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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)JointDEquilibriumequationsDF5FABC2m2m12345
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