弯曲应力纯弯曲

1、 材料力学Professor Shibin WANG （）7.1 Examples under Bending Loading:Ch.7 Stresses in Beams受弯杆件的简化 - 悬臂梁Ch.7 Stresses in Beams7.1 Examples under Bending Loading:受弯杆件的简化 - 简支梁Ch.7 Stresses in Beams7.1 Examples under Bending Loading:受弯杆件的简化 - 外伸梁Ch.7 Stresses in Beams7.1 Examples under Bending Loading:Ch.7

2、 Stresses in BeamsAFMModulesDimension 3100 AFM The worlds best selling SPM silicon-Si(111)2nm?7.1 Examples under Bending Loading:Ch.7 Stresses in BeamsScanning Probe Microscopy( SPM )Components?7.1 Examples under Bending Loading:Ch.7 Stresses in BeamsCantileverSubstrateProbesThe properties and dimen

3、sions of the cantilever play an important role in determining the sensitivity and resolution of the AFM. Cantilever - AFM Probe Tip7.1 Examples under Bending Loading:Ch.7 Stresses in BeamsParis 1889 H=320 mE =15000 s G =70000 KN 70km sightLa Tour Eiffer7.1 Examples under Bending Loading:Pure Bending

4、: 纯弯曲Ch.7 Stresses in Beams7.2 Loading TypesOther Loading TypesEccentric Loading: 偏心加载Transverse Loading: 横向力作用Ch.7 Stresses in BeamsOther Loading TypesCh.7 Stresses in Beams7.3 Normal Stresses in BeamsPCh.7 Stresses in BeamsdxxsxMMsxdx7.3 Normal Stresses in BeamsCh.7 Stresses in Beams应力分布应力公式变 形应变分

5、布平面假定物性关系静力方程7.3.1 The Engineering Beam TheoryCh.7 Stresses in BeamsBending Deformations平面假设？MM7.3.1 The Engineering Beam TheoryxyMMABCDCompressionTensionNo StressNANeutral Axis中性轴zyyyABCDyRdqdxsx=0 on the Neutral Axis. In general we must find the position of the Neutral Axis.Ch.7 Stresses in BeamsM

6、M7.3.1 The Engineering Beam TheoryxyMMABCDCompressionTensionNo StressNANeutral Axis中性轴yABCDyRdqdxCh.7 Stresses in BeamsAssumptionsBeam material is elasticPlane surfaces remain planeand only1Geometry of Deformation:Hookes Law:andABCDyRdqMM7.3.1 The Engineering Beam Theory1xyyNANeutral Axis中性轴0+ve-veL

7、inear Distribution of sx(Eqn )1dxNote:E is a Material Propertyis Curvature 曲率xdxyMMsx7.3.1 The Engineering Beam Theory7.3.1 The Engineering Beam TheoryDeformation in a Transverse Cross SectionRRR=R/zyxEquilibrium:xzyydAsxMArea, ALetButFirst Moment of Area面矩（静矩）Then y is measured from the centroidal

8、axis of the beam cross-section.“Neutral Axis” coincides with the XZ plane through the centroid.yyNANeutral AxisCentroid2Equilibrium:as1Let=The 2nd Moment of Area about Z-axis 惯性矩 THE SIMPLE BEAM THEORY:12&xzyydAsxMArea, A- Applied Bending Moment - Property of Cross-Sectional Area - Stress due to M -

9、 Distance from the Neutral Axis - Youngs Modulus of Beam Material - Radius of Curvature due to M - N.m - m4 - N/m2 or Pa- m - N/m2 or Pa- m zyyyNANeutral Axisxo7.3.1 The Engineering Beam Theoryzyo7.3.2 Properties of AreaydAsxMxzyoy is measured from the Centroidal or Neutral Axis, z. Iz is the 2nd Mo

10、ment of Area about the Centroidal or Neutral Axis, z.Position of Centroidal or Neutral Axis:yCentroidal AxiszoyArea, Ayn(Definition)dAdAEquilibrium:as1The 2nd Moment of Area about y-Z-axis 惯性积 THE SIMPLE BEAM THEORY:xzyydAsxMArea, Az y 轴为对称轴xzyyMArea, ASummaryThe Engineering Beam Theory determines t

11、he axial stress distribution generated across the section of a beam. It is applicable to long, slender load carrying devices. Calculating properties of beam cross sections is a necessary part of the analysis. Neutral Axis Position, y 2nd Moments of Area, Iy, Iz, Ip 7.3.2 The Engineering Beam TheoryS

12、ummaryIt is applicable to long, slender load carrying devices. 7.3.2 The Engineering Beam Theory横力弯曲横截面翘曲纵向截面上出现挤压应力平面假设不成立l/h 比较大时，误差很小， 能满足工程需要7.4.1 Strength and DeformationCh.7 Stresses in BeamsWZ : 抗弯截面模量 弯曲时的强度条件7.4.2 Sample ProblemsCh.7 Stresses in BeamsAPCBl /2l /2简易天车，P=68 kN， l = 9.5 m, 40c

13、 型工字钢的自重为q，=140MPa, 校核安全性Pl/4解：作弯矩图（叠加法）q=801N/m查表查表安全 ？7.4.2 Sample ProblemsCh.7 Stresses in Beams7.4.2 Sample ProblemsCh.7 Stresses in Beams静矩和形心厚度 t 极小的薄片如图 在图形所在平面内建立坐标系yzo重心y z dA 比重 G 薄片重量Cy cz cdA 微面积薄片图形的形心yzo形心y z dAC形心坐标定义图形对 z 轴的静矩图形对 y 轴的静矩mm 3讨论 :坐标轴通过形心时1) y 轴通过形心时2) z 轴通过形心时坐标轴通过图形的形心

14、 图形对该轴的 静矩 等于零ynExample:zoCentroidal Axis200102012512060(Dimensions in mm)Example:zyo2nd Moment of Area:Definition:zyydydAoyzThe Parallel Axis Theorem:zyoDefinition:Example:ydynnzoyExample:(Dimensions in mm)zyo200102012089.630.489.6202030.420010123 What is Iz? What is maximum sx?35.4Example:(Dimensions in mm)zyo20010201