英文弹性力学课件

2 5 3STRESSONOBLIQUEPLANETHROUGHAPOINT Nowitispossibletodeterminethestressesonanyotherobliqueplanepassingthroughthepointusingthedirectioncosines Nowitispossibletodeterminethestressesonanyotherobliqueplanepassingthroughthepointusingthedirectioncosines Inpractice thestateofstressatapointwithreferencetothegeneralCartesianco ordinatesystemisnotverysignificant Becauseofthat Thefailureofastructureorabodyisduetofractureorplasticyielding Thefailuremayoccuronadifferentplane Thefailureplaneisalwaysinclinedtothethreeco ordinateaxes Therefore inthissectionthemethodofdeterminingthestressesonanobliqueplanewillbedescribed ConsideringtheequilibriumofthetetrahedronOABCinFig 2 8 Letsx txy txzbestressesonplaneOBCLetsy tyx tyzbestressesonplaneOACLetsz tzx tzybestressesonplaneOABandsnx sny snzbestressesonplaneABC 2 24 wheredAistheareaofABC DividingEq 2 24 bydA 2 25 Fig 2 8 Similarexpressionsforsnyandsnzcanbeobtained Expressingtheserelationsinmatrixform 2 26 2 27 Eq 2 26 aregivestheCartesiancomponentsofstressonaninclinedplane Toobtainnormalandshearstressesonthisinclinedplane considerasetofaxesx y z Letx axiscoincidewiththenormalntotheplaneABCinFig 2 9 FIG 2 9 Thedirectioncosinescanbewrittenasfollows Usingthesedirectioncosines 2 29 Substitutingforsnx snyandsnzfromEq 2 26 2 30 Similarly 2 31 or 2 32 and 2 34 EXAMPLE2 2 Thestateofstressatapointisgivenasfollows sx 800kPa sy 1200kPa sz 400kPa txy 400kPa tyz 600kPaandtzx 500kPa Determine a thestressesonaplanewhosenormalhasdirectioncosinesl1 1 4andl2 1 2 and b thenormalandshearingstressesonthatplane SOLLUTION Fromtherelation get fromEq 2 26 469644 3 2 5 3STRESSTRANSFORMATION Rotatingtheoriginalaxesforstresses toaxesx y z forstresses thetransformationcanbeobtainedintermsoftheoriginalstresscomponentsandthedirectioncosinesas EXAMPLE2 3 Letthestateofstressatapointbegivenbysx 1000kPa sy 600kPa sz 400kPa txy 800kPa tyz tzx 0 Consideranothersetofco ordinateaxesx y z inwhichz coincideswithzandx isrotatedby30oanticlockwisefromxaxis Determinethestresscomponentsinthenewco ordinatesystem Solution Thesedirectioncosinescanbewritteninthematrixformas Thestateofstressinxyzcanbewrittenas Thus 2 6PRINCIPALSTRESSES Theangleofinclinationofaplanetotheco ordinateaxesdeterminesthevaluesofstresscomponentsonthatinclinedplane Findthemaximumnormalandshearstressesinordertodeterminethesafetyagainstfailureofastructure Inengineeringapplicationsthemajorinterestisto Thereforethemostusefulwayofdescribingastateofstressatapointistoobtainuniquelythelimitingvaluesofnormalandshearstresses UsingEq 2 36 itispossibletoobtainasetofdirectioncosineswhichwillreduceallshearstressestozeroandprovidelimitingvaluesofnormalstresses 2 36 Thenthesedirectioncosineswilldefinethreemutuallyperpendicularplanescalledprincipalplanes Thecorrespondingnormalstressesareknownastheprincipalstresses 2 37 wheres1 s2ands3arethreeprincipalstresses FromEq 2 26 get 2 38 2 39 InordertoobtainanontrivialsolutiontothethreehomogeneouslinearequationsgiveninEq 2 39 thedeterminantofthecoefficientsofthedirectioncosinesshouldbezero 2 40 InEq 2 38 ifwesubstitutevaluesforsnx snyandsnzfromEq 2 26 then ExpandingEq 2 40 toget 2 42 ellipsoid Where 2 43 Therootsofthecubicequation 2 42 s1 s2 s3 arenamedtheprincipalstresses 2 7STRESSINVARIANTS ThecoefficientsI1 I2andI3mustbeinvariant ThethreecoefficientsI1 I2andI3areknownastheFirst SecondandThirdInvariantofStress Fortheparticularcase wherethedirectionsofco ordinateaxescoincidewithprincipalstressdirections sincetheshearstressesonprincipalplanesarezero thestressinvariantreduceto 2 45 2 8DEVIATORICSTRESSES Incertaincases especiallyinthetheoryofplasticity itisoftenconvenienttodividethestressstateintotwocomponents thehydrostaticorvolumetriccomponentthedeviatoricordistortionalcomponent Duringplasticdeformation thechangeinvolumesisusuallyconsideredtobenegligiblehenceonlythedeviatoricstresscomponentsissignificant Thehydrostaticstresscanbewrittenas 2 55 Thusthedeviatoricstress s isgivenby s s I1 3 2 56 Forinstance Similarlysy andsz canbewritten However txy txy tyz tyz tzx tzxNamely 2 57 2 58 TheinvariantofdeviatoricstressesaredenotedbyJ1 J2 andJ3 2 59 or Nowusingthedeviatoricstress theequationforprincipalstressescanbewrittenas and and Then SolutionforEllipsoidEquation 2 42 Mosttime wehavetogetthecertainsolutionofprincipalstressfrom 2 42 Lets s I1 3ors s I1 3 2 46 SubstitutingthisvalueinEq 2 42 or s 3 J2s J3 0 2 47 Let 2 50 Then substitutingEq 2 50 intoEq 2 47 2 51 DividingEq 2 51 byr3 2 52 2 52 Eq 2 52 issimilartothetrigonometricallyexpression 2 53 Ifwelet Thus 2 54 Substitutingthevaluesofr andqbackintoEq 2 50 2 50 s canbeobtainedandinturnsfromEq 2 46 2 46 EXAMPLE2 4 Thestateofstressatapointisgivenbysx 100kPasy 200kPa sz 100kPa txy 200kPa tyz 100kPaandtzx 300kPa Determine a thestressinvariant b theinvariantofdeviatoricstresses c theprincipalstresses d thedirectioncosinesoftheprincipalplanes 2 9MAXMUMSHEARSTRESSES Similartothelimitingnormalstresses theshearstressesalsohavelimitingvalues Theymaybedeterminedinasimilarmannertothatofprincipalstresses Let snx snyandsnzbethethreestresscomponentsontheplaneABCwhosenormalisn sRbetheresultantofthesethreestresses snbethenormalstressontheplanealongntnbetheshearstressontheplanealongn asshowninFig 2 10 FIG2 10 Whentheaxesx yandzaretheprincipalaxes wehave 2 60 ThenormalstresssncanbeexpressedusingEq 2 60 as 2 62 Sincetheshearstressesonprincipalplanesarezero theresultantstresssRcanbewrittenas 2 61 InEq 2 64 thevalueofl32canbesubstitutedas 1 l12 l22 sinceonlytwoofthethreedirectioncosines