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1、CHAPTERTHREEOptimalPlasticDesignofBeamswithUnspecifiedActionsorReactions3.1PRELIMINARYREMARKSOptimizationofunspecifiedforcesillustratessomeinterestingaspectsofop•timaldesign.Evenwhenthespecificcostfunctionisconvex,aproblemwithmovablesupportsbecomesusual
2、lynon-convexandhenceallsolutionssatis•fyingthenecessaryconditionsforoptimalitymustbeinvestigated.Moreover,somesegmentsofthebeammaytakeonazerocross-sectionintheoptimalsolution(seeFig.3.4e,forexample)andthusthedesignerisfacedwithasimpletypeoflayoutproblem
3、.Ifthemagnitudeofanexternalactionisatthechoiceofthedesigner,thenitobviouslyaffectstheoptimizationprocedure.Thesameappliestore•dundantreactionsbecausetheycanbefreelychoseninplasticdesign.Theseunspecifiedactionsandreactionsmayinfluenceouroptimaldesignintw
4、odifferentways.First,thecostofsuchforces(e.g.costofsupportsdepend•ingonreactionforces,costofballastonabeametc.)mustbetakenintoconsideration.Second,theoptimallocationofsupports(oractions)istobedetermined.3.2EXTERNALACTIONS(REACTIONS)ATPRESCRIBEDLOCATIONS
5、LetahorizontalbeambesubjecttosomeunspecifiedexternalactionsorreactionsR(r,m)atgivenpointsB;(i=1,2,...,n)ofthecentroidalaxiswhererisaverticalforceandmisacoupleinaverticalplanecontainingthe115G.I.N.Rozvany,StructuralDesignviaOptimalityCriteria©KluwerAcade
6、micPublishers1989ChapterThreeOptimalPlasticDesign:UnspecifiedActionscentroidalaxis.Moreover,letthecostofsuch(re)actionsbegivenby0.=0.(r,m)whichistermeda(re}actioncostfunction.TheoptimalsolutionforthebeammustthensatisfytheusualPrager-Shieldconditionin(2.
7、6)or(2.31)and(2.32),aswellasthefollowingoptimalityconditions:*atB.U=9"[0(r,m)],(3.1)u'=9,m[0(r,m)],(3.2)wheresubscriptsindicatethattheG-gradientmustbetakenwithrespecttothereactioncomponentmorr.ThepointsetB.(i=1,2,...,n)maycontainaninfinitenumberofpoints
8、andformalinesegment(seeProblemebelow).3.3EXTERNALACTIONSORREACTIONSOFUNSPECIFIEDLOCATION(A)Moment-DependentSpecificCostFunction:!b(M)(i)SimplesupportsofzerocostwithouthingeLetthecostofasimplesupport(overwhichthebeamiscontinuous)b
