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HindawiPublishingCorporationMathematicalProblemsinEngineeringVolume2013,ArticleID649857,[e-mail]29November2012;Revised29December2012;Accepted30December2012AcademicEditor:FeiKangCopyright©2013S.Talataharietal.ThisisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense,whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycited.Acceleratedparticleswarmoptimization(APSO)isdevelopedforfindingoptimumdesignofframestructures.APSOshowssomeextraadvantagesinconvergenceforglobalsearch.ThemodificationsonstandardPSOeffectivelyacceleratetheconvergencerateofthealgorithmandimprovetheperformanceofthealgorithminfindingbetteroptimumsolutions.TheperformanceoftheAPSOalgorithmisalsovalidatedbysolvingtwoframestructureproblems.1.IntroductionOptimumdesignofframestructuresareinclinedtodeter-minesuitablesectionsforelementsthatfulfillalldesignrequirementswhilehavingthelowestpossiblecost.Inthisissue,optimizationprovidesengineerswithavarietyoftech-niquestodealwiththeseproblems[1].Thesetechniquescanbecategorizedastwogeneralgroups:classicalmethodsandmetaheuristicapproaches[2].Classicalmethodsareoftenbasedonmathematicalprogramming,andmanyofmeta-heuristicmethodsmakeuseoftheideasfromnatureanddonotsufferthediscrepanciesofmathematicalprogramming[3–8].Particleswarmoptimization(PSO),oneofmeta-heuristicalgorithms,isbasedonthesimulationofthesocialbehaviorofbirdflockingandfishschooling.PSOisthemostsuc-cessfulswarmintelligenceinspiredoptimizationalgorithms.However,thelocalsearchcapabilityofPSOispoor[9],sinceprematureconvergenceoccursoften.InordertoovercomethesedisadvantagesofPSO,manyimprovementshavebeenproposed.ShiandEberhart[10]introducedafuzzysystemtoadapttheinertiaweightforthreebenchmarktestfunctions.Liuetal.[11]proposedcenterparticleswarmoptimizationbyaddingacenterparticleintoPSOtoimprovetheperfor-mance.Animprovedquantum-behavedPSOwasproposedbyXietal.[12].Jiaoetal.[13]proposedthedynamicinertiaweightPSO,bydefiningadynamicinertiaweighttodecreasetheinertiafactorinthevelocityupdateequationoftheoriginalPSO.Yangetal.[14]proposedanotherdynamicinertiaweighttomodifythevelocityupdateformulainamethodcalledmodifiedparticleswarmoptimizationwithdynamicadaptation.AnumberofstudieshaveappliedthePSOandimprovedittobeusedinthefieldofstructuralengineering[15–21].Inthisstudy,wedevelopedanimprovedPSO,so-calledacceleratedparticleswarmoptimization(APSO)[22],tofindoptimumdesignofframestructures.Theresultedmethodisthentestedbysomenumericalexamplestoestimateitspotentialforsolvingstructuraloptimizationproblems.2.StatementofStructuralOptimizationProblemOptimumdesignofstructuresincludesfindingoptimumsec-tionsformembersthatminimizesthestructuralweight𝑊.2MathematicalProblemsinEngineeringThisminimumdesignshouldalsosatisfyinequalitycon-straintsthatlimitdesignvariablesandstructuralresponses.Thus,theoptimaldesignofastructureisformulatedas[23]minimize𝑊({𝑥})=𝑛∑𝑖=1𝛾𝑖⋅𝐴𝑖⋅𝑙𝑖,subjectto:𝑔min≤𝑔𝑖({𝑥})≤𝑔max,𝑖=1,2,3,...,𝑚,(1)where𝑊({𝑥})istheweightofthestructure;𝑛and𝑚arethenumberofmembersmakingupthestructureandthenumberoftotalconstraints,respectively;maxandmindenoteupperandlowerbounds,respectively;𝑔({𝑥})denotesthecon-straintsconsideredforthestructurecontaininginteractionconstraintsaswellasthelateralandinterstorydisplacements,asfollows.Themaximumlateraldisplacement:𝑔Δ=Δ𝑇𝐻−𝑅≥0.(2)Theinterstorydisplacements:𝑔𝑑𝑗=𝑑𝑗ℎ𝑗−𝑅𝐼≥0,𝑗=1,2,...,𝑛𝑠,(3)whereΔ𝑇isthemaximumlateraldisplacement;𝐻istheheightoftheframestructure;𝑅isthemaximumdriftindex;𝑑𝑗istheinter-storydrift;ℎ𝑗isthestoryheightofthe𝑗thfloor;𝑛𝑠isthetotalnumberofstories;𝑅𝐼istheinter-storydriftindexpermittedbythecodeofthepractice.LRFDinteractionformulaconstraints(AISC2001[24,EquationH1-1a,b]):𝑔𝐼𝑖=𝑃𝑢2𝜙𝑐𝑃𝑛+(𝑀𝑢𝑥𝜙𝑏𝑀𝑛𝑥+𝑀𝑢𝑦𝜙𝑏𝑀𝑛𝑦)−1≥0for𝑃𝑢𝜙𝑐𝑃𝑛0.2,𝑔𝐼𝑖=𝑃𝑢𝜙𝑐𝑃𝑛+89(𝑀𝑢𝑥𝜙𝑏𝑀𝑛𝑥+𝑀𝑢𝑦𝜙𝑏𝑀𝑛𝑦)−1≥0for𝑃𝑢𝜙𝑐𝑃𝑛≥0.2,(4)where𝑃𝑢istherequiredstrength(tensionorcompression);𝑃𝑛isthenominalaxialstrength(tensionorcompression);𝜙𝑐istheresistancefactor(𝜙𝑐=0.9fortension,𝜙𝑐=0.85forcompression);𝑀𝑢𝑥and𝑀𝑢𝑦aretherequiredflexuralstrengthsinthe𝑥and𝑦directions,respectively;𝑀𝑛𝑥and𝑀𝑛𝑦arethenominalflexuralstrengthsinthe𝑥and𝑦directions(fortwo-dimensionalstructures,𝑀𝑛𝑦=0);𝜙𝑏istheflexuralresistancereductionfactor(𝜙𝑏=0.90).Fortheproposedmethod,itisessentialtotransformtheconstrainedoptimizationproblemtoanunconstraintone.Adetailedreviewofsomeconstraint-handlingapproachesispresentedin[25].Inthisstudy,amodifiedpenaltyfunctionmethodisutilizedforhand