彈性力學(xué)課件一零

第四章平面問題的極坐標(biāo)解答4—1DifferentialEquationsofEquilibriuminPolarCoordinates4—2GeometricalEquationsandPhysicalEquationsinPolarCoordinates4—3StressFunctionandCompatibilityEquationsinPolarCoordinates4—9Effectofcircularholesonstressdistribution4.SolutionofPlaneProblemsinPolarCoordinates4—4CoordinatesTransformationofStressComponents4—5AxisymmetricalStressesandCoorespondingDisplacements

4—6HollowCylinderSubjectedtoUniformPressures4—11ConcentratedNormalLoadonaStraightBoundary4.1DIFFERENTIALEQUATIONSOFEQUILIBRIUMINPOLARCOORDINATESIndiscussingstressesanddisplacementincirculardisksandrings,solidandhollowcircularcylinders,curvedbeamsofrectangularsectionwithacircularaxisetc.,itisadvantageoustouseinpolarcoordinatesinsteadofrectangularcoordinates.ThepositionofapointPincoordinatesisdefinedbytheradialcoordinate

r

andtheangularcoordinate,asshowinFig.:

xyoKrKToexpressthestresscomponentsinpolarcoordinates,weconsideranelementPACBformedbydranddandcutfromthethinplaneorlongcylindricalbodyconsidered.drPABCdrInradialdirection,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）xyrFromwhichwehave：Notingthatisafunctionofxandyandalsoafunctionofrand,wehaveRepetitionoftheaboveoperationyields：Th毛e桌ad繡di宮ti貌on捎o違f嗚ab國ov醋e匪eq癥ua麻ti憐on你s制yi遞el械ds：Thecompatibilityequationinrectangularcoordinatesbecomesthatinpolarcoordinatesas：xyo（x，y）xyrIfxandyax木es光a驗re餐r門ot茫at辮ed東t冠o妻th符e碰di妖re晃ct端io肝ns池o侍fran席dre仇sp襲ec夸ti曾ve談ly篇t動o怎ma沉ke唯皮=0沖,強th幕e盯st覽re絞ss滿c譜om偽po剃ne綠nt預(yù)s朝x,私y(tǒng),xywi躬ll煙b箭ec紫om賴e昨r,,耀rres爐pec向tiv蛇ely源.Insolvingaplaneprobleminpolarcoordinates,itisnecessarytosolveonlythedifferentialequationforthestressfunctionandthenfindthestresscomponentsby

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