1Example50TitleP-DeltaeffectanalysisofabeamDescriptionAsimplysupportedbeamissubjectedtothreedifferenttypesoftransverseloadsinconjunctionwithaxialforces(mid-spanpointload,uniformloadandtwoequalendmoments)PerformaP-Deltaeffectanalysisandfindthedisplacementsandinternalbendingmomentsforeachloadcase.2StructuralgeometryandanalysismodelExample503ModelAnalysisType2-D,P-DeltaeffectanalysisUnitSystemin,lbfDimensionLengthL=100.0inCriticalloadPcr=2383.0lbfRelativedisplacementtolerance0.001ElementBeamElementMaterialModulusofelasticityE=29×106psiSectionPropertyAreaA=1.0in2MomentofinertiaIyy=1/12in4BoundaryConditionNodes1,4and7;ConstrainDxandDz.(Hingesupports)Nodes3,6and9;ConstrainDz.(Rollersupports)AnalysisCaseInordertoconsidertheP-DeltaeffectassumePisthe50percentofthePcr.P=2383.0/2lbf.Load1–Aconcentratedloadisappliedatthemid-spanofthebeaminthe-Zdirection,Po=100.0lbfLoad2–Auniformloadisdistributedovertheentirespanofthebeaminthe–Zdirection,ω=0.2lbf/inLoad3–BendingmomentsareappliedtobothendsabouttheY-axis,Mo=±200lbf-inMaximumnumberofiterationfortheP-Deltaeffectanalysis=54ResultsDisplacementsMemberForcesExample505ComparisonofResultsUnit:in,lbf-inLoadtypeResultSAP2000MIDAS/GenVerticaldisplacement-1.7058-1.7049ConcentratedloadBendingmoment4532.4284531.377Verticaldisplacement-0.2141-0.2140UniformloadBendingmoment505.127505.010Verticaldisplacement-0.2090-0.2090EndmomentsBendingmoment449.045448.988ReferencesLivesley,R.K.,andChandler,D.B.,“StabilityFunctionsforStructuralFrameworks”,ManchesterUniversityPress,UK,1956.“SAP90,ASeriesofComputerProgramsfortheFiniteElementAnalysisofStructures,StructuralAnalysisVerificationManual”,ComputerandStructures,Inc.,1992,Example25.
