轴向拉伸与压缩课件教学

1、Bilingual EditionreturnexitChapter 2Chapter 2 Axial tension and compression Shear21 Concepts and practical examples of axial tension and compression 轴向拉伸和压缩的概念和实例22 Internal forces and stresses on the section under axial tension and compression 轴向拉伸和压缩时横截面上的内力和应力23 Stresses in the inclined section o

2、f the rod in tension or compression 直杆轴向拉伸和压缩时斜截面上的应力拉伸、压缩与剪切GOGOGOChapter 224 Mechanical properties of materials in axial tension and compression 材料在拉伸和压缩时的力学性能25 Failure, safety factor and strength conditions 失效、安全因数和强度计算26 Deformation of the rod in axial tension and compression 轴向拉伸和压缩时的变形27 Elas

3、tic strain energy of the rod in axial tension and compression 轴向拉伸和压缩的应变能GOGOGOGOChapter 228 Statically indeterminate problems of axial tension and compression 拉伸、压缩超静定问题29 Temperature stress and assemble stresses 温度应力和装配应力210 The concept of stress concentration 应力集中的概念GOGOGOChapter 221 Concepts and

4、 practical examples of axial tension and compression轴向拉伸和压缩的概念和实例1. Concepts 概念Characteristic of the external force: The acting line of the resultant of external forces is coincided with the axis of the rod.轴向拉压的外力特点：外力的合力作用线与杆的轴线重合。Axial tensionAxial compressionChapter 2Mechanical models are shown

5、in the figures力学模型如图:In axial tension, the corresponding force is called tensile force.轴向拉伸，对应的力称为拉力。In axial compression, the corresponding force is called compressive force.轴向压缩，对应的力称为压力。Chapter 2Characteristic of the deformation: Deformation of the rod is mainly elongation or contraction along th

6、e axis of the rod and companied with lateral reduction or enlargement. 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。Axial tension: Deformation of the rod is axial elongation and lateral shortening.轴向拉伸：杆的变形是轴向伸长，横向缩短。Axial compression: Deformation of the rod is axial shortening and lateral enlargement.轴向压缩：杆的变形是轴向缩

7、短，横向变粗。Chapter 2ABCABBCAxial compression member Axial tension memberChapter 2Axial tension and compression take place very often in enineering applications.2. Practical examples in engineering 工程实例Chapter 2The tower bears compression load, cables bear tension.Return to contents of this chapterChapte

8、r 2towercablepier22 Internal forces & stresses on the section under axial tension and compression轴向拉伸和压缩时横截面上的内力和应力1. Internal force 内力Internal force is the resultant of internal forces, which is acting mutually between two neighbour parts inside the body, caused by the external forces.OxyzChapter 2

9、2. Method of section axial force 截面法 轴力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.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) Basic steps of the method of s

10、ection: Cut off: Substitute: Equilibrium:mmmmxChapter 2(2) Axial forceinternal force of the rod in axial tension or compression, designated by .轴力轴向拉压杆的内力，用 表示。mmSign conventions for the axial force 轴力的正负规定: (tensile force) is positive when its direction point to the outward direction of the normal

11、line of the section. 与外法线同向,为正轴力(拉力)与外法线反向,为负轴力(压力)(compressive force)Chapter 2Example 1 Determine the axial force FN in section 1-1, 2-2, 3-3Solution:112233112233Take the free body as shown in the figure.xChapter 2O3. Diagram of the axial force sketch expression of FN (x) 轴力图 FN(x)的图象表示112233mmx+x+

12、-OChapter 2Meaning 意义: Reflected the variety relation between the corresponding axial force and the position of the section. 反映出轴力与横截面位置变化关系，较直观。 Find out value of the maximum axial force and the position of the section in which the maximum axial force act. That is to determine the position of the c

13、ritical section and supply the information for the calculation of strength.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。Chapter 2Example 2 A rod is subjected to the force system as shown in the following Fig. Try to plot the diagram of the axial force of the rod.ABCDSolution:Determine the internal forc

14、e FN in segment AB.Take the free body as shown in the figure.11BCD11x22D22Chapter 2Similarly, we get the internal forces in segment CD :The diagram of the axial force is shown in the right figure .xABD+-D22ABCD1122OChapter 2xCharacteristic of the diagram of the axial force: Value of sudden change =

15、concentrated load 轴力图的特点：突变值 = 集中载荷 If meeting the force to the right ，the increase of the axial force FN is negative.遇到向右的力 ， 轴力FN 增量为负。+-OChapter 2If meeting the force to the left ，the increase of the axial force FN is positive.遇到向左的力, 轴力FN 增量为正。x+-If meeting the force to the right ，the increase o

16、f the axial force FN is negative.遇到向右的力 ， 轴力FN 增量为负。Chapter 2From the left to the right:Simple method to plot the diagram of axial force.轴力(图)的简便求法DExample 3 The forces with 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

17、 the diagram of the axial force of the rod.ABCDOSolution:3P2P5PPOxABC+-+Chapter 2Example 4 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 diagram of axial force of the the rod.Solution:xq(x)x-Lq(x)k

18、LqxOTake the segment oflength x on the left of point x,its internal force is Chapter 24. Stresses on the section 横截面上的应力Bring forward the problem:问题的提出 The magnitude of the internal force can not scale the strength of the structure member.内力大小不能衡量构件强度的大小。 Intensity of the distributed internal forces

19、 in the section stress.内力在截面的分布集度应力。Definition: Intensity of the internal force due to the external forces. 定义：由外力引起的内力集度Chapter 2MMmay be decomposed into:normal stressshearing stressWhole stressAUnit:Under most cases distribution of the internal force inside engineering members is not uniform. Chap

20、ter 2because breakage or failure often begins from the point at which intensity of the internal force is maximum.Definition of intensity is neither accurate and important Stress in the cross section of the rod in tension or compression: 拉（压）杆横截面上的应力Experiment on the law of deformation and the hypoth

21、esis of plane section：变形规律试验及平面假设：abcdBefore deformationAfter loading d ac bDeformations of longitudinal fibers are the same.纵向纤维的伸长量是相同的。Cross section横截面Chapter 2abcdBefore deformation d ac bAfter loadingHypothesis of plane section: Cross sections remain planes before and after deformations .平面假设：原

22、为平面的横截面在变形后仍为平面。The material is homogeneous and, its deformation is uniform, so the internal force is distributed uniformly.均匀材料、均匀变形，内力当然均匀分布。Chapter 22) Tensile stress: 拉伸应力 d ac bNormal stress due to the axial forces : distributes uniformly in the cross section.轴力引起的正应力 ： 在横截面上均布。s静力学关系：Chapter 2

23、3) Application conditions of the formula:公式的应用条件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.直杆、杆的截面无突变、截面到载荷作用点有一定的距离。lxA(x)22113344112233441 - 1.5 times the width of the memberChapter 24) Saint-Ven

24、ant principle:圣维南原理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. 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。杆端加载方式对正应力分布的影响。Chapter 2DExample 5 Determine the maximum normal stress of

25、 the rod. AAB = ABC = 500mm 2, ACD = 200mm 2ABCDSolution:1. determine the axial force of the rod.2. determine the stress of the rod.20KNx+-ABC10KNin segment AB:OChapter 2ABCDB10KN20KNxACDO+-x4050AB20CDin segment BC:+-Chapter 2OExample 6 Two members support a block of weight Q = 10KN shown in figure.

26、 The cross-section areas of the members are respectively 100mm2 for BC and 200mm2 for AB , determine the stress of each rod.BABCSolution:1. determine theinternal force of each rod.Take point B as our study object as shown in the figureChapter 22. determine the stress of each rod.The rod BC:CBBABABCw

27、e get:BC is under tensionAB is compressedThe rod AB:Return to contents of this chapterChapter 2According to the equilibrium equation:23 Stresses in the inclined section of the straight rod in tension or compression 直杆轴向拉伸和压缩时斜截面上的应力kkArea of the inclined sectionkk: stress in the inclined section. 斜截

28、面上全应力Chapter 2Assume a straight rod is subjected to a tensile force . Determine the stress in the inclined section k-k .kkDecomposition 分解:pa = normal stress 正应力stress in the cross section横截面上的正应力shearing stress 剪应力a is positive if it is along counter- clockwiseIt indicates the change of stresses in

29、 different sections through a point.反映：通过构件上一点不同截面上应力变化情况 。Chapter 2As = 0,As =90,( The maximum normal stress exists in the cross section. 横截面上存在最大正应力 )As = 45,( The maximum shearing stress exists in the inclined section of 45 . 45 斜截面上剪应力达到最大 )kknormal stress:shearing stressAs = 0 , 90,Chapter 2Com

30、plementary 补充：1. State of stress at a point 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。2. Element 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，常用的是正六面体(regular hexahedron)。properties of an element:平行面上,应力均布且相等3. stress element at a point M in the rod in tension or compression 拉压杆内一点 M 的应力单元体MsssssChapter 2MComple

31、mentary ：1. State of stress at a point:2. ElementChapter 2section through a point is called the state of stress at this point.Element delegate of a point inside the member, infinitesimal geometric body which envelops the study point. The element in common use is just perties of an elem

32、enta. stress is distributed uniformly in an arbitrary parallel plane； b. stresses in the parallel plane are equal.There are countless sectionssssthrough a point.Sum of stresses in the differentExample 7 A rod, which the diameter d =1 cm is subjected to a tensile force P =10kN. Determine the maximum

33、shearing stress , the normal stress and shearing stress in the inclined section of an angle 30about the cross section.Solution:1. determine the stress in the cross section2. the maximum shearing stress, stresses in the inclined section of the rod in tension or compression can be determined directly

34、by the formula.Chapter 2Return to contents of this chapterChapter 224 Mechanical properties of materials in axial tension and compression材料在拉伸和压缩时的力学性能Mechanical properties: Some properties about strength and deformation shown by materials under the action of external loads . 力学性能：材料在外力作用下表现的有关强度、变形

35、方面的特性。低碳钢 和 铸铁 在 拉伸压缩 时的力学性能在工程上使用最广泛，力学性能最典型Chapter 2low-carbon steelcasting-ironthe ductile(塑性) materials, the brittle(脆性) materials25 Mechanical properties of materials in axial tension and compressionMechanical properties: Some properties about strength and deformation shown by materials under t

36、he action of external loads . Chapter 2To obtain the mechanical properties of a material , one usually conducts a tensile test or a compressive test on a specimen of the material.A specimen made of casting-ironCategory of materialsA specimen made of low-carbon steelThe ductile materialsThe brittle m

37、aterials1. Testing conditions and instruments试验条件及试验仪器Testing conditions: 试验条件圆柱形长试祥l10d 或短试样l 5ddStandard specimen实验试样：normal temperature(20);static load ( loaded gradually) 静载（缓慢地加载）Chapter 22) Experimental implement: 试验仪器 Universal testing machine 万能材料试验机引伸仪 Extensometer(对试件施加载荷)upper clamper上夹头l

38、ower clamper下夹头(测量试样变形)液压式试验机电子拉力试验机Chapter 22. Tensile diagram of the low carbon steel specimen低碳钢试件的拉伸图 (F- l diagram)实验记录：横坐标l，纵坐标FlFOChapter 23. Stress-strain curve of the low carbon steel specimen低碳钢试件的应力-应变曲线 ( - diagram)OOStress-strain curvelFTensile diagramChapter 2Stress-strain diagram of t

39、he low carbon steel specimen in tensile testObafccc1efhChapter 2The curve can be divided into four periods.ruptureFour periods：Proportional limit1) Elastic range 弹性阶段Oa 比例段Proportional segment:ab 曲线段Curved segment:e Elastic limit2) Yielding(flowing) range 屈服(流动）阶段Sliding lines:滑移线s Yield limitFailur

40、e stress 失效应力Chapter 2Elastic modulus弹性模量(Ob)3) Hardening range 强化阶段 -Strength limitUnloaded law 卸载定律Cold hardening冷作硬化利： to raise the loading capacity in the elastic range利之用：用冷加工的方法来提高材料的强度 。弊之屏：进行退火处理 。Chapter 2弊： 材料的塑性(ductility).cold drawing after being loaded and unloaded:3) Strain hardening r

41、ange-Strength limitUnloaded law 卸载定律Chapter 2is applied to raise the loading capacity in the elastic rangeCold-Work Hardening:冷作硬化The ductilitycold drawing 4) Necking(crack) stage 颈缩（断裂）阶段 Chapter 2in neckingafter breakingruptureMagnified pictureIndicate: shear is primarily responsible for the failu

42、re of ductile material .4) Necking(crack) stage Percent elongation 断后延伸率brittle materialductile materialChapter 2 Percent reduction in area断面收缩率Obpesbafccc1eddgfhProportional segment:elasticyieldStrain-hardeningneckingChapter 2ruptureCold-Work Hardening青铜拉伸应力应变图Oabed4. Mechanical properties of other

43、 plastic material in tension 其他塑性材料在拉伸时的力学性能 无明显屈服阶段without obvious yieldNominal yield stress 0.2 名义屈服极限Chapter 2is the failure stress of this type of material.0.2 即为此类材料的失效应力。A secant for the curve to determine the elastic modulus5. Mechanical properties of the cast iron in tensionRupture occurs wi

44、thout any noticeable prior change in rate of elongation.OStress-strain diagram for casting-iron in tensile test Tensile strength limit of cast iron (failure stress)Chapter 2No linear portionrupture铸铁（脆性材料）在拉伸时的力学性能OStress-strain diagram for fiber reinforced plasticsTensile strength limit (failure st

45、ress)Only one stressChapter 2Failure of brittle materials: normal stresses.Rupture occurs along a surface perpendicular to the load.rupturerupture玻璃钢材料在压缩时的力学性能6.Mechanical properties of the material in compressionStandard specimen:can be a short cylinder or a blockChapter 2The specimen becomes a dr

46、um-shaped during the test and finally, a cake-shaped. And necking never occurs.Stress-strain curve of low-carbon steelElastic period yielding period:Chapter 2is unavailablethe curve is similar to that in tensile test.OStress-strain curve of for casting-ironChapter 2 Compressive strength limitFor mos

47、t brittle materials the curve in compressive test is similar to that in tensile test, ruptureShear stress is responsible for failure.but铸铁 Indices that measures the mechanical properties of materials: Analysis of failure： 1) ductile materials in tension or compressionyield , 2) Brittle materials in

48、tension:3) Brittle materials in compression:, due to maxrupture occurs along a cross section ,due to maxdue to maxetc.7. SummaryChapter 2strength indices plasticity indiceselastic index sliding occurs along an oblique surfaceIndex 指标 塑性材料和脆性材料力学性能比较塑性材料脆性材料断裂前有很大塑性变形断裂前变形很小抗压能力与抗拉能力相近抗压能力远大于抗拉能力延伸率

49、5%延伸率 独立的静平衡方程个数Chapter 22. Method to solve statically indeterminate problems: Equilibrium equations, compatibility equation of deformation and physical equations must be combined together to solve these problems .超静定问题的处理方法：平衡方程、变形协调方程、物理方程相结合, 进行求解。（变形比较法）3. Method for solving statically indetermi

50、nate problems：超静定问题的处理方法（1）静力平衡方程（2）变形协调方程（3）物理关系方程（4）补充方程联立求解elastic lawComplementary equationChapter 2Example 17 Rods 1, 2 and 3 are connected together with a pin as shown in the figure. Knowing the length of each rod is: L1=L2, L3 =L, and the area of each rod is A1=A2=A, A3, modulus of elasticity

51、 of each rod is: E1=E2=E, E3. External force is along the upright direction. Determine the internal force of each rod.CABD123 Equilibrium equations:Solution:AChapter 2Geometric equationcompatibility equation of deformation:Physical equationelastic law:Complementary equation: A1CABD123 Solving the eq

52、uilibrium equations and complementary equation we get:Chapter 2Example 18 Four angles of a wooden pole are reinforced with four equal leg angle steel of 40404. The allowable stresses of steel and wood are respectively st=160MPa and w=12MPa, module of elasticity of them are Est=200GPa and Ew=10GPa re

53、spectively. Determine the permissible load P.Solution: Geometric equation : Equilibrium equations: Physical equation and complementary equation:250Chapter 24040Section area of the angle steel may be obtained from the table of hot-rolled steel:250 Solving equilibrium equations and the complementary e

54、quation we get:型钢表Chapter 2 Determine the permissible load of the structure:250Additionally: What about would it the like if area of the angle steel is increased as 5 times？另外：若将钢的面积增大5倍，怎样？ The maximum load of the structure is controlled by the steel. st=160MPaw=12MPaEst=200GPaEw=10GPaIn the case o

55、f lst=lw , the angle steel will reach the limit state first.What about would it like if the area of the wood is changed into 25mm2 ?若将木的面积变为25mm2，又怎样？Return to contents of this chapterChapter 23A12AChapter 21. Assemble stressesIn engineering applications dimensional errors members in manufacturing a

56、re inevitable.the errors will cause additional internal forces in the structure.assemble stresses (initial stresses 预应力)assemble internal force29 Temperature stress and assemble stresses温度应力和装配应力As these members are assembled into statically indeterminate structures, in There is the assemble stress

57、in statically indeterminate structure. There is no the assemble stress in statically determinate structure.312AChapter 2AProblem 19 Consider a three-member truss as shown in the figure with rod 1 and rod 2 made of the same material . Rod 3 is slightly shorter than the design length of the rod by . A

58、fter the ends of the three members are connected, determine the assemble internalforce of each rod. Geometric equation :Solution: Equilibrium equations: Solving equilibrium equations and the complementary equation we get: Physical equation and complementary equation:312AAdChapter 2312AChapter 22. Te

59、mperature stressDeformation induced by temperature changes will cause additional internal forces in the members either.temperature internal forces temperature stress No temperature stress in statically determinate structure. There is the temperature stress in statically indeterminate structure .Chap

60、ter 2We shall now consider various situations involving changes in temperature.All of the members and structures that we have considered so far were assumed to remain at the same temperature while they were being loaded. Example 20 Materials and dimensions of rod 1 and 2 are all the same as shown in