《弹性力学》课程教学课件（英文讲稿）Chapter 7 The Basic Theory of the Plane Problem

ElasticityChapter 7The Basic Theory of the Plane Problem

1 Elasticity

The Basic Theory of the Space ProblemChapter7TheBasicTheory of the SpaceProblemS 7-1 Equilibrium DifferentialEquationS 7-2 The Stress State at a Point in a Plane ProblemS 7-3 The Principle Stress.The Maximum and the Minimum StressS 7-4 Geometrical Equation. Physical EquationS 7-5 The Basic Equation of the Axisymmetric Problem2

2 Chapter 7 The Basic Theory of the Space Problem §7-4 Geometrical Equation. Physical Equation §7-3 The Principle Stress. The Maximum and the Minimum Stress §7-2 The Stress State at a Point in a Plane Problem §7-1 Equilibrium Differential Equation §7-5 The Basic Equation of the Axisymmetric Problem

The Basic Theory of the Space ProblemS 7-1 Equilibrium Differential Equation20idzazatsyTay+daAt any point P in theaatdzobject, take a smallparallelepiped shownatyeOtxas the figure. TheTya+-dytx+dxayaxatrVExa0stress components onTry+dxaxatyadyeach surfaceof aaortyx+TyaayOdxaxsmall parallelepipedshown in thearefigure..图7-13

3 §7-1 Equilibrium Differential Equation At any point P in the object, take a small parallelepiped shown as the figure. The stress components on each surface of a small parallelepiped are shown in the figure

The Basic Theory of the Space ProblemIf the line connecting the front and back00zof the hexahedron is AB, from mAs = OdzO.OzOydzwe can get:Tzy +O2at二atydydzattaxdydyOzyzdxdzoydydxdzBT+TyzL22ayaoydOyoOtaydzdzdz0dxdy+TTyAzy22OzdyTux+2Simplifying and removing the higher order smallamount, we have: Ty, = Tay

4 If the line connecting the front and back of the hexahedron is AB, from we can get: 0 mAB = d d d d d d d 2 2 yz yz yz y y y x z x z y         + +    d d d d d 0 2 2 zy zy zy z z z x y dxdy z       − + − =      Simplifying and removing the higher order small amount, we have: yz zy   = A B

The Basic Theory of the Space Problemaadz2ats-1a-atd-atdrTie+odyTrdxaxatnYtxtdadCadyaxarda.diSelayCd2T.Tx= Tx=TIn the same way:xzxvyxThe shear stress reciprocal relationship in general space are provedagain.5

5 In the same way: The shear stress reciprocal relationship in general space are proved again. zx xz xy yx = =    