MATERIAL AND GEOMETRIC NONLINEAR ANALYSIS OF FUNCTIONALLY GRADED PLATE-SHELL TYPE STRUCTURES

Transcription

1 1h Inrnaional Confrn on Coposi Sin and Thnolog ICCST/1 A.L. Araújo, J.R. Corria, C.M. Moa Soars, al. (Ediors) IDMEC 15 MATERIAL AND GEOMETRIC NONLINEAR ANALYSIS OF FUNCTIONALLY GRADED PLATE-SHELL TYPE STRUCTURES José S. Moia *, Aurélio L. Araújo, Crisóão M. Moa Soars, Carlos A. Moa Soars * FANOR, Fauldads Nords, A. Sanos Duon, 78, Forala, Brail joia5@gail.o LAETA, IDMEC, Insiuo Suprior Ténio, Unirsidad d Lisboa A. Roiso Pais, Lisboa, Porugal aurlio.araujo@nio.ulisboa.p This or is ddiad in honour of Profssor J.N. Rdd on his 7 h birhda and for his onribuion and ipa o rsarh and duaion on hanis of adand oposi arials and sruurs. Th auhors also prss hir graiud for his frindship and sinifi adis. K ords: Funionall Gradd Marials, Fini Eln, Nonlinar Analsis. Suar: A nonlinar forulaion for gnral Funionall Gradd Marial (FGM) plashll p sruurs is prsnd. Th forulaion aouns for gori and arial nonlinar bhaiour of hs sruurs. Using h Non-Raphson inrnal-irai hod, h inrnal quilibriu pah is obaind, and in as of snap-hrough ourrn h auoai ar-lngh hod is usd. This sipl and fas ln odl is a non-onforing riangular fla pla-shll ln ih 4 dgrs of frdo for h gnralid displans. I is bnhard in h soluion of so illusrai pla- shll apls and h rsuls ar prsnd and disussd ih nurial alrnai odls. 1 INTRODUCTION In an ffor o dlop h supr ha rsisan arials, Koiui [1] firs proposd h onp of Funionall Gradd Marial (FGM). Tpial FGM pla-shll p sruurs ar ad of arials hih ar hararid b a oninuous ariaion of h arial propris or h hinss dirion b iing o diffrn arials, al and rai. Th al rai FGM plas and shlls ar idl usd in airraf, spa hils, raor ssls, and ohr nginring appliaions. Sruurs ad of oposi arials ha bn idl usd o saisf high prforan dands. In suh sruurs srss singulariis a our a h inrfa bn o diffrn arials. In onras, in FGM pla-shll sruurs h sooh and oninuous ariaion of h propris fro on surfa o h ohr liinas abrup hangs in h srss and displan disribuions. In rain appliaions, sruurs an prin larg lasi dforaions and fini roaions. Gori nonlinari plas a signifian rol in h bhaiour of a pla or shll sruurs, spiall hn i undrgos larg dforaions. Also arial nonlinari has a signifian rol in h bhaiour of hs sruurs. Hor a ri of h liraur shos

2 José S. Moia, Aurélio L. Araújo, Crisóão M. Moa Soars,Carlos A. Moa Soars ha f sudis ha bn arrid ou in nonlinar bnding rspons of plas and shlls. This is h ain rason o h prsn or, hih inlud h bhaiour rspons of his p of sruurs du o h arial and gori nonlinariis. Rsarh in FGM sruurs has bn don in h rn ars. Aong ohrs, hr i h folloing ors: Rdd and Chin [] anald h dnai hro lasi rspons of funionall gradd lindrs and plas. Pran and Rdd [3] arrid ou a nonlinar hroplasi analsis of funionall gradd rai al plas using a fini ln odl basd on h Firs-Ordr Shar Dforaion Thor (FSDT). Rdd [4] sudid h bnding and ibraion analss of FGMs plas. Woo and Mguid [5] proidd an analial soluion for larg dflion of FGM plas and shlls undr hanial and hral loading. Yang and Shn [6, 7] arrid ou h analss of nonlinar bnding and pos buling bhaior for FGM plas undr hro hanial load and ih arious boundar ondiions. Ma and Wang [8] insigad h nonlinar hral bnding and pos-buling of irular funionall gradd plas using FSDT and h Third -Ordr Shar Dforaion Thoris (THSDT). Na and Ki [9], aind h ff of hral loading and unifor prssur on h bnding rspons of FGM plas. Rdd and Ariniga [1, 11] prsnd h hro-hanial buling, as ll as bnding and fr ibraion analsis, of FGM plas. Th sa auhors [1] arrid ou h larg dforaion analsis of FGM shlls. Ki al. [13] prsnd h gori nonlinar analsis of funionall gradd arial plas and shlls using a four-nod quasi-onforing shll ln. Barbosa and Frrira [14] prford h gori nonlinar analsis of funionall gradd (FGM) plas and shlls, using h Margurr shll ln. Th plasii forulaion follos a oposi odl proposd b Taura al. [15] rfrrd as h TTO odl. Naaura al. [16] and Gu al. [17] proposd an inrs analsis produr basd on h Kalan filr and insrund iro-indnaion. Jin al. [18] sudid h fraur issus in lasi plasi FGMs. A oprhnsi ri of h arious hods plod o sud h sai, dnai and sabili bhaiour of FGM plas as rnl prsnd b Sainahan al. [19] onsidring analial and nurial hods. In his papr, a larg dforaion analsis for funionall gradd pla-shll sruurs is prsnd. Th forulaion inluds arial and gori nonlinariis and is iplnd in a fini ln odl basd on a non-onforing riangular fla pla-shll fini ln in onjugaion ih h Rdd s hird-ordr shar dforaion hor. FORMULATION OF FGM MODEL A FGM is ad b iing o disin isoropi arial phass, for apl a rai and a al. Th arial propris of an FGM pla/shll sruurs ar assud o hang oninuousl hroughou h hinss, aording o h olu fraion of h onsiun arials. Por-la funion [] and ponnial funion [1] ar oonl usd o dsrib h ariaions of arial propris of FGM. Hor, in boh por-la and ponnial funions, h srss onnraions appar in on of h inrfas in hih h arial is oninuousl bu rapidl hanging. Thrfor, Chung and Chi [] proposd a sigoid FGM, hih as oposd of o por-la funions o dfin a n olu fraion. Chi and Chung [3] indiad ha h us of a sigoid FGM an signifianl rdu h srss innsi faors of a rad bod. To dsrib h olu fraions, h por-la funion, and sigoid funion ar usd hr. Thus, in his or, o odls for siaing h ffi arial propris of h

3 José S. Moia, Aurélio L. Araújo, Crisóão M. Moa Soars,Carlos A. Moa Soars FGM a a poin ar onsidrd..1 Por-la funion: P-FGM Th olu fraion of h rai phas is dfind aording o h por-la []: V () p.5 (1) bing (-h/; h/), h h hinss of sruur, and h ponn p a parar ha dfins gradaion of arial propris aross h hinss dirion. Thus, h al olu fraion is. Sigoid funion: S-FGM h V () 1. V () Th olu fraion uss o por-la funions hih nsur sooh disribuion of srsss []: V V () () p 1 h 1. for h h (3a) p 1 h h for (3b) h In h prsn or, h oninuous ariaion of h arials iur is approiad b h using a rain nubr of lars hroughou h hinss dirion (irual lar approah). In his sns, h prious quaions an b rin for ah lar as follos (P- FGM): V p.5 ; h V 1. V hr is h hinss oordina of id-surfa of ah lar (4) On h olu fraion V and V ha bn dfind, h arial propris (H) of ah lar of a FGM an b drind b h rul of iurs. H V H V H (5) hr H sands for Young s odulus E, h Poisson s raio υ, h ass dnsi ρ, or an ohr hanial propr. Figur 1 sho h ariaion of Young s odulus E hrough h hinss, obaind using h sigoid funion and lars. 3 NONLINEAR ANALYSIS THEORY Th obji of his or is o insiga h goriall non-linar bhaiour of 3

4 /h José S. Moia, Aurélio L. Araújo, Crisóão M. Moa Soars,Carlos A. Moa Soars FGM pla-shll p sruurs undr hanial loading. Th prsn hor onsidrs larg displans ih sall srains. 3.1 Displan Fild and Srains Figur 1. Variaion of Young s odulus hrough h hinss. Th displan fild is basd on h Rdd s hird-ordr shar dforaion hor [4] u p=. p=.5 p=1. p= E+1 9.E+1 1.1E E E+11 Young's odulus 1 3,,, 1 (6),, 3,, u,,,,,, hr u,, ar displans of a gnri poin in h iddl plan of h or lar rfrrd o h loal as -,, dirions, θ, θ, ar h roaions of h noral o h iddl plan, abou h ais (lois) and ais (anilois),, h ar h slops of h angns of h dford id-surfa in, dirions, and, ih h dnoing h oal hinss of h sruur. To h goriall nonlinar bhaiour, h Grn s srain nsor is hr onsidrd. Is oponns ar onninl rprsnd in rs of h linar and nonlinar pars of h srain nsor as L NL (7) Th linar srain oponns assoiad ih h displan fild dfind abo, an b rprsnd in a snhi for as [5]: 4

5 José S. Moia, Aurélio L. Araújo, Crisóão M. Moa Soars,Carlos A. Moa Soars 5 * s s b 3 b L + (8) or in d dlopd for, as: 1 u = u (9) hr h 4. Th nonlinar srain oponns ar gin b NL u 1 = NL u 1 = u u = NL u u = NL u u = NL (1) 3. Consiui Rlaions of FGM Sruurs Th srss-srain rlaions for ah lar, an b rin as follos Q (11)

6 José S. Moia, Aurélio L. Araújo, Crisóão M. Moa Soars,Carlos A. Moa Soars T hr is h srss or and T srain or, is h Q is h lasii ari, hos non-ro lns ar gin b: Q Q 11 Q Q E 1 1 Q1 Q11 E 44 Q55 Q66 G (1) (1 ) For ah lar, linar lasi onsiui quaion is gin b [5]: hr for insan ij ij 4 N A B D M B C E * M D E G Q As C s * Q C s E s 4 D Q H, ih H Equaion (13) an b rin in h folloing for hr * b b * s s (13) ˆ (14) D ˆ ar h rsulan fors and ons. 4 ELASTO-PLASTIC FORMULATION Th ild ondiion an b rin in h for hr h ild ll,, f F Y (15) Y, an b a funion of h hardning parar κ. Fro h Hubr-Miss ririon, for h as of an isoropi arial h ffi srss is gin b f (16) 4.1 Srss-srain rlaions in lasoplasii Th laso-plasi inrns of oal srains an b alulad b suing up h lasi and h plasi srain oponns, 4. Flo rul 1 p d d d (17) Th plasi srain inrn is dfind as proporional o h srss gradin of a plasi ponial Q, hih is an qual o h ild surfa ondiion for an assoiad flo rul:

7 José S. Moia, Aurélio L. Araújo, Crisóão M. Moa Soars,Carlos A. Moa Soars d ε p dq d hr is a proporional onsan ha drins h agniud of h plasi srain inrn and dq d dfins is dirion o b prpndiular o h ild surfa. Inroduing h flo or a dq d and A H d d hr h hardning parar A an b an fro uniaial sing, i an b obaind ' p (18) a T Q d A a T (19) Q a d ε p T a Q a dε () T A a Q a Th inrnal onsiui lasoplasi rlaion is gin b, [6, 7] p p d Q (d d ) Q d (1) Inroduing h quaions (11) and () ino quaion (1), os: T Q a a Q d Q d () T A a Q a Fro his quaion h lasoplasi onsiui ari is obaind as: Q p T Q a a Q Q (3) T A a Q a p p p A B D p p p B C E p D p p p D E G (4) As Cs Cs E s p p 3 3 hr for insan C ij Q ij H3, ih H 3 1 Th prsn or uss an ndd Taura Tooa Oaa (TTO) odl o dsrib h lasi plasi bhaiour of rai/al FGM. Crai arials ar, in gnral, bril arials of rlail highr lasi odulus and srngh han hos of alli arials, hih ha piall duil propris. Thus h rai onsiun in FGM is assud o b lasi hn dforaion as pla. Th laso-plasi dforaion ours ainl b h plasi floing of h alli onsiun. I is adopd h odifid rul-of-iur hih uss h srss-srain ransfr parar q, hih dpnds on h onsiun arial propris and h irosruural inraion ihin h oposi, and is gin b [18]: 3 q (5) 7

8 José S. Moia, Aurélio L. Araújo, Crisóão M. Moa Soars,Carlos A. Moa Soars Using his parar, h hinss ariaion of Young s odulus and h ild srss, Y, a b obaind as follos [18]: q E q E E VE V E V V (6) q E q E Y Y q E E V V (7) q E E 5 VIRTUAL WORK PRINCIPLE Th gorning quaions of h nonlinar probl ar obaind fro h prinipl of irual or in onjugaion ih an updad Lagrangian forulaion [8, 9]. A rfrn onfiguraion is assoiad ih i and h aualid onfiguraion is assoiad ih h urrn i '. Thus ha: V S ij ij d V (8) hr h firs and sond br ar h inrnal and rnal irual or, rspil, and ij and S ij ar h Grn-Lagrang srain oponns and h sond Piola-Kirhhoff srss oponns rspil. Taing h Virual Wor Prinipl quaion (8) and doing h inrnal doposiion of h srss and srain nsors, obain a n quaion in an inrnal for, and linarid. V hr C ijrs srss nsor, rs ij d V V ij ij d V C ijrs is h onsiui nsor, ijar h inrnal oponns of h Cauh ij V ar h inrnal oponns of h linar srain or, and ij inrnal oponns of h nonlinar srain or. 5.1 Fini Eln Forulaion ij ij d V (9) ar h In h prsn ors i is usd a non-onforing riangular pla/shll fini ln odl haing hr nods and igh dgrs of frdo pr nod: h displans,,, and h roaionsθ i, θi, θi. Th u, h slops i i i i i roaion i is inrodud o onsidr a fiiious siffnss offiin K Z o liina h probl of a singular siffnss ari for gnral shap sruurs [7]. Th ln loal displans, roaions and slops, ar prssd in rs of nodal ariabls hrough shap funions j Ni gin in rs of ara o-ordinas L i, [7]. Th displan fild an b rprsnd in ari for as: 3 u = Z ( N d )= Z N a ; i=1 i i d = 3 i= 1 i N d = N a (3) i 8

9 José S. Moia, Aurélio L. Araújo, Crisóão M. Moa Soars,Carlos A. Moa Soars hr h appropria ari Z onaining pors of, and h ln and nodal displan ors, a and d i rspil, ar gin in [5,3]. Th bran, bnding and shar srains, as ll h highr ordr bnding and shar srains an b rprsnd b: B a ; hr h B, B, B, B, ar gin pliil in [3]. b *b s b b B a ; s s B a ; b *b B a ; * s *s B a (31) *s B, ar oponns of h srain-displan ari B, and Th Virual Wor Prinipl quaion (9) applid o a fini ln an b rin in h for: L L NL Q d da δ N δ 1 A h1 A h 1 Δ N δ 1 A h 1 L L d da Fro his quaion obain h ln linar siffnss ari K, h ln gori siffnss ari in L d K, as ll as ln rnal load or da L (3) F and h ln inrnal for or F, and hir dfiniion ar pliil gin in [3]. For h ln gori siffnss ari K, T K G G da (33) A h aris G and τ ar obaind ih a full dlopn, in h sa a as shon in [31] for h as of igh nod isoparari fini ln. Ths aris and ors ar iniiall opud in h loal oordina ss aahd o h ln. To sol gnral sruurs, loal - global ransforaions ar ndd [7]. Doing hs ransforaions, and afr adding h onribuions of all h lns in h doain, h ss quilibriu quaions ar obaind as: i1 i i 1 K q F F L in K (34) Using h Non-Raphson inrnal-irai hod [8,9], h inrnal quilibriu pah is obaind, and in as of snap-hrough ourrn h auoai ar-lngh hod is usd [5,3]. Eq. (34) is hn rin in h for: i1 i i1 i i 1 K q F F L in K (35) and an addiional quaion is plod o onsrain h lngh of a load sp: q i l i T q (36) 9

10 José S. Moia, Aurélio L. Araújo, Crisóão M. Moa Soars,Carlos A. Moa Soars hr F is a fid (rfrn) load or, is a load faor, is h inrnal load faor ihin h load sp, and l is h ar-lngh. 6 APPLICATIONS 6.1 Sipl Suppord Squar S- FGM Pla.Linar Analsis A sipl suppord squar (aa) pla undr uniforl disribud load of innsi q 1. N /, prsnd b Ki al. [13], is anald using h prsn odl. Th hanial propris of h isoropi rai arial ar E = N/, ν =.3 and for isoropi al arial ar E = N/, ν =.3. Th sid o hinss raio is a/h=1. Th S-FGM pla is odlld b using diffrn fini ln shs fro 44 (3 riangular lns) o 44 (115 riangular lns). In his appliaion and all h h FGM ass, h lard approah as arrid ou ih 1 lars. This alu for h nubr of lars has bn hosn basd on a onrgn sud. 3 4 Th non-dinsional nr dflions *= E h q a ar shon in Tabl 1. Fro his abl i an b obsrd ha h rsuls gin b h prsn fini ln FGM odl using a 44 ln sh ha an lln agrn ih h ons obaind b a FSDT Nair analial soluion [13]. For h sa appliaion, Figur shos h nr dflion as a funion of h porla ind. Th disrpanis bn h rsuls of prsn odl and h analial soluion ar abou of.4%. Also is obsrd ha undr h sa load, h as of p=1 has h largs nr dflion. This is du o h fa ha i has h los Young s odulus. Msh Noralid dflion Disrpan [%] Analial.671 Tabl 1. Noralid nr dflion of S-FGM pla (ind p=1). 6. Goriall nonlinar analsis of a funionall gradd pla srip A anilrd P-FGM pla srip subjd o a bnding disribud on on h ohr nd is onsidrd. Th arial propris for h rai and al onsiuns ar E = N/, ν =.3, E = N/, ν =.3. Th gor of h pla is L =1., b = 1., h =.1 and h oal bnding on M Rf = 65, N.. Th srip is odlld ih a 34 fini ln sh ih 56 riangular lns, ih n lars, haing a oal of 64 fr Dgrs OF Frdo (DOF). Figur 3 sho h rsuls for h ip dflion du o a bnding disribud on M for h diffrn por-la p-ind onsidrd, obaind b h prsn odl (PM). In h sa figur ar prsnd disr rsuls an fro Ariniga and Rdd [1] ih a 18Q5 ln sh ih a oal of 1

11 M / Mrf Noralid nr dflion José S. Moia, Aurélio L. Araújo, Crisóão M. Moa Soars,Carlos A. Moa Soars 11 fr DOF. I is obsrd a good agrn bn boh soluions. In Figur 4 ar shon h iniial and dford shaps of h pla srip for spifi load lls, in h as of p-ind p= Analial soluion [13] Prsn odl S-FGM por la ind Figur. Cnr dflion rsus por la ind CERAMIC (P M) CERAMIC Rf. [1] p=. (PM) p=. p=.5 (PM) p=.5 p=1. (PM) p=1. p=. (PM) p=. p=5. (PM) p=5. METAL (PM) METAL Dflion a h ip Figur 3. Tip-dflion W s. nd on M for h FGM pla srip. 6.3 Goriall nonlinar analsis of a hingd funionall gradd lindrial panl A S-FGM lindrial shll panl rprsnd in Figur 5 has h sraigh sids siplsuppord (hingd) and h urd sids fr. Th gor is dfind b: R=.54, L =.58 and subndd angl =. rad, h=.16, and i is odlld b a 1616 fini ln sh, onsidring n lars. Th onsiuns ar ironia and aluiniu and sral por-la p-inds ar onsidrd. Figur 6 shos h nr dflion du o a nr poin applid load P for h diffrn p-ind onsidrd, and as pd, h as of pur al has h largs nr dflion. This is du o h fa ha i has h los Young s odulus. Also, hn h loaddisplans urs obaind ih h prsn odl ar opard ih hos obaind of Ki al. [13], a r good agrn is obsrd. Hr prsn h rsuls oparison 11

12 Cnral poin load (N) José S. Moia, Aurélio L. Araújo, Crisóão M. Moa Soars,Carlos A. Moa Soars for rai and p-ind=1. ass, Figurs 6a and 6b, rspil M/Mrf=. M/Mrf=.5 M/Mrf=.4 M/Mrf= Figur 4. Iniial and dford shaps of pla srip. Figur 5. Clindrial panl Mal p=. Crai p=. p= Cnral dflion W () 1

13 Cnr poin load (N) Cnr poin load (N) José S. Moia, Aurélio L. Araújo, Crisóão M. Moa Soars,Carlos A. Moa Soars Figur 6. Load-dflion ur for diffrn por la ind Crai Prsn odl Ki al. [13] Cnr dflion W () Figur 6a. Load-dflion ur for rai p=1. Prsn odl Ki al. [13] Cnr dflion W () Figur 6b. Load-dflion ur for por la ind p= Nonlinar analsis of a sipl-suppord squar pla Th linar and nonlinar bnding bhaiour of a squar (aa) pla ih h four sids sipl suppord is anald using h prsn odl. This appliaion is proposd b Supriono and Aliabadi [33] and prsnd b Waida [34]. Th hanial propris of h isoropi arial ar E = N/, ν =.316 and uniaial ild srss Y 15. N/. Th sid is a=1., and h hinss is h=.1. Th pla is odlld b using a 1616 fini ln sh, orrsponding o 51 riangular fini lns. Figur 7 shos h nr dflion *=/h du o a uniforl disribud prssur load q* q a / E h

14 q* José S. Moia, Aurélio L. Araújo, Crisóão M. Moa Soars,Carlos A. Moa Soars I an b obsrd r good orrlaions bn soluions obaind ih h prsn odl (PM) and h odl basd in boundar lns usd b Waida [34], for Linar, Goriall Nonlinar (G_NLinar), Marial Nonlinar (M_NLinar) and Marial and Goriall Nonlinar (G+M_NLinar) analss. Folloing h onlusions an fro Figur 7, o siilar appliaions ar ndd for a FGM sruur, and proposd for bnhar purposs in sions 6.5 and Linar (PM) Linar [34 ] G_NLinar (PM) G_NLinar [34 ] M_NLinar (PM) M_NLinar [34 ] G+M_NLinar (PM) G+M_NLinar [34 ] W* Figur 7. Cnr poin dflion for diffrn analss. 6.5 Sipl Suppord Squar P- FGM Pla A sipl suppord squar (aa) ih sid dinsion a=1. and sid o hinss raio a/h=5, is undr a uniforl disribud load of innsi q 1 N /. Th arial propris for h rai and al onsiuns ar E = 35 GPa, ν =.4, E = 7 GPa, ν =.3. Th ild srss for pur al is Y 5 MPa, h raio q/e is an as.8, basd on h iro- indnaion prins b Gu al. [17], and is onsidrd q=9 GPa. This P-FGM pla ih p-ind por-la funion p=1., is anald onsidring four diffrn arial bhaiours: lasi dforaion, ariall nonlinar dforaion, goriall nonlinar dforaion and goriall and arial nonlinar dforaion. In Figur 8 h rsuls obaind ih h prsn odl ar shon, onsidring n lars. 6.6 Goriall nonlinar analsis of a lapd funionall gradd lindrial panl A squar lapd P-FGM lindrial shll panl ih onsiuns silion nirid and sainlss sl, sid-o-hinss raio a/h=1, sid-o-radius raio a/r=.1, and sral por la ponns p ar prsnd. Th arial propris for h rai and al onsiuns ar E = 3.7 GPa, ν =.4, E = 7.78 GPa, ν = Th panl is odlld b a 11 fini ln sh (88 riangular lns), and onsidring n lars. Using h prsn odl, Figur 9 shos h nr dflion du o a nr poin applid load P for h diffrn por-la p-ind onsidrd, and as pd, h as of 14

15 Cnr poin load (N) prssur load (Pa) José S. Moia, Aurélio L. Araújo, Crisóão M. Moa Soars,Carlos A. Moa Soars pur al has h largs nr dflion, and h as of pur rai has h salls nr dflion. Also is obsrd ha h dflion urs of h funionall gradd arial panls ar load bn hos of h al and rai panls. N, h sa squar lapd P-FGM lindrial panl, ih ild srss for pur al Y 5 MPa, and onsidring q=8 GPa, and p-ind p=1., is anald onsidring four diffrn arial bhaiours: linar lasi, arial nonlinar, goriall nonlinar and gori and ariall nonlinar dforaions. In Figur 1 h soluions gin b h prsn odl ar shon Linar M_NLinar G_NLinar G + M_NLinar W () Figur 8. Cnr poin dflion of a squar P-FGM pla for diffrn analss Crai p=. p=. p=1. Sl Cnr dflion W () Figur 9. Load-dflion urs of a P-FGM lapd lindrial panl for diffrn por la ind. 15

16 Cnr Poin Load (N) José S. Moia, Aurélio L. Araújo, Crisóão M. Moa Soars,Carlos A. Moa Soars por la p-ind=1. Linar G_NLinar M_NLinar G+M_NLinar Cnr dflion W () Figur 1. Cnr poin dflion of a P-FGM lapd lindrial panl for diffrn analss. 7 CONCLUSIONS A fini ln odl for h sai analsis of funionall gradd arial pla-shll sruurs is ndd for h arial and goriall nonlinar analsis. Th odl is basd on h Rdd s hird ordr shar dforaion hor, iplnd in a nononforing fla riangular pla/shll ln ih 4 dgrs of frdo for h gnralid displans. Th oninuous ariaion of h hanial propris hrough h hinss is approiad b onl n lars for all FGM appliaions, hih prsrs h rdud opuaional i of h odl. Fro h rsuls obaind b h prsn odl, a r good aura is found ih h aailabl oparing soluions obaind b alrnai odls. To apls ar also proposd hih an b usd as bnhar ss for FGM pla-shll sruurs ih arial and goriall nonlinar bhaiour. ACKNOWLEDGEMENTS: This or as suppord b FCT, Fundação para a Ciênia Tnologia, hrough IDMEC, undr LAETA, proj UID/EMS/5/13. REFERENCES [1] M. Koiui, Th onp of FGM Cra. Trans. Fun. Grad. Mar., 34, 3 1, [] J.N. Rdd, C.D. Chin, Throhanial analsis of funionall lindrs and plas. J. Thral Srss. 1, , [3] G.N. Pran, J.N. Rdd, Nonlinar ransin hrolasi analsis of funionall gradd rai al plas. In. J. Solids Sru., 35, , [4] J.N. Rdd, Analsis of Funionall Gradd Plas, Inrnaional Journal for Nurial Mhods in Enginring, 58, 177,. [5] J. Woo, S.A. Mguid, Nonlinar analsis of funionall gradd plas and shallo shlls. In. J. Solids Sru., 38, , 1. 16

17 José S. Moia, Aurélio L. Araújo, Crisóão M. Moa Soars,Carlos A. Moa Soars [6] J.Yang, H.S.Shn, Nonlinar bnding analsis of shar dforabl funionall gradd plas subjd o hrohanial loads undr arious boundar ondiions. Copos. Par B: Eng., 34, , 3. [7] J. Yang, H.S. Shn, Nonlinar analsis of funionall gradd plas undr ransrs and in-plan loads. In. J. Nonlinar Mh., 38, , 3. [8] L.S. Ma, T.J. Wang, Nonlinar bnding and posbuling of funionall gradd irular plas undr hanial and hral loadings. In. J. Nonlinar Mh. 4, , 3. [9] K.S. Na, J.H. Ki, Nonlinar bnding rspons of funionall gradd plas undr hral loads. J. Thral Srss., 9, 45 61, 5. [1] J.N. Rdd, R.A. Ariniga, Mhanial and hral buling of funionall gradd rai al plas. In: Analsis and Dsign of Plad Sruurs: Sabili. Woodhad Publishing, Cabridg, UK, 6. [11] J.N. Rdd, R.A. Ariniga, Fr ibraion analsis of funionall gradd plas. In: Analsis and Dsign of Plad Sruurs: Dnais. Woodhad Publishing, Cabridg, UK, 6. [1] R.A.Ariniga, J.N. Rdd, Larg dforaion analsis of funionall gradd shlls. Inrnaional Journal of Solids and Sruurs, 44, 36 5, 7. [13] K.D. Ki, G.R. Lobo, S.C. Han, Goriall nonlinar analsis of funionall gradd arial (FGM) plas and shlls using a four-nod quasi-onforing shll ln. Journal of Coposi Marials, 4, 5, , 8. [14] A.T. Barbosa, A.J.M. Frrira, Goriall nonlinar analsis of funionall gradd plas and shlls. Mhanis of Adand Marials and Sruurs, 17, 4 48, 1. [15] I. Taura, Y. Tooa, H. Oaa, Srngh and duili of F Ni C allos oposd of ausni and arnsi ih arious srngh.prodings of h Third Inrnaional Confrn on Srngh of Mals and Allos, ol. 1, Cabridg: Insiu of Mals, 611 5, [16] T. Naaura, T. Wang, S. Sapah, Drinaion of propris of gradd arials b inrs analsis and insrund indnaion. Aa Mar., 48, ,. [17] Y. Gu, T. Naaura, L. Prhli, S. Sapah, J. Walla, Miro-indnaion and inrs analsis o harari lasi plasi gradd arials. Marials Sin and Enginring A, 345, 3 33, 3. [18] J.H.Jin, G.H. Paulino, R.H.Dodds Jr, Cohsi fraur odling of lasi plasi ra groh in funionall gradd arials. Eng Fra Mh, 7 (14), , 3. [19]K. Sainahan, D.T. Nanuar, A.M. Znour, E. Carrra. Srss, ibraion and buling analss of FGM plas: A sa-of-h-ar ri. Coposi Sruurs, 1, 1 31, 15. [] G.Bao, L. Wang. Mulipl Craing in Funionall Gradd Crai/Mal Coaings, Inrnaional Journal of Solids and Sruurs, 3, , [1] F. Dlal, F. Erdogan, Th ra probl for a nonhoognous plan, ASME Journal of Applid Mhanis, 5, , []Y.L. Chung, S.H. Chi, Th rsidual srss of funionall gradd arials, Journal of Chins Insiu of Ciil and Hdrauli Enginring, 13, 1 9, 1. [3]S.H.Chi, Y.L. Chung.Craing in Sigoid funionall gradd oaing, Journal of Mhanis, 18, 41 53,. [4] J.N.Rdd. Mhanis of Lainad Coposi Plas and Shlls, nd Ediion, CRC Prss, 4. 17

18 José S. Moia, Aurélio L. Araújo, Crisóão M. Moa Soars,Carlos A. Moa Soars [5] J.S. Moia, J.I.Barbosa, C.M. Moa Soars, C.A. Moa Soars, Snsiii analsis and opial dsign of gori nonlinar lainad plas and shlls.copurs and Sruurs, 76, 47-4,. [6] E. Hinon, D.R.J On. Fini Eln Sofar for Plas and Shlls. Sansa, UK: Pinridg Prss, [7] O.C. Zinii, Th Fini Eln Mhod. N Yor, USA: MGra-Hill, [8] K.J. Bah, Fini Eln Produrs in Enginring Analsis. Prni-Hall In, Englood Cliffs, N Jrs, 198. [9]K.J. Bah, L.W. Ho, A sipl and ffi ln for analsis of gnral shll sruurs. Copurs and Sruurs, 13, , [3]J.S. Moia, A.L. Araújo, P.G. Marins, C.M. Moa Soars, C.A. Moa Soars, Analsis of ai-passi pla sruurs using a sipl and ffiin fini ln odl. Mhanis of Adand Marials and Sruurs, 18, , 11. [31]J.S. Moia, C.M. Moa Soars, C.A. Moa Soars, Buling bhaiour of lainad oposi sruurs using a disr highr-ordr displan odl. Coposi Sruurs 35, 75-9, [3]M.A. Crisfild, A fas inrnal/irai soluion produr ha handls snaphrough. Copurs and Sruurs, 6, 13-55, 198. [33]Suprioo, M.H. Aliabadi, M. H., Boundar ln hod for shar dforabl plas ih obind gori and arial nonlinariis. Enginring Analsis ih Boundar Elns, 3, 31-4, 6. [34]L.Waida, Forulação do éodo dos lnos d onorno para plaas nrigidas onsidrando-s não-linaridads físia goéria. Ts d Douorado, Unirsidad d S. Paulo, Esola d Engnharia d S. Carlos, 8 (In Porugus). hp://.ss.usp.br/ss/disponiis/18/18134/d /n.php 18