东南大学：《建筑力学 Architectural Mechanics》课程教学课件（英文讲稿）A09 Axial Loading Concept of Stress & Strain

Axial LoadingConcept of Stress & Strainmi@se.edl.cn

Axial Loading Concept of Stress & Strain mi@seu.edu.cn

Contents·Introduction（拉压变形简介）·DiagramofAxialForces（轴力图）·ConceptofStresses（应力的概念）·GeneralStressStateofaPoint（点的一般应力状态）·StressesActingonCrossSections（拉压杆横截面上的应力）·Saint-VenantsPrinciple（圣维南原理）·StressesActingonObliqueSections（拉压杆斜截面上的应力）·DeformationofAxiallyLoadedBars（拉压杆的变形）·ElasticConstantsofEngineeringMaterials（常见工程材料的弹性常数）·NonuniformTension/compression（非均匀拉压）·StrainEnergy（应变能）·StrainEnergyDensity（应变能密度）·MechanicalBehaviorofMaterials（材料的力学性能）·NominalStress-strainCurve（名义应力应变曲线）·StressandDeformationIndicesofLow-carbonSteel（低碳钢的应力和变形指标）2

• Introduction（拉压变形简介） • Diagram of Axial Forces（轴力图） • Concept of Stresses（应力的概念） • General Stress State of a Point（点的一般应力状态） • Stresses Acting on Cross Sections（拉压杆横截面上的应力） • Saint-Venant’s Principle（圣维南原理） • Stresses Acting on Oblique Sections（拉压杆斜截面上的应力） • Deformation of Axially Loaded Bars（拉压杆的变形） • Elastic Constants of Engineering Materials（常见工程材料的弹性常数） • Nonuniform Tension/compression（非均匀拉压） • Strain Energy（应变能） • Strain Energy Density（应变能密度） • Mechanical Behavior of Materials（材料的力学性能） • Nominal Stress-strain Curve（名义应力应变曲线） • Stress and Deformation Indices of Low-carbon Steel（低碳钢的应力和变形指标） Contents 2

Contents:Yield Stress,Ultimate Stress andPercentElongation ofEngineering Materials（常见工程材料的屈服应力、强度极限和断后伸长率）·Strain（Work)Hardening（冷作硬化）·Mechanical Behavior ofGeneral DuctileMaterials under Tension（塑性材料的拉伸力学性能）·MechanicalBehaviorofBrittleMaterialsunderTension（脆性材料的拉伸）·MechanicalBehaviorofLow-carbonSteelunderCompression（低碳钢的压缩）·MechanicalBehaviorofCastIronunderCompression（铸铁的压缩力学性能）·MechanicalBehaviorofCreet（混凝土的力学性能）·MechanicalBehaviorofWood（木材的力学性能）·MechanicalBehaviorofCompositeMaterials（复合材料的力学性能）·MechanicalBehaviorofViscoelasticMaterials（粘弹性材料的力学性能）·StrengthCondition（强度条件）·FailureofBrittlevs.DuctileBarsunderTension（脆性和塑性杆件的拉伸失效）·StressConcentration（拉压杆中的应力集中现象）3

• Yield Stress, Ultimate Stress and Percent Elongation of Engineering Materials （常见工程材料的屈服应力、强度极限和断后伸长率） • Strain (Work) Hardening（冷作硬化） • Mechanical Behavior of General Ductile Materials under Tension（塑性材料的 拉伸力学性能） • Mechanical Behavior of Brittle Materials under Tension（脆性材料的拉伸） • Mechanical Behavior of Low-carbon Steel under Compression（低碳钢的压缩） • Mechanical Behavior of Cast Iron under Compression（铸铁的压缩力学性能） • Mechanical Behavior of Creet（混凝土的力学性能） • Mechanical Behavior of Wood（木材的力学性能） • Mechanical Behavior of Composite Materials（复合材料的力学性能） • Mechanical Behavior of Viscoelastic Materials（粘弹性材料的力学性能） • Strength Condition（强度条件） • Failure of Brittle vs. Ductile Bars under Tension（脆性和塑性杆件的拉伸失效） • Stress Concentration（拉压杆中的应力集中现象） Contents 3

IntroductionBrake cable,L=460mmHand brakepivotAh=660mmh.=490mmW37.5mmP(Resultantofdistributedpressure)4550mm-Uniformhand-100mm-brakepressure4

Introduction 4

IntroductionERH: Load: equal and opposite forces along bar axis: Deformation: extension (contraction) along bar axis andcontraction (extension) transversely5

F F F F • Load: equal and opposite forces along bar axis • Deformation: extension (contraction) along bar axis and contraction (extension) transversely Introduction 5

Internal Forces Illustrated by Method of SectionsInternal Force. The change of interaction force among various parts of asolid body, introduced by external loadingThe Method of SectionFFnVnmII6

Internal Force • The change of interaction force among various parts of a solid body, introduced by external loading. The Method of Section F1 F2 m I F1 F2 m I R M o F1 Fn F2 m I II m Internal Forces Illustrated by Method of Sections 6

Sign Convention of Axial Forces: Coordinate: a right-handed systemMO: Centroid of the cross-sectionAx: Cross-section normal (Bar axis)F. Forces: Fn, Fsy, Fs?: Moments: T, My, M,: Axial Force (Fn)Fo Along bar axis+o Extends or contracts the bar along bar axis一F Positive for tension; Negative for compression(-)o Assuming positive for all unknown axial forces7

F1 F2 m I o x y z FN T M FSy y FSz Mz • Coordinate: a right-handed system O: Centroid of the cross-section x: Cross-section normal (Bar axis) • Forces: FN , FSy , FSz • Moments: T, My , Mz • Axial Force (FN ) o Along bar axis o Extends or contracts the bar along bar axis o Positive for tension; Negative for compression o Assuming positive for all unknown axial forces FN (+) FN (-) Sign Convention of Axial Forces 7

Procedure of Method of Sections: Sectioning the member: Taking either portion: Substituting the other portion with internal forces·EquilibratingmmFFmmFFFFNFn = FFn = -F8

• Sectioning the member • Taking either portion • Substituting the other portion with internal forces • Equilibrating Procedure of Method of Sections F F m m F FN FN = -F F F m m F FN FN = F 8

Diagram of Axial Forces: Abscissa: position of cross sections: Ordinate: axial force: Positive for tension; negative for compressionmm+X9

• Abscissa: position of cross sections • Ordinate: axial force • Positive for tension; negative for compression. F F m m FN x Diagram of Axial Forces 9

Sample Problem. Plot the diagram of axial forceB 2 kN/m CD 1 kN3kN A. Solution2 m—/-2 m—/2 m—1. Internal force in AB、CD3 kNFnFNAB = 3 kNxFNcD = -1 kN2m- x2. Internal force in BCFn(x)=3-2x(0<x<2)33. Diagram of axial forceF (kN)C4. Maximum internal force[FN,max | = 3 kN110

• Plot the diagram of axial force 3 kN A B 2 kN/m C D 1 kN 2 m 2 m 2 m 1. Internal force in AB、CD N N 3 kN 1 kN AB CD F F    2. Internal force in BC 3 kN 2 m x FN x FN,max  3 kN 3. Diagram of axial force FN kN 3 1 4. Maximum internal force Sample Problem F x x x N       3 2 0 2   • Solution 10