梁彎曲的強度Strengthofbeambending主講人：王雅君123目錄CONTENTS彎曲時的應力梁的強度計算提高彎曲強度的措施StressduringbendingStrengthcalculationofbeamsMeasurestoimprovebendingstrength1彎曲時的應力StressduringbendingONE一、彎曲時的應力Stressduringbending截面只有彎矩M，而無剪力Q，稱為純彎曲。ThesectionwithonlybendingmomentMandnoshearforceQiscalledpurebending.縱向線m-m、n-n仍保持直線，但相對轉過一個角度dφ。Thelongitudinallinesm-mandn-nstillmaintainastraightline,butrelativelyrotatebyanangledφ。橫向線變為弧線，a-a縮短，b-b伸長，但仍與m-m、n-n線正交Thehorizontallinebecomesanarc,a-ashortens,b-belongates,butremainsorthogonaltothem-mandn-nlines變形后Afterdeformation一、彎曲時的應力Stressduringbending橫截面在純彎曲變形后仍保持為平面，只是繞截面內某一軸轉動了一個角度。Thecross-sectionremainsflatafterpurebendingdeformation,butrotatesbyananglearoundacertainaxiswithinthecross-section假想梁由許多縱向纖維組成，變形后上面纖維縮短，下面纖維伸長，中間必有一層纖維既不伸長又不縮短，此層稱為中性層。Ahypotheticalbeamiscomposedofmanylongitudinalfibers.Afterdeformation,theupperfibersshorten,thelowerfiberselongate,andtheremustbealayeroffibersinthemiddlethatneitherelongatesnorshortens.Thislayeriscalledtheneutrallayer.中性層與橫截面的交線稱為中性軸。Theintersectionlinebetweentheneutrallayerandthecross-sectioniscalledtheneutralaxis.一、彎曲時的應力Stressduringbending中性層neutrallayer中性軸NeutralAxis一、彎曲時的應力Stressduringbending純彎曲的兩個推論:Twocorollariesofpurebending:●平面假設:橫截面變形后仍為平面，只是繞中性軸發生轉動，距中性軸等高處，變形相等。Planarassumption:Thecross-sectionremainsflatafterdeformation,butonlyrotatesaroundtheneutralaxis,withequaldeformationataheightequaltotheneutralaxis.●橫截面上只有正應力。Thereisonlynormalstressonthecross-section.一、彎曲時的應力Stressduringbending梁橫截面上的正應力大小，與該點至中性軸的距離成正比。Themagnitudeofnormalstressonthecross-sectionofthebeamisproportionaltothedistancefromthatpointtotheneutralaxis.正應力沿截面寬度均勻分布，沿高度呈線性分布。Thenormalstressisuniformlydistributedalongthewidthofthesectionandlinearlydistributedalongtheheight.中性軸把截面分為受拉區和受壓區，最大拉應力和最大壓應力都發生在上、下邊緣處。Theneutralaxisdividesthecross-sectionintotensionandcompressionzones,withthemaximumtensilestressandmaximumcompressivestressoccurringattheupperandloweredges.一、彎曲時的應力Stressduringbending橫截面上的應力分布Stressdistributiononthecross-section(a)M>0(b)M<0一、基本概念Basicconcepts梁的橫截面上的最大正應力:對整個等截面桿件來說，最大正應力發生在彎矩最大的截面上，其值為Themaximumnormalstressonthecross-sectionofthebeam:Fortheentireuniformcross-sectionmember,themaximumnormalstressoccursonthesectionwiththehighestbendingmoment,anditsvalueis

Wz——抗彎截面系數，與橫截面的幾何尺寸和形狀有關(單位:mm3或m3)Wz-Bendingsectioncoefficient,relatedtothegeometricsizeandshapeofthecross-section(unit:mm3Orm3)一、基本概念BasicconceptsRectangularcross-section矩形截面圓形截面circularcrosssectionCircularcross-section圓環形截面

2梁的強度計算StrengthcalculationofbeamsTWO二、梁的強度計算Strengthcalculationofbeams梁的最大正應力不超過材料的許用正應力，最大切應力不超過材料的許用切應力。梁的正應力強度占主要地位。Themaximumnormalstressofthebeamshallnotexceedtheallowablenormalstressofthematerial,andthemaximumshearstressshallnotexceedtheallowableshearstressofthematerial.Thenormalstressintensityofthebeamdominates.二、梁的強度計算Strengthcalculationofbeams梁彎曲的強度條件:Strengthconditionsforbeambending:

二、梁的強度計算Strengthcalculationofbeams

一、基本概念Basicconcepts例:簡支梁如圖所示。已知F=10kN，作用點為梁的中點C，梁的長度為4m，材料的許用應力[σ]=160MPa。試求:梁A、B端的約束力；畫出彎矩圖；確定橫截面的尺寸，a≥?Example:Asimplysupportedbeamisshowninthefigure.ItisknownthatF=10kN,thepointofactionisthemidpointCofthebeam,thelengthofthebeamis4m,andtheallowablestressofthematerial[σ]=160MPa.Trytofind:thebindingforceattheendsofbeamAandB;Drawabendingmomentdiagram;Determinethesizeofthecross-section,a≥?(提示:正方形截面軸的抗彎截面系數w=a3/6

，a為正方形邊長。)(Reminder:Thebendingsectioncoefficientofthesquaresectionw=a2/6,aisthesidelengthofthesquare）FAyFBy(1)求約束力FindbindingforceΣMB=0FAyx4-Fx2=0ΣFY=0FAy-F+FBy=0Fay=5KN，FBy=5KN解:Solution:一、基本概念Basicconcepts(2)畫彎矩圖Drawabendingmomentdiagram彎矩方程:Bendingmomentequation:

kN·m0≤x<2kN·m2≤x≤4(3)計算截面尺寸Calculatecross-sectionaldimensions

3提高彎曲強度的措施MeasurestoimprovebendingstrengthTHREE三、提高彎曲強度的措施Measurestoimprovebendingstrength

彎曲強度是材料力學中一條非常的重要的性質，在工程問題中，常常采取一些措施來提高構件的彎曲強度，以提高構件的利用率，節約生產成本。Bendingstrengthisaveryimportantpropertyinmaterialmechanics.Inengineeringproblems,measuresareoftentakentoimprovethebendingstrengthofcomponentstoimprovetheirutilizationrateandsaveproductioncosts.三、提高彎曲強度的措施Measurestoimprovebendingstrength

以強度條件作為梁設計的主要依據，控制梁彎曲強度的主要因素是彎曲正應力，即:Thestrengthconditionisthemainbasisforbeamdesign,andthemainfactorcontrollingthebendingstrengthofthebeamisthebendingnormalstress,namely:●Wz盡可能的大Wzaslargeaspossible●Mmax盡可能的小Mmaxassmallaspossible三、提高彎曲強度的措施Measurestoimprovebendingstrength圖1Figure1圖2Figure21.降低MmaxReduceMmax(1)合理布置支座Reasonablyarrangesupports三、提高彎曲強度的措施Measurestoimprovebendingstrength(a)門式起重機大梁Gantrycranebeam(b)鍋爐筒體BoilerSimplified合理布置支座位置減小最大彎矩Reasonablyarrangethesupportpositiontoreducethemaximumbendingmoment三、提高彎曲強度的措施Measurestoimprovebendingstrength(2)合理布置載荷Reasonablyarrangetheload圖1Figure1圖2Figure2圖3Figure3三、提高彎曲強度的措施Measurestoimprovebendingstrength2.增大WzIncreaseWz梁承受的最大彎矩，與Wz成正比。ThemaximumbendingmomentthatthebeambearsisdirectlyproportionaltoWz.一、基本概念Basicconcepts2.增大WzIncreaseWz(1)合理放置截面Reasonablyplacethecross-sectionWz=bh2/6Wz=b2h/6b<h(2)合理設計截面Reasonabledesignofcross-section越大，截面的形狀就較為經濟、合理。Thelargerthesize,themoreeconomicalandreasonablethe