材料力學(xué)課件 1

Example1-1Theforceswithmagnitudes5P、8P、4PandPactrespectivelyatpointsA、B、C、Doftherod.Theirdirectionsareshowninthefigure.Trytoplotthediagramoftheaxialforceoftherod.[例1-1]圖示桿的A、B、C、D點(diǎn)分別作用著大小為5P、8P、4P、P的力，方向如圖，試畫出桿的軸力圖。ABCDPAPBPCPDOABCDPAPBPCPDN11Solution：DeterminetheinternalforceN1insegmentOA.Takethefreebodyasshowninthefigure.解：求OA段內(nèi)力N1：設(shè)置截面如圖ABCDPAPBPCPDOABCDPAPBPCPDN12Similarly，wegettheinternalforcesinsegmentAB、BC、CD.Theyarerespectively：

同理，求得AB、BC、CD段內(nèi)力分別為：

N2=–3P

N3=5PN4=PBCDPBPCPDN2CDPCPDN3DPDN43Thediagramoftheaxialforceisshownintherightfigure.軸力圖如右圖Nx2P3P5PP++–Characteristicofthediagramoftheaxialforce：Valueofsuddenchange=concentratedload軸力圖的特點(diǎn)：突變值=集中載荷

4Simplemethodtoplotthediagramofaxialforce：Fromthelefttotheright:軸力(圖)的簡便求法：自左向右:IfmeetingtheforcePtotheleft，theincreaseoftheaxialforceNispositive；遇到向左的P，軸力N

增量為正；5kN8kN3kN+–3kN5kN8kNIfmeetingtheforcetotheright，theincreaseoftheaxialforceNisnegative.遇到向右的P，軸力N增量為負(fù)。5Example1-2LengthoftherodshowninthefigureisL.Distributedforceq=kxisactedonit,directionoftheforceisshowninthefigure.Trytoplotthediagramofaxialforceofthetherod.[例1-2]圖示桿長為L，受分布力q(x)=kx作用，方向如圖，試畫出桿的軸力圖。Lq(x)ox6Solution：Thefreeendoftherodistheoriginofthecoordinateandcoordinatextotherightispositive.Takethesegmentoflengthxontheleftofpointx,itsinternalforceis解：x坐標(biāo)向右為正，坐標(biāo)原點(diǎn)在自由端。取左側(cè)x段為對象，內(nèi)力N(x)為：N(x)xq(x)NxO–K

LxOq7Example1-3AcircularrodissubjectedtoatensileforceP=25kN.Itsdiameterisd=14mmanditsallowablestressis[]=170MPa.Trytocheckthestrengthoftherod.[例1-3]已知一圓桿受拉力P=25kN，直徑d=14mm，許用應(yīng)力[]=170MPa，試校核此桿是否滿足強(qiáng)度要求。8Solution：解：(2)Stress：應(yīng)力：(3)Checkthestrength：強(qiáng)度校核：(4)Conclusion：Thestrengthoftherodsatisfiesrequest.Therodcanworknormally.結(jié)論：此桿滿足強(qiáng)度要求，能夠正常工作。(1)Axialforce：N=P=25kN軸力：N=P=25kN9

Example1-4Athree-pinhouseframeonwhichaverticaluniformload,withtheindensityintensityisq=4.2kN/misappliedisshowninthefigure.Diameterofthesteeltensilerodintheframeisd=16mmanditsallowablestressis[]=170MPa.Trytocheckthestrengthoftherod.[例1-4]已知三鉸屋架如圖，承受豎向均布載荷，載荷的分布集度為：q=4.2kN/m，屋架中的鋼拉桿直徑d=16mm，許用應(yīng)力[]=170MPa。試校核鋼拉桿的強(qiáng)度。Tiebar4.2m8.5m10(1)Determinethereactionsfirstaccordingtotheglobalequilibrium整體平衡求支反力Solution：解：Tiebar8.5m4.2mRARBHA11(3)Stress

應(yīng)力：(2)Determinetheaxialforceaccordingtothepartialequilibrium：

局部平衡求軸力：RAHARCHCN12(4)Strengthcheckandconclusion：

強(qiáng)度校核與結(jié)論：Thisrodsatisfiestherequestofstrength.Itissafe.此桿滿足強(qiáng)度要求，是安全的。RAHARCHCN13hCxLqPABDExample1-5Asimplecraneisshowninthefigure.ACisarigidbeam，sumweightofthehoistandheavybodythatisliftedisP.Whatshouldbetheanglesothattherod

BDhastheminimumweight？Theallowablestressoftherod[]isknown.[例1-5]簡易起重機(jī)構(gòu)如圖，AC為剛性梁，吊車與吊起重物總重為P，為使BD桿最輕，角應(yīng)為何值？已知BD桿的許用應(yīng)力為[]。14Analysis：分析：CxLhqPABDSolution：解：(1)Internalforce

N((q)oftherodBD：TakeACasourstudyobjectasshowninthefigure.

BD桿內(nèi)力N(q)：

取AC為研究對象，如圖15(2)Thecross-sectionareaAoftherodBD：

BD桿橫截面面積A：LNBDYAXAqxPABC16LYAXAqNBDxPABC(3)DeterminetheminimumvalueofVBD

：

求VBD的最小值：174.Stressesintheinclinedsectionoftherodintensionorcompression

拉(壓)桿斜截面上的應(yīng)力PPkka

PkkaPaAssumeastraightrodissubjectedtoatensileforceP.Determinethestressintheinclinedsectionk-k.

設(shè)有一等直桿受拉力P作用。求：斜截面k-k上的應(yīng)力。18Aa:Areaoftheinclinedsection；

Aa：斜截面面積；PPkkaPkkaPaPa=PSolution:Adoptthemethodofsection.解：采用截面法Accordingtotheequilibriumequation：由平衡方程：

Then則Pa：Internalforceintheinclinedsection.Pa：斜截面上內(nèi)力。19

PPkkaPkkaPaFromgeometricrelation由幾何關(guān)系：Substitutingitintotheaboveformulaweget：代入上式，得：Wholestressintheinclinedsection：斜截面上全應(yīng)力：20Decomposition：分解：pa=Itindicatesthechangeofstressesindifferentsectionsthroughapoint.反映：通過構(gòu)件上一點(diǎn)不同截面上應(yīng)力變化情況。Wholestressintheinclinedsection：斜截面上全應(yīng)力：PPkkaPkkapatasaa21As=90°，As=0,90°，As=0°，(Themaximumnormalstressexistsinthecrosssection)(橫截面上存在最大正應(yīng)力)PPkkaPkkapatasaaAs=±45°，(Themaximumshearingstressexistsintheinclinedsectionof45°)當(dāng)=0°時(shí)，當(dāng)=90°時(shí)，當(dāng)=±45°時(shí)，(45°斜截面上剪應(yīng)力達(dá)到最大)當(dāng)=0,90°時(shí)，22Example1-6Arod,whichthediameterd=1cmissubjectedtoatensileforceP=10kN.Determinethemaximumshearingstress,thenormalstressandshearingstressintheinclinedsectionofanangle30°aboutthecrosssection.[例1-6]直徑為d=1cm桿受拉力P=10kN的作用，試求最大剪應(yīng)力，并求與橫截面夾角30°的斜截面上的正應(yīng)力和剪應(yīng)力。23Solution：Stressesintheinclinedsectionoftherodintensionorcompressioncanbedetermineddirectlybytheformula：解：拉壓桿斜截面上的應(yīng)力，直接由公式求之：24Example1-7

Atensilerodasshowninthefigureismadefromtwopartsgluedmutuallytogetheralong

mn.Itissubjectedtotheactionofforce

P.Assumethattheallowablenormalstressis[]=100MPaandallowableshearingstressis[]=50MPafortheadhesive.AreaofcrosssectionoftherodisA=4cm2.Ifstrengthoftherodiscontrolledbytheadhesivewhatistheangle(:between00~600)togetthelargesttensileforce?[例1-7]圖示拉桿沿mn由兩部分膠合而成,受力P，設(shè)膠合面的許用拉應(yīng)力為[]=100MPa；許用剪應(yīng)力為[]=50MPa，并設(shè)桿的強(qiáng)度由膠合面控制,桿的橫截面積為A=4cm2，試問:為使桿承受最大拉力,角值應(yīng)為多大?(規(guī)定:在0~60度之間)。25Combine(1)、(2)andget：聯(lián)立(1)、(2)得：Solution：解：PPmnaPa600300B0026Thecurvesofformula(1)and、(2)areshownintheTig.(2).ObviouslythestrengthoftherodontheleftofpointBiscontrolledbythenormalstress，thatontherightofpointBiscontrolledbytheshearingstress.Asa=60°，fromformula(2)wecanget(1)、(2)式的曲線如圖(2)，顯然，B點(diǎn)左側(cè)由正應(yīng)力控制桿的強(qiáng)度，B點(diǎn)右側(cè)由剪應(yīng)力控制桿的強(qiáng)度，當(dāng)a=60°時(shí)，由(2)式得Pa600300B100Discussion：As討論：若27Solution:

Atthepointofintersectionofcurves(1)and(2)：解:(1)、(2)曲線交點(diǎn)處：Pa600300B100281.Howtoplottheenlargementsketchofthesmalldeformation1.怎樣畫小變形放大圖？2)Accuratemethodtoplotdiagramofdeformation，thearclineasshowninthefigure；

變形圖嚴(yán)格畫法，圖中弧線；1)Determinedeformation△LiofeachrodasshowninFigl.

求各桿的變形量△Li，如圖1；3)Approximatemethodtoplotthediagramofdeformation;thetangentofthearclineshowninthefigure.

變形圖近似畫法，圖中弧之切線。Example1-8Enlargementsketchofthesmalldeformationtheandmethodtodeterminedisplacements[例1-8]小變形放大圖與位移的求法。C'ABCL1L2PC"292.WritetherelationbetweenthedisplacementofpointBshowninFig.2anddeformationsoftworods.寫出圖2中B點(diǎn)位移與兩桿變形間的關(guān)系A(chǔ)BCL1L2B'PFig.2圖230Solution：ThediagramofdeformationisshownintheFig.2.PointBmovestopointB'，F(xiàn)romthediagramofdisplacementwemayknow：解：變形圖如圖2，B點(diǎn)位移至B'點(diǎn)，由圖知：ABCL1L2B'PFig.2圖231800400400DCPAB60°60°Example1-9

SupposethecrossbeamABCDisrigid.Asteelcablewiththecross-sectionarea76.36mm2isaroundapulleywithoutfriction.KnowingP=20kN，E=177GPa.DeterminethestressofthesteelcableandtheuprightdisplacementofpointC.[例1-9]設(shè)橫梁ABCD為剛梁，橫截面面積為76.36mm2的鋼索繞過無摩擦的定滑輪。設(shè)P=20kN，試求鋼索內(nèi)的應(yīng)力和C點(diǎn)的垂直位移。設(shè)鋼索的E=177GPa。321)Determinetheinternalforceofthesteelcable:TakeABDasourstudyobject：求鋼索內(nèi)力：以ABCD為研究對象2)Stressandelongationofthesteelcable鋼索的應(yīng)力和伸長分別為：

PABCDTTYAXA800400400DCPAB60°60°Solution：method1Enlargementsketchmethodofthesmalleformation.

解：方法1

：小變形放大圖法：333）Deformationisshowninthefigure.

UprightdisplacementofpointCis：變形圖如左圖,C點(diǎn)的垂直位移為DCPAB60°60°800400400AB60°60°DB'D'C拉壓AxialTensionandCompression34800400400CPAB60°60°PABCDTTYAXAExample1-9SupposethecrossbeamABCDisrigid.Asteelcablewiththecross-sectionarea76.36mm2isaroundapulleywithoutfriction.KnowingP=20kN，E=177GPa.DeterminethestressofthesteelcableandtheuprightdisplacementofpointC.[例1-9]設(shè)橫梁ABCD為剛梁，橫截面面積為76.36mm2的鋼索繞過無摩擦的定滑輪。設(shè)P=20kN，試求鋼索內(nèi)的應(yīng)力和C點(diǎn)的垂直位移。設(shè)鋼索的E=177GPa。35800400400CPAB60°60°PABCDTTYAXASolution：Method2：Energymethod:(Workofexternalforcesisequaltothestrainenergy)解：方法2：能量法：（外力功等于變形能）1)Determinetheinternalforceofthesteelcable：TakeABCDasourstudyobject：求鋼索內(nèi)力：以ABCD為研究對象：362)Stressofthesteelcableis：鋼索的應(yīng)力為：3)Displacementofpoint

Cis：

C點(diǎn)位移為：Energymethod：Themethodbywhichtheproblemsrelativetotheelasticdeformationofthestructuremembersaresoloedaccordingtotheconceptofstrainenergyiscalledtheenergymethod.

能量法：利用應(yīng)變能的概念解決與結(jié)構(gòu)物或構(gòu)件的彈性變形有關(guān)的問題，這種方法稱為能量法。800400400CPAB60°60°37CPABD123PAN1N3N2Example1-10Rods1，2and3areconnectedtogetherwithapinasshowninthefigure.Knowingthelengthofeachrodis：L1=L2、L3=L；andtheareaofeachrodisA1=A2=A,A3；modulusofelasticityofeachrodis：E1=E2=E、E3.Externalforceisalongtheuprightdirection.Determinetheinternalforceofeachrod.[例1-10]設(shè)1、2、3三桿用鉸鏈連接如圖，已知：各桿長為：L1=L2、L3=L；各桿面積為A1=A2=A、A3；各桿彈性模量為：E1=E2=E、E3。外力沿鉛垂方向，求各桿的內(nèi)力。38

Equilibriumequations:

平衡方程:CPABD123PAN1N3N2Solution：

解：39Geometricequation—compatibilityequationofdeformation：

幾何方程——變形協(xié)調(diào)方程：Physicalequation—elasticlaw：

物理方程——彈性定律：CABD123A140Solvingtheequilibriumequationsandcomplementaryequationweget:

解由平衡方程和補(bǔ)充方程組成的方程組，得:CABD123A1Complementaryequation：detainingfromthegeometricequationandphysicalequation.

補(bǔ)充方程：由幾何方程和物理方程得。41P1mPExample1-11Fouranglesofawoodenpolearereinforcedwithfourequalleganglesteelof40404.Theallowablestressesofsteelandwoodarerespectively[]1=160MPaand[]2=12MPa，moduleofelasticityofthemareE1=200GPaandE2=10GPa.RespectivelyDeterminetheallowablepermissibleloadP.[例1-11]木制短柱的四角用四個(gè)40404的等邊角鋼加固，角鋼和木材的許用應(yīng)力分別為[]1=160MPa和[]2=12MPa，彈性模量分別為E1=200GPa和E2=10GPa；求許可載荷P。42Geometricequation

幾何方程Physicalequationandcomplementaryequation：

物理方程及補(bǔ)充方程：Equilibriumequations:

平衡方程:PPy4N1N2Solution：解：43PPy4N1N2Solvingequilibriumequationsandthecomplementaryequationweget:

解平衡方程和補(bǔ)充方程，得:Determinethepermissibleloadofthestructure：

Method1:

求結(jié)構(gòu)的許可載荷：方法1:44A1=3.086cm2PPy4N1N2Sectionareaoftheanglesteelmaybeobtainedfromthetableofhot-rolledsteel:角鋼截面面積由型鋼表查得:45Inthecaseof△1=△2，theanglesteelwillreachthelimitstatefirst,thatisthemaximumloadisdeterminedbytheanglesteel.

所以在△1=△2的前提下，角鋼將先達(dá)到極限狀態(tài)，即角鋼決定最大載荷。Determinethepermissibleloadofthestructure：

求結(jié)構(gòu)的許可載荷：

Method2:方法2:46

aaaaN1N2Example1-12

Theupperandlowerendsofaladder-likesteelshaftarefixedattemperatureT1=5℃asshowninthefigure.Areasoftheupperandlowersegmentsarerespectively1=cm2and

2=cm2.WhenitstemperaturereachesT2=25℃,determinethetemperaturestressofeachrod.(Linearthermalexpansioncoefficient

=

12.5×10-61/0C；modulusofelasticity

E=200GPa)

[例12]如圖，階梯鋼桿的上下兩端在T1=5℃時(shí)被固定,桿的上下兩段的面積分別=cm2，

=cm2，當(dāng)溫度升至T2=25℃時(shí),求各桿的溫度應(yīng)力。(線膨脹系數(shù)=；彈性模量E=200GPa12.5×１０－６/℃

)47

aaaaN1N2Geometricequation：

幾何方程：

Equilibriumequation：

平衡方程：Physicalequation:物理方程Solution：解：48Solvingequilibriumequationsandthecomplementaryequationweget:

解平衡方程和補(bǔ)充方程，得:Complementaryequation:

補(bǔ)充方程Temperaturestresses:

溫度應(yīng)力49Example1-13Diameterofacopperwireisd=2mmanditslengthisL=500mm.Tensilecurveofcopperisshowninthefigure.Tomakeelongationofthecopperwireis30mmwhetistheforcePthatwemustact

？[例1-13]銅絲直徑d=2mm，長L=500mm，材料的拉伸曲線如圖所示。如欲使銅絲的伸長量為30mm，則大約需加多大的力P？

s(MPa)e(%)50Solution：Deformationmayexceedtherangeoflinearelasticity,thereforetheelasticlawisnotappliedhere.Calculationshouldbedoneinthefollowing:解：變形量可能已超出了“線彈性”范圍，故，不可再應(yīng)用“彈性定律”。應(yīng)如下計(jì)算：Fromthetensilecurve：由拉伸圖知：s(MPa)e(%)511.KnowingtheelasticmodulusofsteelisE＝200GPa，theelasticmodulusofAluminumisE＝71GPa.Trytocompare:①whichmaterialproducesalargerstrainwhentheyaresubjectedtothesamestress？②whichmaterialcorrespondstoalargestresswhentheyhavethesamestrain？

鋼的彈性模量E＝200GPa，鋁的彈性模量E＝71GPa。試比較在同一應(yīng)力作用下，那種材料的應(yīng)變大？在產(chǎn)生同一應(yīng)變的情況下，那種材料的應(yīng)力大？Chapter1Exercises第一章練習(xí)題522.Forthedifferentmembersmadeofasamematerial,dotheyhavethe