真空-堆载联合预压蠕变机理及微观结构特征试验研究课件

CHAPTER4

INTERNALFORCESMechanicsofMaterialsCHAPTER4MechanicsofMater1第四章弯曲内力材料力学第四章弯曲内力材料力学2§4–1Conceptsofplanarbendingandcalculationsketchofthebeam§4–2Theshearingforceandbendingmomentofthebeam§4–3Theshearing-forceandbending-momentequations·theshearing-forceandbending-momentdiagrams§4–4Relationsamongtheshearingforce、thebendingmomentandthedensityofthedistributedloadandtheirapplications§4–5Plotthebending-momentdiagrambythetheoremofsuperpositiom§4–6Theinternal-forcediagramsoftheplanarrigidtheoremframesandcurvedrodsExerciselessonsabouttheinternalforceofbendingCHAPTER4INTERNALFORCESINBENDING

§4–1Conceptsofplanarbend3§4–1平面弯曲的概念及梁的计算简图§4–2梁的剪力和弯矩§4–3剪力方程和弯矩方程·剪力图和弯矩图§4–4剪力、弯矩与分布荷载集度间的关系及应用§4–5按叠加原理作弯矩图§4–6平面刚架和曲杆的内力图弯曲内力习题课第四章弯曲内力§4–1平面弯曲的概念及梁的计算简图第四章4

§4–1

CONCEPTSOFPLANARBENDINGANDCALCULATIONSKETCHOFTHEBEAM1、CONCEPTSOFBENDING1).BENDING:Theactionoftheexternalforceorexternalthecouplevectorperpendiculartotheaxisoftherodmakestheaxisoftherodchangeintocurvefromoriginalstraightlines,thisdeformationiscalledbending.2).BEAM：Thememberofwhichthedeformationismainlybendingisgenerallycalledbeam.INTERNALFORCESINBENDING§4–1CONCEPTSOFPLANARBEN5弯曲内力§4–1平面弯曲的概念及梁的计算简图一、弯曲的概念1.弯曲:杆受垂直于轴线的外力或外力偶矩矢的作用时,轴线变成了曲线，这种变形称为弯曲。2.梁：以弯曲变形为主的构件通常称为梁。弯曲内力§4–1平面弯曲的概念及梁的计算简图一、弯曲63).PracticalexamplesinengineeringaboutbendingINTERNALFORCESINBENDING3).Practicalexamplesinengin73.工程实例弯曲内力3.工程实例弯曲内力8INTERNALFORCESINBENDINGINTERNALFORCESINBENDING9弯曲内力弯曲内力104).Planarbending：Afterdeformationthecurvedaxisofthebeamisstillinthesameplanewiththeexternalforces.

Symmetricbending（asshowninthefollowingfigure）—a specialexampleoftheplanarbending.TheplaneofsymmetryMP1P2qINTERNALFORCESINBENDING4).Planarbending：Afterdeform11弯曲内力4.平面弯曲：杆发生弯曲变形后，轴线仍然和外力在同一平面内。对称弯曲（如下图）——平面弯曲的特例。纵向对称面MP1P2q弯曲内力4.平面弯曲：杆发生弯曲变形后，轴线仍然和外力在同12Unsymmetricalbending—ifabeamdoesnotpossessanyplaneofsymmetry,ortheexternalforcesdonotactinaplaneofsymmetryofthebeamwithsymmetricplanes,thiskindofbendingiscalledunsymmetricalbending.Inlaterchapterswewillmainlydiscussthebendingstressesanddeformationsofthebeamundersymmetricbending.INTERNALFORCESINBENDINGUnsymmetricalbending—ifabe13弯曲内力非对称弯曲——若梁不具有纵对称面，或者，梁虽具有纵对称面但外力并不作用在对称面内，这种弯曲则统称为非对称弯曲。下面几章中，将以对称弯曲为主，讨论梁的应力和变形计算。弯曲内力非对称弯曲——若梁不具有纵对称面，或者，梁虽具有纵142、Calculationsketchofthebeam

Ingeneralsupportsandexternalforcesofthebeamareverycomplex.Weshoulddosomenecessarysimplificationforthemforourconvenientcalculationandobtainthecalculationsketch.1).Simplificationofthebeams

Ingeneralcasewetaketheplaceofthebeambyitsaxis.2).Simplificationoftheloads

Theloads(includingthereaction)actingonthebeammaybereducedintothreetypes：concentratedforce、concentratedforcecoupleanddistributedforce.3).SimplificationofthesupportsINTERNALFORCESINBENDING2、Calculationsketchofthebe15弯曲内力二、梁的计算简图梁的支承条件与载荷情况一般都比较复杂，为了便于分析计算，应进行必要的简化，抽象出计算简图。1.构件本身的简化通常取梁的轴线来代替梁。2.载荷简化作用于梁上的载荷（包括支座反力）可简化为三种类型：集中力、集中力偶和分布载荷。3.支座简化弯曲内力二、梁的计算简图梁的支承条件与载荷情16①Fixedhingedsupport2constraints，1degreeoffreedom.Suchasthefixedhingedsupportunderbridges，thrustballbearingetc.②Movablehingedsupport1constraint，2degreeoffreedom.Suchasthemovablehingedsupportunderthebridge，ballbearingetc.INTERNALFORCESINBENDING①Fixedhingedsupport②Movable17弯曲内力①固定铰支座2个约束，1个自由度。如：桥梁下的固定支座，止推滚珠轴承等。②可动铰支座1个约束，2个自由度。如：桥梁下的辊轴支座，滚珠轴承等。弯曲内力①固定铰支座②可动铰支座18③Rigidlyfixedend

3constraints，0degreeoffreedom.Suchasthesupportofdivingboardattheswimmingpool，supportofthelowerendofawoodenpole.XAYAMA4)Threebasistypesofbeams①Simplebeam(orsimplysupportedbeam)M—Concentratedforcecoupleq(x)—Distributedforce②CantileverbeamINTERNALFORCESINBENDINGA③RigidlyfixedendXAYAMA4)Th19弯曲内力③固定端3个约束，0个自由度。如：游泳池的跳水板支座，木桩下端的支座等。XAYAMA4.梁的三种基本形式①简支梁M—集中力偶q(x)—分布力②悬臂梁弯曲内力③固定端XAYAMA4.梁的三种基本形式①简支20③Overhangingbeam—ConcentratedforcePq—Uniformlydistributedforce5).StaticallydeterminateandstaticallyindeterminatebeamsStaticallydeterminatebeams：Reactionsofthebeamcanbedeterminedonlybystaticequilibriumequations，suchastheabovethreekindsofbasicbeams.Staticallyindeterminatebeams：Reactionsofthebeamcannotbedeterminedoronlypartofreactionscanbedeterminedbystaticequilibriumequations.INTERNALFORCESINBENDING③Overhangingbeam—Concentrated21弯曲内力③外伸梁—集中力Pq—均布力5.静定梁与超静定梁静定梁：由静力学方程可求出支反力，如上述三种基本形式的静定梁。超静定梁：由静力学方程不可求出支反力或不能求出全部支反力。弯曲内力③外伸梁—集中力Pq—均布力5.静定梁与超静定22Example1Astocktankisshowninthefigure.Itslengthis

L=5m，itsinsidediameterisD=1m，thicknessofitswallist=10mm.Densityofsteelis7.8g/cm³.Densityoftheliquidis1g/cm³.Heightoftheliquidis0.8m.Lengthofoverhangingendis1m.Trytodeterminethecalculationsketchofthestocktank.Solution：q—UniformlyDistributedforceINTERNALFORCESINBENDINGExample1Astocktanki23弯曲内力[例1]贮液罐如图示，罐长L=5m，内径D=1m，壁厚t=10mm，钢的密度为：7.8g/cm³，液体的密度为：1g/cm³,液面高

0.8m，外伸端长1m，试求贮液罐的计算简图。解：q—均布力弯曲内力[例1]贮液罐如图示，罐长L=5m，内径D=1m，24q—UniformlyDistributedforceINTERNALFORCESINBENDINGq—UniformlyINTERNALFORCESIN25弯曲内力q—均布力弯曲内力q—均布力26§4–2THESHEARINGFORCEANDBENDINGMOMENTOFTHEBEAM1、Internalforceinbending：Example

KnowingconditionsareP，a，l,asshowninthefigure.DeterminetheinternalforcesonthesectionatthedistancextotheendA.PaPlYAXARBAABBSolution：①Determine externalforcesINTERNALFORCESINBENDING§4–2THESHEARINGFORCEAN27§4–2梁的剪力和弯矩一、弯曲内力：弯曲内力[举例]已知：如图，P，a，l。

求：距A端x处截面上内力。PaPlYAXARBAABB解：①求外力§4–2梁的剪力和弯矩一、弯曲内力：弯曲内力[举28ABPYAXARBmmx②Determineinternalforces— methodofsectionAYAQMRBPMQInternalforcesofthebeaminbendingShearingforceBendingmoment1).Bendingmoment：M

Momentoftheinternalforcecouplewiththeactingplaneinthecross-sectionperpendiculartothesectionwhenthebeamisbending.CCINTERNALFORCESINBENDINGABPYAXARBmmx②Determineinterna29ABPYAXARBmmx弯曲内力②求内力——截面法AYAQMRBPMQ∴弯曲构件内力剪力弯矩1.弯矩：M

构件受弯时，横截面上其作用面垂直于截面的内力偶矩。CCABPYAXARBmmx弯曲内力②求内力——截面法AYAQM302).Shearingforce：Q

Internalforcewhichtheactinglineinthecross-sectionparalleltothesection,whenthebeamisbending.3).Signconventionsfortheinternalforces:①Shearingforce

Q:

Itispositivewhenitresultsinaclockwiserotationwithrespecttotheobjectunderconsideration,otherwiseitisnegative.②BendingmomentM：Itispositivewhenittendstobendtheportionconcaveupwards,otherwiseitisnegative.Q(+)Q(–)Q(–)Q(+)M(+)M(+)M(–)M(–)INTERNALFORCESINBENDING2).Shearingforce：Q3).Signco31弯曲内力2.剪力：Q

构件受弯时，横截面上其作用线平行于截面的内力。3.内力的正负规定:①剪力Q:绕研究对象顺时针转为正剪力；反之为负。②弯矩M：使梁变成凹形的为正弯矩；使梁变成凸形的为负弯矩。Q(+)Q(–)Q(–)Q(+)M(+)M(+)M(–)M(–)弯曲内力2.剪力：Q3.内力的正负规定:①剪力Q:绕研32Example2：Determinetheinternalforcesactingonsections1—1and2—2sectionasshowninfig.(a).Solution：Determineinternalforces bythemethodofsection.

Freebodydiagramoftheleftportionofsection

1—1isshowninfig.（b）.Fig.（a）2、ExamplesqqLab1122qLQ1AM1Fig.（b）x1INTERNALFORCESINBENDINGExample2：Determinetheintern33[例2]：求图（a）所示梁1--1、2--2截面处的内力。xy解：截面法求内力。

1--1截面处截取的分离体

如图（b）示。图（a）二、例题qqLab1122qLQ1AM1图（b）x1弯曲内力[例2]：求图（a）所示梁1--1、2--2截面处的内力。x34Freebodydiagramoftheleftportionof

section2—2isshowninfig.（b）.xy图（a）qqLab1122qLQ2BM2x2图（c）INTERNALFORCESINBENDINGFreebodydiagramoftheleft352--2截面处截取的分离体如图（c）xy图（a）qqLab1122qLQ2BM2x2弯曲内力图（c）2--2截面处截取的分离体如图（c）xy图（a）qqLab1361.Internal-forceequations:Expressionsthatshowtheinternalforcesasfunctionsofthepositionxofthesection..2.Theshearing-forceandbending-momentdiagrams:)(xQQ=Shearingforceequation)(xMM=Bendingmomentequation)(xQQ=Shearing-forcediagramsketchoftheshearing-forceequation)(xMM=BendingMomentdiagramsketchofthebending-momentequation§4–3THESHEARING-FORCEANDBENDING-MOMENTEQUATIONS THESHEARING-FORCEANDBENDING-MOMENTDIAGRAMSINTERNALFORCESINBENDING1.Internal-forceequations:E37弯曲内力1.内力方程：内力与截面位置坐标（x）间的函数关系式。2.剪力图和弯矩图：)(xQQ=剪力方程)(xMM=弯矩方程)(xQQ=剪力图的图线表示)(xMM=弯矩图的图线表示§4–3剪力方程和弯矩方程·剪力图和弯矩图弯曲内力1.内力方程：内力与截面位置坐标（x）间的函数关系38Example3Determinetheinternal-forceequationsandplotthediagramsofthebeamshowninthefollowingfigure.Solution：①Determinethe reactionsofthesupports②Writeouttheinternal- forceequationsP③Plottheinternal- forcediagrams

Q(x)M(x)xxP–PLYOLM(x)xQ(x)MOINTERNALFORCESINBENDING⊕○Example3Determinetheinter39弯曲内力[例3]求下列各图示梁的内力方程并画出内力图。解：①求支反力②写出内力方程PYOL③根据方程画内力图M(x)xQ(x)Q(x)M(x)xxP–PLMO⊕○弯曲内力[例3]求下列各图示梁的内力方程并画出内力图。解：40Solution：①Writeoutthe internal-forceequations②Plottheinternal- forcediagramLqM(x)xQ(x)Q(x)x–qLINTERNALFORCESINBENDINGM(x)x⊕○Solution：①Writeoutthe inte41弯曲内力解：①写出内力方程②根据方程画内力图LqM(x)xQ(x)Q(x)xM(x)x–qL⊕○弯曲内力解：①写出内力方程②根据方程画内力图LqM(x)xQ42Solution：①Determinethe reactionsofthesupports②Writeouttheinternal- forceequationsq0RA③Plottheinternal- forcediagrams

RBLxQ(x)xM(x)INTERNALFORCESINBENDING⊕⊕○Solution：①Determinethe react43弯曲内力解：①求支反力②内力方程q0RA③根据方程画内力图RBLxQ(x)xM(x)⊕⊕○弯曲内力解：①求支反力②内力方程q0RA③根据方程画内力图R441、Relationsamongtheshearingforce、 thebendingmomentandthethe distributedloadByanalysisoftheequilibriumoftheinfinitesimallength

dx，wecanget§4–4

RELATIONSAMANGTHESHEARINGFORCE,THEBENDINGMOMENTANDTHEINDENSITYOFTHEDISTRIBUTEDLOADANDTHEIRAPPLICATIONS

dxxq(x)q(x)M(x)+dM(x)Q(x)+dQ(x)Q(x)M(x)dxAyINTERNALFORCESINBENDINGSlopeofthetangentiallineatapointintheshearing-forcediagramisequaltotheintensityofthedistributedloadatthesamepoint.1、Relationsamongtheshearing45弯曲内力一、剪力、弯矩与分布荷载间的关系对dx

段进行平衡分析，有：§4–4剪力、弯矩与分布荷载集度间的关系及应用dxxq(x)q(x)M(x)+dM(x)Q(x)+dQ(x)Q(x)M(x)dxAy剪力图上某点处的切线斜率等于该点处荷载集度的大小。弯曲内力一、剪力、弯矩与分布荷载间的关系对dx段进行平衡46q(x)M(x)+dM(x)Q(x)+dQ(x)Q(x)M(x)dxAySlopeofthetangentiallineatapointinthebending-momentdiagramisequaltothemagnitudeoftheshearingforceatthesamepoint.Relationbetweenthebendingmomentandtheindensityofthedistributedload：INTERNALFORCESINBENDINGq(x)M(x)+dM(x)Q(x)+dQ(x)Q(x)47弯曲内力q(x)M(x)+dM(x)Q(x)+dQ(x)Q(x)M(x)dxAy弯矩图上某点处的切线斜率等于该点处剪力的大小。弯矩与荷载集度的关系是：弯曲内力q(x)M(x)+dM(x)Q(x)+dQ(x)482、Relationsbetweentheshearingforce、thebendingmomentandtheexternalloadExternalforceNoexternal-forcesegmentUniform-loadsegmentConcentratedforceConcentratedcoupleq=0q>0q<0CharacteristicsofQ-diagramCharacteristicsofM-diagramCPCmHorizontalstraightlinexQQ>0QQ<0xInclinedstraightlineIncreasingfunctionxQxQDecreasingfunctionxQCQ1Q2Q1–Q2=PSuddenchangefromthelefttorightxQCNochangeInclinedstraightlinexMIncreasingfunctionxMDecreasingfunctioncurvesxMTomb-likexMBasin-likeFlexfromthelefttotheright

SuddenchangefromthelefttotherightOppositetomxMFlexoppositetoPMxM1M2INTERNALFORCESINBENDING2、Relationsbetweenthesheari49二、剪力、弯矩与外力间的关系外力无外力段均布载荷段集中力集中力偶q=0q>0q<0Q图特征M图特征CPCm水平直线xQQ>0QQ<0x斜直线增函数xQxQ降函数xQCQ1Q2Q1–Q2=P自左向右突变xQC无变化斜直线xM增函数xM降函数曲线xM坟状xM盆状自左向右折角

自左向右突变与m反弯曲内力xM折向与P反向MxM1M2二、剪力、弯矩与外力间的关系外力无外力段均布载荷段集中力集中50Simplemethodtoplotthediagram:Themethodtoplotthediagramsbyusingtherelationbetweentheinternalforcesandtheexternalforcesandvaluesoftheinternalforcesatsomespecialpoints.Example4

Plottheinternalforcediagramsofthebeamsshowninthefollowingfiguresbythesimplemethodtoplotthediagram.Solution:Specialpoints:aaqaqAPlotthediagrambyusingtherelationbetweentheinternalforcesandtheexternalforcesandtheinternalforcevaluesatsomespecialpointsofthebeam.Endpoint、partitionpoint（thepointatwhichexternalforceschanged）andstationarypoint

etc.INTERNALFORCESINBENDINGSimplemethodtoplotthediag51弯曲内力简易作图法:利用内力和外力的关系及特殊点的内力值来作图的方法。[例4]

用简易作图法画图示梁的内力图。解:利用内力和外力的关系及特殊点的内力值来作图。特殊点:端点、分区点（外力变化点）和驻点等。aaqaqA弯曲内力简易作图法:利用内力和外力的关系及特殊点的内力值来52aaqaqALeftend：Shapeofthecurveisdeterminedaccordingto

；；Andthelawofthepointactedbyconcentratedforce.Partitionpoint

A：StationarypointofM

：Rightend：Qxqa2–qa–xMINTERNALFORCESINBENDINGaaqaqALeftend：Shapeofthecu53弯曲内力aaqaqA左端点：线形：根据；；及集中载荷点的规律确定。分区点A：M的驻点：右端点：Qxqa2–qa–xM弯曲内力aaqaqA左端点：线形：根据；；及集中载荷点的规律54Example5

Plottheinternal-forcediagramsofthebeamsshowninthefollowingfiguresbythesimplemethodtoplotthediagram.Solution：DeterminereactionsLeftendA：

RightofpointB：LeftofpointC：StationarypointofM：RightofpointC：RightendD：qqa2qaRARDQxqa/2qa/2qa/2––+ABCDqa2/2xMqa2/2qa2/23qa2/8–+

LeftofpointB：INTERNALFORCESINBENDINGExample5Plottheinternal-fo55弯曲内力[例5]用简易作图法画下列各图示梁的内力图。解：求支反力左端点A：B点左：B点右：C点左：M的驻点：C点右：右端点D：qqa2qaRARDQxqa/2qa/2qa/2––+ABCDqa2/2xMqa2/2qa2/23qa2/8–+弯曲内力[例5]用简易作图法画下列各图示梁的内力图。解：求56§4–5PLOTTHEDIAGRAMOFBENDINGMOMENTBYTHE THEOREMOFSUPERPOSITIOM1、Theoremofsuperposition：

Internalforcesinthestructureduetosimultaneousactionofmanyforcesareequaltoalgebraicsumoftheinternalforcesduetoseparateactionofeachforce.Applyingcondition：Relationbetweentheparameters(internalforces、stresses、displacements）andtheexternalforcesmustbelinear,thatistheysatisfyHooke’slaw.INTERNALFORCESINBENDING§4–5PLOTTHEDIAGRAMOFBEN57弯曲内力§4–5按叠加原理作弯矩图一、叠加原理：

多个载荷同时作用于结构而引起的内力等于每个载荷单独作用于结构而引起的内力的代数和。适用条件：所求参数（内力、应力、位移）必然与荷载满足线性关系。即在弹性限度内满足虎克定律。弯曲内力§4–5按叠加原理作弯矩图一、叠加原理：

582、Structuralmembersinmechanicsofmaterialisofsmalldeformationandlinearelasticity,andmustobeythisprinciple——methodofsuperpositionSteps：①Plotrespectivelythediagramofthebendingmomentofthebeamundertheseparateactionofeachexternalload；②Sumupthecorrespondinglongitudinalcoordinates(Attention:donotsimplypiecetogetherfigures.）INTERNALFORCESINBENDING2、Structuralmembersinmechan59弯曲内力二、材料力学构件小变形、线性范围内必遵守此原理——叠加方法步骤：①分别作出各项荷载单独作用下梁的弯矩图；②将其相应的纵坐标叠加即可（注意：不是图形的简单拼凑）。弯曲内力二、材料力学构件小变形、线性范围内必遵守此原理步骤：60Example6Plotthediagramofbendingmomentbytheprincipleofsuperposition.

(AB=2a，forcePisactingatthemiddlepointofthebeamAB.）PqqP=+AAABBBxM2xM1xM

+++=+INTERNALFORCESINBENDINGExample6Plotthediagramof61弯曲内力[例6]按叠加原理作弯矩图(AB=2a，力P作用在梁AB的中点处）。qqPP=+AAABBBxM2xM1xM

+++=+弯曲内力[例6]按叠加原理作弯矩图(AB=2a，力P作用在梁62

3、Applicationsofsymmetryandantisymmetry：

ForthesymmetricstructureundertheactionofsymmetricloadsthediagramofitsshearingstressQisantisymmetricandthediagramofthebendingmomentMissymmetric.ForthesymmetricstructureundertheactionofantisymmetricloadsthediagramofitsshearingstressQissymmetricandthediagramofthebendingmomentMisantisymmetric.

INTERNALFORCESINBENDING3、Applicationsofsymmetryan63弯曲内力

三、对称性与反对称性的应用：

对称结构在对称载荷作用下，Q图反对称，M图对称；对称结构在反对称载荷作用下，Q图对称，M图反对称。弯曲内力三、对称性与反对称性的应用：

对称64Example7Plotinternal-forcediagramsofthebeamsshowninthe followingfigure.PPLPPLLLLLLL0.5P0.5P0.5P0.5PP0QxQ1xQ2x–0.5P0.5P0.5P–+–PINTERNALFORCESINBENDINGExample7Plotinternal-for65弯曲内力[例7]作下列图示梁的内力图。PPLPPLLLLLLL0.5P0.5P0.5P0.5PP0QxQ1xQ2x–0.5P0.5P0.5P–+–P弯曲内力[例7]作下列图示梁的内力图。PPLPPLLLLL66PPLPPLLLLLLL0.5P0.5P0.5P0.5PP0MxM1xM2x0.5PLPL0.5PL–++0.5PL+INTERNALFORCESINBENDINGPPLPPLLLLLLL0.5P0.5P0.5P0.5PP067弯曲内力PPLPPLLLLLLL0.5P0.5P0.5P0.5PP0MxM1xM2x0.5PLPL0.5PL–++0.5PL+弯曲内力PPLPPLLLLLLL0.5P0.5P0.5P0.68Example8Correctthemistakesinthefollowinginternal-forcediagrams.a2aaqqa2ABQxxM––++qa/4qa/43qa/47qa/4qa2/449qa2/323qa2/25qa2/4INTERNALFORCESINBENDINGRARBExample8Correctthemistak69弯曲内力[例8]改内力图之错。a2aaqqa2ABQxxM––++qa/4qa/43qa/47qa/4qa2/449qa2/323qa2/25qa2/4弯曲内力[例8]改内力图之错。a2aaqqa2ABQxxM70Example9KnowingQ-diagram,determineexternalloadsandM-diagram(Thereforenoconcentratedforcecouplesactedonthebeam).M(kN·m)Q(kN)x1m1m2m2315kN1kNq=2kN/m+–+x+111.25–INTERNALFORCESINBENDINGExample9KnowingQ-diagram,71弯曲内力[例9]已知Q图，求外载及M图（梁上无集中力偶）。Q(kN)x1m1m2m2315kN1kNq=2kN/m+–+M(kN·m)x+111.25–弯曲内力[例9]已知Q图，求外载及M图（梁上无集中力偶）。72§4–6

THEINTERNAL-FORCEDIAGRAMSOFTHEPLANARRIGIDFRAMESANDCURVEDRODS1、Planarrigidframe1).Planarrigidframe：Structuremadefromrodsofdifferentdirectionthataremutuallyconnectedinrigidityattheirendsinthesameplane.Characteristics：ThereareinternalforcesQ,MandNineachrod.2).Conventionstoplotdiagramofinternalforces：Bending-momentdiagram：Plotitatthesidewherefibersareelongatedandnotmarkthesignofpositiveornegative.

Shearing-forceandaxial-forcediagrams：Maybeplottedatanysideoftheframe（Incommonthediagramwithpositivevalueisplottedoutsidetheframe），butmustmarkthesignsofpositiveandnegative.INTERNALFORCESINBENDING§4–6THEINTERNAL-FORCEDIA73弯曲内力§4–6平面刚架和曲杆的内力图一、平面刚架1.平面刚架：同一平面内，不同取向的杆件，通过杆端相互刚性连接而组成的结构。特点：刚架各杆的内力有：Q、M、N。2.内力图规定：弯矩图：画在各杆的受拉一侧，不注明正、负号。

剪力图及轴力图：可画在刚架轴线的任一侧（通常正值画在刚架的外侧），但须注明正、负号。弯曲内力§4–6平面刚架和曲杆的内力图一、平面刚架174Example10Trytoplottheinternal-forcediagramsoftherigidframeshowninthefigure.P1P2alABC–N-diagramQ-diagramP2+P1+P1M-diagramP1aP1aP1a+P2lINTERNALFORCESINBENDINGExample10Trytoplotthei75弯曲内力[例10]试作图示刚架的内力图。P1P2alABC–N图P2+Q图P1+P1P1aM图P1aP1a+P2l弯曲内力[例10]试作图示刚架的内力图。P1P2alAB76

Example11Asshowninthefigure,PandRareknown,trytoplotinternalforcediagramsofQ,MandN.OPRqmmxSolution：setuppolarcoordinates，OisthepoleandOBis

polaraxis，qdenotesthepositionofthesectionm-m.AB2、Planarrod：Rodthattheaxisisofaplanarcurve.

Methodtoplotinternal-forcediagramofacurvedrodisthesameasthatoftheplanarrigidframe.INTERNALFORCESINBENDINGExample11Asshowninth77弯曲内力二、平面曲杆：轴线为一平面曲线的杆件。

内力情况及绘制方法与平面刚架相同。[例11]已知：如图所示，P及R

。试绘制Q、M、N图。OPRqmmx解：建立极坐标，O为极点，OB

极轴，q表示截面m–m的位置。AB弯曲内力二、平面曲杆：轴线为一平面曲线的杆件。[例11]78OPRqmmxABABOM图OO+Q图N图2PRPP–+INTERNALFORCESINBENDINGOPRqmmxABABOM图OO+Q图N图2PRPP–+IN79弯曲内力OPRqmmxABABOM-diagramOO+Q-diagramN-diagram2PRPP–+弯曲内力OPRqmmxABABOM-diagramOO+Q801、Methodtodeterminedirectlytheinternalforces：

Whenwedeterminetheinternalforcesinanarbitrarysection

A,

wecantaketheleftpartofsectionAasourstudyobjectandusethefollowingformulastocalculateinternalforces.where

PiandPj

arerespectivelyupwardanddownwardexternalforcesactedontheleftpart.DIAGRAMSOFSHEARINGSTRESSESANDBENDINGMOMENTS

EXERCISELESSONSABOUTINTERNALFORCESOFBENDINGINTERNALFORCESINBENDING1、Methodtodeterminedirectly81弯曲内力一、内力的直接求法：

求任意截面A上的内力时，以

A

点左侧部分为研究对象，内力计算式如下，其中Pi、Pj均为A

点左侧的所有向上和向下的外力。剪力图和弯矩图弯曲内力习题课弯曲内力一、内力的直接求法：剪力图和弯矩图弯曲内力习题课82

Relationsamongtheshearingforce、thebendingmomentandtheexternalload：q(x)2、Simplemethodtoplotthediagram:

Themethodtoplotthediagramsbyusingtherelationbetweentheinternalforcesandtheexternalforcesandusingvaluesoftheinternalforcesatsomespecialpoints.INTERNALFORCESINBENDINGRelationsamongtheshearing83弯曲内力剪力、弯矩与分布荷载间的关系：q(x)二、简易作图法:

利用内力和外力的关系及特殊点的内力值来作图的方法。弯曲内力剪力、弯矩与分布荷载间的关系：q(x)二、简易作843、Principleofsuperposition：

Internalforcesinthestructureduetosimultaneousactionofmanyforcesareequaltothealgebrasumoftheinternalforcesduetoseparateactionofeachforce.4、Applicationsofsymmetryandantisymmetry：

Forthesymmetricstructureundertheactionofsymmetricloadsthediagramofitsshearingstressisantisymmetricandthediagramofbendingmomentissymmetric.ForthesymmetricstructureundertheactionofantisymmetricloadsthediagramofitsshearingstressissymmetricandthediagramofbendingmomentisantisymmetricINTERNALFORCESINBENDING3、Principleofsuperposition：

85弯曲内力三、叠加原理：

多个载荷同时作用于结构而引起的内力等于每个载荷单独作用于结构而引起的内力的代数和。四、对称性与反对称性的应用：

对称结构在对称载荷作用下，Q图反对称，M图对称；对称结构在反对称载荷作用下，Q图对称，M图反对称。弯曲内力三、叠加原理：

多个载荷同时作用于结构而86INTERNALFORCESINBENDING5、Relationsbetweentheshearingforce,thebendingmomentandtheexternalloadExternalforceNoexternal-forcesegmentUniform-loadsegmentConcentratedforceConcentratedcoupleq=0q>0q<0CharacteristicsofQ-diagramCPCmHorizontalstraightlinexQQ>0QQ<0xInclinedstraightlineIncreasingfunctionxQxQDecreasingfunctionxQCQ1Q2Q1–Q2=PSuddenchangefromthelefttorightxQCNochangeInclinedstraightlinexMIncreasingfunctionxMDecreasingfunctioncurvesxMTomb-likexMBasin-likeFlexfromthelefttotheright

SuddenchangefromthelefttotherightOppositetomxMFlexoppositetoPMxM1M2INTERNALFORCESINBENDING5、Re87五、剪力、弯矩与外力间的关系外力无外力段均布载荷段集中力集中力偶q=0q>0q<0Q图特征M图特征CPCm水平直线xQQ>0QQ<0x斜直线增函数xQxQ降函数xQCQ1Q2Q1–Q2=PxQC自左向右突变无变化斜直线xM增函数xM降函数xMxMxMxM曲线坟状盆状自左向右折角折向与P反向M1

M2自左向右突变与m反弯曲内力五、剪力、弯矩与外力间的关系外力无外力段均布载荷段集中力集中88Example1

Plotthebending-momentdiagramsofthebeamshowninthefollowingfigure.2PaaP=2PP+xMxM1xM2=+–++2Pa2PaPa(1)INTERNALFORCESINBENDINGExample1Plotthebending-m89弯曲内力[例1]绘制下列图示梁的弯矩图。2PaaP=2PP+xMxM1xM2=+–++2Pa2PaPa(1)弯曲内力[例1]绘制下列图示梁的弯矩图。2PaaP=2P90(2)aaqqqq=+xM1=xM+–+–xM23qa2/2qa2/2qa2INTERNALFORCESINBENDING(2)aaqqqq=+xM1=xM+–+–xM23qa2/291弯曲内力(2)aaqqqq=+xM1=xM+–+–xM23qa2/2qa2/2qa2弯曲内力(2)aaqqqq=+xM1=xM+–+–xM23q92(3)PL/2L/2PL/2=+PxM2xM=+PL/2PL/4PL/2xM1–+–PL/2INTERNALFORCESINBENDING(3)PL/2L/2PL/2=+PxM2xM=+PL/2PL93弯曲内力(3)PL/2L/2PL/2=+PxM2xM=+PL/2PL/4PL/2xM1–+–PL/2弯曲内力(3)PL/2L/2PL/2=+PxM2xM=+PL94(4)50kN2m2m20kNm=+xM2xM=+20kNm50kNmxM120kNm50kN20kNm20kNm++–20kNm30kNm20kNmINTERNALFORCESINBENDING(4)50kN2m2m20kNm=+xM2xM=+20kNm95弯曲内力(4)50kNaa20kNm=+xM2xM=+20kNm50kNmxM120kNm50kN20kNm20kNm++–20kNm30kNm20kNm弯曲内力(4)50kNaa20kNm=+xM2xM=+20k96yzhbSolution:（1）ShearingstressonthecrosssectionisExample2Thestructureisshowninthefigure.Trytoprove：(1）resultantoftheshearingstressesinanarbitrarycrosssectionisequaltotheshearingforceinthesamesection；（2）Resultantmomentofthenormalstressesinanarbitrarycrosssectionisequaltothebendingmomentinthesamesection；（3）whichforcecanbalancetheresultantoftheshearingstressinthelongitudinalsectionatmiddleheightbalanced?.

qINTERNALFORCESINBENDINGNormalstressonthecrosssectionisyzhbSolution:（1）Shearingstres97弯曲内力yzhb解：（1）横截面的剪应力为：[例2]结构如图，试证明：（1）任意横截面上的剪应力的合力等于该面的剪力；（2）任意横截面上的正应力