材料力學(xué) Professor Shibin WANG （王世斌）,7.1 Examples under Bending Loading:,Ch.7 Stresses in Beams,受彎桿件的簡(jiǎn)化 -- 懸臂梁,Ch.7 Stresses in Beams,7.1 Examples under Bending Loading:,受彎桿件的簡(jiǎn)化 -- 簡(jiǎn)支梁,Ch.7 Stresses in Beams,7.1 Examples under Bending Loading:,受彎桿件的簡(jiǎn)化 -- 外伸梁,Ch.7 Stresses in Beams,7.1 Examples under Bending Loading:,Ch.7 Stresses in Beams,AFM,,,Modules,7.1 Examples under Bending Loading:,Ch.7 Stresses in Beams,Scanning Probe Microscopy( SPM )Components,,7.1 Examples under Bending Loading:,Ch.7 Stresses in Beams,,Cantilever,The properties and dimensions of the cantilever play an important role in determining the sensitivity and resolution of the AFM.,Cantilever -- AFM Probe,Tip,7.1 Examples under Bending Loading:,Ch.7 Stresses in Beams,,Paris 1889 H=320 m E =15000 s G =70000 KN 70km sight,7.1 Examples under Bending Loading:,Pure Bending: 純彎曲,Ch.7 Stresses in Beams,7.2 Loading Types,Other Loading Types,Eccentric Loading: 偏心加載,Transverse Loading: 橫向力作用,Ch.7 Stresses in Beams,Other Loading Types,Ch.7 Stresses in Beams,7.3 Normal Stresses in Beams,,Ch.7 Stresses in Beams,7.3 Normal Stresses in Beams,,Ch.7 Stresses in Beams,應(yīng)力分布,應(yīng)力公式,變 形,應(yīng)變分布,7.3.1 The Engineering Beam Theory,Ch.7 Stresses in Beams,Bending Deformations,平面假設(shè)？,,7.3.1 The Engineering Beam Theory,Compression,Tension,No Stress,,Ch.7 Stresses in Beams,,,,,,,,,,M,M,7.3.1 The Engineering Beam Theory,Compression,Tension,No Stress,y,,Ch.7 Stresses in Beams,Assumptions,,Geometry of Deformation:,,Hookes Law:,and,7.3.1 The Engineering Beam Theory,,1,7.3.1 The Engineering Beam Theory,7.3.1 The Engineering Beam Theory,Deformation in a Transverse Cross Section,Equilibrium:,Let,But,First Moment of Area 面矩（靜矩）,Then y is measured from the centroidal axis of the beam cross-section.,Equilibrium:,Let,=The 2nd Moment of Area about Z-axis 慣性矩,THE SIMPLE BEAM THEORY:,,,- Applied Bending Moment,- Property of Cross-Sectional Area,- Stress due to M,- 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,,z,y,,y,,,,,,y,,,NA,Neutral Axis,,,x,,,,,,o,,7.3.1 The Engineering Beam Theory,,,7.3.2 Properties of Area,,,,Position of Centroidal or Neutral Axis:,,(Definition),,Equilibrium:,The 2nd Moment of Area about y-Z-axis 慣性積,THE SIMPLE BEAM THEORY:, y 軸為對(duì)稱軸,Summary,The Engineering Beam Theory determines the 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.,7.3.2 The Engineering Beam Theory,,Summary,It is applicable to long, slender load carrying devices.,7.3.2 The Engineering Beam Theory,橫力彎曲,7.4.1 Strength and Deformation,Ch.7 Stresses in Beams,WZ : 抗彎截面模量,彎曲時(shí)的強(qiáng)度條件,7.4.2 Sample Problems,Ch.7 Stresses in Beams,簡(jiǎn)易天車，P=68 kN， l = 9.5 m, 40c 型工字鋼 的自重為q，=140MPa, 校核安全性,解：作彎矩圖（疊加法）,查表,安全 ？？,7.4.2 Sample Problems,Ch.7 Stresses in Beams,,7.4.2 Sample Problems,Ch.7 Stresses in Beams,靜矩和形心,厚度 t 極小的薄片如圖,在圖形所在平面內(nèi)建立坐標(biāo)系,重心,, 比重 G 薄片重量,dA 微面積,,,形心坐標(biāo),定義,圖形對(duì) z 軸的靜矩,圖形對(duì) y 軸的靜矩,mm 3,討論 :,坐標(biāo)軸通過形心時(shí),1) y 軸通過形心時(shí),2) z 軸通過形心時(shí),坐標(biāo)軸通過圖形的形心,圖形對(duì)該軸的 靜矩 等于零,,Example:,(Dimensions in mm),,,,2nd Moment of Area:,Definition:,,The Parallel Axis Theorem:,,z,y,,,o,Definition:,Example:,(Dimensions in mm),,,z,y,,,o,200,10,20,120,,What is Iz? What is maximum sx?,Example:,(Dimensions in mm),,,z,y,,,o,,200,10,20,120,89.6,,30.4,,,,,89.6,20,20,30.4,200,10,,,35.4,1,2,3,What is Iz? What is maximum sx?,,,,,y,,,NA,,x,Maximum Stress:,,(N/m2 or Pa),,The Perpendicular Axis Theorem:,,z,y,,,o,,From Symmetry,,