材料力学 第二章 剪切

CHAPTER2

SHEAR

MechanicsofMaterials材料力学第二章剪切CHAPTER2SHEAR§2－1STRENGTHCALCULATIONABOUTSHEAR ANDBEARINGOFCONNECTINGMEMBERSEXCERCISELESSONSOFSHEARANDBEARING第二章剪切§2－1连接件的剪切与挤压强度计算拉压和剪切习题课ProductionoftheshearingstressSHEAR剪切剪应力的产生§2－1STRENGTHCALCULATIONABOUTSHEARAND

BEARINGOFCONNECTINGMEMBERS1、Characteristicsofloadsanddeformationofconnectingmembers:1)Connectingmember

Thestructurememberthatconnectsonemembertoanotheriscalledtheconnectingmember.

Suchas：bolts、rivets、keysetc.Theconnectingmemberissmall，butitplaystheroleofpassingloads.

Characteristic：Itcanpassgeneralloadsandcanbedismounted.

PPboltSHEAR§2－1连接件的剪切与挤压强度计算一、连接件的受力特点和变形特点：1、连接件剪切

在构件连接处起连接作用的部件，称为连接件。例如：螺栓、铆钉、键等。连接件虽小，却起着传递载荷的作用。特点：可传递一般力，可拆卸。PP螺栓PPrivetCharacteristic:Itcanpassgeneralloads,butcannotbedismounted.forexample,thetrussinabridgeisconnectedbyit.nogapCharacteristic:Itcanpasstorques.SHEARmShaftKeyGearmPP剪切铆钉特点：可传递一般力，不可拆卸。如桥梁桁架结点处用它连接。无间隙m轴键齿轮特点：传递扭矩。m2)Characteristicsofloadsanddeformation：nn（Resultant）（Resultant）PPUsearivetasanexample：①Characteristicofloads：

Therivetissubjectedtotwoequalandoppositeforces.Theactinglinesofthesetwoforcesareveryclose.②Characteristicofdeformation:

Twopartssubjectedtotwoequalandoppositeforcestendtoshiftoveroneanotheralongthejunctionplaneoftwoforces.SHEAR2、受力特点和变形特点：剪切nn（合力）（合力）PP以铆钉为例：①受力特点：构件受两组大小相等、方向相反、作用线相距很近（差一个几何平面）的平行力系作用。②变形特点：构件沿两组平行力系的交界面发生相对错动。nn（Resultant

）（ResultantPP③Shearingplane：

Theplanealongwhichtwopartsofthemembertendtoshiftoveroneanother.Suchasn–n

.④Internalforceonshearingplane：

Internalforce—ShearingforceQ

，itsactinglineisparalleltotheshearingplane.PnnQShearingplaneSHEAR剪切nn（合力）（合力）PP③剪切面：构件将发生相互错动的面，如n–n

。④剪切面上的内力：内力—剪力Q

，其作用线与剪切面平行。PnnQ剪切面nn(Resultant)(Resultant

）PP3)Threekindsofbreakageatjoint：

①Failureduetoshear

Snipalongtheshearingplaneoftherivet,

suchasalongsectionn–n.

②Breakageduetobearing

Failduetomutualbearingbetweentherivetandthesteelplateintheirconnectingplane.

③BreakageduetotensionPnnQShearingplaneThesteelplateisweakenedinthesectionin whichtherivetholesexistandstressintheweakenedsectionincreasessothatthesteelplateiseasilybrokenduetotensionattheconnectingposition.

SHEAR剪切nn（合力）（合力）PP3、连接处破坏的三种形式：

①剪切破坏沿铆钉的剪切面剪断，如沿n–n面剪断

。

②挤压破坏铆钉与钢板在相互接触面上因挤压而使溃压连接松动，发生破坏。

③拉伸破坏PnnQ剪切面钢板在受铆钉孔削弱的截面处，应力增大，易在连接处拉断。2、PracticalcalculationofshearMethodofthepracticalcalculation：Accordingtopossibilityofbreakageofthemembersomeassumptionsbywhichbasiccharacteristicsubjectedtoforceactionscanbereflectedandcalculationscanbesimplifiedareused.Thencalculateitsnominalstress,determinethecorrespondingpermissiblestressinaccordancewiththeresultofdirecttest.Atlastdothestrengthcalculation.Applyingrange：volumeofthememberisnotlargeandrealstressisquitecomplex.Suchastheconnectingpiecesetc.Assumptionofpracticalcalculation：Assumethatshearingstressisdistributeduniformlyintheshearingplaneandequaltotheaverageshearingstress.SHEAR剪切二、剪切的实用计算实用计算方法：根据构件的破坏可能性，采用能反映受力基本特征，并简化计算的假设，计算其名义应力，然后根据直接试验的结果，确定其相应的许用应力，以进行强度计算。适用：构件体积不大，真实应力相当复杂情况，如连接件等。实用计算假设：假设剪应力在整个剪切面上均匀分布，等于剪切面上的平均应力。1)Shearingplane--AQ：Shiftingplane.

Shearingforce--Q：Internalforceontheshearingplane.2)Nominalshearingforce--：3)Strengthconditionofshear：nn（Resultant

）(Resultant

）PPPnnQShearingplaneWorkingstressshouldnotexceedthepermissiblestress.SHEAR,where剪切1、剪切面--AQ：错动面。

剪力--Q：剪切面上的内力。2、名义剪应力--：3、剪切强度条件（准则）：nn（合力）（合力）PPPnnQ剪切面工作应力不得超过材料的许用应力。3、Practicalcalculationsofbearing1)Bearingforce―Pjy

：Theresultantforceactingonthetouchingplane.Bearing：Thephenomenonthatthereispressureonthepartialareaofamember.Bearingforce：Theresultantforceactingonthebearingplane，designatedbyPjy.Assumption：Bearingstressesaredistributeduniformlyovertheeffective bearingplane.SHEAR三、挤压的实用计算1、挤压力―Pjy

：接触面上的合力。剪切挤压：构件局部面积的承压现象。挤压力：在接触面上的压力，记Pjy。假设：挤压应力在有效挤压面上均匀分布。2)Bearingarea：Areaoftheprojectionplaneofthetouchingplaneinthedirectionperpendicularto

Pjy

3)Strengthconditionofbearing：

WorkingbearingstressshouldnotexceedthepermissiblebearingstressBearingareaSHEAR2、挤压面积：接触面在垂直Pjy方向上的投影面的面积。3、挤压强度条件（准则）：

工作挤压应力不得超过材料的许用挤压应力。剪切挤压面积4、ApplicationsSHEAR1)Checkthestrength2)Selectthecrosssectionarea3)Determinetheexternalload四、应用剪切Example1

Awoodentenonjointisshowninthefigure.Knowingthatthequantitiesarea=b

=12cm，h=35cm，c=4.5cmand

P=40KN.Trytodeterminetheshearingstressandbearingstressforthejoint.Solution：：Freebodydiagramis showninthefigure：ShearingstressandbearingstressShearingforceisBearingforceisSHEARPPPPPPbachh[例1]

木榫接头如图所示，a=b

=12cm，h=35cm，c=4.5cm,

P=40KN，试求接头的剪应力和挤压应力。解：受力分析如图∶剪应力和挤压应力剪切面和剪力为∶挤压面和挤压力为：剪切PPPPPPbachhSolution：Freebodydiagramofthekeyisshown inthefigureExample2Agearandashaftareconnectedbyakey（b×h×L=20×12×100).Thetorquethatthekeycantransmitism=2KNm.Knowingthediameteroftheshaftisd=70mm，thepermissibleshearingstressandthepermissiblebearingstressofthekeyarerespectively[]=60MPaand[jy]=100MPa.Trytocheckthestrengthofthekey.

mbhLSHEARmdP解：键的受力分析如图[例2]

齿轮与轴由平键（b×h×L=20×12×100）连接，它传递的扭矩m=2KNm，轴的直径d=70mm，键的许用剪应力为[]=60M

Pa，许用挤压应力为[jy]=100MPa，试校核键的强度。剪切mbhLmdPAccordingtotheabovecalculation,strengthconditionsofthekeyaresatisfied.Checkthestrengthofshearandbearing

bhLSHEARdmQ综上，键满足强度要求。剪应力和挤压应力的强度校核剪切bhLdmQSolution：Freebodydiagramofthekeyis showninthefigure[Example3]Agearandashaftareconnectedbyakey（b=16mm,h=10mm).Thetorquethatthekeycantransmitism=1600Nm.Knowingthediameteroftheshaftisd=50mm，thepermissibleshearingstressandthepermissiblebearingstressofthekeyarerespectively[]=80MPaand[jy]=240MPa.Trytodesignthelengthofthekey.

bhLSHEARmmmdP解：键的受力分析如图[例3]

齿轮与轴由平键（b=16mm，h=10mm，）连接，它传递的扭矩m=1600Nm，轴的直径d=50mm，键的许用剪应力为[]=80MPa，许用挤压应力为[jy]=240MPa，试设计键的长度。剪切bhLmdPmmbhLStrengthconditionsoftheshearingstressandthebearingstressAccordingtotheabovecalculationSHEARdmQ剪切bhL剪应力和挤压应力的强度条件综上dmQSolution：Freebodydiagram isshowninthefigure[Example4]Arivetedtie-inactedbyforceP=110kNisshowninthefigure.Knowingthethicknessist=1cm，widthofitisb=8.5cm.Thepermissiblestressis[

]=160MPa.Diameteroftherivetisd=1.6cmandthepermissibleshearingstressis[]=140MPa，thepermissiblebearingstressis[jy]=320MPa.Trytocheckthestrengthoftheriveting.(Assumetheforceactedoneachrivetisequal.）bPPttdPPP112233P/4SHEAR解：受力分析如图[例4]

一铆接头如图所示，受力P=110kN，已知钢板厚度为t=1cm，宽度

b=8.5cm，许用应力为[

]=160MPa；铆钉的直径d=1.6cm，许用剪应力为[]=140MPa，许用挤压应力为[jy]=320MPa，试校核铆接头的强度。（假定每个铆钉受力相等。）剪切bPPttdPPP112233P/4Thesections2—2and3—3ofthesteelplatearethecriticalsections.ThestrengthconditionsofshearandbearingTherefore，thetie-inissafe.ttdPPP112233P/4SHEAR钢板的2--2和3--3面为危险面剪应力和挤压应力的强度条件综上，接头安全。剪切ttdPPP112233P/41、Internal-forceand

axial-forcediagramsof

therodintensionand

compression1)ExpressionofAxialforce?2)Methodtodetermineaxialforce?3)Positiveandnegativeofaxialforce?ExerciselessonsoftensionandcompressionandshearWhydoweplottheaxial-forcediagram?Whatshouldwepayattentionwhenwedoit?4)Axial-forcediagram：

ExpressedbythediagramofN=N(x)?PANASimplesketchBCPPNxP+SHEAR一、轴向拉压杆的内力及轴力图1、轴力的表示?2、轴力的求法?3、轴力的正负规定?剪切拉压和剪切习题课为什么画轴力图?

应注意什么?4、轴力图：N=N(x)的图象表示?PANBC简图APPNxP+Simplemethodtodetermineaxialforces:①Taketheleftpartofthesectionxastheobject,theaxialforceonthesectionxcanbecalculatedbyfollowingformula:

Where“P()”and“P()”expressthesumofleftdirectionforcesandthesumofrightdirectionforcesoftheleftpartofthesectionx.

②Taketherightpartofthesectionxastheobject,theaxialforceN(x)ofpointxcanbecalculatedbythefollowingformulate:Where“”and“”denotethesumofrightdirectionforcesandthesumofleftdirectionforcesoftherightpartofsectionx.SHEAR轴力的简便求法:①以x点左侧部分为研究对象,x点的轴力N(x)由下式计算:

其中“P()”与“P()”为x点左侧向左的所有外力与向右的所有外力。②以x点右侧部分为研究对象，x点的轴力N(x)由下式计算：

其中“P()”与“P()”为x点右侧向右的所有外力与向左的所有外力。剪切[Example1]Forces5P,8P,4PandPareactedatpointsA,B,CandDontherodrespectively，theirdirectionsareshowninthefigure.Trytoplottheaxial-forcediagramoftherod.ABCDO5P4PP8PNx–3P5PP2PSHEAR⊕⊕⊕○[例1]图示杆的A、B、C、D点分别作用着5P、8P、4P、P的力，方向如图，试画出杆的轴力图。剪切ABCDO5P4PP8PNx–3P5PP2P⊕⊕⊕○Positiveandnegativeofstress?1)Stressonthecrosssection:

2、Stressoftherodintensionorcompression

Criticalsectionandmaximumworkingstress?

2)Stressontheinclinedsection

Saint-Venantprinciple?

Stressconcentrations?sN(x)Ptasaxs0SHEAR应力的正负规定?1、横截面上的应力:

二、拉压杆的应力危险截面及最大工作应力?

2、拉压杆斜截面上的应力Saint-Venant原理?

应力集中?剪切sN(x)Ptasaxs03、Strengthdesigncriterion：

1)Strengthdesigncriterion

①Checkstrength：②Designthecrosssectionarea：③Designtheload：SHEAR三、强度设计准则（StrengthDesignCriterion）：

1、强度设计准则?

①校核强度：②设计截面尺寸：③设计载荷：剪切1)Elasticlawoftherodwithequalaxialforces

2)Elasticlawwithvariableinternalforces

3)Elasticlawinuniaxialstressedstate

4、Deformationandstrainoftherodintensionorcompression

N(x)dxxPPSHEAR1、等轴力拉压杆的弹性定律

2、变内力拉压杆的弹性定律3、单向应力状态下的弹性定律四、拉压杆的变形及应变剪切N(x)dxxPP4)Possion’sratio（orlateraldeformationfactor）5)Theenlarged-deformationdiagramandthemethodtodeterminethedisplacementC'ABCL1L2PC"SHEAR4、泊松比（或横向变形系数）剪切5、小变形放大图与位移的求法C'ABCL1L2PC"Assemblestress——initialstress

Temperaturestress①Equilibriumequations；

②Geometricequations—compatibilityequationsofdeformation；

③Physicalequations—elasticlaws；

④Complementaryequations：gettingfromgeometricequationsandphysicalequations；

⑤Solvingthecombinedequationsincludingofequilibriumequationsandcomplementequations.6)Thestepstosolvethestaticallyindeterminateproblem

SHEAR装配应力——预应力温度应力剪切①平衡方程；

②几何方程——变形协调方程；

③物理方程——弹性定律；

④补充方程：由几何方程和物理方程得；

⑤解由平衡方程和补充方程组成的方程组。6、超静定问题的处理方法步骤：5、Mechanicalpropertiesofthematerialsintensionandcompression3)Unloadedlaw；Coldhardening；Cold-drawntimeeffect.1)Elasticlaw4)Residualrelativeelongation5)PermanentrelativereductionofareaSHEAR2)Limitstress6)Permissiblestress五、材料在拉伸和压缩时的力学性能3、卸载定律；冷作硬化；冷拉时效。1、弹性定律剪切4、延伸率5、面缩率1)Practicalcalculationofshear6、Practicalcalculationofshearandbearingoftheconnectingmemberoftherodsintensionandcompressionnn（Resultant）（Resultant

）PPPnnQShearingplane2)PracticalcalculationofbearingSHEAR1、剪切的实用计算六、拉(压)杆连接部分的剪切与挤压强度计算剪切nn（合力）（合力）PPPnnQ剪切面2、挤压的实用计算BearingareaSHEARCheckthestrengthDesignthedimensionDesigntheexternalload剪切挤压面积[Example2]

TherodsAB、CD、EFandGHinthestructureareshowninthefigureEachofthemconsiststwoofrolledunequal-legssteels.Knowing[]=170MPa，E=210GPa.RodsACandEGmaybeseenasrigidrod.TrytoselectthesectiondimensionofeachrodandtodeterminethedisplacementsofpointsA、DandC.P=300kN0.8m3.2m1.8m1.2m2m3.4m1.2mABCDFHq0=100kN/mSolution：①Determinetheinternalforce.Thefreebodydiagramisshown inthefigure.EGSHEAR[例2]

结构如图，AB、CD、EF、GH都由两根不等边角钢组成，已知材料的[]=170MPa

，E=210GPa。

AC、EG可视为刚杆，试选择各杆的截面型号和A、D、C点的位移。剪切P=300kN0.8m3.2m1.8m1.2m2m3.4m1.2mABCDFHq0=100kN/m解：①求内力，受力分析如图EGDq0=100kN/mEGACNGNCNANEND=NDP=300kN②DeterminetheareabystrengthconditionSHEAR剪切Dq0=100kN/mEGACNGNCNANEND=NDP=300kN②由强度条件求面积③Determinethetypesoftherodsreferencetothetable④DeterminethedeformationSHEAR剪切③试依面积值查表确定型钢号④求变形⑤Determinethedisplacement.Thedeformationisshowninthefigure.ABDFHEGCC1A1E1D1G1SHEAR⑤求位移,变形图如图剪切ABDFHEGCC1A1E1D1G1[Example3]

DiametersofrodACandBDinthestructureshowninthefigurearerespectivelyd1=25mm,d2=18mm.Knowing[]=170MPa，E=210GPa.RodABmaybeseenasarigidrod.Tryto(1)checkthestrengthofeachrodandtodeterminethedisplacements△Aand△BofpointsAanB.(2)determinethedisplacement△F′ofpointFwhentheforcePisactedonpointA.△F′=△Aisagenerallaw,whichiscalledtheoremofconjugatedisplacement.BNBP=100kNNAAABCDP=100kN1.5m3m2.5mFSolution：①DeterminetheinternalforceThefreebodydiagramisshowninthefigure.

SHEAR[例3]

结构如图，AC、BD的直径分别为:d1=25mm,d2=18mm,已知材料的[]=170MPa

，E=210GPa,AB可视为刚杆，试校核各杆的强度;求A、B点的位移△A和△B。(2)求当P作用于A点时,F点的位移△F′，△F′=△A是普遍规律：称为位移互等定理。剪切BNBP=100kNNAAABCDP=100kN1.5m3m2.5mF解：求内力，受力分析如图②Checkthestrength③Determinethedeformation