材料力学(双语)Bending弯曲

REVIEW REVIEWBending shear stressQS τ = bI* z zQ zτy A*τ max = kτ m = 1.5Q AStrength condition of shear stressτ maxQ = k ≤ [τ ] APart one: Basic DeformationsChapter 6. Bending DeformationContaining Sections: Containing Sections:6.1 Introduction 6.1 Introduction 6.2 Equation of Elastic Curve 6.2 Equation of Elastic Curve 6.3 Direct Integration Method 6.3 Direct Integration Method 6.4 Method of Superposition 6.4 Method of Superposition 6.5* Moment-Area Method 6.5* Moment-Area Method 6.6 Stiffness Condition of Beams and Ways to Improve the Stiffness of 6.6 Stiffness Condition of Beams and Ways to Improve the Stiffness of Beams Beams 6.7 Statically Indeterminate Beams 6.7 Statically Indeterminate BeamsChapter 6. Bending Deformation6.1 IntroductionTensional deformation Torsional deformationNL ∆l = EAM nL ϕ= GI pϕxBending deformationdx vDeflection Deflection 截面形心的竖向位移 挠度 vv 挠度θElastic CurveSlope Slope 转角 θ 转角 θ截面绕中性轴的转角Chapter 6. Bending Deformation6.2 Equation of Elastic CurveWhen deduce the bending normal stress formula, we get : Consider the small deformation : From mathematical relation :1 M (x) = ρ (x) EI z1 ρ (x)Curvature 曲率 Flexural rigidity 抗弯刚度EI zdv θ ≈ tan θ = = v' ( x) dx1ρ=v" ≈ v" 2 3 2 (1 + v ′ )Deduce further, we can get the approximate differential equation of elastic curve (挠曲线近似微分方程):M (x) v ′′ ( x ) = θ ′ = E IZ公式的适用条件： 1) 材料服从虎克定律； 2) 小变形，忽略剪力对挠度的影响。

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。

材料力学Mechanics of Materials构件Structure member强度strength刚度rigidity稳定性stability变形固体solid deformation body基本假设fundamental assumption各向同性Isotropy内力internal force外力external force二力杆two force bar刚体rigid body轴向拉伸Axial tension轴向压缩Axial compression拉力tensile force内力Internal force轴力axial force轴力图diagram of axial force集中载荷concentrate load应力stress平均应力Average stress全应力whole stress平面假设Hypothesis of plane section拉应力Tensile stress应力集中stress concentration剪切shearing正应力normal stress剪应力shearing Stress线应变linear strain剪应变angular strain剪力shearing force弹性应变能Elastic strain energy静定问题statically determinate problem 静不定问题statically indeterminate problem 平衡方程equilibrium equations比例极限Proportional limit弹性极限Elastic limit屈服极限Yielding limit塑性材料ductile material 强度极限Strength limit铸铁cast iron轴Shaft扭转Torsion扭转角The angle of twist扭矩Internal torque扭矩图Internal torque diagram剪应力互等定Theorem of complementary shearingstresses弯曲bending梁Beam固定铰支座Fixed hinged support可动铰支座Movable hinged support固定端Rigidly fixed end简支梁simply supported beam悬臂梁Cantilever beam外伸梁Overhanging beam静定梁Statically determinate beam超静定梁Statically indeterminate beam弯曲内力Internal force in bending弯矩Bending moment内力方程Internal force equation剪力图shearing force diagram弯矩图bending moment diagram分布荷载the distributed load荷载集度the density of the distributed load平面刚架Planar rigid frame曲杆Curved rod纯弯曲Pure Bending中性轴Neutral axis曲率半径the radius of curvature挠度Deflection转角Angle of rotation挠曲线deflection curve位移边界条Boundary conditions of thedisplacement支点位移条件Displacement conditions at thesupports连续条件Continuity conditions 光滑条件Sliding conditions积分常数integral constant 一点的应力状态Stressed state at a point原始单元体Original element 主单元体Principal element主面principal plane 主应力principal stress三向应力状态Three Dimensional State of Stress二向应力状态Plane State of Stress 应力圆Stress Circle单向应力状态Unidirectional State of Stress 主应力迹线Principal stress trajectories应变能密度strain-energy density 强度理论Theories of strength最大拉应力（第一强度）理论：Theory of maximum tensile stress（first strength）最大伸长线应变（第二强度）理论：Theory of maximum tensile strain（second strength）最大剪应力（第三强度）理论Theory of maximum shearing stress(third strength)形状改变比能（第四强度）理论Theory of maximum torsional (shearing)strain energy (fourth strength）莫尔强度理论Mohr’s strength theory相当应力Equivalent stress组合变形Composite deformation 斜弯曲Skew bending中性轴方程Equation of the neutral axis 不稳定平衡instable equilibrium拉(压)弯组合变形Bending and tension or compression稳定平衡Stable equilibrium 压杆失稳Loss of stability of the column临界压力the critical pressure of the column 理想压杆Ideal compressive columns长度系数Length coefficient 约束系数constraint coefficient临界应力Critical stress 柔度Flexibility应变能Strain energy 卡氏定理Catigliano’S Theorem莫尔定理(单位力法) Mohr’S theorem(Method of unit force)。