CHAPTER11

ENERGYMETHODS

材料力學(xué)第十一章能量方法CHAPTER11ENERGYMETHOD

§11–1GENERALEXPRESSIONSOFTHESTRAINENERGY§11–2MOHR’STHEOREM(METHODOFUNITFORCE)§11–3CATIGLIANO’STHEOREM第十一章能量方法

§11–1

變形能的普遍表達(dá)式§11–2莫爾定理(單位力法)§11–3卡氏定理§11–1

GENERALEXPRESSIONSOFTHESTRAINENERGY1、Principleofenergy：2、Calculationofthestrainenergy

ofrods：1).Calculationofthestrainenergyofrodsintensionorcompression：

Strainenergystoredintheelasticbodyisequaltotheworkdonebyexternalforces，thatis:

Methodtoanalyzeandcalculatedisplacements、deformationsandinternalforcesofdeformablebodiesbythiskindofrelationiscalledenergymethod.ENERGYMETHODorDensityofthestrainenergy:§11–1

變形能的普遍表達(dá)式一、能量原理：二、桿件變形能的計算：1.軸向拉壓桿的變形能計算：能量方法

彈性體內(nèi)部所貯存的變形能，在數(shù)值上等于外力所作的功，即

利用這種功能關(guān)系分析計算可變形固體的位移、變形和內(nèi)力的方法稱為能量方法。2.Calculationofthestrainenergyofrodsintorsion：3.Calculationofstrainenergyofrodsinbending：ENERGYMETHODorDensityofthestrainenergy:orDensityofthestrainenergy:2.扭轉(zhuǎn)桿的變形能計算：3.彎曲桿的變形能計算：能量方法3、Generalexpressionsofthestrainenergy：

Strainenergyisindependentoftheorderofloading.Deformationsduetomutuallyindependentloadmaybesummedupeachother.Forslendercolumns，thestrainenergyduetoshearingforcesmaybeneglected.ENERGYMETHODDeflectionfactorofshear三、變形能的普遍表達(dá)式：

變形能與加載次序無關(guān)；相互獨立的力（矢）引起的變形能可以相互疊加。細(xì)長桿，剪力引起的變形能可忽略不計。能量方法Solution：Inenergymethod（workdonebyexternalforcesisequaltothestrainenergy）①DetermineinternalforcesAENERGYMETHODBendingmoment:Torque:Example1

Asemicirclerodasshowninthefigureislieinhorizontalplane.AverticalforcePactatitspointA.

DeterminethedisplacementofpointAinverticaldirection.PROQMNMTAAPNBjTOMN[例1]圖示半圓形等截面曲桿位于水平面內(nèi)，在A點受鉛垂力P的作用，求A點的垂直位移。解：用能量法（外力功等于應(yīng)變能）①求內(nèi)力能量方法APROQMTAAPNBjTO③Workdonebyexternalforcesisequaltothestrainenergy②Strainenergy：ENERGYMETHODLetthen③外力功等于應(yīng)變能②變形能：能量方法Example2DeterminethedeflectionofpointCbytheenergymethod,wherethebeamisofequalsectionandstraight.

Solution：WorkdonebyexternalforcesisequaltothestrainenergyByusingsymmetryweget：Thinking：Forthedistributedload,canwedeterminethedisplacementofpointCbythismethod？qCaaAPBfENERGYMETHODLet[例2]用能量法求C點的撓度。梁為等截面直梁。解：外力功等于應(yīng)變能應(yīng)用對稱性，得：思考：分布荷載時，可否用此法求C點位移？能量方法qCaaAPBf

§11–2

MOHR’STHEOREM(METHODOFUNITFORCE)

DeterminethedisplacementfAofanarbitrarypointA.1、Provementofthetheorem：aAFigfAq(x)Figc

A0P=1q(x)fAFigb

A=1P0ENERGYMETHOD§11–2

莫爾定理(單位力法)求任意點A的位移fA。一、定理的證明：能量方法aA圖fAq(x)圖c

A0P=1q(x)fA圖b

A=1P0

Mohr’stheorem(methodofunitforce)2、GeneralformofMohr’stheoremENERGYMETHOD

莫爾定理(單位力法)二、普遍形式的莫爾定理能量方法3、WhatwemustpayattentiontoasweapplyMohr’stheorem：④CoordinateofM0(x)mustbecoincidewiththatofM(x).Foreachsegmentthecoordinatemaybesetupfreely.⑤Mohr’sintegrationmustbethroughthewholestructure.②M0:Theinternalforceofthestructureasweactageneralizedunitforcealongthedirection,ofthegeneralizeddisplacementthatistobedetermined,wheretheappliedforceistakenout.①M(x)：Theinternalforceofthestructureactedbyoriginalloads.③Theproductoftheappliedgeneralizedunitforceandthegeneralizeddisplacementtobedetermineddeterminedmustbeofthedimensionofwork.

ENERGYMETHOD三、使用莫爾定理的注意事項：④M0(x)與M(x)的坐標(biāo)系必須一致，每段桿的坐標(biāo)系可自由建立。⑤莫爾積分必須遍及整個結(jié)構(gòu)。②M0——去掉主動力，在所求廣義位移

點，沿所求

廣義位移

的方向加廣義單位力

時，結(jié)構(gòu)產(chǎn)生的內(nèi)力。①M(x)：結(jié)構(gòu)在原載荷下的內(nèi)力。③所加廣義單位力與所求廣義位移之積，必須為功的量綱。能量方法Example3

DeterminethedisplacementandtheangleofrotationofpointCbytheenergymethod.Solution：①Plotthediagramofthestructureactedbytheunitload

②DeterminetheinternalforceBAaaCqBAaaC0P=1xENERGYMETHOD[例3]用能量法求C點的撓度和轉(zhuǎn)角。梁為等截面直梁。解：①畫單位載荷圖②求內(nèi)力能量方法BAaaCqBAaaC0P=1xSymmetry③DeformationBAaaC0P=1BAaaCqxENERGYMETHOD()③變形能量方法BAaaC0P=1BAaaCqx()④Determinetheangleofrotation.Setupthecoordinateagain(asshowninthefigure）

qBAaaCx2x1BAaaCMC0=1

d)()(

)()()(00)(00òò+=aBCaABxEIxMxMdxEIxMxMENERGYMETHOD=0④求轉(zhuǎn)角，重建坐標(biāo)系（如圖）

能量方法qBAaaCx2x1BAaaCMC0=1

d)()(

)()()(00)(00òò+=aBCaABxEIxMxMdxEIxMxM=0Solution：①Plotthediagramofthestructureactedbyaunitload

②Determinetheinternalforce510

20A300P=60NBx500Cx1510

20A300Bx500C=1P0ENERGYMETHODExample4Afoldingrodisshowninthefigure.AbearingisatpositionAandtherodmayrotatefreelyinthebearingbutcannotmoveupanddown.Knowing：E=210Gpa，G=0.4E，DeterminetheverticaldisplacementofpointB.[例4]拐桿如圖，A處為一軸承，允許桿在軸承內(nèi)自由轉(zhuǎn)動，但不能上下移動，已知：E=210Gpa，G=0.4E，求B點的垂直位移。解：①畫單位載荷圖②求內(nèi)力能量方法510

20A300P=60NBx500Cx1510

20A300Bx500C=1P0③DeterminethedeformationENERGYMETHOD()③變形能量方法()§11–3CATIGLIANO’STHEOREMGive

Pn

anincrementdPn

，then：1)FirstapplyforcesP1、

P2、???、Pn

onthebody，then：2).FirstapplytheforcedPn

onthebody，then：1、Provementofthetheorem

dnENERGYMETHOD§11–3

卡氏定理給Pn

以增量dPn

，則：1.先給物體加P1、

P2、???、

Pn

個力，則：2.先給物體加力dPn

，則：一、定理證明

能量方法dnAgainapplyforces

P1、

P2、???、Pn，then：

dnn=nPU??dSecondCastigliano’stheoremItalianengineer—AlbertoCastigliano，1847～1884ENERGYMETHOD再給物體加P1、

P2、???、Pn個力，則：

能量方法dnn=nPU??d第二卡氏定理

意大利工程師—阿爾伯托·卡斯提安諾(AlbertoCastigliano，1847～1884)2、whatwemustpayattentiontoasweapplyCatigliano’stheorem：①U—Linearelasticstrainenergyofthewholestructureactedbyexternalloads②

Pnisconsideredasavariable.ThereactionsandthestrainenergyofthestructureandsoonmustbeallexpressedasthefunctionofPn.③

nisthedeformationofthepointactedbyPn

anditisalongthedirectionofPn.④Ifthereisno

Pn

correspondingto

nwemay firstactaPn

along

nanddeterminethepartialderivativeandthenletPn

bezero.dnENERGYMETHOD二、使用卡氏定理的注意事項：①U——整體結(jié)構(gòu)在外載作用下的線彈性變形能②

Pn視為變量，結(jié)構(gòu)反力和變形能等都必須表示為Pn的函數(shù)③

n為Pn

作用點的沿Pn

方向的變形。④當(dāng)無與

n對應(yīng)的Pn

時，先加一沿

n

方向的Pn

，求偏導(dǎo)后，再令其為零。能量方法dn3、Castigliano’stheoremforspecialstructures(rods)：ENERGYMETHOD三、特殊結(jié)構(gòu)（桿）的卡氏定理：能量方法Example5Thestructureisshowninthefigure.DeterminethedeflectionandtheangleofrotationofthesectionA

byCatigliano’stheorem.③Determinethedeformation①DeterminetheinternalforceSolution：Determinethedeflection.Setupthecoordinate②Determinethepartialderivativeoftheinternalforcewithrespectto

PAALPEIxO

ENERGYMETHOD()[例5]結(jié)構(gòu)如圖，用卡氏定理求A

面的撓度和轉(zhuǎn)角。③變形①求內(nèi)力解：求撓度，建坐標(biāo)系②將內(nèi)力對PA求偏導(dǎo)能量方法ALPEIxO

()Determinetheangle

Aofrotation①DeterminetheinternalforceThereisnothegeneralizedforcecorrespondingto

A.wemayactone.“Negativesign”expressesthat

AiscontrarytothedirectionoftheactedgeneralizedforceMA()②DeterminethepartialderivativeoftheinternalforceM(x)withrespecttoMAandletMA=0.③Determinethedeformation（Note：MA=0）LxO

APMAENERGYMETHOD求轉(zhuǎn)角

A①求內(nèi)力沒有與

A向相對應(yīng)的力（廣義力），加之?！柏?fù)號”說明

A與所加廣義力MA反向。()②將內(nèi)力對MA求偏導(dǎo)后，令M

A=0③求變形（注意：MA=0）能量方法LxO

APMAExample

6

DeterminethedeflectioncurveofthebeamshowninthefigurebyCastigliano’stheorem.Solution：Determinethedeflectioncurve—thedeflectionofanarbitrarypointonthebeamf（x）.①Determinetheinternalforces②DeterminethepartialderivativeoftheinternalforceM(x)withrespecttoPxandletPx=0.

Thereisnothegeneralizedforcecorrespondingtof（x）.wemayactone.

PALxBPx

CfxOx1ENERGYMETHOD[例6

]結(jié)構(gòu)如圖，用卡氏定理求梁的撓曲線。解：求撓曲線——任意點的撓度f（x）①求內(nèi)力②將內(nèi)力對Px求偏導(dǎo)后，令Px=0沒有與f（x）相對應(yīng)的力，加之。能量方法PALxBPx

CfxOx1③Determinethedeformation（Note：Px=0）ENERGYMETHOD③變形（注意：Px=0）能量方法Example7

Abeamwithequalsectionisshowninthefigure.Determinethedeflectionf(x)ofpointBbyCatigliano’stheorem.②determineinternalforcesSolution：1.Determineredundantreactionsaccordingto③DeterminethepartialderivativeoftheinternalforcewithrespecttoRC.

①TakeaprimarybeamasshowninthePCAL0.5LBfxOPCAL0.5LBRCENERGYMETHODfigure.[例7]等截面梁如圖，用卡氏定理求B

點的撓度。②求內(nèi)力解：1.依求多余反力，③將內(nèi)力對RC求偏導(dǎo)①取靜定基如圖能量方法PCAL0.5LBfxOPCAL0.5LBRC④DeformationENERGYMETHODSo④變形能量方法2.Determine②Determinethepartialderivativeoftheinternalforcewithrespect①DeterminetheinternalforcesENERGYMETHODtoP.2.求②將內(nèi)力對P求偏導(dǎo)①求內(nèi)力能量方法③DeformationENERGYMETHOD()③變形能量方法()③DeterminethedeformationSolution：①Plotthediagramofthestructureactedbyunitload②DeterminetheinternalforceExample

8

Aframeisshowninthefigure.DeterminethedistancebetweensectionAandsectionB

afterthedeformation.PPAB11ENERGYMETHOD③變形解：①畫單位載荷圖②求內(nèi)力[例8

]結(jié)構(gòu)如圖，求A、B兩面的拉開距離。PPAB能量方法1159

Chapter11Exercises1.Astraightrodwiththetension(compression)rigidityEIissubjectedforcesshowninthefigure.Maythestrainenergybeexpressedas