弹性力学课件35

3.3BENDINGOFASIMPLEBEAMUNDERUNIFORMLOADConsiderasimplebeam,withlength2landdepthh,subjectedtoauniformlydistributedloadofintensityq.forconvenience,onlyaunitwidthofthebeamisconsidered,sothereactionateachendwillbeql.xh1yh/2h/2oqqqThesemi-inversemethodwillbeemployedhere.Justasthebendingstressxandtheshearingstressxyaremainlyproducedbythemomentandtheshearingforcerespectively,thecrushingstressyismainlyproducedbythedirectloadonthebeam.Sincethedirectloadqdoesnotvarywithx,wemayassumethaty

doesnotvarywithxeitherandconsequentlyitisonlyafunctionofy：y=f（y）Todeterminethefunctionf(y),f1(y)andf2(y),wesubstitutetheexpressionforintocompatibilityequation,obtainingwheref1(y)andf2(y)arearbitraryfunctions.Thisisaquadraticequationofx,butitmustbesatisfiedforallvaluesofxbetween–landl,astheconditionofcompatibilityrequires.Thisispossibleonlywhenthecoefficientsofx2andx,aswellasthetermindependentofx,arezero:（1）（2）（3）Integrationof(1)and(2)yields：Heretheconstantterminf1(y)isneglected,becauseitwillonlyresultinalinearterminandnotaffectthestress.Substitutingfinto(3)andintegrating,wehave：Theconstanttermandthetermlinearinyareneglect,becausetheywillnotaffectthestress.Thestresscomponentswillbe:Theseexpressionssatisfythedifferentialequationsofequilibriumandthecompatibilityequation.Hence,ifthearbitraryconstantsA,B,,Kcanbechosentosatisfyalltheboundaryconditions,theseexpressionswillbetherightsolutionoftheproblem.xyh/2h/2oqqq(1)considertheconditionsofsymmetryThus,theexpressionsforx、ymustbeevenfunctionofx,whilethatforxymustbeoddfunctionofx.ThisrequiresSincetheyzplaneisaplaneofsymmetryofthebeamandtheloading,thestressdistributionmustbesymmetricwithrespecttotheplane.E=0F=0G=0xyh/2h/2oqqq(2)considerboundaryconditionsSubstitutingthestresscomponentsexpressionsintotheseequationsandnotingthatE=F=G=0,wehaveNowinordertodetermineHandK,wecanconsidertheboundaryconditionsattheendsofthebeam(theleftandrightendsofthebeamareonlysmallportionsoftheboundary).Iftheboundaryconditionstherecannotbesatisfiedexactly,wemayapplytheSaint-Venant’sprincipletohavetheconditionsapproximatelysatisfied.xyh/2h/2oqqqBoundaryconditionsatends：（1）（2）（3）From(1),wehave：K=0From(2),wehave：(3)issatisfiedNowtheexpressionsforthestresscomponentsare：ThestressdistributiononatypicalcrosssectionisapproximatelyshowninFig.：xyxyComparethesolutionobtainedhereandthatgiveninmechanicsofmaterials,Forthebeamofunitwidth,wehave：So,thestresscomponentscaberewrittenas：Weseethatthebendingstressxgiveninmechanicsofmaterialsmustbesupplementedwithacorrectiontermwhiletheshearingstressxyneedsnocorrection.Astothecrushingstressy,itisonlyconsideredinelasticityandnotinmechanicsofmaterialsatall.Itshouldbenotedthattheaboveequationsrepresenttheexactsolutiononlyifattheends(x=l)therearenormalforcesdistributedaccordingtothelawAndtheshearingforcesaredistributedaccordingtothelaw：However,ateachend,thenormalforcesareequivalenttozero(havingzeroresultantforceandzeroresultantmoment)whiletheshearingforcesareequivalenttoanupwardforceql.Hence,bysaint-Venant’sprinciple,thestresscomponentsrepresenttherightsolutionforthebeamexceptneartheends,eveniftherearenonormalforcesontheendsortheshearingforcesaredistributedinanymannerdifferentfromthatshownabove.3.4TRIANGULARGRAVITYWALLxyggyConsideradamoraretainingwallwithtriangularsectionsubjectedtotheactionofgravityandthepressureofimpoundedliquid.Letthedensityofthewallmaterialbeandthatoftheliquidbe.Atanypointinthewall,eachofthestresscomponentsmusyconsistoftwoparts：xyggy(1)producedbygravity：isproportionaltog(2)producedbythepressure：isproportionaltogHence,ifthestresscomponentscanbeexpressionsintheformofpolynomials,theymustbecombinationsoftheexpressionsintheformsofAgx,Bgy,Cgx,Dgy.WemayassumethestressfunctionisapolynomialofthirddegreeX=0Y=gTheseexpressionshavealreadysatisfiedthedifferentialequationsofequilibriumandthecompatibilityequation.Itremainstoinspectwhethertheboundaryconditionscanalsobesatisfiedbypropervaluesofthearbitraryconstantsa,b,c,d.xyggyBoundaryconditions：(1)theverticalsurface