Finite Element Modelling for Static and Free Vibration Response of Functionally Graded Beam

Transcription

1 69 Finit Elmnt Modlling for Static and Fr Vibration Rspons of Functionally Gradd Bam Abstract A 1D Finit Elmnt modl for static rspons and fr vibration analysis of functionally gradd matrial (FGM) bam is prsntd in this wor. h FE modl is basd on fficint ig-ag thory (ZIG) with two nodd bam lmnt having four dgrs of frdom at ach nod. Linar intrpolation is usd for th aial displacmnt and cubic hrmit intrpolation is usd for th dflction. Out of a larg varity of FGM systms availabl, Al/SiC and Ni/AlO3 mtal/cramic FGM systm has bn chosn. Modifid rul of mitur (MROM) is usd to calculat th young s modulus and rul of mitur (ROM) is usd to calculat dnsity and poisson s ratio of FGM bam at any point. h MALAB cod basd on 1D FE igag thory for FGM lastic bams is dvlopd. A D FE modl for th sam lastic FGM bam has bn dvlopd using ABAQUS softwar. An 8-nod biquadratic plan strss quadrilatral typ lmnt is usd for modling in ABAQUS. hr diffrnt nd conditions namly simply-supportd, cantilvr and clampd- clampd ar considrd. h dflction, normal strss and shar strss has bn rportd for various modls usd. Eign Valu problm using subspac itration mthod is solvd to obtain un-dampd natural frquncis and th corrsponding mod shaps. h rsults prdictd by th 1D FE modl hav bn compard with th D FE rsults and th rsults prsnt in opn litratur. his provs th corrctnss of th modl. Finally, mod shaps hav also bn plottd for various FGM systms. Atb Ahmad Khan a M. Naushad Alam b Najb ur Rahman c Mustafa Wajid d a, b, c, d Dpartmnt of Mchanical Enginring, ZHCE, Aligarh Muslim Univrsity, Aligarh. a hanaatb@gmail.com b naushad7863@rdiffmail.com c najbalig@rdiffmail.com d mustfawajidali@gmail.com Rcivd In rvisd form Accptd Availabl onlin Kywords Finit Elmnt, MROM, ZIG-ZAG hory, Functionally Gradd Bam, Abaqus. 1 INRODUCION In cas of composit matrials whr two distinct matrials (.g. mtal and cramic) with significant diffrnc in matrial proprtis ar bondd togthr, larg jump in th in-plan normal strsss and

2 A.A. Khan t al. / Finit Elmnt Modlling for Static and Fr Vibration Rspons of Functionally Gradd Bam 691 high-transvrs shar strsss occur at th intrfac during fabrication and opration. his lads to dlamination at th intrfac and poor load-baring prformanc. Dvlopmnts in matrials nginring lad to a nw typ of composits with smoothly and continuously varying thrmos-mchanical proprtis that ar calld functionally gradd matrials (FGMs). Functionally gradd matrials (FGMs) provid an lgant solution to this problm, whrin th abrupt chang in composition of matrial from on layr to anothr is rplacd by layrs of gradually varying microstructur and composition. FGM structurs hav various advantags ovr th composit laminats, such as smallr strss concntrations, smallr thrmal rsidual strsss and th possibility to achiv spcific for diffrnt applications. his concpt has found many potntial applications li thrmal and corrosion barrirs, mdical implants, lightwight armour matrial with high-ballistic fficincy, tc. FGMs ar mad by combining diffrnt matrials through compl procsss such as powdr mtallurgy mthods, physical and chmical vapor dposition, solid frform fabrication tc. Dtaild procss for manufacturing FGM s and various aras of application ar prsntd by Rashdat M. t al. Among th various modls which hav bn proposd to prdict th ffctiv lastic proprtis of two-phas matrials, th slf-consistnt modl of Hill (1965), th Mori and anaa (1973), th linar rul of miturs and in Jons (1999) th intrmdiat rul of miturs provid simpl and convnint ways for prdicting th ovrall rspons. For inmatic modlling, svral on-dimnsional (1D) bam and D plat thoris such as Cho and Odn () applid CL (Classical Laminat hory) for thrmal strss analysis of infinit functionally gradd bams and plats with layr-wis compositional chang, Nguyn t al. (8) found th ffct of transvrs shar strsss by nrgy quivalnc using FGM modls basd on FSD thory. h transvrs shar strsss ar obtaind using th mmbran strsss and various quilibrium quations and Sina t al. (9) dvlopd a nw bam thory diffrnt from th traditional first-ordr shar dformation bam thory is usd to analy fr vibration of functionally gradd bams. h bam proprtis is varid through th thicnss according to simpl powr law distribution in trms of volum fraction of matrial constitunts. It is assumd that th latral normal strss is ro and th govrning quations of motion ar drivd using Hamilton s principl. Rddy (1983) dvlopd a highr ordr shar dformation thory of composits. his thory accounts for th parabolic distribution of th transvrs strains through th thicnss of th structur. his thory has bn found to b prdicting dflction and strsss mor accuratly than FSD. Chng and Batra () ploitd Rddy s third-ordr plat thory to study bucling and stady stat vibrations of a simply supportd functionally gradint isotropic polygonal plat rsting on a Winlr Pastrna lastic foundation and subjctd to uniform in-plan hydrostatic loads. Sanar (1) prsntd an lasticity solution for simply supportd FG bams subjctd to sinusoidal transvrs loading. Kapuria t al. (8) hav prsntd an fficint igag modl and its primntal validation for thrmo-lastic static and fr vibration rspons of Al/SiC and Ni/AlO3 FGM bams. Amal t al. (11) gav fr vibration charactristics of functionally gradd bam with matrial graduation in aially or transvrsally through th thicnss basd on th powr law and th obtaind rsults ar compard with prviously publishd wor. Nuttawit and Variddhi (1) applid th diffrntial transformation mthod (DM) to invstigat fr vibration of functionally gradd bams supportd by arbitrary boundary conditionsand th matrial proprtis of bams ar assumd to oby th powr law distribution. H. Yaghoobi t al. (1) invstigatd Bnding analysis of a functionally gradd simply supportd bam for varid nutral surfac subjctd to a uniformly distributd load.

3 69 A.A. Khan t al. / Finit Elmnt Modlling for Static and Fr Vibration Rspons of Functionally Gradd Bam Rsults indicat that position of nutral surfac is vry important in functionally gradd matrials. Kadoli t al. (8) implmntd displacmnt fild basd on highr ordr shar dformation thory to study th static bhavior of functionally gradd mtal cramic (FGM) bams undr ambint tmpratur. FGM bams with variation of volum fraction of mtal or cramic basd on powr law ponnt ar considrd. Najb and Alam (1) prsntd a on dimnsional finit lmnt modl using an fficint layrwis (igag) thory for th dynamic analysis of laminatd bams intgratd with piolctric snsors and actuators. Mohanty t al. (1) prsntd th valuation of static and dynamic bhavior of functionally gradd ordinary (FGO) bam and functionally gradd sandwich (FGSW) bam for pind pind nd condition. h variation of matrial proprtis along th thicnss is assumd to follow ponntial and powr law. A finit lmnt mthod is usd assuming first ordr shar dformation thory for th analysis. Furqan and Naushad (13) assssd highr ordr thory of laminatd bams undr static mchanical loads. h hird ordr thory and First ordr shar dformation thory ar assssd by comparison with th act two-dimnsional lasticity solution of th simply-supportd bam. Mhta t al. (13) usd finit lmnt mthod in modlling th dynamic bhavior of FGM to dtrmin its natural frquncy. h proprtis in th functionally gradd matrial ar assumd to vary according to powr law. h natural frquncis wr obtaind for FG bams undr various boundary conditions including Clampd-Fid, Simply supportd-fid, Clampd-Clampd, Simply supportd-simply-supportd, and Clampd-Simply supportd. Shi-rong Li t al. (14) studid th fr vibration of functionally gradd matrial (FGM) bams basd on both th classical and th first-ordr shar dformation bam thoris. Nguyn t al. (14) prsntd th analytical solutions for th static analysis of th transvrsly or aially functionally gradd bams with taprd cross-sction. h lastic modulus of th bam varis according to th powr form. h prsnt study dals in dvloping a 1D FE modl basd on Zigag thory for FGM lastic bams using MALAB as a mathmatical tool. h modl is usd to comput th dflction, strss and shar strss at various valus of sid to thicnss ratio and upto first fiv fundamntal frquncy. A D FE modl for th sam lastic FGM bam has bn dvlopd using ABAQUS softwar. h rsults prdictd by th 1D FE modl hav bn compard with th rsults obtaind using D FE modl. hs prdictd rsults hav also bn compard with th rsults prsntd in th opn litratur. his is don to prov th corrctnss of th FE modl dvlopd. A thr layr Al/SiC bam, fiv layr Ni/AlO3 and tn layr Ni/AlO3 bam is modlld for th analysis purpos. Mod shaps has also bn plottd for ths FGM systm undr various boundary conditions vi. simply supportd, clampd-clampd and cantilvr. MAERIAL MODELLING For th bi-matrial FGM systms, th volum fractions Vc and Vm of th cramic and th mtal as tan by Kapuria t. al (6) ar assumd to vary along -dirction according to following powr law: 1.5/, 1 1.5, 1 () = 1 () (1)

4 A.A. Khan t al. / Finit Elmnt Modlling for Static and Fr Vibration Rspons of Functionally Gradd Bam 693 Figur 1: Variation of volum fraction of mtal across th non-dimnsional thicnss. Whr M is a non-ngativ ral numbr calld th inhomognity paramtr. h variation of Vm across is plottd for M = 4. h volum fraction for a layr is computd at th cntr of th rspctiv layr. h ffctiv matrial proprtis for a layr ar computd using diffrnt avraging mthods basd on volum fraction at th cntr of layr. h ffctiv matrial lastic modulus E of a layr is computd using th modifid rul of miturs (MROM), which was originally proposd for cmntd carbids by omota t al (1976) and latr adoptd for FGM by many rsarchrs Cho and Odn (). According to this approach, th two phas matrial is tratd as an isotropic composit for which uniaial strss σ, strain ɛ, ar rlatd to constitunt strsss, andɛ, ɛ as, σ = Vm + Vc, ɛ = ɛ Vm + ɛ Vc, () and q = [( )/ (ɛ ɛ )] (3) Whr q is Strss to Strain transfr ratio btwn two phass. h valu of q for Al/SiC has bn primntally dtrmind by Kapuria t. al (8) to b 91.6 GPa, which is usd in this wor for computing th lastic modulus of Al/SiC FGM sampls. For Ni/AlO3 systm, th valu of q is tan as 4.5 GPa as usd by Kapuria t. al (8). Now th Young s modulus E of th FGM matrial can b calculatd plicitly using MROM as, E = [ (q+ )/(q+ )+(1 ) ]/[ (q+ )/(q+ )+(1 )] (4) For this wor, othr matrial proprty i.. Poisson s ratio ν and dnsity hav bn calculatd using linar rul of mitur (ROM) as: ν = Vm + Vc = Vm + Vc (5)

5 694 A.A. Khan t al. / Finit Elmnt Modlling for Static and Fr Vibration Rspons of Functionally Gradd Bam 3 YPES OF BEAM Figur : Gomtry and Volum fraction variation through thicnss for th FGM bams. hr Functionally gradd bams ar considrd in this papr, namly bam (a), bam (b) and bam (c). All thr bams ar combination of mtal and cramic with bottom of th bam bing mtal rich and top of th bam bing cramic rich. Bam (a) is a thr layrd fgm bam of Al/SiC and th volum fraction ach layr is pr-dfind as shown in figur. Bam (b) is a 5 layrd fgm bam of Ni/AlO3 and in this bam also th pr-dfind volum fraction is shown in th figur. Bam (c) is a 1 layrd fgm bam of Ni/AlO3 and is bit diffrnt from th othr two bams. In bam (c) ach layr is of qual thicnss i...1h and th bottom most layr is of pur mtal. Morovr, th volum fraction of th rmaining nin layrs is computd using quation 1. Young s modulus and Dnsity of ach layr is calculatd using MROM and ROM rspctivly. 4 ZIG-ZAG HEORY FOR FGM BEAM Considr a functionally gradd rctangular solid bam. h width, thicnss, and lngth of FGM bam ar dnotd by b, h, and a, rspctivly. h FGM substrat is modlld as a laminat of a numbr of prfctly bondd isotropic layrs with diffrnt matrial proprtis, which vary as pr givn

6 A.A. Khan t al. / Finit Elmnt Modlling for Static and Fr Vibration Rspons of Functionally Gradd Bam 695 gradint of volum fractions of th constitunts. h as along th lngth, width, and thicnss ar, y,, rspctivly. h loads givn to th bam do not vary along thicnss dirction. Figur 3: Gomtric Dscription of Layrd Elastic FGM Bam. and h mid plan of th bams is chosn as th y-plan with = = (h/) = =(h/) bing th bottom and top surfacs. h -co-ordinats of th bottom surfac of th -th layr (numbrd from bottom) is dnotd as. h rfrnc plan = ithr passs through or is th bottom surfac of th layr. For bam of small width, assumptions for mathmatical simplifications of 1-D modl ar- (i) Stat of plan strss ( = = =) (ii) ransvrs normal strss nglctd ( =) (iii) Aial and transvrs displacmnt considrd indpndnt of y. For ZIG, w is approimatd by intgrating th constitutiv quations for ɛ by nglcting th contribution of. hus, constitutiv quation for ɛ is According to th basic assumption of th igag thory, ɛ =, = (6) w (,, t ) = (, t) (7) For ZIG th aial displacmnt u is assumd as a combination of a third ordr variation across th laminat thicnss with a layrwis linar variation. So, for layr, u(,,t) = (,t) - (,t) + (,t) + ξ(,t) + 3η(,t) (8) Whr and dnot th translation and rotation variabls of layr Now,

7 696 A.A. Khan t al. / Finit Elmnt Modlling for Static and Fr Vibration Rspons of Functionally Gradd Bam =, +,,=, + +ξ +3 η +, = + ξ +3 η (9) = = ( +ξ + 3η) (1) u (,,t) = (,t), (,t) + () (,t) (11) 4.1 Equation of Motion h Hamilton s principl is rprsntd as blow, δ δɛ )dv + δ )dƭ= (1) Lt, b th normal forcs pr unit ara on th bottom and top surfacs of th bam in dirction. h tndd Hamilton s principl for th lastic bam can b prssd as, using notation L... = 1 And intgrating th trm u i δ u i by parts for now a bam of lngth a undr abov mntiond loading condition. h principl quation givn by Eq. (1) rducs to th form 1 bd a [ u δu + w δw + δ + δ > b p 1 δw(,, t) + b < u δw > a = p δw(, L, t)] (13) δ u, δ w, δ. his variational quation is prssd in trms of δ u, δ w, δ to yild quations of motions and boundary conditions. In th abov quation, u and w can b prssd as u = f () 1 u 1 δu = f () 1 u1 = δ u 1 f 1 () (14) with u 1 = f 1 () = u w, 1 R (15) h inrtia trms can b prssd as

8 A.A. Khan t al. / Finit Elmnt Modlling for Static and Fr Vibration Rspons of Functionally Gradd Bam 697 < u δu + w δw > = < δ u1 f1 whr inrtia I and I ar dfind as () f () u δ w w > = δ u 1 I u 1 + δ w I w (16) I = I I I I I I I I I = < f 1 () f ()> = 1 I (17) with lmnts plicitly givn as, [ I 11, I 1, I 13 ] = < [1,, [ I, I 3 ] = < [ I 33 ] = < [, R ()] > R ()] > R () [ R ()] > (18) h strain incrmnts δ and ar rlatd to virtual displacmnts by, δ = δ u, + w δ, w, = f ()δ w δ, w, = δ u, + δ w, = R, δ (19) () with 1 = u w,,, (1) h strain nrgy trm in variational quation Eq. (13) prssd as δ + δ > = <[δ 1 f 1 ()+ w δ, w, ] + δ R, = δ 1 F1 + δ Q + δ w, N w, > () Whr bam strss rsultants ar dfind as F 1 = N M P (3) [ N, M, P, Q ] = < [1,, h trms du to load on bottom and top surfacs ar prssd as R (), R, () ] > (4)

9 698 A.A. Khan t al. / Finit Elmnt Modlling for Static and Fr Vibration Rspons of Functionally Gradd Bam F = b ( 1 p + At last, th trms du to load at th nd of th bam ar prssd as, p ) (5) δu + δw > = < δ u 1 f 1 () + δ w > = [ δ u 1 F 1 + V δ w ] * * = [ * * * * N u V w - M w + * * P ], (6) Whr th suprscript * rprsnts th man valu at th nds and V = < > Substituting all abov calculatd sub-parts in th original quation Eq. (13), w hav a [ δ 1 u I 1 u + δ w I w + u, N δ w, M + P +δ, F w ] d [ * * * * N u V w * * w, M + * * P ] = Q + w, N w, (7) 5 FINIE ELEMEN MODEL FOR FGM BEAM A two nodd lmnt, having four dgrs of frdom at ach nod, finit lmnt modl is dvlopd for th analysis of lastic layrd FGM bams undr various inds of mchanical loading mploying 1-D igag thory prsntd arlir. h strss rsultants N, M, P, Q in q. (4) can b rlatd to u, w, as: N M P Q = A A A , A1 A A3 w, A13 A3 A33, A u (8) whr, if A rprsnts th bam stiffnss matri, and is givn by, A = A A A A A A A A A = A (9) Now, th bam strss rsultants from Eq. (3) can b prssd as blow, with F 1 = A 1 Q = A (3) h lmnts of abov matrics ar: [ A 11, A 1, A 13 ] = < ˆQ 11 [ 1,, [ A, A 3 ] = < ˆQ 11 [ R () ] >, R () ]>, (31)

10 A.A. Khan t al. / Finit Elmnt Modlling for Static and Fr Vibration Rspons of Functionally Gradd Bam 699 [ A 33 ] = < ˆQ 11 [ A ] = < 55 R () [ ˆQ, R () ] >, R [ R, () ] >, W can dfin strsss, strains, and strss rsultant in a condnsd form and call thm as gnralisd strss,gnralisd strain and gnralisd strss rsultant rspctivly as, û u w, w ; ˆ = [ u, w,, ]; F = [ N M P Q ]; (3) also, ˆ F = ˆ D, and D = A A (in condnsd form) (33) 5.1 Intrpolation Function for FG Bam Elmnt As dscribd abov, a two nodd lmnt, having four dgrs of frdom at ach nod, has bn usd for intrpolating th mchanical variabls for igag thory. Sinc th highst drivativ of u, w, in th variational quation ar u,, w,,, thrfor to mt th convrgnc rquirmnt, of finit lmnt mthod, u and must b C continuous and w should b C 1 -continuous at th lmnt boundaris. hus, u and hav bn intrpolatd using Lagrang intrpolation function and w has bn intrpolatd using Hrmit cubic intrpolation function. Also, it is vidnt that as is dpndnt on only and not on w, this intrpolation schm will not lad to shar locing., Dnoting th valus of an ntity (.) at nod i by (.)i, u, w, ar intrpolatd in an lmnt of lngth a as Whr th lmntal variabl matrics u = N u, = N, w = Nw (34) u, w, ar givn by u = u u 1, = 1, w = w w w w 1 1 (35),, h intrpolation function N and N ar givn by N = [ N1 N ] N N1 N N N 3 4

11 7 A.A. Khan t al. / Finit Elmnt Modlling for Static and Fr Vibration Rspons of Functionally Gradd Bam whr, N 1 =1- (/a) N = (/a) N1 13 / a 3 / a 3 N ( / a) ( / a ) 3 3 N 3 / a /a 4 3 N / a / a (36) Now if w dfin th gnralid displacmnt vctor at lmnt lvl as U u w (37) h gnralid displacmnt vctor û and gnralid strain vctor ˆ can b prssd in trms of U as: û = ˆ NU, ˆ ˆ BU, (38) whr, ˆB = N N N ˆ, = N N N, N, (in comprssd form) N, N (39) 5. Elmnt Inrtia Stiffnss and Load Vctor h contribution of on lmnt of lngth a to th intgral is prssd as a = [ uˆ Iˆ uˆ + ˆ Dˆ uˆ f u ] d (4) whr,

12 A.A. Khan t al. / Finit Elmnt Modlling for Static and Fr Vibration Rspons of Functionally Gradd Bam 71 Î = f = u I, I F (41) If w substitut th prssions for û and ˆ from Eq. (38) in th prssion for w gt, a = [ ˆ ˆˆ ˆ ˆ ˆ ˆ U N INU B DBU N fu ] d = δu M U K U P (4) whr, M K a ˆ N INd ˆˆ, a Bˆ Dˆ ˆB d, a ˆ P N fud (43) h distributd prssur loads 1 givn by p 1 and p, rspctivly, as p, p ar linarly intrpolatd in trms of thir nodal valus 1 p = N 1 p p, =N p, (44) with 1 p = p = On substituting th prssions for N ˆ and f u in th intgral for P, it yilds, p p p p ,, whr, a a ˆ P N fud = N F d (45)

13 7 A.A. Khan t al. / Finit Elmnt Modlling for Static and Fr Vibration Rspons of Functionally Gradd Bam F = bn ( 1 p + p ), Until hr th lmntal dgrs of frdom wr arrangd as U = u u w w w w 1 1 (46) 1 1,, Howvr, for convninc, ths lmntal dgrs of frdom hav bn arrangd li blow, U = u w w u w w 1 1,, 1 1 (47) Accordingly th lmntal matrics hav also bn rarrangd by changing th indics from [1,,3,4,5,6,7,8] to [1,5,,3,6,7,4,8] rspctivly. 5.3 Systm Equations and Boundary Conditions as Summation of all th lmnts of to th variational intgral, th systm quation is obtaind MU + KU = P, (48) In which th M, K, P matrics ar assmbld from thir lmntal matrics rspctivly. In cas of point loads applid at nods ar addd to P, at locations corrsponding to thir dgrs of frdom numbrs. h variationally consistnt boundary conditions at th bam nds ar obtaind as, u = u or N = w = w or N, V = V, w =, w or, M = = or P = M, P, (49) h gomtric boundary conditions for various nd conditions ar as follows, Simply supportd: Clampd: Fr: N =, (movabl) u = (immovabl) w =, M =, P = u =, =, w =, w =, N = N P = P V = V M = M

14 A.A. Khan t al. / Finit Elmnt Modlling for Static and Fr Vibration Rspons of Functionally Gradd Bam 73 6 ABAQUS MODELLING h structur of various bams that ar usd ar shown in Figur. h thicnss of th bam is tan as unity and th lngth of th bam is tan corrsponding to sid-to thicnss ratio. h global -coordinat is tan along th lngth of th bam and th global y-coordinat is tan along th thicnss. h modl is dvlopd using lmnts along th aial dirction and 1 lmnts in ach layr along th thicnss dirction. hr diffrnt boundary conditions ar tan into account vi. simply-supportd, cantilvr and clampd- clampd. h bam is analyd for dflctions and strsss undr various boundary condition whn th bam is subjctd to load woring along th Y- dirction for various sid-to-thicnss ratios (a/h). h figur blow shows th mshd modl of bam (c). h cntr dflction and strsss ar prsntd hr in non-dimnsional form. Figur 4: Mshing of 1 layr Ni/AlO3 bam (c) FGM bam. 7 RESULS AND DISCUSSION h validation of igag thory FE modl dvlopd is prsntd for static and fr vibration rspons of FGM bams. h matrial proprtis and strss to strain transfr ratio for Al /SiC and Ni /AlO3 systms is tan up from Kapuria t al. (8). his ratio is usd to prdict ffctiv lastic modulus of both th FGM systms using MROM. h static analysis and fr vibration rspons of layrd FGM bams, with cramic contnt varying from to 4%, has bn prsntd hr. h matrial proprtis of th basic constitunts Al, SiC, Ni and AlO3 of th FGM bams, considrd for computing th ffctiv proprtis of th layrs, as usd by Kapuria t. al (8) ar listd in abl 1. h young s modulus of th diffrnt layrs of th bam ar computd using MROM whil dnsity of th diffrnt layrs of th bam ar computd using ROM. Proprty Unit Ni AlO3 Al SiC E GPa ν Kg /m abl 1: Constitunt Proprtis of matrials usd. An assssmnt of 1D Zigag modl for th static rspons of lastic functionally gradd bams mad of multipl layrs of isotropic matrials of layr-wis constant composition is prsntd in this chaptr.

15 74 A.A. Khan t al. / Finit Elmnt Modlling for Static and Fr Vibration Rspons of Functionally Gradd Bam h following mchanical load cass ar considrd: 1. Uniformly distributd load at th top surfac of th bam (load cas 1) p = p,. Sinusoidally varying load ovr th top surfac of th bam (load cas ) p = p sin (π/a), whr givs th location of any point on th bam and a is th total span of th bam. h p is a constant with magnitud 1. h rsults that ar prsntd hav bn non-dimnsionalisd using following prssions with, w = (1*w* Y )/hs 4 p, = ( /S p ), = (1 )/S p, S = a/h, Y = GPa (For Ni/AlO3) = GPa (For Al/SiC) h dimnsionlss ntitis ar chosn such that thir valus ar almost indpndnt of S for diffrnt bams. 7.1 Static Rspons h D FE rsults for w,, and at points across th thicnss ar givn in abl and abl 3 for lastic bam (c) and for inhomognity paramtr M = 4. abl shows th rsult for load cas for span to thicnss ratio S = 5, 1 and 4. h load is applid at th top surfac of th bam. h non-dimnsionalid dflction, w is rportd at th top surfac. h obsrvd rror for simply supportd bam (c) with load cas on comparison with Kapuria t al. (6) is within 3% for thic bam with S = 5. For modratly thic bam, S = 1 rror is within 7%. h 1D FE basd on ig-ag thory and D FE Abaqus rsults ar prsntd in abl 3 for bam (c) with cantilvr and clampd-clampd nd conditions for aspct ratios S = 5, 1 and 4 and inhomognity paramtr M = 4. abl shows th rsults in bnding for both th load cass i.. sinusoidally varying and uniformly distributd load. Furthr, it can b sn from th tabl that th rsults ar in clos agrmnt with ach othr, which vntually shows th corrctnss of th FE modl prsntd. h abls 4 and abl 5 prsnt th rsults for 3 layrd bam (a) with Al/SiC FGM systm undr sinusoidally varying load and uniformly distributd load for diffrnt boundary conditions. h inhomognity constant for ach cas is M = 4. A diffrnt FGM systm is considrd in abls 6 and abl 7. h rsults ar prsntd for 5 layrd Ni / AlO3 FGM bam (b) undr sinusoidally varying load and uniformly distributd load on th top surfac for diffrnt nd conditions. h lngth to thicnss ratio for both th load cass is tan as S = 1 with inhomognity paramtr M = 4.

16 A.A. Khan t al. / Finit Elmnt Modlling for Static and Fr Vibration Rspons of Functionally Gradd Bam 75 Finally, it can b obsrvd that th rsults obtaind using 1D FE modl ar in good agrmnt with th rsults obtaind using -D FE abaqus, which provs th corrctnss of th 1-D ig-ag thory basd FE modl. Load cas Entity S Kapuria t al. (6) D FE w abl : Static rspons of 1 layr FGM bam (c) for simply supportd nd condition with M=4. Load cas 1 Load cas Entity w S cantilvr clampd-clampd cantilvr clampd-clampd 1D FE D FE 1D FE D FE 1D FE D FE 1D FE D FE abl 3: Static rspons of 1 layr Ni/AlO3 FGM bam (c) with M=4. Load cas 1 Load cas Entity 1D FE D FE 1D FE D FE w abl 4: Static rspons of 3 layr Al/SiC FGM bam (a) for simply supportd nd condition with M=4.

17 76 A.A. Khan t al. / Finit Elmnt Modlling for Static and Fr Vibration Rspons of Functionally Gradd Bam Load cas 1 Load cas Entity w cantilvr clampd-clampd cantilvr clampd-clampd 1D FE D FE 1D FE D FE 1D FE D FE 1D FE D FE abl 5: Static rspons of 3 layr Al/SiC FGM bam (a) with M=4. Load cas 1 Load cas Entity 1D FE D FE 1D FE D FE w abl 6: Static rspons of FGM bam (b) for simply supportd nd condition with M=4. Load cas 1 Load cas Entity w cantilvr clampd-clampd cantilvr clampd-clampd 1D FE D FE 1D FE D FE 1D FE D FE 1D FE D FE abl 7: Static rspons of 5 layr Ni/AlO3 FGM bam (b) with M=4. 7. Fr Vibration Rspons h convrgnt FE rsults for natural frquncis ar obtaind by modlling th FGM bams for both of th considrd FGM systms vi. Al/SiC and Ni/AlO3 and for diffrnt boundary conditions. h prdictd natural frquncis, n of first fw mods (using MROM and linar ROM basd proprty stimats) obtaind from 1D FE igag modl ar compard with D FE rsults. All th frquncy valus ar givn with unit H. An 8-nod biquadratic plan strss quadrilatral typ lmnt, CPS8R is usd for modling in ABAQUS. Along th bam lngth 8 lmnts and 15 lmnts wr sdd along thicnss dirction. h rsults for natural frquncis ar validatd with rsults prsntd in Kapuria t al. (8). hs rsults ar prsntd in abls 8 to abl 9. It can b sn that prsnt bam modl in conjunction with MROM with th valu of q as 91.6 GPa (Kapuria t al. (8)) prdicts fundamntal natural frquncis of 3 layr Al/SiC cantilvr bams vry accuratly with maimum rror of 1.35% and for clampd-clampd boundary condition with maimum rror of 5.6%. In cas of 5 layr Ni/AlO3 FGM systm, bam (b) has bn modlld using 1D FE as wll as D FE. Sam typ of discrtisation lmnt (CPS8R) has bn usd with 8 lmnts along th lngth and 5 lmnts along thicnss dirction. h rsults hav bn found to b in clos agrmnt with ach othr. h fundamntal frquncis ar compard with rsults of Kapuria t al. (8) and found

18 A.A. Khan t al. / Finit Elmnt Modlling for Static and Fr Vibration Rspons of Functionally Gradd Bam 77 to hav maimum rror of.86% in cas of cantilvr and 1.9% maimum rror in cas of bam clampd at both th nds. Hr th q valu is tan to b 4.5 GPa (Kapuria t al. (8)). Clos agrmnt of th thortical modl rsults with th rsults prsntd in Kapuria t al. (8) provs consistncy and robustnss of 1D FE modl and thus using it, th fundamntal frquncis of th two forsaid FGM systms for simply supportd boundary conditions ar prsntd in abl 11. From abl 11 it can b sn that th natural frquncis obtaind using 1D FE modl ar in good agrmnt with D FE rsults for all thr diffrnt bams. Also, a 1 layr Ni/AlO3 bam modl has bn considrd for frquncy analysis. h rsults for 1 layr Ni/AlO3 bam ar prsntd in abl 1. All th 1D FE rsults ar in good agrmnt with D FE rsults obtaind using ABAQUS. Frquncy (H) MODE Cantilvr Clampd-Clampd Kapuria t al. (8) 1D FE D FE Kapuria t al. (8) 1D FE DFE abl 8: Natural Frquncis of 3 layr Al / SiC FGM bam undr various nd conditions for M =4. Frquncy (H) MODE Cantilvr Clampd-Clampd Kapuria t al. (8) 1D FE D FE Kapuria t al. (8) 1D FE DFE abl 9: Natural Frquncis of 5 layr Ni / AlO3 FGM bam undr various nd condition for M = 4. Frquncy (H) MODE Cantilvr Clampd-Clampd 1D FE D FE 1D FE DFE abl 1: Natural Frquncis of 1 layr Ni / AlO3 FGM bam undr various nd condition for M = 4.

19 78 A.A. Khan t al. / Finit Elmnt Modlling for Static and Fr Vibration Rspons of Functionally Gradd Bam Frquncy (H) MODE Bam (a) Bam (b) Bam (c) 1D FE DFE 1D FE DFE 1D FE DFE abl 11: Natural Frquncis of simply supportd FGM bams for M = 4. h mod shaps for mid-surfac dflction for th first thr flural mods (n = 1,, 3) hav bn plottd using D FE modl. h D FE mod shaps hav bn plottd using th valus that ar normalid with th maimum dflction valus. hy hav also bn obtaind by using igag 1D FE analysis and ar compard with th D FE rsults. hy hav bn prsntd in Fig 5 to Fig 13 for modratly thic bam (S = 1), rspctivly for diffrnt boundary conditions. his comparison shows thir clos agrmnt with ach othr. hrfor thy can b trustfully usd for futur rfrncs. Figur 5: First thr flural mod shaps of 3 layr Al/SiC FGM bam for cantilvr nd condition. Figur 6: First hr Flural mod shaps of 3 layrd Al/SiC FGM bam for simply supportd nd condition.

20 A.A. Khan t al. / Finit Elmnt Modlling for Static and Fr Vibration Rspons of Functionally Gradd Bam 79 Figur 7: First thr flural mod shaps of 3 layrd Al/SiC FGM bam for clampd-clampd nd condition. Figur 8: First thr flural mod shaps of 5 layr Ni/AlO3 FGM bam for cantilvr nd condition. Figur 9: First thr flural mod shaps of 5 layrd Ni/AlO3 FGM bam for clampd- clampd nd condition.

21 71 A.A. Khan t al. / Finit Elmnt Modlling for Static and Fr Vibration Rspons of Functionally Gradd Bam Figur 1: First thr flural mod shaps of 5 layr Ni/AlO3 FGM bam for simply-supportd nd condition. Figur 11: First thr flural mod shaps of 1 layr Ni/AlO3 FGM bam for cantilvr nd condition. Figur 1: First thr flural mod shaps of 1 layr Ni/AlO3 FGM bam for simply-supportd nd condition.

22 A.A. Khan t al. / Finit Elmnt Modlling for Static and Fr Vibration Rspons of Functionally Gradd Bam 711 Figur 13: First thr flural mod shaps of 1 layr Ni/AlO3 FGM bam for clampd-clampd nd condition. abl 1 shows snsitivity analysis i.. th ffct of dividing th FG bams into diffrnt numbr of layrs on natural frquncis using th dvlopd 1-D FE modl. Non-dimnsional natural frquncis up to third mod of vibration ar computd for span-to-thicnss ratios 5 and. h bam usd by hai and Vo (1) is dividd into 8-layrs, 1-layrs and 1-layrs of qual thicnss and th rsults ar compard with hai and Vo (1). h matrial proprtis and boundary conditions ar tan from hai and Vo (1). Young s modulus and dnsity for ach layr ar calculatd using th linar rul of mitur at th cntr of ach layr. Poisson s ratio is assumd to b constant. h inhomognity paramtr (M) is tan as 5. S MODE 8-layr 1-layr 1-layr hai and Vo (1) abl 1: Non-dimnsional natural frquncis of simply supportd FGM bams for M = 5. h sam rsults ar also plottd in Figurs 14 and15. It is vidnt from th graph that with th incras in numbr of layrs th rsults approach to th act solution prsntd by hai and Vo (1). Morovr, it is intrsting to s that th variation in frquncy for diffrnt layrs incrass slightly with incras in th mod of vibration

23 71 A.A. Khan t al. / Finit Elmnt Modlling for Static and Fr Vibration Rspons of Functionally Gradd Bam Figur 14: Snsitivity analysis of frquncy in trms of numbr of layrs for span to thicnss ratio, S=5 of FG bam. Figur 15: Snsitivity analysis of frquncy in trms of numbr of layrs for span to thicnss ratio, S= of FG bam. 8 CONCLUSIONS h dvlopd 1D FE modl yild fairly accurat rsults for static as wll as fr vibration rspons of functionally gradd (FG) bam. In th assssmnt for static rspons, gnrally th rsults of 1D FE ig-ag thory basd FE modl and D FE ABAQUS ar vry clos for th transvrs dflction but for th strsss som fluctuation in th rsults is obsrvd. In cas of fr vibration, natural frquncis and mod shaps hav bn obtaind from both 1D FE as wll as D FE sourcs, for diffrnt nd constraints. h natural frquncis of all th FGM bam systms hav bn found to b in good agrmnt with th rsults from litratur, which provs th corrctnss of th dvlopd modls. Mod shaps hav bn dtrmind using ABAQUS and MALAB. hs ar vry accurat and almost coincid with ach othr. h natural frquncis in gnral incras with th incras of mods of vibration. Also that th matrial constitution has ngligibl ffct on th gnral dflctd shap of th bam undr a particular nd condition. h Snsitivity analysis in trms of numbr of

24 A.A. Khan t al. / Finit Elmnt Modlling for Static and Fr Vibration Rspons of Functionally Gradd Bam 713 layrs of FG bam shows that th computd rsults approach to th act rsult as th numbr of layrs ar incrasd. FGM bam can also b modlld as layr wis using th dvlopd 1D FE modl. Rfrncs ABAQUS/Standard Usr's Manual, Vrsion 6.1, Vol. 1 (1). Alshorbagy, A.E., Eltahr, M.A., Mahmoud, F.F., (11). Fr vibration charactristics of a functionally gradd bam by finit lmnt mthod, Applid Mathmatical Modlling, 35: Chng, Z.Q., Batra, R.C., (). Eact corrspondnc btwn ignvalus of mmbrans and functionally gradd simply supportd polygonal plats, J. Sound and Vibration. 9: Cho, J.R., insly, O.J., (). Functionally gradd matrial: a paramtric study on thrmal-strss charactristics using th Cran-Nicolson-Galrin schm, Computr Mthods in Applid Mchanics and Enginring, 188: Furqan, M., Alam, M.N., (13). Modlling and Analysis of Laminatd Composit Bams using Highr Ordr hory, AMAE, Int. J. on Manufacturing and Matrial Scinc, 3:8-13. Hadi, A., Danshmhr, A.R., Mhrian, S.M.N., Hossini, M. and Ehsani, F., (13). Elastic Analysis of Functionally Gradd imoshno Bam Subjctd o ransvrs Loading, chnical Journal of Enginring and Applid Scincs, 3: Hill, R., (1965). A slf-consistnt mchanics of composit matrial, Journal of Mchanics and Physics of Solids, 13:13. Jons, R.M., (1999). Mchanics of Composit Matrial, Scond Edition. Kadoli, R., Ahtar, K., Gansan, N., (8). Static analysis of functionally gradd bams using highr ordr shar dformation thory, Applid Mathmatical Modlling, 3: Kapuria, S., Bhattacharyya, M., Kumar, A.N., (6). Assssmnt of coupld 1D modls for hybrid piolctric layrd functionally gradd bams, Composit structurs, 7: Kapuria, S., Bhattacharyya, M., Kumar, A.N., (8). Bnding and fr vibration rspons of layrd functionally gradd bams: A thortical modl and its primntal validation, Composit Structurs 8:39 4. Li, S., Wan, Z., Zhang, J., (14). Fr vibration of functionally gradd bams basd on both classical and first-ordr shar dformation bam thoris, Applid Mathmatics and Mchanics, 35: Mahamood, R.M., Ainlabi, E.., Shula, M. and Pityana, S., (1). Functionally Gradd Matrial: An Ovrviw, Procdings of th World Congrss on Enginring. MALAB and Statistics oolbo Rlas 1a, h MathWors, Inc., Natic, Massachustts, Unitd Stats. Mhta, R., Balaji, P. S., (13). Static and Dynamic Analysis of Functionally Gradd Bam, PARIPEX Indian Journal of Rsarch, :8-85. Mohanty, S. C., Dash, R. R., Rout,., (1). Static and Dynamic Stability Analysis of a Functionally Gradd imoshno Bam, 1 [33 pags]. Mori,., anaa, K., (1973) Avrag strss in matri and avrag lastic nrgy of matrials with misfitting inclusions, Acta Mtallurgica, 1: Nguyn, N.., Kim, N.I., Cho, I., Phung, Q.. and L, J., (14). Static analysis of transvrsly or aially functionally gradd taprd bams, Matrials Rsarch Innovations, 18:6-64. Nguyn,.K., Sab, K., Bonnt, G., (8). First ordr shar dformation plat modls for functionally gradd matrials. Composit Structurs, 83:5-36. Rahman, N. and Alam, M. N., (1). Activ vibration control of a piolctric bam using PID controllr: Eprimntal study, Latin Amrican Journal of Solids and Structurs, 9: Rahman, N. and Alam, M. N., (1). Dynamic Analysis of Laminatd Smart Bams using Zigag hory, Intrnational Journal of Mchanics Structural, 3:35-46.

25 714 A.A. Khan t al. / Finit Elmnt Modlling for Static and Fr Vibration Rspons of Functionally Gradd Bam Rddy, J.N., (1984). A simpl highr ordr thory of laminatd composit plats. Journal of Applid Mchanics, 51: Rddy, J.N., (1997). Analysis of Laminatd Composit Plats and Shlls, Scond Edition, CRC Prss. Sanar, B.V., (1). An lasticity solution for functionally gradd bams, Composit Scinc and chnology 61: Sina, S.A., Navai, H.M., Haddadpour, H., (9). An analytical mthod for fr vibration analysis of functionally gradd bams, Matrials and Dsign, 3: hai, H.., Vo,.P., (1). Bnding and fr vibration of functionally gradd bams using various highr-ordr shar dformation bam thoris, Intrnational Journal of Mchanical Scincs, 6: omota, Y., Kuroi, K., Mori,., and amura, I., (1976). nsil Dformation of wo-ductil-phas Alloys: Flow Curvs of α γ F Cr Ni Alloys. Matrial Scinc Enginring A, 4: Wattanasaulpong, N., Ungbhaorn, V., (1). Fr Vibration Analysis of Functionally Gradd Bams with Gnral Elastically End Constraints by DM, World Journal of Mchanics, : Yaghoobi, H. and Fridoon, A., (1). Influnc of nutral surfac position on dflction of functionally gradd bam undr uniformly distributd load, World Applid Scinc Journal, 1: