弹性力学课件三五

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SolutionofPlaneProblemsinPolarCoordinates4.1DifferentialEquationsofEquilibriuminPolarCoordinates4.2GeometricalandPhysicalEquationsinPolarCoordinates4.3StressFunctionandCompatibilityEquationinPolarCoordinates4.4CoordinatesTransformationofStressComponents4.6HollowCylinderSubjectedtoUniformPressures4.9EffectofCircularHolesonStressDistribution4.11ConcentratedNormalLoadonaStraightBoundary4.5AxisymmetricalStressesandCoorespondingDisplacements

4.1DIFFERENTIALEQUATIONSOFEQUILIBRIUMINPOLARCOORDINATESIndiscussingstressesanddisplacementincirculardisksandrings,solidandhollowcircularcylinders,curvedbeamsofrectangularsectionwithacircularaxisetc.,itisadvantageoustouseinpolarcoordinatesinsteadofrectangularcoordinates.ThepositionofapointPincoordinatesisdefinedbytheradialcoordinate

r

andtheangularcoordinate,asshowinFig.:

drPABCxyoKrKToexpressthestresscomponentsinpolarcoordinates,weconsideranelementPACBformedbydranddandcutfromthethinplaneorlongcylindricalbodyconsidered.rPInradialdirection,weobtaintheequilibriumequation：Sincedissmall,wehaveSimplifyingtheequation,dividingitbyrdrdandthenneglectingtheinfinitesimalterms,weobtain：（1）Similarly,inthetangentialdirection：whichreducesto（2）So,thedifferentialequationsofequilibriuminpolarcoordinatesare：Whichcontainthreeunknownfunctions：r,

andr=r.4.2GEOMETRICALANDPHYSICALEQUATIONSINPOLARCOORDINATESForthestrainsinpolarcoordinates,wedenotetheradialstrain(normalstrainintheradialdirection)byrandthecircumferentialstrain(normalstraininthecircumferentialdirection)by

andtheshearingstrain(thedecreaseoftherightanglebetweennormalandcircumferentiallineelements)byr.Fordisplacements,wedenotetheradialandcircumferentialcomponentsby

urandu

respectively.AtpointP,wetakeradialandcircumferentiallineelementsPAandPB：(1)assumethatonlytheradialdisplacementtakesplacePABxyodrP`B`A`ThusthenormalstrainoftheradiallineelementPAwillbe：ThatofthecircumferentiallineelementPBwillbe：PABxyodP’B’A’rTheangleofrotationofPAwillbe：ThatofPBwillbexAPByodr(2)assumethatonlythecircumferentialdisplacementtakesplaceA”P”B”Hence,theshearingstrainis：ThenormalstrainofPA：ThatofPB：TheangleofrotationofPA：Thatof

PB：Hence,theshearingstrain：Whenboththeradialandcircumferentialdisplacementstakeplace,wecanobtainthetotalstrainsbysuperposition.Geometricequationsinpolarcoordinates.Sincethepolarcoordinatesrandareorthogonal,justastherectangularcoordinatesxandyare,thephysicalequationsinthetwocoordinatesystemsmusthavethesameform,butwithrand

inplaceofxandy,respectively.ForaplanestressproblemForaplanestrainproblem4.3STRESSFUNCTIONANDCOMPATIBILITYEQUATIONINPOLARCOORDINATESWhenthebodyforcesarenotconsidered,thestresscomponentsinpolarcoordinatescanbeexpressedintermsofastressfunction(r,).Theseexpressionmaybederivedfromthoseinrectangularcoordinatesbymeanofcoordinatetransformation.Therelationsbetweenpolarandrectangularcoordinatesare：xyo（x，y）xyrFr筐om乎w答hi英ch粪w冬e美ha稳ve：No蓬ti罢ng害t疮ha野t庸及is聪a父f湾un蓬ct启io论n肝of垮x踢a律nd器y炒a宁nd修a日ls汇o南a但fu斜nc混ti型on聋o贵f用r请an拥d铁,覆w煎e催ha仙veRe扁pe书ti赵ti菠on静o遥f脂th负e懒ab楚ov冷e竞op纪er乱at推io回n生yi帜el泄ds：Th纯e孝ad候di们ti缎on党o忘f贪ab瞧ov毙e蛛eq隆ua恶ti混on推s爽yi荷el绢ds：Thecompatibilityequationinrectangularcoordinatesbecomesthatinpolarcoordinatesas：xyo（x，y）xyrIfxan障dyax哄es浊a消re通r挪ot变at萍ed五t袋o甘th高e阔di就re巡寿ct进io硬ns慨o誓fran狡dre防sp详ec桑ti炭ve芽ly烦t剪o劫ma跟ke呼促=0壳,牺th板e充st靠re梅ss让c茧om啄po院ne叶nt泥s滨x,错y,xywi目ll俘b退ec妨om盾e衡r,,rres俱pec填tiv充ely虽.Insolvingaplaneprobleminpolarcoordinates,itisnecessarytosolveonlythedifferentialequationforthestressfunctionandthenfindthestresscomponentsby

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