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1、Chapter 7 Analysis of Stress and Strain Mechanics of MaterialsIntroductionWelcome to mechanics of materials.This chapter is going to discuss the ananlysis of the state of stress or strain at points in structural members.Firstly,an introduction of this chapter is given in this section.Stresses on cro
2、ss sectionsFlexure and shear FormulaTorsion formula Axial stress formulaHow to determine stresses on inclined sections?In the preceding chapters,stresses on cross sections of*bars,*shafts,and*beams have been discussed.*Those stresses on cross sections of memebers can be calculated directly using the
3、 basic formulas.For instance,*the stresses on the cross section in a bar are given by the axial stress formula,*the stresses in a shaft given by the tortion formula,and*the stresses in a beam given by the flexure and shear formulas.However,on inclined sections of memebers,there are also stresses exi
4、sting.And larger stresses may occur on inclined sections,which may define the failure of memebers.Therefore,the following question comes up:*how to determine stresses on inclined sections,based on stresses known on cross sections?To answer this question,we will start the disscussion of the stress an
5、alysis in the following couple of sections.Stresses on Inclined sectionsBarsUniaxial stress stateUniaxial stress stateComplex stress stateStress transformationTransformation equations?For stresses on inclined sections,we have already given a simple introdution in the preceding sections.Lets recall,s
6、tresses on inclined sections in*bars,which is axially loaded*.Expressions for the*normal and shear stresses acting on inclined sections were presented.And*the manner in which the stresses vary as the inclined section at various angles is shown in the figure.As shown in the graph,the maximum normal s
7、tress occurs at=0 degree,that is,on cross sections,as shown by*the stress element oriented at=0 degree,which is called an*unaxial stress state.*And,the maximum shear stresses occur on planes inclined at 45 to the axis.*For this particular bar,if point A happens to be the weakest point of the member,
8、and the material is brittle,therefore,it has a lower resistance to normal stress,it is likely,it will fail along*this plane,in accordance to this orientation.On the other hand,if point B happens to be the weakest point of the member,and the material is ductile,meaning that it is much weaker in shear
9、,it is likely,the material will fail along*this plane,according to this orientation.*Therefore,as you can see,in this uniaxial case,it is important to determine the stresses acting on inclined planes based on known stresses on cross sections.Especially the maximum normal stress and maximum shear str
10、ess.Then we can better understand how the material behaves under loadings.Another question is,*how to determine the stresses on inclined sections in members,which has a more complex stress state than the unaxial case,for example,beams.Therefore,it is nessessay to find a way to determine stresses on
11、inclined section based on a more general stress state,not just uniaxial,but a complex stress state,maybe biaixal or triaxial stress state.This process is known as*stress transformation.Begin the analysis,by considering an element on which the stresses are known from the basic stress formulas,and the
12、n derive the*transformation equations that give the stresses acting on the sides of a new element at a different orientation.Based on the transformation equations,we can determine the maximum normal stress and the maximum shear stress,and their corrsponding orientation.In the next section,we will di
13、scuss how to derive the transformation equations.Normal stress:x,y,zTriaxialShear stress:xy,yz,zxNormal stress:x,yShear stress:xy BiaxialzxyzzxyzyxyxyxyxStress elementOnly one intrinsic state of stress exists at a point.And as you can see,normally,in the discussion of a general stress state,we use s
14、tress elements to represent the state of stress at a point in a body.For example,a general*three dimensional state of stress for a particle,can be fully characterized by six components,including three normal stresses x,y and z,and three shear stresses xy,yz and zx.*While,for a planar state of stress
15、,there are two normal stresses x and y and one shear stress xy.Keep in mind that*only one intrinsic state of stress exists at a point in a stressed body,regardless of the orientation of the element being used to portray that state of stress.We will start with the plane stress analysis first,and then
16、 move on to the triaxial stress analysis.Chapter Outline7.1 Introduction7.2 Plane Stress 7.3 Principal Stresses and Maximum Shear Stresses 7.4 Mohrs Circle for Plane Stress7.5 Hookes Law for Plane Stress7.6 Triaxial Stress7.7 Plane Strain Chapter Summary and ReviewF FF FAQuestion 1:Check the stress
17、element(both the 3D and 2D elements)for each point marked on the members below.A AF FMF FB BB MFFC C C3D:2D:MMABCACBQuestion 2:Check the stress element(both the 3D and 2D elements)for each point marked on the members below.ABCM2M1D DD 3D:2D:FL/4L/4F123456F+_F+FL/4 1 3 4 6123456Question 3:Check the stress element for the points marked on the beam.EndEnd of this section.