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第七章应力和应变分析强度理论Chapter Seven Stress and Strain Analysis Strength Theories§7–1 应力状态概述§7–1 Concepts of the State of Stress1．一点的应力状态：过一点有无数的截面，这一点的各个截面上应力情况的集合，称为这点的应力状态。
The state of stress at a point: There are countless sections through a point. The gathering of stresses in all sections is called the state of stress at this point.2．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。
Element: Delegate of a point in the member. It is an infinitesimal geometric body enveloping the studied point. In common use it is a correctitude cubic body.* 3．主单元体：各侧面上剪应力均为零的单元体。
Principal element :The element in which the shearing stresses in side planes are all zero.* 4．主平面：剪应力为零的截面。
Principal Planes:The planes on which the shearing stresses are zero.* 5．主应力：主平面上的正应力。
Principal stresses: Normal stresses acting on the principle planes.6．主应力排列规定；按代数值大小Convention of the order for three principal stresses: In magnitude of the algebraic value.7．三向应力状态：三个主应力都不为零的应力状态。
材料力学中英对照词汇A安全因数safety factorB半桥接法half bridge闭口薄壁杆thin-walled tubes比例极限proportional limit边界条件boundary conditions变截面梁beam of variable cross section 变形deformation变形协调方程compatibility equation 标距gage length泊松比Poisson’s ratio补偿块compensating blockC材料力学mechanics of materials冲击荷载impact load初应力，预应力initial stress纯剪切pure shear纯弯曲pure bending脆性材料brittle materialsD大柔度杆long columns单位荷载unit load单位力偶unit couple单位荷载法unit-load method单向应力，单向受力uniaxial stress等强度梁beam of constant strength低周疲劳low-cycle fatigue电桥平衡bridge balancing电阻应变计resistance strain gage电阻应变仪resistance strain indicator 叠加法superposition method叠加原理superposition principle 动荷载dynamic load断面收缩率percentage reduction in area多余约束redundant restraintE二向应力状态state of biaxial stressF分布力distributed force复杂应力状态state of triaxial stress复合材料composite materialG杆，杆件bar刚度stiffness刚架，构架frame刚结点rigid joint高周疲劳high-cycle fatigue各向同性材料isotropical material功的互等定理reciprocal-work theorem工作应变计active strain gage工作应力working stress构件structural member惯性半径radius of gyration of an area惯性积product of inertia惯性矩，截面二次轴距moment of inertia广义胡克定律generalized Hook’s lawH横向变形lateral deformation胡克定律Hook’s law滑移线slip-linesJ基本系统primary system畸变能理论distortion energy theory畸变能密度distortional strain energydensity极惯性矩，截面二次极矩polar moment of inertia极限应力ultimate stress极限荷载limit load挤压应力bearing stress剪力shear force剪力方程equation of shear force剪力图shear force diagram剪流shear flow剪切胡克定律Hook’s law for shear剪切shear交变应力，循环应力cyclic stress截面法method of sections截面几何性质geometrical properties ofan area截面核心core of section静不定次，超静定次数degree of astatically indeterminate problem静不定问题，超静定问题staticallyindeterminate problem静定问题statically determinate problem静荷载static load静矩，一次矩static moment颈缩neckingK开口薄壁杆bar of thin-walled open crosssection抗拉强度ultimate stress in tension抗扭截面系数section modulus in torsion抗扭强度ultimate stress in torsion抗弯截面系数section modulus inbendingL拉压刚度axial rigidity拉压杆，轴向承载杆axially loaded bar理想弹塑性假设elastic-perfectly plasticassumption力法force method力学性能mechanical properties连续梁continuous beam连续条件continuity condition梁beams临界应力critical stress临界荷载critical loadM迈因纳定律Miner’s law名义屈服强度offset yielding stress莫尔强度理论Mohr theory of failure敏感栅sensitive gridN挠度deflection挠曲轴deflection curve挠曲轴方程equation of deflection curve挠曲轴近似微分方程approximatelydifferential equation of the deflectioncurve内力internal forces扭力矩twisting moment扭矩torsional moment扭矩图torque diagram扭转torsion扭转极限应力ultimate stress in torsion扭转角angel of twist扭转屈服强度yielding stress in torsion扭转刚度torsional rigidityO欧拉公式Euler’s formulaP疲劳极限，条件疲劳极限endurancelimit疲劳破坏fatigue rupture疲劳寿命fatigue life偏心拉伸eccentric tension偏心压缩eccentric compression平均应力average stress平面弯曲plane bending平面应力状态state of plane stress平行移轴定理parallel axis theorem平面假设plane cross-section assumptionQ强度strength强度理论theory of strength强度条件strength condition切变模量shear modulus切应变shear strain切应力shear stress切应力互等定理theorem of conjugate shearing stress屈服yield屈服强度yield strength全桥接线法full bridgeR热应力thermal stressS三向应力状态state of triaxial stress三轴直角应变花three-element rectangular rosette三轴等角应变花three-element delta rosette失稳buckling伸长率elongation圣维南原理Saint-Venant’s principle实验应力分析experimental stress analysis塑性变形，残余变形plastic deformationductile materials塑性材料，延性材料塑性铰plastic hingeT弹簧常量spring constant弹性变形elastic deformation弹性模量modulus of elasticity体积力body force体积改变能密度density of energy ofvolume change体应变volume strainW弯矩bending moment弯矩方程equation of bending moment弯矩图bending moment diagram弯曲bending弯曲刚度flexural rigidity弯曲正应力normal stress in bending弯曲切应力shear stress in bending弯曲中心shear center位移法displacement method位移互等定理reciprocal-displacementtheorem稳定条件stability condition稳定性stability稳定安全因数safety factor for stabilityX细长比，柔度slenderness ratio线性弹性体linear elastic body约束扭转constraint torsion相当长度，有效长度equivalent length相当应力equivalent stress小柔度杆short columns形心轴centroidal axis形状系数shape factor许用应力allowable stress许用应力法allowable stress method许用荷载allowable load许用荷载法allowable load methodY应变花strain rosette应变计strain gage应变能strain energy应变能密度strain energy density应力stress应力速率stress ratio应力比stress ratio应力幅stress amplitude应力状态state of stress应力集中stress concentration应力集中因数stress concentration factor应力－寿命曲线，S-N曲线stress-cyclecurve应力－应变图stress-strain diagram应力圆，莫尔圆Mohr’s circle for stressesZ正应变normal strain正应力normal stress中面middle plane中柔度杆intermediate columns中性层neutral surface中性轴neutral axis轴shaft轴力axial force轴力图axial force diagram轴向变形axial deformation轴向拉伸axial tension轴向压缩axial compression主平面principal planes主应力principal stress主应力迹线principal stress trajectory主轴principal axis主惯性矩principal moment of inertia主形心惯性矩principal centroidalmoments of inertia主形心轴principal centroidal axis转角angel of rotation转轴公式transformation equation自由扭转free torsion组合变形combined deformation组合截面composite area最大切应力理论maximum shear stresstheory最大拉应变理论maximum tensile straintheory最大拉应力理论maximum tensile stresstheory最大应力maximum stress最小应力minimum stress。
Asafety factorBhalf bridgethin-walled tubes proportional limit boundary conditionsbeam of variable cross section deformationcompatibility equationgage lengthPoisson’s ratio compensating blockCmechanics of materials impact loadinitial stresspure shearpure bendingbrittle materialsDlong columnsunit loadunit coupleunit-load methoduniaxial stressbeam of constant strength low-cycle fatiguebridge balancing resistance strain gage resistance strain indicator superposition method superposition principle dynamic load percentage reduction in arearedundant restraintEstate of biaxial stressFdistributed forcestate of triaxial stresscomposite materialGbarstiffnessframerigid jointhigh-cycle fatigueisotropical materialreciprocal-work theoremactive strain gageworking stressstructural memberradius of gyration of an areaproduct of inertiamoment of inertiageneralized Hook’s lawHlateral deformationHook’s lawslip-linesJprimary systemdistortion energy theorydistortional strain energy densitypolar moment of inertiaultimate stresslimit loadbearing stressshear forceequation of shear forceshear force diagramshear flowHook’s law for shearshearcyclic stressmethod of sectionsgeometrical properties of an areacore of sectiondegree of a statically indeterminateproblemstatically indeterminate problemstatically determinate problemstatic loadstatic momentneckingKbar of thin-walled open cross sectionultimate stress in tensionsection modulus in torsionultimate stress in torsionsection modulus in bendingLaxial rigidityaxially loaded barelastic-perfectly plastic assumptionforce methodmechanical propertiescontinuous beamcontinuity conditionbeamscritical stress临界荷载critical loadMMiner’s lawoffset yielding stressMohr theory of failuresensitive gridNdeflectiondeflection curveequation of deflection curveapproximately differential equation of thedeflection curveinternal forcestwisting momenttorsional momenttorque diagramtorsionultimate stress in torsionangel of twistyielding stress in torsiontorsional rigidityOEuler’s formulaPendurance limitfatigue rupturefatigue lifeeccentric tensioneccentric compressionaverage stressplane bendingstate of plane stressparallel axis theoremplane cross-section assumptionQstrengththeory of strengthstrength conditionshear modulusshear strainshear stresstheorem of conjugate shearing stress yieldyield strengthfull bridgeRthermal stressSstate of triaxial stressthree-element rectangular rosette three-element delta rosette bucklingelongationSaint-Venant’s principle experimental stress analysis plastic deformationductile materialsplastic hingeTspring constantelastic deformationmodulus of elasticitybody forcedensity of energy of volume change volume strainWbending momentequation of bending moment bending moment diagrambendingflexural rigiditynormal stress in bendingshear stress in bendingshear centerdisplacement methodreciprocal-displacement theoremstability conditionstabilitysafety factor for stabilityXslenderness ratiolinear elastic bodyconstraint torsionequivalent lengthequivalent stressshort columnscentroidal axisshape factorallowable stressallowable stress methodallowable loadallowable load methodYstrain rosettestrain gagestrain energystrain energy densitystressstress ratiostress ratiostress amplitudestate of stressstress concentrationstress concentration factorstress-cycle curvestress-strain diagramMohr’s circle for stressesZnormal strainnormal stressmiddle planeintermediate columnsneutral surfaceneutral axisshaftaxial forceaxial force diagramaxial deformationaxial tensionaxial compressionprincipal planesprincipal stressprincipal stress trajectoryprincipal axisprincipal moment of inertiaprincipal centroidal moments of inertiaprincipal centroidal axisangel of rotationtransformation equationfree torsioncombined deformationcomposite areamaximum shear stress theorymaximum tensile strain theorymaximum tensile stress theorymaximum stressminimum stress。
Chapter 4 Internal Forces in BendingMECHANICS OF MATERIALS 材料力学（双语）Content§4.1 Concept of symmetrical bending and calculation sketch of the beam §4.2 The shearing force and bending moment of the beam §4.3 The shearing-force and bending-moment equations · the shearing-force and bending-moment diagrams §4.4 Relations among the shearing force、the bending moment and the density of the distributed load and their applications §4.5 Plot the bending-moment diagram by the theorem of superposition §4.6 The internal-force diagrams of the planar rigid frames and curved rods2目录§4.1 对称弯曲的概念及梁的计算简图 §4.2 梁的剪力和弯矩 §4.3 剪力方程和弯矩方程 ·剪力图和弯矩图 §4.4 剪力、弯矩与分布荷载集度间的关系及应用 §4.5 按叠加原理作弯矩图 §4.6 平面刚架和曲杆的内力图34.1 Concept of symmetrical bending and calculation sketch of the beam1. Concepts of Bending 1). BENDING(弯曲): The action of the external force or the external couple vector perpendicular to the axis of the rod makes the axis of the rod change into curve from original straight lines, this deformation is called bending. 2). BEAM（梁): The member of whose deformation is mainly bending is generally called beam.43). Practical example in engineering about bending564). Symmetric bending(平面弯曲) After deformation the curved axis of the beam is still in the same plane with the external forces. P1 q P2MThe plane of symmetry7Unsymmetrical bending： If a beam does not possess any plane of symmetry, or the external forces do not act in a plane of symmetry of the beam with symmetric planes, this kind of bending is called unsymmetrical bending.82. Calculation sketch of the beamIn general supports and external forces of the beam are very complex. We should do some necessary simplification for them for our convenient calculation and obtain the calculation sketch.1). Simplification(简化) of the beamsIn general case we take the place of the beam by its axis.2). Simplification of the loads (荷载)The loads (including the reaction) acting on the beam may be reduced into three types：concentrated force、concentrated force couple and distributed force.3). Simplification of the supports (支座条件)9①Fixed hinged support固定铰支座2 constraints，1 degree of freedom. Such as the fixed hinged support under bridges.②Movable hinged support可动铰支座1 constraint，2 degree of freedom. Such as the movable hinged support under the bridge.10θA BDB R RA A AB B B0.50.50.5P0.5PP0.50.50.5P0.5PP 0。
