IJRET: International Journal of Research in Engineering and Technology eissn: pissn:

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1 IJRET: Iteatioal Joual of Reseac i Egieeig ad Tecology eissn: 9-6 pissn: -708 AN AXISYMMETRIC BENDING AND SEAR STRESS ANALYSIS OF OF FUNCTIONALLY GRADED CIRCULAR PLATE BASED ON UNCONSTRAINED TIRD ORDER SEAR DEFORMATION TEORY VIA DIFFERENTIAL QUADRATURE METOD amadm.. F. Taloca Cete fo Iovatio ad Desig College of Egieeig UivesitiTeagaNasioal 4009 Selago Abstact I tis study based o te ucostaied tid ode sea defomatio teoy (UTSDT) umeical aalysis of a axisymmetic bedig ad stesses of cicula plate ae ivestigated. Te mateial popeties ae cosided to gaded toug te tickess of te veticlecoodiate ad follow a simple powe of volume factio of te costituets.goveig equatios ae deived ad DQM is used as a efficiet umeical metod fo solvig te diffeetial equatios.two types of bouday coditios ude te ifluece of te bedig ad body foce ae studied. Te validatio of te esults is doe by a compaiso wit aote study wic available i te liteatue ad foud good ageemet betwee two studies. Idex Tems:bedigseastessciculaplateUTSDTGDQM *** INTRODUCTION Tick ad ti Cicula disk i stuctued compoetsplays a majo ole i egieeig applicatios elated to tis aea is te static aalysis tetypes of plates wic aeotably cucial i tei desig agig fom automotive ailway bake systems to disks wic costitute vital compoets paticulaly i tubo macies. Fuctioallygaded mateials (FGMs) wee fist itoduced i 989 [] weeby a umbe of eseaces because iteested to study tem. I te past decade may of te studies wic caied out o te FGMs disks cocetated o te covetioal plate ad te fist ode sea defomatio teoies. Te covetioal plate teoy (CPT) fuises accuate ad eliableaalysis fo tis plate. As te disk tickess iceases CPT ove pedicts stesses espose because te tasvese sea defomatio ad otay ietia effects ae eglected.so tee a umbe of sea defomatio teoies used to aalyze modeately tick computatio effot. Futemoe te use of te (TSDT) by Reddy is costaied because it cosides te sea stess vaises o te top ad bottom sufaces of te plate but tis limitatio is solved by te ucostaiedtid ode sea defomatio teoy (TSDT) by Leuy []. I past decades seveal studies publised o te static aalysis of FGMs cicula disks Reddy et al. [] Study elates to ax symmetic bedig of fuctioally gaded cicula ad aula plates weeby te fist ode sea defomatio plate teoy was used. Ma ad Wag [4] aalyzed fute by discussig te elatiosip betwee axisymmetic bedig ad bucklig solutios of FGM cicula plates. Tid-ode plate teoy ad classical plate teoy wee demostated ad discussed i detail i tei study. I additio asymmetic flexual vibatio ad a additioal stability aalysis of FGM cicula plates was icluded i temal eviomet by usig fiite elemet teciquespeseted by Pakas ad Gaapati [5]. Also Tee-dimesioal fee vibatio of fuctioally gaded aula plates weeby bouday coditios wee diffeet usig a Cebysev-Ritz metod was also studied by Dog [6]. Malekzade et al. [7] Also sowed ow i temal eviomet i-plae fee vibatio aalysis of FGM ti-tomodeately tick deep cicula aces. Tid-ode sea defomatio teoy was used by Saidi et al. [8] To aalyze axisymmetic bedig ad bucklig of tick fuctioally gaded cicula plates. Subsequetly fout-ode sea defomatio teoy was eseaced by Saaee ad saidi [9] to study axisymmetic bedig of tick fuctioally gaded plate fist ode teoy ad tid ode teoy wee developed to icopoate te sea defomatio effects i te fist ode sea defomatio teoy (FSDT) te costat sea stess coditio toug tickesses violates te statically coditio of zeo sea stess at te fee suface. So its eed fo sea coectio facto to modify te sea foces.te tid ode sea defomatio teoy (TSDT )pedicts paabolic vaiatio of sea stess toug te tickess. Altoug te use of ige ode plate teoy leads to moe a accuate pedictio of te global espose quatities suc as sea foces deflectios stai ad stesses it equies muc Volume: 0 Issue: 07 Jul-0 ttp:// 7

2 IJRET: Iteatioal Joual of Reseac i Egieeig ad Tecology eissn: 9-6 pissn: -708 cicula plates. Besides tis Sepai et al. [0] Aalyzed te effects of big deflectio of temo-mecaical loaded aula FGM plates o oliea elastic foudatio usig te diffeetial quaatue metod. Geometically oliea postbucklig of a impefect cicula FGM plate was studied by Li et al. [] wo foud bot mecaical load ad tasvese o-uifom tempeatue ise. A study coducted by Malekzade et al. [] esulted i tee-dimesioal fee vibatio of tick fuctioally gaded aula plate i temal eviomet by diffeetial quaatue tecique. A ivestigatio of oliea aalysis of fuctioally gaded cicula plates was coducted by Nosie ad Falla [] It was elated to asymmetic tasvese loadig accodig to te fist-ode sea defomatio plate teoy based o vo Kama o-lieaity. I additio Sbulati ad Badella [4] studied tee-dimesioal elasticity solutio of fuctioally gaded tick cicula plates. Coespodigly Golmakai ad Kadkodaya [5] studied axisymmetic oliea bedig aalysis of aula fuctioally gaded plate. Te study used tid-ode sea defomatio teoy. A pecise closed fom aswe fo fee vibatio of cicula ad aula modeately tick fuctioally gaded plates of fist-ode sea defomatio teoy was studied by osseii-asemi et al. [6] Nie ad Zog s study [7] was o fequecy aalysis of multi-diectioal fuctioally gaded aula plates usig state space diffeetial quaatue metod. It was based o te tee-dimesioal teoy of elasticity. It must be oted tat diect displacemet metod was coducted by Ya et al. [8] wit te aim to epeset te axisymmetic bedig of FG cicula plates ude tasvese loads tat wee abitay. Aote teoy usig Midli s plate teoy about fee vibatio was ivestigated by Ebaimi et al [9]. Tis study was coceed about modeately tick sea defomable aula fuctioally gaded by Pilate. Te effects of couplig betwee i-plae ad out-of-plae vibatig modes of smat fuctioally gaded cicula/aula plates was examied by asemi et al. [0]. Noliea bedig ad post-bucklig of a fuctioally gaded cicula plate was examied by Ma ad Wag [] weeby te coditios wee mecaical ad temal loadig. Bayat et al [ ]used fist-ode sea defomatio teoy to study te temal elastic espose of otatig disk wit small ad lage deflectios ad peseted te esults wit aalytical solutios. ad te umeical tecique pesetig static aalysis fo te cicula disks based o ig ucostaied tid ode sea defomatio teoy is quite poo.as well as te use of te sea fuctio model wic is used by [4] ca by applied to te displacemet field of te cicula disk. I te peset studyucostaiedtid ode sea defomatio teoy is used fo axisymmetic static aalysis of fuctioally gaded clamped ad olatigcicule plate.te cicula plate is subjected to two types of loadig bedig ad body foce. Te mecaical popeties ae assumed to be gaded i te tickess diectio accodig to a simple powe law distibutio i tems of te volume factio of te costituet. By te piciple of miimum total eegy te goveig equatios of equilibium ae obtaied accodig to te ucostaied tid ode sea defomatio teoy. By employig te diffeetial quaatue metod as a simple but accuate ad fast coveget metod to discetizete equilibium equatios ad to implemet te bouday coditios. Te effects of body foce paametes te mateial costat ad te geometic paametes of cicula plate o te stesses ad deflectio espose ae is studied i detail.. PROBLEM STATEMENT Coside a FG cicula disk wit tickess( ) ad adius ( a ) axisymmetic wit espect to te z-axis as sow i fig. ad it is subjected to uifom tasvese pessue i case of clamped coditiowilei case of olle suppot coditio it subjected to bot uifom pessue ad body foce Te mid plae of te plate efes to te cyliical coodiate system ( θ z) i te adial cicumfeetial ad axial diectios espectively. Viola et al. [] Used ad ucostaied tid ode sea defomatio teoy (UTSDT) i static aalysis of modeately tick fuctioally gaded cyliical sells subjected to mecaical loadigs. Also Viola et al. [4] Employed (UTSDT) fo aalyzig tedyamicbeaviou of completely doubly cuved lamiated sells ad paels. It is clea fom te above liteatue most of te studies wic caied out o te static aalysis of te cicula disks based o te fist ode sea defomatio teoy ad few of tem based o te tid ode defomatio teoy wic is solved aalytically fo limited bouday coditios. O te ote Figue :FG cicula disk wit tickess () ad adius (a). - Mecaical Popeties OfFGCicula Plate Types Typically FGMs ae made of a mixtue of te two costituets. I tis eseac it is assumed tat te FGMs ae made of a mixtue of ceamic ad metal costituets.te mateial popeties of te FG plate vay cotiuously ad smootly i tickess diectio z ad ae fuctios of volume factio of costituet mateials Volume: 0 Issue: 07 Jul-0 ttp:// 8

3 IJRET: Iteatioal Joual of Reseac i Egieeig ad Tecology eissn: 9-6 pissn: -708 P z Pm Pc Vm + Pc ( ) Wee: P( z ) :mateial popety at locatio z toug tickess.m ad c deotes te metallic ad ceamic costituets espictivlly. Vm :volume factio of metal Wee: V m z ( ) z: tickess coodiates ( z ) p : mateial costat. As te mateial costat is equl to zeo ( p 0) ( p ) ifiity p o equal to te omogeeous isotopic mateial is obtaied as a specializ case of fuctioaly gaded mateial. i fact fom equatio () it possible to obtaia : p 0 V V 0 p z p m c m p V 0 V p z p m c c Fig. sow tat mateial pofile toug te FG plate fo vaious of p. Accodig to elutio () te elastic modulu E ad desity ρ to be vaied accodig to te above equatio. adpoisso s atio υ is assumed to be costat. E z Em EC + Vm + Ec ( ) ( z) ρ ρm ρc Vm + ρc ( 4 ). GOVERING EQUATIONS Based o te ucostaied tied ode sea defomatio teoy (UTSDT) displacemet field i te cyliical coodiate system ca be witte as : ( ) + φ + φ U z u f z g z ( ) w( ) w z ( 6 ) ( 5 ) φ :smalltasvese omal otatio about te θ -axises φ : smalltasvese omal ige ode otatio about θ - axises f ( z) g ( z) ae sea fuctios. Fom te pevious[] te displacemet field as bee impoved by takig ito cosideatio sea fuctios alog te tickess. ideed te model fo te sea fuctio i tis study as take fom pevious wok[4]. Stai- displacemet elatios: Stess-stai elatios Wee : du d d z z φ φ ε + α α z (7a) εθθ u + z α z φ α z φ (7b) ( z ) dw γ z α φ α z φ + (7c) γ 0 γ 0 ε 0 θ θ z z σ Q Q 0 ε σθ Q Q 0 εθ τ z 0 0 Q 66 γ z ( 8 ) Q Q E z Q υq ( + υ ) Q 66 E ( z) ( + υ ) Te total potetial eegy of cicula plate: Π U + V f ( 9 ) a / U π σ ε + σθεθ + τzγ z dz 0 / ( 0 ) Wee : u w ae te displacemets of poits o te middle plae (z0)i te adial ad vetical diectio espictivly. Volume: 0 Issue: 07 Jul-0 ttp:// 9

4 IJRET: Iteatioal Joual of Reseac i Egieeig ad Tecology eissn: 9-6 pissn: -708 a 0 V π ( ρ ω δ u + ρ ω δφ f αρ ω δφ αρ w δφ + qδ w) () Wee : Π U V f ae te total potetial eegy stai eegy ad te potetial eegy of te body foce ad pessue load espectively. By te piciple of miimum total eegy δ Π 0 a 0 d ( N ) + N θ ρ ω u δ + d d ( ( M ) + α ( p ) + M θ α pθ + φ α R + 0 ( αρ ρ ) ω ) δφ + d α ( p ) α p θ α R + αρ ω δφ d ( φ ) + q δ w Wee: N N θ :Stess esultats M M θ :Stess couples p p θ :ige ode stess couples φ :tasvese sea esultat () R : ige ode sea esultat ρ ρ ρ : costat popotioal to te mea fist ad tid momet of te desity alog te tickess. / ( N M P ) σ z z dz / (6a) / ( θ θ θ ) σθ ( ) N M P z z dz / (6b) / ( φ ) τ z ( ) R z dz / (6c) / ( ρ ρ ρ ) ρ ( ) z z z dz / (6d) Fom eqs. (7)(8) ad (6) oe ca obtai te followig elatios: du N A + υ u + dφ dφ ( B α E ) + υ φ α E υ φ + (7a) du Nθ A υ + u dφ dφ + ( B α E ) υ + φ α E υ φ + (7b) du M B + υ u dφ dφ + ( D α F ) + υ φ α F υ φ + (7c) du Mθ B υ + u dφ dφ + ( D αf ) υ + φ αf υ φ + (7d) du P E + υ u + dφ dφ ( F α ) + υ φ α υ φ + (7e) dw Q A44 φ + α D44 φ + φ dw R D44 φ + α F44 φ + φ (7) (7f ) Wee: A B D E F : ae te ciculae disk stiffess coefficiets Volume: 0 Issue: 07 Jul-0 ttp:// 40

5 IJRET: Iteatioal Joual of Reseac i Egieeig ad Tecology eissn: 9-6 pissn: -708 ( A B D E ) / / E z ( 4 6 ) υ / z z z z z dz E z ( 4 ) υ (8a) A44 D44 F44 z z z dz / (8b) Fom equatio () te equilibium equatios ae δ u : δφ : d ( ρ αρ ) ω δφ : δ w d ( N ) + N θ ρ ω 0 (9a) d M α P Mθ + α Pθ Q + α R + 0 : d α ( P ) α P θ α R + αρ ω 0 (9c) d Q q + 0 (9d) Equilibium equatios i tems of displacemets: (9b) ( α ) ( α ) ( α ) ( D α F + α ) ( α + α ) d φ d u du B E + B E B E u + dφ ( D α F + α ) + ( A44 6α D44 + 9α F44 ) φ F dφ α F + α + ( F ) ( D44 9 F44 ) α + α + α + α φ dw ( A44 α D44 ) + ( ρ ρ ) ω 0 (0b) d u du d φ + ( + α ) E E E u F dφ + ( F + α ) ( F α ) ( D44 9α F44 ) φ + + d φ dφ 9 44 α α + α + α E dw φ D44 + αρ ω 0 (0c) dφ α + α φ ( A D ) ( A D ) dφ d w α D44 α D44 φ + A44 + q 0 (0d) ( ) ( ) Usig te followig dimesioless paametes fo simplicity. d u du d φ ( ) ( α ) A + A A u + B E + dφ + ( B α E ) ( B α E ) d φ dφ φ α E α E + α E φ + ρ ω 0 (0a) w R W a E η 4 u U a ϕ φ δ B η F η 5 5 D η A44 η 6 6 a 4 6 A η D44 η 7 4 F44 η 8 Volume: 0 Issue: 07 Jul-0 ttp:// 4

6 IJRET: Iteatioal Joual of Reseac i Egieeig ad Tecology eissn: 9-6 pissn: -708 Q : pessue paamete 6 * q Q ζ : bodyfoec paametes 7 ρ aω ζ 8 ( ρ αρ) ω ζ α dϕ 6 7 dϕ η 4η + η6 4η 7 ϕ 4η 7R dr dr d w dw 4 η7ϕ + η7δ R + δη 6 QR dr dr R Bouday Coditios Clamped cicula plate At R 0 dw 0 U 0 ϕ 0 ϕ 0 dr Q R 0 At R (a) (d) U 0 ϕ 0 ϕ 0 W 0 (b) 4 4ρ ω ζ a te equatios of motio i dimesioless fom ae: Rolle suppot cicula plate At R 0 dw U 0 ϕ 0 ϕ 0 0 dr At R a (c) d U du 4 d ϕ 4 dϕ η δ R + η δ ηδ U + η η4r + η η4 dr dr 4 4 d ϕ 4 dϕ 4 η η4 ϕ η4r η 4 η4 ϕ R ζ + R dr dr (a) 4 d U 4 du 4 η η4δ R + η η4δ η η4δ U + dr dr 8 6 η η d ϕ 8 6 dϕ 9 R 8 6 η η5 R η η ϕ η5 9 dr 9 dr 9 ( η6 8η 7 + 6η 8) R δ d ϕ 8 6 dϕ R η η5 + ( 4η 7 + 6η 8) R ϕ ( η6 4η 7) dr 9 dr 9 R + δ dw R ζ R δ dr (b) d U du 4 d ϕ 4 dϕ η 4δ R + η 4δ η4δ U η5 R + η5 dr dr R dr dr 4 4 d ϕ 4 dϕ η 5 + η 7 η 8 R ϕ R R δ dr dr 4 dw + + η 8 R ϕ η 7 R ζ R R δ δ dr (c) W 0 N 0 M 0 P 0 (d) 4. IMPLEMENTATION OF GDQ METOD Te geealized diffeetial quaatue (DQ) metod is adopted to solve te diffeetial equatios of te aula plate. Te coe of te DQ metod is tat te deivative of a ( 0 x L) fuctio i a domai is appoximated as a weigted liea summatio of a fuctio values at all discete poits i tat domai. Tus DQ metod cages te goveig diffeetial equatios ito a set of coespodig simultaeous equatios. To demostate te DQ metod coside te t deivative of a fuctio f(x) ca be estimated as f ( x) x X Dikf x i k k i. D ik Wee xi ae te discete poits i te vaiable domai ad f( xk ) ae te weigtig coefficiet ad te fuctio value at te discete poits.tus fo te fist-ode deivatives te weigtig coefficiets ca be calculated as [5] D φ ( xi ) ik ( x x ) φ ( x ) i k k i k. i k Volume: 0 Issue: 07 Jul-0 ttp:// 4

7 IJRET: Iteatioal Joual of Reseac i Egieeig ad Tecology eissn: 9-6 pissn: -708 Wee φ ( x ) ( x x ) i i k i i k.. i k Afte tat te domai of te aula plate is divided ito gade poits i diectio. Cebysev polyomial is te best metod to evaluate te gid poits i te domai of te plate[5 ]: i i 0.5* cos π i.. Te goveig eqs. () ca be discetized accodig to te GDQ metod as follows: 4 i ( ) j ηδ ( ) j ηδ i 4 i ( ) j R η η ϕ i R a i j u + a i j u u + R a i j ( ) 4 a i j j j R i η η ϕ η η ϕ 4 Ri a ( i j 4a ( i j + 4 j Ri R i η ϕ η ϕ η ϕ ζ (a) η η4δ Ri a ( i u j + η η4δ a ( i u j η η4δ ui + Ri η η5 + Ri a ( i ϕ j + η η5 + a ( i ϕ j η η5 + + ( η6 8η 7 + 6η 8) R i ϕ j η5 Ri a ( i ϕ j 9 Ri δ η5 a ( i ϕ j + η5 + ( 4η 7 6η 8) R 9 9 i Ri δ ϕ j ( η6 4 η7 ) Ri a ( i wj ζ Ri δ (b) 4 4 η4δ Ri a ( i uj + η4δ a ( i uj η4δ ui + η5 Ri a ( i ϕj + η5 Ri a ( i ϕj η5 + ( η 7 η8 ) R ( ) ( ) i ϕj Ri a i j ϕj a i j ϕj Ri δ η 8 R i ϕj η7 R Ri a i j wj ζ Ri i δ δ (c) 6 7 i j 6 7 j ( 4 ) R a ( i ( 4 ) η η ϕ + η η ϕ ( 4 η ) ( ) ϕ 4 η ϕ η δ ( ) δη ( ) R a i j + R a i j w + a i j w QR 7 i j 7 j 6 i j 6 j (d) Te bouy coditios ca be disaetized by te DQ o: Clampled cicula plate. a 0 U ϕ ϕ ϕ ϕ W i j w 0 U 0 R (4b) Rolle suppot cicula plate. 0 U ϕ ϕ W 0 at R (5b) a i j w 0 j j at R 0 (4a) at R 0 (5a) 4 a ( u j ( ) U 4 a ( j η δ + υη δ + η η ϕ υη υη4 ϕ η4 4 a ( ϕ j υη4ϕ (5c) 0..at R 4 a ( u j ( ) U 5 a ( j η δ + υη δ + η η ϕ υη υη5 ϕ η5 4 a ( ϕ j υη5ϕ (5d) 0...at R Volume: 0 Issue: 07 Jul-0 ttp:// 4

8 IJRET: Iteatioal Joual of Reseac i Egieeig ad Tecology eissn: 9-6 pissn: a ( u j ( 4 ) U 5 a ( j η δ + υη δ + η ϕ ( ) j a j υη υ ϕ ϕ 4 υϕ 0...at R (5e) Te cogegatio of te goveig equatios ad te elated bouday coditios lead to a set of simultaeous liea algebaic equatios wic ca be wite i matix fom as: k bb ( ) ( ) kbb kbd b 0 k k q (6) db dd d k bd Wee ad ae te stiffess matices of bouday of te bouday coditios ad te size of it ae 8 8 ad espectively. k db ad k dd ae stiffess matices of goveig equatios ad ave size of ad espectively. Te vecto b cotais te displacemets coespods to te bouday poits ad is elimiated usig te static codesatio tecique. Te stiffess matix eq.6 ca be educed ito te fom of { } { } kdbkbbkbd + k dd d q (7) Fom te above equatio te vecto of domai displacemets d ca be evaluated. 5. VALIDATION OF TE RESULTS I ode to examie te accuacy ad efficiecy of te esults of tis pape a compaative study made wit aote study to implemet tis a axisymmetic bedig of a clamped ad olle-suppot fuctioally gaded cicula plate ude uifomly distibuted load q. metallic volume factio powe law distibutio toug plate tickess ad all mateial popeties ae gettig fom Reddy et al. [] As sow i tetable. Te evaluatio of te compaiso.betwee te peseted umeical aalysis ad Reddy s exact esult ae illustated i table fo dimesioless maximum deflectio 64WDc 4 qa D E c ( ν ) Wit c Wee ad a ae te tickess ad adius of te cicula plate. Te coclusio fom tese compaisos a excellet ageemet betwee tese esults. Table. Mecaical popeties of ceamic ad metal of cicula FGM plate [ ] mateial Youg s Poiso s atiov modulus( Gpa) ceamic metal Table.Compaisos of te esult got i te peset pape to te esult got by Reddy etal.[] fo maximum dimatio less diflactio of FGM cicula plate fo diffeet values of p. Mateial costat P Reddy [] Clamped plate Rolle suppot plate Peset Clamped plate Rolle suppot plate RESULTS AND DISCUSSION I ode to demostate te bedig ad stess aalysis of FG cicula plate umeically by ucostaied tid ode sea defomatio teoy via a geealized diffeetial quaatue metod two cases ae studied i tis study clamped cicula plate ad olle suppot cicula disk. 6. Clamped Cicula Plate I te below te esults ae peseted i dimesioless fom. w Fig () sows te dimesioless deflectio alog te dimesioless adius (R ) ude bedig load wit diffeet values of mateial costat P.As it expected te deflectio of te metallic plate (p0) ige ta tat of te FG plate (p>0) because of te FG plate moe stiffe ta tat of te pue metal plate. Volume: 0 Issue: 07 Jul-0 ttp:// 44

9 IJRET: Iteatioal Joual of Reseac i Egieeig ad Tecology eissn: 9-6 pissn: -708 Figue :Bedig distibutio (w/) fo clamped cicula platewit (R )fo diffeet value of p. Fig () peset te vaiety of te dimesioless sea stess * τ τ E c toug te tickess of te plate wit te z dimesioless tickess coodiate vaiable fo diffeet a values of mateial costat P ad plate geomety. It is see tat te values of sea stess decease wit iceasig i te tickess to te adius atio ( / a) As well as it is obsevable tat tee is iceasig i te sea stess wit te iceasig of te mateial costat P moeove it ca be see tat te maximum value of sea stess does ot occu i z 0 te mid plae te easo of tis case belogs to te atue of o-omogeous of te mecaical popeties of te FGMs. Figue : Sea stess distibutio toug tikess fo clamped cicula plate (a) a0.p0.5 (b) a0.p (c) a0.p0.5 (d) a0.p Volume: 0 Issue: 07 Jul-0 ttp:// 45

10 IJRET: Iteatioal Joual of Reseac i Egieeig ad Tecology eissn: 9-6 pissn: Rolle Suppot Cicula Disk Fig (4) sows te adial stess paamete body foce paamete ζ * σ E c σ wit te fo diffeet values of te mateial costatp.it is clea tat te edial stess paamete vayig liealy wit te body foce paamete moeove te adial stess i te FG disk ige ta tat of pue metallic disk ( P 0). Fig (5) illustates te vaiatio of te adial stess paamete z z 0.5 wit R 0.5 At ad.it is clea tat ude uifom distibuted load ad body foce te adial stesses fo FG plate ige ta tatofte pue metal plate ( P 0) because of te FG plate as a ige desity. Figue 5: Radial stess σ cofigalio wit (R) at 0.5 fo olle suppot cicula disk fo diffeet value of p. w Fig (6) peset te dimesioless deflectio wit ( R) of te disk ude bedig load wit diffeet values of mateial costat p.it is clea tatte deflectio of te pue metal disk (p0) ige ta tat of FG plate because of te igidty of FG plate. * Fig (7) sows te distibutio of sea stess paamete ( τ ) toug te tickess of te plate fo diffeet value of tickess to adial vaiatio ( a ) ad mateial costat P.It is clea tat te beavios of te sea stess toug te tickess of plate i tis case of olle suppot coditio simila to te case of te clamped bouday coditio. Figue 6: Radial stessσ cofigalio wit (R) at 0.5 fo olle suppot cicula disk fo diffeet value of p. Figue 4: Radial stessσ distibutio body foce (ζ) fo olle suppot cicula disk fo diffeet value of p. Volume: 0 Issue: 07 Jul-0 ttp:// 46

11 IJRET: Iteatioal Joual of Reseac i Egieeig ad Tecology eissn: 9-6 pissn: -708 Figue 8: Bedig distibutio ( ) fo olle suppot cicula plate wit (R )fodiffeet value of p. CONCLUSIONS A axismmatic bedig ad stess aalysis of fuctioally gaded cicula plate ude uifom body foce ad uifom distibuted load by ucostateed tid ode sea defomatio teoy via geealized diffeetial quaatue metod (DQM) te umeical solutio of te ucostateed tid ode sea defomatio teoy ca be applied to diffeet case of bouday coditio as well as it ca be applied to diffeet loadig coditio i cotast to te aalytical solutio limited to bedig load. Figue 7: Sea stess distibutio toug tikess fo olle suppot cicula plate (a) a 0. p 0.5 (b) a0.p (c) a0.p0.5 (d) a0.p REFERENCES [] Koizumi M.99. te cocept of FGM. Ceam Tas Fuct Gad Mate;4:0. [] Leug Ayt. A ucostaied tid ode plate teoy.computstuct 99;40: [] ReddyJ.N. WagC.M.999. Axisymmetic bedig of fuctioally gaded cicula ad aula plates. Eu.J. Mec A/solids 8: [4] Ma L.S. WagT.S004. elatiosips betwee ax symmetic bedig ad bucklig solutios of FGM cicula plates based o tid-ode plate teoy ad classical plate teoy. Iteatioal joual of solids ad stuctues 4:85-0. [5] PekesY Geepeti M. Axisymmetic flexual vibatio ad temoelastic stability of fuctioally gaded mateial cicula plate usig fiite elemet metod. Compos Pat B:Eg 008 7: [6] Dog CI. Tee dimesioal fee vibatio aalysis of FG aula plates usig te cebbysev-ritz metod. Mate Des [7] Malekzade P Atasi MM Kaami G. I-plae fee vibatio of FG cicula aces wit tempeatue-depedet Volume: 0 Issue: 07 Jul-0 ttp:// 47

12 IJRET: Iteatioal Joual of Reseac i Egieeig ad Tecology eissn: 9-6 pissn: -708 popeties ude temal eviomet. J soud vib 0096: [8] SaidiA.R Rasouli A. Saaee S. Axisymmetic bedig ad bucklig aalysis of FG cicula plates usig tid-ode sea defomatio teoy. Compos stuct 00989:0-9. [9] SaaeeS. SaidiA. R. Ax symmetic bedig aalysis of tick fuctioally gaded cicula plates usig fout-ode sea defomatio teoy. Euopea joual of mecaics A/solids : [0] Sepai O. Foouza M. R. Malekzade P. Lage deflectio aalysis of temo-mecaical loaded aula FGM plates o oliea elastic foudatio via DQM. composstuct 009: [] Li S. Zag J Zao Y. Noliea temo mecaical post-bucklig of cicula fuctioally gaded mateial plate wit geometic impefectio. Ti-wall stuct 007;45: [] MalekzadeP. Sapai S.A. Ziaee.R. Teedimesioal fee vibatio of tick fuctioally gaded aula plates i temal eviomet. J.of soud ad vibatio 00;9: [] Nosie A. Falla F. Noliea aalysis of fuctioally gaded cicula plates ude asymmetic tasvese loadig.it J o limec 009;44: [4] SbulatiR Badella L. Tee-dimesioal elastic solutios fo FG cicula plates. Eu J Mec A/solids 0;0:9-5. [5] GalmekaeEM Kedkodeya M. Axisymmetic oliea bedig aalysis of aula fuctioally gaded plate usig tid-ode sea defomatio teoy. Compos Stuct 0;9: [6] osseii-asemis Fedaee M Es agi M. A exact closed fom solutio fo fee vibatio of cicula ad aula modeately tick fuctioally gaded plates based o fistode sea defomatio teoy.it J Mecsci 00;5: [7] NieGJ Zog Z. Fequecy aalysis of multi-diectioal fuctioally gaded aula plates by usig state space diffeetial quaatue metod based o tee-dimesioal teoy of elasticity. Appl Mat Model 00;4: [8] Yu W Rogqiao aojiag D. axisymmetic bedig of FG cicula plates ude abitay tasvese loads. Compos Stuct 00;9: [9] Ebaimi E Rastgoo A Atai AA. Fee vibatio of modeately tick sea defomable aula fuctioally gaded plate by usig Midli s plate teoy.eu J mec A /Solids 009;8: [0] osseiis Azimzade M Roki. A -D Ritz solutio fo fee vibatio of cicula/aula fuctioally gaded plates itegated wit piezoelectic layes. It J of eg sciece 00;48: [] L.S Ma T.J. Wag. Noliea bedig ad postbucklig of fuctioally gaded cicula plate ude mecaical ad temal loadigs. Solids ad Stuct 00; 40:-0. [] Medi Bayat M. Saleem B.B.Sai A.M.S amuda. Temoelastic aalysis of fuctioally gaded otatig disk wit small ad lage deflectio. Ti-WalStuct 007; 45: [] Easmo Viola Luigi Rossetti Nicolas Fatuzzi. Numeical ivestigatio of fuctioally gaded cyliical sells ad paels usig te geealized ucostaied tid ode teoy coupled wit te stess ecovey. Comp stuct0 ; 94: [4] Easmo Viola Facesco Toabe Nicolas Fatuzzi.Geeal ige-ode sea defomatio teoies fo te fee vibatio aalysis of completely doubly-cuved lamiated sells ad paels. Comp stuct0 ; 95: [5] SuC.Diffeetial quaatue ad its applicatio i egieeig.loo;spige 000. BIOGRAPIES F. Talocawas bo i Malaysia. e obtaied is Bacelos i Mecaical Egieeig ad Mastes i Biomedical Egieeig fom Pudue Uivesity USA. is PD was fom Uivesiti Puta Malaysia. e is cuetly a Associate Pofesso at UNITEN ad eads te Cete fo Iovatio ad Desig. faistaloca@gmail.com amad M. was bo i Iaq e obtaied is Bacelos i Mecaical Egieeig ad Mastes i Egieeig fom IRAQ. e is cuetly pusuig is PD at UNITEN i te field of applied mecaics. Volume: 0 Issue: 07 Jul-0 ttp:// 48