工程基础力学Ⅰ 静力学 课件 Chapter 3 Coplanar General Force System、Chapter 4 Space Force System

StaticsStaticsofrigidChapter3CoplanarGeneralForceSystemMaincontents3.1Reductionofcoplanargeneralcase3.2Equilibriumofcoplanargeneral3.3EquilibriumofCompositeBody3.4PlanetrussThetranslationofforcetheoremABABddAB§3.1Resultantofcoplanargenernalcase§3.1Resultantofcoplanargenernalcase●EachforcecanbemovedtoanarbitrarychosenpointObyintroducingacouple.●TheforcesarenowconcurrentforcesatO.Theycanbeaddeduptoaresultantforce,whichistheprinciplevectoroftheoriginalforcesystem

●Thecouplescanbeaddeduptoaresultantcouple,whichistheprinciplemomentoftheoriginalforcesystem.Acoplanarforcesystemonarigidbodycanbereducedtoanequivalentforce-couplesystemconsistingofaforceactingonanarbitrarypoint,plusacouple.Note●Anycoplanarforcesystemcanbereducedtoanequivalentforce-couplesystem.●Theprinciplevectorissimplythevectorsumofalltheforces.ItisnotaffectedbythelocationofpointO.●ThemagnitudeoftheprinciplemomentisthealgebraicsumofthemomentsofalltheforcesaboutO.ItdoesdependonthelocationofpointO.●Whenboththeprinciplevectorandprinciplemomentarezero,thegivenforcesystemexertsnoactionontherigidbody,andthesystemissaidtobeequilibrium.●Whentheprinciplevectorisequaltozero,thegivenforcesystemisreducedtoasinglecouple,calledtheresultantcoupleofthesystem.●Asweknow,acoplanarforcesystemcanbereducedtoanequivalentsystemconsistingoftheprinciplevectoractingatOand

the

principle

moment.Usually,thisprocedurehasfourpossibleoutcomes:●Thegivenforcesystemisreducedtoasingleforce(resultantforce)actingthroughpointO.●Thesystemcanbereducedfurthertoasingleforce(calledtheresultantforceofthesystem)actingatapointdifferentfromO.TheperpendiculardistancebetweenOandF

isThemomentofthisforceaboutOmustbethesameasthe

principle

moment.F§3.2Equilibriumequationsofcoplanargeneralcase

●Sinceagenernalcoplanarforcesystemcanalwaysbereducedtoaprinciplevector,passingthroughanarbitrarychosenpoint,andaprinciplemoment,thesufficientandnecessaryconditionsfortheequilibriumarethreeconditions(2)Verticalforceequilibrium:

(1)Horizontalforceequilibrium:(3)Momentequilibrium:

Example3-1

DeterminetheforcesatA.

Solution:

(1)DrawtheFBDofthebeamABqABxyMFlFAyMAFAx

(2)Equilibriumequations

(3)SolvingtheseequationsThreestepsintheequilibriumanalysisofabody(1)Drawalloftheforceandmomentsthatactonit.(2)Writetheequilibriumequationsintermsoftheforcesandmomentsthatappearonthefree-bodydiagram.(3)Solvetheequilibriumequationsfortheunknowns.§3.3EquilibriumofCompositeBody●Acompositebodymeanstwoormorerigidbodiesinteractingwitheachother:

●Whenacompositebodyisinequilibrium,everybodyofthebodysystemisalsoinequilibrium.●Inadditiontoexternalreactionsatsupports,someforcesduetointeractionofonebodywithanotheralsoneedtobedeterminedinequilibriumanalysisofcompositebody.●Theprimarydifferencebetweenone-bodyandcomposite-bodyproblemsisthatthelateroftenrequirestoanalyzemorethanonefree-bodydiagramandsolvemorethanonesetofequilibriumequations.BEistwo-forcebody,F=20kN，q=10kN/m，M=20kN•m，l=1m.DeterminethereactionsatpointsAandB.Example3-2

EBCDAqllllFMAFAyFBMFAx(1)Objectofstudy:CD

Solution:FCxFCyEquilibriumequations:CBDqFFB(2)Objectofstudy:wholestructure

Solution:Equilibriumequations:BCDAqllllFMAFAyFBMFAx●Foranequilibriumanalysisofcompositebodies,usuallythebodiesneedtobeseparated.Sometimestheanalysisofthewholestructureorcombinationofsomeofthebodiescanmakethesolutionsimpler.ImportantPoints●

Thereareatmostthreeindependentequilibriumequationsforeachfree-bodydiagram.●

Trytotakemomentsaroundsomepointsthattheactionlinesofsomeunknownforcespassthrough.Thiscanresultinmuchsimplerequationstosolve.§3.3EquilibriumofCompositeBody●Atrussisastructurecomposedofstraight,slendermembers

joinedtogetherattheirends.Trussesareusuallydesignedtotransmitforcesoverrelatinglongspans.§3.4Planetruss●

Assumptionsfortrussanalyses:(1)Theweightsofthemembersareneglected.(2)Themembersareconnectedattheirendsbyfrictionlesshinges.(3)Theappliedloadsandreactionsactatthehingesonly.●Basedontheseassumptions,allmembersofatrussonlysustainaxialforces,eitherintensionorincompression.●Twotechniquestocalculatetheinternalforcesofthemembers:(1)TheMethodofJoints(2)TheMethodofSections●

TheMethodofJoints●

TheprocedurefortrussanalysesbytheMethodofJoints

Inthetruss,F=10kN.Determinetheinternalforcesineachmember.ABC2m2m12345FDExample3-3

ABC2m2m12345FD(1)Objectofstudy:wholestructure

EquilibriumequationSolution:FAyFByFBx(2)JointAF2F1FAyAEquilibriumequationABC2m2m12345FDF3F4C(3)JointCEquilibriumequationABC2m2m12345FD(4)JointDEquilibriumequationsDF5FABC2m2m12345FD●

TheMethodofSections●

TheProcedureforTrussAnalysesbytheMethodofSectionsInthetruss,FE=10kN,FG=7kN,AC=AE=CE=CD=DF=ED=EG=GB=DG=GF=FB=1m,Determinetheinternalforces1,2and3.Example3-4

xyABCDEFGFEFG123(1)Objectofstudy:wholestructureEquilibriumequationSolution:xyABCDEFG123FByFAxFEFGFAyxyABCDEFGFByFAxFEFGFAy123mm(2)choosesectionm-mandstudytheleftpart.

EquilibriumequationsFAxFAyF1F2FEF3yACDExImportantPoints§3.4Planetruss

TheEnd

StaticsStaticsofrigidChapter4SpaceForceSystemMaincontents4.1Concurrentforcesysteminspace4.2Systemofcouplesinspace4.3Themomentofaforceaboutapointoranaxis4.4Reductionofaforcesysteminspace4.5Equilibriumequationsofgeneralforcesysteminspace1.Howtorepresentaforceinspace(1)directprojection(2)directprojectionAforcecanbedescribedas4.1Concurrentforcesysteminspace2.Thecompositionandequilibriumofaconcurrentforcesysteminspace(1)CompositionTheresultantforceofconcurrentforcesysteminspaceisequaltovectorsumofthecomponents.

4.1ConcurrentforcesysteminspaceThelawofprojectionoftheresultantforce:Theprojectionoftheresultantforceofaforcesysteminspaceontoanaxisisequaltothealgebraicsumoftheprojectionsofthecomponentforcesontothesameaxis.(2)EquilibriumTheanalyticalmethodforequilibriumequations:4.1ConcurrentforcesysteminspaceThenecessaryandsufficientconditionofequilibriumisthattheresultantforceiszero:1.Coupleinspace(1)Couplemomentvector(2)Coupleinspaceisafreevector4.2Systemofcouplesinspace(3)Thenecessaryandsufficientconditionoftheequivalenceoftwospacecouplesisthatthecouplemomentvectorsofthetwocouplesareequal.2.Compositionandequilibriumofasystemofcouplesinspace(1)CompositionTheresultantmomentofcoupleisthesumofthemomentsofthecomponentcouples.Theprojectionsare

4.2Systemofcouplesinspace(2)Equilibrium1.Themomentofaforceaboutapoint(1)Definition:aforce

acting

ontheApoint,themomentofaforceaboutOpointinspaceisisvector4.3Themomentofaforceaboutapointoranaxis(2)analysisformula4.3Themomentofaforceaboutapointoranaxis(1)Definition:themomentofaspaceforceaboutanaxisisascalar,whichisequaltothemomentoftheprojectionofthisforceonaplaneperpendiculartothisaxisabouttheintersectingpointofthisplaneandthisaxis.4.3Themomentofaforceaboutapointoranaxis2.Themomentofaforceaboutaaxis4.3Themomentofaforceaboutapointoranaxis(2)Analysisformula4.3Themomentofaforceaboutapointoranaxis3.TherelationshipbetweenthemomentoftheforceaboutapointandthemomentoftheforceaboutanaxisthroughthepointTheorem:theprojectionofthevectorofthemomentofaforceaboutapointonanaxisthroughthispointisequaltothemomentoftheforceabouttheaxis.

ThehandleABCEisintheplaneAxy,FactingontheD

pointlocatesinaplanenormaltoyaxis,and

theangleis

α.If

CD=b,rodBCparallelsto

xaxis,rod

CEparallelstoyaxis,thelenghtofABandBCare

l,respectively.DeterminethemomentofFaboutx，yandzaxis,respectively.4.3ThemomentofaforceaboutapointoranaxisExample4-1AsweknownTheprojectionofforce

F

alongeverycoordinateaxis：thecoordinateofpointD:thenSolution：

1.

Theoremoftranslationofaforce

4.4ReductionofaforcesysteminspaceAforceactingonarigidbodycanbemovedparalleltoanyspecifiedpointintherigidbody,butacouplemustbeaddedatthesametime,andthemomentvectorofthecoupleisequaltothemomentvectoroftheoriginalforceaboutthespecifiedpoint.PrinciplevectorPrinciplemoment4.4Reductionofaforcesysteminspace

2.

Reduction

Aspacegeneralforcesystemcanbereducedtoaforceandcouple.Theforceactsonthesimplifiedcenter,

theforcevectorisequaltotheprincipalvectoroftheforcesystem;

themomentvectorofthecoupleisequaltotheprincipalmomentoftheforcesy