Engineering Basic Mechanics I Statics 工程基礎(chǔ)力學(xué)Ⅰ 靜力學(xué) 課件 Chapter 3 Coplanar General 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)JointDEquilibriumequationsDF5FABC2m2m12345