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1、Chapter 4 Shear forces andd bending momentsShear-Forces and Bending-Moments diagramsWelcometoMechanicsofMaterias,inthisvideo,wearegoingtodiscussShearForce,andBendingMomentdiagrams.Shear-force and bending-moments diagramsStaticanalysisreactionforcesPlotV,MdiagramsDividabeamintosegmentsthedifferential
2、andintegralrelationshipsexpressionsofV,M(methodofsections)superpositionmethodShearforceandbendingmomentdiagramsshowhowthesequantitiesvarythroughoutthelengthofthebeam,andgivethemaximumandminimumvaluesofthem.*Thesediagramsareusuallyplottedwiththeshearforceandbendingmomentasordinatesandthedistancexalon
3、gtheaxisofthebeamastheabscissa.Toconstructthesediagrams,generally,makea*staticanalysisofabeamfirstly,tofindout*reactionforcesatsupports,then*divideabeamintosegmentsbetweenthepointsofloadapplication.Toconstructshearforceandbendingmomentdiagrams,wecanuseexpressionsfortheshearforcesandbendingmomentsfou
4、ndbythemethodofsections,ortherelationshipsamongexternalforces,shearforceandbeningmoments,orsuperpositionmethodifapplicable,toconstructthesediagrams.Examples-Basic loading conditionsConcentratedload 1 2Segment1()Segment2()Toprovideaclearunderstandingofthesediagrams,andlearnhowtoconstructandinterprett
5、hesediagrams,letslookatthreeexampleswithbasicloadingconditionsasingleconcentratedload,auniformload,andseveralconcentratedloads.*Inthefirstcaseofconcentratedload,*considerasimplebeamABwithaconcentratedloadPactingonit.Usingthefreebodydiagramoftheentirebeam,*summingforceinydirectionand*summingmomentsat
6、pointA,*RBand*RAcanbefound.*Thendividethebeamintosegments,*segment1and*segment2,eachsegmentbetweenthepointsofexternalloadapplication.*Segment1,rangingfrom0toa,Tofindshearforcesandbendingmoments,methodofsectionsisusedhere.CutthroughCutthroughsegment1atadistancexfromthesupportatA,*selecttheleftportion
7、anddrawtheFBDforit,*summingforcesinydirectionand*momentsatthecutsection,*shearforceVand*bendingmomentMatdistancexfromthesupportcanbeobtained.Forsegment2,whichrangesfromatoL,dothesimilaranalysis,theexpressionsforthe*shearforceVand*bendingmomentcanbeobtained.Usingtheseexpressionstoconstructthe*shearfo
8、rceand*bendingmomentdiagrams.Characteristics of V,M diagrams 1 2(1)SlopeofVdiagram:SlopeofMdiagram:InaccordancewiththerelationshipRegion0 xaandaxLRegion0 xa=VinthisregionInaccordancewiththerelationshipNodistributedload,q=0Region0 xaCertaincharacteristicsoftheshear-forceandbendingmomentdiagramsmaynow
9、beseen.First,*theslopeofshearforcediagramiszeroforbothsegments,horizontallines,as*nodistributedloadactsonthebeam,whichisinaccordancewith*dv/dx=-q.While,*theslopeofbendingmomentdiagramequalstoshearforceforbothregions,alsoinaccordancewiththerelationship*dM/dx=V.Characteristics of V,M diagrams 1 2(2)At
10、theapplicationpointofP(),anabruptchangeintheVdiagramP=-themagnitudeofload P()AndacorrespondingchangeintheslopeoftheMdiagram:PositiveslopenegativeslopeSecond,*atthepointofapplicationoftheloadP,whichisdownward,*thereisanabruptchangeintheshear-forcediagram,*valueofthechangeequaltothemagnitudeofloadPand
11、*acorrespondingchangeintheslopeofthebending-momentdiagram,frompositivepb/Ltonegativepa/L.Characteristics of V,M diagrams 1 2(3)Areaofloadingdiagram:In accordance with the relationship Region0 x a=area of loading diagram=0,for q=0A2Areaofshearforcediagram:=areaofshearforcediagramRegiona x LIn accorda
12、nce with the relationship Third,*theshearforceatx=0andx=abothareequaltopb/L,sothedifferencebetweenthesetwoshearforcesiszero,whichisalsoequaltothe(-)areaofloadingdiagraminthisregion.While,*forthemomentdiagram,differenceofMbetweenpointx=aandx=L,isequaltotheareaofshearforcediagram,asdemonstrated.Charac
13、teristics of V,M diagrams 1 2(4)MaximmumorminimumM:Atthosepointswheretheshearcurvecrossesthereferenceaxis(V=0),thevalueofthemomentontheMdiagramisalocalmaximumorminimum.Atthepoint,Momentreachesitsmaximumvalue:Last,atx=a,curveoftheshearforcecrossesthereferenceaxis,withV=0,thevalueofthemomentatthispoin
14、tisPab/L,Whichisalocalmaximumvalue,anditisalsothemaximumlaodfortheentirebeamforthisexample.Examples-Basic loading conditionsDistributedloadRAq*Inthecaseofdistributedload,similarly,*methodofsectionsisusedtofind*expressionsforshearforceVand*bendingmomentM.Then,usingtheseexpressiontoconstruct*shearforc
15、eand*bendingmomentdigrams.Examples-Basic loading conditionsSeveralconcentratedloads4123(L-x)RBxRA xRAL-x(L-b3-x)RB1324*Whenseveralconcentratedloadsactonabeam.Itisneccessarytodividethebeaminto*segments.Eachsegmentliesbetweenthepointsofloadapplication.Andexpressionsfortheshearforcesandbendingmomentssh
16、ouldbedeterminedforeachsegmentofthebeam.Again,tofindtheshearforceandbendingmomentforeachsegment,methodofsectionsisapplied.*Herearethefree-bodydiagramsusedinthisstep.Ascanbeseeen,forthefirstandsecondsegment,left-handfreebodydiagramisselected,whileforthethirdandforthsegment,right-handfreebodydiagramis
17、used,becausefewerloadsactonthecorrespondingfreebodydiagram.Then,*expressionsforshearforceVandbendingmomentMforeachsegmentcouldbeobtainedfromtheequationsofequilibrium.Usingthoseequationstoconstructthe*shear-forceand*bending-momentdiagrams.SuperpositionApplicabletostaticallydeterminatebeamsinsmalldefo
18、rmationandlinearelasticity.Steps：PlottheV/Mdiagramsforthebeamundereachoftheexternalloads,P1,P2,P3respectively；SumupalltheV/Mdiagramsundereachload.Summationofthediagramsobtainedforeachoftheloadsactingseparately.Whenseveralloadsactonabeam,theshear-forceandbending-momentdiagramscanbeobtainedbysuperposi
19、tionmethod.*Tousethismethod,firstconstructshearforceorbendingmomentdiagramundereachload,*andthensumthesediagramstogettheone.*NotedthatSuperpositionofshear-forceandbending-momentdiagramsispermissiblebecauseshearforcesandbendingmomentsinstaticallydeterminatebeamsarelinearfunctionsoftheappliedloads.EndThatistheendofthisvideo,nextvideoreviewsthischaptershearforcesandbendingmoments.