MINIMUM SLAB THICKNESS OF RC SLAB TO PREVENT UNDESIRABLE FLOOR VIBRATION Submitted by Mohammad Rakibul Islam Khan Student No: 0704046 Submitted to the DEPARTMENT OF CIVIL ENGINEERING BANGLADESH UNIVERSITY OF ENGINEERING AND TECHNOLOGY In partial fulfillment of requirements for the degree ofBACHELOR OF SCIENCE IN CIVIL ENGINEERING February, 2013 ii DECLARATION This is hereby declared that the studies contained in this thesis is the result of research carried out by the author except for the contents where specific references have been made to the research of others. The whole thesis has been done under the supervision of Professor Dr. Khan Mahmud Amanat (Department of Civil Engineering, BUET) and no part of this thesis has been submittedtoanyUniversityoreducationalestablishmentforaDegree,Diplomaor other qualification (except for publication). Signature of author (Mohammad Rakibul Islam Khan) iii ACKNOWLEDGEMENT ThankstoAlmightyAllahforHisgraciousness,unlimitedkindnessandwiththe blessings of Whom the good deeds are fulfilled. TheauthorwishestoexpresshisdeepestgratitudetoDr.KhanMahmudAmanat, Professor,DepartmentofCivilEngineering,BUET,Dhaka,forhiscontinuous supervision all through the study. His systematic guidance, invaluable suggestions and affectionateencouragementateverystageofthisstudyhavehelpedtheauthor greatly. Averyspecialdebtofdeepgratitudeisofferedtotheauthorsparentsandhis younger sister for their continuous encouragement and cooperation during this study. iv ABSTRACT Aninvestigationbasedon3Dfiniteelementmodelingofareinforcedconcrete building, is carried out to study the natural floor vibration. The objective of this work istodeterminetheminimumslabthicknessofareinforcedconcretebuildingto preventundesirablevibrationwhichmaycausediscomforttooccupants.The developedFEmodelhasbeensuccessfullyappliedtosimulatepastexperiments whichvalidatedtheapplicabilityofthemodelforfurtherparametricstudy.The variation of the floor vibration is studied for different slab thickness, span length and floorpanelaspectratio.Formthestudyithasbeenfoundthat,thefloorfrequency increasewithincreasingslabthicknessanddecreasingpartitionwallloadand decreasing column height. The floor frequency decreases with increasing span length and increasing floor panel aspect ratio. Based on findings, two empirical equations are suggestedwhichprovideminimumslabthicknessofashortspanRCbuildingto preventundesirablefloorvibration,forbothconsideringandwithoutconsidering partition wall load. v TABLE OF CONTENTS Page No. DECLARATION ii ACKNOWLEDGEMENTiii ABSTRACTiv Chapter 1INTRODUCTION 1.1 GENERAL2 1.2 BACKGROUND2 1.3 OBJECTIVES AND SCOPES OF THE STUDY3 1.4 ASSUMPTION4 1.5 ORGANIZATION OF THE THESIS4 Chapter 2LITERATURE REVIEW 2.1 GENERAL 6 2.2 TERMINOLOGY6 2.3 FLOOR VIBRATION BACKGROUND INFORMATION10 2.3.1 Sources Of Building Vibration11 2.3.2 Path Of Transmission Of Vibration 12 2.3.3 Receiver12 2.4 CLASSIFICATION OF FLOOR VIBRATION12 2.5 HUMAN RESPONSE TO BUILDING VIBRATION13 2.6 FLOOR VIBRATION PRINCIPAL16 2.7 PAST RESEARCH ON FLOOR VIBRATION18 2.8 A PRACTICAL EXAMPLE OF RESONANCE19 2.9 2.9 REMARKS21 Chapter 3DEVELOPMENT OF FINITE ELEMENT MODEL 3.1 GENERAL23 3.2 SOFTWARE USED FOR FINITE ELEMENT ANALYSIS23 vi 3.3 TYPES OF ANALYSIS OF STRUCTURES 24 3.4 CHARACTERIZATION OF STRUCTURAL COMPONENT IN MODEL25 3.4.1 BEAM4 (3-D Elastic Beam) For Beam, Column25 3.4.2 SHELL43 For Slab27 3.4.3 MASS21 (Structural Mass) For Load Application29 3.4.4 Support Condition30 3.5 LOAD APPLICATION30 3.6 ANALYSIS METHOD31 3.7 FINITE ELEMENT MODEL MESH 32 3.8 MODEL CHARACTERISTIC FOR ANALYSIS ANDTYPICAL RESULT33 3.8.1 Different Parameters33 3.8.2 Different Mode Shapes36 3.9 VERIFICATION OF THE MODEL40 3.9.1 Verification Of The Building Total Reaction Load With Hand Calculation40 3.9.2 Verification Of Modeling Frequency With A Laboratory Test Frequency42 3.10 REMARKS48 Chapter 4PARAMETRIC STUDY AND RESULTS 4.1 GENERAL50 4.2 SELECTED PARAMETERS50 4.3 VARIATION OF FLOOR VIBRATION WITH CHANGE OF SPAN LENGTH52 4.3.1 Analysis Of Floor Vibration WithPartition Wall Load53 4.3.1.1 Discussion68 4.3.1.2 Minimum slab thickness determination69 4.3.2 Analysis Of Floor Vibration Without Partition Wall Load73 4.3.2.1 Discussion88 vii 4.3.2.2 Minimum slab thickness determination89 4.4 REMARKS92 Chapter 5CONCLUSION AND RECOMMENDATION 5.1 GENERAL94 5.2 FINDINGS OF THE INVESTIGATION94 5.3 PROPOSED EQUATIONS94 5.4 RECOMMENDATION FOR FURTHER STUDY95 REFERENCES97 APPENDIX 100 Chapter - 1 Introduction Introduction2 1.1 GENERAL Inrecenttimes,fromsomepracticalobservations,ithasbeenacknowledgedthat, building designers try to increase slenderness and lightness of the structural system. In some cases, building owners try to increase clear span length to increasethe amount of available space. Which consequently decrease the stiffness of the structural system. Oftenthedynamicserviceabilityofstructuralsystemisignored.Structuralvibration isoneoftheimportantelementsofdynamicserviceability.Althoughmostofthe timesdynamicserviceabilitydoesnotcreateanysafetyhazardstothebuilding structure or its occupants. Building structure designed without considering or accurately measuring its dynamic propertiessuchasvibrationcancreatediscomforttooccupantsofthebuilding.Itis best practice to solve the problem, which may create discomfort, before construction of the structure.It will consume more moneyand timeto retrofit a built structure, to make it serviceable to the users. Modernbuildingsarenowbecomingmoreflexible,slenderandirregularshape. Those are more susceptible to undesirable vibration. Most people are sensitive to the frequencyofvibrationintherangeof8Hzto10Hz;becausesomehumanorgans havethefrequencyrangesfrom4Hzto8Hz(Wilson,1998).Ifthestructureshave thefrequencybellow10Hz,thenitcreatesresonancewithhumanbody.This resonance may cause undesirable vibration to the people (Murray, 1997). 1.2 BACKGROUND Floorvibrationwhichisnotadesignproblem,butitisadynamicserviceability problem.Particularlyinthecaseofdynamicloading(humanjumping,earthquake, windetc),thismaycausemanyproblems.Buildingvibrationmaybecauseddueto traffic movement, ground vibration, wind etc. Building exposes with strong wind may causeexcessivedeformationtotheupperfloor.Inthedancingclub,movementsof peoplecreateundesirablefloorvibration.Vibrationmaycausesdamagetostructure, Introduction3 when it creates resonance of the structure. Many structures have been damaged due to resonance. Tacoma Narrows Bridge has been destroyed (November 7, 1940) because ofresonancecausedbywind.InLondonMillenniumfootoverbridgebecame unusable(10June,2000)becauseofresonance.Manybuildingstructuresespecially steelstructuresareverymuchsusceptibletovibration.RCbuildingfloorwithless slabthicknesswithrespecttospanlengthmaycreateuncomfortablevibration.The causeofvibrationmaybestiffnessofthestructure.Nocodesprovidespecific guidelines for slab thickness to prevent undesirable floor vibration. The easiest way to avoidthebuildingfloorvibrationistodesignthefloorwithphysicalunderstanding. Forthisreasonfiniteelementmethodisdeveloptopredictthedynamicpropertyof thefloors.Theobjectiveofthisstudyistopreventundesirablefloorvibrationby providing required slab thickness.

1.3 OBJECTIVES AND SCOPES OF THE STUDY Theobjectiveofthecurrentstudyistoenlargeknowledgeaboutthevibration response of a RC building floor with change of different parameters. The variation of floor vibration is analyzed with change of slab thickness, floor panel aspect ratio and span length. From this study author tries to suggest a minimum slab thickness which may prevent undesirable floor vibration. The principle features of this thesis are summarized as follows; To develop a 3D finite element model of three storied building. Verify the built model with hand calculation and experimental data. To perform modal analysis to determine mode shape and frequency. Analysisthevariationofnaturalfloorfrequencywithvariationofslab thickness. Analysisthenaturalfloorfrequencywithvariationofspanlengthandfloor panel aspect ratio. Build a relation among slab thickness, span length and aspect ratio. Determineminimumslabthicknesstopreventundesirablefloorvibrationfor bothcases,withpartitionwallload(2.4 10-3 N/mm2)andwithoutpartition wall load. This slab thickness is for short span (3m-6m). Introduction4 1.4 ASSUMPTION Theinvestigationisbasedonsomeassumptions,toavoidcomplexityincalculation. These assumptions are as follows; Material is linearly elastic and isotropic. No lateral load is considered in this study. Liveloadisonlyconsideredtodeterminebeamandcolumnsize,andisnot used in determining modal frequency. Floor finished load is assumed to be 1.2 10-3 N/mm2 1.5 ORGANIZATION OF THE THESIS Fiveessentialchaptersareorganizedsystematicallytodescribethewholethesis. Chapter 1 is the current chapter, which introduces the entire study which is performed in this thesis. Chapter 2 submits the provisions of different code specifications as well asalltherecentandpastpublicationsrelevanttothetopic.Inchapter3,the descriptionoftheactualworkregardingthefiniteelementmodelingrelatedtothe subjectisgiven.Inthischaptersystematicdescriptionsareprovidedtohavebetter understanding about 3D building modeling by suitable program package.Parametric study is done in Chapter 4. In this chapter, analysis and output of the result based on various parameters are illustrated. Finally, the summary of the organized outcomes of thethesisisprovidedandfutureresearchworksrecommendationsareproposedin Chapter 5. Flow Chart of the whole thesis; Conclusion And RecommendationParametric Study And ResultsDevelopment of Finite Element ModelLiterature ReviewIntroduction Chapter - 2 Literature Review Literature Review 6 2.1 GENERAL Building structures are more and more becoming modern in urban as well asin rural areas.Designersseemtocontinuouslymovethesafetyborder,inordertoincrease slendernessandlightnessoftheirstructuralsystem(Silva2009).Moderntechniques arealsousedtohighstrengthmaterialtocreateflexibleandlongspanfloor,large column free floors or long cantilever structures. Sometimes the owners need for long spancolumnsfreespaceswithfewnonstructuralelements.Suchfloorssometimes resultinannoyinglevelofvibrationunderordinaryloadingsituation.Whilethis vibration does not present any threat to the structural integrity of the floor, in extreme cases they can cause the building unusable to the occupants (Murray, 1991). Thebuildingdesignerhasnotabletoreliablyestimatethedynamicpropertyof structural system which will not susceptible to annoying vibration (Setareh, 2010). A wide variety of scales and prediction technology are available to the engineer is only tomeasureorestimatethecomplexnatureandbehaviorsoffloorvibration.Since eachmethodmakesnumerousassumptionsaboutthestructure,notallmethodsare equallyavailabletoallsituations.Remedyforannoyingfloorvibrationisalways cumbersome and expensive. It is far better to design a floor properly at the first time, rather than retrofit the structure once the problem arises. The easiest way to avoid the buildingfloorvibrationistodesignthefloorwithphysicalunderstanding.Forthis reason finite element method is develop to predict the dynamic property of the floors. 2.2 TERMINOLOGY Definition of terminology relevant to this study, Vibration:Oscillationofasystemaboutitsequilibriumpositioniscalledvibration. Vibrationcanbefreeorforced.Freevibrationoccurswhenthesystemisexcited andallowedto vibrateatanaturalfrequencyofthesystem. Forcedvibrationoccurs whenasystemiscontinuallyexcitedataparticularfrequencyandthesystemis forcedtooscillateatthat frequency. Literature Review 7 Amplitude: Magnitudeofthevibration. Dependingonwhatisbeingmeasured, the amplitude (figure 2.1) can be a displacement, a velocity, or acceleration. Period:Time for one cycle, usually measured in seconds. Damping:Lossofenergypercycleduringthevibrationofasystem,usuallydueto friction.Viscousdamping,dampingproportionaltovelocity,isgenerallyassumed. Dampingisanimportantparameterinmitigationexcessivevibrationinfloor structures. A precise value for the damping of reinforced concrete structure is mostly unknown.Theusedofpartitionwallinfinishedfloor,increasedampingofthe structure. Murray (1993) used damping of 3% for an office building without partition wall. Cycle: Mo t i o n ofthesystemfromthetimeitpassesthroughagivenpoint travellinginagivendirectionuntilitreturnstothatsamepointandinthesame direction. FundamentalFrequency:Thelowestfrequencyatwhichasystemwilltendto under- gofree vibration. Thisfrequencyisafunctionofthemassinasystemand Peri od

Amplitude Time Figure 2.1:Amplitude and PeriodR ES P O NS E Literature Review 8 the system stiffness. Floors that oscillate at frequencies in the range between 4 and 8 Hz are of particular concern (Sladki, 1999). CriticalDamping:Thedampingrequiredtopreventoscillationofthesystem. Dampingisusuallypresentedasaratioofactualdampingdividedbycritical damping.Logdecrementdampingisdeterminedbytakingthenaturallogarithmof theratioofsuccessivepeaksintheresponsecurve.Modaldampingisdetermined fromananalysisoftheFourierspectrumoftheresponseormodalanalysis.Modal dampingtendstobesmallerthanlogdecrementdampingandtendstomore accuratelymatchtheactual damping present in a structure.

Figure 2.2: Types of dynamic loadings (Murray et al., 1997) Harmonic Load Periodic Load Transient Load Impulsive Load Literature Review 9 DynamicLoading:Dynamicloadingcanbeclassifiedasharmonic,periodic, transient,andimpulse.Sinusoidalorharmonicloadsareusuallyassociatedwith rotatingmachinery.Periodicloadsarecausedbyhumanrhythmicactivitysuchas jumping,dancing.Transientloadscausedduetomovementofpeopleincluding walking, running (figure 2.2). Resonance:Ifthefrequencycomponentofanexistingforceisequaltothenatural frequencyofthestructure,resonancewilloccur.Atresonancetheamplitudeofthe motion tends to become larger. Mode Shape: When a floor vibrates freely in a particular mode, it moves up and down with a certain configuration or mode shape. Each natural frequency has a mode shape, associated with it (figure 2.3 and 2.4). First ModeSecond ModeThird Mode Figure 2.3: Mode Shape of a Building Figure 2.4: Tilting and Twisting Mode Shape of a Building Literature Review 10 Modal Analysis: Modal analysis refers to a computational, analytical or experimental method for determining the natural frequency and mode shape of a structure as well as response of individual modes, to a given excitation. 2.3 FLOOR VIBRATION BACKGROUND INFORMATION Inanygivensituationinvolvingexcessiveorannoyingvibrationtherearealways three factors involves. a)Source : Where the dynamic forces are generated, b)Path: How this energy is transmitted, c)Receiver: How much vibration can be tolerated. 2.3.1 Sources Of Building Vibration Buildingvibrationcanhaveseveralsources.Earthquake,explosioncancauselarge vibrationwithstructuraldamageandsometimestotalbuildingfailure.Slenderand irregularshapedbuildingcanalsobecomewindsensitiveandhavelargelateral movementwhensubjectedtohighwind.Occupantsofthebuildingcanfeelthis vibrationwhenthisvibrationisatlowfrequency(Setareh,2010).Anothersourceof buildingaswellasfloorvibrationisthebuildingoccupantsmovementssuchas walking, jogging, running, dancing. These vibrations are generally small in amplitude but large enough to cause problem such as annoying and discomfort to the occupants (Setareh, 2010). Excessive vibration of building due to normal human activities such as walking needs special attention to engineers and building owners as their occurrences have recently become more common. Accurateprediction,evaluationandassessmentofvibrationcangreatlyassist engineers and architectures to design cost-effective structure without such problems. Literature Review 11 Figure 2.5: Sources of Building Vibration 2.3.2 Path Of Transmission Of Vibration The path is how the vibration is transmitted to the receiver. In this case the path is the building structure from which the vibration is transmitted. 2.3.3 Receiver Thereceiverishewhoreceivesthevibration.Ifthevibrationsourceisthebuilding itself,thenthereceiverwillbeitsoccupants.Ontheotherhand,ifthesourceis human, then the receiver will be the building. 2.4 CLASSIFICATION OF FLOOR VIBRATION Murray(1975)classifiedthehumanperceptionoffloorvibrationinfourcategories, such as Vibration is not noticed by the occupants, Vibration is noticed but do not disturb the occupants, Vibration is noticed and disturb the occupants, Vibration can compromise the security of the occupants. Literature Review 12 Vibration of floors with frequencies below about 10 Hz is much more likely to cause discomfort of the occupants than in floor with frequencies above 10 Hz (Murray et al., 1997;Smithetal.,2007).Basedonthislimit,floorscanbeclassifiedaslow-frequency or high-frequency; Inlow-frequencyfloors,awalkingforceharmonicfrequencycanmatcha natural frequency, resulting in resonant response. Inhigh-frequencyfloors,awalkingforceharmonyfrequencydonotmatch with natural frequency, so no resonant response will occur. ISO classify human response to vibration into three categories. Limit beyond which comfort is reduced, Limit beyond which the working efficiency is impaired, Limit beyond which the safety is endangered. 2.5 HUMAN RESPONSE TO BUILDING VIBRATION Floor vibration induced by human rhythmic activitieslike walking, running, jumping consistonaverycomplexproblem.Thisisduetothefactthatthedynamical excitationcharacteristicsgeneratedduringtheseactivitiesaredirectlyrelatedtothe individualbodyadversitiesandtothespecificwayinwhicheachhumanbeing executesacertainrhythmictask.Alltheseaspectdonotcontributeforaneasy mathematical or physical characterization of this phenomenon. Humanbeingshavealwaysanalyzedthemostapparentdistinctionsofthevarious activitiestheyperformed.Howeverthefirstpioneerondeterminingthehuman induced frequency was Fisher (1895), a German mathematician. Inordertodeterminethedynamicbehavioroffloorstructuralsystemssubjectedto excitations from human activities, various studies have tried to evaluate the magnitude of these rhythmic loads. These loads type caused by human walking motion, produce atransientresponse.Silvaetal.(2009),foundasempiricalformulafordetermining human induced dynamic loads on floor; Literature Review 13

()=

0.5 0.5cos 2

,

(2.1) Where, F (t): dynamic loads, in N t: time, in s T: activity period, (s) Tc: activity contact period, (s) P: weight of the individual (N) Kp: impact coefficient CD: phase coefficient (Figure 2.6) Human response to vibration generated in buildings depends on several factors which can be divided into direct and indirect effects (Setareh, 2010); Directeffectsincludethefrequencies,magnitude,duration,directionof vibration,andintervalbetweenvibrationeventsorexposureofhumansto vibration. Indirect effects on the subjective response to vibration. Designcodesdonotcoveradequatelytodesignabuildingwithoutvibrationthatis caused by human activities such as walking, running. Figure 2.6: Determination of Phase Co-efficient (Faisca, 2003) Aerobic Activities Free Jumps Literature Review 14 Severalmorefactorsinfluencethelevelofperceptionandsensitivityofvibrationto people. Among them; a)Position of the human bodyb)Excitation sources characteristics c)Exposure time d)Floor system characteristic (stiffness, mass, damping) e)Types of activity Currentlytherearesomestandardsandguidesonhumanvibrationevaluationand assessment.ThemostwidelyusedstandardsarepublishedbyISO.Thecurrent editionsarefollowingISO2631-1(ISO1997),whichcontainsthegeneral requirements on the evaluation of human vibration on floor. ISO 2631-2 (ISO 2003), which include evaluation of the human exposure to vibrations of a building in 1-80 Hz range;andISO10137(ISO2007),whichcontainsbasesforserviceabilitydesignof building structures and walkways subjected to vibration. It also includes a multiplying factor to define an acceptable vibration level for walking. Figure 2.7: Positions of human body in floor Literature Review 15 Figure 2.8: Variation of the frequency weighting versus frequency (Setareh, 2010) TheothermajorstandardsrelatedtotheseproblemsarepublishedbytheBritish Standards Institution. BS 6472-1 (BSI 2008), provides guidance on the evaluations of human exposure to building vibrations in 1-80 Hz ranges. Allthosestandardsexplainthehumanbodyvibrationduetobuildingvibration.But there is no enough standard of building vibration due to human activities. ISO 2631-2 (ISO 2003), do not provide an acceptable limit for vibration (acceleration, frequency). Themainimpactofhumanbehavioronthefloororbuildingisthefrequencyinthe vertical(Wk)andhorizontal(Wd)direction.Thefollowinggraphwillexplainthe human walking frequency level.

Thefigure2.8explainsthathumanwalkingfrequencymostlyvariesfrom1.6Hzto 8.8 Hz (Setareh, 2010). So building floor modes with natural frequencies in excess of 10 Hz is not usually excited by people walking. 2.6 FLOOR VIBRATION PRINCIPAL Althoughhumanannoyancecriteriaforfloorvibrationhavebeenknownformany years,ithasonlyrecentlybecomepracticaltoapplysuchcriteriatodesignafloor. Thereasonforthisisthat,theproblemiscomplex.Theloadingiscomplexand Literature Review 16 responseiscomplicated,involvingalargenumberofmodesofvibration.This problem can simplify sufficiently by introducing practical design criteria. Mostfloorvibrationprobleminvolvesrepeatedforcecausedbymachineryorby humanactivitysuchasjumpinganddancing.Althoughwalkingcauselittlemore complicated as force change step by step. Natural frequency of a SDOF system can be calculated from the following formula, Natural frequency, =12

(2.2) Which describe that, with increase stiffness of the structure, the natural frequency of that structure will also increase. And frequency decrease with increasing mass of the structure. Experiencehasshownthathighermodesofvibrationneednottobeconsidered,as theydieoutquicklyanddonotcausediscomfort.Thedegreeofdiscomfortcanbe calculatedbyestimatingpeakaccelerationduetowalkingwiththosecorresponding value in the following figure 2.9 (Murray, 1993) Although such procedure could be used in performance test of a build floor system, it is not amenable to design calculation. There is not enough design code for designing slab thickness of reinforced concrete structure that will not cause excessive vibration totheoccupantsaswasgivenfortheslabtoresistdeflectionbyACI,whichisas follows; Minimum Slab Thickness, =

(0.8+

200000)36+9(2.3) Where, t= Slab thickness, in Ln= Clear span length, ft = Floor panel aspect ratio Fy= Steel yield strength, psi Literature Review 17 Figure 2.9: Human response criteria for floor vibration for human activities(Murray, 1993) The equation 2.3 is for resisting excessive deflection of a two way floor system which is from static serviceability and should be greater than 90 mm. Theobjectiveofthisstudyistodeterminetheminimumslabthicknessbydynamic analysisofthebuildingusingcomputerprogrammingsoftware.Thisstudyhasdone for both type of building with partition wall and without partition wall. 2.7 PAST RESEARCH ON FLOOR VIBRATION Murray et al., (2011) studied on floor vibration characteristic of long span composite slab system. They have done their experiment on a laboratory floor specimen and on afullscalefloormockupandonlaboratoryfootbridge.Inlaboratoryfloor,they used 9.14m 9.14m single bay, single storied floor in experimental setup. They used 222mmcompositeslab of24MPanormalweightconcrete.Thefloorwassupported only in the perimeter with W53066 girder and W36032.9 beam which framed in to Literature Review 18 W31060 column. Then the natural floor vibration is calculated, and found 4.98 Hz. Thelaboratoryfloortestindicatesthatlongspancompositedeckhasverygood resistance to floor vibration due to walking. The floor had natural frequencies that are intherangeofthosemeasuredforcompositeslabandcompositebeamfloorsystem and far above the 3 Hz limit to avoid vandal jumping. PetrovicandPavic(2011),studiedoneffectofnonstructuralpartitionwallon vibration performance of floor structure. One of the most promising research topics in vibrationserviceabilityhasbeenconcernedwithquantifyingtheeffectsofdifferent non-structuralelementsondynamicbehaviouroffloorsystems.Ignoringnon-load bearingcomponents isgenerallyconservativeandtodaythereisaneedfordynamic floor analysis whichwill enablemore realistic and accurate assessment of the modal parameters.Hehaschosenahistoricalapproachasthemostsuitableoneduetothe facttherewasntenoughsystematicresearchinthisfieldinthepast.Therefore, seriousresearchandextensivesimulationandverificationstudiesareneededinthis area. Silva and Thambiratnam (2011), studied about the vibration characteristic of concrete steel composite multi panel floorstructure. Their result showed that the potential for anadversedynamicresponsefromhigherandmultimodalexcitationinfluencedby humaninducedpatternloads.Intheirresearch,theyhavefoundthat,underpattern loading,thesecondandthirdmodesofthestructurecanbeexcitedbyhigher harmonicsoftheactivityfrequency.Thesetypesofconcretesteelcompositemulti-panelfloorstructuresoftenexhibithigherandmultimodalvibrationunderpattern loads,hencethesimplifiedguidanceforvibrationmitigationinthepresentcodes. Under normal jumping activity, there was additional peaks frequency. Setareh (2010), studies the result of the model testing conducted on an office building floorandanalysisofthecollectedvibrationmeasurements.Itcomparestheresults with the structural response using computer analysis. It also studies a sensitivity study toassesstheimportanceofvariousstructuralparametersonfloordynamicresponse. Fromtheresultitconcludedthat,forthestructureusedinthisstudytheraised flooring and non structural element act mainly as added mass and did not contribute to Literature Review 19 floor damping. The addition of non structural element such as partition wall increases the stiffness of the building. Guilhermeetal.(2009),studiedaboutvibrationanalysisoflongspanjoistfloors submittedtohumanrhythmicactivities.Intheirworktheyinvestigatedonthe dynamicbehaviorofcompositefloorswhensubjectedtotherhythmicactivities correspondingtoaerobicsanddancingeffects.Thedynamicloadswereobtained through experimental tests conducted with individuals carrying out rhythmic and non-rhythmicactivitiessuchasstimulatedornonstimulatedjumping.Thisstudyalso present that, dynamic loads can even generated considerable perturbation on adjacent areas, where there is no human rhythmic activities of such kind. WilliamsandWaldron(2009),studiesonstructuralbehaviorofconcretesteel composite floors, subjected to human induced loads. Some of these research findings areusedindevelopingthepracticeguideonsteelstructurevibration.Theyalso provide some simplified formula to be used in to limit the steel structural vibration. Murrayetal.(2003),studiedonfloorvibrationduetohumanactivity.Intheirwork they provide basic principles and simple analytical tools to evaluate steel framed floor systemsandfootbridgesforvibrationsusceptibilitytohumanactivity.Bothhuman comfortandtheneedtocontrolmovementforsensitiveequipmentsareconsidered. Then developed a remedial measures for problematic floor. 2.8 A PRACTICAL EXAMPLE OF RESONANCE The TacomaNarrowsBridge isapairof twin suspensionbridges thatspanthe TacomaNarrows straitof PugetSound in PierceCounty, Washington.Thebridges connectsthecityof Tacoma withthe KitsapPeninsula andcarry StateRoute 16(known as Primary State Highway 14 until 1964) over the strait. TheoriginalTacomaNarrowsBridgeopenedonJuly1,1940.Itreceivedits nickname"GallopingGertie" because of the vertical movement of the deck observed byconstructionworkersduringwindyconditions.Thebridgebecameknownforits pitchingdeck,andcollapsedintoriverofNovember7,1940,underhighwind Literature Review 20 conditions(Figure2.10).ItsmainspancollapsedintotheTacomaNarrowsfour monthslateronNovember7,1940,at11:00AM(PacificTime)duetoaphysical phenomenonknownas resonance.Thebridgecollapsehadlastingeffectsonscience andengineering.Inmanyundergraduatephysicstextstheeventispresentedasan example of elementary forced resonance with the wind providing an external periodic frequency that matched the natural structural frequency. This phenomenon is called resonance. For instance, a company of soldiers can bring a bridgeinadestructivevibrationifthesoldierscrossthebridge,marchinginphase witheachotherandthefrequencywithwhichtheyputdowntheirfeetfitsaneigen frequency of the bridge. As a troop of soldiers walk across a bridge, their movement causes the bridge to vibrate. If the soldiers are in step and if the frequency of their step isthesameasthenaturalvibrationofthebridge,thebridgecouldcollapse.Thisis because each step that the troop makes in unison will cause the amplitude (size) of the bridge'svibrationstobecomehigherandhigher.Iftheamplitudeofthevibrations becomeshighenough,thebridgecancollapse.Topreventthis,soldierspurposely march out of step when crossing a bridge. Figure 2.10: Collapse Of Tacoma Narrows Bridge Literature Review 21 2.9 REMARKS Alotofresearchworkshasbeendoneonhumanactivityoncompositestructural vibration.ThereisnotsuchworkonRCbuilding,toestimateitsvibrationbehavior. Computermodelingandanalyticalrepresentationofbuildingstructuralpropertiesto predict the floor response subjected to excitation due to human activity are important issues that require further study. Vibration analysis of building structure can assist in more accurateestimation ofstructural dynamic properties.Estimates of the stiffness, mass and damping of building systems are needed to predict their dynamic responses. Itisdifficultinmanyinstancestodeterminethestiffnessofthestructuralsystem accurately,particularlyinconcretebuildings,andwherethecontributionofnon-structural elements is required. Chapter - 3 Development of Finite Element Model Development of Finite Element Model23 3.1 GENERAL ThedevelopmentofFiniteElementmethodasananalysistoolwasessentially initiatedwiththeadvancementofdigitalcomputer,whichreplacingtheanalytical methods,especiallywhenitisrequiredtodoaspecifictask.UsingFiniteelement methods,itispossibletoestablishandsolvethegoverningequationwithcomplex form in an easy and effective way. Finite element methods can be used to accurately predict the dynamic properties of reinforced concrete structure. Thischapterdescribesthefiniteelementmodelingofareinforcedbuildingwith differentappropriateedge,corner,interiorcolumnandbeamdimension. Representation of various physical elements with the FEM (Finite Element Modeling) elements, properties assigned to them, boundary condition, material behavior, analysis typehavealsobeendiscussed.Thevariousmaterialbehaviorusedanddetailsof finiteelementmeshingarealsodiscussedindetail.Averificationofthismodelboth for dynamic and static have been also done with a previous work. 3.2 SOFTWARE USED FOR FINITE ELEMENT ANALYSIS A number of goodfinite element analysis tools or packages are readilyavailable in thecivilengineeringfield.Theyvaryindegreeofcomplexity,usabilityand versatility.SuchpackagesareABAQUS,DIANA,PATRAN,SDRC/I-DEAS, ANSYS,EASE,COSMOC,NASTRAN,ALGOR,ANSR,MARC,ETABS, STRAND,ADINA,SAP,FEMSKI,SAFEandSTAADetc.Someofthese programs are intended for special type of structures. Of these the packages ANSYS 11.0isusedinthisstudyforitsrelativeeaseofuse,detaileddocumentation, flexibility andvastness of its capabilities. ANSYS 11.0 is one of the most powerful and versatile packages available for finite element structural analysis. ANSYS Finite element analysis software enables engineer to perform the following tasks Development of Finite Element Model24 BuildcomputermodelsorCADmodelsofstructures,products,components and systems. Apply operating loads and other design performance conditions. Studythephysicalresponses,suchasstresslevels,temperaturedistributions, or the impact of electromagnetic fields. Optimizeadesignearlyinthedevelopmentprocesstoreduceproduction costs. Do prototype testing in environments where it otherwise would be undesirable or impossible (for e.g., biomedical applications). It is a finite element modeling package for numerically solving a wide variety of structural and mechanical problems. This problems include static and dynamic analysis of the structure (both linear and non linear) TheANSYSprogramhasacomprehensiveGraphicalUserInterface(GUI)that gives user easy, interactive access to program functions, commands, documentation and reference materials. Hence the ANSYS program is user friendly. 3.3 TYPES OF ANALYSIS OF STRUCTURES Structure can be analyzed for small deflection and elastic material properties (Linear analysis),smalldeflectionandplasticmaterialproperties(materialnonlinearity), largedeflectionandelasticmaterialproperties(geometricnonlinearity),andfor simultaneous large deflection and plastic material properties. Byplasticmaterialproperties,thestructureisdeformedbeyondyieldofthe material,andthestructuralwillnotreturntoitsoriginalshape,whentheapplied loadsareremoved.Theamountofpermanentdeformationmaybeslightand inconsequential, or substantial and disastrous. Bylargedeflectiontheshapeofthestructurehasbeenchangedenoughthat,the relationshipbetweenappliedloadanddeflectionisnolongerasimplestraightline relationship.Thismeandoublingtheloadwillnotdoublethedeflection,the material properties can however still be elastic.Development of Finite Element Model25 ForthecurrentstudyitisassumedthattheRCframedislinearlyelastic,and materialsarehomogenousandisalwayssteelreinforcedinreality.Accordingto ACIrecommendation,theanalysisresultforRCframeisaccurateenoughforthis assumption. 3.4 CHARACTERIZATION OF STRUCTURAL COMPONENT IN MODEL For modeling this building slab, beam, column separate elements are used. For slab modeling SHELL43 and for beam, column BEAM4 (3-D elastic beam) is used. 3.4.1 BEAM4 (3-D Elastic Beam) For Beam, Column Element Description For modeling beams and columns used element BEAM4, is an elastic, uniaxial, 3-dimensionalelementwhichcanwithstandtension,compressions,torsionand bending.Theelementhastwonodeswithsixdegreesoffreedomateachnode; translations in the nodalx, y, and z axes and rotations about the nodalx, y, z axes. Thegeometry,nodelocations,andthecoordinatesystemforthiselementare shown in Fig. 3.1 The element is defined by two or three nodes, the cross-sectional area,twoareamomentsofinertia(IzzandIyy),anangleoforientation(or) about the element x-axis, and the material properties. Input Summary Element Type : BEAM4 Nodes: I, J, K (K, orientation node is optional) Degrees of Freedom: UX, UY, UZ, ROTX, ROTY, ROTZ Real Constant : AREA, IZZ, IYY, HEIGHT, WIDTH, IXX Material properties: EX, PRXY, DENS Output Data The solution output associated with the element is in twoforms: Nodal displacement include in the overall nodal solution, Additional element output Development of Finite Element Model26 Figure 3.1: BEAM4 Geometry Assumptions and Restrictions Thebeammustnothaveazerolengthorarea.Themomentsofinertia, however, may be zero if large deflections are not used. The beam can have any cross-sectional shape for which the moments of inertia can be computed. The stresses, however, will be determined as if the distance between the neutral axis and the extreme fiber is one-half of the corresponding thickness. Theelementthicknessesareusedonlyinthebendingandthermalstress calculations. The applied thermal gradients are assumed to be linear across the thickness in both directions and along the length of the element. If use the consistent tangent stiffness matrix take care to userealistic (that is, "toscale")elementrealconstants.Thisprecautionisnecessarybecausethe consistentstress-stiffeningmatrixisbasedonthecalculatedstressesinthe Development of Finite Element Model27 element.Ifuseartificiallylargeorsmallcross-sectionalproperties,the calculated stresses will become inaccurate, and the stress-stiffening matrix will suffer corresponding inaccuracies. (Certain components of the stress-stiffening matrixcouldevenovershoottoinfinity.)Similardifficultiescouldariseif unrealisticrealconstantsareusedinalinearprestressedorlinearbuckling analysis. Eigenvaluescalculatedinagyroscopicmodalanalysiscanbeverysensitive to changes in the initial shift value, leading to potential error in either the real or imaginary (or both) parts of the Eigen values. 3.4.2 SHELL43 For Slab Element Description SHELL43iswellsuitedtomodellinear,warped,moderately-thickshellstructures. Theelementhassixdegreesoffreedomateachnode:translationsinthenodalx,y, andzdirectionsandrotationsaboutthenodalx,y,andzaxes.Thedeformation shapesarelinearinbothin-planedirections.Fortheout-of-planemotion,itusesa mixed interpolation of tensorial components. The element has plasticity, creep, stress stiffening, large deflection, and large strain capabilities. Input Summary Element Type : SHELL43 Nodes: I, J, K, L Degrees of Freedom: UX, UY, UZ, ROTX, ROTY, ROTZ Real Constant : TK(I), TK(J), TK(K), TK(L) Material properties: EX, PRXY, DENS Output Data The solution output associated with the element is in two forms:Nodal displacements included in the overall nodal solution, Additional element output. Development of Finite Element Model28 Figure 3.2: SHELL43 Geometry Assumptions and Restrictions Zeroareaelementsarenotallowed.Thisoccursmostoftenwheneverthe elements are not numbered properly. Zero thickness elements or elements tapering down to a zerothickness at any corner are not allowed. Underbendingloads,taperedelementsproduceinferiorstressresultsand refined meshes may be required. Useofthiselementintriangularformproducesresultsofinferiorquality comparedtothequadrilateralform.However,underthermalloads,whenthe elementisdoublycurved(warped),triangularSHELL43elementsproduce more accurate stress results than do quadrilateral shaped elements. QuadrilateralSHELL43elementsmayproduceinaccuratestressesunder thermal loads for doubly curved or warped domains. Theappliedtransversethermalgradientisassumedtovarylinearlythrough the thickness. Theout-of-plane(normal)stressforthiselementvarieslinearlythroughthe thickness. Thetransverseshearstresses(SYZandSXZ)areassumedtobeconstant through the thickness. Development of Finite Element Model29 Shear deflections are included.Elasticrectangularelementswithoutmembraneloadsgiveconstantcurvature results, i.e., nodal stresses are the same as the centroidal stresses.ForlinearlyvaryingresultsuseSHELL63(nosheardeflection)orSHELL93 (with midside nodes).Triangular elements are not geometrically invariant and the element produces a constant curvature solution. Only the lumped mass matrix is available. 3.4.3 MASS21 (Structural Mass) For Load Application MASS21isapointelementhavinguptosixdegreesoffreedom:translationsinthe nodal x, y, and z directions and rotations about the nodal x, y, and z axes. A different mass and rotary inertia may be assigned to each coordinate direction. Figure 3.3: MASS21 Geometry Input Summary Element Type : MASS21 Nodes: I Degrees of Freedom: UX, UY, UZ, ROTX, ROTY, ROTZ Real Constant : MASSX, MASSY, MASSZ Material properties: DENS Output Data Nodaldisplacementsareincludedintheoveralldisplacementsolution.Thereisno elementsolutionoutputassociatedwiththeelementunlesselementreactionforces and/or energies are requested. Development of Finite Element Model30 Assumptions and Restrictions 2-D elements are assumed to be in a global Cartesian Z = constant plane. The mass element has no effect on the static analysis solution unless acceleration or rotation is present, or inertial relief is selectedThe standard mass summary printout is based on the average of MASSX, MASSY, and MASSZ. In an inertial relief analysis, the full matrix is used. All terms are used during the analysis. 3.4.4 Support Condition Atfoundationlevelallcolumnendsareconsideredtoactunderfixedsupport condition with all degrees of freedom of the support being restrained. 3.5 LOAD APPLICATION The x-z plane is acting as the horizontal plane in global co-ordinate system. To find the dynamic behavior of multistoried RC framed building, basic unfactored load case is considered as DL (self weight, partition wall, floor finish). In calculation of column size determination factored dead and live load is used. In determination of beam column size factored Dead load, Live load is used; as,

= 1.4 +1.7 3.1 Dead Load:Deadloadistheverticalloadduetotheweightofpermanentstructural and non structural components of a building. For the present study only self weight of beams,columnsandslabsandnonstructuralload(Partitionwall,floorfinish)are considered as dead load case of the structure. All vertical loads except self weight of beams, columns and slab are applied as mass onthestructure.Totalverticalloadappliedonthestructureis1.1962510-3N/mm2 (25psf)and2.39510-3N/mm2(50psf)forfloorfinishandpartitionwallload Development of Finite Element Model31 respectively. These two loads are applied as mass element of the structure and applied at the nodes. Live Load: Live load (LL) considered in the analysis is the load due to fixed service equipment etc. Total live load applied on the structure is 4.79 10-3 N/mm2 (100 psf). This live load will only used in determining the beam, column size. In modal analysis live load has not used. LateralLoad:Nolateralload(windload,earthquakeload)isconsideredinthis research. 3.6 ANALYSIS METHOD InthisstudyauthorprovideanextensivestudyonfloorvibrationofaRCbuilding withandwithoutpartitionwall(LargeHallRoom),throughANSYSmodeling software. In this study author made two reinforced building model with ANSYS 11. The model wasanalyzedwithmodalanalysistogetthedynamicbehaviorsuchasfrequencyof floorfordifferentspanlengthandfloorpanelaspectratio.Themodeltypeand degree of freedom depends on the complexity of the actualstructure and theresults desired in the analysis modeling a building for a dynamic analysis is currently more anartthanascience.Theoverallobjectiveistoproduceamathematicalmodelthatwillrepresentthedynamiccharacteristicsofthestructureandproducerealisticconsistent with the inputparameters. The building should be modeled as a threedimensionalspaceframewithjointsandnodesselectedtorealisticallymodel the stiffness and inertia effects of the structure. Each joints or nodes should have six degreesoffreedom,threetranslationalandthreerotational.Thestructuralmass should be with a minimum of three translational inertia terms. Points to Remember for modeling: OnlylinearbehaviorisvalidinModalanalysis.Nonlinearelements,ifany, are treated as linear.Development of Finite Element Model32 BothYoungsmodulus(andstiffnessinsomeform)anddensity(ormassin someform)mustbedefined.Materialpropertiescanbelinear,isotropicor orthotropic,andconstantortemperature-dependent.Nonlinearproperties,if any, are ignored. The modal solution natural frequencies and mode shapes is needed to calculate in this study. Type of modal analysis; Subspace Block LanczosPower Dynamics Reduced Unsymmetric Damped QR Damped But in all of the method the subspace, Block Lanczos, or reduced method are used to extractthemodes.Thenumberofmodesextractedshouldbeenoughtocharacterize the structure's response in the frequency range of interest. 3.7 FINITE ELEMENT MODEL MESH FEM is the outlines of the elements used to model the object of interest. To outline of the mesh should give the appropriate view of the object being modeled. Mesh size can varyin theanalysisofa single structure.Different mesharrangementsgenerallygive slightly varying solutions. In fact in real life problems mesh is constantly refined to get a consistent and representative solution. According to the literature of FEM the finer the mesh in an idealization, the smaller are theelementsandthebetterthesolution.Butthishaspracticalconstraintsbecausea veryfinemeshrequirestremendouscomputationaleffortwhichmayjustifiedas difficult to deliver even by the mainstream desktop computers. Another point is that, which element having higher order shape functions the degree of gain in accuracy diminishes after a certain level of fineness in mesh discretization. In Development of Finite Element Model33 factthereareafewelementswhichgiveexactsolutionevenforonlyoneelement discretizationsuchasbeamelements:Thus,itisnotalwaysnecessarytouseavery fine mesh at the expanse of huge computer memory and computational time. It is to be noted here that no mesh is usually the ultimate one, giving exact solution. A refinementofthemeshiswithinthescopeoffurtherstudiesandmaybeselectedon the basis of its approximation of the true result. 3.8 MODEL CHARACTERISTIC FOR ANALYSIS AND TYPICAL RESULT 3.8.1 Different Parameters Inthepresentstudytheobjectiveistodeterminetheminimumslabthicknessfor natural vibration of floor for 10 Hz limit (Murray, 1997). The variation of frequency withvariationofslabthicknessisstudied.Forthisstudyamulti-storiedbuilding having3floors,with3span 3bayhasbeenanalyzedwithANSYSpackage.This studyhasbeenanalyzed3m,4m,5m,6m,7mand8mspanlength,withfloorpanel aspectratioas1,1.2,1.4,1.6,and1.8.Theslabthicknessis takenasthefractionof shortspanlength.Thebeamandcolumnsizeiscalculatedfromtheloadthatis imposed on the building. Slab thickness Slab thickness is considered as, span/60, span/55, span/50, span/45, span/40, span/35, span/30, span/25, span/20, and span/15 for this study. Here span is the length of short span in mm. Column Columns are taken as square. Three type columns (figure 3.4) are used in this study;a)Interior column b)Edge column and c)Corner column Interior tied columns size area calculated from the following equation, Development of Finite Element Model34

= 0.65 0.8

0.85

+(

0.85

) (3.2) Where,

= 1.4 +1.7

(3.3) Ac = Contributory Area, mm2 Gross area of interior column,

is taken 25% greater for beam column self weight. Interior column size (integer) =

Beam width Figure 3.4: Column Position Figure 3.5: Beam Position Corner Column Edge Column Interior Column Edge Beam Interior Beam X Z Bay width Span length Development of Finite Element Model35 Edge Column size = 85% of Interior Column. Corner Column size = 75% of Interior Column. BeamTwo types of Beam (figure 3.5) are used in this study, as a)Edge beam b)Interior beam Beam depth = 3 slab thickness Beam width= (beam depth) 3 250 mm Detail Model View Figure 3.6: Plan View Of The Model Development of Finite Element Model36 Figure 3.7: Elevation Of The Model Figure 3.8: 3-D View Of The Model 3.8.2 Different Mode Shapes This studyisaboutbuildingvibration.Modalanalysisisused tocalculatedynamic behaviorofthebuilding(frequency).Inthecaseofmodalanalysisdifferentmode shapesforprobablevibrationpatternareencountered.Differentmodeshapeshave X Y Development of Finite Element Model37 differentfrequenciesofvibration.Someofthemodeshapeswiththererespect frequencysarefeaturedtogivesomeideasaboutthedifferentmodeshapesof vibrationindynamicanalysis.Themodeshapesareshownin3-Dviewandin elevation view.

3-D ViewElevation Figure 3.9:1st Mode Shape (Frequency 1.3 Hz) 3-D ViewElevation Figure 3.10:5th Mode Shape (Frequency 4.2 Hz) Development of Finite Element Model38 3-D ViewElevation Figure 3.11:7th Mode Shape (Frequency 5.5 Hz) 3-D View Elevation Figure 3.12:10th Mode Shape (Frequency 15.4 Hz) Development of Finite Element Model39 3-D ViewElevation Figure 3.13:15th Mode Shape (Frequency 21 Hz) 3-D View Elevation Figure 3.14:17th Mode Shape (Frequency 22.9 Hz) Development of Finite Element Model40 3-D ViewElevation Figure 3.15:25th Mode Shape (Frequency 27.7 Hz) 3.9 VERIFICATION OF THE MODEL This section is forvalidity of the model as if it gives the accurate result or not with hand calculation or experimental values in the laboratory. Two checks arediscussed here, one is the static load calculation with hand calculation and ANSYS calculation andanotheristhedynamiccalculation,shown asthefrequencyofastructurefrom modal analysis. 3.9.1 Verification Of The Building Total Reaction Load With Hand Calculation IntheverificationofthestaticanalysisresultgivenfromANSYSathreestoried buildingismodeledwith3spansand3bays.Thetotalreactionofthewhole building given the ANSYS is verified with hand calculation. The following data are used in the calculation. No of span= 3 No of bay= 3 Span length= 3000 mm Development of Finite Element Model41 Bay width = 3000 mm No of floor= 3 Floor height= 3000 mm Beam size= 450 mm 300 mm Column size= 300 mm 300 mm Slab thickness = 150 mm Floor finish= 1.19625 10-3 N/mm2 Partition wall= 2.395 10-3 N/mm2 Beam area= 135000 mm2 Column area= 90000 mm2 Slab self weight= Area Slab Thickness Density g No of floor = (3300033000) 150 2.410-9 9810 3 = 858179 N Load from partition wall and floor finish = Load No of floor Area = (1.1962510-3 + 2.39510-3) 3 (3300033000)= 872674 N Load from Beam= Area Beam length No of floor Density g = (450300) (433000+433000) 3 2.410-99810 = 686543 N Load from Column= Area No of columnNo of floor floor heightDensit g = (300300) 16330002.410-99810 = 305130 N Total load from hand calculation= 2722526 N Total load from ANSYS calculation = 2726298 N Almost meet the result from those two calculations with 0.13% error The ANSYS script has been enclosed in Appendix A. Development of Finite Element Model42 3.9.2 Verification Of Modeling Frequency With A Laboratory Test Frequency DynamicresultgivenfromANSYSisverifiedwithanexperimentalresult.M. Murray and T.Sanchez (2011), in theirresearch onFloor Vibration Characteristic OfLongSpanCompositeSlabSystemhasdoneanexperimentonalaboratory floorspecimen.Theymeasuredthefloornaturalvibrationwithavibratometer. Theyuseacompositefloorsystemwithsinglebayandspan.Theymeasuredthe vibrationofthefloorinthemidsection.Thedetailsexperimentalsetupisgiven bellow. Description Of The Laboratory Specimen The laboratory floor, Figure 3.16, was a single 9.14 m by 9.14 m (30 ft by 30 ft) bay constructedandtestedattheVirginiaTechThomasM.MurrayStructural Engineering Laboratory in 2006. The 222 mm (8-3/4 in.) composite slab consisted of a 117 mm (4-5/8 in.) steel deck covered with 105 mm (4-1/8 in.) of 24 MPa (nominal 3.5ksi)normalweightconcrete.Thefloorwassupportedonlyattheperimeter withW53066(W2144)girdersandW360 32.9(W1422)beamswhich framedintoW31060(W1240)steelstubcolumns.Temporaryshoringwas providedatthethirdpointsoftheslabtolimitthedeadloaddeflectionsduring the concrete placement. The measured frequency was 4.98 Hz through their experimental setup. Figure 3.16:3-D View of the Floor (Murray, 2011) Development of Finite Element Model43 Figure 3.17:Section View of the Floor (Murray, 2011) Modeling Procedure Of The Laboratory Specimen By ANSYS Intheexperimentthefloorsystemwascomposite.Butincurrentstudythefloor system is taken as uniform. In the modeling of this floor system the slab thickness is takenasuniform.Tomakethislaboratoryfloorequivalenttomodelfloor,some adjustments are used. Using transformed section method the steel part of the floor system is converted into concrete. The beam size is different in span and bay direction. To make the stiffness ofthemodeledfloorequivalentwiththelaboratoryfloorsystemtwomodulusof elasticity in two directions is used. The calculation of the new modulus of elasticity is given in table 3.1. In the calculation of determining new modulus of elasticity A1, A2, A3 are the area of transformed steel section. The figure 3.17 provides the detail information about the transformedsection.Theequationofdeterminingmomentofinertiaaboutcgcand about bottom is given in equation number 3.4 and 3.5 respectively. Moment Of Inertia about cgc =

312 (3.4) Moment Of Inertia about bottom =

33(3.5) Inthemodelingthesectionpropertiesandmaterialpropertiesisgivenintableno 3.1. After the modeling the modal analysis of the model is done.Development of Finite Element Model44 Table 3.1: Values and dimension of the parameters and structural components of laboratory floor ParameterValues/ Dimension Span length9140 mm Bay width9140 mmFloor height1000 mm No. of floor1 No of span1 No of bay1 Slab thickness 178 mm Live load1.916 10-3 N/mm2 Side Beam width178 mm Side Beam depth 178 mm Front Beam width203 mm Front Beam depth 500 mm Column 203 191 mm Gravitational acceleration9810 mm/sec2

Line mesh size 5 Beam element BEAM4 Slab elementSHELL43 Concrete properties Modulus of elasticity (X direction)20000 N/mm2 Modulus of elasticity (Z direction)5322 N/mm2 Poissons ratio0.15 Density2.4045 10-9 Ton/ mm3 Unit weight2.36 10-5 N/mm2 Development of Finite Element Model45 Figure 3.18: Transformed Slab Thickness Of The Model Figure 3.19:ANSYS modeling view of Laboratory SpecimenDevelopment of Finite Element Model46 Figure 3.20: ANSYS result of the laboratory Floor. Table 3.2: Determination Of Equivalent Modulus Of Elasticity Material Properties Steel poison ratio RHOs0.25 Concrete poison ratio RHOc 0.15 Concrete Modulus of Elasticity, Ec , N/mm2 20000 Steel Modulus of Elasticity, Es, N/mm2 200000 Modular Ratio, n (=Es/Ec)10 Main Section Dimension Width of the Deck , A (mm)305 Height of the Deck, B (mm)105 Width of the Girder , D (mm)100 Height of the Girder, C (mm)117 Over Hang Length, E (mm)103 Thickness of Steel Section, t1 (mm)2 Development of Finite Element Model47 Transformed Section Thickness of section(variable), T (mm)10 Length of Bottom Slab, h1 (mm)206 Height of the Girder, h2 (mm)234 Length of Bottom Girder, h3 (mm)200 Transformed Area Calculation Area of Bottom Slab, A1 (mm2)2060 Area of Perpendicular Girder, A2 (mm2)2340 Area of Bottom Girder, A3 (mm2)2000 Section Properties Calculation X-direction Area of the whole Section, (mm2)54525 Center Of Gravity from top flange, (mm) 96.03 Inertia about X-axis, Ix (mm4) 645989838.8 Average Inertia, (mm4) 143172029.3 Section Properties Calculation Z-direction Transformed Thickness(calculated), (mm) 20 Transformed Area (mm2)6100 Area of whole Section (mm2)38125 Inertia about Z-axis, Iz (mm4) 538020000 Equivalent Plate Thickness Calculation Plate Thickness in X-direction, t (mm)177.93 Calculation for 'D1=

312(12)' in X-direction 9604429451 Calculation for 'D2' in Z-direction2555826279 Transformed Modulus of Elasticity, E25322.18242 ThenaturalFrequencyofthemodelisgivenbyANSYSisat2ndmode.The frequency of the floor specimen is 4.72 Hz. This is close to theexperimental value; 4.98Hz(Murray,2011).TheANSYSscriptingofthislaboratoryspecimenis enclosed in Appendix B. Development of Finite Element Model48 3.10 REMARKS Thischapterprovidesextensiveinformationonhowtobuilda3-Dbuildingframe model,inANSYSandhowtouseandchangedifferentparametersofthemodel. Alsoprovidesinformationaboutthevalidationproceduresofthemodelwith previouswork.Modeshapesareenclosedtoprovideclearunderstandingabout building deformation. The built model is used to do the analysis of determining the minimum slab thickness for 10 Hz (Murray, 1997) vibration. This study is not from design consideration. It is for dynamic serviceability of a building structure. Chapter - 4 Parametric Study And Results Parametric Study And Results 50 4.1 GENERAL Inthischaptertheanalysisofdeterminingminimumslabthicknessfor10Hzlimit (Murray, 1997) floor vibration of a three storied building modeled in previous chapter willbediscussed.Theparametersaretakenbasedonpracticalvaluessothatthe actual building behavior under vibration will be same as the 3-D modeling frame. Current study is actually involved with large number of variables and parameters but onlythevariationoffloorvibrationwithrespectvariationofslabthicknessand variation of span length with different aspect ratio is studied here. Our focused is only confinedwithshortspanbuildingframe.Thematerialpropertiesareassumedtobe linearlyelastic.Noeffectofwindloadandearthquakeloadisstudiedhere;only natural vibration of floor is studied. 4.2 SELECTED PARAMETERS The study has performed for determining the minimum slab thickness which does not createundesirablefloorvibration.Thelimitingfloorvibrationfrequencyis10Hz given by Murray (1997). The parameter describe in table 4.1 used in the analysis. Theanalysisprocedureisfordeterminingthefrequencyforvariousslabthickness. Thisprocessisdoneforvariousspanlengthandfloorpanelaspectratio.Fromthe studycorrespondingslabthicknessfor10Hz(Murray,1997)frequencyisplotted against span length. From these studies the minimum slab thickness is determined. Fromequationnumber2.2,thenaturalfrequencyofsingledegreeoffreedomis dependent on stiffness and mass. The stiffness of beam and column can be calculated from the following equation. Stiffness, =123 (4.1) Where, E= Modulus of elasticity, N/mm2 Parametric Study And Results 51 Table 4.1: Values and dimension of the parameters and structural components ParameterValues/ Dimension Span length3, 4, 5.8 m Aspect ratio1, 1.2, 1.4, 1.6, 1.8 Bay widthas per aspect ratioFloor height3 m No. of story3 No of span3 No of bay3 Slab thickness as per requirement Floor finish load1.210-3 N/mm2 Partition wall load2.410-3 N/mm2 Live load4.7910-3 N/mm2 Beam widthas per requirement Beam depth 3 slab thickness Interior Columnas per requirement Corner Column75% of interior column Edge Column85% of interior column Gravitational acceleration9810 mm/sec2

Line mesh size 5 Beam ElementBEAM4 Slab ElementSHEEL43

Concrete properties Modulus of elasticity20000 N/mm2 Poissons ratio0.15 Density2.4045 10-9 Ton/ mm3 Unit weight2.36 10-5 N/mm2 Parametric Study And Results 52 I= Moment of inertia, mm4 h= Height of the column or length of beam, mm As per IS1893:2002 The approximate fundamental natural period of vibration (Ta ), in seconds, of a moment-resisting frame building without brick infillpanels may be estimated by the following empirical expression :

=0.0750.75 4.2 Natural Frequency, = 1

(4.3) Where,H= Total height of building, m 4.3VARIATIONOFFLOORVIBRATIONWITHCHANGEOFSPAN LENGTH The variation of floor vibration with respect to span length and aspect ratio is studied here.Thisstudyisdividedintwoparts.Oneisvariationoffloorvibrationwith partitionwallload(2.4 10-3N/mm2)andtheotheriswithoutpartitionwallload. Theanalysisforwithpartitionwallisdonefornormalresidentialbuildingwhere average number of population is expected to walk. On the other hand the analysis for without partition wall is done for theater hall room, where large numberof people is expected.Inbothcases,itistriedtofindouttheminimumslabthicknesswhich producevibrationmorethan10Hz.Asifthestructuralvibrationislargerthanthe vibrationproducebyhumanwalking.Murray(1997)hasprovedthatnatural frequencylessthan10Hzwillcausemuchdiscomforttotheoccupants.Because humanswalkingfrequencyisintherangesof2-10Hz(Setareh,2010).Ifthefloor naturalvibrationislessthan10Hz,itmaycreateresonance,whichcreatemuch vibration and cause discomfort for the occupants. 4.3.1 Analysis Of Floor Vibration With Partition Wall Load Theanalysisforvariationoffloorvibrationfordifferentspanlength,slabthickness andaspectratiowillbestudiedinthissectionofabuildingstructureincluding Parametric Study And Results 53 partition wall load. The modeled structure which has done in the previous chapter has shown that the beam and column size is depended on the self weight of the structure andthesuperimposedload(liveload).Withincreasingload(forincreaseofslab thickness)thebeamandcolumnsizeofthestructurealsoincreaseandthereby increasethemomentofinertia.Henceincreasethestiffnessofthestructure.From equation4.1,thestiffnessofthestructuralelementdecreasewithincreasinglength. Hence decrease the natural vibration frequency of the structure. Theselfweightandthepartitionwallloadareimposedonthemodeledstructureas mass element (discussed in section 3.4.3) which is a point element. And it is imposed on a node of the structure. The graphs on variation of floor frequency with respect to slab thickness for different span length and aspect ratio are as follows, Figure 4.1: Frequency Vs Slab thicknesses for 3000 mm span and aspect ratio 1 05101520253050 100 150 200 250Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1Span length=3000 mmBay width= 3000 mm 10 Hz Limit (Murray)ACI Serviceability LimitParametric Study And Results 54 Figure 4.2: Frequency Vs Slab thicknesses for 4000 mm span and aspect ratio 1 Figure 4.3: Frequency Vs Slab thicknesses for 5000 mm span and aspect ratio 1 05101520253050 100 150 200 250 300Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1Span length=4000 mmBay width= 4000mm10 Hz Limit (Murray)ACI Serviceability Limit05101520253050 100 150 200 250 300 350Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1Span length=5000 mmBay width= 5000 mm 10 Hz Limit (Murray)ACI Serviceability LimitParametric Study And Results 55 Figure 4.4: Frequency Vs Slab thicknesses for 6000 mm span and aspect ratio 1 Figure 4.5: Frequency Vs Slab thicknesses for 7000 mm span and aspect ratio 1 051015202530100 150 200 250 300 350 400 450Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1Span length=6000 mmBay width= 6000 mm10 Hz Limit (Murray)ACI Serviceability Limit051015202530100 150 200 250 300 350 400 450 500Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1Span length=7000 mmBay width= 7000 mm10 Hz Limit (Murray)ACI Serviceability LimitParametric Study And Results 56 Figure 4.6: Frequency Vs Slab thicknesses for 8000 mm span and aspect ratio 1 Figure 4.7: Frequency Vs Slab thicknesses for 3000 mm span and aspect ratio 1.2 051015202530100 200 300 400 500 600 700Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1Span length=8000 mmBay width= 8000 mm10 Hz Limit (Murray)ACI Serviceability Limit05101520253050 70 90 110 130 150 170 190 210Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 :1.2Span length=3000 mmBay width= 3600 mm 10 Hz Limit (Murray)ACI Serviceability LimitParametric Study And Results 57 Figure 4.8: Frequency Vs Slab thicknesses for 4000 mm span and aspect ratio 1.2 Figure 4.9: Frequency Vs Slab thicknesses for 5000 mm span and aspect ratio 1.2 05101520253050 100 150 200 250 300Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 :1.2Span length=4000 mmBay width= 4800 mm10 Hz Limit (Murray)ACI Serviceability Limit05101520253050 100 150 200 250 300 350Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 :1.2Span length=5000 mmBay width= 6000 mm10 Hz Limit (Murray)ACI Serviceability LimitParametric Study And Results 58 Figure 4.10: Frequency Vs Slab thicknesses for 6000 mm span and aspect ratio 1.2 Figure 4.11: Frequency Vs Slab thicknesses for 7000 mm span and aspect ratio 1.2 051015202530100 150 200 250 300 350 400 450Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1:1.2Span length=6000 mmBay width= 7200 mm10 Hz Limit (Murray)ACI Serviceability Limit051015202530100 200 300 400 500 600Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1:1.2Span length=7000 mmBay width= 8400 mm10 Hz Limit (Murray)ACI Serviceability LimitParametric Study And Results 59 Figure 4.12: Frequency Vs Slab thicknesses for 8000 mm span and aspect ratio 1.2 Figure 4.13: Frequency Vs Slab thicknesses for 3000 mm span and aspect ratio 1.4 051015202530100 200 300 400 500 600Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1:1.2Span length=8000 mmBay width= 9600 mm10 Hz Limit (Murray)ACI Serviceability Limit05101520253050 100 150 200 250Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1:1.4Span length=3000 mmBay width= 4200 mm10 Hz Limit (Murray)ACI Serviceability LimitParametric Study And Results 60 Figure 4.14: Frequency Vs Slab thicknesses for 4000 mm span and aspect ratio 1.4 Figure 4.15: Frequency Vs Slab thicknesses for 5000 mm span and aspect ratio 1.4 05101520253050 100 150 200 250 300Natural Frequency (Hz)Slab Thickness (mm)Span Length:Bay width= 1:1.4Span length=4000 mmBay width= 5600 mm10 Hz Limit (Murray)ACI Serviceability Limit05101520253050 100 150 200 250 300 350Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1:1.4Span length=5000 mmBay width= 7000 mm10 Hz Limit (Murray)ACI Serviceability LimitParametric Study And Results 61 Figure 4.16: Frequency Vs Slab thicknesses for 6000 mm span and aspect ratio 1.4 Figure 4.17: Frequency Vs Slab thicknesses for 7000 mm span and aspect ratio 1.4 05101520253050 150 250 350 450Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1:1.4Span length=6000 mmBay width= 8400 mm10 Hz Limit (Murray)ACI Serviceability Limit05101520253050 150 250 350 450 550Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1:1.4Span length=7000 mmBay width= 9800 mmACI Serviceability Limit10 Hz Limit (Murray)Parametric Study And Results 62 Figure 4.18: Frequency Vs Slab thicknesses for 8000 mm span and aspect ratio 1.4 Figure 4.19: Frequency Vs Slab thicknesses for 3000 mm span and aspect ratio 1.6 05101520253050 150 250 350 450 550 650 750Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1:1.4Span length=8000 mmBay width= 11200 mmACI Serviceability Limit10 Hz Limit (Murray)05101520253050 100 150 200 250Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1.6Span length=3000 mmBay width= 4800mm10 Hz Limit (Murray)ACI Serviceability LimitParametric Study And Results 63 Figure 4.20: Frequency Vs Slab thicknesses for 4000 mm span and aspect ratio 1.6 Figure 4.21: Frequency Vs Slab thicknesses for 5000 mm span and aspect ratio 1.6 05101520253050 100 150 200 250 300Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1.6Span length=4000 mmBay width= 6400mm10 Hz Limit (Murray)ACI Serviceability Limit05101520253050 100 150 200 250 300 350Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1.6Span length=5000 mmBay width= 8000mm10 Hz Limit (Murray)ACI Serviceability LimitParametric Study And Results 64 Figure 4.22: Frequency Vs Slab thicknesses for 6000 mm span and aspect ratio 1.6 Figure 4.23: Frequency Vs Slab thicknesses for 7000 mm span and aspect ratio 1.6 051015202530100 200 300 400 500 600Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1.6Span length=6000 mmBay width= 9600mm10 Hz Limit (Murray)ACI Serviceability Limit051015202530100 300 500 700 900 1100Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1.6Span length=7000 mmBay width= 11200mmACI Serviceability Limit10 Hz Limit (Murray)Parametric Study And Results 65 Figure 4.24: Frequency Vs Slab thicknesses for 8000 mm span and aspect ratio 1.6 Figure 4.25: Frequency Vs Slab thicknesses for 3000 mm span and aspect ratio 1.8 051015202530100 300 500 700 900 1100Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1.6Span length=8000 mmBay width= 12800mm10 Hz Limit (Murray)ACI Serviceability Limit05101520253050 70 90 110 130 150 170 190 210Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1.8Span length=3000 mmBay width= 5400mm10 Hz Limit (Murray)ACI Serviceability LimitParametric Study And Results 66 Figure 4.26: Frequency Vs Slab thicknesses for 4000 mm span and aspect ratio 1.8 Figure 4.27: Frequency Vs Slab thicknesses for 5000 mm span and aspect ratio 1.8 05101520253050 100 150 200 250 300Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1.8Span length=4000 mmBay width= 7200mm10 Hz Limit (Murray)ACI Serviceability Limit05101520253050 100 150 200 250 300 350 400 450Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1.8Span length=5000 mmBay width= 9000mm10 Hz Limit (Murray)ACI Serviceability LimitParametric Study And Results 67 Figure 4.28: Frequency Vs Slab thicknesses for 6000 mm span and aspect ratio 1.8 Figure 4.29: Frequency Vs Slab thicknesses for 7000 mm span and aspect ratio 1.8 051015202530100 200 300 400 500 600 700Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 :1.8Span length=6000 mmBay width= 10800 mm10 Hz Limit (Murray)ACI Serviceability Limit051015202530100 300 500 700 900 1100Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1.8Span length=7000 mmBay width= 12600mm10 Hz Limit (Murray)ACI Serviceability LimitParametric Study And Results 68 Figure 4.30: Frequency Vs Slab thicknesses for 8000 mm span and aspect ratio 1.8 4.3.1.1 Discussion From the above figure 4.1 figure 4.30, the change of frequency with change of slab thickness,spanlengthandaspectratioisshown.Herespanlengthistheshortest distancealongtheX-direction,andthebaywidthisthelargestdistancealongZ- direction. From the above figure the following points are observed, The natural frequency of floor is increasing with increasing of slab thickness. Becausewithincreasingslabthicknessthebeamandcolumnsizeisalso increased, due to self weight of the slab. Hence increase the moment of inertia ofstructuralelementsandconsequentlyincreasethenaturalfrequencyofthe whole structure. Withincreaseofspanlengththestiffnessofbeamdecrease,sothebuilding vibration is reduced.Withincreaseoffloorpanelaspectratiothenaturalfrequencyofflooris decreasing.Becauseincreasedaspectratioincreasethebaywidthofthe buildingstructure.Hencedecreasethestiffnessofthebuildingelement (equation 4.1). And decrease the floor frequency.051015202530100 300 500 700 900 1100Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1.8Span length=8000 mmBay width= 14400mm10 Hz Limit (Murray)ACI Serviceability LimitParametric Study And Results 69 Increase of slab thickness means the increase of weight of structure and which indicatetheincreaseofmass.Increaseofmassmeansdecreaseofnatural frequency of the structure. Increasingmassdecreasethefloorfrequencyandincreasebeamandcolumn size. Which increase the moment of inertia and increases the floor frequency. Forthesetwocontradictoryconditions,theinitialpartofthecurvemention above in the figure is steeper, and then it becomes flatter.Atlowaspectratio,thepointofchangingslopeisachievedearlier(atsmall slabthickness),butwithincreaseofaspectratiothispointgraduallyshifted towardthelargerslabthickness.Thisconditionisalsoseenwithincreaseof span length. This condition prevails, because initially the moment of inertia is high enough to ignore the effect of reducing frequency due to increasing mass. Thengraduallyreducedfrequencyduetoincreasemassbecomeprominent. And those both affect cause to flatter the curve. Duetoincludingthepartitionwallload,theobservedfrequencyislesserto someextentwithcomparedtowithoutpartitionwallload.Becauseincrease partition wall load increase the mass of the structure. Hence decrease the floor vibration frequency. 4.3.1.2 Minimum slab thickness determination Minimumslabthicknesswouldbesuchthicknesswhichproduces10Hzfrequency for a given set of aspect ratio and span length. In this current study our focused will be confined for short span floor (3m-6m) only. In determination of minimum slab thickness a graph is plotted with the value of slab thickness and span length corresponding to 10 Hz vibration in a slab thickness (mm) Vs span length (m) graph. This procedure is repeated for the aspect ratio of 1, 1.2, 1.4, 1.6, and 1.8. In the same graph paper minimum slab thickness corresponding to span length given by ACI (equation 2.3) is also presented for visualize the deviation of the thickness for 10 Hz limit (Murray, 1997) with the ACI provided thickness. Here ACI limitmeansACIServiceabilityLimit,and10Hzlimitmeans10Hzlimitgivenby Murray (1997). Parametric Study And Results 70 Figure 4.31: Comparison Of Slab Thickness For Aspect Ratio 1 Figure 4.32: Comparison Of Slab Thickness For Aspect Ratio 1.2 020040060080010000 2 4 6 8 10 12Slab Thickness(mm)Span Length (m)ACI Limit10 Hz LimitFormula020040060080010000 2 4 6 8 10 12Slab Thickness(mm)Span Length (m)ACI Limit10 Hz LimitFormulaParametric Study And Results 71 Figure 4.33: Comparison Of Slab Thickness For Aspect Ratio 1.4 Figure 4.34: Comparison Of Slab Thickness For Aspect Ratio 1.6 020040060080010000 2 4 6 8 10 12Slab Thickness(mm)Span Length (m)ACI Limit10 Hz LimitFormula020040060080010000 2 4 6 8 10 12Slab Thickness(mm)Span Length (m)ACI Limit10 Hz LimitFormulaParametric Study And Results 72 Figure 4.35: Comparison Of Slab Thickness For Aspect Ratio 1.8 Fromtheabovevariationoftheslabthicknesswithspanlength,apolynomial equation is formed to determine the tread line ofthe analysis value. Those tread line for different aspect ratio is analyzed to give a common equation which would give the slab thickness for around 10 Hz vibration as a function of span length and aspect ratio. Thefollowingequation(equationnumber4.4)forshortspanlengthisprovidedfor satisfying the above requirement, and the value given from the equation is also plotted in the above figure. To give as much as practical value from the equation no 4.4, floor vibration frequency around 10 Hz is used. = 3.5414+7535582+1795 2050 (4.4) Where,t= Minimum slab thickness for around 10 Hz floor vibration, mm x= Short span length, m = aspect ratio. 020040060080010000 2 4 6 8 10 12Slab Thickness(mm)Span Length (m)ACI Limit10 Hz LimitFormulaParametric Study And Results 73 4.3.2 Analysis Of Floor Vibration Without Partition Wall Load Theanalysisforvariationoffloorvibrationfordifferentspanlength,slabthickness andaspectratiowillbestudiedinthissectionofabuildingstructureexcluding partition wall load. The modeled structure which has done in the previous chapter has shown that the beam and column size is depended on the self weight of the structure andthesuperimposedload(liveload).Withincreasingload(forincreaseofslab thickness)thebeamandcolumnsizeofthestructurealsoincreaseandthereby increasethemomentofinertia.Henceincreasethestiffnessofthestructure.From equation4.1,thestiffnessofthestructuralelementdecreasewithincreasinglength. Hence decrease the natural vibration frequency of the structure. Thisanalysisisforthelargehallroom,theater,andauditoriumetc,wherealarge amount of people is expected to gather and where the clear span length is larger than the normal residential building. The graphs on variation of floor frequency with respect to slab thickness for different span length and aspect ratio are as follows for the floor without partition wall load. Figure 4.36: Frequency Vs Slab thicknesses for 3000 mm span and aspect ratio 1 05101520253050 100 150 200 250Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1 Span length=3000 mmBay width= 3000 10 Hz Limit (Murray)ACI Serviceability LimitParametric Study And Results 74 Figure 4.37: Frequency Vs Slab thicknesses for 4000 mm span and aspect ratio 1 Figure 4.38: Frequency Vs Slab thicknesses for 5000 mm span and aspect ratio 1 05101520253050 100 150 200 250 300Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1Span length=4000 mmBay width= 4000mm10 Hz Limit (Murray)ACI Serviceability Limit05101520253050 100 150 200 250 300 350Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1Span length=5000 mmBay width= 5000 mm 10 Hz Limit (Murray)ACI Serviceability LimitParametric Study And Results 75 Figure 4.39: Frequency Vs Slab thicknesses for 6000 mm span and aspect ratio 1 Figure 4.40: Frequency Vs Slab thicknesses for 7000 mm span and aspect ratio 1 051015202530100 150 200 250 300 350 400 450Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1Span length=6000 mmBay width= 6000 mm 10 Hz Limit (Murray)ACI Serviceability Limit051015202530100 200 300 400 500Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1 Span length=7000 mmBay width= 7000 mm 10 Hz Limit (Murray)ACI Serviceability LimitParametric Study And Results 76 Figure 4.41: Frequency Vs Slab thicknesses for 8000 mm span and aspect ratio 1 Figure 4.42: Frequency Vs Slab thicknesses for 3000 mm span and aspect ratio 1.2 051015202530125 225 325 425 525 625Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1Span length=8000 mmBay width= 8000 mm 10 Hz Limit (Murray)ACI Serviceability Limit05101520253050 70 90 110 130 150 170 190 210Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1.2Span length=3000 mmBay width= 3600 mm 10 Hz Limit (Murray)ACI Serviceability LimitParametric Study And Results 77 Figure 4.43: Frequency Vs Slab thicknesses for 4000 mm span and aspect ratio 1.2 Figure 4.44: Frequency Vs Slab thicknesses for 5000 mm span and aspect ratio 1.2 05101520253050 100 150 200 250 300Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1.2 Spanlength=4000 mmBay width= 4800 mm 10 Hz Limit (Murray)ACI Serviceability Limit05101520253050 100 150 200 250 300 350Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1:1.2Span length=5000 mmBay width= 6000 mm 10 Hz Limit (Murray)ACI Serviceability LimitParametric Study And Results 78 Figure 4.45: Frequency Vs Slab thicknesses for 6000 mm span and aspect ratio 1.2 Figure 4.46: Frequency Vs Slab thicknesses for 7000 mm span and aspect ratio 1.2 051015202530100 150 200 250 300 350 400 450Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1.2Spanlength=6000 mmBay width= 7200 mm 10 Hz Limit (Murray)ACI Serviceability Limit051015202530100 200 300 400 500 600Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1:1.2Span length=7000 mmBay width= 8400 mm 10 Hz Limit (Murray)ACI Serviceability LimitParametric Study And Results 79 Figure 4.47: Frequency Vs Slab thicknesses for 8000 mm span and aspect ratio 1.2 Figure 4.48: Frequency Vs Slab thicknesses for 3000 mm span and aspect ratio 1.4 051015202530100 200 300 400 500 600Natural Frequency (Hz)Slab Thickness (mm)SpanLength : Bay width= 1:1.2Span length=8000 mmBay width= 9600 mm ACI Serviceability Limit10 Hz Limit (Murray)05101520253050 100 150 200 250Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 :1.4Span length=3000 mmBay width= 4200 mm 10 Hz Limit (Murray)ACI Serviceability LimitParametric Study And Results 80 Figure 4.49: Frequency Vs Slab thicknesses for 4000 mm span and aspect ratio 1.4 Figure 4.50: Frequency Vs Slab thicknesses for 5000 mm span and aspect ratio 1.4 05101520253050 100 150 200 250 300Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1:1.4Span length=4000 mmBay width= 5600 mm 10 Hz Limit (Murray)ACI Serviceability Limit05101520253050 100 150 200 250 300 350Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1:1.4Span length=5000 mmBay width= 7000 mm 10 Hz Limit (Murray)ACI Serviceability LimitParametric Study And Results 81 Figure 4.51: Frequency Vs Slab thicknesses for 6000 mm span and aspect ratio 1.4 Figure 4.52: Frequency Vs Slab thicknesses for 7000 mm span and aspect ratio 1.4 051015202530100 150 200 250 300 350 400 450Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1:1.4Span length=6000 mmBay width= 8400 mm 10 Hz Limit (Murray)ACI Serviceability Limit05101520253050 150 250 350 450 550Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1:1.4Span length=7000 mmBay width= 9800 mm ACI Serviceability Limit10 Hz Limit (Murray)Parametric Study And Results 82 Figure 4.53: Frequency Vs Slab thicknesses for 4000 mm span and aspect ratio 1.4 Figure 4.54: Frequency Vs Slab thicknesses for 3000 mm span and aspect ratio 1.6 05101520253050 150 250 350 450 550 650Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1.4Span length=8000 mmBay width= 11200 mm ACI Serviceability Limit10 Hz Limit (Murray)05101520253050 100 150 200 250Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1.6 Span length=3000 mmBay width= 4800mm10 Hz Limit (Murray)ACI Serviceability LimitParametric Study And Results 83 Figure 4.55: Frequency Vs Slab thicknesses for 4000 mm span and aspect ratio 1.6 Figure 4.56: Frequency Vs Slab thicknesses for 5000 mm span and aspect ratio 1.6 05101520253050 100 150 200 250 300Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1.6Span length=4000 mmBay width= 6400mm10 Hz Limit (Murray)ACI Serviceability Limit05101520253050 100 150 200 250 300 350Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1.6Span length=5000 mmBay width= 8000mm10 Hz Limit (Murray)ACI Serviceability LimitParametric Study And Results 84 Figure 4.57: Frequency Vs Slab thicknesses for 6000 mm span and aspect ratio 1.6 Figure 4.58: Frequency Vs Slab thicknesses for 7000 mm span and aspect ratio 1.6 051015202530100 200 300 400 500 600Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1.6Span length=6000 mmBay width= 9600mm10 Hz Limit (Murray)ACI Serviceability Limit051015202530100 200 300 400 500 600 700 800Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1.6 Span length=7000 mmBay width= 11200mmACI Serviceability Limit10 Hz Limit (Murray)Parametric Study And Results 85 Figure 4.59: Frequency Vs Slab thicknesses for 8000 mm span and aspect ratio 1.6 Figure 4.60: Frequency Vs Slab thicknesses for 3000 mm span and aspect ratio 1.8 051015202530100 300 500 700 900 1100Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1.6Span length=8000 mmBay width= 12800mmACI Serviceability Limit10 Hz Limit (Murray)05101520253050 100 150 200 250Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1.8Span length=3000 mmBay width= 5400mm10 Hz Limit (Murray) ACI Serviceability LimitParametric Study And Results 86 Figure 4.61: Frequency Vs Slab thicknesses for 4000 mm span and aspect ratio 1.8 Figure 4.62: Frequency Vs Slab thicknesses for 5000 mm span and aspect ratio 1.8 05101520253050 100 150 200 250 300Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1.8Span length=4000 mmBay width= 7200mm10 Hz Limit (Murray)ACI Serviceability Limit05101520253050 100 150 200 250 300 350 400 450Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1.8Span length=5000 mmBay width= 9000mmACI Serviceability Limit10 Hz Limit (Murray)Parametric Study And Results 87 Figure 4.63: Frequency Vs Slab thicknesses for 6000 mm span and aspect ratio 1.8 Figure 4.64: Frequency Vs Slab thicknesses for 7000 mm span and aspect ratio 1.8 051015202530100 200 300 400 500 600 700Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1.8Span length=6000 mmBay width= 10800mmACI Serviceability Limit10 Hz Limit (Murray)051015202530100 300 500 700 900Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1.8Span length=7000 mmBay width= 12600mmACI Serviceability Limit10 Hz Limit (Murray)Parametric Study And Results 88 Figure 4.65: Frequency Vs Slab thicknesses for 8000 mm span and aspect ratio 1.8 4.3.2.1 Discussion Fromtheabovefigures,therelationbetweenslabthicknessandfloorfrequencyis found for the hall room (without partition wall). The following points are observed, Thefloorfrequencywithoutpartitionwallloadishigherthanthefloor frequency with partition wall load. Asthereisnopartitionwallload,sothefloorvibrationishigher.Because mass is inversely proportional to the floor vibration. Floorfrequencydecreaseswithaspectratio.Aswithaspectratio,oneway system becomes sustain rather two way system. The change of slope of the curve is achieved later, because of reduced load. For the same combination of span length and aspect ratio, 10 Hz frequency of the floor is achieved in less slab thickness then that of with partition wall load.Heavyfloorreducetheamplitudeofthevibration.Havingafloorvibrateat lowamplitude,increasethefrequency.Thatswhywithincreasingslab thickness vibration is also increased. 051015202530100 300 500 700 900 1100Natural Frequency (Hz)Slab Thickness (mm)Span Length : Bay width= 1 : 1.8Span length=8000 mmBay width= 14400mmACI Serviceability Limit10 Hz Limit (Murray)Parametric Study And Results 89 4.3.2.2 Minimum slab thickness determination Theminimumslabthickness,whichisrequiredtopreventvibrationwhichcauses annoyingtotheoccupants,willbedeterminedhere.Itisknownthatmosthumans organs have a natural frequency of about 4-8 Hz (Murray 1991). So the floor vibrating intherangeitisperceivedasuncomforted.Theslabthicknessshouldbesuchthat, which causes vibration greater than 8 Hz. Humanwalkingvibrationisintherangeof8-9Hz.Sothefloorvibrationshouldbe greaterthanthese,asifresonanceisnotcreated.Forallthesereasonsthefloor vibrationiskeptabove10Hz.Indeterminingtheseslabthickness,slabthickness correspondingto10Hzlimitisplottedagainstspanlength.Herespanlengthisthe smaller length in mm. This procedure is repeated for different aspect ratio such as 1, 1.2,1.4,1.6,and1.8.MinimumslabthicknessgivenbyACIisalsoplottedagainst span length to visualize the deviation. Fromthefigure4.66-4.70variationofslabthicknesswithrespecttospanlengthis observed.Apolynomialequationisformedfromtheobservation.HereACIlimit meansACIServiceabilityLimit,and10Hzlimitmeans10HzlimitbyMurray (1997). Figure 4.66: Comparison Of Slab Thickness For Aspect Ratio 1 020040060080010000 2 4 6 8 10 12Slab Thickness(mm)Span Length (m)ACI Limit10 Hz LimitFormulaParametric Study And Results 90 Figure 4.67: Comparison Of Slab Thickness For Aspect Ratio 1.2 Figure 4.68: Comparison Of Slab Thickness For Aspect Ratio 1.4 020040060080010000 2 4 6 8 10 12Slab Thickness(mm)Span Length (m)ACI Limit10 Hz LimitFormula020040060080010000 2 4 6 8 10 12Slab Thickness(mm)Span Length (m)ACI Limit10 Hz LimitFormulaParametric Study And Results 91 Figure 4.69: Comparison Of Slab Thickness For Aspect Ratio 1.6 Figure 4.70: Comparison Of Slab Thickness For Aspect Ratio 1.8 Thisequationisafunctionofspanlengthandaspectratio.Thisequationgivesthe minimumslabthicknesswhichproduceslessvibration.Thisequationisforslab thickness of floor without partition wall load. The equation is, 020040060080010000 2 4 6 8 10 12Slab Thickness(mm)Span Length (m)ACI Limit10 Hz LimitFormula020040060080010000 2 4 6 8 10 12Slab Thickness(mm)Span Length (m)ACI Limit10 Hz LimitFormulaParametric Study And Results 92 = 2.583450.483+370.821160 +1349 (4.5)

Where,t= Minimum slab thickness for around 10 Hz floor vibration, mm x= Short span length, m = aspect ratio. 4.4 REMARKS Fromtheabovestudy,theempiricalequation4.4and4.5canbesuggestedfor determinationoftheslabthicknessforreinforcedconcretebuildinghavingfloor finish1.2 10-3N/mm2.Theproposedslabthicknesscanbeusedtopreventthe undesirable natural floor vibration. Equation 4.4 is for normal residential building and 4.5isforhallroomwherepartitionwallload(2.4 10-3N/mm2)isignored.This formulaisonlyforshortspanbuildingrangesfrom3m6mandconsideringfloor finishload1.2 10-3N/mm2.Inthisstudydampingisnotusedasthisisnatural vibration. Chapter - 5 Conclusion And Recommendation Conclusion And Recommendation 94 5.1 GENERAL Inthepresentstudyaninvestigationhasbeendonetodeterminetherequired minimum slabthickness fromdynamicserviceability.ACIprovidesaminimumslab thickness(equation2.3)topreventexcessivedeflection(staticserviceability)ofthe slab.Thereisnosuchrecommendedslabthicknesswhichcanpreventundesirable floor vibration. This study provides a guide line about slab thickness with function of spanlengthandaspectratio,whichpreventexcessiveuncomfortablevibration.This slabthicknessisdeterminedfortwocases;oneisforwithpartitionwallloadand anotheriswithoutpartitionwallload.Thisstudyisdoneforthreestoriedbuilding with different floor panel aspect ratio. 5.2 FINDINGS OF THE INVESTIGATION The general output and findings from the previous chapters are summarized below: Themodalanalysisofthebuildinghasdoneandvariousmodesofvibration, frequency and time period were determined by this analysis. This will help us to understand the behavior of the building structure under vibration. The floor frequency decrease with increasing floor panel aspect ratio. For span length higher than 5m, slab thickness provided by ACI for deflection may not be good for vibration. The floor frequency decrease with increasing column height. Whenthemassofthestructureincrease,theamplitudeofthatstructure decrease.The increasing of slab thickness causes the increase of frequency of the floor. With increasing span length the floor frequency is reduced.The floor frequency is higher for floor without partition wall load. 5.3 PROPOSED EQUATIONS Basedontheabovefindingsauthorsuggestempiricalequationsforreinforced concretebuildingsslabthicknessfromdynamicserviceability.Thosesuggested Conclusion And Recommendation 95 equations can be used to determine slab thickness to prevent undesirable natural floor vibration. Those equations are only valid for short span building ranges from 3m 6m (smaller length). Those equations are, Considering Partition wall load (2.4 10-3 N/mm2), = 3.5414+7535582+1795 2050 Without considering partition wall load, = 2.583450.483+370.821160 +1349 Where, t= Minimum slab thickness, mm x= Small span length, m = Aspect ratio For all cases, it is assumed that floor finish is 1.2 10-3 N/mm2 5.4 RECOMMENDATION FOR FURTHER STUDY The current study has been carried under limited scope. The results are not sufficient to apply as so many other factors have not considered. Advancement of current study can be done combining some other variables. The following fields related to this study can be considered for the further analysis;

The model was considered to be linearly elastic. To be more realistic with the resultsafiniteelementanalysiswithnonlinearlymaterialproperti