Engineering Basic Mechanics I Statics 工程基础力学Ⅰ 静力学 课件 Chapter 9 Geometric Properties of Cross Sections

StaticsStaticsofdeformablebodyChapter9

GeometricPropertiesofCrossSections9.1Staticmomentofareaandcentroid9.2Momentofinertia,productofinertiaandradiusofgyration9.3Parallelaxistheorem9.4TransformationequationandprincipalmomentofinertiaContents

9.1Staticmomentofareaandcentroid1.Staticmoment

thestaticmomentforthezandyaxesrespectively·cdAyyy0zzzNote：Staticmomentsareforcertainaxesandwillvaryfordifferentaxesforthesamesection.Thevalueofthestaticmomentcanbepositive,negative,orzero.Itsdimensionis[length]3.Thestaticmomentofacombinedsectiontoanaxisisequaltothealgebraicsumofthestaticmomentsofeachsimplefigureinthecombinedsectiontothataxis. Ifthecrosssectionisahomogeneousandequal-thicknesssheet,itscentreofgravityisthecentroid:·cdAyyy0zzz

Whenthepositionofthecentroidisknown,thestaticmomentcanbeobtainedasfollows：2.RelationshipbetweenstaticmomentofareaandcentroidThecoordinateaxispassingthroughthecentroidwithinaplaneiscalledthecentroidalaxis.·cdAyyy0zzzIfthestaticmomentofasectiontoanaxisiszero,thisaxismustbethecentroidalaxisofthecenterofthesection.Conversely,thestaticmomentofthecross-sectionalpatterntoitscentroidalaxisisequaltozero.SolutionSincethezaxisistheaxisofsymmetryandmustpassthroughthecentroidofthesection,wehave

zdAqRy0dzz0y=(2)ForthepurposeofcalculatingSytakeanarrowstripparalleltotheyaxisasthemicroareaExampleTrytofindthestaticmomentsSyandSzofthesemicircularsectionshowninfollowingpictureandthecoordinatesofcentroid.zdAqRy0dzzExampleTrytofindthestaticmomentsSyandSzofthesemicircularsectionshowninfollowingpictureandthecoordinatesofcentroid.3.StaticmomentandcentroidofcompositeareaAsectionconsistingofsimplepictures(e.g.circles,rectangles,triangles,etc.)iscalledacompositesection.Thestaticmomentofacompositesectionforagivenaxisisequaltothealgebraicsumofthestaticmomentsofthesimplepictures,i.e.substituting

intoWecanobtainfollowingequationforcalculatingthecoordinatesofthecentroidofthecompositesectionSolutioncentroidofthecrosssection:

ExampleCalculatethecentroidalaxisycandzcforaT-section.100cIIIcz20ycyz14020

9.2Momentofinertia,productofinertiaandradiusofgyration1.Momentofinertia

Itisdefinedasthemomentofinertiaorsecondmomentforthesectionwithrespecttothezandyaxes,respectively.

zdAyy0zNote：Themomentofinertiaofthesamesectionvariesfordifferentaxes,butitisalwayspositive.Thedimensionofthemomentofinertiais[length]4.Themomentofinertiaofacompositesectionforanaxisshouldbeequaltothesumofthemomentsofinertiaofeachconstituentsectionforthataxis2.ProductofinertiaItisdefinedastheproductofinertiaforthesectionwithrespecttothez,yaxes.Note：Productofinertiavaryfordifferentaxesforthesamesection.Thevalueoftheproductofinertiacanbepositive,negative,orzero.Itsdimensionis[length]4.zdAyy0z3.Radiusofgyration

or

whereiy,izisdefinedastheradiusofgyrationofthesectionwithrespecttothezandyaxis,respectively,withthedimensionoflength.4.PolarmomentsofinertiaDefinitionItisthepolarmomentofinertia

ofthesectionwithrespecttotheoriginofthecoordinates.because

Soweget

zdAyyOzconclusion：(1)Thesamesectionhasdifferentmomentsofinertiaandproductsofinertiafordifferentcoordinateaxes.(2)ThesumofthemomentsofinertiaIyandIzisconstantlyequaltothemomentofinertiaIpofthesectionattheintersectionofthesetwoaxes.①②③conclusion：(3)Iy,IzandIpareconstantlypositive,whiletheproductofinertiaIyzmayhaveanegativeorzerovalue.Theyallhavethedimensionof[length]4.(4)Theproductofinertiawithrespecttotwoorthogonalaxesisequaltozeroiftheaxesistheaxisofsymmetryofthesection.(5)Themomentofinertiaofacompositesectionforanaxisshouldbeequaltothesumofthemomentsofinertiaofeachconstituentsectionforthesameaxis.①②③ExampleCalculatethemomentofinertiaofarectangularsectionshowninpicturewithrespecttotheyandzaxesofsymmetry.czbyhdzzsolution

Takeanarrowrectangleparalleltotheyaxistobethemicro-areadA.Then

dA=bdzThemomentofinertiaofthesectionwithrespecttotheyaxisis

AsimilarmomentofinertiatothezaxisisczbyhdzzExampleCalculatethemomentofinertiaofarectangularsectionshowninpicturewithrespecttotheyandzaxesofsymmetry.DczyExampleCalculatethemomentofinertiaofacircularsectionwithrespecttoitscentroidalaxis.solutionDczyzycDdExampleCalculatethemomentofinertiaofahollowcircularsectionwithrespecttoitscentroidalaxis.solutionThemomentofinertiaofacompositesectionforanaxisshouldbeequaltothesumofthemomentsofinertiaofeachconstituentsectionforthesameaxis.soForahollowcircularsection,we

can

get9.3Parallelaxistheorem

ycandzcareorthogonalaxespassingthroughthecentroidofthesection,andareparalleltotheyandzaxesrespectively.zcdAy0cyczybcyzczaSimilarly

Theaboveequationisknownastheparallelaxistheorem.

zcdAy0cyczybcyzczaThemomentofinertiaofagraphwithrespecttoanyaxisisequaltothemomentofinertiaofthegraphwithrespecttotheaxisparalleltothataxisplustheproductoftheareaofthegraphandthesquareofthedistancebetweenthetwoparallelaxes;Theproductoftheinertiaofagraphwithrespecttoanypairofright-angledaxesisequaltotheproductoftheinertiaofthegraphwithrespecttothecentroidalaxesparalleltothataxisplustheproductoftheareaofthegraphandthespacingofthetwopairsofparallelaxes;Themomentofinertiaofthefigureisminimumforcentroid,andtheproductofinertiaobtainedbyparallelaxistheoremmayincreaseordecrease.Solution

Positionofthesection'scentroidTocalculatethemomentsofinertiaIycandIzc,firstlycalculatethemomentsofinertiaofthetworectanglesIandIIfortheycandzcaxesrespectivelyExampleCalculatethemomentofinertiaoftheT-shapedsectionshowninthepicturefortheycandzccentroidalaxes.100cIIIcz20ycyz14020100cIIIcz20ycyz14020Sothemomentofinertiaofthewholesectionwithrespecttotheycandzcaxesis100cIIIcz20ycyz14020ExampleCalculatethemomentofinertiaoftheT-shapedsectionshowninthepicturefortheycandzccentroidalaxes.9.4Transformationequationandprincipalmomentofinertia1.TransformationequationFortransformationofcoordinatesbytheaxisofrotation,wehavethen

0aaazzdAy1yz1zyy1because

Wecanget

Thesearetransformationequations.2.PrincipalaxisofinertiaandprincipalmomentofinertiaIftheproductofinertiatoapairoforthogonalaxesiszero,thispairofaxesiscalledtheprincipalaxis.Themomentofinertiaaboutaprincipalaxisiscalledtheprincipalmomentofinertia.Whentheprincipalaxisisthecentroidalaxis,itiscalledthecentroidalprincipalaxis.Themomentofinertiawithrespecttothecentroidalprincipalaxisisreferredtoasthecentroidalprincipalmomentofinertia.Ifthesectionhasanaxisofsymmetry,thisaxisisoneoftheprincipalaxesofthecentroid.Theotherprincipalaxisofthecentroidistheaxispassingthroughthesectioncentroidandperpendiculartothisaxisofsymmetry.Forthecrosssectionwithoutaxisofsymmetry,thepositionoftheprincipalaxisisdeterminedbycalculation.Substitutingα=α0

intoWehaveLet，wecangetBysubstitutingaboveequationintothefollowingtwoequationsweobtainIy0andIz0.Substitutingsin2α0

andcos2α0

intoaboveequationandfinallyobtainingthegeneralformulafortheprincipalmomentofinertiais

Iftheoriginofthezoycoordinatesystemisthecenterofthesection,theprincipalmomentofinertiacalculatedbytheaboveequationisalsothecentroidalprincipalmomentofinertiaofthesectionFurthermore,sinceIy1andIy2arecontinuousfunctionsofα,theirextremescanbefoundbyderivingtheirderivatives.wegetand

Comparisonshowsthat

Description:

Theprincipalaxisp