弯曲应力-纯弯曲.ppt

材料力学 Professor Shibin WANG （王世斌）,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 Beams,7.1 Examples under Bending Loading:,受弯杆件的简化 - 简支梁,Ch.7 Stresses in Beams,7.1 Examples under Bending Loading:,受弯杆件的简化 - 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: 纯弯曲,Ch.7 Stresses in Beams,7.2 Loading Types,Other Loading Types,Eccentric Loading: 偏心加载,Transverse Loading: 横向力作用,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,应力分布,应力公式,变 形,应变分布,7.3.1 The Engineering Beam Theory,Ch.7 Stresses in Beams,Bending Deformations,平面假设？,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（静矩）,Then y is measured from the centroidal axis of the beam cross-section.,Equilibrium:,Let,=The 2nd Moment of Area about Z-axis 惯性矩,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 惯性积,THE SIMPLE BEAM THEORY:, y 轴为对称轴,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,横力弯曲,7.4.1 Strength and Deformation,Ch.7 Stresses in Beams,WZ : 抗弯截面模量,弯曲时的强度条件,7.4.2 Sample Problems,Ch.7 Stresses in Beams,简易天车，P=68 kN， l = 9.5 m, 40c 型工字钢 的自重为q，=140MPa, 校核安全性,解：作弯矩图（叠加法）,查表,安全 ？,7.4.2 Sample Problems,Ch.7 Stresses in Beams,7.4.2 Sample Problems,Ch.7 Stresses in Beams,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,重心, 比重 G 薄片重量,dA 微面积,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,图形对该轴的 静矩 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,