FINAL REPORT Colorado Advanced Software Institute NONLINEAR PUSHOVER ANALYSIS OF REINFORCED CONCRETE STRUCTURES Principal Investigator:Enrico Spacone, Ph.D. Assistant Professor Department of Civil, Env. and Arch. Engineering University of Colorado, Boulder Graduate StudentRussel Martino, MS student Department of Civil, Env. and Arch. Engineering University of Colorado, Boulder Collaborating CompanyGreg Kingsley, Ph.D., P.E. Principal KL&A of Colorado Golden, Colorado COLLABORATING COMPANY RELEASE PAGE Project Title:NONLINEAR PUSHOVER ANALYSIS OF REINFORCED CONCRETE STRUCTURES Principal Investigator:Enrico Spacone, Ph.D. University: University of Colorado, Boulder Collaborating Company:KL&A of Colorado Collaborating Company Representative:Greg Kingsley, Ph.D., P.E. As authorized representative of the collaborating company, I have reviewed this report and approve it for release to the Colorado Advanced Software Institute. __________________________________ _________________ SignatureDate i TABLE OF CONTENTS CHAPTERS IINTRODUCTION ............................................................................................... 1 I-ABackground ................................................................................................. 1 I-BObjectives.................................................................................................... 2 IITHE NON LINEAR STATIC PUSHOVER ANALYSIS PROCEDURE.... 4 II-ADefinition of the Non Linear Static Procedure......................................... 4 II-BPerforming the Non Linear Static Procedure............................................ 6 II-B-1Vertical Distribution of Lateral Loads .......................................... 6 II-B-2Building Performance Level ......................................................... 8 II-B-3Calculation of the Seismic Hazard................................................ 9 II-B-4Calculation fo the Target Displacement........................................ 15 II-CReasons for Performing the Non Linear Static Procedure........................ 18 IIILIMITATIONS OF THE NON LINEAR STATIC PROCEDURE............. 20 III-ADesign of Three Reinforced Concrete Moment Resisting Frames .............. 20 III-A-1Formulation of Gravity Loads Used in Design ............................. 22 III-A-2Formulation of Wind Loads Used in Design ................................ 24 III-A-3Formulation of Earthquake Loads Used in Design ....................... 24 III-A-4Total Design Loads and Section Determination............ 28 III-BPerforming the Pushover Analysis On the Moment Frames ....................... 34 III-B-1Period Determination .................................................................... 34 III-B-2Vertical Distribution of Lateral Loads .......................................... 36 III-B-3Element Reduction for Analysis of Frames............... 41 III-B-4Determination of Seismic Hazard for Analysis............. 42 III-B-5Calculation of Target Displacement for Analysis............. 44 III-CComplete Non Linear Dynamic Analyses for Frames.......... 46 III-DComparisons of Full Dynamic Results with Pushover Results........... 49 ii III-EDependence of Target Displacement on Choice of Vy......... 53 III-FConclusions - Limitations and Accuracy of the Pushover Analysis........ 54 IVFORMULATION OF ELEMENT SHEAR RESPONSE 55 IV-AReview of Timoshenko Beam Theory......... 56 IV-BNon Linear Force Based Timoshenko Beam Element........... 58 IV-CSection V Constitutive Law............ 63 IV-C-1Shape of Shear Hysteretic Law.............. 63 IV-C-2Theoretical Values of Shear Hysteretic Law............. 65 IV-C-3Values for Actual Sections Shear Hysteretic Law.......... 74 IV-DObservations on Element Shear Response Formulation........ 76 VNUMERICAL VERIFICATION OR PROPOSED SHEAR MODEL78 V-AColumn Dimensions and Testing Conditions.......... 78 V-BCalculated Shear Strength............ 80 V-CNumerical vs. Experimental Column Response........... 89 V-DConclusions.......... 94 VISHEAR WALL EXAMPLE............. 95 VI-AWall Configuration........... 95 VI-BPerforming the Pushover Analysis on the Shear Wall.. ....... 97 VI-CComplete Non-Linear Dynamic Analysis of the Shear Wall....... 98 VI-DComparisons of Pushover and Dynamic Analysis. ...... 9 VI-EVerification of Flexure Shear Interaction at Element Level........ 100 VI-FConclusions...... 101 VII CONCLUSIONS AND FUTURE WORK.......................................... 102 VIIIBIBLIOGRAPHY..104 APPENDICES IMODIFICATIONS TO PROGRAM FEAP TO PERFORM NON-LINEAR PUSHOVER ANALYSIS.......................................................... 106 AI-AFEAP Pushover Routines ........ 106 iii AI-A-1PUSH Mesh Command ...........106 AI-A-2VvsD Macro Command...........110 AI-BFEAP Shear Element Modifications ........113 IIADDITIONAL CASI REQUIREMENTS ......115 AII-AEvaluation.............................115 AII-BTechnology Transfer.............115 AII-CNetworking ...........................116 AII-DPublications ..........................116 AII-EFunding.................................116 iv ABSTRACT:ThisreportsummarizestheresultsofaresearchconductedattheUniversityofColorado, Boulder,aimedatdevelopingaPC-basedsoftwaretoolforperformingnonlinearpushover analysis of reinforced concrete buildings. The program links two libraries to an existing finite elementprogram,FEAP,developedattheUniversityofCalifornia,Berkeley.Thetwo librariesarea)aframeelementlibrary(whichincludesbeam,beam-columnandshearwall elements);andb)alibraryofuniaxialmateriallaws.Theprojectfirstmodifiedtheexisting programtoperformnonlinearpushoveranalysesonaroutinebasis.Currentseismiccode suggestedproceduresfornonlinearpushoveranalyseswerethenreviewed.Theapplicability ofnonlinearpushoveranalysestotheseismicdesignofreinforcedconcreteframeswas evaluated by studying the response of frames of different heights. The responses of static and dynamicnonlinearanalysesonthesamebuildingswerecompared.A new shear element was thenintroducedandatypicalshearwallofautilitycorewasanalyzedwithapush-over analysis.DetailsonthefeaturesaddedtoprogramFEAPandonthenewcommandsare documented in the appendices. 1 CHAPTER I INTRODUCTION I-ABackground As the United States, Japan, and Europe move towards the implementation of Performance Based Engineeringphilosophiesinseismicdesignofcivilstructures,newseismicdesignprovisionswillrequire structural engineers to perform nonlinear analyses of the structures they are designing.These analyses can take the form of a full, nonlinear dynamic analysis, or of a static nonlinear Pushover Analysis.Because of thecomputationaltimerequiredtoperformafull,nonlineardynamicanalysis,thePushoverAnalysis,if deemedapplicabletothestructureathand,isaveryattractivemethodforuseinadesignofficesetting.Forthisreason,thereisaneedforeasytouseandaccurate,nonlinearPushoverAnalysistoolswhichcan easily be applied in a design office.Even though recent years have seen a great amount of research in the developmentofsuchnonlinearmodelsandtechniques,thereisstillagreatdealofknowledgemissingfor reinforcedconcretestructures.Inparticular,thefollowingmodelingissuesstillneedtobethoroughly addressed: bond slip, structural walls and shear deformations, joint response, and non structural members. IntheUnitedStates,thereferencedocumentforperformingtheNonlinearStaticProcedure,or PushoverAnalysis,iscurrentlytheFederalEmergencyManagementAgencyDocument273(FEMA273) [6].According to this procedure, a vertical distribution of static, monotonically increasing, lateral loads is appliedtoamathematicalmodelofthestructure.Theloadsareincreaseduntilthepeakresponseofthe structureisobtainedonabaseshearvs.roofdisplacementplot.Fromthisplot,andotherparameters representingtheexpected,ordesign,earthquake,themaximumdeformationsthestructureisexpectedto undergoduringthedesignseismiceventcanbeestimated.Becausethemathematicalmodelmust capture theinherentmaterialnonlinearitiesofthestructure,andbecausetheloadappliedtothestructureis increasedmonotonically,detailedmemberinformationcanbeobtained.Thisprocedureismoreinvolved thanapplyingtheapproximatestaticlateralloadallatonce,asisdoneincurrent seismic design codes, in thattheloadsare applied in increments. This allows the deformations of structural members (for example, the plastic-hinge sequence) to be monitored throughout the nonlinear pushover analysis. 2 TheNonlinearStaticProceduremuststillbeusedwithcaution.ThePushoverAnalysisis meant torepresentastaticapproximationoftheresponseastructurewillundergowhensubjectedtodynamic earthquakeloads.Thekeywordinthis definition is approximation.There is a great saving in time when performingthePushoverAnalysisascomparedwiththefullnonlineardynamicanalysis.Butthereare boundtobedrawbackstothemethod.Inparticular,themaximum displacement achieved will be directly relatedtotheshape of the lateral load distribution applied to the structure.If the shape of the lateral load differsfromtheshapethestructureattainswhenloadeddynamically,thecalculatedmaximum displacement could grossly overestimate what the dynamic analysis would predict. WhiletherearecurrentlysomeprogramsavailabletoperformthePushoverAnalysison ReinforcedConcretestructures,theprocedureneedstoberefinedandmoreexperienceisneededtofully access its applicability.One of the several issues still open is modeling the shear deformations in reinfoced concrete columns and structural walls. Shear deformations in Reinforced Concrete members are difficult to model because of the complex mechanisms that govern them.I-AObjectives The main objective of this project was to develop an easy to use and accurate nonlinear Pushover Analysis Software tool for civil structures following the procedures outlined in FEMA 273[6]. Even though theprocedureisgeneral,thefocusofthisstudyisreinforcedconcreteframes.The objective is to develop anaccuratethougheasytounderstandtoolthatcanberoutinelyusedinadesignofficebyastructural engineerthatisfamiliarwithboththePushoverAnalysisprocedureandwithbasicnonlinearstructural analysis techniques. The following are the main tasks of the projects: a)A critical study of the Pushover Analysis procedure as defined by FEMA 273[6]. Comparisons between NonlinearPushoverAnalysesandNonlinearDynamicAnalysesarekeytounderstandingthe limitations of the proposed Pushover Analyses. b)DevelopmentofSoftwareToolforNonlinearPushoverAnalyses.Thisisachievedbymodifyingthe existingFiniteElementAnalysisProgram(FEAP)developedbyProfessorRobertTayloratthe UniversityofCaliforniaatBerkeley[16].Special steps need to be implemented to perform Nonlinear Pushover Analyses following FEMA 273[6] 3 c)Development of a family of models for Nonlinear Pushover Analysis of Reinforced Concrete structures. Some of these models already exist and need to be linked to program FEAP (in particular, fiber beam columnelementswithinteractionbetweenaxialandnormalforces).Othermodels,inparticular elements for reinforced concrete members with shear deformations, need to be developed.d)Verification of the new tool via comparisons between experimental and analytical results. e)Application of the new tool to studies of Reinforced Concrete structural systems. With all of the foregoing arguments in mind, the organization of this report is as follows.Chapter II,TheNonLinearStaticPushoverProcedure,describesthestepsfollowedinperformingtheNon LinearStaticProcedureasgivenbyFEMA273[6].ChapterIII,ApplicationsoftheNonLinearStatic Procedure (Pushover Analysis), discusses the applicability and shortcomings of the procedure. Chapter IV, FormulationofElementShearResponse,describesthesheardeformationformulationforaforcebased beamelement.ChapterV,NumericalVerificationofProposedShearModel,determinestheapplicability andshortcomingsoftheshearformulationdevelopedinchapterIVbycomparingnumericalresultswith test data obtained from Reinforced Concrete columns failing in shear tested at the University of California atSanDiego.ChapterVI,Conclusions,summarizestheresultsandpointstoareasforfuturework.AppendixI,ModificationstoFEAPtoPerformPushoverAnalysis,describesthechangesmadetothe FiniteElementAnalysisProgramtoincludesheardeformationsandtorunthePushoverAnalysis.AppendixIpresentstheModificationstoProgramFEAPtoperformNon-LinearPushoverAnalyses.. Finally, Appendix II includes Additional CASI Requirements for the Poject Report. 4 CHAPTER II THE NON LINEAR STATIC PUSHOVER ANALYSIS PROCEDURE II-A) Definition of the Non Linear Static Procedure (Pushover Analysis) - FEMA 273 [6] The Non Linear Static Procedure or Pushover Analysis is defined in the Federal Emergency ManagementAgencydocument273(FEMA273)[6]asanonlinearstaticapproximationofthe responseastructurewillundergowhensubjectedtodynamicearthquakeloading.Thestatic approximation consists of applying a vertical distribution of lateral loads to a model which captures the material non linearities of an existing or previously designed structure, and monotonically increasing thoseloadsuntilthepeakresponseofthestructureisobtainedonabaseshearvs.roofdisplacement plot as shown in figure II-1. Figure II-1: Static Approximation Used In the Pushover Analysis Thedesiredconditionofthestructureafterarangeofgroundshakings,orBuildingPerformance Level,isthendecideduponbytheowner,architect,andstructuralengineer.TheBuilding PerformanceLevelisafunctionoftheposteventconditionsofthestructuralandnonstructural components of the structure.Some common Building Performance Levels are shown in figure II-2. Figure II-2: Building Performance Level Lateral Loads Structural Model Structural Response Roof DisplacementBase Shear Base Shear Roof Disp Owner, Architect, Engineer Collapse Prevention Immediate Occupancy Life Safety Operational 5 BasedonthedesiredBuildingPerformanceLevel,theResponseSpectrumforthedesign earthquake may be determined.The Response Spectrum gives the maximum acceleration, or Spectral ResponseAcceleration,astructureislikelytoexperienceunderthedesigngroundshakinggiventhe structures fundamental period of vibration, T.This relation is shown qualitatively in figure II-3. Figure II-3: Response Spectrum FromtheResponseSpectrumandBaseShearvs.RoofDisplacementplot,theTarget Displacement,t,maybedetermined.TheTargetDisplacementrepresentsthemaximum displacementthestructurewillundergoduringthedesignevent.Onecanthenfindthemaximum expecteddeformationswithineachelementofthestructureattheTargetDisplacementandredesign them accordingly.The Target Displacement is shown qualitatively in figure II-4. Figure II-4: Target Displacement Response Spectrum Spectral Response Accel, Sa Roof Displacement Structural Response Target Displacement, tBase Shear 6 II-B) Performing the Non Linear Static Procedure (Pushover Analysis) The steps in performing the Non Linear Static Procedure or Pushover Analysis are: 1)Determine the gravity loading and the vertical distribution of the lateral loads. 2)Determine the desired Building Performance Level. 3)Calculate the Seismic Hazard. 4)Compute the maximum expected displacement or Target Displacement, t. Each of these steps are described in the sections following. II-B-1) Determine the Vertical Distribution of the Lateral Loads Inadditiontothegravityloads,thefirstthingthatcanbedeterminedisthevertical distributionofthelateralloads.ThegravityloadstobeusedinthePushoverAnalysis are calculated byequationII.1,whiletheverticaldistributionoflateralloadsisgivenbytheFEMA273[6]Cvx loading profile reproduced as equation II.2. ) ( 1 . 1S L D GQ Q Q Q + + = (II.1) Where, QG is equal to the total gravity force, QD is equal to the total dead load effect, QL is equal to the effective live load effect, defined as 25% of the unreduced live load, and QS is equal to 70% of the full design snow load except where the design snow load is less than thirty pounds per square foot in which case it is equal to 0.0. ==niki ikx xvxh wh wC1(II.2) TheCvxcoefficientrepresentsthelateralloadmultiplicationfactortobeappliedatfloorlevelx,wx representsthefractionofthetotalstructuralweightallocatedtofloorlevel x,hxistheheightoffloor level x above the base, and the summation in the denominator is the sum of these values over the total number of floors in the structure, n.These values are shown schematically in figure II-5. 7 Figure II-5: Values for Determining the Vertical Distribution of the Lateral Loads The parameter k varies with the structural fundamental period, T.k is 1.0 for T less than or equal to 0.5 secondsand2.0for Tgreaterthanorequalto2.5seconds.Inbetweenthese values, k varies linearlyas shown in figure II-6.The effect that the parameter k has on the Cvx loading profile is also shown in figure II-6.For shorter, stiffer structures, the fundamental period will be small and the variation of the lateral loading over the height of the building will approach the linear distribution shown in figure II-6 forakvalueequalto1.0.Fortaller,moreflexiblestructures,thefundamentalperiodwillbegreater andthevariationofthelateralloadingovertheheightofthestructurewillapproachthenonlinear distribution shown in figure II-6 for k equal to 2.0.The implication of this is that for stiffer structures thehighermoderesponseofthestructurewillbelesssignificantandthelateralloadingcanenforce purelyfirstmoderesponse. As the structure becomes more flexible however, the higher mode effects become much more important and the k value attempts to account for this by adjusting the lateral load distribution. Figure II-6: Variation of k with Fundamental Period T, and Effect of k on Lateral Load Cvnwn Cv4 Cv3 Cv2 Cv1 w4 w3 w2 w1 h1 h2 h3 h4 hn Fundamental Period, T (sec) 0.51.52.5 2.0 1.0 Determination ofk Effect of k on CFloor 1 Floor 4 Floor 3 Floor 2 Floor 5 0.10.30.40.50.2 k = 2 k = 1 Cvx 8 II-B-2) Building Performance Level Determination ThenextthingthatmaybedeterminedistheBuildingPerformanceLevel.TheBuilding Performance Level is the desired condition of the building after the design earthquake decided upon by the owner, architect, and structural engineer, and is a combination of the Structural Performance Level and the Non Structural Performance Level.The Structural Performance Level is defined as the post eventconditionsofthestructuralbuildingcomponents.Thisisdividedintothreelevelsandtwo ranges.Thelevelsare,S1:ImmediateOccupancy,S3:LifeSafety,andS5:Collapse Prevention.TherangesareS2:whichisarangebetweenS1andS3,andS4:whichisa rangebetweenS3andS5.Therangesareincludedtodescribeanybuildingperformancelevel whichmaybedecideduponbytheowner,architect,andstructuralengineer.TheNonStructural Performance Level is defined as the post event conditions of the non - structural components.This is dividedintofivelevels.TheyareNA:Operational,NB:ImmediateOccupancy,NC:Life Safety,ND:HazardsReduced,andNE:NonStructuralDamageNotLimited.Bycombining thenumberfromtheStructuralPerformanceLevelwiththesecondletterfromtheNonStructural Performance Level, one can attain the total Building Performance Level.The combinations to achieve the most common Building Performance Levels, 1 A: Operational, 1 B: Immediate Occupancy, 3 C: Life Safety, and 5 E: Collapse Prevention, are shown in figure II-7. Figure II-7: Determination of Building Performance Level Building Performance Level S - 1Immediate Occupancy S - 2Range Between S-1 & S-3 S - 5Collapse Prevention S - 3Life Safety S - 4Range Between S-3 & S-5 Structural Level 1 - A 1 - B 3 - C 5 - E N - A Operational N - E Damage NotLimited N - C Life Safety N - D Hazards Reduced Non - Structural Level N - B Immediate Occupancy 9 Theowner,architect,andstructuralengineercannowdecidewhatBuildingPerformance Level they want their building to achieve after a range of ground shakings which are expected to occur atagivendesignlocation.ReferringtofigureII-8,AwouldcorrespondtoaBuildingPerformance levelofOperationalaftera50%probabilityofexceedancein50yearseismicevent,Fwould correspondtoaBuildingPerformanceLevelofImmediateOccupancyaftera20%probabilityof exceedancein50yearseismiceventandsoon.ThevaluesKandPshowninboldinfigureII-8 correspondtotheperformanceoneachieveswhendesigningbytheUniformBuildingCode(UBC) [17].ThiscorrespondstoLifeSafetyaftera10%probabilityofexceedancein50yeareventand CollapsePreventionaftera2%probabilityofexceedancein50yearevent,respectively.Onecan easilyseethatthenewdesignapproachallowsthedesignertoadvancethestateoftheartfromthe UBCcodebygivingmanymoredesignoptionsandallowingtheowner,architect,andengineerto predict the post event conditions of the structure for a wide range of ground motions. Figure II-8: Building Performance Level for Given Seismic Event II-B-3) Calculation of the Seismic Hazard An important parameter that must be determined for the Pushover Analysis is the Seismic Hazard of a given location.The Seismic Hazard is a function of: 1)The Building Performance Level2)The Mapped Acceleration Parameters (found from contour maps included with FEMA 273) 3)The Site Class Coefficients (which account for soil type) 4)The effective structural damping Seismic Event Building Performance Level H DB F J MN I E L PO K G CA50% / 50 years 20% / 50 years 10% / 50 years 2% / 50 years 1 - A1 - B3 - C5 - E 10 5)The Fundamental Structural Period TheBuildingPerformanceLevelentersintotheSeismicHazardthroughthereturnperiodofthe earthquake under consideration.The return period for the design earthquake, PR, is defined as: ) 1 ln( 02 . 05011EPReP= (II.3) Where PE50 is the probability of exceedance in 50 years under consideration.Referring to figure II-8, iftheowner,architect,andstructuralengineerdeterminethatconditionA,K,andPmustbemet, corresponding to Operational after a 50% probability of exceedance in 50 years event, Life Safety after a 10% probability of exceedance in 50 years event, and Collapse Prevention after a 2% probability of exceedancein50yearseventrespectively,thentheReturnPeriodwouldbecalculatedthreeseparate times with PE50 equal to 0.5, 0.1, and 0.02 respectively.Since the Seismic Hazard is a function of this ReturnPeriod,aswillbe shown subsequently, the Pushover Analysis would need to be run separately foreach%exceedanceconsideredandtheendresultscomparedwiththeacceptancecriteriagivenin FEMA 273 [6] for the Building Performance Level at each % exceedance. OncetheReturnPeriodforthe%exceedanceunderconsiderationhasbeendetermined,the mappedaccelerationparametersareusedtodeterminethemodifiedmappedshortperiodresponse accelerationparameter,SS,andthemodifiedmappedaccelerationparameteratonesecondperiod, S1.These parameters are found from: If SS2/50 is less than 1.5g and PE50 is between 2% in 50years and 10% in 50 years then [ ] [ ] 73 . 3 ) ln( 606 . 0 * ) ln( ) ln( ) ln( ) ln(50 / 10 50 / 2 50 / 10 + =R i i i iP S S S S (II.4) WhenSS2/50 isgreaterthanorequalto1.5gorSS2/50 islessthan1.5gandPE50isgreaterthan10% probability of exceedance in 50 years then nRi iPS S =47550 / 10(II.5) ThesubscriptiintheaboveequationsisequaltoSifthemodifiedmappedshortperiodresponse accelerationparameterisbeingdeterminedanditisequalto1ifthemodifiedmappedresponse acceleration parameter at a one second period is being determined.The parameter Si2/50 in equation II.4 is the mapped short period acceleration parameter (i =S) or the mapped acceleration parameter at a one 11 secondperiod(i=1)fora2%probabilityofexceedancein50yearsevent.TheparameterSi10/50 in equationsII.4andII.5isthemappedshortperiodaccelerationparameter(i=S)orthemapped accelerationparameterataonesecondperiod(i=1)fora10%probabilityofexceedancein50years event.Theseparametersarefoundfromcontourmapswhichmaptheshortperiodresponse acceleration and the response acceleration at a one second period at probabilities of exceedance of 2% in 50 years and 10% in 50 years for the for the entire United States and are included with FEMA 273.The value n in equation II.5 is a parameter which depends on the mapped parameter SS2/50 and PE50 and is tabulated in FEMA 273. These tables are reproduced in Table II-1. Table II-1:Values for exponent n for use in equation II.5 Now that the modified mapped short period response acceleration parameter and the modified mappedresponseaccelerationparameterataonesecondperiodhavebeendetermined,these parametersmustbefurtheradjustedtoaccountforthesoiltypeatthesite.Thefinaldesignshort periodspectralresponseaccelerationparameter,SXS,andthefinaldesignspectralresponse acceleration parameter at a one second period, SX1, shall be determined from: Region Value of n for use with SSValue of n for use with S1 California Pacific Northwest Mountain Central US Eastern US 2% = 1.5g PE50 > 10% & SS2/50 < 1.5g PE50 > 10% & SS2/50 >= 1.5g 2% = 1.5g PE50 > 10% & SS2/50 < 1.5g PE50 > 10% & SS2/50 >= 1.5g 0.29 0.54 0.54 0.890.670.59 0.440.440.290.44 0.54 1.09 0.93 0.98 0.50 0.56 0.600.590.59 0.96 0.44 1.050.80 0.800.77 1.25 0.89 1.25 0.89 0.77 12 S a XSS F S = ( II.6) 1 1S F Sv X=(II.7) Fa is a function of the soil class at the site and the modified mapped short period response acceleration parameter,SS,andFvisafunctionofthesoilclassatthesiteandmodifiedmappedresponse accelerationparameterataonesecondperiod,S1.ValuesofFaandFvaretabulatedinFEMA273.These tables are reproduced in tables II-2 and II-3 respectively.Linear interpolation shall be used for valuesofSSorS1betweentabulatedvaluesandthe*representsaconditioninwhichsitespecific geotechnical investigation and dynamic site response analyses should be performed. Table II-2: Values for Site Class coefficient, Fa, for use in equation II.6 Table II-3: Values for Site Class coefficient, Fv, for use in equation II.7 Definitions and classifications of soil type are included in FEMA 273 and are as follows: Class A: Hard rock with measured shear wave velocity, vs > 5,000 ft/sClass B: Rock with 2,500 ft/s < vs < 5,000 ft/s, where vs is the measured shear wave velocity. Site ClassS= 1.25 A B C D E F 0.80.8 1.01.0 0.80.80.8 1.21.21.1 1.1 1.0 1.0 1.0 1.0 1.0 1.0 1.61.2 0.9*1.72.5 1.4 1.2 ***** Site ClassS= 0.5 A B C D E F 0.80.8 1.01.0 0.80.80.8 1.61.71.5 1.6 1.0 1.4 1.0 1.5 1.3 1.0 2.41.8 2.4*3.23.5 2.0 2.8 ***** 13 ClassC:Verydensesoilandsoftrockwithshearwavevelocity,1,200ft/s 2,000 psf. ClassD:Stiffsoilwithshearwavevelocity,600ft/s 40%, and undrained shear strength, su < 500 psf or a soil profile with shear wave velocity, vs < 600 ft/s.If insufficient data are available to classify a soil profile as type A through D, a type E profile should be assumed. ClassF:Soilsrequiringsitespecificevaluationsarethosesoilsthatarevulnerableto potentialfailureorcollapseunderseismicloading,suchasliquefiablesoils,quickandhighly sensitiveclaysandcollapsibleweaklycementedsoils,peatsand/orhighlyorganicclayswitha thicknessgreaterthan10ft,veryhighplasticityclaysthathaveaplasticityindex,PI,greaterthan75 andwithathicknessgreaterthan25ft,andsoftormediumclayswhichhaveathicknessgreaterthan 120 ft. Intheaboveclassifications,theshearwavevelocity,vs,theStandardPenetrationTestblow count, N, and the undrained shear strength, su, are average values over a 100 ft depth of soil. Basedonthedesignspectralresponseaccelerationparameters,SXSandSX1,theGeneral ResponseSpectrumcanbeformulated.TheGeneralResponseSpectrumgraphicallyrelatesthe Spectral Response Acceleration, Sa, as a function of Structural Fundamental Period, T.The relation is defined as: ) / 3 4 . 0 ( * ) / (0T T B S SS XS a+ = for 02 . 0 0 T T < (II.8) S XS aB S S / = for 0 02 . 0 T T T < (II.9) ) /(1 1T B S SX a= for0T T > ( II.10) The values BS and B1 in equations II.8 to II.10 are parameters which account for the effective dampingcoefficientofthestructureandaretabulatedinFEMA273.Thesevaluesarereproducedin tableII-4andlinearinterpolationshallbeusedforintermediatevaluesoftheeffectivedamping coefficient, . 14 Table II-4: Damping Coefficients BS and B1 to be used in equations II.8 to II.10 ThevalueT0inequationsII.8toII.10isthecharacteristicperiodoftheresponsespectrum, definedastheperiodassociatedwiththetransitionfromtheconstantaccelerationsegmentofthe spectrum to the constant velocity segment of the spectrum.It is calculated from: ) /( ) (1 1 0B S B S TXS S X= (II.11) WiththeapplicationofequationsII.3throughII.11theGeneralResponseSpectrumcanbe formulatedforthedesigneventbeingconsidered.TheGeneralResponseSpectrumisshown qualitatively in figure II-9. Figure II-9: General Response Spectrum 0.4SXS/BS SX1/B1 Sa = SXS/BS Fundamental Structural Period, T Spectral Response Acceleration, Sa Sa = SX1/(B1T) 0.2T0T01.0 Sa = (SXS/BS)(0.4+3T/T0) Effective Damping, (% of critical) BS B1 < 2 5 10 20 30 > 50 40 0.80.8 1.01.0 1.31.2 1.81.5 2.31.7 2.71.9 3.02.0 15 TheGeneralResponseSpectrumisafunctionofthemanysiteanddesigneventspecific parameterswhicharerelatedbyacomplicatedsystemofequations.However,onceithasbeen developed,sinceitisafunctiononlyofsitelocationparametersandthedesigneventunder consideration, it becomes a very useful tool as it describes the maximum acceleration a structure, with a given fundamental period, must endure during the design event.II-B-4)Calculation of the Target Displacement The Target Displacement, i.e. the maximum displacement the structure is expected to undergo duringthedesignevent,cannowbeobtained.Thetargetdisplacementiscalculatedfromthe following equation: gTS C C C Cea t223 2 1 04 = (II.12) WherethevalueC0isamodificationfactorthatrelatesspectraldisplacementandlikely building roof displacement.Values for C0 are tabulated in FEMA 273 as a function of the total number of stories of the structure and are included in table II-5. Table II-5: Values for modification factor C0 for use in equation II.12 C1isamodificationfactorwhichrelatesexpectedmaximuminelasticdisplacementsto displacements calculated for linear elastic response.Values for C1 are obtained from: 0 . 11 = C for0T Te (II.13) [ ] R T T R Ce/ / ) 1 ( 0 . 10 1 + = for0T Te < (II.14) Number of StoriesModification Factor C1 1 2 3 5 10 + 1.0 1.2 1.3 1.4 1.5 1. Linear Interpolation should be used for intermediate values 16 TeistheeffectivefundamentalperiodofthestructureandisdefinedasgiveninequationII.17.Tois thecharacteristicperiodoftheresponsespectrum,definedasthe period associated with the transition fromtheconstantaccelerationsegmentofthespectrumtotheconstantvelocitysegmentofthe spectrumandiscalculatedasshowninequationII.11.Ristheratioofelasticstrengthdemandto calculated yield strength coefficient.Values for R are obtained from: 01/ C W VSRya= (II.15) Sa is the Response Spectrum Acceleration, in gs, ( where g must be in consistent units, usually in/s2) at the effective fundamental period and damping ratio of the building in the direction under consideration asdescribedinsectionII-B-3andobtainedfrom equations II.8 through II.10.Vy is the yield strength calculatedusingtheresultsofthePushoverAnalysis,wherethenonlinearforcedisplacement curveofthebuildingischaracterizedbyabilinearrelationasshowninfigureII-10.Wisthetotal deadloadandanticipatedliveload,ascalculated by equation II.1.C0 is as defined above and values are tabulated in table II-5. C2isamodificationfactorthatrepresensttheeffectofhysteresisshapeonthemaximum displacement response of the structure.Values for C2 are tabulated in FEMA 273 and are a function of BuildingPerformanceLevel,framingtype,andthefundamentalperiodofthestructure.Theyare included in table II-6. Table II-6: Values for modification factor C2 used in equation II.12 Building Performance Level Framing Type 11 Framing Type 22 Framing Type 22 Framing Type 11 T = 0.1 secondT >T second Immediate Occupancy Life Safety Collapse Prevention 1.0 1.0 1.01.51.2 1.1 1.0 1.0 1.01.01.0 1.3 1.Structures in which more than 30% of the story shear at any level is resisted by components or elements whose strength and stiffness may deteriorate during the design earthquake.Such elements and components include: ordinary moment resisting frames, concentrically braced frames, frames with partially restrained connections, tension only braced frames, unreinforced masonry walls, shear critical walls and piers, or any combination of the above. 2.All frames not assigned toFraming Type 1. 17 C3 is a modification factor to represent increased displacements due to dynamic P effects.Forbuildingswithpositivepostyieldstiffness,C3shallbesetequalto1.0.Forbuildingswith negative post yield stiffness, C3 shall be calculated from: eTRC2 / 33) 1 (0 . 1+ =(II.16) Values for R and Te are obtained from equations II.15 and II.17 respectively, and is the ratio of post yieldstiffnesstoeffectiveelasticstiffness,wherethenonlinearforcedisplacementrelationis characterized by a bilinear relation as shown in figure II-10. The effective fundamental period of the structure in the direction under consideration, Te may be calculated from: eieKKT T = (II.17) WhereTistheelasticfundamentalperiodofthestructure(inseconds)inthedirectionunder consideration calculated by elastic dynamic analysis.Ki is the elastic lateral stiffness of the building in thedirectionunderconsiderationandisfoundfromtheinitialstiffnessofthenon linearbaseshear vs. roof displacement curve as shown in figure II-10.Ke is the effective lateral stiffness of the building in the direction under consideration and is defined as the slope of the line which connects the point of intersectionofthepostyieldstiffnesslinewiththehorizontallineattheyieldbaseshearvalueto zero,whileintersectingtheoriginalbaseshearvs.roofdisplacementcurveat60%oftheyieldbase shear value.Ki and Ke are shown in figure II-10. Figure II-10:Bilinear Relation of Base Shear vs. Roof Displacement Plot Ki Non Linear Structural Response Roof Displacement Base Shear Vy 0.6 Vy Ke K 18 II-C)Reasons for Performing the Non Linear Static Procedure (Pushover Analysis) TheproceduretoperformthePushoverAnalysiswasthoroughlyoutlinedintheprevious section.It is easily seen that it is by no means an easy procedure.This brings up the question of why shouldone perform the pushover analysis, especially when it was defined as a static approximation to anactualdynamicanalysis?Also,sincetheanalysisisappliedtopreviouslydesignedorexisting structures,whymustoneperformamoredetailedanalysisthanjustdesigningbyanappropriate code such as the UBC? TherearetworeasonswhythePushoverAnalysismaybepreferredtoafulldynamic analysis.The first reason is computational time.To run a full dynamic, non linear analysis on even a simple structure takes a long time.If the Pushover Analysis is deemed applicable (see chapter III for applicabilityconditions)tothestructureathand,accurateresultscanbeobtainedinfractionsofthe timeitwouldtaketogetanyusefulresultsfromthefullydynamicanalysis.Sinceoneofthemain goalsofthisresearchwastodevelopacomputationaltoolwhichcouldbeeasilyappliedinadesign office, time is a very important parameter.This makes the Pushover Analysis much more applicable in a design office. Thesecondreasonhastodowithearthquakeunpredictability.Whenperformingadynamic analysis, it is best to use a series of earthquakes.This further increases the computational time.If we were to redesign a structure based on a maximum displacement achieved from a full dynamic analysis basedononeparticularearthquake,itiseasytoimaginethattherecouldbeanearthquakewhichhad thesameprobabilityofexceedancepercentagebuthadadifferentfrequencycontent.Basedonthe fundamentalperiodofthestructure,thiswouldincreaseordecreasethemaximumresponse.So,one wouldnotknowifthedesignwasthemaximumthatcouldbeexpecteduntilagreatnumberof earthquakegroundmotionrecordsweretested.ThePushoverAnalysisnaturallyaccountsforall earthquakeswiththesameprobabilityofexceedancebypredictingthemaximumdisplacementthat canbeexpectedintheformoftheTargetDisplacement.Now,computationaltimehasbeenfurther reduced,sinceonlyoneanalysismustberunforeachexceedanceprobabilitythatthedesigneris interestedin,strengtheningtheideathatthePushoverAnalysisismuchmorepracticalinadesign office. 19 TherearealsotworeasonswhythePushoverAnalysismaybepreferredtodesigning accordingtoanexistingcode,suchastheUBC.Thefirstisthatitadvancesthestate of the art from code design.The Pushover allows the designer to determine the buildings performance under a range ofgroundshakingswhilethecurrentcodedesignjustdeterminesthatthebuilding wont fall down or threatenlifeundertheworstpossibleshaking.Thisallowsownerstochooseinadvancewhatthe conditionoftheirbuildingwillbeafteragiveneventwhichinturnlimitstheircostsinpurchasing earthquakeinsurance.Also,byknowingtheresultingconditionofthebuildingafteranyground motion,includingsmallgroundmotionswhichmaybejustlargeenoughtocausesomenon structuraldamage,thedesignerscanmodifytheirdesigntoprotectexpensivearchitecturalfixturesor tolimittheinconveniencethatcanbecausedtobuildingoccupantswhenmechanicalorplumbing componentsaredamaged.Thisincreasestheoveralleffectivenessofthestructurefurtheringits applicability in a design office. Thesecondreasonisthatsincethemodeldirectlyincorporatestheactualmaterial nonlinearitiesofeachmember,andthestructureismonotonicallyforcedintotheinelasticresponse range, the designer is able to get detailed member information at displacements up to and including the maximumdisplacement.From this information, sections of members which will be most damaged by thegroundshakingcanbelocatedandthesesectionscanberedesignedtodevelopthestrengthor ductilitythatwillberequiredofthem.Incomparison,whendesigningbyanappropriatecode,the maximumloadsareapplieddirectlytothestructureandonlythemaximumresponseisdetermined.Therelationatspecificloadingvaluesbeforethemaximumislostandtheinterrelationamong contributing elements is not available.So, the designer has no idea of what the effect of increasing the strengthorductilityatonesectionwillhaveupontheother.Thisrequiresthatbothsectionsobtain theirmaximumstrengthorductility,whilethePushoverAnalysisallowsthedesignertomodifyone sectionwhichinturncouldhaveabeneficialresultontheothersectionloweringthemaximum response it would have to endure.So, the Pushover Analysis increases the effectiveness and efficiency of the design. 20 CHAPTER III APPLICATIONS OF THE NON LINEAR STATIC PROCEDURE (PUSHOVER ANALYSIS) ThePushoverAnalysiswasdefinedinchapterIIasanonlinearstaticapproximationofthe responseastructurewillundergowhensubjectedtodynamicearthquakeloading.Becauseweare approximatingthecomplexdynamicloadingcharacteristicofgroundmotionwithamuchsimpler monotonicallyincreasingstaticload,thereareboundtobelimitationstotheprocedure.Theobjectiveof thischapteristoquantifytheselimitations.ThiswillbeaccomplishedbyperformingthePushover Analysisandafullnonlineardynamicanalysisonreinforcedconcretemomentresistingframesofsix, twelve,andtwentystories.TheresultingTargetDisplacementobtainedfromthePushoverAnalysismay then be compared with the maximum displacement at the roof of each structure obtained from the dynamic analysis.ThePushoverAnalysiswillfollowthestepsoutlinedinchapterII,whilethestepsnecessaryto perform the dynamic analysis will be described as they are evaluated. III-A)Design of Three Reinforced Concrete Moment Resisting Frames The design of each frame will be carried out according to the 1997 Uniform Building Code (UBC) [17] and the American Concrete Institute (ACI) structural concrete building code requirements 318-95 [1].The frames are located in the Los Angeles, California area which falls under UBC earthquake zone 4.The frames to be designed are each one of four moment resisting frames in the structure and have common bay widths,storyheights,andfloorplans.ThetypicalfloorplanandsectionareshowninfigureIII-1,while theframedimensionsaregiveninfigureIII-2.Commonfloorarealoadswillbeusedforeachframeas givenbytheUBCcodeandarerepresentativeofatypicalofficebuilding. These loads are also shown in figureIII-1.Thedesignproceduredescribedherewillshowtheformulationofthegravityloads,wind loads, and earthquake loads used in design. 21 Figure III-1:Typical Floor Plan, Floor Section, and Loads Common to All Frames 22 Figure III-2:Design Frame Dimensions and Member Sizes III-A-1)Formulation of Gravity Loads Used in the Design of Frames. The floor loads typical to each frame are shown in figure III-1.They consist of dead loads which arethepartitionload,ceilingload,slabweight,andtransversebeamweight,andafloororroofliveload.In addition to these loads, the self weight of the girders and columns must be added.However, because the girdersandcolumnsmustbedesigned,theirweightisnotknownatthestartofthedesignprocess.Throughaniterativeprocedure,therequiredsectionsforeachmembercanbefoundandtheirweight included in the gravity loads.The concrete sections for use in the formulation of gravity loads are shown in figure III-2, while the gravity loads are determined in table III-1. 23 Table III-1:Formulation of Gravity Loads Used in Design Self - Weight Dead LoadsDescription t (in) h (ft) L (ft)Conc wght, wc (k/ft3)P1 (kips)P2 (kips) P (kips)Slab 6 12 18 0.15 8.1 16.2 16.2Transverse Beam 9 0.67 18 0.15 1.35 1.35 1.35Superimposed LoadsDescription W (ft) L(ft)P1 (kips)P2 (kips) P (kips)Ceiling (DL) 12 18 1.1 2.2 2.2Partition Walls (DL) 12 18 2.2 4.3 4.3Floor Load (LL) 12 18 5.4 10.8 10.8Roof Load (LL) 12 18 3.2 6.5 6.5DescriptionH1 (in)W (in) L (ft)Conc wght, wc (k/ft3)P1 (kips)P2 (kips) P (kips)26 x 18 (Girder) 26 18 12 0.15 2.9 5.9 5.928 x 20 (Girder) 28 20 12 0.15 3.5 7.0 7.032 x 22 (Girder) 32 22 12 0.15 4.4 8.8 8.8C - 1 (Column) 20 20 14 0.15 5.8 5.8 0.0C - 2 (Column) 24 24 14 0.15 8.4 8.4 0.0C - 3 (Column) 28 28 14 0.15 11.4 11.4 0.0C - 4 (Column) 32 32 14 0.15 14.9 14.9 0.01 All Values Are Illustrated BelowNote : The Force at a Given Location Is the Sum of the Forces Corresponding to thePoint Loads at That Location Due to Column Size, Beam Size, and Typical Floor Loads.The Change In Dead Load at the Roof Is Due to 1/2 Column Length There.Typical Floor LoadsDesign Section LoadsApplied Surface Load (psf)10205030 24 III-A-2) Formulation of Wind Loads for Use in Design Thecalculationofthewindloadingtobeappliedtoeachframewillbecarriedoutbasedonthe UBC wind loading profile.The wind pressure associated with each floor level is given by: w s q eI q C C P = (III.1) P is equal to the design wind pressure and is based on the basic wind speed at the design location and the exposure condition.For the Los Angeles area, the basic wind speed is 70 mph as given on the UBC wind map, and the exposure for a structure which has surrounding buildings is exposure C.From these two conditions, the remaining parameters can be determined. Ce is equal to the combined height, exposure and gust factor, and is a function of the exposure of the building and height of each floor level.Values for this coefficient are tabulated in the UBC. Cqisequaltothepressurecoefficientforthestructureorportionofthestructureunder consideration and is tabulated in the UBC. qsisequaltothewindstagnationpressureatastandardheightof33atthedesignlocationas tabulated in the UBC. Iw is equal to the importance factor of the structure also laid out in the UBC. The total wind force acting at each floor level on a frame is the design wind pressure multiplied by the floor height and the tributary width of the frame.Calculations for each frame are detailed in table III-2. III-A-3)Formulation of Earthquake Loads for Use in Design Thecalculationofearthquakeloadstobeappliedtoeachframewillbecarriedoutbasedonthe UBCearthquakeloadingprofile.Theforcecausedbyanearthquaketo be applied at each floor level is a function of the Design Base Shear, V, which is given by: WRTI CVv= (III.2) However, the Design Base Shear need not exceed: WRI CVa5 . 2= (III.3) Further, the Design Base Shear must not be less than the least of the following: 25 Table III-2: UBC Wind Load Calculations Used in Design Data Common To All FramesWind Importance Factor, Iw = 1.0Wind Stagnation Pressure @33', qs = 12.6 psfPressure Coefficient, Cq = 1.4Six Story FrameHeigth 1 - UBC tbl 16-G (ft)Heigth 2 - UBC tbl 16-G (ft)Heighth of Floor, H (ft)Ce associated w/ H1Ce associated w/ H2Ce associated w/ FloorTributary Width, W (ft)Wind Force, P*WH (kips)0 15 14 1.06 1.06 1.06 18 4.7125 30 28 1.19 1.23 1.214 18 5.4040 60 42 1.31 1.43 1.322 18 5.8840 60 56 1.31 1.43 1.406 18 6.2560 80 70 1.43 1.53 1.48 18 6.5880 100 84 1.53 1.61 1.546 18 6.87Twelve Story FrameHeigth 1 - UBC tbl 16-G (ft)Heigth 2 - UBC tbl 16-G (ft)Heighth of Floor, H (ft)Ce associated w/ H1Ce associated w/ H2Ce associated w/ FloorTributary Width , W (ft)Wind Force, P*WH (kips)0 15 14 1.06 1.06 1.06 18 4.7125 30 28 1.19 1.23 1.214 18 5.4040 60 42 1.31 1.43 1.322 18 5.8840 60 56 1.31 1.43 1.406 18 6.2560 80 70 1.43 1.53 1.48 18 6.5880 100 84 1.53 1.61 1.546 18 6.8780 100 98 1.53 1.61 1.602 18 7.12100 120 112 1.61 1.67 1.646 18 7.32120 160 126 1.67 1.79 1.688 18 7.50120 160 140 1.67 1.79 1.73 18 7.69120 160 154 1.67 1.79 1.772 18 7.88160 200 168 1.79 1.87 1.806 18 8.03Twenty Story FrameHeigth 1 - UBC tbl 16-G (ft)Heigth 2 - UBC tbl 16-G (ft)Heighth of Floor, H(ft)Ce associated w/ H1Ce associated w/ H2Ce associated w/ FloorTributary Width, W (ft)Wind Force, P*WH (kips)0 15 14 1.06 1.06 1.06 18 4.7125 30 28 1.19 1.23 1.214 18 5.4040 60 42 1.31 1.43 1.322 18 5.8840 60 56 1.31 1.43 1.406 18 6.2560 80 70 1.43 1.53 1.48 18 6.5880 100 84 1.53 1.61 1.546 18 6.8780 100 98 1.53 1.61 1.602 18 7.12100 120 112 1.61 1.67 1.646 18 7.32120 160 126 1.67 1.79 1.688 18 7.50120 160 140 1.67 1.79 1.73 18 7.69120 160 154 1.67 1.79 1.772 18 7.88160 200 168 1.79 1.87 1.806 18 8.03160 200 182 1.79 1.87 1.834 18 8.15160 200 196 1.79 1.87 1.862 18 8.28200 300 210 1.87 2.05 1.888 18 8.39200 300 224 1.87 2.05 1.9132 18 8.50200 300 238 1.87 2.05 1.9384 18 8.62200 300 252 1.87 2.05 1.9636 18 8.73200 300 266 1.87 2.05 1.9888 18 8.84200 300 280 1.87 2.05 2.014 18 8.95 26 IW C Va11 . 0 = (III.4) WRI ZNVv8 . 0= (III.5) In the above equations: Z is equal to the seismic zone factor.For the Los Angeles area, the seismic zone is Zone 4 and the seismic zone factor is equal to 0.4. CvandCaareseismiccoefficientsandarefunctionsofthesoiltypeandtheseismiczonefactor. A stiff soil profile will be the basis for design corresponding to soil type SD. I is equal to the seismic importance factor for the structure under consideration. W is equal to the total seismic dead load equal to the total dead load of each structure. Nv is equal to the near source factor of the structure to known faults.Under consideration will be a site with a known fault greater than 15 km away and Nv equals 1.0. Risequaltotheoverstrengthfactorbasedonthelateralforceresistingsystemofthestructure.UnderconsiderationwillbeasystemofreinforcedconcreteOrdinaryMomentResistingFramessinceno special designs will be considered (i.e. plastic hinges etc.). T is the elastic fundamental period of the structure (seconds) in the direction under consideration.For UBC calculations, this period is defined as: 4 / 3) (n th C T =( III.6) Ct is a numerical coefficient equal to 0.03 for reinforced concrete moment resisting frames. hn is equal to the height in feet from the base to the roof of the structure. Oncethebaseshear,V,foreachframe has been determined, the lateral force to be applied at the top of the structure, Ft, can be calculated.This is given in the UBC as: TV Ft07 . 0 = (III.7) Where T and V are as described above.Fromthebaseshear,V,andthefundamentalperiod,T,asgivenbytheUBC,thelateralforce distribution along the height of the structure, Fx, can be determined.This is given as: 27 ==nii ix x txh wh w F VF1) ((III.8) Fx is equal to the lateral force to be applied at floor level x, wx is the fraction of the total structure weight, W, allocated to floor level x, hx is the height of floor level x above the base of the structure, and the summationinthedenominatoristhesumofthesevaluesoverthetotalnumberoffloors,n.The calculationsoftheaboveequationsleadingtothelateralforcedistributionoverthetotalheightofeach structure is illustrated in table III-3. Table III-3: UBC Earthquake Loads Used in Design 0.41.01.00.40.41.03.5Seismic Coefficient , Ca = Importance Factor, I =Overstrength Factor, R =Data Common to All FramesNear Source Factor, Nv=Near Source Factor, Na=Seismic Coefficient , Cv = Seismic Zone Factor, Z = Fundamental Period, T (s) = 0.832 UBC 30-8Design Base Shear, V = 162 UBC 30-4 - 30-7Concentrated Force at Top, Ft= 9.4 UBC 30-14Floor #Floor Weight, Wi (kips)Wi/Wtotal wi (kips)Floor Height, hi (ft) wi * hiQuake Force, Fx (kips)1 198 0.168 14 2.35 7.42 198 0.168 28 4.70 14.73 198 0.168 42 7.05 22.14 198 0.168 56 9.40 29.45 198 0.168 70 11.75 36.8Roof 189 0.160 84 13.48 51.6Wtotal = 1180 wi*hi = 48.74Fundamental Period, T (s) = 1.400 UBC 30-8Design Base Shear, V = 226 UBC 30-4 - 30-7Concentrated Force at Top, Ft= 22.1 UBC 30-14Floor #Floor Weight, Wi (kips)Wi/Wtotal wi (kips)Floor Height, hi (ft) wi * hiQuake Force, Fx (kips)1 213 0.086 14 1.21 2.82 213 0.086 28 2.41 5.53 213 0.086 42 3.62 8.34 213 0.086 56 4.82 11.05 213 0.086 70 6.03 13.86 213 0.086 84 7.23 16.57 205 0.083 98 8.13 18.68 205 0.083 112 9.30 21.29 198 0.080 126 10.10 23.110 198 0.080 140 11.23 25.611 198 0.080 154 12.35 28.2Roof 189 0.077 168 12.88 51.5Wtotal = 2471 wi*hi = 89.31Twelve Story FrameSix Story FrameFundamental Period, T (s) = 2.053 UBC 30-8Design Base Shear, V = 402 UBC 30-4 - 30-7Concentrated Force at Top, Ft= 57.8 UBC 30-14Floor #Floor Weight, Wi (kips)Wi/Wtotal wi (kips)Floor Height, hi (ft) wi * hiQuake Force, Fx (kips)1 243 0.055 14 0.77 1.92 243 0.055 28 1.55 3.83 243 0.055 42 2.32 5.74 233 0.053 56 2.96 7.25 233 0.053 70 3.70 9.06 233 0.053 84 4.44 10.87 233 0.053 98 5.18 12.68 223 0.051 112 5.69 13.99 223 0.051 126 6.40 15.610 223 0.051 140 7.11 17.311 223 0.051 154 7.82 19.112 223 0.051 168 8.53 20.813 223 0.051 182 9.25 22.614 205 0.047 196 9.14 22.315 205 0.047 210 9.79 23.916 205 0.047 224 10.44 25.517 198 0.045 238 10.72 26.118 198 0.045 252 11.35 27.719 198 0.045 266 11.98 29.2Roof 189 0.043 280 12.05 87.2Wtotal = 4400 wi*hi = 141.19Twenty Story Frame 28 III-A-4)Total Design Loads and Section Determination The total design loads are factored combinations of the gravity loads (dead and live), wind loads, andearthquakeloadsdeterminedpreviously.ThetotalunfactoreddesignloadsareshowninfigureIII-3, figure III-4, and figure III-5 for the six, twelve, and twenty story frames respectively.Figure III-3: Gravity, Wind and Earthquake Loads for the Six Story Frame. Fromtheseunfactoredloads,wecanapplytheACIloadcombinationsandfindthefactoredloadswhich must be designed for.The ACI load combinations which must be checked are: LL DL 7 . 1 4 . 1 + (III.9a) WL LL DL 28 . 1 28 . 1 05 . 1 + (III.9b) WL DL 3 . 1 9 . 0 (III.9c) EL LL DL 4 . 1 28 . 1 05 . 1 + + (III.9d) EL DL 43 . 1 9 . 0 + (III.9e) 29 Figure III-4: Gravity, Wind, and Earthquake Loads for the Twelve Story Frame 30

Figure III-5: Gravity, Wind, and Earthquake Loads for the Twenty Story Frame 31 InequationsIII.9,DLisequaltothedeadload,LLisequaltotheliveload,WLisequaltothe wind load, and EL is equal to the earthquake load.Once the loads have been factored, the members can be designed.Theresultingsectionlocationsareshownin figure III-6, while the resulting sections are shown in figures III-7 and III-8. Figure III-6:Design Section Locations for Six, Twelve and Twenty Story Frames 32 Figure III-7:Final Design Sections Beams 33 Figure III-8:Final Design Sections Columns 34 III-B)Performing the Pushover Analysis on the Moment Resisting Frames Oncetheframeshavebeendesigned,thePushoverAnalysiscanbeperformedoneachframeto determineitsmaximumexpectedresponse.AsdescribedinchapterII,thefirstthingthatneedstobe calculatedistheverticaldistributionofthelateralloads.However,sincethisdistributionisafunctionof thefundamentalperiodofthestructurethroughtheexponentk,theperiodmustfirstbedetermined.The Finite Element Analysis Program (FEAP) [16] will be used to perform the Pushover as well as the dynamic analyses,sothiswillalsobeusedtofindthefundamentalperiodofthestructures.Oncethisperiodhas beendetermined,thelateralloaddistributioncanbecalculated,followedbytheSeismicHazardandthe TargetDisplacement.TheBuildingPerformanceLevelwillnotbeincludedheresinceonlytheTarget Displacementistobecomparedwiththemaximumdynamicdisplacementfoundattheroofofeach structure.So,anearthquakethathasa10%probabilityofexceedancein50yearswillbeusedasthe Building Performance Level requirement. III-B-1) Period Determination The fundamental period of a structure is a function of its mass and stiffness, so the masses at each floorlevelmustbecalculated.Thestiffnessofeachstructureisafunctionofthemembersizes,which weredesignedintheprevioussection,andFEAPcalculatesthisinternally.Themassofthestructureisa function of the total gravity load to be used in the Pushover Analysis.The total gravity load to be used in the Pushover Analysis was given in equation II.1 and is reproduced here as: ) ( 1 . 1S L D GQ Q Q Q + + = (III.10) Recall that the live load to be used is 25% of the unreduced live load for the structure and the dead load is the total dead load effect.Since the structures are located in Los Angeles, the design snow load is less than 30 psf and therefore equal to zero.So the total gravity load to be used in the Pushover Analysis is equal to thedeadloadshowninfigures III-3 to III-5 plus 25% of the live load shown in these figures all times the factor1.1.Notethatthedeadloadandliveloadusedherearetheunfactoreddesignloads.Forthe determinationofthetotalmassonthestructure,FEAPrequiresaninputmassperunitvolumeforeach element.Since the total gravity load of equation III.10 must be included in the mass, it must be converted 35 into a mass per unit volume for the input.This will be accomplished by including the total load at a floor level,minusthecolumnweight,astheinputmassperunitvolumeforthebeams.So,themassperunit volumeinputforeachbeamisthetotalloadateachfloorlevel,minusthecolumnweight,dividedbythe volumeofthemember.Thevolumeofeachmemberisitslengthtimesitswidthtimesitsheight.Since thisdivisiongivesaloadperunitvolume,thisvaluemustalsobedividedbythegravitationalconstant.Theinputmassperunitvolumeforthecolumnswilljustbetheunitweightofconcrete.Themassesper each member as a function of the total Pushover Analysis gravity loads are formulated in table III-4. Table III-4: Calculation of Mass Per Unit Volume for Each Member Once the applicable masses have been included in the input, FEAP can determine the fundamental period of each frame.The fundamental, second, and third periods of each frame are shown for comparison in table III-5. Table III-5:Fundamental, Second, and Third Periods for Each Frame # Stories in Frame Fundamental Period, T (sec)Second Period, T2 (sec) Third Period, T3 (sec)6 1.61 0.517 0.29412 2.66 0.954 0.55020 3.59 1.39 0.797Sect # P (kips)P = 6*P (kips)Width, W (in)Heigth, H (in)Length, L (in) Distributed Mass inputBeam 1 - 1a 35.9 215.4 18 26 864 1.378634002E-06Beam 2 - 2a 37.1 222.6 20 28 864 1.190655912E-06Beam 3 - 3a 39.1 234.6 22 32 864 9.981699435E-07Sect # Distributed Mass inputAll Sections 2.246520589E-07Beam mass per unit volume = (P1 + P2 + P3 + 4P) / (Beam L * W * H * g)Where P1 = P3 = 1/2 P, and P2 = P because column mass is calculated seperatelyColumn mass per unit volume = (concrete unit weight) / g0.15 8.68056E-05Column Distributed MassesBeam Distributed MassesConc Unit Weight (k/ft3) Conc Unit Weight (k/in3) 36 Thesecondandthirdperiodsforeachframeareshownbecauseit is interesting to notice that the secondperiodofthetwentystoryframeisclosetothefirstperiodofthesixstoryframe.So,itcanbe expectedthatthetwentystoryssecondmodewillcontributeinitsdynamicresponseasmuchasthesix storys first mode will to its dynamic response.A similar statement can be made about the second mode of thetwelvestorybuildingsinceitssecondperiodis about 60% of the six storys first period.This may be importantwhencomparingthePushoverAnalysiswiththedynamicanalysisbecausetheCvxload distributionisenforcingsomevariationofthefirstmoderesponseineachofthestructures.So,ifafirst moderesponseisaccurateforthesixstoryframe,itiseasytoseethatfortheotherframesitwillbeless accurate as higher modes contribute more to the dynamic response. III-B-2) Calculation of the Vertical Distribution of Lateral Loads Nowthatthefundamentalperiodforeachframestructurehasbeendetermined,thevertical distribution of the lateral loads for the Pushover Analysis can be calculated.The vertical distribution of the lateral loads is given by the FEMA 273 Cvx loading profile as given by equation II.2.This equation will be reproduced here for convenience and is: ==nikx ikx xvxh wh wC1(III.11) The Cvx loading coefficient is the value to be applied at each floor level of the frame.wx is the fraction of thetotalweight,W,allocatedtotheflooratlevelx,hxistheheightoftheflooratlevel xabovethebase, andthesummationinthedenominatoristhesumofthesevaluesoverthetotalnumberofstories,six, twelve, or twenty for this example.All of these parameters are illustrated in figure II-5.The exponent k is afactorwhichvarieswithfundamentalperiodasshowninfigureII-6.TheseCvxloadingcoefficientsare inputintoFEAPandtheprogrammultipliesthembythemonotonicallyincreasingloadtogetthelateral loads at each increasing value.The formulation of the Cvx loading coefficients for each frame is detailed in tableIII-6.OnceallofthesevalueshavebeendeterminedtheycanbecombinedwiththePushover Analysis gravity loads determined in equation III.10 to get the total frame loading for analysis as shown in figures III-9, III-10, and III-11 for the six, twelve, and twenty story frames, respectively. 37 Table III-6:Determination of the Vertical Distribution of Lateral Loads for Each Frame FEAP Fundamental Period, T = 1.61 secPeriod Dependent Factor, k = 1.55Floor #Weight, QGi (kips)QGi / QGtotal, wi Height of floor, hi (ft) wi * hik Lateral Load Factor, Cvx 1 236 0.170 14 10.2 0.02252 236 0.170 28 30.0 0.06593 236 0.170 42 56.3 0.12374 236 0.170 56 88.0 0.19345 236 0.170 70 124.5 0.2735Roof 209 0.150 84 146.1 0.3210QGtotal = 1388 wi * hik = 455FEAP Fundamental Period, T = 2.66 secPeriod Dependent Factor, k = 2.00Floor #Weight, QGi (kips)QGi / QGtotal, wi Height of floor, hi (ft) wi * hik Lateral Load Factor, Cvx 1 252 0.0864 14 16.9 0.00172 252 0.0864 28 67.7 0.00663 252 0.0864 42 152.4 0.01494 252 0.0864 56 270.9 0.02665 252 0.0864 70 423.4 0.04156 252 0.0864 84 609.6 0.05977 243 0.0836 98 802.4 0.07868 243 0.0836 112 1048.1 0.10279 236 0.0810 126 1285.6 0.126010 236 0.0810 140 1587.1 0.155511 236 0.0810 154 1920.4 0.1882Roof 209 0.0716 168 2020.0 0.1979QGtotal = 2913 wi * hik = 10205FEAP Fundamental Period, T = 3.59 secPeriod Dependent Factor, k = 2.00Floor #Weight, QGi (kips)QGi / QGtotal, wi Height of floor, hi (ft) wi * hik Lateral Load Factor, Cvx 1 285 0.0551 14 10.8 0.00042 285 0.0551 28 43.2 0.00163 285 0.0551 42 97.2 0.00374 274 0.0528 56 165.7 0.00635 274 0.0528 70 258.9 0.00986 274 0.0528 84 372.9 0.01427 274 0.0528 98 507.5 0.01938 264 0.0509 112 638.7 0.02439 264 0.0509 126 808.3 0.030710 264 0.0509 140 997.9 0.037911 264 0.0509 154 1207.5 0.045912 264 0.0509 168 1437.0 0.054613 264 0.0509 182 1686.5 0.064114 243 0.0470 196 1805.4 0.068615 243 0.0470 210 2072.5 0.078816 243 0.0470 224 2358.0 0.089617 236 0.0455 238 2578.9 0.098018 236 0.0455 252 2891.2 0.109919 236 0.0455 266 3221.3 0.1224Roof 209 0.0403 280 3156.1 0.1199QGtotal = 5179 wi * hik = 26315Twenty Story FrameTwelve Story FrameSix Story Frame 38 Figure III-9: Gravity Loads and Cvx Loading Coefficients for Six Story Frame 39 Figure III-10:Gravity Loads and Cvx Loading Coefficients for Twelve Story Frame 40 Figure III-11:Gravity Loads and Cvx Loading Coefficients for Twenty Story Frame 41 III-B-3)Element Reduction for Analysis of Frames Because of the length of time required to run a complete non linear dynamic analysis of these structures, an effort was made to limit the numbers of elements and nodes in each frame.Since a comparison is to be made between dynamic and Pushover analyses, this also applies to the Pushover.The elements and nodes finally adopted for the FEAP input files are shown in figure III-12, while the transformation of the midspan point loads shown in figures III-9 to III-11 are detailed in table III-7. Figure III-12:Element and Node Configuration Used in FEAP Input 212019181716151413121110987654321424140393837363534333231302928272625242322636261605958575655545352515049484746454443 1 8 15234567 14131211109212019181716131211109876543212625242322212019181716151439383736353433323130292827 42 Table III-7:Transformation of Midspan Point Loads to Fixed End Forces and Moments III-B-4)Determination of Seismic Hazard for Analysis ThecalculationoftheseismichazardwasdetailedinsectionII-B-3andisafunctionofthe probability of exccedance in 50 years under consideration, the mapped short period acceleration parameter, themappedaccelerationparameterataonesecondperiod,thesoilsiteconditions,theeffectivestructural damping, and the fundamental period of the structure.The current analysis considers an earthquake with a 10%probabilityofexceedancein50years,aLosAngeleslocation,stiffsoilconditionsand5%effective structuraldamping.TheapplicableequationsareequationsII.3throughII.11tocalculatethemodified floor #'sP1 = P3 (kips) P (kips)P2 (kips)P1+P = P3+P (kips)P2+2*P (kips)M = 96*P (kip-in) +/-6 -20.4 -32.9 -36.1 -53.3 -101.9 3158.45 to 1 -25 -35.9 -42.2 -60.9 -114 3446.4floor #'sP1 = P3 (kips) P (kips)P2 (kips)P1+P = P3+P (kips)P2+2*P (kips)M = 96*P (kip-in) +/-12 -20.4 -32.9 -36.1 -53.3 -101.9 3158.411 to 9 -25 -35.9 -42.3 -60.9 -114.1 3446.48 to 7 -25.7 -37.1 -43.6 -62.8 -117.8 3561.66 to 1 -28.5 -37.1 -46.3 -65.6 -120.5 3561.6floor #'sP1 = P3 (kips) P (kips)P2 (kips)P1+P = P3+P (kips)P2+2*P (kips)M = 96*P (kip-in) +/-20 -20.4 -32.9 -36.1 -53.3 -101.9 3158.419 to 17 -25 -35.9 -42.2 -60.9 -114 3446.416 to 14 -25.7 -37.1 -43.6 -62.8 -117.8 3561.613 to 8 -29.5 -39.1 -48.3 -68.6 -126.5 3753.67 to 4 -32.8 -39.1 -51.7 -71.9 -129.9 3753.63 to 1 -36.7 -39.1 -55.6 -75.8 -133.8 3753.6Six Story FrameTwelve Story FrameTwenty Story Frame 43 mappedshortperiodandonesecondperiodaccelerationparameters,SSandS1,attheprobabilityof exceedanceunderconsideration,theReturnPeriod,PR,attheprobabilityofexceedanceunder consideration, the design acceleration parameters, SXS and SX1, based on the soil type parameters, Fa and Fv, and the Spectral Response Acceleration, Sa, based on structural damping coefficients, BS and B1, the period at which the constant acceleration region of the spectrum intersects with the constant velocity region of the spectrum,T0,andthefundamentalstructuralperiod,T.Thesecalculationsarecarriedout,andvaluesfor theparametersarelistedintableIII-8,whiletheGeneralResponseSpectrumisshowninfigureIII-13.Shown on the General Response Spectrum curve is where the six, twelve, and twenty story frames lie. Table III-8: Formulation of the General Response Spectrum for Analysis Figure III-13:General Response Spectrum forAnalysis Return PeriodSS (g's) S1 (g's) SS (g's) S1 (g's)@ HL, PR (yrs) SS (g's) S1 (g's)2 0.75 1.25 0.5 475 1.25 0.50Intersect Period1Fa FvSxs (g's) Sx1 (g's) BS B1 T0 (sec)1.0 1.5 1.25 0.75 1.0 1.0 0.61 Period at Which the Constant Velocity and Acceleration Regions of the Design Spectrum IntersectAccelerations @ HLSoil Site Class = D Design Accelerations Damping Coeff10Mapped 2% / 50yrs Mapped 10% / 50yrs Hazard Level, HL (% / 50yrs)00.250.50.7511.251.50 0.5 1 1.5 2 2.5 3 3.5 4Fundamental Structural Period, T (sec)Spectral Response Acceleration, Sa (g's)T0 = 0.6 s0.2 T0 = 0.12 s6 Story Frame, T = 1.61 s12 Story Frame, T = 2.66 s20 Story Frame, T = 3.59 s 44 III-B-5)Calculation of Target Displacement for Analysis Finally the Target Displacement can be calculated for each of the analysis frames.Recall from equation II.12 that the Target Displacement, t, is equal to: gTS C C C Cea t223 2 1 04 = (III.12) All parameters in equation III.12 were defined in chapter II-B-4 and these parameters plus the values to determine these parameters are shown in table III-9.The Effective Stiffness, the Elastic Stiffness, and the post yield stiffness, which enter into the parameters in equation III.12, must be calculated from the Base Shear vs. Roof Displacement curve as defined in equation II.17 and illustrated for each of the analysis frames in figures III-14 to III-16. Table III-9:Values Used in Obtaining Target Displacements for Analysis Figure III-14:Base Shear vs. Roof DisplacementWith Target Displacement for Six Story Frame # stories R T0 (sec) Ki (k/in) Ke (k/in) Ke (k/in)Vy (kips)6 3.27 0.6 14.1 12.6 1.35 0.107 14012 3.22 0.6 8.68 8.26 0.748 0.091 17020 2.94 0.6 8.11 7.40 0.198 0.027 245# stories C0C1C2C3Sa (g's) Te(sec) t (in)6 1.42 1.0 1.0 1.0 0.469 1.70 18.812 1.50 1.0 1.0 1.0 0.282 2.73 30.820 1.50 1.0 1.0 1.0 0.209 3.76 43.3Internal ValuesEquation Values0204060801001201401600 5 10 15 20 25Roof Displacement (in)Base Shear (kips)Ki = 14.1 kips/inKe = 12.6 kips/inKe = 1.35 kips/in t = 18.8 inOriginal Non - Linear CurveVy = 140 kips0.6 Vy = 84 kips 45 Figure III-15:Base Shear vs. Roof Displacement With Target Displacement for Twelve Story Frame Figure III-16:Base Shear vs. Roof Displacement With Target Displacement for Twenty Story Frame 0204060801001201401601802000 5 10 15 20 25 30 35 40 45Roof Displacement (in)Vy = 170 kips0.6 Vy = 102 kipsKi = 8.68 kips/inKe = 8.26 kips/in t = 30.8 inOriginal Curve Ke = 0.748 kips/in0501001502002503000 20 40 60 80 100 120Roof Displacement (in)Base Shear (kips)Vy = 245 kips0.6 Vy = 147 kips t = 43.3 in Ke = 0.198 kips/inKe = 7.40 kips/inKi = 8.11 kips/inOriginal Curve 46 III-C)Complete Non Linear Dynamic Analyses for Each Frame The full non linear dynamic analysis will now be run on each frame.The dynamic loading will be a ground motion record as provided by the SAC joint venture [13].SAC provides a number of ground motions on their web page which [14] are scaled so that the mean response spectrum matches that given by FEMA273.TobeconsistentwiththePushoverAnalysis,thegroundmotionmustbethatforaLos Angelessite,onstiffsoil,witha5%structuraldampingratio,anda10%probabilityofexceedancein50 years.This happens to be the 1940 El Centro ground motion record as shown in figure III-17. Figure III-17:1940 El Centro Ground Motion Used in Dynamic Analysis FEAP allows dynamic analysis for frames through the frame analysis additions added by Spacone etal[15].Thegroundmotionisinputasafileandtheaccelerationsareconvertedtoforcesateachtime step by the effective force form of the equation of motion which is: [ ]{ } [ ]{ } [ ]{ } [ ]{ } { }eff gp v M v K v C v M = = + + ' ' ' ' ' (III.13) Wherev,v,andvaretherelativeacceleration,velocity,anddisplacementvectorsofeachelement respectively, and vg is the acceleration vector due to the ground motion.M is equal to the mass matrix of each element as formulated in table III-4, K is equal to the stiffness matrix of each element, and C is equal -300-200-10001002003000 2 4 6 8 10 12 14 16 18 20Time (s)Acceleration (in/s2) 47 to the element damping matrix formulated in FEAP through Rayleigh Damping coefficients.The Rayleigh damping coefficients are given by Chopra [4] as: j ij i +=2, j i +=2 (III.14 a) K M C + = (III.14 b) Whereisequaltothestructuraldampingratio,5%inthiscase,andiandjareequaltothefirstand secondfrequencies(rad/s)ofthestructurerespectivelyforFEAPinput.Thesevaluesareformulatedin table III-10. Table III-10:Rayleigh Damping Coefficients Used in Dynamic Analysis TheresultingdynamicdisplacementsattheroofofeachframeareshowninfiguresIII-18toIII-20.Includedinthesefiguresarethemaximumandminimumdynamicdisplacementsobtainedatthe roof of each structure. Figure III-18:Dynamic Roof Displacement for Six Story Frame -20-15-10-5051015200 5 10 15 20 25Time (s)Roof Displacement (in)Max Disp = 15.0 inMin Disp = -15.0 in1 2 1 = 2 2.5111956 8.0407019 0.05 0.1913568 0.0094771 2 1 = 2 1.4880482 4.206531 0.05 0.1099207 0.01756061 2 1 = 2 1.1086985 2.8633657 0.05 0.0799234 0.0251758Six Story FrameTwelve Story FrameTwenty Story Frame 48 Figure III-19:Dynamic Roof Displacement for Twelve Story Frame Figure III-20:Dynamic Roof Displacement for Twenty Story Frame -20-15-10-5051015200 5 10 15 20 25Time (s)Roof Displacement (in)Max Disp = 14.2 inMin Disp = -12.7 in-20-15-10-5051015200 5 10 15 20 25Time (s)Roof Displacement (in)Max Disp = 10.7 inMin Disp = -12.5 in 49 III-D)Comparisons of Full Dynamic Results With Pushover Analysis Results The Target Displacements obtained from the Pushover Analysis and the maximum and minimum displacementsobtainedattheroofofeachstructurefromthecompletedynamicanalysisaretabulatedin table III-11. Table III-11:Target Displacement and Maximum and Minimum Dynamic Displacements for Frames Incomparingtheseresults,onecanseethatforthesixstoryframeweget results which compare verywell.OnethingtonoteisthateventhoughtheTargetDisplacementisgreaterthanthedynamic displacement,theresultsforthesixstoryframearegoodbecausethis Target Displacement is a maximum displacementexpectedforanyearthquakewitha10%probabilityofexceedancein50years.Thismeans thatadifferentearthquakewithadifferentfrequencycontentcouldgiveahighermaximumdisplacement underdynamicloading.Toclarifytheseresults,theloaddistributionforthePushoverAnalysiswillbe comparedtothedynamicdisplacedshapeatseveraltimes.ThesetimesaredescribedinfigureIII-21(a) whenthemaximumdynamicroofdisplacementoccursbeforetheminimumorIII-21(b)whenthe minimum dynamic roof displacement occurs before the maximum.Figure III-21:Times at Which Dynamic Displaced Shapes Will be Plotted # StoriesTarget Displacement (in)Maximum Dynamic Roof Displacement (in)Minimum Dynamic Roof Displacement (in)6 18.8 15.0 -15.012 30.8 14.2 -12.720 43.3 10.7 -12.5 50 With the definition of these times, the Pushover static load distribution and the dynamic displaced shape at the defined times can be plotted for the six story frame (figure III-22). Figure III-22: Pushover Load Distribution and Dynamic Displaced Shapes Six Story Frame It can be seen in the above figure, that even though the displaced shape changes with time, there is adefiniteparticipationofthesecondmodeasisapparentbythenonlinearityofthedisplacedshapes.Furthermore,itcanbeseenthat at the maximum and minimum displaced shapes the middle portion of the buildinghasavelocitywhichisoppositetothetopportion.Thisisapparentbythedisplacedshapesat times t5, t6, t2, and t3 in which the middle portion of the building has a greater displacement than the roof.So, at the maximum and minimum displacements, when the velocity of the roof is equal to zero, the middle portion of the building must be traveling in the opposite direction.This limits the maximum and minimum displacements at the roof that will be obtained from the dynamic analysis because the middle portion of the building cancels out some of the roof displacement.So, while the Pushover Analysis loading is enforcing a first mode response in the structure, the El Centro earthquake excites the first and second modes of this six story frame, however an earthquake with a different frequency content could excite only the first mode and the maximum and minimum displacements would approach the Target Displacement. Incomparingtheresultsforthetwelvestoryframe,amuchlargerdiscrepancybetweenthe PushoverAnalysisspredictionandtheactualdynamicdisplacementisrecognized.Again,plottingthe Pushoverloaddistributionandthedynamicdisplacedshapeatseveraltimes(figureIII-23),itisseenthat again the first mode response is being enforced in the Pushovers static load, while the dynamic loading is excitingthesecond,third,andevenhighermodes.Thisaccountsforthelargediscrepancy.12345670.00 0.10 0.20 0.30 0.40Cvx Load FactorFloor Number01234567-20 -10 0 10 20Displacement (in)Floor Levelmint2t1t7t4t6 t5t3 max 51 Figure III-23: Pushover Load Distribution and Dynamic Displaced Shapes Twelve Story Frame Finally,comparingtheresultsforthetwentystoryframe,againaverylargediscrepancyis apparentbetweenthePushoverAnalysisspredictedresultandtheresultobtainedfromthedynamic analysis.InviewingthecomparisonofthePushoversstaticloadwiththedynamicdisplacedshapeat varioustimes(figureIII-24),itisseenthatonceagainthePushoverisenforcingafirstmoderesponsein the structure while the ground motion is exciting the higher modes. Figure III-24: Pushover Load Distribution and Dynamic Displaced Shapes Twenty Story Frame

1357911130.00 0.05 0.10 0.15 0.20 0.25Cvx Load FactorFloor Number02468101214-20 -10 0 10 20Displacement (in)Floor Levelmint5t6t4t7t1t2t3 max161116210.00 0.05 0.10 0.15Cvx Load FactorFloor Number0510152025-20 -10 0 10 20Displacement (in)Floor Levelmin t5 t6 t3t2maxt7t4t1 52 Theconclusionsthatcanbedrawnfromthesecomparisonsisthatforstiffer,lowerperiod structures,suchasthesixstoryframe,thepushoveranalysisisaccurateandincludesexpected displacements that could be caused by a multitude of earthquakes.However, as the fundamental period of the structure increases, there is less likely to be an earthquake that excites primarily the first mode response, and the higher modes are more likely to be excited by every earthquake.This was illustrated in the twelve andtwentystoryframeresults.Tofurtherillustratethisstatement,figureIII-25showsthefrequency contentofthe1940ElCentrogroundmotionusedintheanalysisplottedasafunctionofperiod.Also shown in the figure are the first three periods of each structure repeated here for convenience. Figure III-25:Frequency Content of 1940 El Centro Ground Motion as a function of Period Whiledifferentearthquakeswillvaryinfrequencycontentplots,itcaneasilybeseenthatasthe fundamental period, and therefore the higher periods, increases, an earthquake will be less likely to have a frequency content distribution which excites primarily the first mode of the structure.As the higher periods increase,theyareenteredintothe range which will be excited by the earthquake, therefore contributing to its overall dynamic response.So, the Pushover Analysis is accurate for stiff structures whose higher modes will have less effect on the overall dynamic response. 0501001502002500 1 2 3 4 5 6 7 8Period, T (s)Absolute Value of Fourier Transform, |C(w)|T1 (s) T3 (s) T2 (s) Frame # Stories120.294 0.517 1.61 60.550 0.954 2.6620 0.797 1.39 3.59 53 III-E)Dependence of Target Displacement on Choice of Vy SincetheaccuracyofthePushoverAnalysisisbeingdetermined,somethingmustbesaidabout the subjective choice in the structural yield level on the base shear vs. roof displacement plot, Vy.Because thechoiceissubjective,theremaybeasignificantchangeinthemaximumexpecteddisplacement,or TargetDisplacement,givendifferentchoicesinthebaseshearvs.roofdisplacementplotyieldlevel.To quantify this, the target displacement is calculated for changes in the choice of Vy in figure III-26 for the six story frame. Figure III-26 aFigure III-26 b Figure III-26 cFigure III-26 d Figure III-26:Target Displacements for Changes in the Choice of Vy InfigureIII-26a,theoriginalchoiceofVyisshownwithitscorrespondingTargetDisplacement.Thisisclearlythebestchoice in Vy as the approximate bilinear post yield line is approximately tangent to 0204060801001201401600 5 10 15 20 25Roof Displacement (in)Base Shear (kips) Vy = 152 kips t = 19.1 in0204060801001201401600 5 10 15 20 25Roof Displacement (in)Base Shear (kips)Vy = 140 kips t = 18.8 in0204060801001201401600 5 10 15 20 25Roof Displacement (in)Base Shear (kips)Vy = 120 kips t = 18.4 in0204060801001201401600 5 10 15 20 25Roof Displacement (in)Base Shear (kips)Vy = 100 kips t = 18.0 in 54 the actual base shear vs. roof displacement curve.However, the graphs III-26 b-d show other choices that couldbepossible.FigureIII-26bshowsanintermediatevalueof the Vy, and the corresponding change in TargetDisplacementisapproximately2%.FigureIII-26cshowstheVyoccurringataverylowvalue whichisclearlynottheyieldvalueofthestructure.TheresultingchangeintheTargetDisplacementis approximately4%.FigureIII-26dshowstheyieldvalueequaltotheultimatevalueattainedbythe structure,meaningthestructurehaszeropostyieldstiffness.Thisisalsoclearlynottheyieldvalue, however the resulting change in the Target Displacement is only about 1.6%.So, the figure shows that for even bad choices of Vy, such as those shown in figures III-26 c and d especially, the resulting change in the Target Displacement is less than 5%.So, the Target Displacement is not greatly affected by changes of the choiceintheyieldlevelonthebaseshearvs.roofdisplacementplot,evenwhenthatchoiceisnotvery reasonable. III-F)Conclusions on the Limitations and Accuracy of the Pushover Analysis FromthecomparisonofthethreereinforcedconcreteframesanalyzedwithboththePushover Analysisandthecompletedynamicanalysis,itisseenthatthePushoverAnalysisisaccurateforshorter, stifferstructureswhosehighermodesdonotcontributesignificantlytotheoveralldynamicresponse.Further,itisseenthatthePushoverAnalysisincorporatesthemaximumexpecteddisplacementthatmay occurfromarangeofground motions with a given percent exceedance in 50 years by assuming the worst casescenerio,whichispurelyfirstmoderesponse.However,asthehighermodeeffectsbecomemore significant,thePushoverAnalysisoverestimatesthemaximumdisplacementexpectedduringthedesign event. Ithasalsobeenshownthat,eventhoughtheTargetDisplacementisafunctionofthesubjective valueofthestructuralyieldlevelonthebaseshearvs.roofdisplacementplot,theresultsarenotgreatly effected by even illogical choices in the value of Vy. 55 CHAPTER IV FORMULATION OF ELEMENT SHEAR RESPONSE ChapterIIIdescribestheapplicationoftheNSPanalysistomomentresistingframeswhose responseisgovernedprimarilybyflexuraldeformation.However,seldom,ifeverinreinforcedconcrete structures,isthelateralforceresistingsystemcomposedentirelyofmomentresistingframes.More commonlyitiscomposedofstructuralwalls,oracombinationofstructuralwallsandmomentresisting frames.Since the primary goal in this research is to supply a Pushover Analysis tool which may easily be appliedinadesignoffice,theprogramdevelopedheremusthavetheabilitytoanalyzestructuralwalls.SomecommonconfigurationsofstructuralwallsusedtoresistlateralforcesareillustratedinfigureIV-1.Theyarecoupledwalls,wallswithopenings,wallswithvaryingdimensionsorthicknesses,elevatorand stair cores, or any combination of these wall systems. Figure IV-1:Common Configurations of Structural Walls EventhoughthelimitationsofthePushoverAnalysiswereoutlinedinChapterIII,itlendsitself welltostructuralwallsbecausetheyareverystiffandhaveveryshortperiods,thustheirresponseis typicallydeterminedbytheirfundamentalperiodofvibration.Thisbenefitisoffsetbythefactthat structuralwalls,oftencalledshearwalls,havealowspantodepthratio,thussheardeformationsmustbe 56 accountedfor.Tocapturethisbehavior,theexistingforcebasedfiberbeamelementpresentedin Spacone et al [15], must be modified to include shear deformations. This chapter will extend the original formulationtoincludetheshearresponseofReinforcedConcreteelements.First,theTimoshenkobeam theory is reviewed, then the changes in the total element flexibility are illustrated, and the section flexibility ismodifiedtoincludesheardeformationsbydevelopingacyclicshearhystereticlawtorelateshear deformationstoshearforceateachsectionintheelement.Alsoincludedarecommentsonpossible enhancements to the proposed formulation. IV-A)Review of Timoshenko Beam Theory InEulerBernoullibeamtheory,sheardeformationsareneglected,andplanesectionsremain planeandnormaltothelongitudinalaxis.IntheTimoshenkobeamtheory,planesectionsstillremain planebutarenolongernormaltothelongitudinalaxis.Thedifferencebetweenthenormaltothe longitudinalaxisandtheplanesectionrotationisthesheardeformation.Theserelationsareshownin figure IV-2. Figure IV-2:Bernoulli and Timoshenko Beam Deformations ItcanbeseeninfigureIV-2thatintheEuler-Bernoullibeamthedeformationatasection, dvo/dx,isjusttherotationduetobendingonly,sincetheplanesectionremainsnormaltothelongitudinal axis.However,intheTimoshenkobeamthesectiondeformationisthesumoftwocontributions:oneis 57 duetobending,dvb/dx,andtheotheristhesheardeformation,dvs/dx.Byconsideringaninfinitetesimal lengthofthebeam,asshowninfigureIV-3,itisseenthatthesheardeformationinTimoshenkobeam theory, dvs/dx, is the same as the shear strain related to pure shear, . Figure IV-3: Infinitesimal Length of Beam Showing Bending and Shear Deformations For linear elastic materials, Hookes law for shear applies and: G = (IV.1) Whereisequalto the shear stress applied to the element and G is the shear modulus of elasticity for the material.In the Timoshenko beam theory, the shear stress is assumed constant over the cross section.The shear force, V, is related to the shear stress through: sA V = (IV.2) where As is equal to the shear area of the section. Combining these two equations: sGA V = (IV.3) Whilethisequationonlyappliestolinearlyelasticmaterials,itwillbethe basis for the formulation of the non-linear shear force - shear strain relation.In this study, it is assumed that V and are interrelated though the shear area of the section multiplied by a value which accounts for the non-linear response of Reinforced Concrete to shear force.The existing force based elements available in FEAP account for the deformation resulting from bending alone, however they do not include the shear deformations.The shear deformations will be added according to the following formulation. = = 58 IV-B)Nonlinear Force Based Timoshenko Beam Element To include shear deformations in the element, the element flexibility must be established. To express clearly the modifications to the existing Bernoulli formulation, both the Timoshenko and Bernoulli formulations will be carried out.The beam elements added by Spacone et al [15] have five degrees of freedom, without rigid body modes, in three dimensions at the element level.They are two bending moments, Q1 Q4, or rotations, q1 q4, at each end and one axial force, Q5, or displacement, q5 (figure IV-4). Figure IV-4:Three Dimensional Flexibility Based Element The capital X, Y, and Z in figure IV-4 are the global degrees of freedom while the lowercase x, y, andzarethelocaldegreesoffreedomfortheelement.Theelementisdividedintotransversesections alongitslengthateachoneoftheclassicGuassorGuassLobattointegrationpoints.Eachofthese transverse sections is further divided into longitudinal fibers.This configuration is shown in figure IV-5. Figure IV-5:Element Divided into Transverse Sections and Longitudinal Fibers x yzXZYQ3, q3Q1, q1Q2, q2Q4, q4Q5, q5 59 Toobtaintheflexuralandaxialresponsealongtheelement,thefibersareanalyzedandthe stresses,,andmoduli,E,aresummedtocomputethesectionresponse.Thesectionresponses(mainly forces and flexibility) are then summed according to weight factors depending on the location of the section andtheintegrationschemeusedtocomputethetotalaxialandflexuralresponsealongtheelement.The fibersectionmodelyieldsinteractionbetweenflexuralandaxialresponses.Theshearresponseforeach elementiscalculatedatthesectionlevel.Thisresponseisthensummedoverallthesections,again according to the weight factors depending on the location of the section and the integration scheme used, to get the total element response due to shear.In the proposed approach shear response is independent of the axialandflexuralresponsesatthesectionlevel.However,sinceforceequilibriumissatisfiedpointwise alongtheelement,interactionamongtheaxialflexuralandshearresponsesisenforcedattheelement level. This will be clarified in the following formulation. Therearethreemajorstepsintheforcebasedformulation.Inthefirststep,equilibrium,the force fields are expressed as functions of the nodal forces: Q x b x D ) ( ) ( = (IV.4) Where D(x) are the section forces, b(x) contain the element force shape functions, and Q contain the nodal forces as illustrated in figure IV-4. The second step is to write the section constitutive law in which the section deformations, d(x), are related to the section forces, D(x), through the section flexibility matrix, f(x): ) ( ) ( ) ( x D x f x d = (IV.5) Thethirdstepistosatisfycompatibility(inanaveragesense).Startingfromacompatiblestateof deformations,theelementrelationbetween forces, Q, and corresponding deformations, q, is obtained with the application of the principle of virtual forces: =LT Tdx x d x D q Q0) ( ) ( (IV.6) Putting together IV.4, IV.5 and IV.6, along with noting the arbitrariness of Q, one obtains: 60 Q dx x b x f x b qLT=0) ( ) ( ) ( (IV.7) or: FQ q =(IV.8a) where: =LTdx x b x f x b F0) ( ) ( ) ( (IV.8b) is the element flexibility. TheaboveequationsapplytotheforcebasedformulationofbothBernoulliandTimoshenko beams.However there are differences in the individual components which enter into the equations.These differenceswillbeillustratedbelow.AsshowninfigureIV-4,thethreedimensionalelementhasfive degrees of freedom, two moments at each end and an axial load, irrespective of whether shear deformations areincludedornot.However,onthesectionlevel,thenumberofnonzerodeformationsdependon whetherornotsheardeformationsareincluded.Thesectiondegreesoffreedomareillustratedinfigure IV-6. Figure IV-6:Three Dimensional Section Degrees of Freedom Bernoulli and Timoshenko Beams , ,, ,, ,,, 61 AscanbeseenfromfigureIV-6,theBernoullibeamsectionhasthreenonzerodeformations whiletheTimoshenkobeamsectionhasfivedeformations.Becausetheshearforcesarerequiredinthe sectionformulation,theequilibriumstatementmustincludetheseforcesandthereforetheshapefunctions fortheTimoshenkobeammustincludetherelationbetweenshearforceappliedatasectiontothenodal forcesoftheelement.AddingthesetotheshapefunctionsincludedintheBernoullibeamformulation, whichassumesconstantaxialforceandlinearmomentalongtheelement,givestheconstantshearforce distributionsoftheTimoshenkobeam.Theconstitutiverelationmustalsobemodifiedbecausethe sections now include shear forces and deformations.The section flexibility, which is obtained by inverting thesectionstiffness,containstheshearflexibilityofthesectiondecoupledfromtheaxialandbending terms.Weightedintegrationoff(x),however,couplesshear,axialandbendingresponsesontheelement level.These relations are detailed thoroughly in figure IV-7. ReferringtofigureIV-7,itisinterestingtonotethatwhiletheconstitutiveequationforthe Bernoulli beam exhibits a full 3x3 section flexibility matrix, the same relation for the Timoshenko beam is a full 3x3 matrix for the flexural and axial responses but is merely diagonal for the shear response resulting in a 5x5 section flexibility matrix.This is due to the fact that, as mentioned earlier, the fiber responses are added over the section to obtain the response at each section.So, at the section level, the axial and flexural responsesarenaturallycoupled.Theshearresponseisnotcalculatedatthefiberlevel,butisdetermined for the entire section as an average response.However, the flexibility equation for both beam types is a full 5x5matrix.Thisisduetothefactthattheelementresponseisfoundbysummingtheresponsesofeach sectionweightedthroughtheshapefunctions,b(x),thatimposelinearmomentsandconstantshearin equilibriumwiththeendmoments.So,theformulationusedherefortheTimoshenkobeamincludes couplingofaxialandflexuralresponsesonthesectionlevel,butshearresponsesareindependentonthe sectionlevel.However,shearandaxialflexuralresponsesarecoupledattheelementlevel.Inother words, because the shear forces are related to the end moments through equilibrium, the shear and bending forces are coupled through equilibrium. 62 Figure IV-7:Comparison of Components in Bernoulli and Timoshenko Element Formulations Element Level 5 dofs Section Level 3 dofs Element Level 5 dofs Section Level 5 dofs Bernoulli BeamTimoshenko Beam ''NMMx Dzy) ('') () () () (xxxx dyz'') () () () () () (x Vx Vx Nx Mx Mx Dzyyz'') () () () () () (xxxxxx dzyyz1) Equilibrium Q x b x D ) ( ) ( ]]]]]]]]

1 0 0 0 00 1 0 00 0 0 1) (LxLxLxLxx b]]]]]]]]]]]

01 10 00 0 01 11 0 0 0 00 1 0 00 0 0 1) (L LL LLxLxLxLxx b2) Constitutive Law ) ( ) ( ) ( x D x f x d ]]]]]

f e ce d bc b ax f ) (]]]]]]]]

hgf e ce d bc b ax f0 0 0 00 0 0 00 00 00 0) (f(x) is section flexibility from fiber section and Nonlinear V- relation.Axial and Bending Deformations Coupled Shear Deformations Uncoupled f(x) is section flexibility from fiber section. Axial and Bending Deformations Coupled 3) Compatibility FQ q [ ] 5 5x FF is full 5x5 Element Flexibility Matrix With Coupling Among Axial and Flexural Deformations F is full 5x5 Element Flexibility Matrix With Coupling Among Axial, Flexural and Shear Deformations [ ] 5 5x FNote: a through h are values yet undefined.They are included to represent non zero flexibility determined values 63 IV-C)Section V Constitutive Law Withtheelementforcebasedformulationcarriedoutintheprevioussection,allthenecessary componentsoftheelementformulationareknownexceptfortheVconstitutivelaw.Inthepresent formulation, a generalized, nonlinear V law is assumed, in the form: ( ) g V =(IV.9) where g indicates the nonlinear function. If the material was still in the linearly elastic range, then: sGA g = ) ( (IV.10) IV-C-1)Shape of Shear Hysteretic Law The shear hysteretic law implemented here will be a modified version of the bilinear law proposed byFilippouetal[7]. This law, shown in figure IV-8, is clearly not readily applicable to this formulation becausethelawrelatessectionmomenttoshearrotationandtheflexibilityformulationdeveloped previouslyrequiresalawwhichrelatesshearforcetoshearstrain.So,thislawwillbemodifiedtobe applicable to the procedure developed previously and only the shape will be retained. Figure IV-8:Filippou et als Moment Shear Rotation Hysteretic Law Referring to figure IV-8, Mcr isthe cracking moment of the section, My is the yield moment of the section,Mmaxisthemaximummomentattainedonthepreviousloadingcycle,maxisthemaximum M, Moment,Shear Rotation M* M max max+ My+ Mcr- Mcr- My 64 previous rotation attained, and M* is a target moment reduced due to section damage.Damage occurs after the section has yielded.The degraded stiffness is due to the closi