东南大学：《建筑力学 Architectural Mechanics》课程教学课件（英文讲稿）A13 Bending Stress

Bending Stressmi@see.cn

Bending Stress mi@seu.edu.cn

Contents·PureBendingvs.NonuniformBending（纯弯曲与横力弯曲）·AssumptionsforPureBending（纯弯曲基本假设）·NeutralSurface&NeutralAxes（中性层与中性轴）·Kinematics（几何关系）Hooke'sLaw（物理关系）·StaticEquivalency（静力等效关系）·PureBendingNormalStressFormula（纯弯曲正应力公式）·NormalStressStrengthCondition（正应力强度条件）·StressConcentrations（应力集中）·Bendingof a Composite Beam（复合梁弯曲）·Bending of a Curved Beam（曲梁弯曲）2

• Pure Bending vs. Nonuniform Bending（纯弯曲与横力弯曲） • Assumptions for Pure Bending（纯弯曲基本假设） • Neutral Surface & Neutral Axes（中性层与中性轴） • Kinematics（几何关系） • Hooke’s Law（物理关系） • Static Equivalency（静力等效关系） • Pure Bending Normal Stress Formula（纯弯曲正应力公式） • Normal Stress Strength Condition（正应力强度条件） • Stress Concentrations（应力集中） • Bending of a Composite Beam（复合梁弯曲） • Bending of a Curved Beam（曲梁弯曲） Contents 2

Contents·Shearing Stresses in a RectangularBeam（矩形梁切应力）·EffectofShearingStress/Strain（切应力和切应变效应）·ShearingStresses inaWide-flangeBeam（宽翼缘梁切应力）·ShearFlowinaThin-walledBeam（薄壁梁剪力流）·ShearingStressesinaCircularBeam（圆截面梁切应力）·Shearing Stresses in an Equilateral Triangular Beam（等边三角梁切应力）·ShearingStress StrengthCondition（切应力强度条件）·Rational DesignofBeams（梁的合理设计）·Nonprismatic and Constant-strength Beams（非等直梁和等强度梁）: Unsymmetric Loading of Thin-Walled Members and Shear Center（薄壁梁的非对称弯曲与剪力中心）3

• Shearing Stresses in a Rectangular Beam（矩形梁切应力） • Effect of Shearing Stress/Strain（切应力和切应变效应） • Shearing Stresses in a Wide-flange Beam（宽翼缘梁切应力） • Shear Flow in a Thin-walled Beam（薄壁梁剪力流） • Shearing Stresses in a Circular Beam（圆截面梁切应力） • Shearing Stresses in an Equilateral Triangular Beam（等边三角梁 切应力） • Shearing Stress Strength Condition（切应力强度条件） • Rational Design of Beams（梁的合理设计） • Nonprismatic and Constant-strength Beams（非等直梁和等强度梁） • Unsymmetric Loading of Thin-Walled Members and Shear Center （薄壁梁的非对称弯曲与剪力中心） Contents 3

Pure Bending vs. Nonuniform BendingP/PB? Pure bending (CD)DC9LPaaFs = O, M = constFsPNonuniform bending (AC & DB)F≠0，M≠0PMPa4

S F M   0 const ， • Pure bending (CD) S F M ≠ ， ≠ 0 0 • Nonuniform bending (AC & DB) Pure Bending vs. Nonuniform Bending 4 P B P P C D A P a a l P P FS Pa M

Deformation Characteristics: Before:MM·After:: Straight longitudinal lines turns into curves.: Longitudinal lines get shortened under compression and lengthenedundertension: Cross-section lines remain straight and perpendicular to longitudinalcurves.5

M M • Before: • After: Deformation Characteristics • Straight longitudinal lines turns into curves. • Longitudinal lines get shortened under compression and lengthened under tension. • Cross-section lines remain straight and perpendicular to longitudinal curves. 5

Assumptions for Pure Bending.Plane assumption: under pure bending, cross-sections of beams remain planar andperpendicular to beam axis and only rotate a smallangle.: Assumption of uniaxial stress state: individuallongitudinal layers are under uniaxialtension/compression along beam axis, withoutstresses acting in between.6

• Assumption of uniaxial stress state: individual longitudinal layers are under uniaxial tension/compression along beam axis, without stresses acting in between. • Plane assumption: under pure bending, cross￾sections of beams remain planar and perpendicular to beam axis and only rotate a small angle. Assumptions for Pure Bending 6

Neutral Surface & Neutral AxesNeutral Axis. Before:Neutral AxisAfter:NeutralSurface: Neutral Surface the longitudinal layer under neither tension norcompression: Neutral Axes: intersecting lines of the neutral surface & crosssections.7

• Neutral Axes: intersecting lines of the neutral surface & cross sections. Neutral Surface & Neutral Axes • Neutral Surface the longitudinal layer under neither tension nor compression. Neutral Axis Neutral Surface z Neutral Surface Neutral Axis z • Before: • After: 7

KinematicsNeutralaxisMRByr(p+y)do-pde12pdep12NeutralOSurfaceThe y-coordinate is02010201measured from the proposedb-aa61neutral axis228

a b 1 2 1 2 o1 o2 y   ( ) y d d y d y             Neutral Surface 1 2 2 ρ dθ o1  o2  a b Kinematics 8 • The y-coordinate is measured from the proposed neutral axis. y z

Hooke's Lawα(y)= Ec(y)= EyPNeutral surface: Normal stress acting on a longitudinal layer is linearly proportionalto its distance from the neutral surface, positive for layers undertension / negative for layers under compression.: Remark: the above equation can only be used for qualitative analysisof stresses in bending beams since it is difficult to measure thecurvature of radius (p) of individual longitudinal layers.9

• Normal stress acting on a longitudinal layer is linearly proportional to its distance from the neutral surface, positive for layers under tension / negative for layers under compression. • Remark: the above equation can only be used for qualitative analysis of stresses in bending beams since it is difficult to measure the curvature of radius (ρ) of individual longitudinal layers. Hooke’s Law 9     y   y E y E   

Static EquivalencyEE0= F =(odA=vdAA1p福6X: Neutral axis passes through the centroid: y = O.M: for an arbitrarily defined y-coordinate:1J=ZAJ./AW0= M.=/z·od1MdAM一EIC1pEM.yU0:C21pEI, : flexural rigidityVy?dA : second moment of cross-section w.r.t. z=10

y z 0 N A A E E F dA ydA Ay                A A y yzdA E M z dA  0  2 1 z Z z A A z z z E E M M y dA y dA I EI E M y y I                    • Neutral axis passes through the centroid: EIz : flexural rigidity 2 z A I y dA   : second moment of cross-section w.r.t. z. i i i y A y A  • for an arbitrarily defined y-coordinate: Static Equivalency 10 y  0