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 connections§§1 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§§4~5 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ρ，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ρ，横截面积为A,长度为Lρ ρgAL例题：gAL例题：FNx+ρ ρxρgAρgA2? ? 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A’B’材料相同， 杆材料相同， 杆A’B’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C’B’ACBA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-Venant's PrincipleSaint-Venant's 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) ①①uniform distribution of stresses over cross section②②no 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0constγ γε ε====0constτ τσ σ====3.横截面正应力横截面正应力Stresses over the cross sectionAFN= =σ σFN :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α=2.8α=2.8o oFFxα αα αα=11.3α=11.3o o锥度2锥度2α α ≤15≤15o o时，时，maxσ σavσ σ与的相对误差与的相对误差<5%3?? Saint-Venant's PrincipleSaint-Venant's 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-Venant's PrincipleSaint-Venant's 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= =σ σStresses 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-Venant's 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?α: 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 pα α∑ ∑00FAFα αcosF2045maxσ σστστα α===== =o∑ ∑= =− −= =0cos , 0FpFxα αα αα ασ σα αα αcoscos0====AFpασασασασαααα20coscos==== pα ασ σαταταααα2sin2sin0==== p00maxσ σσ σσ σα α===== =3. σ σα 、α 、τ τα αand maximum stressand maximum stress4?? ExamplesExamplesKnown：：F = 50 kN，，A = 400 mm2Ask for：：Stresses on m-m planeS l tiSolution:FF−= −=N263N0m10400N1050− −××−=−==××−=−==AFAFσ σ81.25 10 Pa125 MPa= −×= −o50= =α αMPa -51.6 50coscos 202050======ooσ σα ασ σσ σMPa -61.6001 sin22 sin 20050======ooσ σα ασ στ τ? ? 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§§4~5 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-Δ-Δl 曲线Tensile test and load-extension curve曲线Tensile test and load-extension curveσ σ?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 steelσ σp-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 σbσ σs-yield stress屈服极限屈服极限线弹性线弹性Linear elasticε εσpoσ σb-ultimate strength强度极限强度极限屈服屈服Yieldσsα αE=tanα α-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 pathεplastic strainσe－－elastic limit弹性极限弹性极限εe－－elastic strain弹性应变弹性应变σ σσ硬化硬化Strain hardeningNeck Slip line屈服屈服Yieldσeσbεp－－plastic 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σ σ e(或或σ σ p)提高的现象提高的现象线弹性线弹性Linear elasticε εσpo12εeεp? ?Plastic Strain: Permanent strain after the stress is removed伸长率？伸长率？001100×−=lllδPlasticity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）l1－－final gauge length of the tensile specimen001100× ×− −= =AAAψ ψ? ?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）：δ δ≥ 5 %≥ 5 %(mild steel, aluminum, copper ,etc.)Brittle Materials（脆性材料）：（脆性材料）：δ δ< 5 % (high-carbon steel, cast iron , concrete, glass, ceramics, bronze, etc.) A1－－final 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εσσpσbσso??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)εσεσσpσeσbεpεe1 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ε ε /%/%硬铝硬铝σp0.2－－Conditional 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 ≈ ≈cstsσ σσ σ≈ ≈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σ σ cb= 3~4σ σ 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 Properties7World 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σ σABC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ε εoA§§6 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 Factornmaxσ σσ σ= =Kσ σmax－最大应力－最大应力Maximum stressσ σn －名义应力－名义应力Nominal Stress8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NFAσ σ= =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 ratiomaxminσσ=RAverage stress2minmaxσσσ+=aFatigue Failure and S-N curveσ σ bσ σ sσ σ r－－endurance limit? ?疲劳破坏疲劳破坏− −破坏时应力低于破坏时应力低于σ σb甚至甚至σ σsUnder cyclic loading, a material may fail at a stress much less than the material's 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−Ν−Ν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（（σ σ 或或τ τ）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σ σb甚至甚至σ σsFail at a stress much less than the material‘s σ σbor yield stress σ σs? ?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 σ σmax＝＝σ σb, stress concentration have significant influence to strength.? ?塑性材料构件塑性材料构件Ductile Materials :当当σ σmax达到达到σ σs后再加载，后再加载，σ σ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 σ σmaxattain σ σs, 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?? Examples9??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）{bsuσσσσσσ=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][σ σσ σ= =n≥ ≥1 Factor of safety安全因数安全因数Materials Brittle-][Materials Ductile-][bbssnnσσσσ==（（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=1.5~2.2，，nb=3.0~5.0?强度条件?强度条件Failure criterion for axially loaded bar ][maxNmaxσ σσ σ≤⎟⎠⎞⎜⎝⎛=≤⎟⎠⎞⎜⎝⎛=AF ][maxN,σ σ≤ ≤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[ [σt] ]和[ [σc] ]一般不相同，需分别校核；若σmax超出[σ] 在5%以内，工程计算中是允许的Strength analysisGiven F, A and [ [σ σ] ]，，To check the strength of the barMost common strength problems常见强度问题常见强度问题：：][][Nσ σAF= =Finding out the safe extent of working loadsGiven A and [ [σ σ] ]，，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 structure][maxN,σ σFA ≥ ≥Design of cross-sectionGiven F and [ [σ σ] ]，，To find A?? ExamplesFinding the internal force: A free-body diagram of pin A allows us to determine the force in each member.α αlA12F0 , 0From==∑∑yxFFFFN1FN2Stress:σ σ1=F/(A1sinα α), σ σ2= - F/(A2tgα α)so that:FN1=F/sinα α(tensile) , FN2=F/tgα α(compressive ),∑∑yx1. Known: α, F, A1, A2, [σ t ], [σ c ] To analysis the strength of the structureα αlA12F ][1N11tAFσσ≤⎟⎟⎠⎞⎜⎜⎝⎛=FN1=F/sinα α(tensile) , FN2= -F/tgα α(compressive ) ][2N22cAFσσ≤⎟⎟⎠⎞⎜⎜⎝⎛=2. Known: F, α, [σ t ], [σ c ] To design: A1 , A2α αlA12F ][1N11tAFσσ≤⎟⎟⎠⎞⎜⎜⎝⎛=A1≥F/([σt]sinα)FN1=F/sinα α(tensile) , FN2= -F/tgα α (compressive ) ][2N22cAFσσ≤⎟⎟⎠⎞⎜⎜⎝⎛=A2≥F/([σc]tgα) 103. Known: α, A1, A2, [σ t ], [σ c ] To find: [F]α αlA12F1N1 ][sinAFFtσα≤= ][1N11tAFσσ≤⎟⎟⎠⎞⎜⎜⎝⎛=])[],([][ 21FFMinF =∴sinαFN1=F/sinα α(tensile) , FN2= -F/tgα α (compressive ) ][2N22cAFσσ≤⎟⎟⎠⎞⎜⎜⎝⎛=2N2 ][AtgFFcσα≤=F≤ [F1]=[σt] A1sinα, F≤ [F2]=[σc] A2tgα4. Known: A2, α, [σ t ], [σ c ] To design: A1 α αlA12FOptimum design of structureIf [F1]<[F2], then [F]= [F1];If [F1]>[F2] , then [F]= [F2].Optimum design:FN1=F/sinα α(tensile) , FN2= -F/tgα α (compressive )pglet [F1]=[F2][F1]=[σt] A1sinα, [F2]=[σc] A2tgαA1=[σc] A2tgα / ([σt] sin α)5. Known: F, [σ t ]= [σ c ]=[σ], To design: α, A1 , A2 , let the structure have minimum weightα αlA12FF=F/sinαA=F/([σ]tgα)l =lDensity γ1=γ2=γA1=F/([σ]sinα) , l1=l/cosαFN1=F/sinαFN2= -F/tgαA2=F/([σ]tgα), l2=lWeight()()[ ][ ]⎟⎠⎞⎜⎝⎛+=+=⎟⎠⎞⎜⎝⎛+=+=ααασγγsincos2sin22211FllAlAW32sin0==ααddW4454′°=α3kN(-)9kN21kN(+)MPa110][=σ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 .1066103423111σπσ≤=×××==AN][6 .11410109423222σπσ≥=×××==AN][4 .104161021423333σπσ≤=×××==AN2σ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，[σ]1=[σ]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/A1≤[σ]1F=3/2A1[σ]1=3/2×400×160=96000N=96kN杆2的强度条件；N2/A2≤[σ]2F=3A2[σ]2=3×300×160=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=A1[σ]1+A2[σ]2=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/A2≤[σ]2,Fx= l A2[σ]2F（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 A2[σ]2/(A1[σ]1+A2[σ]2)=3 l /7杆1的强度条件N1/A1≤[σ]1 (l-x)F= l A1[σ]1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, σ σs=850MPa，套管，套管σ σ’’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’][ ][]()()nDDndDsts442222σπσπ′−=−∴Example: Two members with uniform cross section are joined by a glued scarf splice at an angle θ. The ultimate stresses for the glued joint are σU= 23 MPa and τU= 11.5 MPa. Determine the range of values of θ 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 10-3m2θσα20cosAF=θθταcossin0AF=Normal and shear stresses on inclined plane:Normal Stress:σασθσnAFU≤=20cosTwo members with uniform cross section are joined by a glued scarf splice at an angle θ. The ultimate stresses for the glued joint are σU= 23 MPa and τU= 11.5 MPa. Determine the range of values of θ for which the factors of safety in shear and normal stresses are 7.19 and 8.45, respectively.Shear Stress:Shear Stress:τατθθτnAFU≤=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):θAxB例：例：ml2=mxB1=35107 . 7mmN−×=γ[ ]MPat150=σ[]MPac100=σBymymB5 . 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): yB,A1,A2目标函数：桁架的重量W(最小)θ2θ1FBClyB12⎪⎩⎪⎨⎧=−+−==−−=∑∑0coscos00sinsin021122112FNNFNNFyxθθθθ()2111sinsinθθθ+=FN()2122sinsinθθθ+−=FN解：1.应力分析由此()ylxFNBB221−+=lyxFNBB222+−=由正弦定理：θ2θ1FABClxByB12lN1l2()1221lAylxFBB−+=σ2222lAyxFBB+−=σ杆1和2横截面上的正应力12()⎟⎠⎞⎜⎝⎛++−+=222221BBBByxAylxAWγ()[ ][]⎪⎪⎭⎪⎪⎬⎫≤+≤−+cBBtBBlAyxFlAylxFσσ2221222.最轻重量设计()mmyB1500500≤≤θ2θ1FABClxByB123.最优解搜索采用直接实验法搜索。

首先在条件所述范围内选取一系列yB值，由强度条件确定A1与A2，最后计算相应W，在yB～W曲线中选取使最小的yB与相应的A1与A2，即为本问题的最优解§§8 Strength of Joints and Connections连接部分的强度连接部分的强度?? IntroductionIntroduction?? Shearing StressShearing Stress剪切应力剪切应力?? ShearingShearing StressStress剪切应力剪切应力?? Bearing Stress挤压应力Bearing Stress挤压应力?? ExamplesExamples? Introduction? IntroductionMany engineering structures and machines consist of components connected through joints, such as bolted joints螺钉, riveted joints铆钉, glued and welded joints.lugPin销钉销钉Bolt螺栓螺栓lug耳片耳片通常连接件的应力场较复杂In general, the stress field of joints are complex;Forms of failure:by shearing, or bearing pressure, or tension.假定计算法(实验依据，经验公式)A mixture of stress analysis and experience of the behaviour of actual joints;破坏形式： 剪断、剪豁（当边距大于钉直径2倍时可避免剪豁）挤压破坏、拉断（拉断可按拉压杆公式计算）。

剪切应力Shearing Stress剪切应力Shearing StressOne of the simplest types of joint between two plates of material is a bolted or riveted lap joint剪切面剪切面][Sτ τ≤ ≤AF－－剪切强度条件剪切强度条件Failure criterionfor shear stress[ [τ τ ] −] −许用切应力许用切应力Allowable shear stress假设：剪切面上的切应力均匀分布假设：剪切面上的切应力均匀分布The shear stress is assumed to be uniformly distributed over the entire sectionAFS= =τ τ13??挤压应力挤压应力Bearing Stress?Bearing Stress?bearing surface挤压面挤压面-连接件间的相互挤压接触面连接件间的相互挤压接触面Two surfaces that are in contact.耳片销钉耳片销钉? ?When one surface is ? ?Failure caused by bearing stress挤压破坏挤压破坏-在接触区的局部范围内，产生显著塑性变形在接触区的局部范围内，产生显著塑性变形Significant deformation in bearing surfacecompressed into another, a bearing stress（挤压应力）（挤压应力）results.ExampledFδσδσbbs≈ ≈Failure criterion for bearing stressMax. Bearing Stress最大挤压应力最大挤压应力][bsbsσ σσ σ≤ ≤g挤压强度条件挤压强度条件[ [σ σbs] −] − Allowable bearing stress钉拉断：钉拉断：Fdτ τπ π δ δ= =224()bsFDdσπσπ=−=−δDdFKnown：：δ δ = =2 mm，，b =15 mm，，d =4 mm，，[τ τ ] ] =100 MPa，，[σ σ bs ]=300 MPa，，[σ σ ]=160 MPaTo find: [F] = ?? Examples? ExamplesSolution：：1.破坏形式破坏形式Forms of failure of the jointShearing failureBearing & tensile failuresWhen a > 2d, this modes of failure can be disregarded142. Allowable load[F]][π42ττττ≤=≤=dFkN 257. 14][π2=≤=≤τ τdFF???][bsbsσδσσδσ≤=≤=dFkN 40. 2][bs=≤=≤σ σδ δdF][)(maxσδσσδσ≤−=≤−=dbFkN 52. 3][)(=−≤=−≤σ σδ δdbFkN 257. 1][ =∴ F??例 已知：例 已知：F = 80 kN, δ δ = = 10 mm, b = 80 mm, d = 16 mm, [τ τ ] = 100 MPa, [σ σ bs ] = 300 MPa, [σ σ ] = 160 MPa试：试：校核接头强度校核接头强度解：解：1. 接头受力分析接头受力分析当各铆钉的材料当各铆钉的材料与直径与直径均相同，均相同，且外力作用线且外力作用线在铆钉群剪切面上的投影，通过铆钉群剪切面形心在铆钉群剪切面上的投影，通过铆钉群剪切面形心时时，可近似认为，可近似认为各铆钉剪切面上的剪力相等各铆钉剪切面上的剪力相等4SFF = =][MPa 5 .99ππ422Sττττ<===<===dFdF2. 强度校核强度校核剪切强度：剪切强度：][MPa 125bsSbbsσδδσσδδσ<===<===dFdF][MPa 125)(1N11σδσσδσ<=−==<=−==dbFAF][MPa 125)2(432N22σδσσδσ<=−==<=−==dbFAF挤压强度：拉伸强度：挤压强度：拉伸强度：对于图示矩形截面杆，已知杆的许用应力[s]、材料比重r、总长L、截面厚度t以及单位杆长上的均匀载荷p。

现将杆件制成阶梯形试确定使杆件重量W为最小时杆件分段处坐标a，并给出相应的横截面高度h1与h2，已知h2=2h1注：略去过渡圆角产生的附加重量1）杆的各段均应满足强度条件由此可得出坐标 与两段截件，由此可得出坐标a与两段截面A1和A2（或h1和h2）的关系式2）将重量W写成坐标a的函数，然后取极小值ThanksThanks。