優(yōu)化結(jié)構(gòu)設(shè)計-外文文獻

SURVEYPAPEROptimizationofStructuralDesignI.~W.PRAGER3Abstract.Typicalproblemsofoptimalstructuraldesignarediscussedtoindicatemathematicaltechniquesusedinthisfield.Anintroductoryexample(Section2)concernsthedesignofabeamforprescribedmaximaldeflectionandshowshowsuitablediscretizationmayleadtoaproblemofnonlinearprogramming,inthiscase,convexprogramming.Theproblemofoptimallayoutofatruss(Section3)isdiscussedatsomelength.Anewmethodofestablishingoptimalitycriteria(Section4)isillustratedbytheoptimaldesignofastaticallyindeterminatebeamofsegmentwiseconstantorcontinuouslyvaryingcrosssectionforgivendeflectionunderasingleconcentratedload.Otherapplicationsofthismethod(Section5)arebrieflydiscussed,andasimpleexampleofmultipurposedesign(Section6)concludesthepaper.IntroductionThemostgeneralproblemofstructuraloptimizationmaybestatedasfollows:fromallstructuraldesignsthatsatisfycertainconstraints,selectoneofminimalcost.Notethatthisstatementdoesnotnecessarilydefineauniquedesign;theremaybeseveraloptimaldesignsofthesameminimalcost.Typicaldesignconstraintsthatwillbeconsideredinthefollowingspecifyupperboundsfordeformationsorstresses,orlowerboundsforload-carryingcapacity,bucklingload,orfundamentalnaturalfrequency.Bothsinglepurposeandmultipurposestructureswillbeconsidered,thatis,structuresthatarerespectivelysubjecttoasingledesignconstraintoramultiplicityofconstraints.Thetermcostinthestatementofthedesignobjectivemayrefertothemanufacturingcostortothetotalcostofmanufactureandoperationovertheexpectedlifetimeofthestructure.Inaerospacestructures,thecostofthefuelneededtocarryagreaterweightfrequentlyovershadowsthecostofmanufacturetosuchanextentthatminimalweightbecomesthesoledesignobjective.Thispointofviewwillbeadoptedinthefollowing.Inthefirstpartofthispaper,typicalproblemsofoptimaldesignwillbediscussedtoillustratemathematicaltechniquesthathavebeenusedinthisfield.Thesecondpartwillbeconcernedwithapromisingtechniqueofwideapplicabilitythathasbeendevelopedrecently.Throughoutthepaper,itwillbeemphasizedthattheclassofstructureswithinwhichanoptimumissoughtmustbecarefullydefinedifmeaninglesssolutionsaretobeavoided.Thefactwillalsobestressedthatcertainintuitiveoptimalitycriteriaofgreatappealtoengineersdonotnecessarilyfurnishtrueoptima.Forgreaterclarityinthepresentationofdesignprinciples,themajorityofexampleswillbeconcernedwithsingle-pruposestructureseventhoughmultipurposestructuresareoffargreaterpracticalimportance.DiscretizationToexplorethemathematicalcharacterofaproblemofstructuraloptimization,itisfrequentlyusefultoreplacethecontinuousstructurebyadiscreteanalog.Consider,forinstance,thesimply-supportedelasticbeaminFig.1.Themaximumdeflectionproducedbythegivenload6Pisnottoexceedagivenvalue Todiscretizetheproblem,replacethebeambyasequenceofrigidrodsthatareconnectedbyelastichinges.InFig.1,onlyFig.1.Discreteanalogofelasticbeam.threehingeshavebeenintroduced;but,tofurnishrealisticresults,thediscretizationwouldhavetouseamuchgreaternumberofhinges.Thebendingmomentitransmittedacrosstheithhingeissupposedtoberelatedtotheangleofflexureibyi=sii(1)wheresiistheelasticstiffnessofthehinge.Sincethebeamisstaticallydeterminate,thebendingmoments iatthehingesareindependentofthestiffnessessi;thus,1=5Ph=s11, 2=3Ph=s22,3=Ph=s33.(2)Inthefollowing,theanglesofflexurei,willbetreatedassmall.InadesignspacewiththerectangularCartesiancoordinates i,i=1,2,3,thenonnegativecharacteroftheanglesofflexureandtheconstraintsonthedeflectionsuiatthehingesdefinetheconvexfeasibledomain1，2，30,51+32+3-6/h0,31+92-33-6/h0,（3）1+32+53-6/h0，Aswillbeshowninconnectionwithalaterexample,thecost(intermsofweight)ofprovidingacertainstiffnessmaybeassumedtobeproportionaltothisstiffness.Thedesignobjectivethusiss1+s2+s3=Minor,by(2),5/1+3/2+1/3=Min(4)Notethat,fortheconvexprogram(3)-(4),alocaloptimumisnecessarilyaglobaloptimum.Thisremarkisimportantbecauseadesignthatcanonlybestatedtobelighterthanall neighboringdesignssatisfyingtheconstraintsisoflittlepracticalinterest.Notealsothattheoptimumwillnot,ingeneral,correspondtoapointofdesignspacethatliesonanedgeorcoincideswithavertexofthefeasibledomain.Thisremarkshowsthattheintuitivelyappealingconceptof competingconstraintsisnotnecessarilyvalid.Suppose,forinstance,thatadesigns1,s2,s3hasbeenfoundforwhichu3<u2<u1=.Ifsdenotesasufficientlysmallchangeofstiffness,thedesigns1+s,s2-s,s3,whichhasthesameweight,mightthenbeexpectedtohavedeflectionu1，u2，u3satisfying u3<u2,u2<u1<u1=,andallthreestiffnessescouldbedecreasedinproportionuntilthedeflectionatthefirsthingehasagainthevalue.Ifthisargumentwerecorrect,thisprocessofreducingthestructuralweightcouldberepeateduntilthedeflectionsatthehinges1and2hadboththevalue&.Insubsequentdesignchanges,s1ands2wouldbeincreasedbythesamesmallamountwhiles3wouldbedecreasedbytwicethisamounttokeeptheweightconstant.Inthisway,itmightbearguedthattheoptimaldesignmustcorrespondtoapointonanedgeoratavertexofthefeasibledomain,thatis,that,fortheoptimaldesign,twoorthreeoftheconstraininginequalitiesmustbefulfilledasequations.Thisconceptofcompetingconstraints,towhichappealisfrequentlymadeintheengineeringliterature,isobviouslynotapplicabletotheproblemonhand.Minimum-weightdesignofbeamswithinequalityconstraintsondeflectionhasrecentlybeendiscussedbyHaugandKirmser(Ref.1).Earlierinvestigations(see,forinstance,Refs.2-4)involvedinequalityconstraintsonthedeflectionataspecificpoint,forinstance,atthepointofapplicationofaconcentratedload.Inspecialcases,wherethelocationofthepointofmaximumdeflectionisknownapriori,forinstance,fromsymmetryconsiderations,aconstraintonthemaximumdeflectioncanbeformulatedinthis way.AsBarnett(Ref.3)haspointedout, however,constrainingaspecificratherthanthemaximumdeflectionmayleadtoparadoxicalresults.Forexample,whensomeloadsactingonahorizontalbeamaredirecteddownwardwhileothersaredirectedupward,itmaybepossibletofindadesignforwhichthedeflectionatthespecifiedpointiszero.Sinceitwillremainzeroasallstiffnessesaredecreasedinproportion,thedesignconstraintiscompatiblewithdesignsofarbitrarilysmallweight.OptimalIntheprecedingexample,thetypeandlayoutofthestructure(simplysupported,straightbeam)weregivenandonlycertainlocalparameters(stiffnessvalues)wereatthechoiceofthedesigner.Amuchmorechallengingproblemariseswhentypeand/orlayoutmustalsobechosenoptimally.Figure2ashowsthegivenpointsofapplicationofloadsPandQthataretobetransmittedtotheindicatedsupportsbyatruss,thatis,astructureconsistingofpin-connectedbars,thelayoutofwhichistobedeterminedtominimizethestructuralweight.Tosimplifytheanalysis,Dorn,Gomory,andGreenberg(Ref.5)discretizedtheproblembyrestrictingtheadmissiblelocationsofthejointsofthetrusstothepointsofarectangulargridwithhorizontalspacinglandverticalspacingh(Fig.2a).Optimizationisthenfoundtorequirethesolutionofalinearprogram.TheoptimallayoutdependsFig.2.OptimallayoutoftrussaccordingtoDorn,Gomory,andGreenberg(Ref.5).onthevaluesoftheratiosh/landP/Q.Figures2bthrough2dshowoptimallayoutsforh/l=1andP/Q=O,,and.Forh/l=1andagivenvalueof P/Q,theoptimallayoutisuniqueexceptforcertaincriticalvaluesof P/Q,atwhichtheoptimallayoutchanges,forinstance,fromtheforminFig.2ctothatinFig.2d.Thenextexample,however,admitsaninfinityofoptimallayoutsthatareallassociatedwiththesamestructuralweight.ThreeforcesofthesameintensityP,withconcurrentlinesofactionthatformanglesof120°witheachother,havegivenpointsofapplicationthatformanequilateraltriangle(Fig.3@Atrussthatconnectsthesepointsistobedesignedforminimalweight,whenanupperbound0isprescribedforthemagnitudeoftheaxialstressinanybar.Figures3band3cshowfeasiblelayouts.Aftertheforcesinthebarsofthesestaticallydeterminatetrusseshavebeenfoundfromequilibriumconsiderations,thecross-sectionalareasaredeterminedtofurnishanaxialstressofmagnitude0ineachbar.Thefollowingargument,whichisduetoMaxwell(Ref.6,pp.175-177),

showsthatthetwodesignshavethesameweight.Imaginethattheplanesofthetrussesaresubjectedtothesamevirtual,uniform,planardilatationthatproducestheconstantunitextensioneforall lineelements.Bytheprincipleofvirtualwork,thevirtualexternalwork WeoftheloadsPonthevirtualdisplacementsoftheirpointsofapplicationFig.3.Alternativeoptimaldesigns.equalsthevirtualinternalworkvirtualelongations~equalsthevirtualinternalworkvirtualelongations~ofthebars.thetypicalbararedenotedbyAandWi=FofthebarforcesFontheIfcross-sectionalareaandlengthofL,thenF=0Aand=L.Thus,WiWi=0AL=0V5)whereVisthetotalvolumeofmaterialusedforthebarsofthetruss.Now,Wedependsonlyontheloadsandthevirtualdisplacementsoftheirpointsofapplicationbutisindependentofthelayoutofthebars;therefore,ithasthesamevalueforbothtrusses.IffollowsfromWe=Wiand(5)thatthetwotrussesusethesameamountofmaterial.Ifallcross-sectionalareasofthetwotrussesarehalved,eachofthenewtrusseswillbeabletocarryloadsofthecommonintensity P/2withoutviolatingthedesignconstraint.SuperpositionofthesetrussesinthemannershowninFig.3dthenresultsinanalternativetrussforthefullloadintensityPthathasthesameweightasthetrussesinFigs.3band3c.Fig.4.AlternativesolutiontoprobleminFig.3a.Figure4showsanothersolutiontotheproblem.Thecenterlinesoftheheavyedgemembersarecirculararcs.Theaxialforceineachofthesemembershasconstantmagnitudecorrespondingtothetensileaxialstress0.Theotherbarsarecomparativelylight.Theyarealsounderthetensileaxialstress 0andareprismatic,exceptforthebars AO,BO,andCO,whicharetapered.Thebarsthatarenormaltothecurvededgemembersmustbedenselypacked.Ifonlyafinitenumberisused,asinFig.4,andtheedgemembersaremadepolygonalratherthancircular,aslightlyhigherweightresults.Thisstatement,however,ceasestobevalidwhentheweightoftheconnectionsbetweenbars(gussetplatesandrivetsorwelds)istakenintoaccount.TheinteriorbarsinFig.4mayalsobereplacedbyawebofuniformthicknessunderbalancedbiaxiattension.Whilefullycompetitiveastoweight,thisdesignhas,however,beenexcludedbytheunnecessarilynarrowformulationoftheproblem,whichcalledforthedesignofatruss.Inthiscase,theexcludeddesigndoesnothappentobelighterthantheothers.However,unlesstheclassofstructureswithinwhichanoptimumissoughtisdefinedwithsufficientbreadth,itmayonlyfurnishasequenceofdesignsofdecreasingweightthatconvergestowardanoptimumthatisnotitselfamemberoftheconsideredclass.Figure5illustratesthisremark.Thediscreteradialloadsattheperipheryaretobetransmittedtothecentralringbyastructureofminimalweight.IfthewordstructureinthisstatementweretobereplacedbytheexpressionFig.5.Optimalstructurefortransmittingperipheralloadsto

centralringistrussratherthandisk

diskofcontinuouslyvaryingthickness, theoptimalstructureofFig.5wouldbeexcluded.NotethatFig.5showsonlytheheavymembers.Betweenthese,therearedenselypackedlightmembersalongthelogarithmicspiralsthatintersecttheradiiat 45oTheproblemindicatedinFig.3ahasaninfinityofsolutions,eachofwhichcontainsonlytensionmembers.Figure6illustratesaproblemthatrequirestheuseofcompressionaswellastensionmembersandhasauniquesolution.ThehorizontalloadPatthetopofthefigureistobetransmittedtothecurved,rigidfoundationatthebottombyatrusslikestructureofFig.6.UniqueoptimalstructurefortransmissionofloadPtocurved,rigidwall.minimalweight,thestressesinthebarsofwhicharetobeboundedby-0and0.Theoptimaltrusshasheavyedgemembers;thespacebetweenthemisfilledwithdenselypacked,lightmembers,onlyafewofwhichareshowninFig.6.Notethatthedisplacementsofthedenselypackedjointsofthestructuredefineadisplacementfieldthatleavesthepointsofthefoundationfixed.Adisplacementfieldsatisfyingthisconditionwiltbecalledkinematicallyadmissible.Thereisakinematicallyadmissibledisplacementfieldthateverywherehastheprincipalstrains 1=0/Eand2=-0/E,whereEis

Young'smodulus.Indeed,ifuandvarethe(infinitesimal)displacementcomponentswithrespecttorectangularaxesxandy,thefactthattheinvariant 1+2vanishesfurnishestherelationux+vy=0，(6)wherethesubscriptsxandyindicatedifferentiationwithrespecttothecoordinates.Similarly,thefactthatthemaximumprincipalstrainhastheconstantvaluee1yieldstherelation24ux*vy-(vx+uy)(vx+uy)=-41 (7)Inviewof(6),thereexistsafunctionx,ysuchthat8)Substitutionof(8)intothefoundationare,u=u=8)Substitutionof(8)intothefoundationare,u=(7)finallyfurnishes2224 xy2+xxyy=412 (9)Alongv=O,whichisequivalentto=0， =0 (10)where isthennderivativeofTalongthenormaltothefoundationare.Thepartialdifferentialequation(9)ishyperbolic,anditscharacteristicsarethelinesofprincipalstrain.TheCauchyconditions(10)onthefoundationarcuniquelydeterminethefunction,andhencethedisplacements(8),inaneighborhoodofthisarc.ThesedisplacementswillnowbeusedasvirtualdisplacementsintheapplicationoftheprincipleofvirtualworktoanarbitrarytrusslikestructurethattransmitstheloadPtothefoundationare(Fig.6)andinwhicheachbarisunderanaxialstressofmagnitude%.WiththenotationsusedaboveinthepresentationofMaxwell'sargmnent,We=Wi=F.Here,｜F｜=0Aand｜｜0/EL,becausenolineelementexperiencesaunitextensionorcontractionofamagnitudeinexcessof 0/E.Accordingly,We=F∑｜F｜｜｜(02/E)V, (11)whereVisagainthetotalvolumeofmaterialusedinthestructure.Next,imagineasecondtrusslikestructurewhosemembersfollowthelinesofprincipalstrainoftheconsideredvirtualdisplacementfieldandundergothecorrespondingstrains.Quantitiesreferringtothisstructurewillbemarkedbyanasterisk.Applyingtheprincipleofvirtual****workasbefore,onehas We=We,butF**=0Aand*=0/ELwithcorrespondenceofsigns.Accordingly,*We=F**=02/EV* (12)InviewofWe*=We,comparisonof(11)and(12)revealsthatthesecondstructurecannotusemorematerialthanthefirst.TheargumentjustpresentedisduetoMichell(Ref.7),who,however,consideredpurelystaticboundaryconditionsand,consequently,failedtoarriveatauniqueoptimalstructure.Theimportanceofkinematicboundaryconditionsfortheuniquenessofoptimaldesignwaspointedoutbythepresentauthor(Ref.8).Figure7illustratesanimportantgeometricpropertyoftheorthogonalcurvesofprincipalstraininafieldthathasconstantprincipalstrainsofequalmagnitudesandoppositesigns.LetABCandDEFbetwofixedcurvesofonefamily.Theanglec~formedbythetangentsofthesecurvesattheirpointsofintersectionwithacurveoftheotherfamilydoesnotdependonthechoiceofthelattercurve.Inthetheoryofplaneplasticflow,orthogonalfamiliesofFig.7.Geometryofoptimallayout.curvesthathavethisgeometricpropertyindicatethedirectionsofthemaximumshearingstresses(sliplines).Inthiscontext,theyareusuallynamedafterHencky(Ref.9)andPrandtl(Ref.10);theirpropertieshavebeenstudiedextensively(see,forinstance,Refs.11-13).Figure8showstheoptimallayoutwherethespaceavailableforthestructureisboundedbytheverticalsthroughdandB.Becausethefoundationarcisastraight-linesegment,therearenobarsinsidethetriangledBC.Hereagain,theedgemembersareheavy,andtheothermembers,ofwhichonlyafewareshown,arecomparativelylight.Thelayoutofthesebarsstronglyresemblesthetrajectoriatsystemofthehumanfemur(see,forinstance,ReL14,p.12,Fig.6).ForfurtherexamplesofMichellstructures,seeRefs.15-16.NewMethodofEstablishingOptimalityCriteriaThebeaminFig.9isbuiltinatAandsimplysupportedbyBandC.ItsdeflectionatthepointofapplicationofthegivenloadPistohavethegivenvalue .ThebeamistohavesandwichsectionofconstantcorebreadthBandconstantcoreheightH.ThefacesheetsaretohavethecommonbreadthB,andtheirconstantthicknesses T1《HandT2《HinthespansL1andL2aretobedeterminedtominimizethestructuralweightofthebeam.Sincethe

Fig.8.OptimallayoutwhenavailablespaceisboundedbyverticalsthroughAandB.dimensionsofthecoreareprescribed,minimizingtheweightofthebeammeansminimizingtheweightofthefacesheets.Moreover,sincetheelasticbendingstiffnesssiofthecrosssectionwithfacesheetthicknessTi,i=1,2, issiEBH2Ti/2,whereEisYoung'smodulus,mayberegardedasthequantitythatistobeminimized.WL1WL1s1L2s2(13)Fig.9.Beamwithspanwiseconstantcrosssection.Letxibethedistanceofthetypicalcrosssectioninthespan LifromtheLeftendofthisspan,anddenotecurvatureandbendingmomentatthiscrosssectionbykiandMisiki.Theprescribedquantity Pmaythenbewrittenas2P= Mikidxi=siki2dxi(14)wheretheintegrationisextendedovertheiispanLiWithintheframeworkoftheproblem,abeamdesignisdeterminedbythevaluesofsi,i=t,2.Ifsiandsiaretwodesignssatisfyingthedesignconstraint(givenvalueofP),andkiandkiarethecurvaturesthattheyassumeunderthegivenload,itfollowsfrom(14)thatsiki2dxi= sikidxi (15)Moreover,sincethecurvatureiikiiskinematicallyadmissible.,derivedfromadeflectionsatisfyingtheconstraintsatthesupport)forthedesign si,itfollowsfromtheprincipleofminimumpotentialenergyforthedesign sithatsikidxi2Piisiki2dxi2P(16)Suppressingtheterms2Pin(16)andusing(15),oneobtainstheinequalitysisiLii0i(17)wherei(1/Li)ki2dxi（18）isthemean-squarecurvatureinthespan Li.If12(19)itfollowsfrom(17)and(13)thatthedesigns~thatsatisfies(19)inadditiontothedesignconstraintcannotbeheavierthananarbitrarydesignsithatsatisfiesonlythedesignconstraint.Thecondition(19)thusissufficientforoptimality;thatitisalsonecessarymaybeshownasfollows.Withthedefinitioni sisiLi(20)theconditionthatthedesignsishouldnotbeheavierthanthedesign sitakestheformi0.(21)iOntheotherhand,theinequality(17),whichfollowedfromtheprincipleofminimumpotentialenergy,becomesii0.(22)iThequantities1,2and1,2willberegardedasthecomponentsofvectorsandwithrespecttothesamerectangularaxes.Theinequality(21)statesthatthevector cannotpointfromtheoriginintothehalf-spacebelowthebisectorsofthesecondandfourthquadrants,andtheinequality(22)demandsthatthescalarproductof andbenonnegative.Now,theoptimaldesignsianditsmeancurvaturesiareunknownbutfixed.Thedesign si,ontheotherhand,isonlysubjecttothedesignconstraint,whichprescribesthevalueofPand,hence,determinesthemagnitudeofthevector)twhenitsdirectionhasbeenchosen.Moreover,intheneighborhoodoftheoptimaldesign si,therearedesigns siofstructuralweightsthatcomearbitrarily closetotheminimumweight.Thecorrespondingvectors arearbitrarilyclosetotheboundaryofthehalf-spacedefinedbytheinequality(21).Ifthescalarproductofandistobenonnegativeforallfeasiblevectors ,thevector mustbedirectedalongtheinteriornormalofthishalf-spaceattheorigin,thatis,(19)isanecessaryconditionforoptimality.

ThisproofofnecessityisduetoSheuandPrager(Ref.17).MultipurposeDesignFigure11illustratesaproblemofmultipurposedesign.Underdifferentconditionsofloading,oneandthesamestructuralelementistoserveastie,beam,orcolumn.Inthefirstcase,itselongationunderthegivenlongitudinalloadLisnottoexceedthegivenvalue Inthesecondcase,itsdeflectionunderthegivencentraltra