东南大学：《建筑力学 Architectural Mechanics》课程教学课件（英文讲稿）A11 Torsion

Torsionmi@ser.eor.cn

Torsion mi@seu.edu.cn

Contents·IntroductiontoTorsion（扭转简介）·ExamplesofTorsionShafts（扭转轴示例）·SignConventionofTorque（扭矩符号规则）·TorqueDiagram（扭矩图）·Power&Torque（功率与扭矩）·InternalTorque&Stress-Static Indeterminacy（内力扭矩和应力一超静定概念的引入）·GeneralRelations Involved inDeformableSolids（分析可变形固体的几大基本关系）·Kinematics（几何关系）·Hooke'sLawforShearingDeformation（剪切变形物理关系）·StaticEquivalency（静力等效关系）·TorsionalStress&AngleofTwist（圆轴扭转的应力与扭转角）2

• Introduction to Torsion（扭转简介） • Examples of Torsion Shafts（扭转轴示例） • Sign Convention of Torque（扭矩符号规则） • Torque Diagram（扭矩图） • Power & Torque（功率与扭矩） • Internal Torque & Stress - Static Indeterminacy（内力扭矩和应力 -超静定概念的引入） • General Relations Involved in Deformable Solids（分析可变形固 体的几大基本关系） • Kinematics（几何关系） • Hooke’s Law for Shearing Deformation（剪切变形物理关系） • Static Equivalency（静力等效关系） • Torsional Stress & Angle of Twist（圆轴扭转的应力与扭转角） Contents 2

Contents·NonuniformTorsionofCircularShafts（圆轴的非均匀扭转）·Strength&StiffnessAnalysis（强度和刚度条件）·StrainEnergyand itsDensity（扭转应变能与应变能密度）·PolarMoments ofInertia&SectionModulus（圆截面的极惯性矩与扭转截面系数）·Theorem of Conjugate Shearing Stress（切应力互等定理）·StressesonObliqueCrossSections（扭转轴斜截面上的应力）·FailureModesofTorsionalShafts（扭转轴的失效模式）·StressConcentrations（扭转轴的应力集中）·TorsionofNoncircularMembers（非圆截面杆的扭转）·MembraneAnalogy（薄膜比拟）·TorsionofThin-walledOpenShafts（薄壁开口截面杆的扭转）·TorsionofThin-walledHollowShafts（薄壁空心截面杆的扭转）3

• Nonuniform Torsion of Circular Shafts（圆轴的非均匀扭转） • Strength & Stiffness Analysis（强度和刚度条件） • Strain Energy and its Density（扭转应变能与应变能密度） • Polar Moments of Inertia & Section Modulus（圆截面的极惯性矩 与扭转截面系数） • Theorem of Conjugate Shearing Stress（切应力互等定理） • Stresses on Oblique Cross Sections（扭转轴斜截面上的应力） • Failure Modes of Torsional Shafts（扭转轴的失效模式） • Stress Concentrations（扭转轴的应力集中） • Torsion of Noncircular Members（非圆截面杆的扭转） • Membrane Analogy（薄膜比拟） • Torsion of Thin-walled Open Shafts（薄壁开口截面杆的扭转） • Torsion of Thin-walled Hollow Shafts（薄壁空心截面杆的扭转） Contents 3

Introduction to TorsionGenerator. Turbine exerts torque T on theshaftRotationTurbine? Shaft transmits the torque tothe generator(a). Generator creates an equal andopposite torque T? Cross section remains planarInterestedin stresses andstrains of circular shafts(b)subjected to twisting couplesor torques4

• Interested in stresses and strains of circular shafts subjected to twisting couples or torques • Generator creates an equal and opposite torque T • Cross section remains planar • Turbine exerts torque T on the shaft Introduction to Torsion • Shaft transmits the torque to the generator 4

Introduction to TorsionTorsional Deformationofa Circular shaft.Radial lines remain straightLongitudinal linesduring deformationremainstraightbut spiral.Noticethedeformation oftherectangularelementwhenitissubjectedtoatorque

Introduction to Torsion 5

Examples of Torsion ShaftsPower generation shaftAutomotive power train shaftShip drive shaftTire shift driveComplex crank shaftScrewdriver6

6 Ship drive shaft Complex crank shaft Automotive power train shaft Screwdriver Power generation shaft Tire shift drive Examples of Torsion Shafts

Sign Convention of TorqueMeMeMe.MeMe: Sign (Positive) convention (right-hand rule): thumb - cross-section normal; rest fingers - torque

• Sign (Positive) convention (right-hand rule): thumb – cross￾section normal; rest fingers – torque Sign Convention of Torque 7

Torque Diagram: Abscissa: cross-section position: Ordinate: torqueMeiMe2Me3Me4-Me4xMe1Mei+Me28

• Abscissa: cross-section position • Ordinate: torque Torque Diagram 8

Power & TorqueMotor. Transformation from electricpower to mechanical powerC motor power: P (kW); generated torqueM. (N-m):: n (rpm, revolutions per minute):1000 ×60 PPM。×2元n = P×1000×60= M。95492元nn: n (rps, revolutions per second):P1000 P159M,×2元n=P×1000=M2元nn· @ (rad/s, radian per second))PM。× =P×1000= M.=100009

• Transformation from electric power to mechanical power Power & Torque 9 1000 60 2π 1000 60 9549 2π e e P P M n P M n n         • motor power: P (kW); generated torque: Me (N·m): • n (rpm, revolutions per minute): 1000 2π 1000 159 2π e e P P M n P M n n       e e 1000 1000 P M P M        • n (rps, revolutions per second): • ω (rad/s, radian per second):

Internal Torque & Stress - Static Indeterminacy. Net of the internal shearing stresses is aninternal torque, equal and opposite to theapplied torque,T=[pdF = p(t dA): Although the net torque due to the shearingstresses is known, the distribution of thedFstresses is not.: Distribution of shearing stresses is(a)statically indeterminate - must considerBshaft deformations.: Unlike the normal stress due to axial loadsthe distribution of shearing stresses due to(b)torsional loads can not be assumed uniform10

    T   dF    dA • Net of the internal shearing stresses is an internal torque, equal and opposite to the applied torque, • Although the net torque due to the shearing stresses is known, the distribution of the stresses is not. • Unlike the normal stress due to axial loads, the distribution of shearing stresses due to torsional loads can not be assumed uniform. • Distribution of shearing stresses is statically indeterminate – must consider shaft deformations. Internal Torque & Stress - Static Indeterminacy 10