CHAPTER2

SHEAR

MechanicsofMaterials材料力學第二章剪切CHAPTER2SHEAR§2－1STRENGTHCALCULATIONABOUTSHEAR ANDBEARINGOFCONNECTINGMEMBERSEXCERCISELESSONSOFSHEARANDBEARING第二章剪切§2－1連接件的剪切與擠壓強度計算拉壓和剪切習題課ProductionoftheshearingstressSHEAR剪切剪應(yīng)力的產(chǎn)生§2－1STRENGTHCALCULATIONABOUTSHEARAND

BEARINGOFCONNECTINGMEMBERS1、Characteristicsofloadsanddeformationofconnectingmembers:1)Connectingmember

Thestructurememberthatconnectsonemembertoanotheriscalledtheconnectingmember.

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

Characteristic：Itcanpassgeneralloadsandcanbedismounted.

PPboltSHEAR§2－1連接件的剪切與擠壓強度計算一、連接件的受力特點和變形特點：1、連接件剪切

在構(gòu)件連接處起連接作用的部件，稱為連接件。例如：螺栓、鉚釘、鍵等。連接件雖小，卻起著傳遞載荷的作用。特點：可傳遞一般力，可拆卸。PP螺栓PPrivetCharacteristic:Itcanpassgeneralloads,butcannotbedismounted.forexample,thetrussinabridgeisconnectedbyit.nogapCharacteristic:Itcanpasstorques.SHEARmShaftKeyGearmPP剪切鉚釘特點：可傳遞一般力，不可拆卸。如橋梁桁架結(jié)點處用它連接。無間隙m軸鍵齒輪特點：傳遞扭矩。m2)Characteristicsofloadsanddeformation：nn（Resultant）（Resultant）PPUsearivetasanexample：①Characteristicofloads：

Therivetissubjectedtotwoequalandoppositeforces.Theactinglinesofthesetwoforcesareveryclose.②Characteristicofdeformation:

Twopartssubjectedtotwoequalandoppositeforcestendtoshiftoveroneanotheralongthejunctionplaneoftwoforces.SHEAR2、受力特點和變形特點：剪切nn（合力）（合力）PP以鉚釘為例：①受力特點：構(gòu)件受兩組大小相等、方向相反、作用線相距很近（差一個幾何平面）的平行力系作用。②變形特點：構(gòu)件沿兩組平行力系的交界面發(fā)生相對錯動。nn（Resultant

）（ResultantPP③Shearingplane：

Theplanealongwhichtwopartsofthemembertendtoshiftoveroneanother.Suchasn–n

.④Internalforceonshearingplane：

Internalforce—ShearingforceQ

，itsactinglineisparalleltotheshearingplane.PnnQShearingplaneSHEAR剪切nn（合力）（合力）PP③剪切面：構(gòu)件將發(fā)生相互錯動的面，如n–n

。④剪切面上的內(nèi)力：內(nèi)力—剪力Q

，其作用線與剪切面平行。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、連接處破壞的三種形式：

①剪切破壞沿鉚釘?shù)募羟忻婕魯?，如沿n–n面剪斷

。

②擠壓破壞鉚釘與鋼板在相互接觸面上因擠壓而使?jié)哼B接松動，發(fā)生破壞。

③拉伸破壞PnnQ剪切面鋼板在受鉚釘孔削弱的截面處，應(yīng)力增大，易在連接處拉斷。2、PracticalcalculationofshearMethodofthepracticalcalculation：Accordingtopossibilityofbreakageofthemembersomeassumptionsbywhichbasiccharacteristicsubjectedtoforceactionscanbereflectedandcalculationscanbesimplifiedareused.Thencalculateitsnominalstress,determinethecorrespondingpermissiblestressinaccordancewiththeresultofdirecttest.Atlastdothestrengthcalculation.Applyingrange：volumeofthememberisnotlargeandrealstressisquitecomplex.Suchastheconnectingpiecesetc.Assumptionofpracticalcalculation：Assumethatshearingstressisdistributeduniformlyintheshearingplaneandequaltotheaverageshearingstress.SHEAR剪切二、剪切的實用計算實用計算方法：根據(jù)構(gòu)件的破壞可能性，采用能反映受力基本特征，并簡化計算的假設(shè)，計算其名義應(yīng)力，然后根據(jù)直接試驗的結(jié)果，確定其相應(yīng)的許用應(yīng)力，以進行強度計算。適用：構(gòu)件體積不大，真實應(yīng)力相當復雜情況，如連接件等。實用計算假設(shè)：假設(shè)剪應(yīng)力在整個剪切面上均勻分布，等于剪切面上的平均應(yīng)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：錯動面。

剪力--Q：剪切面上的內(nèi)力。2、名義剪應(yīng)力--：3、剪切強度條件（準則）：nn（合力）（合力）PPPnnQ剪切面工作應(yīng)力不得超過材料的許用應(yīng)力。3、Practicalcalculationsofbearing1)Bearingforce―Pjy

：Theresultantforceactingonthetouchingplane.Bearing：Thephenomenonthatthereispressureonthepartialareaofamember.Bearingforce：Theresultantforceactingonthebearingplane，designatedbyPjy.Assumption：Bearingstressesaredistributeduniformlyovertheeffective bearingplane.SHEAR三、擠壓的實用計算1、擠壓力―Pjy

：接觸面上的合力。剪切擠壓：構(gòu)件局部面積的承壓現(xiàn)象。擠壓力：在接觸面上的壓力，記Pjy。假設(shè)：擠壓應(yīng)力在有效擠壓面上均勻分布。2)Bearingarea：Areaoftheprojectionplaneofthetouchingplaneinthedirectionperpendicularto

Pjy

3)Strengthconditionofbearing：

WorkingbearingstressshouldnotexceedthepermissiblebearingstressBearingareaSHEAR2、擠壓面積：接觸面在垂直Pjy方向上的投影面的面積。3、擠壓強度條件（準則）：

工作擠壓應(yīng)力不得超過材料的許用擠壓應(yīng)力。剪切擠壓面積4、ApplicationsSHEAR1)Checkthestrength2)Selectthecrosssectionarea3)Determinetheexternalload四、應(yīng)用剪切Example1

Awoodentenonjointisshowninthefigure.Knowingthatthequantitiesarea=b

=12cm，h=35cm，c=4.5cmand

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

木榫接頭如圖所示，a=b

=12cm，h=35cm，c=4.5cm,

P=40KN，試求接頭的剪應(yīng)力和擠壓應(yīng)力。解：受力分析如圖∶剪應(yīng)力和擠壓應(yīng)力剪切面和剪力為∶擠壓面和擠壓力為：剪切PPPPPPbachhSolution：Freebodydiagramofthekeyisshown inthefigureExample2Agearandashaftareconnectedbyakey（b×h×L=20×12×100).Thetorquethatthekeycantransmitism=2KNm.Knowingthediameteroftheshaftisd=70mm，thepermissibleshearingstressandthepermissiblebearingstressofthekeyarerespectively[]=60MPaand[jy]=100MPa.Trytocheckthestrengthofthekey.

mbhLSHEARmdP解：鍵的受力分析如圖[例2]

齒輪與軸由平鍵（b×h×L=20×12×100）連接，它傳遞的扭矩m=2KNm，軸的直徑d=70mm，鍵的許用剪應(yīng)力為[]=60M

Pa，許用擠壓應(yīng)力為[jy]=100MPa，試校核鍵的強度。剪切mbhLmdPAccordingtotheabovecalculation,strengthconditionsofthekeyaresatisfied.Checkthestrengthofshearandbearing

bhLSHEARdmQ綜上，鍵滿足強度要求。剪應(yīng)力和擠壓應(yīng)力的強度校核剪切bhLdmQSolution：Freebodydiagramofthekeyis showninthefigure[Example3]Agearandashaftareconnectedbyakey（b=16mm,h=10mm).Thetorquethatthekeycantransmitism=1600Nm.Knowingthediameteroftheshaftisd=50mm，thepermissibleshearingstressandthepermissiblebearingstressofthekeyarerespectively[]=80MPaand[jy]=240MPa.Trytodesignthelengthofthekey.

bhLSHEARmmmdP解：鍵的受力分析如圖[例3]

齒輪與軸由平鍵（b=16mm，h=10mm，）連接，它傳遞的扭矩m=1600Nm，軸的直徑d=50mm，鍵的許用剪應(yīng)力為[]=80MPa，許用擠壓應(yīng)力為[jy]=240MPa，試設(shè)計鍵的長度。剪切bhLmdPmmbhLStrengthconditionsoftheshearingstressandthebearingstressAccordingtotheabovecalculationSHEARdmQ剪切bhL剪應(yīng)力和擠壓應(yīng)力的強度條件綜上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：受力分析如圖[例4]

一鉚接頭如圖所示，受力P=110kN，已知鋼板厚度為t=1cm，寬度

b=8.5cm，許用應(yīng)力為[

]=160MPa；鉚釘?shù)闹睆絛=1.6cm，許用剪應(yīng)力為[]=140MPa，許用擠壓應(yīng)力為[jy]=320MPa，試校核鉚接頭的強度。（假定每個鉚釘受力相等。）剪切bPPttdPPP112233P/4Thesections2—2and3—3ofthesteelplatearethecriticalsections.ThestrengthconditionsofshearandbearingTherefore，thetie-inissafe.ttdPPP112233P/4SHEAR鋼板的2--2和3--3面為危險面剪應(yīng)力和擠壓應(yīng)力的強度條件綜上，接頭安全。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、軸向拉壓桿的內(nèi)力及軸力圖1、軸力的表示?2、軸力的求法?3、軸力的正負規(guī)定?剪切拉壓和剪切習題課為什么畫軸力圖?

應(yīng)注意什么?4、軸力圖：N=N(x)的圖象表示?PANBC簡圖APPNxP+Simplemethodtodetermineaxialforces:①Taketheleftpartofthesectionxastheobject,theaxialforceonthesectionxcanbecalculatedbyfollowingformula:

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

②Taketherightpartofthesectionxastheobject,theaxialforceN(x)ofpointxcanbecalculatedbythefollowingformulate:Where“”and“”denotethesumofrightdirectionforcesandthesumofleftdirectionforcesoftherightpartofsectionx.SHEAR軸力的簡便求法:①以x點左側(cè)部分為研究對象,x點的軸力N(x)由下式計算:

其中“P()”與“P()”為x點左側(cè)向左的所有外力與向右的所有外力。②以x點右側(cè)部分為研究對象，x點的軸力N(x)由下式計算：

其中“P()”與“P()”為x點右側(cè)向右的所有外力與向左的所有外力。剪切[Example1]Forces5P,8P,4PandPareactedatpointsA,B,CandDontherodrespectively，theirdirectionsareshowninthefigure.Trytoplottheaxial-forcediagramoftherod.ABCDO5P4PP8PNx–3P5PP2PSHEAR⊕⊕⊕○[例1]圖示桿的A、B、C、D點分別作用著5P、8P、4P、P的力，方向如圖，試畫出桿的軸力圖。剪切ABCDO5P4PP8PNx–3P5PP2P⊕⊕⊕○Positiveandnegativeofstress?1)Stressonthecrosssection:

2、Stressoftherodintensionorcompression

Criticalsectionandmaximumworkingstress?

2)Stressontheinclinedsection

Saint-Venantprinciple?

Stressconcentrations?sN(x)Ptasaxs0SHEAR應(yīng)力的正負規(guī)定?1、橫截面上的應(yīng)力:

二、拉壓桿的應(yīng)力危險截面及最大工作應(yīng)力?

2、拉壓桿斜截面上的應(yīng)力Saint-Venant原理?

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

1)Strengthdesigncriterion

①Checkstrength：②Designthecrosssectionarea：③Designtheload：SHEAR三、強度設(shè)計準則（StrengthDesignCriterion）：

1、強度設(shè)計準則?

①校核強度：②設(shè)計截面尺寸：③設(shè)計載荷：剪切1)Elasticlawoftherodwithequalaxialforces

2)Elasticlawwithvariableinternalforces

3)Elasticlawinuniaxialstressedstate

4、Deformationandstrainoftherodintensionorcompression

N(x)dxxPPSHEAR1、等軸力拉壓桿的彈性定律

2、變內(nèi)力拉壓桿的彈性定律3、單向應(yīng)力狀態(tài)下的彈性定律四、拉壓桿的變形及應(yīng)變剪切N(x)dxxPP4)Possion’sratio（orlateraldeformationfactor）5)Theenlarged-deformationdiagramandthemethodtodeterminethedisplacementC'ABCL1L2PC"SHEAR4、泊松比（或橫向變形系數(shù)）剪切5、小變形放大圖與位移的求法C'ABCL1L2PC"Assemblestress——initialstress

Temperaturestress①Equilibriumequations；

②Geometricequations—compatibilityequationsofdeformation；

③Physicalequations—elasticlaws；

④Complementaryequations：gettingfromgeometricequationsandphysicalequations；

⑤Solvingthecombinedequationsincludingofequilibriumequationsandcomplementequations.6)Thestepstosolvethestaticallyindeterminateproblem

SHEAR裝配應(yīng)力——預應(yīng)力溫度應(yīng)力剪切①平衡方程；

②幾何方程——變形協(xié)調(diào)方程；

③物理方程——彈性定律；

④補充方程：由幾何方程和物理方程得；

⑤解由平衡方程和補充方程組成的方程組。6、超靜定問題的處理方法步驟：5、Mechanicalpropertiesofthematerialsintensionandcompression3)Unloadedlaw；Coldhardening；Cold-drawntimeeffect.1)Elasticlaw4)Residualrelativeelongation5)PermanentrelativereductionofareaSHEAR2)Limitstress6)Permissiblestress五、材料在拉伸和壓縮時的力學性能3、卸載定律；冷作硬化；冷拉時效。1、彈性定律剪切4、延伸率5、面縮率1)Practicalcalculationofshear6、Practicalcalculationofshearandbearingoftheconnectingmemberoftherodsintensionandcompressionnn（Resultant）（Resultant

）PPPnnQShearingplane2)PracticalcalculationofbearingSHEAR1、剪切的實用計算六、拉(壓)桿連接部分的剪切與擠壓強度計算剪切nn（合力）（合力）PPPnnQ剪切面2、擠壓的實用計算BearingareaSHEARCheckthestrengthDesignthedimensionDesigntheexternalload剪切擠壓面積[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]

結(jié)構(gòu)如圖，AB、CD、EF、GH都由兩根不等邊角鋼組成，已知材料的[]=170MPa

，E=210GPa。

AC、EG可視為剛桿，試選擇各桿的截面型號和A、D、C點的位移。剪切P=300kN0.8m3.2m1.8m1.2m2m3.4m1.2mABCDFHq0=100kN/m解：①求內(nèi)力，受力分析如圖EGDq0=100kN/mEGACNGNCNANEND=NDP=300kN②DeterminetheareabystrengthconditionSHEAR剪切Dq0=100kN/mEGACNGNCNANEND=NDP=300kN②由強度條件求面積③Determinethetypesoftherodsreferencetothetable④DeterminethedeformationSHEAR剪切③試依面積值查表確定型鋼號④求變形⑤Determinethedisplacement.Thedeformationisshowninthefigure.ABDFHEGCC1A1E1D1G1SHEAR⑤求位移,變形圖如圖剪切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]

結(jié)構(gòu)如圖，AC、BD的直徑分別為:d1=25mm,d2=18mm,已知材料的[]=170MPa

，E=210GPa,AB可視為剛桿，試校核各桿的強度;求A、B點的位移△A和△B。(2)求當P作用于A點時,F點的位移△F′，△F′=△A是普遍規(guī)律：稱為位移互等定理。剪切BNBP=100kNNAAABCDP=100kN1.5m3m2.5mF解：求內(nèi)力，受力分析如圖②Checkthestrength③Determinethedeformation