弹性力学lecture (11)

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1、Chapter4SolutionofPlaneProblemsinPolarCoordinatesӰ໿ॏ༰Å4-1&Å4-2DifferentialEquationsofEquilibriumTheoryofElasticityGeometricalEquationsPhysicalEquations㄀क䆆Å4-3&Å4-4TransformationofstresscomponentsStressfunctionandcompatibilityequation2009ń2010༰ભóԛ༰୙৴ںඁ2010ᑈ4᳜26᮹ᔍᗻ࡯ᄺTheoryofEl

2、asticity1of31ܟ઒सٝӖ༰ිઁူ߬ඹ༰ၝිઁٝёຂBasicEquationsinPolarCoordinatesStresstransformationbetweenrectangularandpolarsystems(1)Equationsofequilibrium(3)Physicalequations{}sstxyx,,yΩ{}sstr,,jrjArotationjfrom(x,y)to(r,j)ïï-strj1rsrsjes=-1()msssjsjtjj=++cos22sin2sincos+++fr=0rErjPlanes

3、tressrxyxyïïrrjr221ïïstt21ssjsjtjjj=+-xyxsincos2ysincosjr+++=jrjf0es=-()msjjjrrjïïrrEtssjj=-()sincos+tjj()cos22-sinrjyxxy21()+mg=t(2)GeometricalequationsrjrEjïur{}sstrjr,,jΩ{}sstxyx,,yArotation-jfrom(r,j)to(x,y)er=ïr1-mm2⊬·Planestraines=-∆¸srrj22uuïE«¹1-mss=+-cosjsjtjjsin2si

4、ncosrj1xrjrje=+222j1-mm⊬·ssjs=++sincosjtjj2sincosrrjïej=-∆¸ssjryrjrjE«¹1-m1ïïuuurjjtssjj=-()sincos+t()cos22jj-sing=+-21()+mxyrjrjrjrïïjrrgtrj=rjEᔍᗻ࡯ᄺTheoryofElasticity2of31ܟ઒सٝӖ༰ිઁူ߬ඹ༰ၝිઁٝёຂᔍᗻ࡯ᄺTheoryofElasticity3of31ܟ઒सٝӖ༰ිઁူ߬ඹ༰ၝිઁٝёຂWithchainruleInrectangularcoordinates,th

5、estressfunctionisused½ïïﻺrj½ïïïrjjsin==cos,j-Stressesareexpressedas°°°°ïïxx…»ïx°°°°ïrïïxxr2®¾=…»®¾ïFïïï…»rjïïïrjjcoss=xx2-fx°°°°==sin,jïyEquilibrium°°¯¿ïïyy…»¬¼ïy°°¯¿ïjïïyyr2ïFequationsares=-fyyy2ïxsatisfiedUsestress2automaticallyïFt=-xyfunctionin222222ïïxyïF2ïF2sincosjjïF

6、sinjïF2sincosjjïFsinjïF=-cosj+++polar22222ïïxrrrïïïjrrrjïïrjFmustsatisfycompatibilityequationcoordinates222222ïF2ïF2sincosjjïFcosjïF2sincosjjïFcosjïF=+sinj+-+422222∞F=0ïïyrrrïïïjrrrjrjïï22222222orïFïFcosjj-sinïFsincosjjïFcosjj-sinïFsincosjjïF=+sincosjj---2222222ïïxyïrrrïïjrr

7、rjrjïïï⊬·ïï∆¸+F=022«¹ïïxyᔍᗻ࡯ᄺTheoryofElasticity4of31ܟ઒सٝӖ༰ිઁူ߬ඹ༰ၝිઁٝёຂᔍᗻ࡯ᄺTheoryofElasticity5of31ܟ઒सٝӖ༰ිઁူ߬ඹ༰ၝිઁٝёຂStresscomponentsÅ4-5䕈ᇍ⿄ᑨ࡯੠Ⳍᑨⱘԡ⿏211ïFïFs=+rrïïrrj22ᅀճыġ2ᄗ๞඘ԅ໹ᅱݧ੶๞स२ಾïFsj=2ఁྡྷᅀճыԅŃඹڶճыïrᅀԅఉੋ՛ಾճыੋÇïï⊬·1Ft=-∆¸rjïïrrj«¹Compatibilityequation222⊬·ïïï11∆¸++F=0222«¹ï

8、ïïrrrrjᔍᗻ࡯ᄺTheoryofElasticity6of31ܟ઒सٝӖ༰ිઁူ߬ඹ༰ၝිઁٝёຂᔍᗻ࡯ᄺTheoryofElasticity7