1、Chapter1 Tension and Compression第一章第一章拉伸与压缩拉伸与压缩Axially Loaded Bar The simplest case to consider at the start is that of an initially straight metal bar of constant cross section,loaded at its ends by a pair of oppositely directed collinear forces coinciding with the longitudinal axis of the bar and
2、 acting through the centroid of each cross section.For static equilibrium the magnitudes of the forces must be equal.If the forces are directed away from the bar,the bar is said to be in tension;if they are directed toward the bar,a state of compression exists.These two conditions are illustrated in
3、 Fig.1-1.轴向受载杆件轴向受载杆件同线的同线的质心质心平衡平衡大小大小不变的不变的 首先我们研究最简单情况：一等截面金属直杆在其两首先我们研究最简单情况：一等截面金属直杆在其两端承受一对共线、反向力的作用。这两个力的作用线与各截端承受一对共线、反向力的作用。这两个力的作用线与各截面形心组成的纵向轴线重合，为了满足静力学平衡条件，这面形心组成的纵向轴线重合，为了满足静力学平衡条件，这两个力的数值必须相等。如果这两个力的方向是离开此杆的，两个力的数值必须相等。如果这两个力的方向是离开此杆的，杆将承受拉伸；如果这两个力是指向此杆的，杆将承受压缩，杆将承受拉伸；如果这两个力是指向此杆的，杆将承
4、受压缩，如图如图1-1所示。所示。Fig.1-1图1-1 Under the action of this pair of applied forces,internal resisting forces are set up within the bar and their characteristics may be studied by imagining a plane to be passed through the bar anywhere along its length and oriented perpendicular to the longitudinal axis of
5、 the bar.Such a plane is designated as a-a in Fig.1-2(a).Fig.1-2 图1-2垂直的，垂直的，正交的正交的 在这样两个力的作用之下，杆的内部将产生抗力。我在这样两个力的作用之下，杆的内部将产生抗力。我们可以用位于杆轴某处、且与杆轴垂直的假想截面来研究们可以用位于杆轴某处、且与杆轴垂直的假想截面来研究杆的内部抗力，这样的截面如图杆的内部抗力，这样的截面如图1-2（a）中的）中的a-a所示。所示。If for purposes of analysis the portion of the bar to the right of this
6、plane is considered to be removed,as in Fig.1-2(b),then it must be replaced by whatever effect it exerts upon the left portion.By this technique of introducing a cutting plane,the originally internal forces now become external with respect to the remaining portion of the body.For equilibrium of the
7、portion to the left this“effect”must be a horizontal force of magnitude P.However,this force P acting normal to the cross-section a-a is actually the resultant of distributed forces acting over this cross section in a direction normal to it.施加（力）施加（力）水平的水平的法线法线合力合力分布式的分布式的为了分析计算，可考虑将此截面右侧的杆段除去，如图为了分
8、析计算，可考虑将此截面右侧的杆段除去，如图1-2（b）所示。因而，必须补充右侧杆段对左侧杆段的作用。用此处所示。因而，必须补充右侧杆段对左侧杆段的作用。用此处引入的截面法，初始的内力便成为保留杆段的外力。为使左引入的截面法，初始的内力便成为保留杆段的外力。为使左侧杆段平衡，这种效应在数值上等于水平力侧杆段平衡，这种效应在数值上等于水平力P。然而，沿截面。然而，沿截面a-a法向作用的力法向作用的力P实际上是截面上法向分布力合成的结果。实际上是截面上法向分布力合成的结果。Fig.1-2 图1-2 Instead of speaking of the internal force acting on
9、 some small element of area,it is better for comparative purposes to treat the normal force acting over a unit area of the cross section.The intensity of normal force per unit area is termed the normal stress and is expressed in units of force per unit area,e,g.,1N/m2.Normal Stress法应力法应力把把称作称作比较的比较的
10、应力应力替代讨论作用在某处小面积上内力，最好转为处替代讨论作用在某处小面积上内力，最好转为处理单位面积上法向力。单位面积上法向力的强度理单位面积上法向力。单位面积上法向力的强度称为法应力，它是用单位面积上的作用力单位表称为法应力，它是用单位面积上的作用力单位表示的，亦即示的，亦即12N/mIf the forces applied to the ends of the bar are such that the bar is in tension,then tensile stresses are set up in the bar;if the bar is in compression w
11、e have compressive stresses.It is essential that the line of action of the applied end forces pass through the centroid of each cross section of the bar.如果杆端的力使杆拉伸，杆内就产生拉应力，如果杆是受压如果杆端的力使杆拉伸，杆内就产生拉应力，如果杆是受压缩的，杆内产生压应力。施加在杆端的力作用线必须通过每缩的，杆内产生压应力。施加在杆端的力作用线必须通过每一个截面形心。一个截面形心。The axial loading shown in Fi
12、g.1-2(a)occurs frequently in structural and machine design problems.To simulate this loading in the laboratory,a test specimen is held in the grips of either an electrically driven gear-type testing machine or a hydraulic machine.Both of these machines are commonly used in materials testing laborato
13、ries for applying axial tension.Test Specimens试样样品样品夹具夹具电力地电力地液压的液压的应用，施加应用，施加 图图1-2（a）所示的轴向载荷经常出现在结构）所示的轴向载荷经常出现在结构和机械设计中，为了在实验室中模拟这种轴向载和机械设计中，为了在实验室中模拟这种轴向载荷，试件应夹持在电子或液压万能试验机的夹头荷，试件应夹持在电子或液压万能试验机的夹头中。这两种试验机通常在材料实验室用做施加轴中。这两种试验机通常在材料实验室用做施加轴向载荷。向载荷。In an effort to standardize materials testing tech
14、niques the American Society for Testing Materials(ASTM)has issued specifications that are in common use.Only two of these will be mentioned here,one for metal plates thicker than in(4.76mm)and appearing as in Fig.1-3,the other for metals over 1.5in(38mm)thick and having the appearance shown in Fig.1
15、-4.163 在材料实验技术规范标准中，美国材料实验协会颁布在材料实验技术规范标准中，美国材料实验协会颁布了详细的使用说明。此处只讲其中两种试件。一种厚度超了详细的使用说明。此处只讲其中两种试件。一种厚度超过过3/16in（4.76mm）的金属板试件如图）的金属板试件如图1-3所示，另外一所示，另外一种直径为种直径为1.5in（38mm）的金属圆棒状试件如图）的金属圆棒状试件如图1-4所示。所示。标准化标准化As may be seen from these figures,the central portion of the specimen is somewhat smaller than
16、 the end regions so that failure will not take place in the gripped portion.The rounded fillets shown are provided so that no stress concentrations will arise at the transition between the two lateral dimensions.The standard gage length over which elongations are measured is 8in(203mm)for the specim
17、en shown in Fig.1-3 and 2in(57mm)for that shown in Fig.1-4.正如图中所看到的，试件的中部区域尺寸略小于两端尺寸。正如图中所看到的，试件的中部区域尺寸略小于两端尺寸。因而，破环不会发生在两端的夹持段，过度圆角保证试件因而，破环不会发生在两端的夹持段，过度圆角保证试件两种尺寸的过渡部位不会发生应力集中。供测量伸长的标两种尺寸的过渡部位不会发生应力集中。供测量伸长的标距长度为距长度为8in（203mm，图，图1-3试件）和试件）和2in（51mm，图，图1-4试件）。试件）。稍微，有些稍微，有些失效，破坏失效，破坏过渡区过渡区伸长伸长标距标距
18、 The elongations are measured by either mechanical or optical extensometers or by cementing an electric resistance-type strain gage to the surface of the material.This resistance strain gage consists of a number of very fine wires oriented in the axial direction of the bar.As the bar elongates,the e
19、lectrical resistance of the wires changes and this change of resistance is detected on a Wheatstone bridge and interpreted as elongation.伸长计伸长计粘贴粘贴电阻应变片电阻应变片细的细的转换转换 试件的伸长既可以使用机械的、光学的伸长试件的伸长既可以使用机械的、光学的伸长计，也可以使用粘贴在试件表面的电阻应变片来计，也可以使用粘贴在试件表面的电阻应变片来测量，这种电阻应变片由若干个沿试件轴向的、测量，这种电阻应变片由若干个沿试件轴向的、很细的电阻丝组成。当试件
20、伸长时，电阻丝的电很细的电阻丝组成。当试件伸长时，电阻丝的电阻就会发生改变，电阻的改变由惠斯通电桥测量阻就会发生改变，电阻的改变由惠斯通电桥测量并转换为伸长量。并转换为伸长量。Let us suppose that one of these tension specimens has been placed in a tension-compression testing machine and tensile forces gradually applied to the ends.The strain is usually expressed in units of inches per
21、inch or meters per meter and consequently is dimensionless.Normal Strain法应变法应变应变应变增量增量 对于每一个事先指定的轴向载荷增量都要测对于每一个事先指定的轴向载荷增量都要测量相应的标距的伸长。由这些数值得到单位长度量相应的标距的伸长。由这些数值得到单位长度的伸长，并将之定义为法应变，用的伸长，并将之定义为法应变，用来表示，可以来表示，可以用标距长度用标距长度L去除总伸长去除总伸长来得到法应变。来得到法应变。As the axial load is gradually increased in increments,t
22、he total elongation over the gage length is measured at each increment of load and this is continued until fracture of the specimen takes place.Knowing the original cross-sectional area of the test specimen the normal stress,denoted by,may be obtained for any value of axial load by the use of the re
23、lationAPStress-Strain Curve应力应力应变曲线应变曲线where P denotes the axial load in pounds or Newtons and A the original cross-sectional area.Having obtained numerous pairs of values of normal stress and normal strain,the experimental data may be plotted with these quantities considered as ordinate and absciss
24、a,respectively.This is the stress-strain curve or diagram of the material for this type of loading.Stress-strain diagrams assume widely differing forms for various materials.Figure 1-5 is the stress-strain diagram for a medium-carbon structural steel,Fig.1-6 is for an alloy steel,and Fig.1-7 is for
25、hard steels and certain nonferrous alloys.For nonferrous alloys and cast iron the diagram has the form indicated in Fig.1-8,while for rubber the plot of Fig.1-9 is typical.量，数量量，数量纵坐标纵坐标横坐标横坐标碳素结构钢碳素结构钢有色金属合金有色金属合金铸铁铸铁P是轴向载荷，是轴向载荷，A是初始截面积。于是，得到由法应是初始截面积。于是，得到由法应力力和法应变和法应变组成的很对数据对，然后分别以法应组成的很对数据对，然后分
26、别以法应力为纵轴，法应变为横轴，用上述实验数据作图，力为纵轴，法应变为横轴，用上述实验数据作图，这就是材料的应力这就是材料的应力应变曲线或拉伸图。图应变曲线或拉伸图。图1-5是碳是碳素结构钢的应力素结构钢的应力应变图，图应变图，图1-6是合金钢，图是合金钢，图1-7是硬质刚或某些有色金属合金的应力是硬质刚或某些有色金属合金的应力应变图，有应变图，有色金属合金或铸铁的应力色金属合金或铸铁的应力应变图如图应变图如图1-8所示，而所示，而对于橡胶，图对于橡胶，图1-9是典型的应力是典型的应力应变图。应变图。1.The concepts of stress and strain can be illu
27、strated in an elementary way by considering the extension of a prismatic bar.As shown in fig.1,a prismatic bar is one that has constant cross section throughout its length and a straight axis,in this illustration the bar is assumed to be loaded at its ends by axial forces P that produce a uniform st
28、retching,or tension,or the bar.2.By making an artificial cut(section mm)through the bar at right angles to its axis,we an isolate part of the bar as a free body see Fig.1(b).At the left hand the tensile force P is applied,and at the other end there are forces representing the action of the removed portion of the bar upon the part that remains.These forces will be continuously distributed over the cross section,analogous to the continuous distribution of hydrostatic pressure over a submerged surface.
