弹性力学1第四章.ppt

弹性力学第四章 1 Chapter4solutionofplaneproblemsinpolarcoordinates 第四章平面问题极坐标解答 弹性力学第四章 2 Chapter4 SolutionofPlaneProblemsinPolarCoordinates 4 1Differentialequationsofequilibriuminpolarcoordinates Indiscussingstressesanddisplacementsincirculardisksandrings solidandhollowcircularcylinders curvedbeamsofrectangularsectionwithacircularaxisetc itisadvantageoustousepolarcoordinatesinsteadofrectangularcoordinates 弹性力学第四章 3 Polarcoordinates极坐标 ThepositionofapointPinpolarcoordinatesisdefinedbytheradialcoordinaterandtheangularcoordinate 一点P的极坐标用径向坐标r和角坐标 表示P r displacements 位移 uru strains 应变 r rr stresses 应力 r r bodyforce 体力 KrK 弹性力学第四章 4 ThepositionofapointPinpolarcoordinatesisdefinedbytheradialcoordinaterandtheangularcoordinate 4 1Differentialequationsofequilibriuminpolarcoordinates 弹性力学第四章 5 4 1Differentialequationsofequilibriuminpolarcoordinates Summingupalltheforcesactingontheelementintheradialdirection weobtaintheequilibriumequation 弹性力学第四章 6 4 1Differentialequationsofequilibriuminpolarcoordinates Replacingby simplifyingtheequation dividingitbyandthenneglectingtheinfinitesimalterms weobtain Similarly theequilibriumconditionoftheelementinthetangentialdirectionwillyield 弹性力学第四章 7 4 1Differentialequationsofequilibriuminpolarcoordinates whichreducesto 弹性力学第四章 8 4 2GeometricalandPhysicalEquationsinPolarCoordinates极坐标中的几何物理方程 4 2 1GeometricalEquationsinPolarCoordinates Inconsideringthedisplacementinpolarcoordinates letusdenotebyandthecomponentsofthedisplacementintheradialandcircumferentialdirections respectively Weuseforthestraininradialdirection forthestraininthecircumferentialdirection forshearingstrain 弹性力学第四章 9 4 2 1GeometricalEquationsinPolarCoordinates Firstly weassumethatonlytheradialdisplacementtakesplace 仅考虑径向位移 弹性力学第四章 10 4 2 1GeometricalEquationsinPolarCoordinates ThedisplacementsofpointsP A andBare ThusthenormalstrainoftheradiallineelementPAwillbe 弹性力学第四章 11 4 2 1GeometricalEquationsinPolarCoordinates andthatofthecircumferentiallineelementPBwillbe 弹性力学第四章 12 4 2 1GeometricalEquationsinPolarCoordinates TheangleofrotationofPAwillbe andthatofPBwillbe 弹性力学第四章 13 4 2 1GeometricalEquationsinPolarCoordinates Hence theshearingstrainis Next weassumethatonlythecircumferentialdisplacementtakesplace 弹性力学第四章 14 4 2 1GeometricalEquationsinPolarCoordinates Next weassumethatonlythecircumferentialdisplacementtakesplace 弹性力学第四章 15 4 2 1GeometricalEquationsinPolarCoordinates ThedisplacementsofpointsP A andBare ThusthenormalstrainofPAwillbe 弹性力学第四章 16 4 2 1GeometricalEquationsinPolarCoordinates andthatofthecircumferentiallineelementPBwillbe 弹性力学第四章 17 4 2 1GeometricalEquationsinPolarCoordinates TheangleofrotationofPAwillbe andthatofPBwillbe 弹性力学第四章 18 4 2 1GeometricalEquationsinPolarCoordinates Hence theshearingstrainis Inthegeneralcase whenboththeradialandcircumferentialdisplacementstakeplace wecanobtainthetotalstrainsbysuperposition 弹性力学第四章 19 4 2 1GeometricalEquationsinPolarCoordinates 弹性力学第四章 20 4 2 2PhysicalEquationsinPolarCoordinates Sincethepolarcoordinatesrand areorthogonal justastherectangularcoordinatesxandyare thephysicalequationsinthetwocoordinatesystemsmusthavethesameform butwithrand inplaceofxandyrespectively 弹性力学第四章 21 4 3StressFunctionandCompatibilityEquationinPolarCoordinates极坐标中的应力函数及相容方程 Theseexpressionsmaybederivedfromthoseinrectangularcoordinatesbymeansofcoordinatetransformationasfollows Whenthebodyforcesarenotconsidered thestresscomponentsinpolarcoordinatescanbeexpressedintermsofastressfunction 4 3 1StressFunctioninPolarCoordinates 弹性力学第四章 22 4 3 1StressFunctioninPolarCoordinates Therelationsbetweenpolarandrectangularcoordinatesare fromwhichwehave 弹性力学第四章 23 4 3 1StressFunctioninPolarCoordinates Notingthatisafunctionofxandyandalsoafunctionofrand wehave Repetitionoftheaboveoperationyields 弹性力学第四章 24 4 3 1StressFunctioninPolarCoordinates 弹性力学第四章 25 4 3 1StressFunctioninPolarCoordinates 弹性力学第四章 26 4 3 1StressFunctioninPolarCoordinates 弹性力学第四章 27 4 3 1StressFunctioninPolarCoordinates Now notingthatas weobtain Also notingthatas wefind 弹性力学第四章 28 4 3 1StressFunctioninPolarCoordinates Similarly wecangettheexpressionforshearingstress Notingthatas wefind Itiseasytoverifythattheseexpressionsdosatisfythedifferentialequationsofequilibrium 4 1 1 and 4 1 2 when 弹性力学第四章 29 4 3 2CompatibilityEquationinPolarCoordinates Ontheotherhand sincetheadditionofEqs 4 3 2 and 4 3 3 yields thecompatibilityequationinrectangularcoordinates becomesthatinpolarcoordinatesas 弹性力学第四章 30 4 4coordinatetransformationofstresscomponents应力分量的坐标变换式 Egs 4 4 1 4 4 12 弹性力学第四章 31 4 5Axisymmetrialstressesandcorrespondingdisplacements轴对称应力和相应的位移 Axisymmetrialstresses 轴对称应力 1 thenormalstresscomponentsareindependentof 2 theshearingstresscomponentsvanish3 hencethestressdistributionissymmetricalwithrespecttoanyplanepassingthroughthezaxis r r r r r 0 弹性力学第四章 32 r r r r r 0 r r r r r r 弹性力学第四章 33 Axisymmetrialstresses轴对称应力 Compatibilityequation 弹性力学第四章 34 弹性力学第四章 35 Thisisanordinarydifferentialequation whichcanbereducedtoalineardifferentialequationwithconstantcoefficientsbyintroducinganewvariabletsuchthat 弹性力学第四章 36 Substitutingthemintothecompatibilityequations wecanget solve Integrationtwice wecangetthegeneralsolution 弹性力学第四章 37 Ifthereisnoholeattheoriginofcoordinates constantsAandBvanish sinceotherwisethestresscomponentsbecomeinfinitewhenr 0 henceforaplatewithoutaholeattheoriginandwithnobodyforces onlyonecaseofstressdistributionsymmetricalwithrespecttotheaxismayexist namely thatwhen r constantandthattheplateisinaconditionofuniformtensionoruniformcompressioninalldirectionsinitsplane 弹性力学第四章 38 将应变代入几何方程 对应第一 二式分别积分 应变通解 SubstitutionofEqs 4 5 3 4 5 5 intophysicalEquationsyieldstheaxisymmetricalstraincomponents 4 求对应的位移 弹性力学第四章 39 分开变量 两边均应等于同一常量F 将代入第三式 弹性力学第四章 40 由两个常微分方程 Whichhasasolution 弹性力学第四章 41 其中 代入 得轴对称应力对应的位移通解 I K 为x y向的刚体平移 H 为绕o点的刚体转动角度 位移通解 弹性力学第四章 42 说明 2 在轴对称应力条件下 形变也是轴对称的 但位移不是轴对称的 3 实现轴对称应力的条件是 物体形状 体力和面力应为轴对称 1 在轴对称应力条件下 式 c d e 为应力函数 应力和位移的通解 适用于任何轴对称应力问题 说明 弹性力学第四章 43 4 轴对称应力及对应的位移的通解 d e 已满足相容方程 它们还必须满足边界条件及多连体中的位移单值条件 并由此求出其系数A B及C 说明 5 轴对称应力及位移的通解 d e 可以用于求解应力或位移边界条件下的任何轴对称问题 6 对于平面应变问题 只须将换为 弹性力学第四章 44 思考题 为什么在轴对称应力下 得出的位移是非轴对称的 如何从数学推导和物理概念上解释这种现象 弹性力学第四章 45 4 6Hollowcylindersubjectedtouniformpressures均布压力作用下的中空圆柱体 弹性力学第四章 46 Stresses应力 theconditionofsingle valueddisplacementsleadstoB 0 弹性力学第四章 47 Boundaryconditions边界条件 弹性力学第四章 48 Thisisthewell knownLame sformulasforthestressesinahollowcircularcylindersubjectedtouniformpressures Theyaretheprinciplestressesatthepoint 弹性力学第四章 49 Onlyinternalpressureacts只有内压qa 0qb 0P66 E Eqs 4 6 3 Fig 4 6 1 b b saint venant sprinciple 弹性力学第四章 50 Iftheouterradiusbismuchlargerthantheinnerradiusa thecylinderbecomesalargebodywithasmallcircularholeofradiusa sothestressesabovebecome 弹性力学第四章 51 Onlyexternalpressureacts只有外压qa 0qb 0P66 E Egs 4 6 5 Fig 4 6 1c 弹性力学第四章 52 单值条件的说明 1 多连体中的位移单值条件 实质上就是物体的连续性条件 即位移连续性条件 2 在连续体中 应力 形变和位移都应为单值 单值条件 按位移求解时 取位移为单值 求形变 几何方程 也为单值 求应力 物理方程 也为单值 弹性力学第四章 53 按应力求解时 取应力为单值 求形变 物理方程 也为单值 求位移 由几何方程积分 常常会出现多值项 所以 按应力求解时 对于多连体须要校核位移的单值条件 单值条件 对于单连体 通过校核边界条件等 位移单值条件往往已自然满足 对于多连体 应校核位移单值条件 并使之满足 弹性力学第四章 54 4 7purebendingofcurvedbeams曲梁的纯弯曲 P67 E Fig 4 7 1 弹性力学第四章 55 curvedbeams曲梁 Acurvedbeamwithaconstantnarrowrectangularcrosssection 矩形断面曲梁Acurvedbeamwithacircularaxis 圆轴曲梁Innerradiusisa outerradiusisb centralangleis 内半径a 外半径b 中心角 ThemomentofeachcoupleperunitwidthofthebeamattheendsisM 单位宽度的端弯矩为M 弹性力学第四章 56 Axisymmetricalstressesofthecurvedbeam曲梁的轴对称应力 Sincethebendingmomentisconstantalongthebeam thestressdistributionisthesameonallcrosssectionsandthesolutionoftheproblemcanthereforebeobtainedbyusingEqs 4 5 2 to 4 5 5 因为弯矩相同 因此各断面上应力分布相同而可用轴对称应力表达式 4 5 2 到 4 5 5 Alnr Br2lnr Cr2 D 4 5 2 r r r A r2 B 1 2lnr 2C 4 5 3 2 r2 A r2 B 3 2lnr 2C 4 5 4 r 0 4 5 5 弹性力学第四章 57 Boundaryconditions边界条件 1 r r a 0 r r b 0 r 0 0and r 0aresatisfiedautomatically2 r r a 0A a2 B 1 2lna 2c 0 4 7 2 r r b 0A b2 B 1 2lnb 2c 0 4 7 3 弹性力学第四章 58 Boundaryconditions边界条件 3 ab 0dr 0 ab 0dr abd2 dr2dr d dr ab r r ab b r r b a r r a 0satisfiedautomaticallyifEqs 4 7 2 4 7 3 aresatisfied 当方程 4 7 2 4 7 3 满足后 ab 0dr 0自动满足 弹性力学第四章 59 Boundaryconditions边界条件 4 ab 0rdr M ab 0rdr abrd2 dr2dr abrd d dr rd dr ab abd drdr r2 r ab ab b2 r r b a2 r r a ab ab r a r b M Alnr Br2lnr Cr2 D 4 5 2 Alna Ba2lna Ca2 D Alnb Bb2lnb Cb2 D M 4 7 6 弹性力学第四章 60 Boundaryconditions边界条件 SolvingforA B CfromEqs 4 7 2 4 7 3 4 7 6 andthensubstitutingABCintoEqs 4 5 3 to 4 5 5 weobtaintheGolovin ssolutionEqs 4 7 7 to 4 7 9 解方程 4 7 2 4 7 3 4 7 6 得ABC 然后将ABC代入方程 4 5 3 至 4 5 5 得Golovin s解答 4 7 7 至 4 7 9 弹性力学第四章 61 Analysisofthesolution解答分析 Thedistributionof rand isapproximatelyshowninFig 4 7 2应力 r和 近似地绘于图4 7 2 弹性力学第四章 62 Analysisofthesolution解答分析 mechanicsofmaterials材料力学 r 0 r 0 0Elasticity弹性力学 r 0 r 0 0theneutralaxis where 0 islocatedalittlenearersurfacethantheoutersurface 中性轴偏内mechanicsofmaterials材料力学Elasticity弹性力学Crosssectionsremainplaneafterbending 弯曲后横截面仍为平面 弹性力学第四章 63 4 9Effectofcircularholesonstressdistribution圆孔对应力分布的影响 Aphenomenonofstressconcentrationintheneighborhoodofahole anelasticbodywithoutholeissubjectedtoacertainstressdistribution Ifasmallholeismadeinsidethebody largeadditionalstressintheneighborhoodoftheholewilltakeplace Howeverthestressdistributionisalmostthesameatdistanceswhicharelargeincomparisonwiththedimensionofthehole Itiscalledthephenomenonofstressconcentration Itisofalocalizedcharacter 弹性力学第四章 64 4 9Effectofcircularholesonstressdistribution圆孔对应力分布的影响 孔边应力集中现象 无孔弹性体中有某种应力分布 在该弹性体中有一小孔后 孔边产生很大的附加应力 而离孔较远处应力基本无变化 这种现象叫孔边应力集中现象 它具有局部性 弹性力学第四章 65 A arectangularplatewithasmallcircularholeofradiusaandsubjectedtouniformtensileforceofintensityqinthexdirection 具有半径为a的小圆孔的矩形板在x方向受均布荷载q p71 E Fig 4 9 1 弹性力学第四章 66 Thestressatanypointonthecircler b awillbethesameasiftherewerenoholeatall 半径为b a的圆周上的应力同无孔时的应力 x r b q y r b 0 yx r b 0p61 E 4 4 4 to 4 4 6 r 0 5q 0 5qcos2 0 5q 0 5qcos2 r 0 5qsin2 弹性力学第四章 67 弹性力学第四章 68 r 0 5q 0 5qcos2 r 0 5qsin2 r 0 5q r 0part1 r 0 5qcos2 part2 r 0 5qsin2 弹性力学第四章 69 Part1r b r 0 5q r 0 P66Eqs 4 6 5 withqb 0 5qanda b 0becomeEqs 4 9 4 p72 弹性力学第四章 70 Part2r b r 0 5qcos2 r 0 5qsin2 Since r r r 2 r2 2 2 r2 r r r weassume f r cos2 4 r r 2 r2 2 2 r2 2 0 4 3 9 cos2 f 4 r 2 rf r 9 r2f 9 r3f 0f 4 r 2 rf r 9 r2f 9 r3f 0f Ar4 Br2 C D r2 cos2 Ar4 Br2 C D r2 p73Eqs 4 9 5 r r 弹性力学第四章 71 Part2 boundarycondition r r a 0 r r a 0 r r b 0 5qcos2 r r b 0 5qsin2 ABCD Solution part1 part2Eqs 4 9 6 弹性力学第四章 72 r a q 1 2cos2 max r a 3qindependentofa两端受拉 产生压应力 弹性力学第四章 73 B arectangularplatewithasmallcircularholeofradiusaandsubjectedtouniformtensileforceofintensityq1andq2inthexandydirectionrespectively 具有半径为a的小圆孔的矩形板在x和y方向分别受到均布荷载q1和q2作用 1 Putq q1inEqs 4 9 6 用q q1代入式 4 9 6 2 putq q2andreplace by 2inEqs 4 9 6 式 4 9 6 中q用q2代 用 2代 3 Addingtheresultstogether weobtainEqs 4 9 7 上述结果相加得本问题解答 4 9 7 弹性力学第四章 74 C Aplateofanyshapeinplanestressorstrainconditionwithasmallcircleholelocatedatsomedistanceawayfromtheboundaryissubjectedtoanyexternalforces Assumethatthereisnohole findthestresscomponentsandthenthemagnitudesanddirectionsoftheprincipalstresses 1and 2 atthepointcorrespondingtothecenterofthehole Placetheoriginofcoordinatesatthecenterofthehole withxandyaxesalong 1and 2respectively andapplyEqs 4 9 7 withq1 1andq2 2 弹性力学第四章 75 C 任意平面问题的板中有一离边界较远的半径为a的小圆孔 受到任意荷载作用 假定无洞 求出洞中心处的应力 主应力 1和 2及方向 置坐标原点于洞中心 x轴在 1方向 y轴在 2方向 将q1 1和q2 2代入式 4 9 7 得欲求的解答 弹性力学第四章 76 4 10wedgeloadedatthevertexorontheedges契形体在契顶或契边受力 P75 E Fig 4 10 1 弹性力学第四章 77 A awedgeissubjectedtoaconcentratedloadatitsvertex 契形体在契顶受集中力 因次分析dimensionalanalysisstress force length 2P force length 1r length dimensionlessstresses P r rf 弹性力学第四章 78 rf 1 substitutionof 1 into 4 r r 2 r2 2 2 r2 2 0yields1 r3 f 4 2f f 0 f 4 2f f 0f Acos Bsin Ccos Dsin r Acos Bsin Ccos Dsin r Ccos Dsin 弹性力学第四章 79 stresses r Ccos Dsin r r r 2 r2 2 2 r Dcos Csin 4 10 3 2 r2 0 4 10 4 r r r 0 4 14 5 弹性力学第四章 80 boundarycondition r 2 r Dcos Csin 0 r 0 2 0 r 2 0aresatisfied 弹性力学第四章 81 boundarycondition r 2 r Dcos Csin freebody0AB Fx 0 2 2 rcos rd Pcos 0 Fy 0 2 2 rsin rd Psin 0 弹性力学第四章 82 boundarycondition r 2 r Dcos Csin freebody0AB Fx 0 2 2 rcos rd Pcos 0 Fy 0 2 2 rsin rd Psin 0solvingforDCandsubstitutingDCintoEq 4 10 3 weobtain r 2P r cos cos sin sin sin sin 0 r 0 4 10 6 弹性力学第四章 83 specialcase1特例1 2 Fig Pyx r 2Psin r 0 r 0 弹性力学第四章 84 Specialcase2特例2 0 4 11concentratednormalloadonastraightboundary Flamant sproblem 直线边界上作用法向集中荷载 符拉芒问题 r 2P