstructure,andDKdenotesthecombinationofthedamagesdonetothesuperstructureandtothesubstructure.
Thesettlement-causeddamagestothesuperstructureareassumedtobedependentfromthesettlement-causeddamagestothesubstructure.
DKshouldbeevaluatedbytakingthemechanicalandfunctionalinteractionsbetweenthesuperstructureandsubstructureintoaccount.
Anexamplewillbepresentedlater.
P(~)istheoccurrenceprobabilityofDK,andL':CF(DK)P(DK;AA.
ub)istheexpectedlossproducedbyDK.
Theoptimumdesignchoiceisgivenby(2}.
.
inwhichAupandAubaretheoptimumdesignvariablesofthesuperstructureandthesubstructureselectedoutofmanydesignaltematives,AupandA.
ub.
3.
OCCURRENCEPROBABILITYOFSETTLEMENTSupposeabridgeshowninFigure3.
Thedifferential(uneven)settlementoisloadPQi1~S;#.
.
0njs;-H(i)thpierr-L-i+1)thpiersoft'piledgroundfoundationoN,+-'·M,.
.
.
.
,.
,.
.
,.
ocdenotestheexpectedlossforthecase@,P(Dsuh.
1)denotestheoccurrenceprobabilityofthedifferentialsettlementDsub.
1,P(Dsup,21Dsub.
1)denotes17.
4*(a)Gsub=l.
l3r~19(b)*~Gsup=l86=218Gsub=O.
1.
6'"-1GsubGsupFigure1OOptimumSolutionsOptimizationoffoundationofbridgeonsoftground119theprobabilityoffailureofthemaingirdersubjectedtotheadditionalstresses.
EachcaseshowninFig.
9ishandledinthesamefashion.
Thesummationoftheexpectedlossesforalithecasesplusconstructioncostistheobjectivefunctionwhichwetrytominimizebyproperlychoosingthedesignaltematives,AsupandAsubFigure1Oshowthefmalresultsoftheabovementionedoptimizationprocedure.
TheabscissaofFigure1O(a)isthesafetyfactorGsubagainstthedifferentialsettlementofthefoundation,whiletheordinateisthetotalexpectedcostE[Gr]plottedagainstGsubwiththesafetyfactorGsupofthemaingirderasaparameter.
TheabscissaofFigure1O(b)isthesafetyfactorGsupofthemaingirderandtheparameteristhesafetyfactoroffoundation.
ThesafetyfactorsatwhichthetotalexpectedcostbecomesminimizedareGsup=1.
86andGsub=1.
13.
Thesetwovaluesaretheoptimumcombinationoftwosafetyfactorsforthesuperandsubstructure.
Itmaybeinterestingtocomparetothesetwovalueswiththesafetyfactorsrequiredbythecurrentconventionaldesigncodes,i.
e.
,Gsup=l.
7andGsub=l.
4.
Thesafetyfactoroffoundationintheoptimumdesignissmallerthanthesafetyfactorinthecurrentdesigncode.
Thesafetyfactorofmaingirderintheoptimumdesignislargerthanthesafetyfactorinthecurrentdesigncode.
Theseresultsareduetothesettlementoffoundationattheoptimumdesignwhichislargerthanthesettlementallowableinthecurrentdesigncode.
5.
CONCLUSIONSTheoptimizationprocedureforthebridgedesignisbrieflyoutlinedandanexampleoftheapplicationoftheoptimizationprocedureispresented.
Astheconclusions,followingsshouldbenoted.
(1)Theuseoftheobjectivefunctionderivedforthetotalsystemincludingboththesuperstructureandthesubstructureleadstotheoptimumdesignmorerationalthanthedesignoptimizedseparatelyforthesuperstructureandsubstructure.
(2)Theexamplepresentedinthispaperresultedthesafetyfactorsforthesuperstructureandsubstructurewhichhappenedtobefairlyclosetothesafetyfactorsrequiredbytheconventionaldesigncodes.
(3)Theproposedmethodseemstobeusefulinseekingthebridgedesignswithmuchharmonyinthewholesystemofthesuperstructureandsubstructure.
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MatsuoandY.
Demura,Proc.
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