優(yōu)化結(jié)構(gòu)設(shè)計(jì)外文文獻(xiàn)翻譯、中英文翻譯、外文翻譯

1、JOURNALOFOPTIMIZATIONTHEORYANDAPPLICATIONS:Vol.6,No.I.1970SURVEYPAPEROptimizationofStructuralDesignI.W.PRAGER3AbstractTypicalproblemsofoptimalstructuraldesignarediscussedtoindicatemathematicaltechniquesusedinthisfield.Anintroductoryexample(Section2)concernsthedesignofabeamforprescribedmaximaldeflect

2、ionandshowshowsuitablediscretizationmayleadtoaproblemofnonlinearprogramming,inthiscase,convexprogramming.Theproblemofoptimallayoutofatruss(Section3)isdiscussedatsomelength.Anewmethodofestablishingoptimalitycriteria(Section4)is川ustratedbytheoptimaldesignofastaticallyindeterminatebeamofsegmentwisecons

3、tantorcontinuouslyvaryingcrosssectionforgivendeflectionunderasingleconcentratedload.Otherapplicationsofthismethod(Section5)arebrieflydiscussed,andasimpleexampleofmultipurposedesign(Section6)concludesthepaper.1. IntroductionThemostgeneralproblemofstructuraloptimizationmaybestatedasfollows:fromallstru

4、cturaldesignsthatsatisfycertainconstraints,selectoneofminimalcost.Notethatthisstatementdoesnotnecessarilydefineauniquedesign;theremaybeseveraloptimaldesignsofthesameminimalcost.Typicaldesignconstraintsthatwillbeconsideredinthefollowingspecifyupperboundsfordeformationsorstresses,orlowerboundsforload-

5、carryingcapacity,bucklingload,orfundamentalnaturalfrequency.Bothsinglepurposeandmultipurposestructureswillbeconsidered,thatis,structuresthatarerespectivelysubjecttoasingledesignconstraintoramultiplicityofconstraints.Thetermcostinthestatementofthedesignobjectivemayrefertothemanufacturingcostortotheto

6、talcostofmanufactureandoperationovertheexpectedlifetimeofthestructure.Inaerospacestructures,thecostofthefuelneededtocarryagreaterweightfrequentlyovershadowsthecostofmanufacturetosuchanextentthatminimalweightbecomesthesoledesignobjective.Thispointofviewwillbeadoptedinthefollowing.Inthefirstpartofthis

7、paper,typicalproblemsofoptimaldesignwillbediscussedto川ustratemathematicaltechniquesthathavebeenusedinthisfield.Thesecondpartwillbeconcernedwithapromisingtechniqueofwideapplicabilitythathasbeendevelopedrecently.Throughoutthepaper,itwillbeemphasizedthattheclassofstructureswithinwhichanoptimumissoughtm

8、ustbecarefullydefinedifmeaninglesssolutionsaretobeavoided.Thefactwillalsobestressedthatcertainintuitiveoptimalitycriteriaofgreatappealtoengineersdonotnecessarilyfurnishtrueoptima.Forgreaterclarityinthepresentationofdesignprinciples,themajorityofexampleswillbeconcernedwithsingle-pruposestructureseven

9、thoughmultipurposestructuresareoffargreaterpracticalimportance.2. DiscretizationToexplorethemathematicalcharacterofaproblemofstructuraloptimization,itisfrequentlyusefultoreplacethecontinuousstructurebyadiscreteanalog.Consider,forinstance,thesimply-supportedelasticbeaminFig.1.Themaximumdeflectionprod

10、ucedbythegivenload6PisnottoexceedagivenvalueTodiscretizetheproblem,replacethebeambyasequenceofrigidrodsthatareconnectedbyelastichinges.InFig.1,onlyFig.1.Discreteanalogofelasticbeam.threehingeshavebeenintroduced;but,tofurnishrealisticresults,thediscretizationwouldhavetouseamuchgreaternumberofhinges.T

11、hebendingmomentitransmittedacrosstheithhingeissupposedtoberelatedtotheangleofflexureby”i=sW(1)wheresistheelasticstiffnessofthehinge.Sincethebeamisstaticallydeterminate,thebendingmomentsMiatthehingesareindependentofthestiffnesses;thus,11=5Ph=s%,12=3Ph=s212,I3=Ph=備.(2)Inthefollowing,theanglesofflexure

12、q,willbetreatedassmall.InadesignspacewiththerectangularCartesiancoordinates,i=1,2,3,thenonnegativecharacteroftheanglesofflexureandtheconstraintsonthedeflectionsatthehingesdefinetheconvexfeasibledomain91,92,%至0,5%+3。2+4-6:/h0,34+9%-3%-66/|1W0,(3)91+32+503-65/h0,Aswillbeshowninconnectionwithalaterexam

13、ple,thecost(intermsofweight)ofprovidingacertainstiffnessmaybeassumedtobeproportionaltothisstiffness.Thedesignobjectivethusissi+s2+ss=Minor,by(2),5/11+3/i2+1/飛=MinNotethat,fortheconvexprogram(3)-(4),alocaloptimumisnecessarilyaglobaloptimum.Thisremarkisimportantbecauseadesignthatcanonlybestatedtobelig

14、hterthanallneighboringdesignssatisfyingtheconstraintsisoflittlepracticalinterest.Notealsothattheoptimumwillnot,ingeneral,correspondtoapointofdesignspacethatliesonanedgeorcoincideswithavertexofthefeasibledomain.Thisremarkshowsthattheintuitivelyappealingconceptolcompetingconstraintsisnotnecessarilyval

15、id.Suppose,forinstance,thatadesign,s2,s3hasbeenfoundforwhichu3u2u1=、.Ifsdenotesasufficientlysmallchangeofstiffness,thedesigns+s,電-s,&,whichhasthesameweight,mightthenbeexpectedtohavedeflectionui,u2,u3satisfyingu3u2,u2uiu1=6,andallthreestiffnessescouldbedecreasedinproportionuntilthedeflectionatthefirs

16、thingehasagainthevalue.Ifthisargumentwerecorrect,thisprocessofreducingthestructuralweightcouldberepeateduntilthedeflectionsatthehinges1and2hadboththevalue&.Insubsequentdesignchanges,sands2wouldbeincreasedbythesamesmallamountwhiles3wouldbedecreasedbytwicethisamounttokeeptheweightconstant.Inthisway,it

17、mightbearguedthattheoptimaldesignmustcorrespondtoapointonanedgeoratavertexofthefeasibledomain,thatis,that,fortheoptimaldesign,twoorthreeoftheconstraininginequalitiesmustbefuelledasequations.Thisconceptofcompetingconstraints,towhichappealisfrequentlymadeintheengineeringliterature,isobviouslynotapplic

18、abletotheproblemonhand.Minimum-weightdesignofbeamswithinequalityconstraintsondeflectionhasrecentlybeendiscussedbyHaugandKirmser(Ref.1).Earlierinvestigations(see,forinstance,Refs.2-4)involvedinequalityconstraintsonthedeflectionataspecificpoint,forinstance,atthepointofapplicationofaconcentratedload.In

19、specialcases,wherethelocationofthepointofmaximumdeflectionisknownapriori,forinstance,fromsymmetryconsiderations,aconstraintonthemaximumdeflectioncanbeformulatedinthisway.AsBarnett(Ref.3)haspointedout,however,constrainingaspecificratherthanthemaximumdeflectionmayleadtoparadoxicalresults.Forexample,wh

20、ensomeloadsactingonahorizontalbeamaredirecteddownwardwhileothersaredirectedupward,itmaybepossibletofindadesignforwhichthedeflectionatthespecifiedpointiszero.Sinceitwillremainzeroasallstiffnessesaredecreasedinproportion,thedesignconstraintiscompatiblewithdesignsofarbitrarilysmallweight.3. OptimalInth

21、eprecedingexample,thetypeandlayoutofthestructure(simplysupported,straightbeam)weregivenandonlycertainlocalparameters(stiffnessvalues)wereatthechoiceofthedesigner.Amuchmorechallengingproblemariseswhentypeand/orlayoutmustalsobechosenoptimally.Figure2ashowsthegivenpointsofapplicationofloadsPandQthatare

22、tobetransmittedtotheindicatedsupportsbyatruss,thatis,astructureconsistingofpin-connectedbars,thelayoutofwhichistobedeterminedtominimizethestructuralweight.Tosimplifytheanalysis,Dorn,Gomory,andGreenberg(Ref.5)discretizedtheproblembyrestrictingtheadmissiblelocationsofthejointsofthetrusstothepointsofar

23、ectangulargridwithhorizontalspacinglandverticalspacingh(Fig.2a).Optimizationisthenfoundtorequirethesolutionofalinearprogram.TheoptimallayoutdependsP2QPFig.2.OptimallayoutoftrussaccordingtoDorn,Gomory,andGreenberg(Ref.5).onthevaluesoftheratiosh/landP/Q.Figures2bthrough2dshowoptimallayoutsforh/l=1andP

24、/Q=O,0.5,and2.0.Forh/l=1andagivenvalueofP/Q,theoptimallayoutisuniqueexceptforcertaincriticalvaluesofP/Q,atwhichtheoptimallayoutchanges,forinstance,fromtheforminFig.2ctothatinFig.2d.Thenextexample,however,admitsaninfinityofoptimallayoutsthatareallassociatedwiththesamestructuralweight.Threeforcesofthe

25、sameintensity3,withconcurrentlinesofactionthatformanglesof120witheachother,havegivenpointsofapplicationthatformanequilateraltriangle(Fig.3Atrussthatconnectsthesepointsistobedesignedforminimalweight,whenanupperbound00isprescribedforthemagnitudeoftheaxialstressinanybar.Figures3band3cshowfeasiblelayout

26、s.Aftertheforcesinthebarsofthesestaticallydeterminatetrusseshavebeenfoundfromequilibriumconsiderations,thecross-sectionalareasaredeterminedtofurnishanaxialstressofmagnitude。ineachbar.Thefollowingargument,whichisduetoMaxwell(Ref.6,pp.175-177),showsthatthetwodesignshavethesameweight.Imaginethattheplan

27、esofthetrussesaresubjectedtothesamevirtual,uniform,planardilatationthatproducestheconstantunitextensioneforalllineelements.Bytheprincipleofvirtualwork,thevirtualexternalworkWeoftheloadsPonthevirtualdisplacementsoftheirpointsofapplicationequalsthevirtualinternalworkWf=ZFofthebarforcesFonthevirtualelo

28、ngationsofthebars.Ifcross-sectionalareaandlengthofthetypicalbararedenotedbyAandL,thenF=；=0Aand，=；L.Thus,WJ=o0EAL=仃0wV(5)whereVisthetotalvolumeofmaterialusedforthebarsofthetruss.Now,Wedependsonlyontheloadsandthevirtualdisplacementsoftheirpointsofapplicationbutisindependentofthelayoutofthebars;therefo

29、re,ithasthesamevalueforbothtrusses.IffollowsfromWe=Wfand(5)thatthetwotrussesusethesameamountofmaterial.Ifallcross-sectionalareasofthetwotrussesarehalved,eachofthenewtrusseswillbeabletocarryloadsofthecommonintensityP/2withoutviolatingthedesignconstraint.SuperpositionofthesetrussesinthemannershowninFi

30、g.3dthenresultsinanalternativetrussforthefullloadintensityPthathasthesameweightasthetrussesinFigs.3band3c.Fig.4.AlternativesolutiontoprobleminFig.3a.Figure4showsanothersolutiontotheproblem.Thecenterlinesoftheheavyedgemembersarecirculararcs.Theaxialforceineachofthesemembershasconstantmagnitudecorresp

31、ondingtothetensileaxialstress0.Theotherbarsarecomparativelylight.Theyarealsounderthetensileaxialstress：.0andareprismatic,exceptforthebarsAO,BO,andCO,whicharetapered.Thebarsthatarenormaltothecurvededgemembersmustbedenselypacked.Ifonlyafinitenumberisused,asinFig.4,andtheedgemembersaremadepolygonalrath

32、erthancircular,aslightlyhigherweightresults.Thisstatement,however,ceasestobevalidwhentheweightoftheconnectionsbetweenbars(gussetplatesandrivetsorwelds)istakenintoaccount.TheinteriorbarsinFig.4mayalsobereplacedbyawebofuniformthicknessunderbalancedbiaxiattension.Whilefullycompetitiveastoweight,thisdes

33、ignhas,however,beenexcludedbytheunnecessarilynarrowformulationoftheproblem,whichcalledforthedesignofatruss.Inthiscase,theexcludeddesigndoesnothappentobelighterthantheothers.However,unlesstheclassofstructureswithinwhichanoptimumissoughtisdefinedwithsufficientbreadth,itmayonlyfurnishasequenceofdesigns

34、ofdecreasingweightthatconvergestowardanoptimumthatisnotitselfamemberoftheconsideredclass.Figure5illustratesthisremark.Thediscreteradialloadsattheperipheryaretobetransmittedtothecentralringbyastructureofminimalweight.IfthewordstructureinthisstatementweretobereplacedbytheexpressionFig.5.Optimalstructu

35、refortransmittingperipheralloadstocentralringistrussratherthandiskdiskofcontinuouslyvaryingthickness,theoptimalstructureofFig.5wouldbeexcluded.NotethatFig.5showsonlytheheavymembers.Betweenthese,therearedenselypackedlightmembersalongthelogarithmicspiralsthatintersecttheradiiat_45Theproblemindicatedin

36、Fig.3ahasaninfinityofsolutions,eachofwhichcontainsonlytensionmembers.Figure6illustratesaproblemthatrequirestheuseofcompressionaswellastensionmembersandhasauniquesolution.ThehorizontalloadPatthetopofthefigureistobetransmittedtothecurved,rigidfoundationatthebottombyatrusslikestructureofFig.6.Uniqueopt

37、imalstructurefortransmissionofloadPtocurved,rigidwall.minimalweight,thestressesinthebarsofwhicharetobeboundedby-and00.Theoptimaltrusshasheavyedgemembers;thespacebetweenthemisfilledwithdenselypacked,lightmembers,onlyafewofwhichareshowninFig.6.Notethatthedisplacementsofthedenselypackedjointsofthestruc

38、turedefineadisplacementfieldthatleavesthepointsofthefoundationfixed.Adisplacementfieldsatisfyingthisconditionwiltbecalledkinematicallyadmissible.Thereisakinematicallyadmissibledisplacementfieldthateverywherehastheprincipalstrains1=，/Eand；2=-o/E,whereEisYoungsmodulus.Indeed,ifuandvarethe(infinitesima

39、l)displacementcomponentswithrespecttorectangularaxesxandy,thefactthattheinvariant；1+；2vanishesfurnishestherelationux+vy=0,(6)wherethesubscriptsxandyindicatedifferentiationwithrespecttothecoordinates.Similarly,thefactthatthemaximumprincipalstrainhastheconstantvalueelyieldstherelation244*vy-(vx+uy)(vx

40、+uy)=-4；1Inviewof(6),thereexistsafunction可x,ysuchthatu=Py,v=Px(8)Substitutionof(8)intofinallyfurnishes2224彳xy+彳xxPyy=41(9)Alongthefoundationare,u=v=O,whichisequivalentto尸中P=0,=0二n(10)whereisthe；nderivativeofTalongthenormaltothefoundationare.Thepartialdifferentialequation(9)ishyperbolic,anditscharact

41、eristicsarethelinesofprincipalstrain.TheCauchyconditions(10)onthefoundationarcuniquelydeterminethefunctionP,andhencethedisplacements(8),inaneighborhoodofthisarc.Thesedisplacementswillnowbeusedasvirtualdisplacementsintheapplicationoftheprincipleofvirtualworktoanarbitrarytrusslikestructurethattransmit

42、stheloadPtothefoundationare(Fig.6)andinwhicheachbarisunderanaxialstressofmagnitude%.WiththenotationsusedaboveinthepresentationofMaxwellsargmnent,We=Wf=1F九.Here,|F|=仃0Aand|九|M(仃0/E)L,becausenolineelementexperiencesaunitextensionorcontractionofamagnitudeinexcessof00/E.Accordingly,(11)whereVisagainthe%

43、=工F九E|F|九|(仃02/E)V,totalvolumeofmaterialusedinthestructure.Next,imagineasecondtrusslikestructurewhosemembersfollowthelinesofprincipalstrainoftheconsideredvirtualdisplacementfieldandundergothecorrespondingstrains.Quantitiesreferringtothisstructurewillbemarkedbyan*asterisk.Applyingtheprincipleofvirtua

44、lworkasbefore,onehasWe=W，,but一*F*=ct0Aand九=(仃0/E)Lwithcorrespondenceofsigns.Accordingly,*2*We=ZF=O0/EV(12)InviewofWe=We,comparisonof(11)and(12)revealsthatthesecondstructurecannotusemorematerialthanthefirst.TheargumentjustpresentedisduetoMichell(Ref.7),who,however,consideredpurelystaticboundarycondit

45、ionsand,consequently,failedtoarriveatauniqueoptimalstructure.Theimportanceofkinematicboundaryconditionsfortheuniquenessofoptimaldesignwaspointedoutbythepresentauthor(Ref.8).Figure7illustratesanimportantgeometricpropertyoftheorthogonalcurvesofprincipalstraininafieldthathasconstantprincipalstrainsofeq

46、ualmagnitudesandoppositesigns.LetABCandDEFbetwofixedcurvesofonefamily.Theanglecformedbythetangentsofthesecurvesattheirpointsofintersectionwithacurveoftheotherfamilydoesnotdependonthechoiceofthelattercurve.Inthetheoryofplaneplasticflow,orthogonalfamiliesofFig.7.Geometryofoptimallayout.curvesthathavet

47、hisgeometricpropertyindicatethedirectionsofthemaximumshearingstresses(sliplines).Inthiscontext,theyareusuallynamedafterHencky(Ref.9)andPrandtl(Ref.10);theirpropertieshavebeenstudiedextensively(see,forinstance,Refs.11-13).Figure8showstheoptimallayoutwherethespaceavailableforthestructureisboundedbythe

48、verticalsthroughdandB.Becausethefoundationarcisastraight-linesegment,therearenobarsinsidethetriangdBC.Hereagain,theedgemembersareheavy,andtheothermembers,ofwhichonlyafewareshown,arecomparativelylight.Thelayoutofthesebarsstronglyresemblesthetrajectoriatsystemofthehumanfemur(see,forinstance,ReL14,p.12

49、,Fig.6).ForfurtherexamplesofMichellstructures,seeRefs.15-16.4.NewMethodofEstablishingOptimalityCriteriaThebeaminFig.9isbuiltinatAandsimplysupportedbyBandC.ItsdeflectionatthepointofapplicationofthegivenloadPistohavethegivenvalue、.ThebeamistohavesandwichsectionofconstantcorebreadthBandconstantcoreheig

50、htH.ThefacesheetsaretohavethecommonbreadthB,andtheirconstantthicknessesT1HandT2HinthespansL1andL2aretobedeterminedtominimizethestructuralweightofthebeam.SincetheFig.8.OptimallayoutwhenavailablespaceisboundedbyverticalsthroughAandB.dimensionsofthecoreareprescribed,minimizingtheweightofthebeammeansmin

51、imizingtheweightofthefacesheets.Moreover,sincetheelasticbendingstiffness2siofthecrosssectionwithfacesheetthicknessTi,i=1,2,iss=EBHTi/2,whereEisYoungsmodulus,W=LsL2s2(13)mayberegardedasthequantitythatistobeminimized.Fig.9.Beamwithspanwiseconstantcrosssection.LetXibethedistanceofthetypicalcrosssection

52、inthespanLifromtheLeftendofthisspan,anddenotecurvatureandbendingmomentatthiscrosectionbyKandMi=siK.TheprescribedquantityP、maythenbewrittenas2P、=_!Mikidx=sikidxi(14)wheretheintegrationisextendedoverthespalniWithintheframeworkoftheproblem,abeamdesignisdeterminedbythevaluesofS,i=t,2.Ifsiandsiaretwodesi

53、gnssatisfyingthedesignconstraint(givenvalueofP、)，andkiandkiarethecurvaturesthattheyassumeunderthegivenload,itfollowsfrom(14)that工siJK2dxi=sJkidx(15)Moreover,sincethecurvaturekiiskinematicallyadmissible(i.e.,derivedfromadeflectionsatisfyingtheconstraintsatthesupport)forthedesig由,itfollowsfromtheprinc

54、ipleofminimumpotentialenergyforthedesignthat.22.ZsiJkidxi-2P6Zsikjdx2P6(16)Suppressingtheterms2P6inii(16)andusing(15),oneobtainstheinequality(ssLi4之0(17)where2H=(1/Li)Jkidxi(18)isthemean-squarecurvatureinthespanLi.If4=$(19)itfollowsfrom(17)and(13)thatthedesignsthatsatisfies(19)inadditiontothedesignc

55、onstraintcannotbeheavierthananarbitrarydesignsithatsatisfiesonlythedesignconstraint.Thecondition(19)thusissufficientforoptimality;thatitisalsonecessarymaybeshownasfollows.Withthedefinition冊(cè)=(si-si)Li(20)theconditionthatthedesignsishouldnotbeheavierthanthedesignsitakestheform%i-0.(21)Ontheotherhand,t

56、heinequality(17),whichfollowedfromtheprincipleofminimumpotentialenergy,becomes“iJi0.(22)Thequantities1,2and1,L2willberegardedasthecomponentsofvectorsandwithrespecttothesamerectangularaxes.Theinequality(21)statesthatthevectorcannotpointfromtheoriginintothehalf-spacebelowthebisectorsofthesecondandfour

57、thquadrants,andtheinequality(22)demandsthatthescalarproductofandbenonnegative.Now,theoptimaldesignsianditsmeancurvaturesareunknownbutfixed.Thedesigns,ontheotherhand,isonlysubjecttothedesignconstraint,whichprescribesthevalueofP、and,hence,determinesthemagnitudeofthevector)twhenitsdirectionhasbeenchose

58、n.Moreover,intheneighborhoodoftheoptimaldesigns,therearedesignssofstructuralweightsthatcomearbitrarilyclosetotheminimumweight.Thecorrespondingvectorsarearbitrarilyclosetotheboundaryofthehalf-spacedefinedbytheinequality(21) .Ifthescalarproductofand二istobenonnegativeforallfeasiblevectors,thevector-mustbedirectedalongtheinteriornormalofthishalf-spaceattheorigin,thatis,(19)isanecessaryconditionforoptimality.ThisproofofnecessityisduetoSheuandPrager(Ref.17).5.MultipurposeDesignFigure11川ustra