《弹性力学》课程教学课件（英文讲稿）Chapter 4 Polar Solutions for Planar Problems

ElasticityCheaoarem

Elasticity 1

POLARSOLUTIONSFORPLANEPROBLEMSChapter 4 Polar Solutions For Plane ProblemsS 4-1 Eguilibrium DifferentialEquations in PolarCoordinatesS 4-2 Geometric and Physical Functions in PolarCoordinatesS 4-3 Stress Functions and Compatibility Equationsin Polar CoordinatesS 4-4 Coordinates Conversion of Stress ComponentsS 4-5 Axisymmetric Stress and Its DisplacementS 4-6 Circular Ring or Cylinder under UniformLoading Pressure

Chapter 4 Polar Solutions For Plane Problems §4-1 Equilibrium Differential Equations in Polar Coordinates §4-5 Axisymmetric Stress and Its Displacement §4-2 Geometric and Physical Functions in Polar Coordinates §4-3 Stress Functions and Compatibility Equations in Polar Coordinates §4-4 Coordinates Conversion of Stress Components §4-6 Circular Ring or Cylinder under Uniform Loading Pressure 2

POLARSOLUTIONSFORPTANEPROBLEMSChapter 4 Polar Solutions For Plane ProblemsS 4-7 Pressure TunnelS 4-8 Stress Concentration at the Hole Edge of theCircular HoleS 4-9 Normal Concentrated Forces on the Boundaryfor Semi-infinitePlane Body 4-1oDistributedForce ontheBoundaryforSemi-infinitePlane BodyExercise

Chapter 4 Polar Solutions For Plane Problems §4-10 Distributed Force on the Boundary for Semi-infinite Plane Body §4-8 Stress Concentration at the Hole Edge of the Circular Hole §4-7 Pressure Tunnel §4-9 Normal Concentrated Forces on the Boundary for Semi-infinite Plane Body Exercise 3

POLARSOLUTIONSFORPLANEPROBLEMSS 4-1Eguilibrium DifferentialEquations in Polar CoordinatesDealing with elasticity problems, what form of coordinatesystem we choose, which can't affect on the essence fordescribing problem, but is relate to the level of difficulty onsolving problem directly. If coordinate is suitable, it cansimplify the problem considerably. For example, for circular,wedged and sector and so on, it will be more convenient byusing polar coordinates than using rectangular coordinates.Considering an differential field PACBin the plate

§4-1 Equilibrium Differential Equations in Polar Coordinates Dealing with elasticity problems, what form of coordinate system we choose，which can’t affect on the essence for describing problem, but is relate to the level of difficulty on solving problem directly. If coordinate is suitable, it can simplify the problem considerably. For example, for circular, wedged and sector and so on, it will be more convenient by using polar coordinates than using rectangular coordinates. Considering an differential field in the plate PACB 4

POLARSOLUTIONSFORPLANEPROBLEMSnormal stress in the r direction is called radial normal stressdenoted by . ; normal stress in the direction is calledtangential normal stress denoted by .; shear stress is denotedby Tro, stipulation of sign of each stress component is similar tothe one in rectangular coordinates. Body force components ofradial direction and hoop are denoted by K,and Ke, respectivelyFig. 4-1.+x0660TerPConsidering equilibrium of an unitTroOdo4element, there are three equilibriumatreedrBequations:rfar1droo,drZF =0,ZF。=0,ZM=0a.or000deOT00adea0TerXtya0Fig.4-15

Considering equilibrium of an unit element, there are three equilibrium equations: r r r r d   +     r r r r r d   +           d   +       d r r   +  d  r r   dr K Kr y x o P A B C Fig.4－1 Fr = 0,F = 0,M = 0 normal stress in the direction is called radial normal stress denoted by ; normal stress in the direction is called tangential normal stress denoted by ; shear stress is denoted by , stipulation of sign of each stress component is similar to the one in rectangular coordinates. Body force components of radial direction and hoop are denoted by and , respectively. Fig. 4-1.  r     r Kr K r 5