Non-Linear Bending Analysis of Moderately Thick Functionally Graded Plates Using Generalized Differential Quadrature Method

Transcription

1 Iteatioal Joual of Aeospace Scieces, (3): DOI:.593/j.aeospace.3.4 No-Liea Bedig Aalsis of Modeatel Thick Fuctioall Gaded Plates Usig Geealized Diffeetial Quadatue Method J. E. Jam *, S. Maleki, A. Adakhshideh Composite Mateials ad Techolog Cete, M UT, Teha, Ia Abstact Liea ad o-liea bedig aalsis of modeatel thick fuctioall gaded (FG) ectagula plates with diffeet bouda coditios ae peseted usig geealized diffeetial quadatue (GDQ) method. The modulus of elasticit of plates is assumed to va accodig to a powe law distibutio i tems of the volume factios of the costituets. Based o the fist-ode shea defomatio theo ad Vo Kama tpe o-lieait, the goveig sstem of equatios iclude a sstem of thitee patial diffeetial equatios (PDEs) i tems of ukow displacemets, foces ad mo mets. To deive liea sstem of equatios, o-liea tems ae omitted i fome equatios. Pesece of all plate vaiables i the goveig equatios povides a simple pocedue to satisf diffeet bouda coditios. Successive applicatio of the GDQ techique to the goveig equatios esulted i a sstem of o-liea algebaic equatios. The Newto Raphso iteative scheme is the emploed to solve the esultig sstem of o-liea equatios. Illustative eamples ae peseted to demostate accuac ad apid covegece of the peseted GDQ techique. Accuac of the esults fo both displacemet ad stess compoets ae veified with compaig the peset esults with those of aaltical ad fiite elemet methods. It is foud that the theo ca pedict accuatel the displacemet ad stess compoets eve fo small umbe of gid poits. Kewods No-liea, Fuctioall Gaded, Geealized Diffeetial Quadatue. Itoductio To avoid the mateial mis match, a special mateial amed fuctioall gaded mateial (FGM) was poposed b a goup of mateial scietists i Japa i[], i which the mateial popeties ae vaied smoothl ad cotiuousl fom oe suface to the othe. The gadatio of the mateial popeties though the thickess elimiates jumps o abupt chages i the stess ad displacemet distibutios. The FGM is suitable fo vaious applicatios, such as themal coatigs of baie fo space stuctues, ceamic egies, optical thi laes, biomateial electoics, uclea fusios, gas tubies, etc. I ealit, ma plate stuctues ae subjected to high load levels that ma udego lage deflectios. The effect of this lage deflectio is to stetch the middle plae of the plate iducig membae stesses. B this membae actio, the load caig capacit of the plate is iceased to a lage etet. Fo plates of this kid, the goveig diffeetial equatios ae o-liea. The o-lieait of the goveig equatios ma be duo to eithe * Coespodig autho: jejam@mail.com (J. E. Jam) Published olie at Copight Scietific & Academic Publishig. All Rights Reseved mateial o-lieait o geometic o-lieait. I this pape ol geometic o-lieait will be cosideed. This o-lieait is due to the fact that the stai displacemet elatios ae o-liea. A cosideable amout of liteatue eists o the o-liea aalsis of ectagula plates. Zekou[] used geealized shea defomatio theo fo bedig aalsis of fuctioall gaded plates. Woo ad Meguid[3] povided a aaltical solutio fo the lage deflectios of fuctioall gaded plates ad shallow shells ude tasvese mechaical loads ad a tempeatue field usig Fouie seies. Yag ad She[4] studied oliea bedig aalsis of shea defomable fuctioall gaded plates subjected to themo-mechaical loads. The also ivestigated the lage deflectio ad postbucklig of fuctioall gaded ectagula plates ude tasvese ad i-plae loads b usig a semi-aaltical appoach[5]. Tsug ad Shukla[6] povided a eplicit solutio fo the oliea static ad damic esposes of the fuctioall gaded ectagula plate usig the quadatic etapolatio techique fo lieaizatio, fiite double Chebshev seies fo spatial discetizatio of the vaiables ad Houbolt time machig scheme fo tempoal discetizatio. Redd[7] developed Navie's solutios fo ectagula plates ad fiite elemet models to stud the oliea damic espose of FG plates usig the highe-ode shea defomatio plate theo. Based

2 5 J. E. Jam et al.: No-Liea Bedig Aalsis of Modeatel Thick Fuctioall Gaded Plates Usig Geealized Diffeetial Quadatue Method o Redd's highe-ode shea defomatio plate theo, She[8] also studied the oliea bedig of a simp l -suppoted fuctioall gaded ectagula plate subjected to mechaical ad themal loads. Navazi et al[9] developed a aaltical solutio fo oliea clidical bedig of a fuctioall gaded plate. I this pape it is show that the liea plate theo is iadequate fo aalsis of fuctioall gaded plate eve i the small deflectio age. Ghaad Pou ad Aliia[] obtaied a aaltical solutio fo lage deflectio of ectagula fuctioall gaded plates ude pessue loads b miimizatio of the total potetial eeg of the plate. Navazi ad Haddadpou[] peseted a eact solutio fo oliea clidical bedig of shea defomable fuctioall gaded plates. Zhao ad Liew[] ivestigated the oliea espose of fuctioall gaded plates ude mechaical ad themal loads usig the mesh-fee method. Babosa ad Feiea[3] used fiite elemet method fo oliea aalsis of fuctioall gaded plates. Hoa et al.[4] peseted a aalsis o oliea damic chaacteistics of a simpl suppoted fuctioall gaded ectagula plate subjected to the tasvesal ad i-plae ecitatios i time depedet themal eviomet. It is well-kow that aaltical methods ae ol applicable to paticula poblems such as o-liea bedig of FG plates with at least two opposite sides simpl suppoted. Thus, umeical techiques, as alteatives to aaltical appoaches, have bee developed to obtai solutios fo FG plates subjected to diffeet tpes of bouda coditios. Amog these umeical studies, oe ca efe to diffeetial quadatue (DQ)[5] ad geealized diffeetial quadatue (GDQ)[6-7]. The DQ ad GDQ techiques wee peseted b Bellma et al[5] ad Shu[6] {} ε = { ε } + z{} κ as efficiet pocedues to obtai solutios of patial diffeetial equatios with elativel small umbe of gid poits ad less computatioal effot. The mai advatage of the GDQ ove DQ is its ease of the computatio of weightig coefficiets without a estictio o the choice of gid poits. I this pape, geealized diffeetial quadatue method is emploed to obtai solutios fo liea ad o-liea bedig aalsis of modeatel thick FG plates. Fo the assessmet of the accuac ad covegece of the method, the peset esults ae compaed with those of othe ivestigatos. It is foud that the GDQ method pedicts accuatel both the displacemet ad stess compoets.. Goveig Equatio Coside a ectagula plate of side A,B ad thickess h, show i Fig.. Accodig to the fist-ode shea defomatio theo, the displacemet field is epessed as[8]: Uz (,, ) = u (, ) + zφ (, ) Vz (,, ) = v (, ) + zφ (, ) () Wz (,, ) = w (, ) whee u,v ad w deote the displacemets i, ad z diectios ad φ, φ ae otatios of z ad z plaes of mid-plae, espectivel. The stai displacemet elatios due to vo Kámá-tpe o-lieait, ca be epessed as follows[8]: u w v w u v w w w w ε = + ( ), ε = + ( ), ε = + + ( )( ), ε z = + φ, ε z = + φ φ φ φ φ κ =, κ =, κ = + () 4 3 B A h Fi gue. Geomet of plate Itegatig the elevat stess compoets though the thickess of the plate, i-plae stess esultats (N, N, N ), couple esultats (M, M, M ), ad the tasvese shea stess esultats (Q, Q ), ae obtaied. Cosequetl costitutive equatios ae stated as follows[8]:

3 Iteatioal Joual of Aeospace Scieces, (3): N A B ε Q A55 A54 ε z =, Ks M B D = κ Q A45 A 44 ε z whee K s is shea coectio facto ad i all peseted esults (K s =5/6) ad h / ( Aij, Bij, Dij ) = ( Qij )(, z, z ) dz i, j =,, 4, 5, 6 h./ (3b) Fo FGMs we have: Ez ( ) ν Ez ( ) Ez ( ) Q = Q =, Q = Q =, Q 44 = Q55 = Q66 = ν ν ( + ν) Q6 = Q6 = Q6 = Q6 = Q45 = Q54 = (4) z Ez ( ) = ( Ec Em) + + Em h whee c, m deote ceamic ad metal, espectivel. Fiall, the costitutive equatios fo FG plates i tems of displacemet ad otatio compoets ca be deived as[8]: u v u u N A A6 A A6 B B6 B B6 A + A + A6 u v v u N A6 A66 A6 A66 B6 B66 B6 B66 A6 + A6 + A66 u v v u N A A6 A A6 B B6 B B6 A + A + A6 u v v u M B B6 B B6 D D6 D D6 B + B + B6 u v v u M B6 B66 B6 B66 D6 D66 D6 D66 B6 + B6 + B66 (3a)

4 5 J. E. Jam et al.: No-Liea Bedig Aalsis of Modeatel Thick Fuctioall Gaded Plates Usig Geealized Diffeetial Quadatue Method u v v u M B B6 B B6 D D6 D D6 B + B + B6 w w Q KA s 55 β KA s 45 β w w + + = Q KA s + β KA s β + = Usig the piciple of miimum total potetial eeg, the oliea equilibium equatios ca be show to be[8]: M N N N N + = + = Q Q N N w w w w N + + N N w N w w N + Q(, θ ) = M M M + Q = + Q = Eight costitutive equatios (5) ad five equatios of equilibium (6) ae the thitee patial diffeetial equatios goveig the oliea bedig of FG plates. Udelied tems i equatios () ae o-liea tems which ae omitted fo liea aalsis... Bouda Coditios Two diffeet bouda coditios ae cosideed at each edge of the plate as[8]: -Clamped (C): u v = w = = φ = -Simpl suppoted (S): = φ (at all edges) (7) u v = w = φ = M = at ( = cte) = = v = w = = M u φ = at ( = cte) (5) (6) (8a) (8b) 3. Applicatio of GDQ I ode to use GDQ techique the plate is divided ito gid poits whee (,) of gid poits is zeos of the well-kow Chebshev polomials[6]. A i B j = [ cos( π)] i =,,...,, = [ cos( π)] j =,,..., i j Accodig to the GDQ method, the patial deivative of a ukow fuctio with espect to a vaiable is appoimated b a weighted sum of fuctio values at all discete poits i that diectio. Cosideig a fuctio f() with discete gid poits, we have[6]: m f ( ) ( m ) = Cij f ( j ) i =,,..., () m j = i (9) whee j is the discete poit i the vaiable domai, ad f( j ) ad C ae the fuctio values at these poits ad elated ( m ) ij

5 Iteatioal Joual of Aeospace Scieces, (3): weightig coefficiets, espectivel. I ode to detemie the weightig coefficiets ( m ) C ij, the Lagage itepolatio basic fuctios ae used as the test fuctios ad eplicit fomulas fo computig these weightig coefficiets ae give b[6] () Φ( k ) Ckj = k, j =,,..., ; k j ( ) Φ( ) k j j () whee Φ ( ) = ( ) k, j =,,..., ; k j () k k j k= Fo the fist-ode deivative (i.e., m=) ad also fo highe-ode deivatives, oe ca use the followig elatios iteativel : ( ) C ( ) ( ) () kj Ckj = Ckk Ckj ; k j ( k j) ( ) ( ) kk kl l= l k C = C k =,,...,, The et step is to discetize the goveig equatios based o the defiitios give i (). Fo eample, the fist equatio i (5) at a sample gid poit ( i, j ) ca be witte as: (3) () () N ( i, j ) + A C iku( k, j ) + C ikw( k, ) j + k= k= () () () () A C jkv( i, k ) + ( ) C jkw( i, k ) B C ikφ ( k, j ) B C jkφ ( i, k ) k= + + = k= k= k= (4) whee ( i, j ) is a gid poit iside the plate with i =,,..., ad j =,,...,. () () C ij, C ae weightig coefficiets fo ij fist ode patial deivatives[6]. Followig the pocedue eplaied above leads to a sstem of 3( ) oliea algebaic equatios with the same umbe of ukows. It should be oted that applig bouda coditios (7, 8) to the obtaied algebaic equatios, five of the thitee ukow paametes at each bouda ode will vaish. At last a icemetal-iteative method should be used to solve the esultig oliea sstem of equatios. I the peset aalsis, the solutio algoithms ae based o the Newto Raphso method. This method ields the followig lieaized sstem of equatios fo the icemetal solutio at (+)th iteatio[8]: whee { δ } = Kˆ T { } { } ˆ T { } ( ) { R} ( ) { } { ˆ} R = K F (5) (6) I Equatios (5) ad (6) { } is the vecto of ukows at the th iteatio, { R } is the esidual ad { F ˆ } is the foce vecto of sstem of equatios. The taget stiffess mati is defied as: ({ } ) { R} { } Kˆ T The total solutio i (+)th iteatio is obtaied fom: { } { } { } δ + (7) = + (8) At the begiig of the iteatio (i.e., =), the ukow vecto is assumed to be zeo (i.e., { } = ). Due to this assumptio, the solutio at the fist iteatio is the liea solutio, sice the oliea stiffess mati educes to the liea oe. The iteatio pocess is cotiued util the diffeece betwee educes to a { } ad { } +

6 54 J. E. Jam et al.: No-Liea Bedig Aalsis of Modeatel Thick Fuctioall Gaded Plates Usig Geealized Diffeetial Quadatue Method peselected eo toleace[8]. I this stud the eo citeio is assumed to be: k= k= + k + k k < ε (9) whee is the total umbe of ukows i a GDQ odal poit ad ε is the eo toleace. 4. Result ad Discussio To demostate the efficiec ad accuac of the peset method, GDQ esults fo both liea ad o-liea bedig of FG ectagula plates ae validated with those of othe umeical ad aaltical methods. I all eamples the plate is cosideed to be costucted fom Alumium-Alu mia FGM ad load (Q) is cosideed to be uifom. Fist eample is liea bedig of full simpl FG plate. Fig. shows omalized deflectio of SSSS ectagula FG plate vesus aspect atio A/B i compaiso with aaltical esults of Zekou[]. As well ca be see i the figue the deflectio is maimum fo the metallic plate ad miimum fo the ceamic plate. Deflectio of FG plates is itemediate to that of the metallic ad ceamic plates. Nomalized deflectio of SSSS squae FG plate vesus side to thickess atio A/h is also show i Fig. 3. It ca be cocluded fom the figue, fo thi plates the effect of thickess o o-dimesioal deflectio is egligible ad esults of fist ode theo is the same as classical theo. The sstem of equatios which is used hee guaatees the same ode of accuac fo pedictios of vaious stess ad displacemet compoets. I ode to pove the idea vaiatio of omalized stess i, diectio of SSSS FG plate though the thickess is demostated i Figs 4, 5, espectivel. The i-plae stesses ae compessive thoughout the lowe half ad tesile thoughout the uppe half of plate. It is illustated that miimum value of zeo fo all i-plae stesses occus at z/h=.53 ad it is idepedet of aspect atio ad side to thickess atio. Secod eample egads to o-liea aalsis fo FG plates. Fig. 6 shows vaiatio of omalized cetal stess of full simpl suppoted FG squae plate of side A=mm ad thickess h=mm though the thickess. Nomalized cetal deflectio of this plate vesus load is also show i Fig. 7. Icluded i these figues ae also aaltical esults of Ghaad Pou ad Aliia[]. It is see that the GDQ esults ae i good ageemet with those obtaied b aaltical solutio. Fi gue. Nomalized deflectio of SSSS FG ect agula plat e vesus A/B (liea aalsis) Figue 3. Nomalized deflectio of SSSS FG squae plate vesus A/h (liea aalsis) Fi gue 4. Vaiatio of omalized stess i diectio of SSSS FG squae plat e (liea aalsis)

7 Iteatioal Joual of Aeospace Scieces, (3): show i Fig.. Last eamples of this stud egads to o-liea aalsis of SSCS ad CCSS plates. Vaiatio of omalized deflectio of SSCS ad CCSS squae FG plate at / B=.5 ae show i Fig. 8, 9 espectivel. I this 4 4 figues q = QA / E ad esults ae validated with mh = ABAQUS fiite elemet esults. Fig. 8 illustate the covegece ate of method with vaious umbe of gid poits. As well ca be see the method is fast covegece ad the pedicted esults with 9 9 gid poits ae ve good. Good accuac of GDQ method is a oticeable poit of all eamples. Fi gue 5. Vaiatio of omalized stess i diectio of SSSS FG ectagula plate though the thickess (liea aalsis) Figue 8. Vaiatio of omalized deflectio of SSCS FG squae plate at / B=.5 (o-liea aalsis) Fi gue 6. Vaiatio of omalized cetal stess of SSSS FG squae plate though the thickess(o-liea aalsis) Fi gue 9. Vaiatio of omalized deflectio of CCSS FG squae plate at / B=.5 (o-liea aalsis) Fi gue 7. Nomalized cetal deflectio of SSSS FG squae plat e vesus load (o-liea aalsis) As oticed befoe pesece of all vaiables i sstem of equatios povides a simple pocedue to appl diffeet bouda coditios. To defie diffeet bouda coditios the edges of the plate ae umbeed fom to 4 as 5. Coclusios The Diffeetial Quadatue (DQ) method is used to obtai umeical solutio fo liea/o-liea bedig of FG plates subjected to uifoml distibuted load ad diffeet bouda coditios. Compaisos of the esults with those available i the liteatue show good ageemet fo both displacemets ad stess compoets. It is show that the sstem equatio which is used i this stud povides a simple pocedue to appl diffeet bouda coditios. Results also evealed that the method is efficiet ad

8 56 J. E. Jam et al.: No-Liea Bedig Aalsis of Modeatel Thick Fuctioall Gaded Plates Usig Geealized Diffeetial Quadatue Method accuate ad theefoe, could be used fo moe complicated poblems. REFERENCES [] Koizumi M. FGM activities i Japa. Composites 994:8, 4. [] Zekou A M. Geealized shea defomatio theo fo bedig aalsis of fuctioall gaded plates. Applied Mathematical Modellig 6:3, [3] Woo J ad Meguid S A. Noliea aalsis of fuctioall gaded plates ad shallow shells. Iteatioal Joual of Solids ad stuctues :38, [4] Yag J ad She H S. Noliea bedig aalsis of shea defomable fuctioall gaded plates subjected to themo-mechaical loads ude vaious bouda coditios. Composites: Pat B 3:34,3 5. [5] Yag J ad She H S. No-liea aalsis of fuctioall gaded plates ude tasvese ad i-plae loads. Iteatioal Joual of No-Liea Mechaics3:38, [6] Tsug-Li W U, Shukla K K ad Hug J H. Noliea static ad damic aalsis of fuctioall gaded plates. Iteatioal Joual of Applied Mechaics ad Egieeig 6:(3), [7] Redd J N. Aalsis of fuctioall gaded plates. Iteatioal Joual fo Numeical Methods i Egieeig :47, [8] She H S. Noliea bedig espose of fuctioall gaded plates subjected to tasvese loads ad i themal eviomets. Iteatioal Joual of Mechaical Scieces :44, [9] Navazi H M, Haddadpou H ad Rasekh M. A aaltical solutio fo oliea clidical bedig of fuctioall gaded plates. Thi-Walled Stuctues 6:44,9 37. [] GhaadPou S A M ad Aliia M M. Lage deflectio behavio of fuctioall gaded plates ude pessue loads. Composite Stuctues 6:75,67 7. [] Navazi H.M, Haddadpou H. Noliea clidical bedig aalsis of shea defomable fuctioall gaded plates ude diffeet loadigs usig aaltical methods. Iteatioal Joual of Mechaical Scieces 8: 5, [] Zhao X, Liew K.M. Geometicall oliea aalsis of fuctioall gaded plates usig the elemet-fee kp-ritz method. Compute Methods i Applied Mechaics ad Egieeig 9: 98, [3] Babosa J A T, Feeia A J M. Geometicall oliea aalsis of fuctioall gaded plates ad shells. Mechaics of Advaced Mateials ad Stuctues : 7,4 48. [4] Hao Y X, Zhag W, Yag J ad Li S. Y. Noliea damic espose of a simpl suppoted ectagula fuctioall gaded mateial plate ude the time-depedet themal mechaical loads, Joual of Mechaical Sciece ad Techolog : 5 (7), [5] Bellma R E ad Casti J. Diffeetial quadatue ad log tem itegatio. Joual of Mathematical Aalsis ad Applicatios 97:34, [6] Shu C ad Du H. Implemetatio of clamped ad simpl suppoted bouda coditios i the GDQ fee vibatio aalsis of beams ad plates. Iteatioal Joual of Solids ad Stuctues 997:34(7), [7] Aghdam M M, Faahai R N, Dasht M ad Rezaei Nia S M. Applicatio of geealized diffeetial quadatue method to the bedig of thick lamiated plates with vaious bouda coditios. Applied Mechaics ad Mateials 6:5-6, [8] Redd J N. Mechaics of lamiated compoet plates ad shells theo ad aalsis, Secod editio, CRC Pess, 4.