(1.9.5)--6.4 Doubly Symmetric Beams with材料力学材料力学.ppt

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1、 Mechanics of MaterialsDoubly Symmetric Beams with Inclined LoadsChapter 6 Stresses in Beams(Advanced Topics)Inthissection,wearegoingtodiscussdoublysymmetricbeamswithInclinedLoads.Doublysymmetric：boththex-yandx-zplanesareplanesofsymmetryDoubly symmetric beams with inclined loadsInclinedloads：loadsdo

2、notactintheplaneofsymmetryInclinedloadsPreviousdiscussionsofbendingdealtwithbeamspossessingalongitudinalplaneofsymmetry,forinstance,thex-yplaneinthisfigureandsupportinglateralloadsactinginthatsameplane.Undertheseconditions,thebendingstressesareobtainedfromtheflexureformula,providedthatthematerialish

3、omogeneousandlinearlyelastic.*questioniswhathappenswhenthebeamisunderinclindloads？Analytical ProcedureSuperposition:Twoplanebendingareanalysedseparatelyandthecorrespondingresultsaresuperposed.Decomposition:DecomposetheloadalongthetwoCentroidalprincipalaxesofthecrosssectiontoobtaintwoorthogonalplaneb

4、ending.yzF Fz zF Fy yF F Generally,tofindthebendingstressesinthebeamunderinclindloads,*decomposetheinclinedloadintotwocomponents,oneactingineachplaneofsymmetry.*Thenthebendingstressesareobtainedfromtheflexureformulaforeachloadcomponentactingseparately.Thefinalstressesareobtainedbysuperposingthesepar

5、atestresses.Sign Conventions for Bending MomentsyzyMyMz:Tension:CompressionAsapreliminarymatter,firstestablishsignconventionsforthebendingmomentsMyandMzactingabouttheyandzaxes,respectively.Themomentsarepositivewhentheirvectorspointinthepositivedirectionsofthecorrespondingaxes,andtheright-handrulefor

6、vectorsgivesthedirectionofrotationindicatedbythecurvedarrowsinthefigure.And*onacrosssection,apositivemomentMyproduces*compressionontheright-handsideofthebeamand*tensionontheleft-handside.Similarly,apositivemomentMzproduces*compressionontheupperpartofthebeamand*tensiononthelowerpart.1.Normal Stresses

7、(Bending Stresses)Iy,Iz:themomentsofinertianormalstressescorrespondingtoMy：normalstressescorrespondingtoMz：The normal stress at point A：Tocalculatenormalstressesatanypointinacrosssection,firstly,wecanobtainthenormalstressesassociatedwiththeindividualbendingmomentsMyandMzfromtheflexureformula,thensup

8、erposethemtofindthefinalstressatthepointproducedbybothmomentsactingsimultaneously.Forinstance,considerthestressesata*pointAinthecrosssection,*apositivemomentMyproducestensionatthispoint,and*apositivemomentMzproducescompression;thus,*thenormalstressatpointAis*obtained.Usethisequationtofindthenormalst

9、ressatanypointinthecrosssectionbysubstitutingtheappropriatealgebraicvaluesofthemomentsandthecoordinates.2.Neutral Axis-determinedbyequatingthenormalstressxtozero.The angle between the neutral axis and the z axis:Anotherquestion,howtofindthenetrualaxisforadoublysymmetricbeamssubjectedtoinclinedLoads?

10、Fromthedefinitionofthenetrualaxis,theequationoftheneutralaxisisdeterminedbyequatingthe*normalstressxto*zero.Thisequationshowsthattheneutralaxis*n-nisastraightlinepassingthroughthecentroidC.Asweknow,thenormalstressesvarylinearlywithdistancefromtheneutralaxis,themaximumnormalstressesoccuratpointslocat

11、edfarthestfromtheneutralaxis.Knowingtheorientationoftheneutralaxisisusefulwhendeterminingthepointswherethenormalstressesarethelargest.Therefore,fromtheaboveequaion,*theanglebetweentheneutralaxisandthezaxiscanbedeterminedas*.Fromthisexpression,theanglemayvaryfrom-90degreeto+90degree,Dependinguponthem

12、agnitudesanddirectionsofthebendingmoments.3.Relationship between the neutral axis and the inclination of the loadsTheanglebetweentheneutralaxisandthezaxis:PsinPcosMomentMyandMz:Ingeneral,Asjustdiscussed,theorientationoftheneutralaxiswithrespecttothezaxisisdeterminedbytheangle*.Todeterminetheorientat

13、ionoftheneutralaxisrelativetotheangleofinclinationoftheloadsactingonthebeam,usethe*cantileverbeamasanexample.ThebeamisloadedbyaforcePactingintheplaneoftheendcrosssectionandinclinedatanangletothepositiveyaxis.TheloadPcanberesolvedintocomponents*Pcosand*Psinintheyandzdirectionrespectively.Therefore,th

14、ecorresponding*bendingmomentsMyandMzactingonacrosssectionatdistancexfromthefixedsupportarefound*.TheratioofthesemomentsMy/Mzisequalto*tan.ItshowsthattheresultantmomentvectorMisattheanglefromthezaxisandisperpendiculartothelongitudinalplanecontainingtheforceP.ThensubstituetheratioofMy/Mzintotheexpress

15、ionoftan,*tanisequalto.whichshowsthat*theangleisgenerallynotequaltotheangle.Thus,theneutralaxisisnotperpendiculartothelongitudinalplanecontainingtheload,exceptinspecialcases.Theneutralaxisisnotnecessarilyperpendiculartotheplaneoftheload,thisfactcangreatlyaffectthestressesinabeam,especiallyiftheratio

16、oftheprincipalmomentsofinertiaisverylarge.Undertheseconditions,thestressesinthebeamareverysensitivetoslightchangesinthedirectionoftheloadandtoirregularitiesinthealignmentofthebeamitself.Exceptions to the general ruleWhentheprincipalmomentsofinertiaareequal(Iy=Iz)Whentheloadliesinthex-y plane(=0or180

17、)Whentheloadliesinthex-z plane(=90)thez axisistheneutralaxisthey axisistheneutralaxisNowletslookatthoesspecialcases,inwhichtheneutralaxisisperpendiculartothelongitudinalplanecontainingtheload.Case1,Whentheloadliesinthex-y plane,thatis=0or180,itmeansthatthez axisistheneutralaxis.Case2:Whentheloadlies

18、inthex-z plane,thatis=90,itmeansthatthey axisistheneutralaxis.Case3:Whentheprincipalmomentsofinertiaareequal,ieIy=Iz.Example:A4-mlongcantileverbeamisconstructedfromanIPN500section.AloadP=45kNactsintheverticaldirectionattheendofthebeam.Becausethebeamisverynarrowcomparedtoitsheight,itsmomentofinertiaa

19、boutthezaxisismuchlargerthanitsmomentofinertiaabouttheyaxis.(a)Determinethemaximumbendingstressesinthebeamifthe yaxisofthecrosssectionisverticalandthereforealignedwiththeloadP.(b)Determinethemaximumbendingstressesifthebeamisinclinedatasmallangle=1totheloadP.Hereisanexampleshowingthestressesinthebeamareverysensitivetoslightchangesinthedirectionoftheloadorinthealignmentofthebeamitself.Endofthissection.End