Combinationofmodalresponses:Aclosed-formformulationforrigidresponsecoefficientOriginalResearchArticle

RecentstudiesconductedbyUSNuclearRegulatoryCommission(USNRC)forcombiningmodalresponsesinaresponsespectrummethodofseismicanalysisanddesignhaveemphasizedthateachmodalresponsequantityshouldbeseparatedintodamped-periodicandrigidpartsbeforecombiningthecontributionsfromdifferentmodes.Thedamped-periodicpartsofmodalresponsesarecombinedusingthedouble-sumequationwhereastherigidpartsarecombinedalgebraically.Aparticularmodalresponsequantityisseparatedintodamped-periodicandrigidpartsusingthe“rigidresponsecoefficient”.TheUSNRCsponsoredstudyrecommendsthecalculationofrigidresponsecoefficientbyeithertheLindley–YowapproachorGuptamethod.WhileLindley–Yowʹsmethodhasaheuristicbasisandgivesincorrectresultsinlowfrequencyregion,Guptaʹsmethodisbasedonnumericalstudiesoffree-fieldearthquakemotionsandworkswellinthefrequencyregionsofinterestforafree-fieldgroundmotion.Aclosed-formsolutionwasdevelopedbyHahnandValentiin1997usingafrequencydomainapproach.Withappropriatesimplifications,theirworkcanbeshowntoresultinanexpressionwhichisverysimilartothatgivenbyGupta.Itmustbenotedthattheearthquakeinputtothesecondarysystemssuchaspipingandequipmentisdefinedbyafloormotionandnotafree-fieldgroundmotion.Thefrequencycharacteristicsofafloormotionareverydifferentfromthoseofafree-fieldgroundmotion.Inthispaper,westudythevalidityofexistingformulationsforthecaseoffloormotionsanddevelopaclosed-formsolutionbasedonatimedomainapproachtoexplainthebehaviorofrigidresponsecoefficient.Theformulationisthenusedtoexplainthenatureofvariationinrigidresponsecoefficientforgroundaswellasfloormotions.Itisshownthattheproposedformulationanditssimplifiedformgivesresultsthatareidenticaltothoseevaluatednumericallyinthecompletefrequencyregionofinterest.