薄板弯曲课件

薄板弯曲*SheetBendingSummarizationSheetisdifferentfromthickboard。Generally,iftheratioofthethicknessoftheboardandtheminimaldimensionoftheboardfacesatisfies:Wecalltheboardsheet.

Choosetheordinateoriginasapointofthemiddleplane,andaxesofxandyinthemiddleplane,zperpendiculartoit,whichareshowninfig..

Wecalltheplanehalvesthethicknessoftheboardmiddleplane.*薄板弯曲概述薄板区别于厚板。通常情况下，板的厚度t与板面的最小尺寸b的比值满足如下条件:则称为薄板。

将坐标原点取于中面内的一点,x和y轴在中面内，z垂直轴向下，如图所示。

我们把平分板厚度的平面称为中面。*

Whensheetacceptscommonload,wealwaysdividetheloadintotwocomponents.Oneistransverseload,whichisperpendiculartomiddleplane,oneislongitudinalload,whichactsinmiddleplane.Forthelatter,weassumeitsdistributingisevenalongthethicknessofthesheet,andtreatitastheplanestressproblem.Inthischapter,wejustdiscussthestress、strainanddisplacementwhensheetisbentbecauseoftransverseload.SheetBending*

当薄板受有一般载荷时，总可以把每一个载荷分解为两个分量，一个是垂直于中面的横向载荷，另一个是作用于中面之内的纵向载荷。对于纵向载荷，可认为它沿薄板厚度均匀分布，按平面应力问题进行计算。本章只讨论由于横向载荷使薄板发生小挠度弯曲所引起的应力、应变和位移。薄板弯曲*§12－1BasicHypothesis

Forsmallbendingproblemofsheet,wegenerallyadopttheseassumptions:（1）fixityoftheboardthicknessNamely:anynormalwhichisperpendiculartomiddleplanehasthesamebending.（2）fixityofnormalofthemiddleplane

Thenormalstrainperpendiculartomiddleplaneisverysmall,thuswecanignoreit.Namely.Fromgeometricequations,wehave,thusweget:SheetBending*§12－1基本假设

薄板小挠度弯曲问题，通常采用如下假设：（1）板厚不变假设即：在垂直于中面的任一条法线上，各点都具有相同的挠度。（2）中面法线保持不变假设

垂直于中面方向的正应变很小，可以忽略不计。即，由几何方程得，从而有：薄板弯曲*

Thebeelinewhichisperpendiculartomiddleplanebeforedeformationisstillabeelineafterdeformation,andisstillperpendiculartothebentmiddleplane.Namely:（3）theboardsurfaceisneutrosphereNamely

Fromthegeometricfunctions:（4）thestresshasverysmalleffectondeformation,thuscanbeignored.Scilicetwethink:SheetBending*

在变形前垂直于中面的直线，变形后仍为直线，并垂直于弯曲后的中面。即（3）中面为中性层假设即由几何方程得（4）应力对变形的影响很小，可以略去不计。亦即认为薄板弯曲*§12－2BasicEquations

Solvingsheetbendingproblemintermsofdisplacement.Choosethesheetbendingasthebasicunknownquantity，andexpresstheotherphysicalquantitieswith.（1）GeometricFunction

FetchasmallrectangleABCDonthemiddleplane,asshowninfig..Itssidelengthsaredx

anddy.

Underactionoftheload,therectangleisbentto

flexuralplaneA’B’C’D’。supposethebendingatpointAis,theobliquitiesofstretchflexuralplanealongdirectionxandyareand.SheetBending*§12－2基本方程

按位移求解薄板弯曲问题。取薄板挠度为基本未知量，把所有其它物理量都用来表示。（1）几何方程

在薄板的中面上取一微小矩形ABCD如图所示。它的边长为dx和dy，载荷作用后，弯成曲面A’B’C’D’。设A点的挠度为，弹性曲面沿x和y方向的倾角分别为和，则薄板弯曲*ThebendingatpointBisThebendingatpointDis

FromandweknowOrcanbewrittenasAfterintegralbyz,andusing

，wegetThenthestraincomponentscanbeexpressedbyasSheetBending*B点的挠度为D点的挠度为

由和可知或写成对z进行积分，并利用，得于是应变分量用表示为:薄板弯曲*

Undersmalldeformation,becauseofweenybending,thecurvatureofstretchflexuralplanealongthecoordinatedirectioncanapproximativelybedenotedbybending:Thusthestraincomponentscanalsobewrittenas:SheetBending*

小变形下，由于挠度是微小的，弹性曲面在坐标方向的曲率可近似地用挠度表示为:所以应变分量又可写成薄板弯曲*（2）PhysicalFunctions

Ignoringthestraincausedby,physicalfunctionsbecome:Expressthestresscomponentswithstraincomponents,wehave:SheetBending*薄板弯曲（2）物理方程

不计所引起的应变，物理方程为：把应力分量用应变分量表示，得：*（3）DifferentialEquationforStretchFlexuralPlane

Ignoringbodyforce,fromthefirsttwoformulasofequilibriumequations,weget:

Theseformulasindicate:themainstresscomponentshavelineardistributingalongboardthickness.Expressstresscomponentswithbending,wehaveSheetBending*薄板弯曲（3）弹性曲面微分方程

在不计体力的情况下，由平衡方程的前二式得：

上式说明，主要的应力分量沿板的厚度线性分布。将应力分量用挠度表示，得：*

Substitutephysicalfunctionswhichstresscomponentsaredenotedbybendingintotheaboveformulasandcancelterms,weget

Becausethebendingdoesn’tchangealongzaxis,andtheboundaryconditionsontopsurfaceandundersurfaceofsheetare:SheetBending*薄板弯曲

将应力分量用挠度表示的物理方程代入上式，并化简得：

由于挠度不随z变化，且薄板在上下面的边界条件为：*Integratingtheabovetwoformulasonz,weget:Fromthethirdformulaofdifferentialequationsofequilibrium,weget:SubstitutingintotheexpressionwhicharedenotedbyBending,andcancelingterms,weget（1）SheetBending*薄板弯曲将前面二式对z进行积分，得：再由平衡微分方程第三式，得：将用挠度表达式代入，并化简得：（1）*Becausethebendingdoesn’tchangealongzaxis,andwehavetheboundarycondition:Integratingformula（1）onz,weget:Supposingtheloadactsonperunitareaontopsurfaceofsheetisq(includingtransversesurfaceforceandtransversebodyforce),andtheboundaryconditiononboardtopsurfaceisSubstitutetheexpressionofintotheaboveformula,wegetthedifferentialequation:SheetBending*薄板弯曲由于挠度不随z变化，且薄板有边界条件:将（1）式对z积分，得:

设在薄板顶面上每单位面积作用的载荷q(包括横向面力和横向体力），板上面的边界条件为:将的表达式代入该边界条件，得薄板挠曲微分方程:*whereWecallDbendrigidityofsheet.Thesheetbendingdifferentialequationisalsocalledstretchflexuralplanedifferentialequation,whichisthebasicdifferentialequationofsheetbendingproblems.SheetBending*薄板弯曲其中称为薄板的弯曲刚度。

薄板挠曲微分方程也称为薄板的弹性曲面微分方程，它是薄板弯曲问题的基本微分方程。*§12－3

InternalForceofCrossSection

Fetchasmallhexahedronatthecrosssectionofthesheet.Itsthreesides’lengtharerespectively,asshowninfig..Atthecrosssectionwhichisperpendiculartoxaxis,therearenormalstressandshearstress.Becausethesumofandequalstozeroalongtheboardthickness,theycanonlybesynthesizedtobendingmomentandtwistingmoment;whilecanonlybesynthesizedtotransverseshearingforce.

Obviously,atthesectionperpendiculartoxaxis,thevaluesofperunitwidthare:SheetBending*薄板弯曲§12－3横截面上的内力

在薄板横截面上取一微分六面体，其三边的长度分别为,如图所示。在垂直于x轴的横截面上，作用着正应力和剪应力。由于和在板厚上的总和为零，只能分别合成为弯矩和扭矩；而只能合成横向剪力。

显然，在垂直于x轴的横截面上，每单位宽度之值如下：*SimilarlySheetBending*薄板弯曲同理*

Substitutingintotheexpressionbetweenstresscomponentsandbending,andintegrating,weget

Theaboveformulasarecalledelasticityequationsbetweeninteriorforceanddeformationofsheetbendingproblems.SheetBending*薄板弯曲将上节给出的应力分量与挠度之间关系代入，并积分得：上式称为薄板弯曲问题中内力与变形之间的弹性方程。*

Accordingtotheexpressionbetweenstresscomponentsandbending,andthedifferentialequationsandtheelasticityequations,canceling,wegettherelationsamongstresscomponents,bendingmoment,twistingmoment,load:SheetBending*薄板弯曲

利用应力分量与挠度之间的关系、薄板挠曲微分方程以及内力与形变之间的弹性方程，消去，可以给出各应力分量与弯矩、扭矩、剪力、载荷之间的关系。*

Obviously,alongthesheetthickness,themax.ofstresscomponentsexistsattheboardsurface,themax.ofandexistatthemiddleplane,whilethemax.ofexistsattheloadactingplane.Moreover,amongthestresscomponentscausedbycertainload,havethebiggernumericalvalue,thusitisthemainstress;andhavetherelativelysmallernumericalvalues,theyarethesecondarystresses;extrusionstresshasthesmallestnumericalvalue,itisthemostunimportantstress.Thus,whenwecalculatetheinteriorforceofsheet,wemainlycalculatebendingmomentandtwistingmoment.SheetBending*薄板弯曲

显然，沿着薄板的厚度，应力分量的最大值发生在板面，和的最大值发生在中面，而之最大值发生在载荷作用面。并且，一定载荷引起的应力分量中，在数值上较大，因而是主要应力；及数值较小，是次要的应力；挤压应力在数值上最小，是更次要的应力。因此，在计算薄板的内力时，主要是计算弯矩和扭矩。*

§12－4BoundaryConditionofSheetTaketherectangleasshowninfig.forexample.1TheFixedSideSupposingOAisthefixedsupportingboundary,atwhichthebendingandthenormalslopeofstretchplaneequaltozero.namely:2TheSimpleSupportingSide

SupposingOCisthesimplesupposingboundary,atwhichthebendingandbendingmomentMySheetBending*薄板弯曲§12－4薄板的边界条件以图示矩形板为例：1固定边

假定OA

边是固支边界，则边界处的挠度和曲面的法向斜率等于零。即：2简支边

假设OC

边是简支边界，则边界处的挠度和弯矩My*equaltozero.

Namely:fromEvenatOCside:namelyThentheboundaryconditionsofsimplesupposingsideOCcanbewrittenas:SheetBending*薄板弯曲等于零。即：由于且在OC上即则简支边OC

边界条件可写成：*3TheFreeSide

CBsideisthefreeboundary,alongwhichthebendingmoment,twistingmomentandtransverseshearingstressequaltozero,namely:Becausethetwistingmomentcanbeswitchedtoequivalentshearingforce,thesecondandthethirdconditionscanbeunitedinto:SheetBending*薄板弯曲3自由边

板边CB为自由边界，则沿该边的弯矩、扭矩和横向剪应力都为零，即：由于扭矩可以变换为等效的剪力，故第二及第三个条件可合并为：*

SubstitutingintotheexpressionsamongMx、Qx、Mxyand

wegettheboundaryconditionsoffreeboundaryCB:SheetBending*薄板弯曲将Mx、Qx、Mxy与的关系代入，得自有边界CB

的边界条件为：*

§12－5

SolutionofSheetBendingunderRectangularCoordinate

Whensolvingsheetbendingproblemsintermsofdisplacement,wealwaysadopthalfconversesolution.Firstweenactaexpressionwhichcoefficientisunderconfirmforsheetbending;thenfromdifferentialfunctionandboundaryconditions,weconfirmthecoefficient;atlast,fromtheexpressionbetweenthebendingandstresscomponents,wegetthestresscomponents.Example1Trytoseekthemaximalbendingandmaximalbendingmomentofthefixedellipticsheetwhichisunderevendistributingloadq.Solution:undercoordinateasshowninfig.,theboundaryfunctionofellipticsheetisSheetBending*

薄板弯曲§12－5薄板弯曲的直角坐标求解

用位移法求解薄板弯曲问题，通常采用半逆解法。首先设定具有待定系数的薄板挠度的表达式；其次利用薄板曲面微分方程和边界条件，确定待定常数；最后由挠度与应力分量的关系，求得应力分量。例1试求边界固定的椭圆形薄板在承受均布载荷q

后的最大挠度和最大弯矩。解：在图示坐标下，椭圆薄板的边界方程为:*

Enactingtheexpressionofbendingis:WhereCisaconstant.Supposingnistheoutsidenormalofsheetboundary,thenatthesheetboundary,wehaveNoticethatObviouslytheexpressionofbendingsatisfiesthefixedboundarycondition.SheetBending*薄板弯曲设挠度的表达式为:其中C为常数。设n为薄板边界外法线，则在薄板的边界上应有:注意到显然所设挠度的表达式满足固定边界条件。*Substitutingtheexpressionofbendingintothedifferentialequation:Weget:thusTheinteriorforceSheetBending*薄板弯曲将挠度的表达式代入弹性曲面微分方程得:从而内力*

Themaximal

deflectionis:Themaximalflexuraltorqueis（supposea>b):whereSheetBending*薄板弯曲最大挠度为:最大弯矩为（设a>b):其中*

Example2

Trytosolvethemaximal

deflectionofrectangularsheet,whichisquadrangularfreely-supportedandbeareduniformload.Solution：selectthecoordinatesystemasthefigureshownAssumingthusattheboundarywherex=0andx=a,theboundaryconditions

issatisfiednaturally.SubstitutingtheexpressionofintothedifferentialequationofelasticcurvedfaceSheetBending*薄板弯曲例2、试求图示四边简支，承受均布载荷的矩形薄板之最大挠度。解：取图示坐标系设则在x=0及x=a边界上，边界条件自然满足。将的表达式代入弹性曲面微分方程*

yieldsExpandingbyFourierserieswheremisevennumbermisoddnumberthusChoosingtheparticularsolutionofdifferentialequationyields:SheetBending*

薄板弯曲得将展为傅立叶级数其中m为偶数m为奇数则取微分方程的特解为：*

andnoticingthedeflectionisevenfunctionofy,thusgeneralsolutionofinhomogeneouslinearordinarydifferentialequationis:yieldsUsingtheboundaryconditions(symmetricalcharacteristic)At

SheetBending*薄板弯曲并注意到挠度是y

的偶函数，则非齐次线性常微分方程的一般解为:利用边界条件(已用对称性）处，得*

Theexpressionofdeflectiongives:ifa=b，thusItisobviousthatwecanachievehighprecisiononlybychoosingtwoterminseries.SheetBending*薄板弯曲挠度的表达式：若a=b，则可见，在级数中仅取两项，就可以达到较高的精度。*

§12－6

AxisymmetricBendingofCircularSheetWhensolvingtheproblemsofbendingofcircularsheet,itwillbemoreconveniencetochoosepolarcoordinates.Ifthetransverseloadofcircularsheetisaboutzaxissymmetry

(axiszisverticaltosheetandtowardsunderside),thusthedisplacementofelasticsheetisalsoaboutzaxissymmetry

,i.e.isonlythefunctionofranddon’tchangeaccordingto.1.DifferentialEquationofElasticCurvedFaceReferringtothedifferentialequationofelasticcurvedfaceinrectangularcoordinate.Inpolarcoordinate,whenitistheaxisymmetricbendofcircularsheet,differentialequationofelasticcurvedfacecanbewrittenas:orSheetBending*

薄板弯曲§12－6圆形薄板的轴对称弯曲

求解圆板弯曲问题时，采用极坐标较方便。如果圆形薄板所受的横向载荷是绕z轴对称的（z轴垂直板面向下），则该弹性薄板的位移也将是绕z轴对称的，即只是r

的函数，不随而变。一、弹性曲面微分方程

参照直角坐标下的弹性曲面微分方程。极坐标下，圆形薄板轴对称弯曲时，曲面微分方程可写成：或*

2.InternalForceexpandingyields:sothegeneralsolutionofdifferentialequationiswhereisaarbitraryparticularsolution.

Fromthesheet,takeoutadifferentialcell,asthefigureshown.Inthecross-sectionwhererisconstant,momentofflexionandtransverseshearareMr

and

respectively;Inthecross-sectionwhereisconstant,theyareand.Becauseitisaxisymmetricproblem,thereisnotmomentoftorsion.SheetBending*

薄板弯曲二、内力展开后得：该微分方程的通解为其中是任意一个特解。

从薄板内取出一个微分单元体，图示。在r为常量的横截面上，弯矩和横向剪力分别为Mr

和；在为常量的横截面上，则为和。由于是轴对称问题，故没有扭矩。*

Evolvexaxisandyaxistothedirectionofrandthedirectionofofthisdifferentialcellrespectively,andusingcoordinatetransformationformula,wehave:SheetBending*

薄板弯曲

把x轴和y轴分别转到这个微分单元体的r和方向，则利用坐标转换公式，有：*

3.StressComponentUsingcoordinatetransformationformula，similarlygives:UsinginternalforcetodenotestresscomponentgivesSheetBending*

薄板弯曲三、应力分量利用坐标转换公式，同理有：将应力分量用内力表示有：*Example3

Givensolidcircularsheet,radiusisa,circumferenceisfixedanditisunderuniformloadandconcentratedforcepwhereitisincentreofcircle.Solution:Accordingtotheknowncondition,thisisaxisymmetricbendofCircularSheet,lineofdeflectionfunctionisspecialsolutioniswefindgeneralsolutionisFortheflexivityofcenterofsolidcircularsheet,weknow

Fromthesheet,takeoutasmallsheetwhoseradiusisr,fortheequilibriumconditionofzdirectiongives

SheetBending*薄板弯曲例3、半径为a的实心圆板，周边固支，受均布载荷及圆心处的集中力P作用，求挠度。解：由题意知，本题为圆板轴对称弯曲，挠曲线方程为：取特解知通解为由实心圆板中心处的挠度应有界知：从板中取出半径为r的部分圆板，由z方向的平衡条件给出*soFurthermorewehavesoFor

yieldsforyieldsSothedeflectionofsheetisSheetBending*薄板弯曲故而又有故由得由得故板的挠度*§12－7SolutionoftheDisplacementofSheetbyCalculusofVariationWhensheetisatbendingofsmalldeflection,ispaucity,whichcanbeomitted.SodeformationenergyofelasticsheetisUsingdeflectiontodenote:SheetBending*薄板弯曲§12－7变分法求薄板的位移

薄板小挠度弯曲时，为微量，可略去不计。此时弹性薄板的变形能:用挠度表示:*

whereAistheareaofsheet.Toarbitraryshapesheetwhoseedgesarefixedandpolygon(notholeinthesheet)whereattheboundaryofsheet,forintegrationbypartsformulasyields:Tofixedsheet,i.e.Torectangularsheetwherealongtheboundary，alwayshaveorthusSheetBending*

薄板弯曲其中A为薄板面积。

对于板边固定的任意形状板，以及板边界处的多边形（板中无孔洞），由分步积分公式得:对于固定板，即对于沿板边的矩形板，总有或因此*sodeformationenergyofelasticsheetissimplifiedasExample4Evaluatethedeflectionofsimplysupportedrectangularsheet,thatisunderuniformload.Solution:UsingRitzmethod.Thedeflectionofplateistrigonometricseriesasfollows:

Itisobvious,everytermofthisseriesissatisfiedwiththeboundaryconditionsofquadrilateralsimplysupported.Theelasticdeformationenergyofplateis：SheetBending*薄板弯曲即弹性板的变形能简化为：例4求四边简支矩形板在均布载荷作用下的挠度。解：用里兹法。取板的挠度为如下重三角级数显然，该级数的每一项都满足四边简支的边界条件。板的弹性变形能：*

Undertheuniformload,potentialenergyVofexternalforcegives

Totalpotentialenergy:FortheconditionsofⅡachievingtheextremevalue,yields（m,nareodd）SheetBending*

薄板弯曲在均布载荷作用下，外力势能V为总位能:由Ⅱ取极值的条件得出：（m,n均为奇数）*

thuswegetso（m,nareodd）(morniseven）SheetBending*

薄板弯曲由此得出故（m,n均为奇数）(m或n为偶数时）*Exercise12.1

Givenrectangularsheet,OAisfixed,OCissimplysupported,ABandBCarefree.AngularpointBissupportedbychainbar,theloadappliedtotheedgesofsheetisasthefigureshown.Trytousedeflectiontodenotetheboundaryconditionoftheedgesofsheet.xyzM0qoACBabSolution：（1）OAedge（2）OCedgethelatterequationisdenotedbydeflection,givesSheetBending*薄板弯曲练习12.1

矩形薄板具有固定边OA，简支边OC及自由边AB和BC，角点B处有链杆支承，板边所受荷载如图所示。试将板边的边界条件用挠度表示。xyzM0qoACBab解：（1）OA边（2）OC边后一式用挠度表示为*（3）ABedgeusedeflectiontodenote,gives