ANALYTICAL SOLUTION FOR NONLINEAR BENDING OF FG PLATES BY A LAYERWISE THEORY

Transcription

1 6 TH ITERATIOAL COFERECE O COMPOSITE MATERIALS AALYTICAL SOLUTIO FOR OLIEAR BEDIG OF FG PLATES BY A LAYERWISE THEORY M. Taha*, S.M. Mrababaee** *Departmet of Mechacal Egeerg, Ferdows Uversty of Mashhad, Mashhad, Ira **Khorasa Research Ceter for Techology Developmet (KRCTD), Qucha Hghway, Mashhad, Ira Keywords: Fuctoally graded materals, Geometrc olearty, Plates, Aalytcal soluto, Layerwse method Abstract A layerwse theory s used to aalye aalytcally dsplacemets ad stresses fuctoally graded (FG) composte plates cyldrcal bedg subected to thermomechacal loadgs. The plates are assumed to have sotropc, two-costtuet materal dstrbuto through the thcess, ad the modulus of elastcty of the plate s assumed to vary accordg to a power-law dstrbuto terms of the volume fractos of the costtuets. The olear stra-dsplacemet relatos the vo Kármá sese are used to study the effect of geometrc olearty. The equlbrum equatos are solved eactly ad also by usg a perturbato techque. umercal results are preseted to show the effect of the materal dstrbuto o the deflectos ad stresses. Itroducto I covetoal lamated composte materals, there s a hgh chace that debodg wll occur at some etreme loadg codtos. O the other had, gradually varyg the volume fracto of the costtuets ca resolve ths problem. Fuctoally graded materals (FGMs) are composte materals whch ehbt a progressve chage composto, structure, ad propertes as a fucto of spatal drecto wth the materal. May studes for thermal stress ad lear thermal bedg of FGM plates are avalable the lterature (e.g, see [-3]). However, vestgatos olear aalyss of FGM plates uder thermal or mechacal loadg are lmted umber. For eample, Pravee ad Reddy [4] vestgated the respose of fuctoally graded ceramc- plates usg a plate fte elemet that accouts for the trasverse shear stras, rotary erta ad moderately large rotatos the vo Kármá sese. Reddy [5] preseted solutos for rectagular fuctoally graded plates based o the thrd-order shear deformato plate theory. The formulato accouted for the thermomechacal couplg, tme depedecy, ad the vo Kármá-type geometrc olearty. Usg a asymptotc method, the three-dmesoal thermo-mechacal deformatos of fuctoally graded rectagular plate were vestgated by Reddy ad Cheg [6] ad the dstrbutos of temperature, dsplacemets ad stresses the plate were calculated for dfferet volume fracto of ceramc costtuet. She [7] aalyed olear bedg of a smply supported, fuctoally graded rectagular plate subected to a trasverse uform or susodal load ad thermal evromets. He obtaed goverg equatos based o Reddy s hgher-order shear deformato plate theory ad solved them by a med Galer-perturbato techque. Based o the vo Kármá theory, Woo ad Megud [8] derved a aalytcal soluto epressed terms of Fourer seres for the large deflecto of fuctoally graded plates ad shallow shells uder trasverse mechacal loadg ad a temperature feld. Yag ad She [9] usg a sem-aalytcal approach aalyed the olear bedg ad post-buclg behavors of FG rectagular plates subected to combed acto of trasverse ad -plae loads. Taha et al. [] aalytcally aalyed fuctoally graded beams subected to thermomechacal loadgs based o a frst-order shear deformato theory. Hseh ad Lee [] solved the verse problem of a fuctoally

2 M. TAHAI, S.M. Mrababaee graded ellptcal plate wth large deflecto ad dsturbed boudary uder uform load. They derved the goverg equatos of a th plate wth large deflecto based o the classcal olear vo Kármá plate theory. The they employed a perturbato techque o dsplacemet terms coucto wth Taylor seres epaso of the dsturbed boudary codtos to solve the oclasscal problem. Agarwal et al. [] used the estg statcally eact beam fte elemet based o the frst order shear deformato theory to study the geometrc olear effects o statc ad dyamc resposes sotropc, composte ad fuctoally graded materal beams. They utled both vo Kármá stra tesor ad Gree s stra tesor the statc case, whereas, for the wave propagato studes oly the vo Kármá stras were used. It s teded here to accurately determe the dsplacemets ad stresses fuctoally graded plates cyldrcal bedg subected to thermomechacal loadgs. To ths ed, based o a layerwse theory the goverg equatos are obtaed. Also, the olear stra-dsplacemet relatos are used to study the effect of geometrc olearty. The equlbrum equatos are solved eactly for FGM plates wth the same boudary codtos ad also by usg a perturbato techque for FGM plates wth geeral boudary codtos. umercal results are preseted to show the fluece of materal propertes, plate geometry, mechacal loadg ad the temperature feld o the resultg trasverse deflecto ad stress state. Theoretcal Formulato. Dsplacemet Feld ad Stras Cosder a fuctoally graded plate of thcess h, legth a the -drecto, ad fte etet the y-drecto. Sce the plate s log, t may safely be assumed that a state of plae stra ests. Hece, the followg dsplacemet feld s assumed: u (, y, ) = u ( ) + U ( ) Φ ( ) u (, y, ) = =,,, + () u (, y, ) = w( ) 3 It s to be oted that a repeated de dcates summato over all values of that de. I Eqs. () u ad w are the dsplacemet of the pots the mddle surface of the plate the - ad -drectos, respectvely, U (=,,, +) s the dsplacemet compoet of all pots located o the th plae the -drecto, ad Φ s a cotuous fucto of the thcess coordate (global terpolato fucto). Also deotes the total umber of umercal layers cosdered a plate. I the preset study we wsh to vestgate the effect of geometrc o-learty o the respose quattes. Therefore, the vo Kármá-type of geometrc o-learty s tae to cosderato the stra-dsplacemet relatos. Substtutg Eqs. () the approprate stra-dsplacemet relatos [3] results : ε = u + ( w) + UΦ ε = ε = γ = γ = y y y γ = w + U Φ () where a prme dcates a ordary dervatve wth respect to the correspodg depedet varable.. Costtutve Relatos We cosder a fuctoally graded plate, whch s made from a mture of ceramcs ad s. It s assumed that the composto propertes of FGM vary through the thcess of the plate. The varato of materal propertes ca be epressed as: ( ) p( ) = pt pb Vt + pb (3) where p deotes a geerc materal property le modulus, p t ad p b deote the correspodg propertes of the top ad bottom faces of the plate, respectvely. Also V t Eq. (3) deotes the volume fracto of the top face costtuet ad follows a smple power-law as: Vt = + (4) h where s the thcess coordate ( h/ h/) ad s a parameter that dctates the materal varato profle through the thcess. Here we assume that modul E ad G, coeffcet of thermal epaso α, ad thermal coductvty vary accordg to Eq. (3) ad the Posso's rato ν s assumed to be a costat. The lear costtutve relatos are: σ Q Q ε σ y Q Q = εy α T σ Q 66 γ y y

3 AALYTICAL SOLUTIO FOR OLIEAR BEDIG OF FG PLATES BY A LAYERWISE THEORY σ Q γ y 44 y = σ Q 55 γ where E( ) Q = Q = ν, ν E( ) Q = ν E ( ) Q44 = Q55 = Q66 = = G( ) ( + ν ) (5) (6) ad T s the temperature chage from a stress-free state that wll be obtaed by solvg the oedmesoal heat trasfer equato..3 Equlbrum Equatos Usg the prcple of mmum total potetal eergy [3], the equlbrum equatos ca be show to be: d δ u : = d (7a) dm δu : Q = d (7b) δ w: dq d dw + + q( ) = d d d (7c) where q() s the trasverse load o the top surface of the plate. I Eqs. (7) the geeraled stress resultats are defed as: h / = σ σ h / h /, Q = σ Φ σ Φ h / (, Q ) (, ) d ( M ) (, ) d (8) The boudary codtos cosst of specfyg the followg quattes at =± a : Geometrc (Essetal) u Force (atural) w w + Q U M (9) Upo substtuto of Eqs. (5) to Eqs. (8), the geeraled stress resultats terms of dsplacemet compoets wll be obtaed whch ca be preseted as follows: = A [ u + ( w ) /] + B U T ( T) = [ + ( ) /] + M B u w D U M Q = A w + A U Q = A w + A U () where as: + () = + () = + () = + () = + () = A = Q d, =,55 A = Q Φ d B = Q Φ d A = Q Φ Φ d D = Q Φ Φ d () The thermal resultats Eqs. () are defed ( ) T T + ( ) = (, M ) = Q (, Φ ) α Td () Lastly, the goverg equatos of equlbrum are obtaed by substtutg Eqs. () to Eqs. (7). 3 Aalytcal Solutos I ths secto a fuctoally graded plate subected to a uform trasverse load o ts top surface ad/or thermal load s cosdered. I order to solve the equlbrum equatos thermal loadgs the temperature feld should be ow. It s assumed that oe value of temperature s mposed o the bottom surface ad the other value o the top surface of the plate. I ths case, the temperature dstrbuto through the thcess ca be obtaed by solvg a smple steady state heat trasfer equato through the thcess of the plate. Ths equato s gve by: d dt ( ) d = d (3) ad the boudary codtos are T = Tb at = h/ ad T = Tt at = h/. It s readly see that the soluto to Eq. (3) s: where ( ξ) ( ξ) ( ) T = c ha + c (4) t b ξ = +, A ( ξ ) = h ( Tt Tb)( t b) c = ha [ () A()] dξ ξ µ 3

4 M. TAHAI, S.M. Mrababaee c TA t () TA b () = A () A () (5) wth µ = b ( b t). It s to be oted that the tegral of A ( ξ ) Eqs. (5) has aalytcal soluto for =., =.5, ad all teger values. For other values of, ths tegral must be solved umercally. I what follows two soluto methodologes for Eqs. (7) are preseted. 3. Eact Solutos I ths secto, the boudary codtos of the plate at =± a / are assumed to be the same. Before dscussg the procedure adopted for solvg Eqs. (7), t s approprate to dcate here that ths layerwse theory two types of smple supports at the edges of the plate (.e., at =± a / ) may be classfed, amely: S: u = M = W = (6a) S3: = M = W = (6b) also two types of clamped supports may be classfed, amely: C: u = U = W = (7a) C3: = U = W = (7b) It s to be oted that these types of boudary codtos are defed smlar to the deftos the equvalet sgle-layer theores. For smplcty, the boudary codtos of a composte plate cyldrcal bedg may be represeted a cocse rule. For eample, a plate wth the edge codtos C at = a / ad S3 at = a / may be called C-S3. I order to obta the eact solutos of equlbrum Eqs. (7), Eq. (7a) s tegrated wth respect to to yeld: = or A u w B U T [ + ( ) /] + = + (8) where s a costat of tegrato. et we solve Eq. (8) to obta: u + ( w ) / = ( + B U )/ A (9) T Subttutg Eq. (9) to Eq. () ad the subsequet results to Eqs. (7b) ad (7c) to yeld: BB D U A55U A55w = A (a) A U + A + w = q (b) 55 ( 55 ) ( ) Eqs. () are + lear ordary dfferetal equatos wth costat coeffcets. It s to be oted that there est repeated ero roots (or egevalues) the characterstc equato of the set of equatos (). I order to ehace the soluto scheme of these equatos, some small artfcal terms wll be added to these equatos so that the characterstc roots become all dstct (see [4]): B B AU A w q w where D U A55U A55w = ε U A 55 + ( 55 + ) = ( ) + ε h h (a) (b) ε = ε Φ Φ d () wth ε beg a prescrbed small umber. et, Eqs. () ca be solved aalytcally for ay sets of boudary codtos terms of the uow costat. After solvg these equatos, we wll use oe more codto to fd the fal solutos. I order to solve Eqs. (), for coveece, the followg state space varables are troduced: { X( )} = { U( )}, { X( )} = { U } = { X } X ( ) = w( ), X ( ) = w = X (3) T where { X} = [ U, U,..., U + ]. Substtuto of Eqs. (3) to Eqs. () results a system of +4 coupled frst-order ordary dfferetal equatos whch, o the other had, may be preseted as: { X } = [ A]{ X} + { F} (4) T T T wth X X X X3 X4 { } = [{ },{ },, ]. I Eq. (4) the coeffcet matr [A] ad vector {F} are preseted Apped. The geeral solutos of Eq. (4) are gve by (e.g. see [5]): { X} = [ U][ Q( λ)]{ K} + [ U][ Q( λ)] [ Q( λ)] [ U] { F} d λ + (5) 4 wth [ Q( )] = dag( e λ, e λ,..., e λ ) ad {K} beg +4 arbtrary uow costats of tegrato to be foud by mposg the boudary codtos. Here, [ U ] ad λ ( =,,..., + 4) are, respectvely, the matr of egevectors ad 4

5 AALYTICAL SOLUTIO FOR OLIEAR BEDIG OF FG PLATES BY A LAYERWISE THEORY egevalues of the coeffcet matr [A] whch, geeral, must be regarded to have comple values. et, order to obta we ote that, for eample, for the C-C ad S-S boudary types we have u = at =± a / whch wll allow us to fd a tral ad error process. Towards ths ed, we ote that tegratg Eq. (9) from to a/ results : T = a/ a / + BU [ u ] = ( w ) d = A (6) Clearly, because of symmetry we have u ()=. Therefore, we coclude that: / A a BU ( ) T = w d a + A (7) By mag the solutos of the Eqs. (5) to satsfy (7) a tral ad error process, we wll obta the eact value of. Fally, u wll be obtaed by tegratg Eq. (9) as: T + u = ( w ) d+ A A B U (8) 3. Perturbato techque I ths secto, Ldstedt-Pocaré method s used to solve the three coupled olear ordary dfferetal equatos appearg (7). To ths ed, we defe W as follows: w() = W (9) Also the uow varables Eqs. (7) are represeted by the followg epasos: 3 u( ) = u( ) W + u( ) W + u3( ) W + (3a) 3 3 U( ) = U( ) W + U( ) W + U( ) W + (3b) 3 w ( ) = w( W ) + w( W ) + w( W ) + (3c) 3 where W s a uow parameter whch wll be foud at the ed of aalyss. et, mechacal loadg ( T = ) we let: q= q W + q W + q W + (3) 3 3 Also thermal loadg ( q ( ) = ) we cosder the chage of temperature of the top surface of the plate as T. Usg Eq. (3) we ca derve the temperature feld terms of T. Fally, ths case, we let: T = TW + TW + TW + (3) 3 3 where q s ad T s are some uow costats whch wll be foud by mposg certa codtos. These codtos are foud by otg that from (9) ad (3c) we must coclude that: w () = (33) w () =, =, 3, Substtutg Eqs. (3) ad (3) to Eqs. (7), results fte sets of coupled ordary lear dfferetal equatos. For eample, three frst sets of equatos for mechacal loadg case are: W : Au + BU = Bu DU AU Aw = A w + A U = q W : Au + BU = Aww 3 W : B u + D U A U A w = B ww A w + A U = q A u w B U w Au + BU = A( ww + ww ) 3 3 B u + D U A U A w = B( ww + w w ) = 3 A u w + u w + w B( U w + U w ) A w A U q [ ( ) / ] (34) (35) (36) After the soluto for these sets of equatos are obtaed, the costats q s (or T s) are foud by mposg the codtos (33). Fally W s foud by umercally solvg the polyomal equato (3) (or (3)). 4 umercal results ad dscusso The soluto procedures outled the prevous secto are appled to fuctoally graded plates subected to a uform dstrbuted load or a steady state thermal load. The total thcess of the plate s deoted by h wth ts legth beg a. The 5

6 M. TAHAI, S.M. Mrababaee legth-to-thcess rato (.e., a/h) s assumed, uless otherwse metoed, to be 5 all umercal eamples. Also the total thcess of the plate (h) s cosdered to be. m. It s assumed that the bottom surface of the plate s rch of (Alumum) ad the top surface s rch of ceramc (Zrcoa). The thermomechacal propertes of Alumum ad Zrcoa are as follows [5]: E = 7GPa, ν =.3, m m 6 o α m = 3 / C, m = 4 W mk E c = 5GPa, ν c =.3, 6 o = / C, =.9W mk. α c c (37) Fg. shows the dstrbuto of the volume fracto of the ceramc phase through the plate thcess for varous values of the power-law de =5 = = V c =.5 =. Fg.. Varato of the volume fracto of the ceramc phase through the thcess of the FGM plate. I all umercal results, the lear Lagraga terpolato fucto through the thcess s used. Also the total thcess of the FGM plate s dvded to twety umercal layers. I what follows, several umercal eamples are preseted for a plate subected to a uform trasverse load or a steady state temperature feld. 4. Mechacal loadg To study the olear bedg behavor of fuctoally graded plates subected to a trasverse uform load, may eamples were solved umercally. I the perturbato techque, thrd order, as gve by Eqs. (34)-(36) were used. I the umercal results the varous o-dmesoal parameters used are: legth = L thcess = h deflecto w = w h logtudal stress σ = σ h /( q a ) 4 4 load parameter q = q a /( Emh ). (38) Here, q deotes the testy of the appled uform load. Whe the edges of the plate have S3 ad C3 supports, t ca be show that the olear ad lear results are detcal. Here, for brevty, we wll oly preset the umercal results for C-C supports. Fg. presets the varato of the ceter deflecto of the FGM plate wth = versus the load parameter q. It s see that for the mamum deflectos greater tha.3h a olear soluto s requred. Fg. 3 llustrate the varato of odmesoal logtudal stress σ alog the top surface of ths plate, whe q = 8. To chec the correctess ad accuracy of the preset method, the results acheved from ths theory are compared wth those obtaed by utlg the commercal fte elemet pacage of ASYS. It s to be oted that the plate has bee modeled ASYS by usg threedmesoal eght-oded structural sold elemets ad the total thcess of the plates has bee subdvded to twelve layers. It ca be see from ths fgure that very good agreemet s obtaed usg the metoed method. Also t s observed that the eact method ad perturbato techque yeld detcal results Lear olear q Fg.. Varato of o-dmesoal ceter deflecto w of the FGM plate wth = versus q. 6

7 AALYTICAL SOLUTIO FOR OLIEAR BEDIG OF FG PLATES BY A LAYERWISE THEORY σ Lear olear, Eact olear, Per. Tech. Lear, FEM olear, FEM.5.5 ceramc =. =.5 = = Fg. 3. Varato of o-dmesoal aal stress alog the top surface of the FGM plate wth = subected to q = 8. σ Fg. 4 llustrates the varato of the odmesoal ceter deflecto of the FGM plate wth dfferet values of the power-law de subected to a uform pressure. Also varato of the odmesoal ceter deflecto w of the FGM plate 6 subected to q = /m versus legth-tothcess rato a/h for varous values of the powerlaw de are dsplayed Fg 5. The results show that a pure plate has hghest deflecto. Ths s epected because the fully plate s the oe wth the lower stffess tha the ceramc ad FGM plates. Fg. 6 shows through the thcess dstrbuto of the o-dmesoal aal stress σ of the FGM plate subected to q = 7 for varous values of the power-law de. Uder the applcato of the pressure loadg, the stresses are compressve at the top surface ad tesle at the bottom surface ceramc =. =.5 = = q Fg. 4. Varato of the o-dmesoal ceter deflecto w of the FGM plate a/h Fg. 5. Varato of o-dmesoal ceter deflecto w of the FGM plate versus legth-to-thcess rato a/h. ceramc =. = = = σ Fg. 6. Through the thcess dstrbuto of odmesoal aal stress σ of the FGM plate subected to q = Thermal loadg Here we preset some umercal results for a represetatve smply supported plate (S-S) whch s subected to a thermal loadg through ts thcess drecto. The temperature of the bottom -rch surface s ept costat at T m = o C ad that of the top ceramc-rch surface s vared form T c = o C to T c =3 o C. A Stress free temperature T = C s assumed. The temperature feld through the thcess of the plate ca be easly obtaed from Eq. (3). Fgure 7 shows the varato of the temperature through the thcess of the FGM plate for varous values of the power-law de. It s see that the temperature the plates wth both ceramc ad 7

8 M. TAHAI, S.M. Mrababaee s always greater tha that correspodg to a fully ceramc or fully plate. Varato of o-dmesoal deflecto w alog the legth of the FGM plate wth = whe T c = o C ad T c =3 o C are show Fg. 8. The results of the lear aalyss are also preseted the fgure to hghlght the dfferece betwee lear ad olear resposes wth creasg the temperature of ceramc-rch surface. From the fgure t ca be observed that at hgher temperature olear effects are predomat ad deflecto values are greater tha the lear oes owg to the decreased structure stffess due to cotrbuto from the olear terms the stra feld. s observed that the effect of olearty creased the ceter deflecto of the plates. Also t s see form these fgures that the fully plate has greater ceter deflecto compared to FGM ad fully ceramc plates. It s to be oted that the deflecto depeds o the product of the temperature ad the thermal epaso coeffcet. Therefore, the respose of the graded plates s ot termedate to the ad ceramc plates. The deflecto of the FGM plate correspodg to =. seems to be a mmum. ote that the temperature profles for the varous plates are close to each other, ad ths probably s the reaso why the deflectos uder temperature feld for the varous graded plates are also close to each other ceramc =. =.5 = = = T( o C) Fg. 7. Temperature profle through the thcess of the FGM plate ceramc =. =.5 = = T C ( o C) Fg. 9. Lear varato of the o-dmesoal ceter deflecto w wth creasg the temperature of the top surface of the FGM plate Lear, T c =3 o C olear, T c =3 o C Lear, T c = o C olear, T c = o C Fg. 8. Varato of o-dmesoal deflecto w alog the legth of the FGM plate wth =. Fgs. 9 ad show, respectvely, the lear ad olear varatos of the o-dmesoal ceter deflecto w wth creasg the temperature of the top surface for the FGM ad fully ad ceramc plates. By comparg Fg. 9 wth Fg., t ceramc =. =.5 = = T C ( o C) Fg.. olear varato of the o-dmesoal ceter deflecto w wth creasg the temperature of the top surface of the FGM plate. Fally, dstrbuto of o-dmesoal deflecto w alog the legth of the C-S FGM plate wth = whe T c = o C ad T c =3 o C are show Fg.. It s see that the olearty 8

9 AALYTICAL SOLUTIO FOR OLIEAR BEDIG OF FG PLATES BY A LAYERWISE THEORY effect s also sgfcat ths d of boudary codto Lear, T c =3 o C olear,t c =3 o C Lear, T c = o C olear, T c = o C Fg.. Varato of o-dmesoal deflecto w alog the legth of the C-S FGM plate wth =. 5 Coclusos I ths study, based o a layerwse theory, FGM plates cyldrcal bedg subected to thermomechacal loadgs are aalyed. The olear stra-dsplacemet relatos are used to study the effect of geometrc olearty. The equlbrum equatos are solved eactly ad also by usg a perturbato techque. The results obtaed from these two methods are preseted for varous loadg ad boudary codtos. The umercal results show that the olearty effect o the plate resposes s sgfcat. O the other had, the results dcate that the effect of olearty s to lower the magtude of the trasverse deflecto mechacal loadg ad s to hgher thermal loadg. Fally, t s show that thermal loadg case deflecto the ceter of fully plate s hgher tha that of fully ceramc ad FGM plates. Refereces [] oda,. ad Tsu, T. Steady thermal stresses a plate of fuctoally gradet materal. Trasactos of Japa Socety of Mechacal Egeers Seres A, Vol. 57, pp. 98 3, 99. [] Tagawa, Y., Aa, T., Kawamura, R. ad Oa,. Traset heat coducto ad thermal stress problems of a ohomogeeous plate wth temperature-depedet materal propertes. Joural of Thermal Stresses, Vol. 9, pp. 77-, 996. [3] Reddy, J.. ad Ch, C.D. Thermoelastcal aalyss of fuctoally graded cylders ad plates. Joural of Thermal Stresses, Vol., pp , 998. [4] Pravee, G.. ad Reddy, J.. olear traset thermoelastc aalyss of fuctoally graded ceramc- plates. Iteratoal Joural of Solds ad Structures, Vol. 35, o. 33, pp , 998. [5] Reddy, J.. Aalyss of fuctoally graded plates. Iteratoal Joural for umercal Methods Egeerg, Vol. 47, pp ,. [6] Reddy, J.. ad Cheg, Z.Q. Three-dmesoal thermo-mechacal deformatos of fuctoally graded rectagular plates. Europea Joural of Mechacs A/Solds, Vol., pp ,. [7] She, H.-S. olear bedg respose of fuctoally graded plates subected to trasverse loads ad thermal evromets. Iteratoal Joural of Mechacal Sceces, Vol. 44, pp ,. [8] Woo, J. ad Megud, S.A. olear aalyss of fuctoally graded plates ad shallow shells. Iteratoal Joural of Solds ad Structures, Vol. 38, pp ,. [9] Yag, J. ad She, H.-S. o-lear aalyss of fuctoally graded plates uder trasverse ad plae loads. Iteratoal Joural of o-lear Mechacs, Vol. 38, pp , 3. [] Taha, M., Torabadeh, M.A. ad Feredoo, A. olear aalyss of fuctoally graded beams. Joural of Achevemets Materals ad Maufacturg Egeerg, Vol. 8, o. -, pp 35-38, 6. [] Hseh, J.-J. ad Lee, L.-T. A verse problem for a fuctoally graded ellptcal plate wth large deflecto ad slghtly dsturbed boudary. Iteratoal Joural of Solds ad Structures, Vol. 43, o., pp , 5. [] Agarwal, S., Charaborty, A. ad Gopalarsha, S. Large deformato aalyss for asotropc ad homogeeous beams usg eact lear statc solutos. Composte Structures, Vol. 7, pp. 9 4, 6. [3] Fug, Y.C. Foudato of sold mechacs. Eglewood Clffs, ew Jersey: Pretce-Hall, 965. [4] Taha, M. ad oser, A. Accurate determato of terlamar stresses geeral cross-ply lamates. Mechacs of Advaced Materals ad Structures, Vol., o., pp. 67-9, 4. [5] Fral, J.. Matr theory. Eglewood Clffs, ew Jersey: Pretce-Hall, 968. Apped The coeffcet matr [ A ] ad vector { F } appearg Eqs. (4) are defed as: 9

10 M. TAHAI, S.M. Mrababaee [ A] [] [ I] {} {} [ a ] [] {} [ a ], = T T {} {} T T {} { b} b {} {} { F} = f where [] ad [I] are ( + 4) ( + 4) square ad ero detty matrces, respectvely, ad {} s a ero vector wth +4 rows. The remag matrces the above equatos are: [ a ] = ([ D ] { B }{ B } / A ) ([ A ] + [ ε ]) T 55 T a = D B B A A55 b = A55 A55 + = ε /( 55 + ) f = q/( A55 + ). [ ] ([ ] { }{ } / ) { } { } { }/( ) b A