Free Vibration Analysis of Thick Functionally Graded Rectangular Plates Using Variable Refined Plate Theory

Transcription

1 Jorl of el Reer d pplto SS: 5-78 ess: 5-79 Vol. o Free Vrto l of Tk Ftoll Grded Retglr Plte Ug Vrle Refed Plte Teor JR Rez lk * d Kj Reeed: 7 g. ; epted: o. trt: t pper free rto of ftoll grded (FG) retglr pl pported tk plte ed o to rle-refed plte teor preeted. ordg to poer-l dtrto te det d eltt odl of te plte re odered to r le Poo rto ott. order to etrt te fe ottte eqto of oto lto prple eploed. Te g r of t teor etgted oprg te et relt reported ger order er deforto teor. Frterore pretr td of o-deol trl freqee rred ot d te flee of geoetrl preter pet rto of te plte o tee freqee re tded. Keord: Free Vrto Ftoll Grded Retglr Plte Sll eforto Refed Plte Teor. trodto Ftoll grded terl (FG) re oel l of opote trtre e gret pplto for egeerg deg d ftre. Frt te re e geerto of opote trtre preeted grop of Jpee ett [ ]. Tee terl e derle properte for pef pplto der tre oetrto prtlrl for rrft el eletro d oter egeerg trtre. order to ole plte prole te oe of te plte teor d te tpe of olto etod re eer. Trog te td of opreee re of ltertre t fod tt reere e ee rred ot o free rto of te FG plte d ot of te epzed erl etod [-7]. Free rto of pl pported d lped retglr t FG plte preeted rte [6] ed o te ll plte teor. oter ork rte [7] lo etgted free rto klg d tt defleto of dfferet pe of FG qre rlr d ke plte t eerl odr odto o te of te ll plte teor frt order er deforto teor d trd order er deforto teor. Zo et l. [8] e preeted free rto l for FG qre d ke plte t dfferet odr odto g te eleet-free kp-rtz etod o te of te FST. Kt d llkrj [9] preeted ger-order teor for free rto l of etrll lted ltlered plte tt te trere er tr trog te tke of te plte d rotr ert effet e ee odered. tg [] preeted trl freqee d klg tree of pl pported FG plte ed o ger-order pprote plte teor. oe e et l. [] lzed e et loed-for proedre to ole free rto l of ftoll grded retglr tk plte g te trd-order er deforto plte teor (TST) for plte t to oppote pl pported edge. oe-e et l. [] lo preeted ltl etod for free rto l of odertel tk retglr FG plte ee pported eter kler or Pterk elt fodto. oe-e et l. [] tded et free rto of Reer dl FG plte t te elp of te Le-tpe olto relted to te e ere to oppote edge re pl pported d te oter edge of te plte e dfferet odr *. Correpodg tor:. S. eprtet of el Egeerg ol Uert of Teolog ol r ( k@oo.o).. S. eprtet of el Egeerg ol Uert of Teolog ol r (kj_d@oo.o)

2 Free Vrto l R. lk d. Kj odto. Seg-Eok et l. [] deeloped te to rle refed plte teor (RPT) for plte re der te to of te trere d -ple fore d oted te tffe d tre. Te opred te o-deol defleto oted ro teore el te ll lte plte teor te frt order er deforto teor te ger order er deforto teor d te refed plte teor. Seg-Eok et l. [5] lo rred ot klg l of otrop d ortotrop plte g te foregog teor. Te prpoe of t pper to deelop te RPT for FG plte. Te preet teor tfe eqlr odto t te top d otto fe of te plte tot oderg te er orreto ftor. Te eqto of oto of te plte re oted t te elp of ll of rto. To lltrte te r of te preet teor te oted relt re opred t tree deol eltt olto oted te frt-order d te ger-order teore.. forlto.. terl properte of P-FG plte flt d odertel tk ftoll retglr plte t legt dt d for tke eql to d repetel o Fg.. Te plte ed to e pl pported odre. Coderg of te ole frto of te ottet terl l [6] te og odl (z) ρ z of E d det FG plte e olded fto of tke oordte z elo:.. Refed plte teor retglr plte t legt dt d tke eql to d repetel odered. order to proeed t te forlto of te prole g te to-rle refed plte teor (RPT) t ed tt te dpleet ( ) of te plte re ll opro t te tke of te plte; ee te tr oled re odered to e ftel. O te oter d te trere orl tre te z- dreto σ z er ll opro t te ple tree σ d σ. So t oe defto te tre tr relto e reded fro 6 6 tr to 5 5 tr rede te oplet of te prole. Te totl dpleet of te plte te z-dreto () ed to e otg of tree opoet of eteo edg d er ter re fto of d te te. Te dpleet te d z- dreto re lo defed []: U( zt ) V( zt ) ( t ) ( t ) z 5 ( t ) z z ( zt ) ( t ) ( t ) z 5 z z ( t ) () ( t ) ( t ) ( t ) E ( z) E ( E E ) z p () ρ ( z) ρ ( ρ ρ ) z p Te rpt d o te etll d er ottet repetel. p te poer l de tke pote le. Te rto of Yog odl te P-FG plte o Fg.. Oter tpe of FG terl S-FG d E- FG e fod [7]. Fg.. C odel of FG retglr plte.

3 Jorl of el Reer d pplto (JR) Vol. o Yog odl P. P. P.5 P P P5 P z/ Fg.. Vrto of Yog odl P-FG plte. ere d re te d-ple dpleet d t te te. Te tr dpleet relto re ge : U V U V U V z z () z z ed o ooke l te tre dpleet relto re defed : σ z E( z ) σ ν ) [ ν ] σ E( z [ ν ] E( z ) σ z z ( ν ) ( ν ) ν. Goerg eqto E( z ) z σ E( z ) ( ν ) () t e erfed tt te frt rto of te Lgrg L T U (.e. lto prple) led to te eqto of oto. t td U deote te totl tr eerg d T te ket eerg of te plte e rtte : δ T U plte plte ( ρ ( z)( U V ) dddz) dt t / t / σ δ σ δ σ δ dddzdt σ zδ z σ zδ z (5) Stttg Eq. () d () to Eq. (5) d olletg te oeffet of δ δ δ δ d δ te follog eqto of oto for FG plte re ge []: δ (6) δ δ δ δ Q z Q z Q Q z z 8 ere re te ert ter d defe elo: ( ) ( z ) ( z ) dz ρ (7) ere dot-opertor oeto o te dfferetto t repet to te te. Te tre reltt Q d Q re ge []: ( ) ( σ σ ) ( ) ( σ σ σ ) ( ) ( σ σ σ ) ( Q Q ) ( σ σ ) z z ( Q Q ) ( σ σ ) z z z σ dz (8) z z z dz 5 5z zdz z 5z ere d Q re te tre reltt d defed te pped. ertg Eq. () to Eq. (6) te goerg eqto of FG plte e rtte : ( ) ( ) ( ) dz dz (9)

4 Free Vrto l R. lk d. Kj odr odto ere odr odto for FG plte t pl pported odr t ll edge re odered : For () For To ole t prole t te ge odr odto er olto proedre eploed for tfg te follog epo: o V o U () ere π π V U d re oeffet. Fll te Egle prole for rto of te plte oted : [ ] [ ] { } { } λ ω K () ere ] [ K ] [ ω d λ re te tffe tre trl freqe d te etor of ko oeffet repetel. For oeee te fdetl freqe preter Ω d te o-deol trl freqe preter Ωˆ of te plte re ge repetel []: E ρ ω Ω E ˆ ρ ω Ω () 5. erl relt d do T eto lltrte te effe d tlt of propoed teor. Ug te preol deeloped ltl olto ed o te FST d ST erl pretr tde ed o RPT re preeted. t td te FG plte tke to e de of to tpe of terl (.e. l O / l d ZrO / l ) te terl properte re Tle. order to etgte te effe d r of te preeted teor te relt re opred t otrop FG plte. ere for geerlt d oeee to o-deol ter tke rto τ d pet rto η re defed d repetel. tter of ft e te poer l de (p) pproe to zero te plte otrop fll er d e t de pproe to te plte otrop d fll etll. To ee free rto of te plte te le of -ple dtrted lod re et to zero. Tle o te opro of te fdetl freqe preter for te oogeo otrop l O / l qre plte for. τ d. τ. Te relt re opred t et ger order er deforto teor (ST) [8] o er good greeet. Tle o te opro of te o-deol trl freqe preter Ωˆ for l O / l qre plte t toe relt oted ed o ST [ 9] (Et olto) d FST [] for 5. p d. T tle o tt tere er good greeet etee te relt of RPT d toe of et ltl olto g ger order er deforto plte teor [9]. t Tle d re te

5 Jorl of el Reer d pplto (JR) Vol. o orrepodg ode pe re defed er of lf e X d Y-dreto repetel. Te frt tree trl freqe preter for dfferet le of te poer l de (.e. p. 5 5 ) re oted d te erl relt re opred t et loed for olto [] Tle ere te odered plte de of l / l o. Coderg te oted relt ll trl freqee deree e te poer l de ree. Te reo te ft tt te trl freqe preter of FG plte re dretl depedet o Yog odl. Terefore reg te poer l de te elt odl of FG plte deree. oreoer e ee t tle ll te trl freqe preter re t ree te tke rto of te plte. T dotedl de to reg te rgdt of te plte. depted Tle 5 te relt of odeol trl freqe ( Ωˆ ) for SSSS l / ZrO qre t d odertel tk plte re opred t toe oted ST [ 9] teor eplog te poer ere etod [] d fte eleet ST etod []. Te flee of te pet rto o te fdetl freqe preter ( Ω ) for ott le of tke rto ( τ.) d poer l de ( ) p of retglr plte l / Zro preeted Tle 6. t oered tt reg te pet rto te fdetl freqe preter deree. Tle. terl properte of te ed FG plte terl Properte E ( GP) ν ρ ( kg ) l ( l ) 7. 7 l ( l O ) 8. 8 Zro ZrO. 57 Tle. Copro of fdetl freqe preter ( Ω ) for terl Fll er Fll etll Tle. Coprte relt of te o-deol trl freqe τ ( ) etod.5 ().. () () () () () () l/l O qre plte ( η ) etod τ.. Propoed ST ( p ) 6 [] ST[8] Propoed p [] ST ( ) ST[8] Ωˆ for l/l O qre plte ( η ) Poer l de (p).5 Propoed Et ST[] FST[7] Propoed Et ST[] ST[9] FST[7] Propoed Et ST[] ST[9] FST[7] Propoed Et ST[] ST[9] FST[7] Propoed Et ST[] ST[9] FST[7] Propoed Et ST[] ST[9] Propoed Et ST[] ST[9]

6 Free Vrto l R. lk d. Kj 7 Tle. Frt tree trl freqe preter ( Ω ) for l/l O retglr plte ( η ) Poer l de (p) τ ( ) etod.5 5 () Propoed Et FST[] () Propoed Et FST[] () Propoed Et FST[] () Propoed Et FST[] () Propoed Et FST[] () Propoed Et FST[] () Propoed Et FST[] () Propoed Et FST[] () Propoed Et FST[] o-deol frt trl frqe /.5 // / /.5 / / Fg.. Vrto of Yog odl retglr P-FG plte. Egefreqe preter /.(*) /.5(*) /.(*) /.(**) /.5(**) /.(**).5 l - - Poer de (p) Fg.. Vrto of te ege-freqe preter ( ) er te p for te FG qre plte ( η ) l/l O (*) l/zro (**). etod Tle 5. Coprte relt of te o-deol trl freqe ( Ωˆ ) for l/zro qre plte p p τ. τ τ. τ. 5 τ. τ. p p p 5 Propoed ST[] ST[9] [] FST[] Tle 6. Vrto of fdetl freqee t dfferet le of pet rto for τ. ( p ) l/zro ( ) etod η.5 /.5 Propoed Et ST[] Error (%) Te flee of tke rto o free rto of te retglr plte t ro pet rto o Fg.. epeted reg te pet rto led to dereg te tffe of te plte d oeqetl dereg te o-deol frt trl freqe. ere o-deol preter ( ω ρ E ) Ω lled te ege-freqe preter

7 Jorl of el Reer d pplto (JR) Vol. o defed ρ d E re odered det d Yog odl of te d-ple ( z ) repetel. T preter defed [] d t o pel rtert. Fg. o te rto of poer l de ( p to ) o te ege-freqe preter ( ω ρ E ) for FG retglr plte t dfferet tke rto. Frt tree egefreqee re for l/l O d te et tree oe re for l/zro. t e ee tt te poer l de gret effet o Ω. tter of ft tot oderg te terl Ω te d le t te pot rod ( p ) d ( p 8 ) repetel. Te l/l O FG plte terl ore flee t l/zro o te ege-freqe preter. etoed erler te reo tt te freqe preter re depedt o eltt odl. Terefore reg te poer l de te elt odl of FG plte deree de to reg te rgdt of te plte. 6. Colo Free rto l of tk ftoll grded retglr plte ddreed. to- rle refed plte teor deeloped for ftoll grded terl plte. Te teor ge prol dtrto of te trere er tr d o er orreto ftor eeded. Te erl relt e proed te effe d r of t teor. Te relt ere opred t te et olto oted ger order er deforto plte teor. Te effet of dfferet pet rto d tke rto o free rto of FG retglr plte t to dfferet terl ere lo etgted. Te olo re lted : - Te relt of RPT re er good greeet t et ltl olto g ger order er deforto plte teor. - ll trl freqee deree e te poer l de ree. - ee of reg te rgdt of te FG plte ll te trl freqe preter re t ree te tke rto of te plte. - reg te pet rto te fdetl freqe preter deree. 5- t oo tt te poer l de effete ftor o ege-freqe preter. 6- Te l/l O FG plte terl ore effet o te ege-freqe preter opro t l/zro FG plte. oeltre j j pped Q Q Q Q S. z z z z pleet ter eteo te z- dreto pleet ter edg te z-dreto pleet ter er te z-dreto Stre te lol oordte Str te lol oordte Poo' rto S ( ) ν 6 6 ν ( ) ν 6 6 ν ( ) ν 6 6 ν ()

8 Free Vrto l R. lk d. Kj 7 5E ν ( ) ν 6 6 5F ν ( ) ν F ν ( ) ν ν ( ) ( ) Fν ( ) ( ) 5 6 ν ( ) ( ) 5 ( EF ) Referee / / E ( z ) 6 ( z z z z z ) ν () [] Yo.; Koz.; r T.; Sot. Proeedg of frt tertol po o ftoll grdet terl Sed Jp; 99. [] Koz.; Te oept of FG. Cer Tr Ft Grdet ter 99. [] Redd J..; P.. "Stlt d rto of otrop ortotrop d lted plte ordg to ger-order er deforto teor" Sod d Vrto []. F.; Le.. " ger order er deforto teor for te rto of tk plte" Sod d Vrto [5] Zo.; Ceg Y. K.; F. T. K.; Lo S.. "Treedeol rto l of tk retglr plte g Cee polol d Rtz etod" Sold d Strtre [6] rte S. "Ftoll grded plte ee lke oogeeo plte" Copote Prt-Egeerg [7] rte S. "Free rto klg d tt defleto of ftoll grded plte" Copote See d Teolog [8] Zo X.; Lee Y. Y.; Le K.. "Free rto l of ftoll grded plte g te eleetfree kp-rtz etod" Sod d Vrto [9] Kt T.; krj. " ger-order teor for free rto of -etrll lted opote d d plte-fte eleet elto". Copter d Strtre [] tg. "Free rto d tlt of ftoll grded plte ordg to - gerorder deforto teor". Copote Strtre [] oe-e S.; Fdee.; tpor S. R. "Std o te free rto of tk ftoll grded retglr plte ordg to e et loed- for proedre" Copote Strtre pre do:.6/ j.je.... [] oe-e S.; Rok d Ter.; k.; Od. "Free rto of ftoll grded retglr plte g frt-order er deforto plte teor". ppled tetl odellg [] oe-e S.; Fdee.; tpor S. R. " e et ltl ppro for free rto of Reer dl ftoll grded retglr plte" el See 5. [] Seg-Eok K.; -T T.; Jeog L. " to rle refed plte teor for lted opote plte" Copote Strtre [5] Seg-Eok K.; -T T.; Lee J. "klg l of plte g te to rle refed plte teor" T lled Strtre [6] o G.; g L. "ltple rkg ftoll grded er/etl otg" Sold d Strtre [7] Sg-o C.; Ye-Lg C. "el eor of ftoll grded terl plte der trere lod-prt : l" Sold d Strtre [8] Sfr.; Eeerger. "Stlt d rto of er deforle plte frt order d ger order

9 Jorl of el Reer d pplto (JR) Vol. o l" Sold Strtre [9] tg. "Free rto d tlt of ftoll grded plte ordg to - gerorder deforto teor" Copote Strtre [] Zo X.; Lee Y. Y.; Le K.. "Free rto l of ftoll grded plte g te eleet-free kp- Rtz etod" Sod Vrto [] Vel S. S.; tr R. C. "Tree-deol et olto for te rto of ftoll grded retglr plte" Sod Vrto [] Prd S.; dopd J. "Free rto l of ftoll grded red pel g ger-order fte eleet forlto" Sod Vrto