真空-堆載聯(lián)合預(yù)壓蠕變機(jī)理及微觀結(jié)構(gòu)特征試驗(yàn)研究課件

CHAPTER4

INTERNALFORCESMechanicsofMaterialsCHAPTER4MechanicsofMater1第四章彎曲內(nèi)力材料力學(xué)第四章彎曲內(nèi)力材料力學(xué)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平面彎曲的概念及梁的計(jì)算簡圖§4–2梁的剪力和彎矩§4–3剪力方程和彎矩方程·剪力圖和彎矩圖§4–4剪力、彎矩與分布荷載集度間的關(guān)系及應(yīng)用§4–5按疊加原理作彎矩圖§4–6平面剛架和曲桿的內(nèi)力圖彎曲內(nèi)力習(xí)題課第四章彎曲內(nèi)力§4–1平面彎曲的概念及梁的計(jì)算簡圖第四章4

§4–1

CONCEPTSOFPLANARBENDINGANDCALCULATIONSKETCHOFTHEBEAM1、CONCEPTSOFBENDING1).BENDING:Theactionoftheexternalforceorexternalthecouplevectorperpendiculartotheaxisoftherodmakestheaxisoftherodchangeintocurvefromoriginalstraightlines,thisdeformationiscalledbending.2).BEAM：Thememberofwhichthedeformationismainlybendingisgenerallycalledbeam.INTERNALFORCESINBENDING§4–1CONCEPTSOFPLANARBEN5彎曲內(nèi)力§4–1平面彎曲的概念及梁的計(jì)算簡圖一、彎曲的概念1.彎曲:桿受垂直于軸線的外力或外力偶矩矢的作用時(shí),軸線變成了曲線，這種變形稱為彎曲。2.梁：以彎曲變形為主的構(gòu)件通常稱為梁。彎曲內(nèi)力§4–1平面彎曲的概念及梁的計(jì)算簡圖一、彎曲63).PracticalexamplesinengineeringaboutbendingINTERNALFORCESINBENDING3).Practicalexamplesinengin73.工程實(shí)例彎曲內(nèi)力3.工程實(shí)例彎曲內(nèi)力8INTERNALFORCESINBENDINGINTERNALFORCESINBENDING9彎曲內(nèi)力彎曲內(nèi)力104).Planarbending：Afterdeformationthecurvedaxisofthebeamisstillinthesameplanewiththeexternalforces.

Symmetricbending（asshowninthefollowingfigure）—a specialexampleoftheplanarbending.TheplaneofsymmetryMP1P2qINTERNALFORCESINBENDING4).Planarbending：Afterdeform11彎曲內(nèi)力4.平面彎曲：桿發(fā)生彎曲變形后，軸線仍然和外力在同一平面內(nèi)。對(duì)稱彎曲（如下圖）——平面彎曲的特例。縱向?qū)ΨQ面MP1P2q彎曲內(nèi)力4.平面彎曲：桿發(fā)生彎曲變形后，軸線仍然和外力在同12Unsymmetricalbending—ifabeamdoesnotpossessanyplaneofsymmetry,ortheexternalforcesdonotactinaplaneofsymmetryofthebeamwithsymmetricplanes,thiskindofbendingiscalledunsymmetricalbending.Inlaterchapterswewillmainlydiscussthebendingstressesanddeformationsofthebeamundersymmetricbending.INTERNALFORCESINBENDINGUnsymmetricalbending—ifabe13彎曲內(nèi)力非對(duì)稱彎曲——若梁不具有縱對(duì)稱面，或者，梁雖具有縱對(duì)稱面但外力并不作用在對(duì)稱面內(nèi)，這種彎曲則統(tǒng)稱為非對(duì)稱彎曲。下面幾章中，將以對(duì)稱彎曲為主，討論梁的應(yīng)力和變形計(jì)算。彎曲內(nèi)力非對(duì)稱彎曲——若梁不具有縱對(duì)稱面，或者，梁雖具有縱142、Calculationsketchofthebeam

Ingeneralsupportsandexternalforcesofthebeamareverycomplex.Weshoulddosomenecessarysimplificationforthemforourconvenientcalculationandobtainthecalculationsketch.1).Simplificationofthebeams

Ingeneralcasewetaketheplaceofthebeambyitsaxis.2).Simplificationoftheloads

Theloads(includingthereaction)actingonthebeammaybereducedintothreetypes：concentratedforce、concentratedforcecoupleanddistributedforce.3).SimplificationofthesupportsINTERNALFORCESINBENDING2、Calculationsketchofthebe15彎曲內(nèi)力二、梁的計(jì)算簡圖梁的支承條件與載荷情況一般都比較復(fù)雜，為了便于分析計(jì)算，應(yīng)進(jìn)行必要的簡化，抽象出計(jì)算簡圖。1.構(gòu)件本身的簡化通常取梁的軸線來代替梁。2.載荷簡化作用于梁上的載荷（包括支座反力）可簡化為三種類型：集中力、集中力偶和分布載荷。3.支座簡化彎曲內(nèi)力二、梁的計(jì)算簡圖梁的支承條件與載荷情16①Fixedhingedsupport2constraints，1degreeoffreedom.Suchasthefixedhingedsupportunderbridges，thrustballbearingetc.②Movablehingedsupport1constraint，2degreeoffreedom.Suchasthemovablehingedsupportunderthebridge，ballbearingetc.INTERNALFORCESINBENDING①Fixedhingedsupport②Movable17彎曲內(nèi)力①固定鉸支座2個(gè)約束，1個(gè)自由度。如：橋梁下的固定支座，止推滾珠軸承等。②可動(dòng)鉸支座1個(gè)約束，2個(gè)自由度。如：橋梁下的輥軸支座，滾珠軸承等。彎曲內(nèi)力①固定鉸支座②可動(dòng)鉸支座18③Rigidlyfixedend

3constraints，0degreeoffreedom.Suchasthesupportofdivingboardattheswimmingpool，supportofthelowerendofawoodenpole.XAYAMA4)Threebasistypesofbeams①Simplebeam(orsimplysupportedbeam)M—Concentratedforcecoupleq(x)—Distributedforce②CantileverbeamINTERNALFORCESINBENDINGA③RigidlyfixedendXAYAMA4)Th19彎曲內(nèi)力③固定端3個(gè)約束，0個(gè)自由度。如：游泳池的跳水板支座，木樁下端的支座等。XAYAMA4.梁的三種基本形式①簡支梁M—集中力偶q(x)—分布力②懸臂梁彎曲內(nèi)力③固定端XAYAMA4.梁的三種基本形式①簡支20③Overhangingbeam—ConcentratedforcePq—Uniformlydistributedforce5).StaticallydeterminateandstaticallyindeterminatebeamsStaticallydeterminatebeams：Reactionsofthebeamcanbedeterminedonlybystaticequilibriumequations，suchastheabovethreekindsofbasicbeams.Staticallyindeterminatebeams：Reactionsofthebeamcannotbedeterminedoronlypartofreactionscanbedeterminedbystaticequilibriumequations.INTERNALFORCESINBENDING③Overhangingbeam—Concentrated21彎曲內(nèi)力③外伸梁—集中力Pq—均布力5.靜定梁與超靜定梁靜定梁：由靜力學(xué)方程可求出支反力，如上述三種基本形式的靜定梁。超靜定梁：由靜力學(xué)方程不可求出支反力或不能求出全部支反力。彎曲內(nèi)力③外伸梁—集中力Pq—均布力5.靜定梁與超靜定22Example1Astocktankisshowninthefigure.Itslengthis

L=5m，itsinsidediameterisD=1m，thicknessofitswallist=10mm.Densityofsteelis7.8g/cm3.Densityoftheliquidis1g/cm3.Heightoftheliquidis0.8m.Lengthofoverhangingendis1m.Trytodeterminethecalculationsketchofthestocktank.Solution：q—UniformlyDistributedforceINTERNALFORCESINBENDINGExample1Astocktanki23彎曲內(nèi)力[例1]貯液罐如圖示，罐長L=5m，內(nèi)徑D=1m，壁厚t=10mm，鋼的密度為：7.8g/cm3，液體的密度為：1g/cm3,液面高

0.8m，外伸端長1m，試求貯液罐的計(jì)算簡圖。解：q—均布力彎曲內(nèi)力[例1]貯液罐如圖示，罐長L=5m，內(nèi)徑D=1m，24q—UniformlyDistributedforceINTERNALFORCESINBENDINGq—UniformlyINTERNALFORCESIN25彎曲內(nèi)力q—均布力彎曲內(nèi)力q—均布力26§4–2THESHEARINGFORCEANDBENDINGMOMENTOFTHEBEAM1、Internalforceinbending：Example

KnowingconditionsareP，a，l,asshowninthefigure.DeterminetheinternalforcesonthesectionatthedistancextotheendA.PaPlYAXARBAABBSolution：①Determine externalforcesINTERNALFORCESINBENDING§4–2THESHEARINGFORCEAN27§4–2梁的剪力和彎矩一、彎曲內(nèi)力：彎曲內(nèi)力[舉例]已知：如圖，P，a，l。

求：距A端x處截面上內(nèi)力。PaPlYAXARBAABB解：①求外力§4–2梁的剪力和彎矩一、彎曲內(nèi)力：彎曲內(nèi)力[舉28ABPYAXARBmmx②Determineinternalforces— methodofsectionAYAQMRBPMQInternalforcesofthebeaminbendingShearingforceBendingmoment1).Bendingmoment：M

Momentoftheinternalforcecouplewiththeactingplaneinthecross-sectionperpendiculartothesectionwhenthebeamisbending.CCINTERNALFORCESINBENDINGABPYAXARBmmx②Determineinterna29ABPYAXARBmmx彎曲內(nèi)力②求內(nèi)力——截面法AYAQMRBPMQ∴彎曲構(gòu)件內(nèi)力剪力彎矩1.彎矩：M

構(gòu)件受彎時(shí)，橫截面上其作用面垂直于截面的內(nèi)力偶矩。CCABPYAXARBmmx彎曲內(nèi)力②求內(nèi)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彎曲內(nèi)力2.剪力：Q

構(gòu)件受彎時(shí)，橫截面上其作用線平行于截面的內(nèi)力。3.內(nèi)力的正負(fù)規(guī)定:①剪力Q:繞研究對(duì)象順時(shí)針轉(zhuǎn)為正剪力；反之為負(fù)。②彎矩M：使梁變成凹形的為正彎矩；使梁變成凸形的為負(fù)彎矩。Q(+)Q(–)Q(–)Q(+)M(+)M(+)M(–)M(–)彎曲內(nèi)力2.剪力：Q3.內(nèi)力的正負(fù)規(guī)定:①剪力Q:繞研32Example2：Determinetheinternalforcesactingonsections1—1and2—2sectionasshowninfig.(a).Solution：Determineinternalforces bythemethodofsection.

Freebodydiagramoftheleftportionofsection

1—1isshowninfig.（b）.Fig.（a）2、ExamplesqqLab1122qLQ1AM1Fig.（b）x1INTERNALFORCESINBENDINGExample2：Determinetheintern33[例2]：求圖（a）所示梁1--1、2--2截面處的內(nèi)力。xy解：截面法求內(nèi)力。

1--1截面處截取的分離體

如圖（b）示。圖（a）二、例題qqLab1122qLQ1AM1圖（b）x1彎曲內(nèi)力[例2]：求圖（a）所示梁1--1、2--2截面處的內(nèi)力。x34Freebodydiagramoftheleftportionof

section2—2isshowninfig.（b）.xy圖（a）qqLab1122qLQ2BM2x2圖（c）INTERNALFORCESINBENDINGFreebodydiagramoftheleft352--2截面處截取的分離體如圖（c）xy圖（a）qqLab1122qLQ2BM2x2彎曲內(nèi)力圖（c）2--2截面處截取的分離體如圖（c）xy圖（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彎曲內(nèi)力1.內(nèi)力方程：內(nèi)力與截面位置坐標(biāo)（x）間的函數(shù)關(guān)系式。2.剪力圖和彎矩圖：)(xQQ=剪力方程)(xMM=彎矩方程)(xQQ=剪力圖的圖線表示)(xMM=彎矩圖的圖線表示§4–3剪力方程和彎矩方程·剪力圖和彎矩圖彎曲內(nèi)力1.內(nèi)力方程：內(nèi)力與截面位置坐標(biāo)（x）間的函數(shù)關(guān)系38Example3Determinetheinternal-forceequationsandplotthediagramsofthebeamshowninthefollowingfigure.Solution：①Determinethe reactionsofthesupports②Writeouttheinternal- forceequationsP③Plottheinternal- forcediagrams

Q(x)M(x)xxP–PLYOLM(x)xQ(x)MOINTERNALFORCESINBENDING⊕○Example3Determinetheinter39彎曲內(nèi)力[例3]求下列各圖示梁的內(nèi)力方程并畫出內(nèi)力圖。解：①求支反力②寫出內(nèi)力方程PYOL③根據(jù)方程畫內(nèi)力圖M(x)xQ(x)Q(x)M(x)xxP–PLMO⊕○彎曲內(nèi)力[例3]求下列各圖示梁的內(nèi)力方程并畫出內(nèi)力圖。解：40Solution：①Writeoutthe internal-forceequations②Plottheinternal- forcediagramLqM(x)xQ(x)Q(x)x–qLINTERNALFORCESINBENDINGM(x)x⊕○Solution：①Writeoutthe inte41彎曲內(nèi)力解：①寫出內(nèi)力方程②根據(jù)方程畫內(nèi)力圖LqM(x)xQ(x)Q(x)xM(x)x–qL⊕○彎曲內(nèi)力解：①寫出內(nèi)力方程②根據(jù)方程畫內(nèi)力圖LqM(x)xQ42Solution：①Determinethe reactionsofthesupports②Writeouttheinternal- forceequationsq0RA③Plottheinternal- forcediagrams

RBLxQ(x)xM(x)INTERNALFORCESINBENDING⊕⊕○Solution：①Determinethe react43彎曲內(nèi)力解：①求支反力②內(nèi)力方程q0RA③根據(jù)方程畫內(nèi)力圖RBLxQ(x)xM(x)⊕⊕○彎曲內(nèi)力解：①求支反力②內(nèi)力方程q0RA③根據(jù)方程畫內(nèi)力圖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彎曲內(nèi)力一、剪力、彎矩與分布荷載間的關(guān)系對(duì)dx

段進(jìn)行平衡分析，有：§4–4剪力、彎矩與分布荷載集度間的關(guān)系及應(yīng)用dxxq(x)q(x)M(x)+dM(x)Q(x)+dQ(x)Q(x)M(x)dxAy剪力圖上某點(diǎn)處的切線斜率等于該點(diǎn)處荷載集度的大小。彎曲內(nèi)力一、剪力、彎矩與分布荷載間的關(guān)系對(duì)dx段進(jìn)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彎曲內(nèi)力q(x)M(x)+dM(x)Q(x)+dQ(x)Q(x)M(x)dxAy彎矩圖上某點(diǎn)處的切線斜率等于該點(diǎn)處剪力的大小。彎矩與荷載集度的關(guān)系是：彎曲內(nèi)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、彎矩與外力間的關(guān)系外力無外力段均布載荷段集中力集中力偶q=0q>0q<0Q圖特征M圖特征CPCm水平直線xQQ>0QQ<0x斜直線增函數(shù)xQxQ降函數(shù)xQCQ1Q2Q1–Q2=P自左向右突變xQC無變化斜直線xM增函數(shù)xM降函數(shù)曲線xM墳狀xM盆狀自左向右折角

自左向右突變與m反彎曲內(nèi)力xM折向與P反向MxM1M2二、剪力、彎矩與外力間的關(guān)系外力無外力段均布載荷段集中力集中50Simplemethodtoplotthediagram:Themethodtoplotthediagramsbyusingtherelationbetweentheinternalforcesandtheexternalforcesandvaluesoftheinternalforcesatsomespecialpoints.Example4

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

etc.INTERNALFORCESINBENDINGSimplemethodtoplotthediag51彎曲內(nèi)力簡易作圖法:利用內(nèi)力和外力的關(guān)系及特殊點(diǎn)的內(nèi)力值來作圖的方法。[例4]

用簡易作圖法畫圖示梁的內(nèi)力圖。解:利用內(nèi)力和外力的關(guān)系及特殊點(diǎn)的內(nèi)力值來作圖。特殊點(diǎn):端點(diǎn)、分區(qū)點(diǎn)（外力變化點(diǎn)）和駐點(diǎn)等。aaqaqA彎曲內(nèi)力簡易作圖法:利用內(nèi)力和外力的關(guān)系及特殊點(diǎn)的內(nèi)力值來52aaqaqALeftend：Shapeofthecurveisdeterminedaccordingto

；；Andthelawofthepointactedbyconcentratedforce.Partitionpoint

A：StationarypointofM

：Rightend：Qxqa2–qa–xMINTERNALFORCESINBENDINGaaqaqALeftend：Shapeofthecu53彎曲內(nèi)力aaqaqA左端點(diǎn)：線形：根據(jù)；；及集中載荷點(diǎn)的規(guī)律確定。分區(qū)點(diǎn)A：M的駐點(diǎn)：右端點(diǎn)：Qxqa2–qa–xM彎曲內(nèi)力aaqaqA左端點(diǎn)：線形：根據(jù)；；及集中載荷點(diǎn)的規(guī)律54Example5

Plottheinternal-forcediagramsofthebeamsshowninthefollowingfiguresbythesimplemethodtoplotthediagram.Solution：DeterminereactionsLeftendA：

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

LeftofpointB：INTERNALFORCESINBENDINGExample5Plottheinternal-fo55彎曲內(nèi)力[例5]用簡易作圖法畫下列各圖示梁的內(nèi)力圖。解：求支反力左端點(diǎn)A：B點(diǎn)左：B點(diǎn)右：C點(diǎn)左：M的駐點(diǎn)：C點(diǎn)右：右端點(diǎn)D：qqa2qaRARDQxqa/2qa/2qa/2––+ABCDqa2/2xMqa2/2qa2/23qa2/8–+彎曲內(nèi)力[例5]用簡易作圖法畫下列各圖示梁的內(nèi)力圖。解：求56§4–5PLOTTHEDIAGRAMOFBENDINGMOMENTBYTHE THEOREMOFSUPERPOSITIOM1、Theoremofsuperposition：

Internalforcesinthestructureduetosimultaneousactionofmanyforcesareequaltoalgebraicsumoftheinternalforcesduetoseparateactionofeachforce.Applyingcondition：Relationbetweentheparameters(internalforces、stresses、displacements）andtheexternalforcesmustbelinear,thatistheysatisfyHooke’slaw.INTERNALFORCESINBENDING§4–5PLOTTHEDIAGRAMOFBEN57彎曲內(nèi)力§4–5按疊加原理作彎矩圖一、疊加原理：

多個(gè)載荷同時(shí)作用于結(jié)構(gòu)而引起的內(nèi)力等于每個(gè)載荷單獨(dú)作用于結(jié)構(gòu)而引起的內(nèi)力的代數(shù)和。適用條件：所求參數(shù)（內(nèi)力、應(yīng)力、位移）必然與荷載滿足線性關(guān)系。即在彈性限度內(nèi)滿足虎克定律。彎曲內(nèi)力§4–5按疊加原理作彎矩圖一、疊加原理：

582、Structuralmembersinmechanicsofmaterialisofsmalldeformationandlinearelasticity,andmustobeythisprinciple——methodofsuperpositionSteps：①Plotrespectivelythediagramofthebendingmomentofthebeamundertheseparateactionofeachexternalload；②Sumupthecorrespondinglongitudinalcoordinates(Attention:donotsimplypiecetogetherfigures.）INTERNALFORCESINBENDING2、Structuralmembersinmechan59彎曲內(nèi)力二、材料力學(xué)構(gòu)件小變形、線性范圍內(nèi)必遵守此原理——疊加方法步驟：①分別作出各項(xiàng)荷載單獨(dú)作用下梁的彎矩圖；②將其相應(yīng)的縱坐標(biāo)疊加即可（注意：不是圖形的簡單拼湊）。彎曲內(nèi)力二、材料力學(xué)構(gòu)件小變形、線性范圍內(nèi)必遵守此原理步驟：60Example6Plotthediagramofbendingmomentbytheprincipleofsuperposition.

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

+++=+INTERNALFORCESINBENDINGExample6Plotthediagramof61彎曲內(nèi)力[例6]按疊加原理作彎矩圖(AB=2a，力P作用在梁AB的中點(diǎn)處）。qqPP=+AAABBBxM2xM1xM

+++=+彎曲內(nèi)力[例6]按疊加原理作彎矩圖(AB=2a，力P作用在梁62

3、Applicationsofsymmetryandantisymmetry：

ForthesymmetricstructureundertheactionofsymmetricloadsthediagramofitsshearingstressQisantisymmetricandthediagramofthebendingmomentMissymmetric.ForthesymmetricstructureundertheactionofantisymmetricloadsthediagramofitsshearingstressQissymmetricandthediagramofthebendingmomentMisantisymmetric.

INTERNALFORCESINBENDING3、Applicationsofsymmetryan63彎曲內(nèi)力

三、對(duì)稱性與反對(duì)稱性的應(yīng)用：

對(duì)稱結(jié)構(gòu)在對(duì)稱載荷作用下，Q圖反對(duì)稱，M圖對(duì)稱；對(duì)稱結(jié)構(gòu)在反對(duì)稱載荷作用下，Q圖對(duì)稱，M圖反對(duì)稱。彎曲內(nèi)力三、對(duì)稱性與反對(duì)稱性的應(yīng)用：

對(duì)稱64Example7Plotinternal-forcediagramsofthebeamsshowninthe followingfigure.PPLPPLLLLLLL0.5P0.5P0.5P0.5PP0QxQ1xQ2x–0.5P0.5P0.5P–+–PINTERNALFORCESINBENDINGExample7Plotinternal-for65彎曲內(nèi)力[例7]作下列圖示梁的內(nèi)力圖。PPLPPLLLLLLL0.5P0.5P0.5P0.5PP0QxQ1xQ2x–0.5P0.5P0.5P–+–P彎曲內(nèi)力[例7]作下列圖示梁的內(nèi)力圖。PPLPPLLLLL66PPLPPLLLLLLL0.5P0.5P0.5P0.5PP0MxM1xM2x0.5PLPL0.5PL–++0.5PL+INTERNALFORCESINBENDINGPPLPPLLLLLLL0.5P0.5P0.5P0.5PP067彎曲內(nèi)力PPLPPLLLLLLL0.5P0.5P0.5P0.5PP0MxM1xM2x0.5PLPL0.5PL–++0.5PL+彎曲內(nèi)力PPLPPLLLLLLL0.5P0.5P0.5P0.68Example8Correctthemistakesinthefollowinginternal-forcediagrams.a2aaqqa2ABQxxM––++qa/4qa/43qa/47qa/4qa2/449qa2/323qa2/25qa2/4INTERNALFORCESINBENDINGRARBExample8Correctthemistak69彎曲內(nèi)力[例8]改內(nèi)力圖之錯(cuò)。a2aaqqa2ABQxxM––++qa/4qa/43qa/47qa/4qa2/449qa2/323qa2/25qa2/4彎曲內(nèi)力[例8]改內(nèi)力圖之錯(cuò)。a2aaqqa2ABQxxM70Example9KnowingQ-diagram,determineexternalloadsandM-diagram(Thereforenoconcentratedforcecouplesactedonthebeam).M(kN·m)Q(kN)x1m1m2m2315kN1kNq=2kN/m+–+x+111.25–INTERNALFORCESINBENDINGExample9KnowingQ-diagram,71彎曲內(nèi)力[例9]已知Q圖，求外載及M圖（梁上無集中力偶）。Q(kN)x1m1m2m2315kN1kNq=2kN/m+–+M(kN·m)x+111.25–彎曲內(nèi)力[例9]已知Q圖，求外載及M圖（梁上無集中力偶）。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彎曲內(nèi)力§4–6平面剛架和曲桿的內(nèi)力圖一、平面剛架1.平面剛架：同一平面內(nèi)，不同取向的桿件，通過桿端相互剛性連接而組成的結(jié)構(gòu)。特點(diǎn)：剛架各桿的內(nèi)力有：Q、M、N。2.內(nèi)力圖規(guī)定：彎矩圖：畫在各桿的受拉一側(cè)，不注明正、負(fù)號(hào)。

剪力圖及軸力圖：可畫在剛架軸線的任一側(cè)（通常正值畫在剛架的外側(cè)），但須注明正、負(fù)號(hào)。彎曲內(nèi)力§4–6平面剛架和曲桿的內(nèi)力圖一、平面剛架174Example10Trytoplottheinternal-forcediagramsoftherigidframeshowninthefigure.P1P2alABC–N-diagramQ-diagramP2+P1+P1M-diagramP1aP1aP1a+P2lINTERNALFORCESINBENDINGExample10Trytoplotthei75彎曲內(nèi)力[例10]試作圖示剛架的內(nèi)力圖。P1P2alABC–N圖P2+Q圖P1+P1P1aM圖P1aP1a+P2l彎曲內(nèi)力[例10]試作圖示剛架的內(nèi)力圖。P1P2alAB76

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

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

Methodtoplotinternal-forcediagramofacurvedrodisthesameasthatoftheplanarrigidframe.INTERNALFORCESINBENDINGExample11Asshowninth77彎曲內(nèi)力二、平面曲桿：軸線為一平面曲線的桿件。

內(nèi)力情況及繪制方法與平面剛架相同。[例11]已知：如圖所示，P及R

。試?yán)L制Q、M、N圖。OPRqmmx解：建立極坐標(biāo)，O為極點(diǎn)，OB

極軸，q表示截面m–m的位置。AB彎曲內(nèi)力二、平面曲桿：軸線為一平面曲線的桿件。[例11]78OPRqmmxABABOM圖OO+Q圖N圖2PRPP–+INTERNALFORCESINBENDINGOPRqmmxABABOM圖OO+Q圖N圖2PRPP–+IN79彎曲內(nèi)力OPRqmmxABABOM-diagramOO+Q-diagramN-diagram2PRPP–+彎曲內(nèi)力OPRqmmxABABOM-diagramOO+Q801、Methodtodeterminedirectlytheinternalforces：

Whenwedeterminetheinternalforcesinanarbitrarysection

A,

wecantaketheleftpartofsectionAasourstudyobjectandusethefollowingformulastocalculateinternalforces.where

PiandPj

arerespectivelyupwardanddownwardexternalforcesactedontheleftpart.DIAGRAMSOFSHEARINGSTRESSESANDBENDINGMOMENTS

EXERCISELESSONSABOUTINTERNALFORCESOFBENDINGINTERNALFORCESINBENDING1、Methodtodeterminedirectly81彎曲內(nèi)力一、內(nèi)力的直接求法：

求任意截面A上的內(nèi)力時(shí)，以

A

點(diǎn)左側(cè)部分為研究對(duì)象，內(nèi)力計(jì)算式如下，其中Pi、Pj均為A

點(diǎn)左側(cè)的所有向上和向下的外力。剪力圖和彎矩圖彎曲內(nèi)力習(xí)題課彎曲內(nèi)力一、內(nèi)力的直接求法：剪力圖和彎矩圖彎曲內(nèi)力習(xí)題課82

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

Themethodtoplotthediagramsbyusingtherelationbetweentheinternalforcesandtheexternalforcesandusingvaluesoftheinternalforcesatsomespecialpoints.INTERNALFORCESINBENDINGRelationsamongtheshearing83彎曲內(nèi)力剪力、彎矩與分布荷載間的關(guān)系：q(x)二、簡易作圖法:

利用內(nèi)力和外力的關(guān)系及特殊點(diǎn)的內(nèi)力值來作圖的方法。彎曲內(nèi)力剪力、彎矩與分布荷載間的關(guān)系：q(x)二、簡易作843、Principleofsuperposition：

Internalforcesinthestructureduetosimultaneousactionofmanyforcesareequaltothealgebrasumoftheinternalforcesduetoseparateactionofeachforce.4、Applicationsofsymmetryandantisymmetry：

Forthesymmetricstructureundertheactionofsymmetricloadsthediagramofitsshearingstressisantisymmetricandthediagramofbendingmomentissymmetric.ForthesymmetricstructureundertheactionofantisymmetricloadsthediagramofitsshearingstressissymmetricandthediagramofbendingmomentisantisymmetricINTERNALFORCESINBENDING3、Principleofsuperposition：

85彎曲內(nèi)力三、疊加原理：

多個(gè)載荷同時(shí)作用于結(jié)構(gòu)而引起的內(nèi)力等于每個(gè)載荷單獨(dú)作用于結(jié)構(gòu)而引起的內(nèi)力的代數(shù)和。四、對(duì)稱性與反對(duì)稱性的應(yīng)用：

對(duì)稱結(jié)構(gòu)在對(duì)稱載荷作用下，Q圖反對(duì)稱，M圖對(duì)稱；對(duì)稱結(jié)構(gòu)在反對(duì)稱載荷作用下，Q圖對(duì)稱，M圖反對(duì)稱。彎曲內(nèi)力三、疊加原理：

多個(gè)載荷同時(shí)作用于結(jié)構(gòu)而86INTERNALFORCESINBENDING5、Relationsbetweentheshearingforce,thebendingmomentandtheexternalloadExternalforceNoexternal-forcesegmentUniform-loadsegmentConcentratedforceConcentratedcoupleq=0q>0q<0CharacteristicsofQ-diagramCPCmHorizontalstraightlinexQQ>0QQ<0xInclinedstraightlineIncreasingfunctionxQxQDecreasingfunctionxQCQ1Q2Q1–Q2=PSuddenchangefromthelefttorightxQCNochangeInclinedstraightlinexMIncreasingfunctionxMDecreasingfunctioncurvesxMTomb-likexMBasin-likeFlexfromthelefttotheright

SuddenchangefromthelefttotherightOppositetomxMFlexoppositetoPMxM1M2INTERNALFORCESINBENDING5、Re87五、剪力、彎矩與外力間的關(guān)系外力無外力段均布載荷段集中力集中力偶q=0q>0q<0Q圖特征M圖特征CPCm水平直線xQQ>0QQ<0x斜直線增函數(shù)xQxQ降函數(shù)xQCQ1Q2Q1–Q2=PxQC自左向右突變無變化斜直線xM增函數(shù)xM降函數(shù)xMxMxMxM曲線墳狀盆狀自左向右折角折向與P反向M1

M2自左向右突變與m反彎曲內(nèi)力五、剪力、彎矩與外力間的關(guān)系外力無外力段均布載荷段集中力集中88Example1

Plotthebending-momentdiagramsofthebeamshowninthefollowingfigure.2PaaP=2PP+xMxM1xM2=+–++2Pa2PaPa(1)INTERNALFORCESINBENDINGExample1Plotthebending-m89彎曲內(nèi)力[例1]繪制下列圖示梁的彎矩圖。2PaaP=2PP+xMxM1xM2=+–++2Pa2PaPa(1)彎曲內(nèi)力[例1]繪制下列圖示梁的彎矩圖。2PaaP=2P90(2)aaqqqq=+xM1=xM+–+–xM23qa2/2qa2/2qa2INTERNALFORCESINBENDING(2)aaqqqq=+xM1=xM+–+–xM23qa2/291彎曲內(nèi)力(2)aaqqqq=+xM1=xM+–+–xM23qa2/2qa2/2qa2彎曲內(nèi)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彎曲內(nèi)力(3)PL/2L/2PL/2=+PxM2xM=+PL/2PL/4PL/2xM1–+–PL/2彎曲內(nèi)力(3)PL/2L/2PL/2=+PxM2xM=+PL94(4)50kN2m2m20kNm=+xM2xM=+20kNm50kNmxM120kNm50kN20kNm20kNm++–20kNm30kNm20kNmINTERNALFORCESINBENDING(4)50kN2m2m20kNm=+xM2xM=+20kNm95彎曲內(nèi)力(4)50kNaa20kNm=+xM2xM=+20kNm50kNmxM120kNm50kN20kNm20kNm++–20kNm30kNm20kNm彎曲內(nèi)力(4)50kNaa20kNm=+xM2xM=+20k96yzhbSolution:（1）ShearingstressonthecrosssectionisExample2Thestructureisshowninthefigure.Trytoprove：(1）resultantoftheshearingstressesinanarbitrarycrosssectionisequaltotheshearingforceinthesamesection；（2）Resultantmomentofthenormalstressesinanarbitrarycrosssectionisequaltothebendingmomentinthesamesection；（3）whichforcecanbalancetheresultantoftheshearingstressinthelongitudinalsectionatmiddleheightbalanced?.

qINTERNALFORCESINBENDINGNormalstressonthecrosssectionisyzhbSolution:（1）Shearingstres97彎曲內(nèi)力yzhb解：（1）橫截面的剪應(yīng)力為：[例2]結(jié)構(gòu)如圖，試證明：（1）任意橫截面上的剪應(yīng)力的合力等于該面的剪力；（2）任意橫截面上的正應(yīng)力