弹性力学课件二八

APPLIEDELASTICITY华中科技大学力学系

1Introduction1.1CONTENTSOFTHEORYOFELASTICITY1.2SOMEIMPORTANTCONCEPTSINTHEORYOFELASTICITY1.3BASICASSUMPTIONS1.1CONTENTSOFTHEORYOFELASTICITYPurposeThetheoryofelasticity,oftencalledelasticityforshort,isthebranchofsolidmechanics.Purposeistoanalysethestressesanddisplacementofstructuralormachineelementswithintheelasticrangeandtherebytocheckthesufficiencyoftheirstrength,stiffnessandstability.Althoughthispurposeisthesameasthatofstudyingmechanicsofmaterialsandstructuralmechanics,thesethreebranchesofsolidmechanicsdodifferfromoneanotherbothintheobjectsstudiedandinthemethodsofanalysisused.Mechanicsofmaterialsdealsessentiallywiththestressesanddisplacementsofastructuralormachineelementintheshapeofabar,straightorcurved,whichissubjectedtotension,compression,shear,bending,ortorsion.Structuralmechanics,onthebasisofmechanicsofmaterials,dealswiththestressesanddisplacementsofastructureofabarsystem,suchasatrussorarigidframe.DifferentTheobjectsstudiedAstothestructuralelementswhicharenotintheformofabar,suchasblocks,plates,shells,dams

andfoundations,theyareanalysedonlyinthetheoryofelasticity.ThemethodsofanalysisusedAbarshapeelementsubjectedtoexternalloads:MechanicsofmaterialsandelasticityElasticity:doesnotneedthoseassumptions.Thustheresultsobtainedaremoreaccurateandmaybeusedtochecktheapproximateresultsobtainedinmechanicsofmaterials.Mechanicsofmaterials:someassumptionsareusuallymadeonthestrainconditionorthestressdistribution.Theseassumptionssimplifythemathematicalderivationtoacertainextent,butofteninevitablyreducethedegreeofaccuracyoftheresultsobtained.Inmechanicsofmaterials,itisassumedthataplanesectionofthebeamremainsplaneafterbending.Thisassumptionleadstothelineardistributionofbendingstresses.nnForexample,theproblemofbendingofastraightbeamundertransverseloads:Inthetheoryofelasticity,however,onecansolvetheproblemwithoutthisassumptionandprovethatifthedepthofthebeamisnotmuchsmallerthanthespanlength,thestressdistributionwillbefarfromlinearvariation,andthemaximumtensilestressisseriouslyundervaluedinmechanicsofmaterials

MutualinfiltrationTheutilizationofvariousmethodsofanalysisinstructuralmechanicsgreatlystrengthenedthetheoryofelasticityandenabledengineerstoobtainthesolutionsofmanycomplicatedproblemsinelasticity.Forexample,usingthefiniteelementmethod,wecansolveaprobleminelasticitybythediscretizationofthebodyconcernedandthentheapplicationofthedisplacementmethod,theforcemethod,orthemixedmethodinstructuralmechanics.Moreover,inthedesignofastructure,wecanutilizethedifferentbranchesofsolidmechanicsfordifferentmembersofthestructure,andevenfordifferentpartsofasinglemember,togetthemostsatisfactoryresultswiththeleastamountofwork.Weshouldnotpaytoomuchattentiontothefuzzyandtemporarydividinglinesbetweenthethreecoursesinsolidmechanics.Onthecontrary,weshouldnoteallthepossibilitiesofthejointapplicationofthethreecourses.1.2SOMEIMPORTANTCONCEPTSINTHEORYOFELASTICITYexternalforces,stresses,deformationsanddisplacementsBodyforces:externalforces,ortheload,distributedoverthevolumeofthebody.Surfaceforces:externalforces,ortheload,distributedoverthesurfaceofthebodyExternalforces(ortheloads):1.Surfaceforces：Suchasthepressureofonebodyonanotherorhydrostaticpressure2.Bodyforces:Suchasgravitationalforces,orinertiaforcesinthecaseofbodyinmotion3.Indicationofexternalforces：a.SurfaceforcesxlimS0yzoYXZSFF：theintensityofsurfaceforceatPTheprojectionsofF

onthex,y,zaxesaredenotedbyX,YandZrespectivelyandcalledthesurfaceforcescomponentsatP.Suchacomponentisconsideredpositiveornegativeaccordingasitactsinthepositiveornegativedirectionofthecoordinateaxis.Itsdimensionis[force]/[length]2.b.

BodyforceslimV0yzoYXZVFxThecomponentis

consideredpositiveornegativeaccordingasitactsinthepositiveornegativedirectionofthecorrespondingcoordinateaxis.Itsdimensionis[force]/[length]3.F：theintensityofbodyforceatP.TheprojectionsofF

onthex,y,zaxesaredenotedbyX,YandZrespectivelyandcalledthebodyforcescomponentsatP.2)Internalforces（internalforcesatacertainpointatP）Thestressisresolvedintoanormalcomponent

calledthenormalstressandatangentialcomponent

calledtheshearingstress.S：thestressonthesectionmnatpointPanditsdirectionisthelimitingdirectionof

Q.limA0yzoAQxmnNote：Generallyspeaking,thestressesondifferentsectionspassingthroughthesamepointinabodyaredifferent.HowtodescribethestressconditionatthepointP？zyoxIsolateanelementaryatpointPfromthebody.yzoxxxyxzxxyxzyyxyzyyxyzzzyzxzzyzxxxyxzxxyxzyyxyzyyxyzzzyzxzzyzx1.Toindicatedtheactingplaneandthedirectionofanormalstress,weassociatethestresswithacoordinatesubscript.x、y、zAstoashearingstress,weshouldassociateitwithtwocoordinatesubscripts.

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