材料力学第二章.pdf

1Chapter 2 Tensile&Compressive Stresses and Strength Properties of Materials轴向拉压应力与材料的力学性能轴向拉压应力与材料的力学性能?Stress in axially loaded bar?Strength Properties of Materials?Strength of axially loaded bar?Strength of Joints and connections1 Introduction引言引言2 Axial Force and Axial Force Diagram轴力与轴力图轴力与轴力图3 Tensile and Compressive Stresses拉压杆的应力拉压杆的应力45 Mechanical Properties of Materials材料的力学性能材料的力学性能6 Stress Concentration 应力集中应力集中7 Failure Criterion for Axially Loaded Bar轴向拉压强度条件轴向拉压强度条件8 Strength of Joints and Connections连接部分的强度连接部分的强度1引言Introduction引言Introductionbar?Tension and compression External Force：along the direction of the axisDeformation：extension or contraction,along the direction of the axis杆件受力特点：外力或其合力的作用线沿杆件轴线杆件变形特点：轴向伸长或缩短杆件受力特点：外力或其合力的作用线沿杆件轴线杆件变形特点：轴向伸长或缩短2轴力与轴力图轴力与轴力图Axial Force and Axial Force Diagram轴力轴力Axial force：通过截面形心并沿杆件轴线：通过截面形心并沿杆件轴线(at the centroid of the cross-section,along the direction of the axis);Sign conventions:Positive in tension拉为正拉为正,Negative in compression压为负压为负.To find the axial forces FFFF=12RFF=N1：AB（F1=F，F2=2F）?轴力图轴力图Axial Force DiagramFFN1FF=N20N2=+=+FF：BC要点：逐段分析轴力要点：逐段分析轴力The axial force is determined from free-body diagrams for each section；设正法求轴力设正法求轴力The axial forces can be assumed in positive direction at first.求任一截面上的轴力，并画出轴力图。考虑自重，密度为，横截面积为A,长度为L。求任一截面上的轴力，并画出轴力图。考虑自重，密度为，横截面积为A,长度为L。gAL例题：gAL例题：FNx+xgAgA2?回顾历史：回顾历史：回顾历史：回顾历史：直杆简单拉伸实直杆简单拉伸实验验伽利略像伽利略像伽利略指出：伽利略指出：1.如果C的重量越来越大，杆件最后总会象绳索一样断开；.如果C的重量越来越大，杆件最后总会象绳索一样断开；2.同样粗细的麻绳、木杆、石条、金属棒的承载能力各不相同；同样粗细的麻绳、木杆、石条、金属棒的承载能力各不相同；验验3.相同材料制成的杆件，承载能力与横截面积成正比，与其长度无关。3.相同材料制成的杆件，承载能力与横截面积成正比，与其长度无关。A思考：思考：杆杆AB与杆与杆AB材料相同，杆材料相同，杆AB的截面积大于杆的截面积大于杆AB的截面积。的截面积。A1、若所挂重物的重量相同，哪根杆危险1、若所挂重物的重量相同，哪根杆危险？2、若C2、若C的重量大于C的重量，哪根杆危险的重量大于C的重量，哪根杆危险？粗杆粗杆CBACBA细杆细杆AFAFNN?3 Tensile and compressive stresses轴向拉压应力轴向拉压应力轴向拉压应力轴向拉压应力?拉压杆横截面上的应力拉压杆横截面上的应力Stresses over the Cross-Section Stresses over the Cross-Section?拉压杆斜截面上的应力拉压杆斜截面上的应力Stresses on an Oblique PlaneStresses on an Oblique Plane?圣维南原理圣维南原理Saint-Venants PrincipleSaint-Venants Principle?例题例题ExamplesExamples1.试验观察Experimentalobservation1.试验观察Experimentalobservation?拉压杆横截面上的应力Stresses over the cross section拉压杆横截面上的应力Stresses over the cross section变形后横线仍为直线,仍垂直于杆件轴线,只是间距增大变形后横线仍为直线,仍垂直于杆件轴线,只是间距增大.Transversal line after deformation:straight;perpendicular to the axis.2.Assumption based on deformation observation(plane assumption)uniform distribution of stresses over cross sectionno shear stress平面假设平面假设:横截面仍保持为平面，且仍垂直于杆件轴线；横截面仍保持为平面，且仍垂直于杆件轴线；正应变沿横截面均匀分布横截面上没有切应变正应变沿横截面均匀分布横截面上没有切应变0const =0const =3.横截面正应力横截面正应力Stresses over the cross sectionAFN=FN:轴力轴力Axial force;A:横截面的面积横截面的面积Cross-sectional area横截面正应力公式横截面正应力公式两端受均匀分布载荷时锥形杆x方向正应力分布情况两端受均匀分布载荷时锥形杆x方向正应力分布情况=2.8=2.8o oFFx =11.3=11.3o o锥度2锥度2 1515o o时，时，max av 与的相对误差与的相对误差5%3?Saint-Venants PrincipleSaint-Venants Principle加载点临域的应力分布加载点临域的应力分布Stress distribution in the vicinity of the applied load圣维南原理圣维南原理问题：杆端作用均布力，横截面应力均布.杆端作用集中力，横截面应力均布吗?圣维南原理圣维南原理Saint-Venants PrincipleSaint-Venants Principle力作用于杆端的分布方式，只影响杆端局力作用于杆端的分布方式，只影响杆端局部范围的应力分布部范围的应力分布，影响区约为距杆端影响区约为距杆端1应力均匀区应力均匀区Uniform-stress area部范围的应力分布部范围的应力分布，影响区约为距杆端影响区约为距杆端1 倍的横向尺寸。在影响区外，应力的分布与外力的作用方式无关（倍的横向尺寸。在影响区外，应力的分布与外力的作用方式无关（At a distance equal to,or greater than,the width of the member,the stress distribution may be assumed independent of the actual mode of application of the loads）。）。杆端镶入底座杆端镶入底座,横向变形受阻横向变形受阻The transversal displacement is restricted.1.等直杆或小锥度杆Straight bar(or stepped bar)with uniform section,or with small taper;2.外力过轴线 The applied force P acts through the 公式的适用范围Necessary conditions for the equation to be validAFN=Stresses over the cross section外力轴线ppgcentroid of the cross section;3.当外力均匀地加在截面上，此式对整个杆件都适用，否则仅适用于离开外力作用处稍远的截面The normal stress distribution in an axially loaded member is uniform,except in the near vicinity of the applied load(known as Saint-Venants Principle).在材料力学的习题中，一般假定外力是均匀地加在截面上。在材料力学的习题中，一般假定外力是均匀地加在截面上。1.1.Stresses on an Oblique Plane?Stresses on an Oblique Plane斜截面应力斜截面应力FF低碳钢拉伸时为什么会沿45出现滑移线？斜截面上有何应力？斜截面上有何应力？What kinds of stresses on an oblique plane?如何分布？如何分布？Distribution of the stress?:Positive when rotate anticlockwise from the x axis to the normal.以以x轴为始边，逆时针转向者为正轴为始边，逆时针转向者为正横截面上横截面上的正应力的正应力横截面间横截面间的纤维变的纤维变斜截面间斜截面间的纤维变的纤维变斜截面上斜截面上的应力均的应力均的正应力的正应力均匀分布均匀分布The stress distribution on an cross-section is uniform的纤维变的纤维变形相同形相同Fibres between two cross-sections have the same deformation的纤维变的纤维变形相同形相同Fibres between two oblique sections have the same deformation的应力均的应力均匀分布匀分布The stress distribution on an oblique section is uniform2.p p 00FAF cosF2045max =o=0cos ,0FpFx coscos0=AFp20coscos=p 2sin2sin0=p00max =3.、and maximum stressand maximum stress4?ExamplesExamplesKnown：F=50 kN，A=400 mm2Ask for：Stresses on m-m planeS l tiSolution:FF=N263N0m10400N1050=AFAF 81.25 10 Pa125 MPa=o50=MPa-51.6 50coscos 202050=oo MPa-61.6001 sin22 sin 20050=oo?思考：思考：思考：思考：FF1 1、变形后两直线的夹角是否改变变形后两直线的夹角是否改变1 1、变形后两直线的夹角是否改变变形后两直线的夹角是否改变2、如果改变，试定性解释为什么改变3、如果改变，试定量分析角度的改变量2、如果改变，试定性解释为什么改变3、如果改变，试定量分析角度的改变量45 Mechanical Properties of Materials?拉伸试验与应力应变图?拉伸试验与应力应变图Tensile Tests and Stress-Strain Diagram?低碳钢拉伸应力应变曲线?低碳钢拉伸应力应变曲线Tensile Stress-Strain Curve for Mild Steel?卸载与再加载路径?卸载与再加载路径Unloading and Reloading Path?名义屈服极限?名义屈服极限Conditional Yield Limit?脆性材料脆性材料拉伸应力拉伸应力应变曲线应变曲线Stress-Strain Curves for Brittle Materials材料的力学性能材料的力学性能?脆性材料脆性材料拉伸应力拉伸应力应变曲线应变曲线Stress Strain Curves for Brittle Materials?复合与高分子材料的力学性能?复合与高分子材料的力学性能Strength Properties of Composite Materials and Polymers?材料压缩时的应力应变曲线?材料压缩时的应力应变曲线Compressive stress-strain curve?温度对力学性能的影响?温度对力学性能的影响Temperature Effect to Strength Properties2.Tensile tests?Test machineTest machine?拉伸试验与应力应变图拉伸试验与应力应变图Tensile tests and stress-strain diagram1.拉伸标准试样拉伸标准试样Test specimen Test specimen 标距GB/T6397-1986金属拉伸试验试样金属拉伸试验试样?拉伸试验与拉伸试验与F-l 曲线Tensile test and load-extension curve曲线Tensile test and load-extension curve?低碳钢拉伸应力应变曲线?低碳钢拉伸应力应变曲线Tensile stress-strain curve for mild steel p-proportional limit比例极限比例极限低碳钢拉伸时的应力应变图低碳钢拉伸时的应力应变图Slip line屈服屈服Yield硬化硬化Strain hardeningNeck b s-yield stress屈服极限屈服极限线弹性线弹性Linear elastic po b-ultimate strength强度极限强度极限屈服屈服Yields E=tan-elastic modulus弹性模量弹性模量屈服极限屈服极限5Tested specimenTested specimenNecking and failure缩颈与断裂缩颈与断裂Slip line滑移线滑移线?卸载与再加载路径?卸载与再加载路径Unloading and reloading pathplastic straineelastic limit弹性极限弹性极限eelastic strain弹性应变弹性应变 硬化硬化Strain hardeningNeck Slip line屈服屈服Yieldebpplastic strain塑性应变塑性应变冷作硬化冷作硬化(Work Hardening)由于预加塑性变形，而使由于预加塑性变形，而使 e(或或 p)提高的现象提高的现象线弹性线弹性Linear elastic po12ep?Plastic Strain:Permanent strain after the stress is removed伸长率？伸长率？001100=lllPlasticity塑性塑性材料能经受较大塑性变形而不破坏的能力材料能经受较大塑性变形而不破坏的能力l试验段原长（标距）试验段原长（标距）initial gauge length of the tensile specimenlfif?Percentage increase in length伸长率（延伸率）伸长率（延伸率）l1final gauge length of the tensile specimen001100 =AAA?断面收缩率断面收缩率Percentage reduction in areaA试验段横截面原面积试验段横截面原面积initial cross-sectional area of the tensile specimen?塑性与脆性材料塑性与脆性材料Ductile and brittle materials Ductile Materials（塑性材料）：（塑性材料）：5%5%(mild steel,aluminum,copper,etc.)Brittle Materials（脆性材料）：（脆性材料）：5%(high-carbon steel,cast iron,concrete,glass,ceramics,bronze,etc.)A1final cross-sectional area of the tensile specimen四个阶段，三个特征点，两个现象：四个阶段，三个特征点，两个现象：四个阶段：四个阶段：线性，屈服，硬化，缩颈线性，屈服，硬化，缩颈三个特征点：三个特征点：比例极限，屈服极限，强度极限比例极限，屈服极限，强度极限两个现象两个现象滑移线滑移线缩颈缩颈总结总结pbso?低碳钢材料拉伸时的力学性能低碳钢材料拉伸时的力学性能两个现象两个现象：滑移线滑移线，缩颈缩颈?低碳钢卸载与再加载时的力学性能低碳钢卸载与再加载时的力学性能弹性极限冷作硬化？弹性应变塑性应变弹性极限冷作硬化？弹性应变塑性应变(残余应变残余应变)pebpe1 2o?Conditional Yield Limit（名义屈服极限）（名义屈服极限）30铬锰硅钢铬锰硅钢50钢钢塑性材料拉伸塑性材料拉伸Tensile stress-strain curve for Ductile materials一般金属材料的力学性能一般金属材料的力学性能 /%/%硬铝硬铝p0.2Conditional yield limit,or proof stress名义屈服极限名义屈服极限6Tensile stress-strain curve for cast iron铸铁拉伸时的应力应变图铸铁拉伸时的应力应变图?脆性材料拉伸应力应变曲线脆性材料拉伸应力应变曲线Stress-Strain Curves for Brittle Materialsfailure surface:flat断口与轴线垂直断口与轴线垂直Fiber-Reinforced Composite Material纤维增强复合材料纤维增强复合材料Synthetic Polymers高分子材料高分子材料?Strength Properties of Composite Materials and Polymers?材料压缩时的应力应变曲线材料压缩时的应力应变曲线Compressive stress-strain curve Typical compressive and tensile curve for mild steel低碳钢压缩低碳钢压缩ctEE csts Compressive stress-strain curve for cast iron铸铁压缩铸铁压缩Tensile stress-strain curve for cast iron铸铁拉伸铸铁拉伸 cb=34 tb断口与轴线约成断口与轴线约成45oThe failure surface is oriented at approximately 45oto the axis of loading.Typical compressive and tensile stress-strain curves for concrete（混凝土）（混凝土）?Temperature Effect to Strength Properties7World Trade Centre据分析，由于大量飞机燃油燃烧，温度高达据分析，由于大量飞机燃油燃烧，温度高达1200 C，组成大楼结构的钢材强度急剧降低，致使大厦铅垂塌毁，组成大楼结构的钢材强度急剧降低，致使大厦铅垂塌毁大厦受撞击后，为什麽沿铅垂方向塌毁？大厦受撞击后，为什麽沿铅垂方向塌毁？()刚度最大；()强度最高；()塑性最好。()刚度最大；()强度最高；()塑性最好。ABC下图为A、B、C三种材料的应力应变曲线下图为A、B、C三种材料的应力应变曲线 oA6 Stress Concentration 应力集中应力集中?应力集中应力集中Stress Concentration?交变应力作用下材料的疲劳交变应力作用下材料的疲劳Fi?交变应力作用下材料的疲劳交变应力作用下材料的疲劳Fatigue under Repeated Stresses?应力集中对构件强度的影响应力集中对构件强度的影响Effect of Stress Concentration?应力集中应力集中Stress Concentration由于截面急剧变化引起应力局部增大现象由于截面急剧变化引起应力局部增大现象Caused by the change of cross section应力集中应力集中Stress Concentration应力集中因素应力集中因素Stress Concentration Factornmax =K max最大应力最大应力Maximum stress n 名义应力名义应力Nominal Stress8例：下面受力杆件，哪个截面上的应力可以采用公式计算例：下面受力杆件，哪个截面上的应力可以采用公式计算NFA=h2h/31234q2h/3?Fatigue under Repeated StressesRepeated cyclic loading交变应力交变应力或或循环应力循环应力构件在交变载荷作用下的力学性能不同于静载时的力学性能构件在交变载荷作用下的力学性能不同于静载时的力学性能循环特征/应力比循环特征/应力比Stress ratiomaxmin=RAverage stress2minmax+=aFatigue Failure and S-N curve b s rendurance limit?疲劳破坏疲劳破坏 破坏时应力低于破坏时应力低于 b甚至甚至 sUnder cyclic loading,a material may fail at a stress much less than the materials ultimate strength or yield stress.?持久极限持久极限Endurance limit-A critical stress,below which cyclic stresses cannot cause a fatigue failure.N?S曲线曲线 在交变应力作用下，应力在交变应力作用下，应力s（或或）与相应应力循环数（或寿命）与相应应力循环数（或寿命）N 的关系曲线的关系曲线Fatigue?破坏时应力低于破坏时应力低于 b甚至甚至 sFail at a stress much less than the materials bor yield stress s?即使是塑性材料，也呈现脆性断裂即使是塑性材料，也呈现脆性断裂A fatigue failure is of a brittle nature,even for materials that are normally ductile.?经历裂纹萌生、逐渐扩展到最后断裂三阶段经历裂纹萌生、逐渐扩展到最后断裂三阶段As the structure have minute cracks or other defects in it,under repeated cyclic loading,the large stresses that occur at these stress concentrations cause the cracks to grow until fracture eventually occursCrack initiationFracture钢拉伸疲劳断裂钢拉伸疲劳断裂Fatigue failure of steel specimencracks to grow,until fracture eventually occurs.?应力集中对构件强度的影响?应力集中对构件强度的影响Effect of Stress Concentration?脆性材料构件脆性材料构件Brittle Materials:failure at max b,stress concentration have significant influence to strength.?塑性材料构件塑性材料构件Ductile Materials:当当 max达到达到 s后再加载，后再加载，分布趋于均匀化，不影响构件静强度。分布趋于均匀化，不影响构件静强度。After maxattain s,the distribution of stresses approach to uniform,stress concentration do not have significant influence to strength.?Fatigue life:应力集中促使疲劳裂纹的形成与扩展，对构件（塑性与脆性材料）的疲劳强度影响极大应力集中促使疲劳裂纹的形成与扩展，对构件（塑性与脆性材料）的疲劳强度影响极大stress concentrationhave significant influence to initiation and growth of cracks.7 Failure criterion for axially loaded bar轴向强度条件?轴向强度条件?Static Failure and Allowable Stress失效与许用应力失效与许用应力?Failure criterion for axially loaded bar轴向拉压强轴向拉压强度条件度条件?Examples9?Static Failure and Allowable Stress 失效与许用应力失效与许用应力Ultimate stress（极限应力）（极限应力）bsu=stress that components are expected to support Static Failure（静荷失效）（静荷失效）Allowable stress:（许用应力许用应力：构件工作应力的最大容许值构件工作应力的最大容许值）Ductile Materials塑性材料塑性材料Brittle Materials脆性材料脆性材料nu =n 1 Factor of safety安全因数安全因数Materials Brittle-Materials Ductile-bbssnn=（许用应力许用应力：构件工作应力的最大容许值构件工作应力的最大容许值）Usually,Usually,ns=1.52.2，nb=3.05.0?强度条件?强度条件Failure criterion for axially loaded bar maxNmax =AF maxN,AFFailure criterion for axially loaded bar 变截面变轴力拉压杆变截面变轴力拉压杆Bar with variable cross-section or axial force 等截面拉压杆等截面拉压杆Uniform Bar强度条件强度条件Most common strength problems常见强度问题常见强度问题：（1）校核强度）校核强度当已知构件的尺寸,材料和外力时，可以检查构件是否满足强度要求。脆性材料的 t 和 c 一般不相同，需分别校核；若max超出 在5%以内，工程计算中是允许的。Strength analysisGiven F,A and ，To check the strength of the barMost common strength problems常见强度问题常见强度问题：N AF=Finding out the safe extent of working loadsGiven A and ，To find FN,max（2）决定承载能力（或决定许用载荷）决定承载能力（或决定许用载荷）如果构件的尺寸和材料已知，可以由下式决定许用载荷：（3）选择横截面尺寸）选择横截面尺寸当构件的外力和材料确定后，可以由下式计算横截面面积：当构件的外力和材料确定后，可以由下式计算横截面面积：（4）优化设计）优化设计Optimum design of structuremaxN,FA Design of cross-sectionGiven F and ，To find A?ExamplesFinding the internal force:A free-body diagram of pin A allows us to determine the force in each member.lA12F0,0From=yxFFFFN1FN2Stress:1=F/(A1sin),2=-F/(A2tg)so that:FN1=F/sin(tensile),FN2=F/tg(compressive),yx1.Known:,F,A1,A2,t,c To analysis the strength of the structure lA12F 1N11tAF=FN1=F/sin(tensile),FN2=-F/tg(compressive)2N22cAF=2.Known:F,t,c To design:A1,A2 lA12F 1N11tAF=A1F/(tsin)FN1=F/sin(tensile),FN2=-F/tg (compressive)2N22cAF=A2F/(ctg)103.Known:,A1,A2,t,c To find:F lA12F1N1 sinAFFt=1N11tAF=),(21FFMinF=sinFN1=F/sin(tensile),FN2=-F/tg (compressive)2N22cAF=2N2 AtgFFc=F F1=t A1sin,F F2=c A2tg4.Known:A2,t,c To design:A1 lA12FOptimum design of structureIf F1F2,then F=F2.Optimum design:FN1=F/sin(tensile),FN2=-F/tg (compressive)pglet F1=F2F1=t A1sin,F2=c A2tgA1=c A2tg/(t sin)5.Known:F,t=c=,To design:,A1,A2,let the structure have minimum weight lA12FF=F/sinA=F/(tg)l=lDensity 1=2=A1=F/(sin),l1=l/cosFN1=F/sinFN2=-F/tgA2=F/(tg),l2=lWeight()()+=+=+=+=sincos2sin22211FllAlAW32sin0=ddW4454=3kN(-)9kN21kN(+)MPa110=例例图示圆截面钢杆,弹性模量E=210GPa，许用应力为。已知载荷P=3kN，试校核强度。P2P2P2P2P61016各杆的强度条件:kNPN31=kNN92=kNN213=解:用截面法分析各杆内力,画出内力图.1.1066103423111=AN6.11410109423222=AN4.104161021423333=AN2虽然大于许用应力，但差别小于5%，所以可认为近似相等。即本题满足强度条件。例题 如图1-10所示的结构，已知各杆的面积和材料为A1=400mm2，A2=300mm2，1=2=160MPa，试计算该结构所能承受的最大载荷。N2N1l/32l/3（1）由平衡条件确定各杆轴力与载荷P之间的关系式：MA=0 N2=F/3;Y=0N1=2/3FF（2）由强度条件计算最大载荷杆1的强度条件；N1/A11F=3/2A11=3/2400160=96000N=96kN杆2的强度条件；N2/A22F=3A22=3300160=114000N=114kN要使结构安全工作应取其较小值，即F=96kN（2）为使该结构安全受力，按杆1的强度取F=96kN。对杆2来说，强度是有富裕的?不经济注意：（1）最大载荷可否写为F=A11+A22=112kN？否！11如图所示，当F力作用在何处时，结构所承担的载荷最大?并求此时的载荷值。N2N1xl-xF解：（1）由平衡条件，得N1=F(l-x)/l N2=Fx/l杆2的强度条件N2/A22,Fx=l A22F（2）由强度条件可见，调整了力F的作用点之后，已做到等强联解2式，得x=l A22/(A11+A22)=3 l/7杆1的强度条件N1/A11 (l-x)F=l A11例：例：管道修理问题合金钢管管道修理问题合金钢管D=30mm,d=27mm,s=850MPa，套管，套管 s=250MPa。求套管的外径。求套管的外径Dt，两种材料取相同的安全系数。两种材料取相同的安全系数。思想：思想：等强原则等强原则F=F()()nDDndDsts442222=Example:Two members with uniform cross section are joined by a glued scarf splice at an angle.The ultimate stresses for the glued joint are U=23 MPa and U=11.5 MPa.Determine the range of values of for which the factors of safety in shear and normal stresses are 7.19 and 8.45,respectively.Solution:Normal and shear stresses on inclined plane:Solution:Where:A0=(56 mm)(42 mm)=2.352 x 10-3m220cosAF=cossin0AF=Normal and shear stresses on inclined plane:Normal Stress:nAFU=20cosTwo members with uniform cross section are joined by a glued scarf splice at an angle.The ultimate stresses for the glued joint are U=23 MPa and U=11.5 MPa.Determine the range of values of for which the factors of safety in shear and normal stresses are 7.19 and 8.45,respectively.Shear Stress:Shear Stress:nAFU=cossin0预定参数(设计中已确定，设计者不能任意修改的量)设计变量(可由设计者调整的量):AxB例：例：ml2=mxB1=35107.7mmN=MPat150=MPac100=BymymB5.15.0已知,F=100kN,的范围：，如何设计使桁架的重量最小？约束条件(对设计变量的限制条件)(1)强度条件约束(截面、杆件的强度);(2)几何条件约束(B点的高度范围)设计变量(可由设计者调整的量)