弯曲正应力公式

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1、StressesinBeamsCHAPTER12STRESSESINBEAMS12.1TheFlexureFormula12.1弯曲正应力公式Thebeamisconsideredtobepurebendingwhenit当梁上只有弯矩而没issubjectedtoabendingmomentwithouteither有剪力和轴力作用时为纯shearingoraxialforces.Further,thebeamhasa弯曲。并且梁的横截面不变uniformcrosssection,thecrosssectionissymmetric化，有竖直方向的对称轴，aboutavert

2、icalaxis,theloadsactintheplaneof加载在对称平面内，弯曲也symmetry,andbendingtakesplaceinthesameplane.发生在同一平面内。Thebeamhasbeenloadedbycouplestoproduce力偶加载在梁上产生purebending.Afterthebendingmomentisdetermined纯弯曲。加载后确定了梁内whentheexternalcoupleswereappliedtothebeam,the的弯矩后，需要确定应力在stressdistributiononthecrosssecti

3、onofabeamneeds梁横截面的分布情况。tobedeveloped.如图12.1(a)所示的未ConsidertheundeformedbarinFigure12.1(a),变形的杆，横截面为矩形，whichhasasquarecrosssectionandismarkedwith杆上标示纵向线和横向线。longitudinalandtransversegridlines.Whenabending当加载一个弯矩后，标示线momentisapplied,ittendstodistorttheselinesintothe如图12.1(b)所示。图中可以patternshow

4、ninFigure12.1(b).Hereitcanbeseen看出，纵向线变为曲线，横thatthelongitudinallinesbecomecurvedandthe向线保持为直线但有一定verticaltransverselinesremainstraightandyetundergo的旋转。arotation.Figure12.1(a)Figure12.1(b)46StressesinBeamsThebehaviourofanydeformablebarsubjectedtoa在弯矩作用下的可变bendingmomentcausesthematerialwithint

5、hebottom形杆件变形特性使得杆件portionofthebartostretchandthematerialwithinthe的下半部材料受拉，上半部topportiontocompress.Consequently,betweenthese材料受压。因此，这两部分tworegionstheremustbeasurface,calledtheneutral材料的中间一定存在着一surface,inwhichlongitudinalfibresofthematerialwill个面，称为中性层，这层上notundergoachangeinlength.纵向的材料纤维的长度没

6、有变化。Fromtheseobservationsthefollowingthreeassumptionscanbemaderegardingthewaythestress由此可得关于应力使deformsthematerial.First,thelongitudinalaxisx,材料产生变形的三个假设。whichlieswithintheneutralsurface,doesnot第一，在中性层上的纵向坐experienceanychangeinlength.Ratherthemoment标x长度保持不变。力矩使willtendtodeformthebeamsothatthi

7、slinebecomesa梁产生弯曲变形，从而直线curvethatliesinthex-yplaneofsymmetry.Second,all变为在x-y对称平面内的曲crosssectionsofthebeamremainplaneand线。第二，变形时所有的横perpendiculartothelongitudinalaxisduringthe截面保持平面并且与纵向deformation.Andthird,anydeformationofthecross线垂直。第三，任何横截面sect