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1、第1页/共84页CHAPTER 2 SHEAR221 STRENGTH CALCULATION ABOUT SHEAR 1 STRENGTH CALCULATION ABOUT SHEAR AND BEARING OF CONNECTING MEMBERSAND BEARING OF CONNECTING MEMBERSEXCERCISE LESSONS OF SHEAR AND BEARINGEXCERCISE LESSONS OF SHEAR AND BEARING第2页/共84页第二章 剪 切221 1 连接件的剪切与挤压强度计算第3页/共84页Production of the she
2、aring stress第4页/共84页剪应力的产生第5页/共84页221 STRENGTH CALCULATION ABOUT SHEAR AND 1 STRENGTH CALCULATION ABOUT SHEAR AND BEARING OF CONNECTING MEMBERSBEARING OF CONNECTING MEMBERS1 1、Characteristics of loads and deformation of connecting members:Characteristics of loads and deformation of connecting member
3、s:1)Connecting member1)Connecting member The structure member that connects one member to another is called the connecting member.Such as：bolts、rivets、keys etc.The connecting member is small，but it plays the role of passing loads.Characteristic：It can pass general loads and can be dismounted.PPbolt第
4、6页/共84页221 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 1、连接件 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螺栓第7页/共84页PPrivetCharacteristic:It can pass general loads,but can not be dismounted.for example,the truss in a bridge is connected by it.no gapCharacteristic:It can pass torques.mShaf
5、tKeyGearm第8页/共84页PP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。m第9页/共84页2)2)Characteristics of loads and deformation：nn（Resultant）（Resultant）PPUse a rivet as an example：Characteristic of loads：The rivet is subjected to two equal and opposite forces.The acting lines of these two forces are very cl
6、ose.Characteristic of deformation:Two parts subjected to two equal and opposite forces tend to shift over one another along the junction plane of two forces.第10页/共84页2 2、受力特点和变形特点：nn（合力）（合力）PP以铆钉为例：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。变形特点：构件沿两组平行力系的交界面发生相对错动。第11页/共84页nn（Resultant）（ResultantP
7、PShearing plane：The plane along which two parts of the member tend to shift over one another.Such as n n.Internal force on shearing plane：Internal force Shearing force Q，its acting line is parallel to the shearing plane.PnnQShearing plane第12页/共84页nn（合力）（合力）PP剪切面：构件将发生相互的错动面，如n n。剪切面上的内力：内力 剪力Q，其作用线与
8、剪切面平行。PnnQ剪切面第13页/共84页nn(Resultant)(Resultant）PP3)Three kinds of breakage at joint：Failure due to shear Snip along the shearing plane of the rivet,such as along section n n.Breakage due to bearing Fail due to mutual bearing between the rivet and the steel plate in their connecting plane.Breakage due
9、 to tensionPnnQShearing plane The steel plate is weakened in the section in which the rivet holes exist and stress in the weakened section increases so that the steel plate is easily broken due to tension at the connecting position.第14页/共84页nn（合力）（合力）PP3、连接处破坏的三种形式：剪切破坏 沿铆钉的剪切面剪断，如 沿n n面剪断。挤压破坏 铆钉与钢
10、板在相互接触面 上因挤压而使溃压连接松动，发生破坏。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。第15页/共84页2 2、Practical calculation of shearMethod of the practical calculation：According to possibility of breakage of the member some assumptions by which basic characteristic subjected to force actions can be reflected and calculation
11、s can be simplified are used.Then calculate its nominal stress,determine the corresponding permissible stress in accordance with the result of direct test.At last do the strength calculation.Applying range：volume of the member is not large and real stress is quite complex.Such as the connecting piec
12、es etc.Assumption of practical calculation：Assume that shearing stress is distributed uniformly in the shearing plane and equal to the average shearing stress.第16页/共84页二、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
13、上均匀分布，等于剪切面上的平均应力。第17页/共84页1)Shearing plane-AQ：Shifting plane.Shearing force-Q：Internal force on the shearing plane.2)Nominal shearing force-：3)Strength condition of shear：nn（Resultant）(Resultant）PPPnnQShearing planeWorking stress should not exceed the permissible stress.,where第18页/共84页1、剪切面-AQ：错动面。
14、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。第19页/共84页3 3、Practical calculations of bearing1)Bearing forcePjy：The resultant force acting on the touching plane.Bearing：The phenomenon that there is pressure on the partial area of a member.Bearing force：The resultant force acti
15、ng on the bearing plane，designated by Pjy.Assumption：Bearing stresses are distributed uniformly over the effective bearing plane.第20页/共84页三、挤压的实用计算1、挤压力Pjy：接触面上的合力。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。第21页/共84页2)Bearing area：Area of the projection plane of the touching plane in the di
16、rection perpendicular to P Pjy 3)Strength condition of bearing：Working bearing stress should not exceed the permissible bearing stressBearing area第22页/共84页2、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。挤压面积第23页/共84页4 4、ApplicationsApplications1)Check the strength2)Select the cross secti
17、on area3)Determine the external load第24页/共84页四、应用四、应用第25页/共84页Example 1 A wooden tenon joint is shown in the figure.Knowing that the quantities are a=b=12cm，h=35cm，c=4.5cm and P=40KN.Try to determine the shearing stress and bearing stress for the joint.Solution：Free body diagram is shown in the figu
18、re：Shearing stress and bearing stressShearing force isBearing force isPPPPPPbachh第26页/共84页 例1 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：PPPPPPbachh第27页/共84页Solution：Free body diagram of the key is shown in the figureExample Example 2 2 A gear and a sha
19、ft are connected by a key（bhL=20 12 100).The torque that the key can transmit is m=2KNm.Knowing the diameter of the shaft is d=70mm，the permissible shearing stress and the permissible bearing stress of the key are respectively=60 MPa and jy=100 MPa.Try to check the strength of the key.mbhLmdP第28页/共8
20、4页解：键的受力分析如图 例2 2 齿轮与轴由平键（bhL=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=100M Pa，试校核键的强度。mbhLmdP第29页/共84页According to the above calculation,strength conditions of the key are satisfied.Check the strength of shear and bearing bhLdmQ第30页/共84页综上，键满足强度要求。剪应力和挤压应力的强度校核bhLdmQ第31页/共84页S
21、olution：Free body diagram of the key is shown in the figureExample 3 A gear and a shaft are connected by a key（b=16mm,h=10mm).The torque that the key can transmit is m=1600Nm.Knowing the diameter of the shaft is d=50mm，the permissible shearing stress and the permissible bearing stress of the key are
22、 respectively =80MPa and jy=240MPa.Try to design the length of the key.bhLmmmdP第32页/共84页解：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=80M Pa，许用挤压应力为 jy=240M Pa，试设计键的长度。bhLmdPmm第33页/共84页bhLStrength conditions of the shearing stress and the bearing stressAccording to the ab
23、ove calculation dmQ第34页/共84页bhL剪应力和挤压应力的强度条件综上dmQ第35页/共84页Solution：Free body diagram is shown in the figureExample 4 A riveted tie-in acted by force P=110kN is shown in the figure.Knowing the thickness is t=1cm，width of it is b=8.5cm.The permissible stress is =160MPa.Diameter of the rivet is d=1.6cm
24、 and the permissible shearing stress is=140MPa，the permissible bearing stress is jy=320MPa.Try to check the strength of the riveting.(Assume the force acted on each rivet is equal.）bPPttdPPP112233P/4第36页/共84页解：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=160M Pa；铆钉的直径d=1.6cm，许用剪应力为=140
25、M Pa，许用挤压应力为jy=320M Pa，试校核铆接头的强度。（假定每个铆钉受力相等。）bPPttdPPP112233P/4第37页/共84页The sections 22 and 33 of the steel plate are the critical sections.The strength conditions of shear and bearingTherefore，the tie-in is safe.ttdPPP112233P/4第38页/共84页钢板的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。ttdPPP112233P/4第39页/共84页1
26、1、Internal-force and axial-force diagrams of the rod in tension and compression1)Expression of Axial force?2)Method to determine axial force?3)Positive and negative of axial force?Why do we plot the axial-force diagram?What should we pay attention when we do it?4)Axial-force diagram：Expressed by the
27、 diagram of N=N(x)?PANASimple sketchBCPPNxP+第40页/共84页一、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?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+第41页/共84页Simple method to determine axial forces:Take the left part of the section x as the object,the axial force on the section x can be calculated by followin
28、g formula:Where“P()”and“P()”express the sum of left direction forces and the sum of right direction forces of the left part of the section x.Take the right part of the section x as the object,the axial force N(x)of point x can be calculated by the following formulate:Where“”and“”denote the sum of ri
29、ght direction forces and the sum of left direction forces of the right part of section x.第42页/共84页轴力的简便求法: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。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。第43页/共84页Example 1 Forces 5Forces 5P,P,8 8P,P,4 4P P and and P P are a
30、cted at points are acted at points A,B,C A,B,C andand D D on the rodon the rod respectively respectively，their directions are shown in the figure.Try to their directions are shown in the figure.Try to plot the axial-force diagram of the rod.plot the axial-force diagram of the rod.ABCDO5P4PP8PNx3P5PP
31、2P第44页/共84页 例例1 1 1 1 图示杆的图示杆的A A、B B、C C、D D点分别作用着点分别作用着5 5P P、8 8P P、4 4P P、P P的力，方向如图，试的力，方向如图，试画出杆的轴力图。画出杆的轴力图。ABCDO5P4PP8PNx3P5PP2P第45页/共84页Positive and negative of stress?1)Stress on the cross section:2 2、Stress of the rod in tension or compressionCritical section and maximum working stress?2)
32、Stress on the inclined sectionSaint-Venant principle?Stress concentrations?N(x)Px第46页/共84页应力的正负规定?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?N(x)Px第47页/共84页3 3、Strength design criterion：1)Strength design criterionCheck strength：Design the cross section area：Design the load：第48页/共8
33、4页三、强度设计准则（Strength Design Criterion）：1、强度设计准则?校核强度：设计截面尺寸：设计载荷：第49页/共84页1)Elastic law of the rod with equal axial forces2)Elastic law with variable internal forces3)Elastic law in uniaxial stressed state4、Deformation and strain of the rod in tension or compressionN(x)dxxPP第50页/共84页1、等轴力拉压杆的弹性定律2、变内
34、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N(x)dxxPP第51页/共84页4)Possions ratio（or lateral deformation factor）5)The enlarged-deformation diagram and the method to determine the displacement CABCL1L2PC第52页/共84页4、泊松比（或横向变形系数）5、小变形放大图与位移的求法CABCL1L2PC第53页/共84页Assemble stressinitial stress Temperature stressEquilib
35、rium equations；Geometric equationscompatibility equations of deformation；Physical equationselastic laws；Complementary equations：getting from geometric equations and physical equations；Solving the combined equations including of equilibrium equations and complement equations.6)The steps to solve the
36、statically indeterminate problem 第54页/共84页装配应力预应力温度应力平衡方程；几何方程变形协调方程；物理方程弹性定律；补充方程：由几何方程和物理方程得；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：第55页/共84页5 5、Mechanical properties of the materials in tension and compression3)Unloaded law；Cold hardening；Cold-drawn time effect.1)Elastic law4)Residual relative elongati
37、on5)Permanent relative reduction of area2)Limit stress6)Permissible stress第56页/共84页五、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4、延伸率5、面缩率第57页/共84页1)Practical calculation of shear6 6、Practical calculation of shear and bearing of the connecting member of the rods in tension and compressionnn（Resultant）（Res
38、ultant）PPPnnQShearing plane2)Practical calculation of bearing第58页/共84页1、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nn（合力）（合力）PPPnnQ剪切面2、挤压的实用计算第59页/共84页Bearing areaCheck the strengthDesign the dimensionDesign the external load第60页/共84页挤压面积第61页/共84页Example Example 2 The rods AB、CD、EF and GH in the structure are sho
39、wn in the figure Each of them consists two of rolled unequal-legs steels.Knowing =170MPa，E=210GPa.Rods AC and EG may be seen as rigid rod.Try to select the section dimension of each rod and to determine the displacements of points A、D and C.P=300kN0.8m3.2m1.8m1.2m2m3.4m1.2mABCDFHq0=100kN/mSolution：D
40、etermine the internal force.The free body diagram is shown in the figure.EG第62页/共84页 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 =170=170 M P a ，E=210=210 G P a。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。P=300kN0.8m3.2m1.8m1.2m2m3.4m1.2mABCDFHq0=100kN/m解：求内力，受力分析如图EG第63页/共84页Dq0=100kN/mEGACNGNCNANEND=NDP=300kNDetermi
41、ne the area by strength condition第64页/共84页Dq0=100kN/mEGACNGNCNANEND=NDP=300kN由强度条件求面积第65页/共84页Determine the types of the rods reference to the tableDetermine the deformation第66页/共84页试依面积值查表确定型钢号求变形第67页/共84页Determine the displacement.The deformation is shown in the figure.ABDFHEGCC1A1E1D1G1第68页/共84页求
42、位移,变形图如图ABDFHEGCC1A1E1D1G1第69页/共84页Example Example 3 Diameters of rod AC and BD in the structure shown in the figure are respectively d1=25mm,d2=18mm.Knowing =170MPa，E=210GPa.Rod AB may be seen as a rigid rod.Try to(1)check the strength of each rod and to determine the displacements A and B of point
43、s A an B.(2)determine the displacement Fof point F when the force P is acted on point A.F=A is a general law,which is called theorem of conjugate displacement.BNBP=100kNNAAABCDP=100kN1.5m3m2.5mFSolution：Determine the internal forceThe free body diagram is shown in the fingure.第70页/共84页 例3 结构如图，AC、BD
44、的直径分别为:d1=25mm,d2=18mm,已知材料的=170 M Pa，E=210 G Pa,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。BNBP=100kNNAAABCDP=100kN1.5m3m2.5mF解：求内力，受力分析如图第71页/共84页Check the strengthDetermine the deformation and the displacement第72页/共84页校核强度求变形及位移第73页/共84页Determine displacement F of point F
45、when force P is acted on point A P=100kN1.5m3m2.5mAFBCD第74页/共84页求当P作用于A点时,F点的位移FP=100kN1.5m3m2.5mAFBCD第75页/共84页Example Example 4 The structure is shown in the figure.Knowing=2MPa，E=20GPa,unit weight of concrete =22kN/m.Try to design the areas of the two parts and to determine the displacement A of p
46、oint ASolution：Determine the area in accordance with the strength condition.P=100kN12m12mA第76页/共84页 例4 结构如图，已知材料的=2MPa，E=20GPa,混凝土容重=22kN/m，试设计上下两段的面积并求A点的位移A。解：由强度条件求面积P=100kN12m12mA第77页/共84页 Chapter 2 Exercises 1、What is the difference between bearing and compression？2、Knowing the relation between
47、 the allowable shearing stressand the allowable tensile stressof the bolt material is=0.6，Try to determine the reasonable ratio of the diameter d of the bolt to the height h of the bolt cap。Solution：第78页/共84页 第二章第二章 练习题练习题 一、挤压与压缩有何区别？一、挤压与压缩有何区别？二、已知螺栓材料的许用剪应力二、已知螺栓材料的许用剪应力 与许用与许用拉应力拉应力 之间的关系为之间的关系
48、为=0.6=0.6，试求螺，试求螺栓直径栓直径d d与螺栓头高度与螺栓头高度h h的合理比值。的合理比值。解：解：第79页/共84页 3、As shown in the figure,when the safe crosshead is over-loaded,the safe pin will be snipped.Knowing for the safe pin,the average diameter is d=5mm and the limit shearing stress is .Try to determine the maximum force couple M that can be transmitted by the crosshead.Solution：第80页/共84页 三、图示安全联轴器超载时，安全销即被剪三、图示安全联轴器超载时，安全销即被剪断。已知安全销的平均直径断。已知安全销的平均直径d=5mmd=5mm，其极限剪应，其极限剪应力力 ,试求该联轴器允许传递的最大试求该联轴器允许传递的最大力偶矩力偶矩M.M.解：解：第81页/共84页第82页/共84页第83页/共84页感谢您的观看！第84页/共84页