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1、Chapter 1 Stress 应力1.1 Introduction 概述statics 静力学Mechanics of Materials 材料力学stress 应力strain 应变strength of the material 材料强度deformation 变形rod 杆beam 梁theory of elasticity 弹性理论theory of plasticity 塑性理论Mechanics of materials is a branch of mechanics that studies the internal effects of stress and strain

2、 in a solid body that is subjected to an external loading.材料力学是力学的一个分支，主要研究受到外力作用的固体，其应力与应变的内部效应。Stress is associated with the strength of the material from which the body is made, while strain is a measure of the deformation of the body.应力主要与该变形体材料的强度联系在一起，而应变则是该变形体变形的表征量。Mechanics of materials inc

3、ludes the study of the bodys stability when a body such as a column is subjected to compressive loading.材料力学还包括对于受到压缩载荷作用的构件（例如压杆）的稳定性问题研究。相关知识：1638年，举世闻名的意大利数学家、天文学家、力学家伽里略（Galileo，15641642）在荷兰莱登出版了世界上第一本材料力学教本两种新的科学，首先提出了材料的力学性质和强度计算的方法。人们认为，材料力学作为一门学科，就从这里开始。18世纪初，实验法开始在法国广泛应用，法国的科学家泊松（S.D. Poiss

4、on，17811840）、法国的力学家圣维南（B de Saint-Venant，17971886）（此后，法国力学家圣维南于19世纪中叶运用弹性力学方法奠定了柱体扭转理论研究的基础，因而学术界习惯将柱体扭转问题称为圣维南问题）、法国数学家、工程师拉梅（Gabriel. Lame，17951870）以及法国的力学家、工程师纳维埃（Navier 17851838）等等都对弯曲理论、扭转理论、稳定理论以及材料实验作出卓越的贡献，丰富、发展和完善了材料力学这门学科。瑞士的数学家、力学家欧拉（L.Euler，17071783），对数学、刚体力学以及材料力学中的弹性线、稳定理论等都有重大贡献。1.2 E

5、quilibrium of a Deformable Body 变形体的平衡问题external loads 外力surface forces 面力concentrated forces 集中力linear distributed loads 线性分布载荷resultant force 合力centroid 形心body forces 体力gravitation field重力场gravity 重力electromagnetic field 电磁场support reactions 支反力（包括力与力偶）support 支座coplanar force system 共面力系translati

6、on 平动rotation 转动couple moment 力偶矩cable 绳索，悬索roller 滚动支座（移动铰支座）smooth support 光滑支承external pin 外部铰链（固定铰支座）internal pin 内部铰链（连接铰链）fixed support 固定支承，固定端equations of equilibrium 平衡方程balance of forces 力的平衡balance of moments 力矩平衡free-body diagram 受力图internal loadings 内力cross section 截面normal force 轴力shea

7、r force 剪力torsional moment (torque) 扭矩bending moment 弯矩vector 矢量scalar 标量coplanar loadings 共面载荷（二维问题）the method of section 截面法right-hand rule 右手法则The resultant force FR of w(s) is equivalent to the area under the distributed loading curve, and this resultant acts through the centroid C or geometric

8、center of this area.分布载荷w(s)的等效合力FR大小等于分布载荷曲线所覆盖的面积大小，力的作用线经过该面积形心或几何中心。If the support prevents translation in a given direction, then a force must be developed on the member in that direction. Likewise, if rotation is prevented, a couple moment must be exerted on the member.如果支座约束了（物体）在某一方向上的平动，则该支

9、座对物体在该方向上施加一个约束力。同样，如果支座约束了（物体）的转动，则该支座对物体施加一个约束力偶。Equilibrium of a body requires both a balance of forces, to prevent the body from translating or having accelerated motion along a straight or curved path, and a balance of moments, to prevent the body from rotating.物体的平衡要求满足合力的平衡（确保物体不会沿某一直线或曲线路径发生

10、平动或加速运动）以及合力矩的平衡（确保物体不会发生转动）。Successful application of the equations of equilibrium requires complete specification of all the known and unknown forces that act on the body, and so the best way to account for all these forces is to draw the bodys free-body diagram.成功应用平衡方程要求明确所有已知和未知的载荷，能够考虑所有载荷的最好办

11、法是绘制该物体的受力图。By the right-hand rule, the thumb gives the arrowhead sense of this vector and the fingers or curl indicate the tendency for rotation (twisting or banding).右手法则中，拇指指向该矢量的箭头方向，四指弯曲方向表明转动趋势（扭转或弯曲）方向。1.3 Stress 应力stress 应力tangent 相切的normal 正交的the quotient of A and B A比B的商normal stress 正应力te

12、nsile stress 拉应力compressive stress 压应力shear stress 切应力subscript 下标prefix 前缀state of stress 应力状态Stress describes the intensity of the internal force acting on a specific plane (area) passing through a point.应力用来描述经过某一点的平面上内力的集度。1.4 Average Normal Stress in an Axial Loaded Bar 轴向载荷作用下的平均正应力prismatic b

13、ar 等截面杆homogeneous material 均匀材料isotropic material 各向同性材料average normal stress 平均正应力uniaxial stress 单向应力normal force diagram 轴力图Homogeneous material has the same physical and mechanical properties throughout its volume, and isotropic material has these same properties in all directions.均匀材料在材料内部各处的物

14、理特性和力学特性均相同；各向同性材料在各个方向上的物理特性和力学特性均相同。As a sign convention, the normal force will be positive if it causes tension in the member, and negative if it causes compression.根据符号规定，如果轴力导致构件伸长，则轴力为正；如果导致构件缩短，则轴力为负。1.5 Average Shear Stress 平均切应力average shear stress 平均切应力simple shear 简单剪切direct shear 直接剪切bol

15、t 螺栓pin 连接销，连接铰pure shear 纯剪切All four shear stresses must have equal magnitude and be directed either toward or away from each other at opposite edges of the element. This is referred to as the complementary property of shear.所有四个切应力必须大小相等，方向则共同指向或共同背离两平面交线，该理论即为切应力互等定理。1.6 Allowable Stress 许用应力allo

16、wable stress 许用应力factor of safety 安全因数1.7 Design of Simple Connections 简单连接件的设计bearing stress 挤压应力The area of the column base plate B is determined from the allowable bearing stress for the concrete.压杆基座B的面积可以根据混凝土材料的许用挤压应力确定。The embedded length l of this rod in concrete can be determined using the

17、allowable shear stress of the bond glue.嵌入混凝土杆子的嵌入长度l可以根据连接胶的许用切应力确定。The area of the bolt for this lap joint is determined from the shear stress, which is largest between the plates.连接搭接板的螺栓面积可以根据其切应力确定，其在两板之间取得最大值。Chapter 2 Strain 应变deformation 变形strain 应变normal strain 正应变shear strain 切应变dimensionl

18、ess quantity 无量纲量parallelepiped 平行六面体Cartesian strain components直角（笛卡尔）坐标应变分量The normal strains cause a change in volume of the element, whereas the shear strains cause a change in its shape. Both of these effects occur simultaneously during the deformation.正应变导致微元体积发生改变，而切应变导致微元形状发生改变。这两种效应在变形中是同时发

19、生的。Chapter 3 Mechanical Properties of Materials 材料力学特性3.1 The Tension and Compression Test 拉伸与压缩实验mechanical properties of materials 材料的力学特性tension test 拉伸实验compression test 压缩实验specimen 实验试件gauge-length 标距ball-and-socket joint 球铰支座caliper 游标卡尺extensometer 伸长计electrical-resistance strain gauge 电阻应变仪

20、The operation of this gauge is based on the change in electrical resistance of a very thin wire or piece of metal foil under strain.应变仪的工作原理基于细导线或金属箔片在变形条件下电阻的改变情况。3.2 The Stress-Strain Diagram 应力-应变曲线stress-stain diagram 应力-应变曲线nominal stress/strain 名义应力/应变elastic behavior 弹性行为linear elastic region

21、 线弹性区域the proportional limit 比例极限the elastic limit 弹性极限yielding 屈服yielding stress 屈服应力plastic deformation 塑性变形upper yield point 屈服上限lower yield point 屈服下限carbon steel 碳钢hot rolled steel 热轧钢strain hardening 应变硬化ultimate stress 极限应力necking 颈缩fracture stress 断裂应力Here the curve is actually a straight li

22、ne throughout most of this region, so that the stress is proportional to the strain. The material in this region is said to be linear elastic.该阶段的曲线大部分为直线，表明应力与应变成正比关系。该阶段材料（行为）称为线弹性。A slightly increase in stress above the elastic limit will result in a breakdown of the material and cause it to defo

23、rm permanently. This behavior is called yielding, and it is indicated by the rectangular dark orange region of the curve.当应力超过弹性极限后材料会失效，导致永久性变形。（材料的）该行为称为屈服，用（应力-应变）曲线中深桔色的矩形区域表示。Notice that once the yield point is reached, then as shown in Fig.3-4, the specimen will continue to elongate (strain) w

24、ithout any increase in load. When the material is in this state, it is often referred to as being perfectly plastic.注意当到达屈服点时，如图3-4所示，试件将在载荷不变的情况下继续伸长。当材料处于该状态下时，被称为完全塑性状态。When yielding has ended, an increase in load can be supported by the specimen, resulting in a curve that rises continuously but

25、becomes flatter until it reaches a maximum stress referred to as the ultimate stress, u.当屈服阶段结束后，试件上的载荷继续增大，导致（应力-应变）曲线继续上升但逐渐平坦直至到达最大应力，即强度极限u。This decrease is fairly uniform over the specimens entire gauge length; however, just after, at the ultimate stress, the cross-sectional area will begin to

26、decrease in a localized form in this region as the specimen elongates further, Fig.3-5a.试件在其标距范围内（横截面尺寸）缩小是均匀的，但是到达强度极限后，会在局部区域内横截面尺寸进一步缩小，如图3-5a所示。3.3 Stress-Strain Behavior of Ductile and Brittle Materials 塑性材料和脆性材料的应力-应变行为ductile material 塑性材料brittle material 脆性材料mild steel 低碳钢percent elongation

27、延伸率percent reduction in area 断面收缩率brass 黄铜molybdenum 钼zinc 锌aluminum 铝off-set method 偏移法yield strength 屈服强度polymer （高分子）聚合物gray cast iron 灰口铸铁concrete 混凝土methacrylate plastic甲基丙烯酸塑料Any material that can be subjected to large strains before it fractures is called a ductile material.任何可以在断裂前承受较大应变的材料称

28、作塑性材料。Actually, this metal often does not have a well-defined yield point, and consequently it is standard practice to define a yield strength using a graphical procedure called the offset method.实际上，这种材料并没有一个非常明确的屈服点，所以通常标准的做法是使用图解的偏移法来确定屈服强度。The characteristics of its stress-strain diagram depend

29、primarily on the mix of concrete (water, sand, gravel, and cement) and the time and temperature of curing.（混凝土）的应力-应变曲线特性取决于混凝土的配比（水、砂、石和水泥）以及固化时间和温度。3.4 Hookes Law 胡克定律Hookes law 胡克定律modulus of elasticity 弹性模量Youngs modulus 杨氏模量slope 斜率rolled steel 轧制钢tool steel 工具钢stiffness 刚度vulcanized rubber 硫化橡

30、胶permanent set 永久变形spring back 回弹interatomic force 原子间力If a specimen of ductile material, such as steel, is loaded into the plastic region and then unloaded, elastic strain is recovered as the material returns to its equilibrium state. The plastic strain remains, however, and as a result the materia

31、l is subjected to a permanent set.如果塑性材料试件，如低碳钢，加载到塑性区域然后卸载，则材料中的弹性应变可以恢复至其平衡状态。材料中的塑性应变将保留，材料中有永久变形。If the load is reapplied, the atoms in the material will again be displaced until yielding occurs at or near the stress A, and the stress-strain diagram continues along the same path as before, Fig.3

32、-14b. It should be noted, however, that this new stress-strain diagram, defined by OAB, now has a higher yield point (A), a consequence of strain-hardening. In other words, the material now has a grater elastic region; however, it has less ductility, a smaller plastic region, than when it was in its

33、 original state.重新加载后，材料内的原子被再一次移位直至发生屈服或者到达A点处，应力应变曲线将沿原有的路径上升，如图3-14b所示。需要注意的是，这条由OAB定义的新应力-应变曲线具有更高的屈服点A，这是应变硬化的结果。换言之，现在材料具有更大的弹性范围，但塑性降低，即塑性区域比原有状态更小。3.5 Strain Energy 应变能strain energy 应变能strain-energy density 应变能密度modulus of resilience 回弹模量modulus of toughness 韧性模量By definition, work is determ

34、ined by the product of the force and displacement in the direction of the force.根据定义，功可以由力与力的方向上的位移之积来确定。Since the force is increased uniformly from zero to its final magnitude F when the displacement z is attained, the work done on the element by the force is equal to the average force magnitude (F

35、/2) times the displacement z.由于力的大小从零均匀增加到F，同时位移大小为z，则该力对微元做功的大小等于力的平均值（F/2）乘以位移z。Physically a materials resilience represents the ability of the material to absorb energy without any permanent damage to the material.从物理意义上说，一种材料的回弹模量体现了其在不导致材料永久性破坏条件下吸收能量的能力。3.6 Poissons Ratio泊松比Poissons ratio 泊松比i

36、n the longitudinal direction 在纵向方向上in the axial direction 在轴向方向上in the lateral direction 在横向方向上in the radial direction 在径向方向上longitudinal elongation 纵向伸长lateral contraction 横向收缩Within the elastic range the ratio of the strains in the longitudinal or axial direction and in the lateral or radial direc

37、tion is a constant.在弹性范围内，纵向（轴向）正应变和横向（径向）正应变之比为常数。The negative sign is included there since longitudinal elongation (positive strain) causes lateral contraction (negative strain), and vice versa.（公式中的）负号表明纵向伸长（正应变）会导致横向收缩（负应变），反之亦然。3.7 The shear Stress-Strain Diagram剪切应力-应变曲线shear stress-strain dia

38、gram 剪切应力-应变曲线pure shear 纯剪切angle of twist 扭转角ultimate shear stress 极限切应力shear modulus of elasticity 弹性剪切模量modulus of rigidity 刚性模量，刚度模量Chapter 4 Axial Load 轴向载荷（轴力）4.1 Saint-Venants Principle 圣维南原理the centroidal axis 形心轴localized deformation 局部变形Saint-Venants principle 圣维南原理The stress and strain pr

39、oduced at points in a body sufficiently removed from the region of load application will be the same as the stress and strain produced by any applied loadings that have the same statically equivalent result, and are applied to the body within the same region.物体上各点处所产生的应力和应变，当其位置距离力的作用点足够远时，与在同一区域内作用

40、等效载荷所产生的应力和应变是相同的。4.2 Elastic Deformation of an Axially Loaded Member 轴向受载构件的弹性变形elastic deformation 弹性变形algebraic addition 代数和sign convention 符号规定Using the method of sections, a differential element (or wafer) of length dx and cross-sctional area A(x) is isolated frome the bar at the arbitrary posi

41、tion x.使用截面法，在杆上任意位置x处截取长度为dx，截面积为A(x)的微元。The displacement of one end of the bar with respect to the other is then found from the algebraic addition of the relative displacements of the ends of each segment.（杆）一端相对于另一端的位移等于每段杆端点相对位移的代数和。Both the force and displacement are positive if they cause tens

42、ion and elongation, respectively; whereas a negative force and displacement will cause compression and contraction, respectively.若（轴）力为拉力，位移为伸长状态时，则（轴）力和位移为正；若（轴）力为压力，位移为收缩状态时，则（轴）力和位移为负。4.3 Principle of Superposition 叠加原理deflect 偏转By subdividing the loading into components, the principle of superpo

43、sition states that the resultant stress or displacement at the point can be determined by algebraically summing the stress or displacement caused by each load component applied separately to the member.通过将载荷分解为若干分量，根据叠加原理，某一点处的应力或位移值可以通过代数求和各载荷分量在该点所产生的应力或位移来确定。4.4 Statically Indeterminate Axially L

44、oaded Member 轴向受载的静不定构件statically determinate problems 静定问题statically indeterminate problems 静不定问题，超静定问题compatibility condition （变形）协调条件kinematic condition 运动学条件4.5 The Force Method of Analysis for Axially Loaded Members 应用力法分析轴向受载构件the force method 力法flexibility method of analysis 柔度分析法redundant su

45、pport 多余约束4.6 Thermal Stress 热应力the linear coefficient of thermal expansion 热膨胀线性系数In a statically indeterminate member, these thermal displacements will be constrained by the supports, thereby producing thermal stresses that must be considered in design.在静不定构件中，热位移由于被支座所限制，会导致（构件内部）产生热应力，这一现象需要在设计中

46、考虑到。4.7 Stress Concentrations 应力集中stress concentration 应力集中grid lines 网格线fillet 圆角stress-concentration factor 应力集中因数fatigue loading 疲劳载荷elastic perfectly plastic material 弹塑性材料Not only do complex stress distributions arise just under a concentrated loading, they can also arise at sections where the

47、members cross-sectional area changes.不仅集中载荷的作用点处会产生复杂的应力分布，在构件横截面尺寸发生变化的位置也会产生复杂的应力分布。Chapter 5 Torsion 扭转5.1 Torsional Deformation of a Circular Shaft 圆截面轴的扭转变形shaft 轴helix 螺旋线warp 翘曲the front and rear faces of the element 微元的前后面circular tube 圆形管Torque is a moment that tends to twist a member about

48、 its longitudinal axis.扭矩是导致构件绕其纵向轴线发生扭转的力矩。Note that twisting causes the circles to remain circles, and each longitudinal grid line deforms into a helix that intersects the circles at equal angles.需要注意到的是扭转变形导致原有的圆形（截面网格）仍然保持圆形，纵向网格则变形为和圆形（截面网格）所成角度相等的螺旋线。Here a radial line located on the cross sec

49、tion at a distance x from the fixed end of the shaft will rotate through an angle(x). The angle(x), so defined, is called the angle of twist.位于距固定端长度为x的横截面上的径向线段将绕着轴线转过角度(x)，该角度称为扭转角。The shear strain within the shaft varies linearly along any radial line, from zero at the axis of the shaft to a maxi

50、mum gmax at its outer boundary.轴上的切应变沿径向方向呈线性变化，在轴心处为零，在外圆周上取得最大值gmax。5.2 The Torsion Formula 扭转公式polar moment of inertia 极惯性矩solid shaft 实心轴annulus 环形面circumference 周长tubular shaft 空心轴geometric property 几何特性torque diagram 扭矩图If the material is linear-elastic, then Hookes law applies, t=Gg, and cons

51、equently a linear variation in shear strain, as noted in the previous section, leads to a corresponding linear variation in shear stress along any radial line on the cross section. 如果材料在线弹性范围内，可以应用胡克定律，t=Gg，则根据上一节讨论的结果，呈线性变化的切应变会导致在横截面的径向方向上，切应力也呈现线性变化（趋势）。Not only does the internal torque T develop

52、 a linear distribution of shear stress along each radial line in the plane of the cross-section area, but also an associated shear-stress distribution is developed along an axial plane.扭矩T不仅导致在横截面沿径向方向上线性分布的切应力，还导致在轴向平面内也存在切应力的分布。As a sign convention, T will be positive if by the right-hand rule the

53、 thumb is directed outward from the shaft when the fingers curl in the direction of twist as caused by the torque.根据符号规定，如果根据右手法则，当四指沿弯矩转动方向弯曲，而拇指指向圆轴截面的外法线方向时，该弯矩T为正。5.3 Power Transmission 功率传递angular speed 角速度instantaneous power 瞬时功率frequency of a shafts rotation 转轴旋转频率5.4 Angle of twist扭转角gear 齿轮

54、bearing 轴承We will use the right-hand rule, whereby both the torque and angle will be positive, provided the thumb is directed outward from the shaft when the fingers curl to give the tendency for rotation.我们使用右手法则，当四指沿旋转趋势方向弯曲，拇指指向轴外法线方向时，扭矩和扭转角均为正值。5.5 Statically Indeterminate Torque-Loaded Members

55、扭转作用下的静不定构件The necessary condition of compatibility, or the kinematic condition, requires the angle of twist of one end of the shaft with respect to the other end to be equal to zero, since the end supports are fixed.必要的协调方程或运动条件要求该轴一端相对于另一端的扭转角为零，因为轴的两端支座处均为固定端。Chapter 6 Bending 弯曲6.1 Shear and Mom

56、ent Diagrams 剪力弯矩图symmetric cross section 对称截面unsymmetric bending 非对称弯曲simply supported beam 简支梁cantilevered beam 悬臂梁overhanging beam 外伸梁the boom of a crane 起重机臂架shear and moment functions 剪力与弯矩方程a clockwise rotation 顺时针转动In general, the internal shear and moment functions of x will be discontinuous

57、, or their slope will be discontinuous, at points where a distributed load changes or where concentrated forces or couple moments are applied.通常情况下，在分布载荷发生变化，或者有集中力或集中力偶矩作用的位置，用x表示的剪力和弯矩方程会不连续，或者方程的斜率会不连续。The positive directions are as follows: the distributed load acts upward on the beam; the internal shear force causes a clockwise rotation of the beam segment on which it acts; and the internal moment causes compression in the top fibers of the seg