彈性力學(xué)課件三五

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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）xyrFromwhichwehave：Notingthatisafunctionofxandyandalsoafunctionofrand,wehaveRepetitionoftheaboveoperationyields：The檢ad艙dit即ion貪of爹ab雁ove仿eq黎uat梁ion眼sy便iel相ds：Thecompatibilityequationinrectangularcoordinatesbecomesthatinpolarcoordinatesas：xyo（x，y）xyrIfxandyaxe藥sa王re憶rot寒a(chǎn)te幅dt烘ot舟he此dir訴ect示ion閘so叮fran債dre嫁sp烤ec芝ti未ve豬ly猶t顯o咐m(xù)a費(fèi)ke齊劈燕=0糖,舊th冒e倒st止re跟ss偉c歇om斷po鵲ne陡nt鋪s很x,y,文xywil殊lb胳eco豈me融r,衣,rres咱pec跨tiv執(zhí)ely螺.Insolvingaplaneprobleminpolarcoordinates,itisnecessarytosolveonlythedifferentialequationforthestressfunctionandthenfindthestresscomponentsby

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