弯曲应力纯弯曲

Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,Bending of Beams,Mechanics of Materials,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 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,Dimension 3100 AFM,The worlds best selling SPM,silicon-Si(111),2nm,?,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,Substrate,Probes,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,La Tour Eiffer,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,P,Ch.7 Stresses in Beams,dx,x,s,x,M,M,s,x,dx,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,平面假定,物性关系,静力方程,7.3.1,The Engineering Beam Theory,Ch.7 Stresses in Beams,Bending Deformations,平面假设？,M,M,7.3.1,The Engineering Beam Theory,x,y,M,M,A,B,C,D,Compression,Tension,No Stress,NA,Neutral Axis,中性轴,z,y,y,y,A,B,C,D,y,R,dq,d,x,s,x,=0 on the,Neutral Axis,. In general we must find the position of the,Neutral Axis,.,Ch.7 Stresses in Beams,M,M,7.3.1,The Engineering Beam Theory,x,y,M,M,A,B,C,D,Compression,Tension,No Stress,NA,Neutral Axis,中性轴,y,A,B,C,D,y,R,dq,d,x,Ch.7 Stresses in Beams,Assumptions,Beam material is elastic,Plane surfaces remain plane,and only,1,Geometry of Deformation:,Hookes Law:,and,A,B,C,D,y,R,dq,M,M,7.3.1,The Engineering Beam Theory,1,x,y,y,NA,Neutral Axis,中性轴,0,+ve,-ve,Linear Distribution of,s,x,(Eqn ),1,d,x,Note:,E is a Material Property,is Curvature,曲率,x,d,x,y,M,M,s,x,7.3.1,The Engineering Beam Theory,7.3.1,The Engineering Beam Theory,Deformation in a Transverse Cross Section,R,R,R=R/,z,y,x,Equilibrium:,x,z,y,y,d,A,s,x,M,Area, A,Let,But,First Moment of Area,面矩（静矩）,Then y is measured from the centroidal axis of the beam cross-section.,“Neutral Axis” coincides with the XZ plane through the centroid.,y,y,NA,Neutral Axis,Centroid,2,Equilibrium:,as,1,Let,=The 2,nd,Moment of Area about Z-axis,惯性矩,THE SIMPLE BEAM THEORY:,1,2,&,x,z,y,y,d,A,s,x,M,Area, A,- 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,- m,4,- N/m,2,or,Pa,- m,- N/m,2,or,Pa,- m,z,y,y,y,NA,Neutral Axis,x,o,7.3.1,The Engineering Beam Theory,z,y,o,7.3.2,Properties of Area,y,d,A,s,x,M,x,z,y,o,y is measured from the Centroidal or Neutral Axis, z.,I,z,is the 2,nd,Moment of Area about the Centroidal or Neutral Axis, z.,Position of Centroidal or Neutral Axis:,y,Centroidal Axis,z,o,y,Area, A,y,n,(Definition),d,A,d,A,Equilibrium:,as,1,The 2,nd,Moment of Area about y-Z-axis,惯性积,THE SIMPLE BEAM THEORY:,x,z,y,y,d,A,s,x,M,Area, A,z,y,轴为对称轴,x,z,y,y,M,Area, A,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.,Neutral Axis Position, y,2,nd,Moments of Area,I,y,I,z,I,p,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,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 Deformation,Ch.7 Stresses in Beams,W,Z,:,抗弯截面模量,弯曲时的强度条件,7.4.2,Sample Problems,Ch.7 Stresses in Beams,A,P,C,B,l,/2,l,/2,简易天车，,P=68,kN，,l =,9.5 m, 40c,型工字钢,的自重为,q,，,=140,MPa,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 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,y,z,o,重心,y,z,dA,比重,G,薄片重量,C,y,c,z,c,dA,微面积,薄片图形的形心,y,z,o,形心,y,z,dA,C,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,y,n,Example:,z,o,Centroidal Axis,200,10,20,125,120,60,(,Dimensions in mm),Example:,z,y,o,2,nd,Moment of Area:,Definition:,z,y,y,d,y,d,A,o,y,z,The Parallel Axis Theorem:,z,y,o,Definition:,Example:,y,d,y,n,n,z,o,y,Example:,(,Dimensions in mm),z,y,o,200,10,20,120,89.6,30.4,89.6,20,20,30.4,200,10,1,2,3,What is I,z,?,What is maximum,s,x,?,35.4,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 I,z,?,What is maximum,s,x,?,y,NA,x,Maximum Stress:,89.6,40.4,M,xz,(N/m,2,or,Pa),Example:,The Perpendicular Axis Theorem:,z,y,o,d,A,z,y,R,The Polar 2,nd,Moment of Area (About the X-axis),d,R,R,From Symmetry,o,y,z,