1、,CHAPTER 1AXIAL TENSION AND COMPRESSION,1,2,材料力学,第一章 轴向拉伸和压缩,CHAPTER 1 AXIAL TENSION AND COMPRESSION,11 Concepts and practical examples of axial tension and compression 12 Internal force、method of section、axial force and its diagram 13 Stresses on the section and strength conditions,1-4 Deformation
2、of the rod in axial tension and compression law of elasticity,1-5 Elastic strain energy of the rod in axial tension and compression,1-6 Statically indeterminate problems and their treatment methods of axial tension and compression,1-7 Mechanical properties of materials in axial tension and compressi
3、on,3,4,11 轴向拉压的概念及实例 12 内力、截面法、轴力及轴力图 13 截面上的应力及强度条件,第一章 轴向拉伸和压缩,1-4 拉压杆的变形 弹性定律,1-5 拉压杆的弹性应变能,1-6 拉压超静定问题及其处理方法,1-7 材料在拉伸和压缩时的力学性能,11 CONCEPTS AND PRACTICAL EXAMPLES OF AXIAL TENSION AND COMPRESSION,Characteristic of the external force：The acting line of the resultant of external forces is coincide
4、d with the axis of the rod.,1、Concepts,Characteristic of the deformation：Deformation of the rod is mainly elongation or contraction along the axis of the rod and companied with lateral reduction or enlargement.,Axial tension ：Deformation of the rod is axial elongation and lateral shortening.,Axial c
5、ompression：Deformation of the rod is axial shortening and lateral enlargement.,AXIAL TENSION AND COMPRESSION,5,6,拉压,11 轴向拉压的概念及实例,轴向拉压的外力特点：外力的合力作用线与杆的轴线重合。,一、概念,轴向拉压的变形特点：杆的变形主要是轴向伸缩，伴随横向缩扩。,轴向拉伸：杆的变形是轴向伸长，横向缩短。,轴向压缩：杆的变形是轴向缩短，横向变粗。,In axial compression，the corresponding force is called compressive
6、 force.,In axial tension，the corresponding force is called tensile force.,Mechanical models are shown in the figures,AXIAL TENSION AND COMPRESSION,7,8,拉压,轴向压缩，对应的力称为压力。,轴向拉伸，对应的力称为拉力。,力学模型如图,Practical examples in engineering,2、,AXIAL TENSION AND COMPRESSION,9,10,拉压,AXIAL TENSION AND COMPRESSION,11,1
7、2,拉压,1、Internal forceInternal force is the resultant of internal forces, which is acting mutually between two neighbour parts inside the body,caused by the external forces.,12 INTERNAL FORCE、METHOD OF SECTION、AXIAL FORCE AND ITS DIAGRAM,AXIAL TENSION AND COMPRESSION,13,14,拉压,一、内力指由外力作用所引起的、物体内相邻部分之间
8、分布内力系的合成（附加内力）。,12 内力 截面法 轴力及轴力图,2、Method of section axial force,Calculation of the internal forces is the foundation to analyze the problems of strength、rigidity、stability etc. The general method to determine internal forces is the method of section.,1). Basic steps of the method of section： Cut of
9、f：Assume to separate the rod into two distinct parts in the section in which the internal forces are to be determined. Substitute：Take arbitrary part and substitute the action of another part on it by the corresponding internal force in the cut-off section. Equilibrium：Set up equilibrium equations f
10、or the remained part and determine the unknown internal forces according to the external forces acted on it.（Here the internal forces in the cut-off section are the external forces for the remained part）,AXIAL TENSION AND COMPRESSION,15,16,拉压,二、截面法 轴力,内力的计算是分析构件强度、刚度、稳定性等问题的基础。求内力的一般方法是截面法。,1. 截面法的基
11、本步骤： 截开：在所求内力的截面处，假想地用截面将杆件一分为二。 代替：任取一部分，其弃去部分对留下部分的作用，用作用在截开面上相应的内力（力或力偶）代替。 平衡：对留下的部分建立平衡方程，根据其上的已知外力来计算杆在截开面上的未知内力（此时截开面上的内力对所留部分而言是外力）。,2. Axial forceinternal force of the rod in axial tension or compression，designated by N .,Such as： Determine N by the method of section.,Cut off:：,Substitute：,
12、Equilibrium：,AXIAL TENSION AND COMPRESSION,17,18,拉压,2. 轴力轴向拉压杆的内力，用N 表示。,例如： 截面法求N。,截开：,代替：,平衡：,Reflected the variety relation betweenthe corresponding axial force and the position of the section. Find out value of the maximum axial force and the position of the section in which the maximum axial fo
13、rce act. That is to determine the position of the critical section and supply the information for the calculation of strength.,3、 Diagram of the axial force sketchexpression of N (x),3). Sign conventions for the axial force:,axial force N (tensile force)is positive when its direction point to the ou
14、tward direction of the normal line of the section, (compressive force)negative inward,x,meaning,AXIAL TENSION AND COMPRESSION,19,20,反映出轴力与横截面位置变化关系，较直观； 确定出最大轴力的数值 及其所在横截面的位置， 即确定危险截面位置，为 强度计算提供依据。,拉压,三、 轴力图 N (x) 的图象表示。,3. 轴力的正负规定:,N 与外法线同向,为正轴力(拉力),N与外法线反向,为负轴力(压力),N,x,P,意义,Example 1 The forces wi
15、th magnitudes 5P、8P、4P and P act respectively at points A、B、C、D of the rod. Their directions are shown in the figure. Try to plot the diagram of the axial force of the rod.,Solution： Determine the internal force N1 in segment OA. Take the free body as shown in the figure.,AXIAL TENSION AND COMPRESSI
16、ON,21,22,拉压,例1 图示杆的A、B、C、D点分别作用着大小为5P、8P、4P、P 的力，方向如图，试画出杆的轴力图。,解： 求OA段内力N1：设置截面如图,Similarly，we get the internal forces in segment AB、BC、CD . They are respectively：,N2= 3P N3= 5P N4= P,The diagramof the axial force is shownin the right figure .,D,PD,N,x,2P,3P,5P,P,AXIAL TENSION AND COMPRESSION,23,24
17、,拉压,同理，求得AB、BC、CD段内力分别为：,N2= 3P N3= 5P N4= P,轴力图如右图,D,PD,N,x,2P,3P,5P,P,Simple method to plot the diagram of axial force： From the left to the right:,Characteristic of the diagram of the axial force：Value of sudden change = concentrated load,If meeting the force P to the left ，the increase of the ax
18、ial force N is positive； If meeting the force to the right ，the increase of the axial force N is negative.,3kN,5kN,8kN,AXIAL TENSION AND COMPRESSION,25,26,拉压,轴力(图)的简便求法： 自左向右:,轴力图的特点：突变值 = 集中载荷,遇到向左的P， 轴力N 增量为正； 遇到向右的P ， 轴力N 增量为负。,3kN,5kN,8kN,Solution：The free end of the rod is the origin of the coo
19、rdinate and coordinate x to the right is positive. Take the segment of length x on the left of point x, its internal force is,q,K L,x,O,Example 2 Length of the rod shown in the figure is L. Distributed force q = kx is acted on it, direction of the force is shown in the figure. Try to plot the diagra
20、m of axial force of the the rod.,L,q(x),N,x,O,AXIAL TENSION AND COMPRESSION,27,28,拉压,解：x 坐标向右为正，坐标原点在自由端。 取左侧x 段为对象，内力N(x)为：,q,q L,x,O,例2 图示杆长为L，受分布力 q = kx 作用，方向如图，试画出杆的轴力图。,L,q(x),N,x,O,1、Concept of stress,13 STRESSES ON THE SECTION AND STRENGTH CONDITIONS,Bring forward the problem：,1). The magnit
21、ude of the internal force can not scale the strength of the structure member. 2). Strength：Intensity of the distributed internal forces in the sectionstress； The load-bearing capacity of the material.,1). Definition：Intensity of the internal force due to the external forces.,AXIAL TENSION AND COMPRE
22、SSION,29,30,拉压,一、应力的概念,13 截面上的应力及强度条件,问题提出：,1. 内力大小不能衡量构件强度的大小。 2. 强度：内力在截面的分布集度应力；材料承受荷载的能力。,1. 定义：由外力引起的内力集度。,Under most cases distribution of the internal force inside engineering members is not uniform. Definition of intensity is neither accurate and important because breakage or failure often b
23、egins from the point at which intensity of the internal force is maximum.,Average stress：,Whole stress（sum stress）：,2). Expression of stress：,AXIAL TENSION AND COMPRESSION,31,32,拉压,工程构件，大多数情形下，内力并非均匀分布，集度的定义不仅准确而且重要，因为“破坏”或“失效”往往从内力集度最大处开始。,平均应力：,全应力（总应力）：,2. 应力的表示：,Whole stress may be decomposed in
24、to:,a. Stress perpendicular to the section is called“normal stress”,b. Stress lying in the section is called“shearing stress”,AXIAL TENSION AND COMPRESSION,33,34,拉压,全应力分解为：,a.垂直于截面的应力称为“正应力” (Normal Stress)；,b.位于截面内的应力称为“剪应力”(Shearing Stress)。,Before deformation,1). Experiment on the law of deformat
25、ion and the hypothesis of plane section：,Hypothesis of plane section：Cross sections remain planes before and after deformations . Deformations of longitudinal fibers are the same,After loading,2、Stress in the cross section of the rod in tension or compression,AXIAL TENSION AND COMPRESSION,35,36,拉压,变
26、形前,1. 变形规律试验及平面假设：,平面假设：原为平面的横截面在变形后仍为平面。纵向纤维变形相同。,受载后,二、拉（压）杆横截面上的应力,The material is homogeneous and , its deformation is uniform , so the internal force is distributed uniformly。,2. Tensile stress：,Normal stress due to the axial forces ： distributes uniformly in the cross section.,Critical section
27、：The section in which internal force is maximum and of which the dimension is smallest. Critical point：The point at which the stress is maximum.,3. Critical section and the maximum working stress ：,AXIAL TENSION AND COMPRESSION,37,38,拉压,均匀材料、均匀变形，内力当然均匀分布。,2. 拉伸应力：,轴力引起的正应力 ： 在横截面上均布。,危险截面：内力最大的面，截面
28、尺寸最小的面。 危险点：应力最大的点。,3. 危险截面及最大工作应力：,Straight rod、cross section of the rod is without sudden change、there is a certain distance from the section to the point at which the load acts.,4). Application conditions of the formula：,6). Stress concentration：,Stress increases abruptly near the cross section w
29、ith a sudden change in dimension,5). Saint-Venant principle：,Distribution and magnitude of the stress in the section at a certain distance from the point at which the load is acted are not affected by the acting form of external loads.,AXIAL TENSION AND COMPRESSION,39,40,拉压,直杆、杆的截面无突变、截面到载荷作用点有一定 的距
30、离。,4. 公式的应用条件：,6. 应力集中（Stress Concentration）：,在截面尺寸突变处，应力急剧变大。,5. Saint-Venant原理：,离开载荷作用处一定距离，应力分布与大小不受外载荷作用方式的影响。,Sketch of Saint-Venant principle and stress concentrations,(Red real lines denote the line before deformation and red dashed lines denote the shape after deformation.),Sketch of deforma
31、tion：,Sketch of the stress distribution：,AXIAL TENSION AND COMPRESSION,41,42,拉压,Saint-Venant原理与应力集中示意图,(红色实线为变形前的线，红色虚线为红色实线变形后的形状。),变形示意图：,应力分布示意图：,7). Criterion of the strength design：,where：allowable stress， max the maximum working stress at the critical point.,Design the dimension of the section
32、：,Three kinds of calculation of strength may be doneaccording to the criterion of strength:,That structure members are ensured not to be wrecked and have certain safe degree.,Check the strength：,Determine the allowable load：,AXIAL TENSION AND COMPRESSION,43,44,拉压,7. 强度设计准则（Strength Design）：,其中：-许用应力
33、， max-危险点的最大工作应力。,设计截面尺寸：,依强度准则可进行三种强度计算：,保证构件不发生强度破坏并有一定安全余量的条件准则。,校核强度：,许可载荷：,Example 3 A circular rod is subjected to a tensile force P =25 kN. Its diameter is d =14mm and its allowable stress is =170MPa. Try to check the strength of the rod.,Solution： Axial force：N = P =25kN,Stress：,Check the st
34、rength：,Conclusion：The strength of the rod satisfies request.The rod can work normally.,AXIAL TENSION AND COMPRESSION,45,46,拉压,例3 已知一圆杆受拉力P =25 k N，直径 d =14mm，许用应力=170MPa，试校核此杆是否满足强度要求。,解： 轴力：N = P =25kN,应力：,强度校核：,结论：此杆满足强度要求，能够正常工作。,Example 4 A three-pin house frame on which a vertical uniform load
35、, with the in density intensity is q =4.2kN/m is applied is shown in the figure. Diameter of the steel tensile rod in the frame is d =16 mm and its allowable stress is =170MPa. Try to check the strength of the rod.,AXIAL TENSION AND COMPRESSION,47,48,拉压,例4 已知三铰屋架如图，承受竖向均布载荷，载荷的分布集度为：q =4.2kN/m，屋架中的钢
36、拉杆直径 d =16 mm，许用应力=170M Pa。 试校核钢拉杆的强度。, Determine the reactions first according to the global equilibrium,Solution：,AXIAL TENSION AND COMPRESSION,49,50,拉压,钢拉杆,8.5m,4.2m,RA,RB,HA,Stress：,Strength check and conclusion：,This rod satisfies the request of strength. It is safe., Determine the axial force
37、according to the partial equilibrium：,AXIAL TENSION AND COMPRESSION,51,52,拉压,应力：,强度校核与结论：,此杆满足强度要求，是安全的。, 局部平衡求 轴力：,HC,Example 5 A simple crane is shown in the figure. AC is a rigid beam ，sum weight of the hoist and heavy body that is lifted is P. What should be the angle so that the rod BD has the
38、minimum weight？ The allowable stress of the rod is known .,Analysis：,x,L,h,q,P,A,B,C,D,AXIAL TENSION AND COMPRESSION,53,54,拉压,例5 简易起重机构如图，AC为刚性梁，吊车与吊起重物总重为P，为使 BD杆最轻，角 应为何值？ 已知 BD 杆的许用应力为。,分析：,x,L,h,q,P,A,B,C,D, The cross-section area A of the rod BD：,Solution： Internal force N(q ) of the rod BD： Ta
39、ke AC as our study object as shown in the figure.,YA,XA,NBD,x,L,P,A,B,C,AXIAL TENSION AND COMPRESSION,55,56,拉压, BD杆横截面面积A：,解： BD杆内力N(q )： 取AC为研究对象，如图,YA,XA,NBD,x,L,P,A,B,C,YA,XA,NBD,x,L,P,A,B,C, Determine the minimum value of VBD ：,AXIAL TENSION AND COMPRESSION,67,58,拉压,YA,XA,NBD,x,L,P,A,B,C, 求VBD 的
40、最小值：,3、Stresses in the inclined section of the rod in tension or compression,Aa:Area of the inclined section；Pa：Internal force in the inclined section.,：,From geometric relation,Substituting it into the above formula we get,Solution: Adopt the method of section. According to the equilibrium equation
41、：Pa=P then,Assume a straight rod is subjected to a tensile force P. Determine the stress in the inclined section k-k .,Whole stress in the inclined section：,AXIAL TENSION AND COMPRESSION,59,60,拉压,三、拉(压)杆斜截面上的应力,设有一等直杆受拉力P作用。 求：斜截面k-k上的应力。,解：采用截面法 由平衡方程：Pa=P,则：,Aa：斜截面面积；Pa：斜截面上内力。,由几何关系：,代入上式，得：,斜截面上
42、全应力：,Decomposition：,It indicates the change of stresses in different sections through a point.,As =90，,As = 0,90，,Whole stress in the inclined section：,AXIAL TENSION AND COMPRESSION,61,62,拉压,斜截面上全应力：,分解：,反映：通过构件上一点不同截面上应力变化情况。,当 = 90时，,当 = 0,90时，,2、Element：Element delegate of a point inside the memb
43、er, infinitesimal geometric body which envelops the study point. The element in common use is just hexahedron properties of an elementa、stress is distributed uniformly in an arbitrary parallel arbitrary plane；b、stresses in the parallel plane opposite planeare equal.,3、stress element at a point M in
44、the rod in tension or compression:,1. State of stress at a point：There are countless sections through a point. Sum of stresses in the different section through a point is called the state of stress at this point.,Complementary：,AXIAL TENSION AND COMPRESSION,63,64,2、单元体：单元体构件内的点的代表物，是包围被研究点的无限小的几何体，常
45、用的是正六面体。单元体的性质a、平行面上，应力均布；b、平行面上，应力相等。,3、拉压杆内一点M 的应力单元体:,1.一点的应力状态：过一点有无数的截面，这一点的各个截面上的应力情况，称为这点的应力状态。,补充：,拉压,Take a free body as shown in the Fig.3. a is positive if it is along countclockwise； t a is positive if it makes the free body rotate clockwise. From the equilibrium of the free body we get：
46、,4、Stress in the inclined section of the rod in tension or compression,AXIAL TENSION AND COMPRESSION,65,66,取分离体如图3， a 逆时针为正； t a 绕研究对象顺时针转为正；由分离体平衡得：,拉压,4、拉压杆斜截面上的应力,Example 6 A rod, which the diameter d =1 cm is subjected to a tensile force P =10kN. Determine the maximum shearing stress , the norma
47、l stress and shearing stress in the inclined section of an angle 30about the cross section.,Solution：Stresses in the inclined section of the rod in tension or compression can be determined directly by the formula：,AXIAL TENSION AND COMPRESSION,67,68,例6 直径为d =1 cm 杆受拉力P =10 kN的作用，试求最大剪应力，并求与横截面夹角30的斜
48、截面上的正应力和剪应力。,解：拉压杆斜截面上的应力，直接由公式求之：,拉压,Example 7 A tensile rod as shown in the figure is made from two parts glued mutually together along mn. It is subjected to the action of force P. Assume that the allowable normal stress is =100MPa and allowable shearing stress is =50MPa for the adhesive. Area of cross section of the rod is A= 4cm. If strength of the rod is controlled by the adhesive what is the angle ( : between 0 060 0) to get the largest tensile force?,
