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1、I9Lateraldeflectionsofcircularplates19.1IntroductionInthischapter,considerationwillbemadeofthreeclassesofplateproblem,namely(i)smalldeflectionsofplates,wherethemaximumdeflectiondoesnotexceedhalftheplatethickness,andthedeflectionsaremainlyduetotheeffectsofflex
2、ure;(ii)largedeflectionsofplates,wherethemaximumdeflectionexceedshalftheplatethickness,andmembraneeffectsbecomesignificant;and(iii)verythickplates,wheresheardeflectionsaresignificant,Platestakemanyandvariousformsfromcircularplatestorectangularones,andfromplat
3、esonships'deckstoonesofarbitraryshapewithcut-outsetc;however,inthischapter,considerationswillbemademostlyofthesmalldeflectionsofcircularplates.19.2Platedifferentialequation,basedonsmalldeflectionelastictheoryLet,wbetheout-of-planedeflectionatanyradiusr,sothat
4、,andd2w-de---dr'drAlsoletR,=tangentialorcircumferentialradiusofcurvatureatr=AC(seeFigure19.1).R,=radialormeridionalradiusofcurvatureatr=BC.Platedifferentialequation,basedonsmalldeflectionelastictheory459Figure19.1Deflectedformofacircularplate.Fromstandardsmal
5、ldeflectiontheoryofbeams(seeChapter13)itisevidentthatRr=1i”-=l/dedr(19.1)dr2or---1-&(19.2)RrdrFromFigure19.1itcanbeseenthatR,=AC=rl8(19.3)or-----1-lh-8(19.4)R,rdrrLetz=thedistanceofanyfibreontheplatefromitsneutralaxis,sothat=-1E~=radialstrain=--E(or-YO,)(19.5
6、)Rr460LateraldeflectionsofcircularplatesandE,=circumferentialstrain=-z1=-(ai-vor)(19.6)R,EFromequations(19.1)to(19.6)itcanbeshownthat(19.7)(19.8)where,a,=radialstressduetobendinga,=circumferentialstressduetobendingThetangentialofcircumferentialbendingmomentpe
7、runitradiallengthisEdez3+‘I*=-(1-v’)(:+7)[TI-,,’--Et312(1-v’)therefored’wMI=D(:+v$)=.I--+.-)1dw(19.9)rdrdr’where,t=platehcknessPlatedifferentialequation,basedonsmalldeflectionelastictheory461andEt3D==flexuralrigidity12(1-Y’)Sdarly,theradialbendingmomentperuni
8、tcircumferentiallength,(19.10)Mr=D($+:)=D[&+&)dr’rdrSubstitutingequation(19.9)and(19.10)intoequations(19.7)and(19.8),thebendingstressescouldbeputinthefollowingform:(s,=12M,xzIt’and(J=12~,
