9-平面问题的极坐标求解Chapter_9.ppt

1、TheoryofElasticity,Chapter9Two-DimensionalSolution,Content,IntroductionMathematicalPreliminariesStressandEquilibriumDisplacementsandStrainsMaterialBehavior-LinearElasticSolidsFormulationandSolutionStrategiesTwo-DimensionalProblemsIntroductiontoFiniteElementMethodThree-DimensionalProblemsBendingofThi

2、nPlates,9,1,Two-DimensionalSolutioninPolarCoordinate,9.1PolarCoordinateFormulation(极坐标下的求解)9.2CoordinateTransformationofStressComponents（应力分量的坐标变换）9.3Axisymmetrialstressesandcorrespondingdisplacements（轴对称应力和位移）9.4Hollowcylindersubjectedtouniformpressures（圆环受均布压力）9.5Stressconcentrationofthecircularho

3、le（圆孔的孔口应力集中）,9,2,Two-DimensionalSolutioninPolarCoordinate,PracticalEngineeringneedsolutionsinpolarcoordinate,Aeroengineanditsrotorsystem,Somesymmetryandcircularstructure,9,3,Two-DimensionalSolutioninPolarCoordinate,StressConcentrationinPracticalEngineering,FirstcivilairlineComet:May2nd,1953,London-

4、JohnstonAirdisaster:Jan.10,1954,andApril8,1954,9,4,Two-DimensionalSolutioninPolarCoordinate,StressConcentrationisthemainreasonthatcausetheairdisasters,9,5,9.1PolarCoordinateFormulation,PolarCoordinates,9,6,9.1PolarCoordinateFormulation,PolarCoordinates,9,7,9.1PolarCoordinateFormulation,BasicEquation

5、sinPolarCoordinates（极坐标下的基本方程）,Stresses,Laplaceoperators,Biharmonicoperators,9,8,9.1PolarCoordinateFormulation,EquilibriumEquations(平衡方程),9,9,9.1PolarCoordinateFormulation,GeometricalEquations几何方程,9,10,9.1PolarCoordinateFormulation,PhysicalEquations物理方程,PlainStress,PlainStrain,9,11,9.1PolarCoordinat

6、eFormulation,BoundaryConditions(边界条件),9,12,9.2CoordinateTransformationofStressComponents,PolarCoordinateCartesianCoordinate,CartesianCoordinatePolarCoordinate,9,13,9.3Axisymmetrialstressesandcorrespondingdisplacements(轴对称应力和位移),axisymmetric,fieldquantitiesareindependentoftheangularcoordinate,X,9,14,

7、9.3Axisymmetrialstressesandcorrespondingdisplacements,axisymmetriccase,axisymmetriccase,9,15,9.3Axisymmetrialstressesandcorrespondingdisplacements,Michellsolutionofthebiharmonicequation,H,I,Kassociatedwiththerigid-bodymotion,Planestresscase,9,16,9.3Axisymmetrialstressesandcorrespondingdisplacements,

8、Functionof“hole”ondistribution,Nohole:,AandBvanish.,Otherwisewhenr=0,stresscomponentbecomeinfinite,Aplatewithoutaholewithnobodyforces(axissymmetrical),Ifthereisholeattheorigin,wewillinvestigateitnext,9,17,9.4Hollowcylindersubjectedtouniformpressures,planestressconditions,Axisymmetricproblem,Boundary

9、Conditions,9,18,9,19,9.4Hollowcylindersubjectedtouniformpressures,3unknowns,2Equations?,formultiplyconnectedregions,thecompatibilityequationsarenotsufficienttoguaranteesingle-valueddisplacements.,Singleormultiplyconnectedregion?,B=0,9.4Hollowcylindersubjectedtouniformpressures,B=0,9,20,9.4Hollowcyli

10、ndersubjectedtouniformpressures,DemonstrationofSaint-VenantsPrinciple,9,21,9.5Stressconcentrationofthecircularhole,Review:,9,22,9.5Stressconcentrationofthecircularhole,Whatisstressconcentration?,Thestressconcentrationnearaholeisacriticalissueconcerningthestrengthofasolidstructure.Thestressconcentrat

11、ioncanbemeasuredbythestressconcentrationcoefficientsthataretheratiosbetweenthemostseverestressatthecriticalpoint(ortermedhotspot)andtheremotestress.,9,23,9.5Stressconcentrationofthecircularhole,Examples:,9,24,9.5Stressconcentrationofthecircularhole,Solution:,Selectionofcoordinate,Toanalysestressconc

12、entrationnearthehole,itisconvenienttousepolarcoordinate.,Probleminpolarcoordinate,Boundaryconditionsinpolarcoordinate:,9,25,9.5Stressconcentrationofthecircularhole,Probleminpolarcoordinate,9,26,9.5Stressconcentrationofthecircularhole,Problem1,Problem2,Probleminpolarcoordinate,9,27,9.5Stressconcentra

13、tionofthecircularhole,SolutionofProblem1,whenba,9,28,9.5Stressconcentrationofthecircularhole,SolutionofProblem2,Problem2,由边界条件可假设：r为r的某一函数乘以cos2；r为r的某一函数乘sin2。,Assume：,9,29,9.5Stressconcentrationofthecircularhole,9,30,9.5Stressconcentrationofthecircularhole,B.C.,9,31,9.5Stressconcentrationofthecircularhole,SuperpositionofSolution1and2,G.KirschSolution,9,32,9.5Stressconcent