Chapter 3 Bilingual Edition return exit

Chapter 3 Practical calculation about shear and bearing of connecting members 连接件的剪切与挤压实用计算 1. Characteristics of loads and deformation of connecting members 连接件的受力特点和变形特点： 1) Connecting member The structure member that connects one member to another is called the connecting member. Such as：bolts, rivets, keys etc. 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。 bolt Chapter 3

For example, the truss in a bridge is connected by it. The connecting member is small, but it plays the role of passing loads. 连接件虽小，却起着传递载荷的作用。 It can pass general loads and can be dismounted. bolt no gap rivet It can pass general loads, but can not be dismounted For example, the truss in a bridge is connected by it. Chapter 3

Key Bind 平键连接 m Key键 单键连接 m Shaft 轴 wheel 轮毂 It can pass torques 特点：传递扭矩。 花键连接 Chapter 3

2) Characteristics of loads and deformation： 受力特点和变形特点 ① Characteristic of loads: The member is subjected to two equal and opposite forces. The acting lines of these two forces are very close. 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。 m direct shear 剪切 Chapter 3

构件沿两组平行力系的交界面发生相对错动。 ② Characteristic of deformation 变形特点 : 构件沿两组平行力系的交界面发生相对错动。 Two parts subjected to two equal and opposite forces tend to shift over one another along the junction plane of two forces. 我们将错位横截面称为剪切面 The plane along which two parts of the member tend to shift over one another. shearing plane shearing plane Chapter 3

3) Internal force on shearing plane 剪切面上的内力 Internal force — Shearing force , its acting line is parallel to the shearing plane. How about the shearing stress？ 剪力 The real stress is quite complex. 剪应力在剪切面上的分布情况是非常 复杂的，工程上往往采用实用计算的方法。 2. Practical calculation of shear Assumption of practical calculation: Assume that shearing stress is distributed uniformly in the shearing plane. 实用计算假设: 假设剪应力在整个剪切面上均匀分布 Chapter 3

名义剪应力-- Shearing force 受剪切螺栓剪切面面积： 2) Nominal shearing stress--： 名义剪应力-- Shearing force The area of shearing plane 受剪切螺栓剪切面面积： 受剪切单键剪切面面积： b l h d Chapter 3

In the case of single shear 3) Single shear and double shear 单剪和双剪 In the case of single shear shearing plane In the case of double shear middle the side Chapter 3

Permissible shear stress 4) Strength condition of shear 剪切强度条件(准则) : Working stress should not exceed the permissible stress. [τ]——许用剪应力 Permissible shear stress [τ]可以通过材料剪切实验来确定 在该试验中, 应使试样的受力尽可能地接近实际联接件的情况,求得试样失效时的极限载荷Fs0，然后根据公式τ0＝Fs0/ A求出名义极限剪应力τ0 ，除以安全系数 n，得许用剪应力[τ]，从而建立强度条件。 对于塑性较好的低碳钢材料，根据实验所积累的数据并考虑安全系数: [τ] =(0.6—0.8)[σ] Chapter 3

The bolt is in double shear. planes to study. two Example 22 For the joint shown in the figure. Knowing that d=20mm, = 8mm, F=15kN, the bolt is made of steel having [ ]=30MPa, try to check the strength of the bolt . Solution: The bolt is in double shear. Take the portion of the bolt between shearing planes to study. two The area of shearing plane Chapter 3

Thus the shearing stress is Conclusion：the strength condition of the key is satisfied. Chapter 3

the area of shearing plane: Example 23 The punch is stamping a hexagonal holes on a metal plate as shown in Fig. Knowing that the thickness of the plate is t, the strength limit of the metal plate in shear is , determine the minimum stamping force P required. stamping force Solution: the shearing force Fs=P the area of shearing plane: Chapter 3

Bearing plane 挤压面—该接触面。 3. Practical calculations of bearing 挤压的实用计算 Bearing: The phenomenon that there is pressure on the partial area of a member. 挤压:连接件和被连接件接触面相互压紧的现象。 接触面上由于挤压力太大而产生塑性变形，形成的破坏称挤压破坏。 bearing plane Bearing plane 挤压面—该接触面。 Bearing force (挤压力):The resultant force acting on the bearing plane. 在挤压面上的压力. Chapter 3

effective bearing area Bearing stress : 由挤压力引起的应力称为挤压应力。 bearing plane 与剪切应力的分布一样，挤压应力的分布也非常复杂，工程上往往采取实用计算的办法，一般假设挤压应力平均分布在有效挤压面上。 Assumption: Bearing stresses are distributed uniformly over the effective bearing plane. effective bearing area 挤压力分布 有效挤压面面积 Chapter 3

挤压面积：接触面在垂直 方向上的投影面的面积。 挤压面积：接触面在垂直 方向上的投影面的面积。 Bearing area: Area of the projection plane of the touching plane in the direction perpendicular to . bearing plane 接触面为平面时，则计算挤压面积为接触面的面积。 d h b l h 键连接 接触面为半圆柱面，则计算挤压面为直径投影面。 Chapter 3

Strength condition of bearing 挤压强度条件: Working bearing stress should not exceed the permissible bearing stress. [σbs]可由有关规范中查到,通常大于许用应力。 对于钢材，一般可取 挤压破坏也是连接的可能破坏方式, 挤压应力是连接件与被连接件之间的相互作用, 因此, 当两者材料不相同时, 应校核挤压许用应力较低的材料的挤压强度。 Chapter 3

diagram of the bolt is shown in the figure. Example 24 For the joint shown in the figure, knowing that [bs] = 100 MPa, d=20mm, = 8mm, F=15kN. Check the strength of bearing of the bolt. Solution: Free body diagram of the bolt is shown in the figure. Check the bearing strength of the middle portion: the most dangerous Conclusion：the bearing strength condition of the bolt is satisfied. Chapter 3

挤压力： Fb =15kN，挤压面积: Abs = 2td ∴ d ≥ 17.8mm。若取标准件，查手册，d =20mm。 Example 25 铆钉和板用同一种材料制成,已知t =8mm，[τ]=30MPa, [σbs]=100MPa, P=15kN, 试选择直径d 。 Solution: 取铆钉中段研究 t 2t ① 剪切强度计算 剪力： ② 挤压强度计算 挤压力： Fb =15kN，挤压面积: Abs = 2td ∴ d ≥ 17.8mm。若取标准件，查手册，d =20mm。 Chapter 3

(2) 若铆钉和板用不同材料制成，计算挤压强度时， 应以铆钉为研究对象还是以板为研究对象？ 思考： t 3t (1) 若中间板的厚度为3t ， 应取哪段研究？ 中段: 两端: (2) 若铆钉和板用不同材料制成，计算挤压强度时， 应以铆钉为研究对象还是以板为研究对象？ Chapter 3

① Free body diagram is shown in the figure Example 26 A wooden tenon joint(木榫接头)is shown in the figure. Knowing a = b =12cm, l=35cm, c=4.5cm and P=40KN. Try to determine the shearing stress and bearing stress for the joint. Solution: ① Free body diagram is shown in the figure shearing plane bearing plane Chapter 3

②Shearing stress and bearing stress Shearing force is Shearing area: Bearing force is Bearing area: ②Shearing stress and bearing stress shearing plane bearing plane Chapter 3

Free body diagram is shown in the figure. Example 27 A gear and a shaft are connected by a key（b×h×L=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 [bs]= 100 MPa. Try to check the strength of the key. M d h b Solution: ① check the M strength of shear O Free body diagram is shown in the figure. Chapter 3

free body diagram is shown in the figure. ② check the strength of bearing take the upper portion of the key to study， free body diagram is shown in the figure. Conclusion： according to the above calculation, strength conditions of the key are satisfied. Chapter 3

4.Three kinds of breakage at joint ② Breakage due to bearing 连接处破坏的三种形式 n ① Failure due to shear 剪切破坏 Snip along the shearing plane of the rivet, such as along section n– n. 沿铆钉的剪切面剪断，如 沿n– n面剪断 。 ② Breakage due to bearing 挤压破坏 Fail due to mutual bearing between the rivet and the plate in their connecting plane. 铆钉与板在相互接触面上因挤压而压溃,使连接松动,发生破坏。 Chapter 3

② Breakage due to bearing ① Failure due to shear ② Breakage due to bearing 剪切破坏 挤压破坏 n ③ Breakage due to tension 拉伸破坏 The 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. 钢板在受铆钉孔削弱的截面处，应力增大，易在连接处拉断。 Chapter 3

Assume the force acted on each rivet is equal. Composite fastener group joints 钉群 钉子分布对称于轴线，则每个钉子平均分担外荷载。 Assume the force acted on each rivet is equal. 铆钉 被连接件 盖板 挤压 (拉和挤压) (挤压和剪切) 拉 Chapter 3

分析：可能的破坏形式有： （1）铆钉剪切 破坏。 （2）铆钉或板的挤压破坏。 （3）钢板拉断。 Example 28 一铆钉接头如图所示，铆钉和板用同一种材料制成，P=110KN, 铆钉的直径d=1.6cm，板厚t=1cm,宽度b=8.5cm ,其[τ]=140MPa, [σbs]=320MPa, [σ]=160MPa，试校核此铆接头的强度。 分析：可能的破坏形式有： b t t （1）铆钉剪切 破坏。 （2）铆钉或板的挤压破坏。 （3）钢板拉断。 Chapter 3

（1）铆钉剪切 破坏。 （2）铆钉或板的挤压破坏。 （3）钢板拉断。 Example 28 The lap joint shown in figure is fastened by four rivets of 16mm diameter, the plate and rivets are made of a same material and [τ]=140MPa, [σbs]=320MPa, [σ]=160MPa, t=1cm, b=8.5cm, assuming that the axial load P=110KN. Check the strength of the lap joint. t b t 分析：可能的破坏形式有： （1）铆钉剪切 破坏。 （2）铆钉或板的挤压破坏。 （3）钢板拉断。 Chapter 3

① Free body diagram is shown in the figure Solution: b ① Free body diagram is shown in the figure ② The strength conditions of shear and bearing t Chapter 3

End of this chapter ③ The strength condition of the steel plate m n ③ The strength condition of the steel plate b The sections m—m and n—n of the steel plate are the critical sections. + O x End of this chapter Return to contents of this chapter Therefore, the lap joint is safe. Chapter 3 exit

教材： 刘鸿文编，《材料力学》（上、下册）高等教育出版社，2010年9月第5版。 第三章 教材上的作业 Page 67 2.56， 2.58 教材： 刘鸿文编，《材料力学》（上、下册）高等教育出版社，2010年9月第5版。 Goodbye