Limit Analysis of FGM Circular Plates Subjected to Arbitrary Rotational Symmetric Loads

Transcription

1 Wold Academy o Sciece, Egieeig ad Techology 36 7 Limi Aalysis o FGM Cicla Plaes Sbjeced o Abiay oaioal Symmeic Loads Kagaovi M.H., Faghidia S. A ad Aghavai J. Absac The limi load cayig capaciy o cioally gaded maeials (FGM) cicla plaes sbjeced o a abiay oaioally symmeic loadig has bee comped. I is povided ha he plae maeial behaves igid peecly plasic ad obeys eihe he Sqae o he Tesca yield cieio. To his ed he ppe ad lowe bod piciples o limi aalysis ae employed o deemie he exac vale o he limiig load. The coecess o he esl ae veiied ad ially limiig loads o wo examples amely; hogh adis ad hogh hickess FGM cicla plaes wih simply sppoed edges ae calclaed, especively ad moeove, he vales o ciical loadig aco ae deemied. Keywods Cicla plae, FGM cicla plae, Limi aalysis, Lowe ad Uppe bod heoems. I. INTODUCTION TUDYING he behavio o sces i he plasic age Sis o gea pacical impoace. Limi aalysis heoems, as powel devices, have bee mosly applied o he case o igid-plasic maeials ad led o he desig o moe eicie sces. Tadiioal elasic desig cao, i geeal, pese ealisic esimaes o he e load cayig capaciy. As a esl, sces ha have bee maaced accodig o elasic desig piciples ae elaively heavy ad expesive. Applicaio o he limi aalysis appoach esls i he desig o lighe sces ha hose obaied hogh elasic desig. Sch sces ae eqely ilized, especially i aeospace aspoaio devices. Limi aalysis is based o lowe ad ppe bod heoems, which esls i coespodig esimaes o he ciical vale o he load (he limi load o he sce). I hese wo bods coicide, he exac solio o he poblem ad/o e limi load o he sce is obaied. Howeve, i has o be emphasized ha he exac solio o he descibed poblem shold saisy all o he ollowigs: (a) kiemaic admissibiliy o he deomaio ield, (b) saic admissibiliy o he sess ield, Mascip eceived Oc. 5, 7. S. A. Faghidia, is a PhD sde ( aghidia@mech.shai.ed) M.H.Kagaovi, is wih Mechaical Egieeig Depame, Shai Uivesiy o Techology, Azadi Ave. P.O.Box , Teha, I.. Ia (coespodig aho phoe: (+98) ; ax: (+98) 66; e- mail: mhkaga@shai.ed). ad (c) saisacio o he yield cieio, all ove he plae. Limi aalysis heoems, ad some o hei applicaios i he scal aalysis, wee is iodced i he 95 s. Amog he mos dameal woks o his sbjec, he coibios o Page [], Hoe [], Masield [3], ad Hodge [4] ae emakable. Some o he mos oable woks o he sbjec have bee eviewed exesively i Save ad Massoe [5] ad Soboka [6]. I a aemp o aalyse he plasic behavio o plaes sbjeced o some geeal kid o loadig, Ghoashi [7] cosideed cosa hickess cicla plaes obeyig eihe he Sqae o Tesca yield cieio ad obaied exac solios o he case o oaioally symmeic load. Also Ghoashi e al. [8] cosideed he vaiable hickess plaes which ae sed exesively. Fhemoe, i aohe aemp, Gowei [9] has deived he limi load o ohoopic cicla plaes. B so a hee is o epoed wok o he limi aalysis o ohe ypes o maeial sch as cioally gaded maeials o FGMs. I he pese pape, he load cayig capaciy o FGM cicla plaes de abiay oaioal symmeic loadigs has bee cosideed. I his way, he simple sppoed cicla plae is cosideed o be hogh adis FGM as well as hogh hickess FGM. The ppe ad lowe bod heoems o limi aalysis ae ilized ad he ciical loadig aco is comped. Ulimaely he ocome o he pape is illsaed hogh seveal case sdies. II. DESCIPTION OF FGMS POPETIES Advaces i maeial syhesis echologies have sped he developme o a ew class o maeials, called cioally gaded maeials (FGMs), wih pomisig applicaios i aeospace, aspoaio, eegy, elecoics ad biomedical egieeig [-]. I applicaios ivolvig sevee hemal gadies (e.g. hemal poecio sysems), FGMs exploi he hea, oxidaio ad coosio esisace ypical o ceamics, ad he segh, dciliy ad oghess ypical o meals. I a ypical FGM, he volme acios o he cosies ae vaied gadally ove a maco scale geomeical dimesio sch as coaig hickess. Wihi a FGM, he diee phases have diee cios o si he eeds o he maeial. The maeial gadies idced by he 33

2 Wold Academy o Sciece, Egieeig ad Techology 36 7 spaial vaiaios o he maeial popeies make FGMs behave dieely om commo homogeeos, isoopic maeials ad adiioal composies. Plasic limi aalysis o hese kids o FGM plaes shold be pdaed based o he yield popeies o he plaes. I o aalysis, o a simply sppoed FGM plae i is assmed ha he composiioal vaiaio o ceamic ad meal phases ca be appoximaed by a idealized powe-law eqaio as Eq. () o he hogh adis FGM V () ad as Eq. () o hogh hickess FGM plaes: z V () Hee, V epeses he volme acio o he ceamic phase, z is he locaio meased om he mid sace, is he oal hickess o he FGM laye, is he adial coodiae, is he adis o he plae ad is he powe expoe (see Fig. ()). Noe ha, a ay locaio, V V, whee V is he volme acio o he meal phase. I o model, he composiio is pe ceamic a he sppo edge (o hogh adis FGM) ad a he saces (o hogh hickess FGM) while i is pe meal a he cee (o hogh adis FGM) ad a he mid sace (o hogh hickess FGM). Whe =, he composiio chages liealy hogh he hickess o adis, while / o coespods o he qadaic o paabolic disibio. Sice we eed o speciy a likely age o he kow vaiable i he ivese aalysis, i is assmed ha ages om /3 o 3. Ay vale oside his age is o sally desied sice sch a FGM wold coai oo mch o oe phase. (Fo example whe = /3 o 3, oe phase has 75% o he oal volme.) I he case o hogh hickess, whe he lowe sace is meal ad ppe sace is ceamic ad he Eq. () shold be modiied ad ca be expessible as z V, b he simila aalysis shold be doe. I he ce aalysis, he so-called modiied le o mixes descibed i [3-6] is adoped. I he composie is eaed as isoopic, is i-axial sess ad sai ca be decomposed io: VV ad V V (3) Whee, ad, ae he sesses ad sais o he ceamic ad meal de i-axial sess ad sai codiios, especively. The omalized aio o he sess o sai ase is he deied by a paamee q o descibe he aio o sess-o-sai ase. q,q Combiig Eqs. (3) ad (4) ad seig E, oe may obai he ollowig expessio o he eecive modls [7]: qe qe E VE V E/ V V (5) qe qe Fo applicaios ivolvig plasic deomaio o ceamic/meal (bile/dcile) composies, i is assmed ha he composie yields oce he meal cosie yields. Wih hese assmpios, he yield sess, Y, o he composie may be obaied as ollows [7]: q E E Y V V V (6) q E E whee E, E ae Yog s modle o he cosie phases, ad deoes he yield sess o he meal (phase ). The above eqaio idicaes ha he yield sess o he composie depeds o he yield sess o he meal, he volme acio o he meal, Yog s modle o he cosie phases, ad he paamee q. III. POBLEM STATEMENT Fig. () shows a FGM cicla plae o adis ad hickess which is sbjeced o a abiay oaioally symmeic loadig () pe i aea. Pimaily, i is assmed ha he FGM popeies o he plae i is deomed shape ae a cio o adis oly, he aohe poblem is cosideed whee he FGM popeies o he plae is a cio o hickess oly. ewiig he loadig cio as (), whee is called he load aco, he mai ppose o his pape is o obai he ciical vale o, i.e., which c geeaes he collapse mechaism i he plae. Fig (). FGM plae wih oaioally symmeic loadig I geeal, he collapse mechaism cosiss o a ew ses o yield lies alog which he plae becomes lly plasic hogh he hickess. The omaio o he collapse mechaism is sally cosideed as he las sage o load cayig cio. (4) 34

3 Wold Academy o Sciece, Egieeig ad Techology 36 7 Theeoe, havig evalaed c, he load cayig capaciy o he plae ca be expessed as c (). As is csomay i limi aalysis, i is assmed ha all deecios ae egligibly small il he collapse mechaism is lly omed. To be compaible, he maeial is spposed o be a igid-peecly plasic oe. This assmpio implies ha he seqece o omaio o he yield lies pio o collapse is o sigiica o he esls. Fhemoe, i is assmed ha he maeial obeys eihe he Sqae o he Tesca yield cieio, as illsaed i Fig.. I his ige, m m ad m sad o he adial, ageial, ad limae (lly plasic) bedig momes pe i legh, especively. I is well kow ha he limae bedig mome pe i legh o a plae wih cosa hickess ad yield sess is: y y y (7) 4 m zdz Fig (). Sqae (coios) ad Tesca (dashed) yield diagams Fo he case o a hogh adis FGM, accodig o Eq. (6) i has bee show ha, he yield bedig sess ca be deied as [9], y a b (8) Hece, he limae bedig mome pe i legh ca be deied as: y() y() (9) 4 m zdz Clealy, o a vaiable hickess plae, m () wold o loge yield a cosa vale. As a esl, i he yield cieio is expessed i ems o he bedig momes pe i legh, he yield diagam clealy wold chage o dieig pois o he plae. This epeseaio is depiced i Fig. (). Accodig o Eq. (9) he omlaio is as he same as he omlaio o he isoopic homogeeos cicla plaes wih vaiable hickess [8]. O he ohe had o he case o hogh hickess FGM, i has bee show ha [9]: z m y a b () So oe ca id he limae bedig mome pe i legh as: y( ) () m z zdz m Whee m is cosa ad is idepede o z. Theeoe, he same omlaio as o he case o isoopic homogeeos maeial ca be sed. IV. CITICAL LOAD FACTO FO THOUGH ADIUS SIMPLY SUPPOTED FGM PLATES Fo a simply sppoed plae de asvese loadig, i may be obseved ha sess saes lie o he lie AB i Fig. (), o which he wo yield cieia ae ideical. Hece, he aalysis wold esl i he same ocome i eihe cieio applied. Accodig o Koie's le [], he deomaio ield ad hece he collapse mechaism shold be compaible wih he assmed yield cieio. I may be easily veiied ha he sm collapse mechaism is compaible wih he segme AB o he wo yield cieia show i Fig. (). Hece, i ca be assmed ha he sm collapse mechaism wold be geeaed i he plae. Fig. (3) illsaes he geomeic paamees o he sm shaped collapse mechaism. I shold be oed ha his mechaism ca be egaded as a geealizaio o he simple coical oe (wih ). Fig (3). The sm collapse mechaics A. Uppe Bod Solio Ude peseed coigaio o he plae collapse i Fig. (3), i he ceal pa o he plae wih adis, has a cosa delecio o w, he whe chages om zeo o, he we have; 35

4 Wold Academy o Sciece, Egieeig ad Techology 36 7 w ( ) w () i which w is he delecio o he plae cee whe we have a ceal hige ad is he adis o he plae. The oal ieal (dissipaive) wok may ow be wie as ollows: w w Wi m( ) m( ) d (3) whee, he is em o he igh-had side is de o he wok a he ieal cicla hige Q. I has bee calclaed by he mliplicaio o he limae bedig mome pe i legh, legh o he cicla hige a Q, ad he agle o oaio a his poi, w. The secod em epeses he ieal wok o he coical pa o he mechaism ad has bee calclaed i a simila way. Fo he exeal wok doe by he applied loadig, oe obais, We () w() d d (4) whee w ( ) epeses he deecio cio o he plae. Eq. (4), o a sm collapse mechaism show i Fig. (3), ca be ewie as: We ( wd ) ( w ) ( ) d (5) o by sig Eq. () We w ( )( ) d ( )( ) d (6) As a esl o he implemeaio o he vial wok piciple ad he ppe bod heoem o limi aalysis, ppe bod esimaio o he ciical load aco, i.e., may be obaied by eqaig he ieal wok, Eq. (3) o he exeal wok, Eq. (6). The ial esl is: m ( ) m( d ) (7) ()( ) d ()( ) d B. lowe bod solio I ode o apply he lowe bod heoem, he eqaio o eqilibim o he plae shold be cosideed. Fhemoe, a compaible sess ield which does o violae he assmed yield cieio shold be iodced. The eqaio o eqilibim o a cicla plae wih oaioally symmeic geomey, loadig, ad boday codiios is as ollows []: m m ( ) d (8) whee is a dmmy vaiable o iegaio. Iegaio o Eq. (8) gives: C m () m d () d d (9) whee, is aohe dmmy vaiable o iegaio, ad is a iegaio cosa. The iieess o he esls a he plae cee saes ha C shold vaish. Fo simply sppoed boday codiios, he sm shaped collapse mechaism ad eihe o he wo yield cieia show i Fig. (). A discoios disibio o m will esl i as: m m () kow Sice, i he egio adial yield lies ae geeaed, m is eqal o m abo he vales o m iside he cicle C b ohig ca be said, excep ha hey shold saisy he yield cieio de cosideaio. Usig Eqs. (9) ad () o, oe obais: () m m d m d d d I he is iegal, m is kow. I ode o evalae his iegal, oe may evalae Eq. () a : m m d m d dd () Theeoe, Sbsaio o Eq. (3) io Eq. () esls i: m d m d d (3) m m dd m d (4) dd Fially, sice he plae is simply sppoed, he adial bedig mome a he plae edge is zeo, i.e. m (5) Uilizig Eqs. (), (4) ad (5), oe obais: m dd m d (6) dd Theeoe, he ollowig lowe bod aswe o he ciical load aco, i.e., may be obaied: l m ( ) m d l (7) d d Cosideig he special case o he coical collapse mechaism wih, Eq. (7) edces o he oe epoed i e. [7]. 36

5 Wold Academy o Sciece, Egieeig ad Techology 36 7 I shold be oed ha Eq. (7) has bee obaied hogh he applicaio o he plae eqaio o eqilibim, Eq. (8). Howeve, i ca be cosideed as a complee lowe bod solio oly i he yield cieio is saisied ad he collapse mechaism wih a igid ie disk is valid. A ecessay codiio o saisacio o he yield cieio, which eses ha Eq. (7) is a lowe bod solio, is ha he iegal o m ove he ie cicla disk o he plae shold o exceed he plae load cayig capaciy i his egio. Hece, sig Eq. (3), m d d m d (8) I he ie cicla disk is igid (coespodig o a sm collapse mechaism) he codiio give i Eq. (8) wih sic ieqaliy sig wold be saisied. I hs peses a ecessay codiio o he igidiy o he ie cicla disk. V. CITICAL LOAD FACTO FO THOUGH THICKNESS SIMPLY SUPPOTED FGM PLATES Sice o a hogh hickess FGM plae, m is cosa a each poi, i behaves simila o a homogeeos isoopic plae. Ths, o a simply sppoed plae, a simple coical collapse mechaism, coespodig o he ollowig kiemaically admissible deomaio ield, ca be assmed: ( W W ) (9) Whee W is he plae delecio a adis, ad W is he delecio a he cee o he plae. I he ollows ha m ad he ageial cvae k ae boh posiive hogho he plae. Theeoe accodig o Koie s le, he compaible sess saes, lie o AB i Fig.. Sice he wo cieia ae coicide i his egio, he same esls o he ciical load aco wold be obaied ad coespodig esls o e. [8] ae applicable o he pese poblem, hece: m c d (3) d A. Solio scheme ad case sdies I his secio, ciical loadig aco is ivesigaed o he ollowig loadig cio: () ab c (3) This loadig cio is expessed i a geeal om ad i icopoaes diee ype o loadigs i.e., iomly disibed wih b=c=, liealy vayig loadigs (c=) as well as paabolic loadig. The maeial popey o each phase is cosideed as show i Table ad q 4.5 GPa[9]. Table Maeial popeies o Ti ad TiB [9] Maeials Yog s modls Yield sess (GPa) (MPa) Ti 7 45 TiB 375 Based o he vales lised i Table ad yieldig model epeseed by Eq. (6), we ca coclde ha: V (3) Y B. simply sppoed hogh adis FGM cicla plae I is evide om Eq. (8) ha m is a cio o adis ad is i he om o: m AB (33) Cosideig Eqs. (7), (7), (3) ad (33) we obai he exac ciical loadig aco o a simply sppoed hogh adis FGM cicla plae obeyig sqae o Tesca yieldig cio as below: 6 AB AB l (34) 5 3 a5b3c a 5b 3c l A B B a5b 3c a 5b 3c (35) The esl give i Eq. (34) is deived o =, wheeas i Eq. (35) =/. Sice we have obaied he aalyical om o he solio, as i is see hee is o eed o se maeial popeies show i Table o compe meical vale. C. simply sppoed hogh hickess FGM cicla plae Cosideig Eqs. () ad (3) we ae able o obai he exac vale o he ciical loadig aco o a simply sppoed hogh hickess FGM cicla plae obeyig sqae o Tesca yield cio as: 6m c (36) a5b3c D. Veiicaio 37

6 Wold Academy o Sciece, Egieeig ad Techology 36 7 Fo a homogeeos isoopic maeial, i is obvios ha yield sess is cosa, heeoe, accodig o Eq. (33) we have: A, B (37) 4 Y Cosideig Eqs. (7), (7) ad (37) ad assmig ha a cosa iom pesse p is applied o he plae, we coclde ha: 3 Y c ; 3 (38) p ( ) The above esl is he same as show i Eq. (5) i [8]. Sice he omla deived o a hogh hickess simply sppoed FGM cicla plae is he same o a homogeeos plae, i is o ecessay o veiy Eq. (3). VI. CONCLUSIONS [3] S. Sesh, ad A. Moese, Fdameals o Fcioally Gaded Maeials, IOC Commicaios Ld, Lodo, 998. [4]. L. Williamso, B. H. abi, ad Dake, Fiie eleme aalysis o hemal esidal sesses a gaded ceamic-meal ieaces, I. Model descipio ad geomeical eecs, J. T. J. Appl. Phys., vol. 74, pp. 3-3, 993 [5] A. Moese, ad S. Sesh, Fcioally gaded meals ad mealceamic composies: Pa, Pocessig, S. I. Mae. ev., vol. 4, pp , 995. [6] S. Sesh, ad A. Moese, Fcioally gaded meals ad mealceamic composies: Pa, Themomechaical Behavio,A. I. Mae. ev., vol. 4, pp. 85-6, 997. [7] I. Tama, Y. Tomoa, H. Ozawa Segh ad dciliy o Fe Ni C alloys composed o aseie ad maesie wih vaios segh, I: Poc. o he Thid I. Co. o Segh o Meals ad Alloys, vol.. Cambidge: Isie o Meals; pp. 6 5, 973. [8] M. H. Kagaovi ad M. Ghoashi, Limi aalysis o a cicla plae sbjeced o a abiay oaioal symmeic loadig, Poc. 3d I. Co. o Compaioal Plasiciy, Baceloa;:pp.49-59, 99 [9] Z. H. Ji.H. Doddas J, Cack gowh esisace behavio o a cioally gaded maeial: compaioal sdies, Egieeig Face Mechaics; vol. 7, pp.65-67, 4. Limi aalysis o simply sppoed cicla FGM plaes wih cosa hickess sbjeced o a abiay oaioal symmeic loadig has bee discssed sig ppe ad lowe bod piciples. I his way, geeal eqaios o he ciical load aco cosideig he Sqae ad Tesca yield cieia have bee obaied. Fo he simply sppoed hogh adis o hogh hickess FGM cicla plaes, i was show ha he esls obaied o eihe o yield cieio ae he same. Fo a paabolic ype o loadig he exac solio is deived o simply sppoed FGM plaes. EFEENCES [] W. Page, A iodcio o plasiciy eadig, MA: Addiso- Wesley; 959. [] M.. Hoe, Plasic heoy o sces. Cambidge, MA: MIT; 97. [3] E. H. Masield, Sdies i collapse aalysis o igid-plasic plaes wih a sqae yield diagam, Poc.. Soc. A; pp.4:3-38, 958. [4] P.G. Hodge, Limi aalysis o oaioally symmeic plaes ad shells, Eglewood Clis, NJ: Peice-Hall; 963. [5] M.A. Save, Massoe CE. Plasic aalysis ad desig o plaes, shells ad disks. Amsedam, Noh-Hollad; 97. [6] Z. Soboka, Theoy o plasiciy ad limi desig o plaes, Amsedam: Elsevie; 989. [7] M. Ghoashi, Limi aalysis o cicla plaes sbjeced o abiay oaioal symmeic loadigs, I. J. Mech. Sci.; vol. 36, o., pp.87-94,994. [8] M. Ghoashi, M. Daeshpazhooh, Limi aalysis o vaiable hickess cicla plaes, Comp. ad Sc.; vol. 79, o, pp ,. [9] M. Gowei, I. Shoji, M. Yaka, D. Hideaki, Plasic limi aalysis o cicla plaes wih espec o iied yield cieios, I. J. Mech. Sci.; vol. 4, o.,pp , 998. [] T. Hiai, Fcioally gadie maeials, I: Book J, edio. Pocessig o ceamics, Pa. Ma. Sci. ad Tech., Weiheim, Gemay: VCH Velagsgesellscha mbh;; vol.7b, pp. 9 34, 996. [] S. Sesh, A. Moese Fcioally gaded maeials, Lodo: The Isie o Maeials, IOM Commicaios Ld.; 998. [] G.H. Palio, Z.H. Ji,.H. J. Dodds Faile o cioally gaded maeials, I: B. Kaihaloo, W.G. Kass, edios. Compehesive Sc. Ieg., vol., Oxod: Elsevie Sciece Limied;, ch