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1、,CHAPTER 2 SHEAR,材料力学,第二章 剪切,CHAPTER 2 SHEAR,21 STRENGTH CALCULATION ABOUT SHEAR AND BEARING OF CONNECTING MEMBERS,EXCERCISE LESSONS OF SHEAR AND BEARING,第二章 剪 切,21 连接件的剪切与挤压强度计算,拉压和剪切习题课,Production of the shearing stress,SHEAR,剪切,剪应力的产生,21 STRENGTH CALCULATION ABOUT SHEAR AND BEARING OF CONNECTING
2、MEMBERS,1、Characteristics of loads and deformation of connecting members:,1)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
3、：It can pass general loads and can be dismounted.,bolt,SHEAR,21 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。,螺栓,rivet,Characteristic: It can pass general loads, but can not be dismounted.for example, the truss in a bridge is connected by
4、 it.,no gap,Characteristic: It can pass torques.,SHEAR,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,2)Characteristics of loads and deformation：,Use 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 close
5、., 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.,SHEAR,2、受力特点和变形特点：,剪切,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。,（ Resultant ）,（Resultant,Shearing plane：
6、 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.,SHEAR,剪切,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 ，其作用线与剪切面平行。,3)Three kinds of breakage a
7、t 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 to tension,The steel plate is weakened in the section in which the riv
8、et holes exist and stress in the weakened section increases so that the steel plate is easily broken due to tension at the connecting position.,SHEAR,剪切,3、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,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、Practical calculation of shear,Met
9、hod 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 calculations can be simplified are used. Then calculate its nominal stress, determine the corresponding permissible stres
10、s 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 pieces etc.,Assumption of practical calculation：Assume that shearing stress is distributed uniformly in the s
11、hearing plane and equal to the average shearing stress.,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)Shearing plane-AQ ： Shifting plane. Shearing force-Q：Internal force on the shea
12、ring plane.,2)Nominal shearing force-：,3)Strength condition of shear：,Working stress should not exceed the permissible stress.,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）：,工作应力不得超过材料的许用应力。,3、Practical calculations of bearing,1)Bearing forcePjy ：The resultant force acting on
13、the touching plane.,Bearing：The phenomenon that there is pressure on the partial area of a member. Bearing force：The resultant force acting on the bearing plane，designated by Pjy.,Assumption：Bearing stresses are distributed uniformly over the effective bearing plane.,SHEAR,三、挤压的实用计算,1、挤压力Pjy ：接触面上的合
14、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)Bearing area：Area of the projection plane of the touching plane in the direction perpendicular to Pjy,3)Strength condition of bearing： Working bearing stress should not exceed the permissible bearing stress,Bearing area,SHEAR,2、挤压面积：接触面
15、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、Applications,SHEAR,1)Check the strength,2)Select the cross section area,3)Determine the external load,四、应用,剪切,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. T
16、ry to determine the shearing stress and bearing stress for the joint.,Solution：Free body diagram is shown in the figure,：Shearing stress and bearing stress,Shearing force is,Bearing force is,SHEAR,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,
17、Solution：Free body diagram of the key is shown in the figure,Example 2 A gear and a shaft 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
18、 respectively = 60 MPa and jy= 100 MPa. Try to check the strength of the key.,SHEAR,解：键的受力分析如图,例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。,剪切,According to the above calculation, strength conditions of the key are satisfied.,Check the strength of
19、shear and bearing,SHEAR,综上，键满足强度要求。,剪应力和挤压应力的强度校核,剪切,Solution：Free body diagram of the key is shown in the figure,Example 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 she
20、aring stress and the permissible bearing stress of the key are respectively = 80MPa and jy = 240MPa.Try to design the length of the key.,SHEAR,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。,剪切,Strength conditions of the shearing stres
21、s and the bearing stress,According to the above calculation,SHEAR,剪切,剪应力和挤压应力的强度条件,综上,Solution：Free body diagram is shown in the figure,Example 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 = 160MP
22、a. Diameter of the rivet is d =1.6cm 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.）,SHEAR,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
23、 Pa ；铆钉的直径d=1.6cm，许用剪应力为= 140M 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。）,剪切,The sections 22 and 33 of the steel plate are the critical sections.,The strength conditions of shear and bearing,Therefore，the tie-in is safe.,SHEAR,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,1、Internal-force and axial-
24、force diagrams of the rod in tension and compression,1)Expression of Axial force?,2)Method to determine axial force?,3)Positive and negative of axial force?,Exercise lessons of tension and compression and shear,Why do we plot the axial-force diagram? What should we pay attention when we do it?,4)Axi
25、al-force diagram： Expressed by the diagram of N = N(x)?,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?,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 fol
26、lowing 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 “ ”denot
27、e the sum of right direction forces and the sum of left direction forces of the right part of section x.,SHEAR,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。,剪切,Example 1 Forces 5P, 8P, 4P and P are acted at p
28、oints A, B, C and D on the rod respectively，their directions are shown in the figure. Try to plot the axial-force diagram of the rod.,A,B,C,D,O,2P,SHEAR,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,A,B,C,D,O,2P,Positive and negative of stress?,1)Stress on the cross section:,2、Stress of the rod
29、 in tension or compression,Critical section and maximum working stress?,2)Stress on the inclined section,Saint-Venant principle?,Stress concentrations?,SHEAR,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,3、Strength design criterion：,1) Strength design criterion,Chec
30、k strength：,Design the cross section area：,Design the load：,SHEAR,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：,剪切,1)Elastic law of the rod with equal axial forces,2)Elastic law with variable internal forces,3)Elastic law in uniaxial stressed state,4、Deformation and strain of th
31、e rod in tension or compression,SHEAR,1、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,4)Possions ratio（or lateral deformation factor）,5)The enlarged-deformation diagram and the method to determine the displacement,SHEAR,4、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,Assemble stressinitial stress,Tempera
32、ture stress,Equilibrium 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
33、 steps to solve the statically indeterminate problem,SHEAR,装配应力预应力,温度应力,剪切,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：,5、Mechanical properties of the materials in tension and compression,3)Unloaded law；Cold hardening；Cold-drawn time effect.,1)Elastic law,4)Residual r
34、elative elongation,5)Permanent relative reduction of area,SHEAR,2)Limit stress,6)Permissible stress,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)Practical calculation of shear,6、Practical calculation of shear and bearing of the connecting member of the rods in tension and compression
35、,2)Practical calculation of bearing,SHEAR,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,剪切,2、挤压的实用计算,Bearing area,SHEAR,Check the strength,Design the dimension,Design the external load,剪切,挤压面积,Example 2 The rods AB、CD、EF and GH in the structure are shown in the figure Each of them consists two of rolled unequal-
36、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=300kN,0.8m,3.2m,1.8m,1.2m,2m,3.4m,A,B,C,D,F,H,q0=100kN/m,Solution：Determine the internal force.The free body diag
37、ram is shown in the figure.,E,G,SHEAR,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 M P a ，E=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。,剪切,P=300kN,0.8m,3.2m,1.8m,1.2m,2m,3.4m,A,B,C,D,F,H,q0=100kN/m,解：求内力，受力分析如图,E,G,D,q0=100kN/m,E,G,A,C,NG,NC,NA,NE,ND,=ND,P=300kN,Determine the area by strength condition,
38、SHEAR,剪切,D,q0=100kN/m,E,G,A,C,NG,NC,NA,NE,ND,=ND,P=300kN,由强度条件求面积,Determine the types of the rods reference to the table,Determine the deformation,SHEAR,剪切,试依面积值查表确定型钢号,求变形,Determine the displacement.The deformation is shown in the figure.,A,B,D,F,H,E,G,C,C1,A1,E1,D1,G1,SHEAR,求位移,变形图如图,剪切,A,B,D,F,H,
39、E,G,C,C1,A1,E1,D1,G1,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 points A an B. (2
40、) 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.,A,B,C,D,P=100kN,1.5m,3m,2.5m,F,Solution：Determine the internal force The free body diagram is shown in the fingure.,SHEAR,例3 结构如图，AC、BD的直径分别为:d1 =2
41、5mm, 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。,剪切,A,B,C,D,P=100kN,1.5m,3m,2.5m,F,解：求内力，受力分析如图,Check the strength,Determine the deformation and the displacement,SHEAR,校核强度,求变形及位移,剪切,Determine displacement F of point F when force P is acte
42、d on point A,P=100kN,1.5m,3m,2.5m,A,F,B,C,D,SHEAR,求当P作用于A点时,F点的位移F,剪切,P=100kN,1.5m,3m,2.5m,A,F,B,C,D,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 point A,Solut
43、ion：Determine the area in accordance with the strength condition.,P=100kN,12m,12m,A,SHEAR,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,剪切,P=100kN,12m,12m,A,Chapter 2 Exercises 1、What is the difference between bearing and compression？ 2、Knowing the relation between the allo
44、wable 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：,SHEAR,第二章 练习题 一、挤压与压缩有何区别？ 二、已知螺栓材料的许用剪应力与许用拉应力之间的关系为=0.6，试求螺栓直径d与螺栓头高度h的合理比值。,剪切,解：,3、As shown in the figu
45、re, 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：,SHEAR,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d=5mm，其极限剪应力 ,试求该联轴器允许传递的最大力偶矩M. 解：,剪切,END,本章结束,