THREE-DIMENSIONAL THERMAL BUCKLING ANALYSIS OF FUNCTIONALLY GRADED CYLINDRICAL PANELS USING DIFFERENTIAL QUADRATURE METHOD(DQM)

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1 JOURNAL OF THEORETICAL AND APPLIED MECHANICS 54, 1, pp , Wasaw 2016 DOI: /jtam-pl THREE-DIMENSIONAL THERMAL BUCKLING ANALYSIS OF FUNCTIONALLY GRADED CYLINDRICAL PANELS USING DIFFERENTIAL QUADRATURE METHODDQM) Seyed A. Ahmadi, Hadi Poushahsavai Babol Univesity of Technology, Depatment of Mechanical Engineeing, Babol, Ian ali Themal buckling analysis of functionally gaded cylindical panels subjected to vaious conditions is discussed in this pape. Buckling govening equations ae solved using the diffeential quadatue method. It is assumed that the mechanical popeties of the panel ae gaded though thickness accoding to a powe function of the thickness vaiable. The panel is assumed to be unde the action of thee types of themal loading including unifom tempeatue ise and vaiable tempeatue ise in the axial and adial diection. In the pesent study, the effects of powe law index, panel angle, diffeent themal load conditions and geometic paametes on the buckling behavio of functionally gaded cuved panels ae studied. The esults obtained though the pesent method ae compaed to the finite element solutions and the epoted esults in the liteatue. A desiable compatibility is concluded. Keywods: themal buckling, cuved panel, functionally gaded mateial, diffeential quadatue method 1. Intoduction Due to special mechanical popeties, cicula cylindical panels ae widely used in engineeing stuctues such as pessue vessels, nuclea eactos, spacecafts and jet engine exhausts. Due to the inceasing demands fo heat-esisting, enegy absobing, light-weight elements and high stuctual pefomance equiements in extemely high tempeatue envionments and highspeed industies such as fusion eactos, aicaft and aeospace stuctues the use of special mateials with high themal and mechanical esistance has gained much populaity by many eseaches. The applications of functionally gaded mateialsfgms) have attacted much attention in the past two decades since they wee fist epoted by Koizumi1993). FGMs ae composite mateials, micoscopically inhomogeneous, in which mechanical popeties vay smoothly and continuously fom one suface to the othe. The main advantage of FGMs is that the ceamic component povides high tempeatue esistance due to its low themal conductivity while the metal component pevents factue induced by themal stesses due to the hightempeatuegadientinaveyshotpeiodoftime.whentheseaesubjectedtoathemal loading, the detemination of themal buckling capacity of these stuctues is impotant to achieve an optimized design in cost and weight. Buckling analyses of vaious stuctues wee caied out by many eseaches. A eview of eseach on the buckling esponse of plates and shells in a tempeatue envionment was pesented bythonton1993).hedidsomeeseachonthemalbucklingofplatesandshells.inhiswok, he descibed elastic themal buckling of metallic as well as composite plates and shells. Muphy and Feeia2001) investigated themal buckling analysis of impefect flat plates based on the enegy consideation. They showed the atio of the citical tempeatue fo a pefect ectangula plate to that of an impefect plate as a function of the initial impefection amplitude. Mahayni 1966) studied themal buckling behavio of doubly cuved isotopic panels using Galekin s

2 136 S.A. Ahmadi, H. Poushahsavai method. Chang and Chui1991) caied out bifucation buckling analysis of composites unde the action of unifom tempeatue change using highe ode tansvese shea defomation theoy and the finite element method. Ealie, the Diffeential Quadatue Method intoduced by Jang et al.1989), was applied only to ectangula plates and lately it was consideed fo shells. Mifakhaei and Redekop1998) used the Diffeential Quadatue Method to study buckling behavio of cicula cylindical shells. Alibeigloo and Kani2010) and Haftchenai et al.2007) used this method to study cylindical shells as well. The study of stuctues of functionally gaded mateials has eceived consideable attention in ecent yeas. Buckling of functionally gaded plates unde themal loads was studied by Javahei and Eslami2002b). They used classical plate theoy and obtained nonlinea equilibium and linea stability equations using vaiational fomulations. Shahsiah and Eslami2003) consideed effects of vaious tempeatue distibutions on themal buckling of simply suppoted FG cylindical shells, using the fist ode shea defomation theoy, howeve the tempeatue dependency of mateial popeties was not included. Themoelastic stability of FG cylindical shells subjected to vaious themal load conditions was studied by Wu et al.2005). Themal buckling analysis of functionally gaded plates consideing simply suppoted bounday conditionsbyusingthefistsheadefomationtheoywascaiedoutbywu2004).heeached the stability equation of functionally gaded shells using Donnell s shell theoy and pesented its closed-fom solution. Buckling analysis of FG plates using a highe ode theoy was pesented by Javahei and Eslami2002a). It was shown that highe ode shea defomation theoy accuately pedicts the buckling behavio, wheeas the classical plate theoy oveestimates the citical loads. Beivik1997) discussed the buckling esponse of composite cylindical panels unde the actionofmechanicalandthemalloading.zhaoetal.2007)andzhaoandliew2010)used the element-fee kp-ritz method fo themal and mechanical buckling analysis of functionally gaded cylindical shells. They obtained thee-dimensional buckling equations of the shell based on the Donnell shell theoy and pesented a closed fom solution to pedict buckling loads caused by themal loads and citical edge displacement in the longitudinal diection. In this pape, buckling analysis of cylindical panels made of a functionally gaded mateial subjected to thee types of themal loading is investigated. To obtain the buckling load of the cylindical panels, the Diffeential Quadatue MethodDQM) is used to discetize diffeential equations obtained based on the second Piola-Kichhoff stess tenso using thee-dimensional theoy of elasticity by Akbai Alashti and Ahmadi2014). The mateial popeties ae assumed to be tempeatue independent and vay continuously along the thickness accoding to a powe law function while Poisson s atio of the mateial is taken to be constant. Effects of vaious paametes including panel cuvatue, gading index, vaious themal load conditions and geometic atios on the buckling behavio of the cuved panels ae investigated. Numeical esults ae validated against finite element calculations and esults that ae available in the offeed liteatue. 2. Govening equation fo buckling Conside a thick cylindical panel made of ceamic and metallic mateials with the inne adiusr 1,mid-sufaceadiusa,thicknesshandlengthL.Thegeometicpaametesandthe cylindical coodinate system. i.e., θ and x-coodinates ae shown in Fig. 1. The components of the displacement field in this coodinate system ae expessed as w, v and u, espectively. Assume that the mateial is isotopic, inhomogeneous with Young s modulus vaying continuously in the thickness diection, i.e. fom ceamic in the inne laye to metallic in the oute laye accoding to the following fomula V m = 2z+h ) K V c +V m =1 2.1) 2h

3 Thee-dimensional themal buckling analysis Fig. 1. Geomety of a cylindical panel wheev c andv m epesentthevolumefactionsoftheceamicandmetallicconstituentand K denotes the volume faction index that indicates the mateial vaiation pofile though the FG shell thickness. Thus, the Young modulus in the adial diection is assumed to vay accoding tothepowelawinthefollowingfoms 2z+h ) K Ez)=E c +E mc E mc =E m E c 2.2) 2h wheee m ande c denotetheelasticmodulusofthemetalandceamic,espectively.themateial compositionvaiessmoothlyfomtheoutesufacez=h/2)oftheshellasmetaltotheinne sufacez= h/2)asceamic.mateialpopetiesoftheshellaeassumedtobeindependentof the tempeatue field and Poisson s atio is consideed to be constant thoughout the thickness of the shell. In ode to calculate buckling loads of panels, the buckling equations obtained by Akbai Alashti and Ahmadi2014) ae used. In this wok, also the finite element linea o bifucation buckling analysis of the cylindical panel using ANSYS suite of pogam is caied out. The eigen buckling analysis pedicts theoeticalbucklingstengthofashellmadeofalineaelasticmateial.thisanalysisisusedtopedict thebifucationpointonanf-udiagamusingalineaizedmodeloftheelasticstuctue.itis a technique used to detemine buckling pessues at which the stuctue becomes unstable and thei coesponding buckling mode shapes. The basic fom of the eigen buckling analysis is Kφ=λ i Sφ 2.3) wheek,φ i,λ i andsaethestuctualstiffnessmatix,eigenvecto,eigenvaluesandstess stiffness matix, espectively. Eight noded quadilateal shell elements, namely Shell281, ae used to model the thick cylindical shell. The elements can handle membane, bending and tansvese shea effects and ae able to fom the cuvilinea suface satisfactoily. The elements ae suitable fo modeling of the laye and have the stess stiffening, lage deflection and lage stain capabilities. Bounday conditions of shell panels ae defined using equilibium equations. Fo the initial and petubed equilibium positions, we have σ a+ h 2,θ ) =σ a h 2,θ ) =0 τ θ a+ h 2,θ ) =τ θ a h 2,θ )=0 τ x a+ h 2,θ ) =τ x a h 2,θ )=0 2.4)

4 138 S.A. Ahmadi, H. Poushahsavai Bounday conditions at the panel edges ae defined as: upanddownedge,x=0,l Simplysuppoted: w=v=σ xx=0 Clamped: w=v=u=0 latealedges,θ=0,β Simplysuppoted: w=σ θθ =u=0 Clamped: w=v=u=0 2.5) 2.6) 3. Calculation of buckling load In this wok, two types of panels ae consideed: Case1.Thepanelisassumedtobesimplysuppotedatlatealedgesandclampedattwoends. Theefoe,themalvaiationcausesnoaxialstessonthepanel,N θ =0. Case2.Weassumethatthepanelhasclampedboundayconditionsatalledges.Fothis casethemalloadingcauseaxialandcicumfeentialstessesatthepanelwalls,n x 0, N θ 0. By substituting the components of the displacement field in the stess-stain and linea staindisplacement equations and the esulted expession in the buckling equations, the equilibium equations ae defined in tems of components of the displacement field. In the pesent wok, a polynomial expansion based on the Diffeential Quadatue Method applied by Bellman and Casti1971) is used to discetize and solve the obtained buckling equations. Accoding to this method, the fist ode deivative of the function fx) can be appoximated as a linea sum of all functional values in the domain df N = w 1) dx ij fx j) fo i=1,2,...,n 3.1) x=xi j=1 wheew 1) ij istheweightingcoefficientandndenotesthenumbeofgidpointsx i inthedomain. Thee ae diffeent methods fo calculation of the weighting coefficients matix, see Shu2000). Hee, the weighting coefficients of the fist ode deivatives ae defined based on the Lagange intepolation polynomials as whee w 1) ij = M 1) x i )= M 1) x i ) x i x j )M 1) x j ) fo i j w 1) ii = M2) x i ) 2M 1) x j ) N x i x k ) Nx i,x j )=M 1) x i )δ ij k=1 k i,j M 2) x)=n 2) x,x k )x x k )+2N 1) x,x k ) and fo highe ode deivatives, we have 3.2) 3.3) w ) ij = w 1) ij w 1) ii w 1) ) ij x i x j w ) ii= N j=1,j i w ) ij fo i,j=1,2,...,n =2,3,...,N 1 3.4)

5 Thee-dimensional themal buckling analysis Now, applying the above fomulation to the buckling equations, we have G 2 N l=1 Gz) Gz) a 2) i,l w l,j,k+ G 2 G 2 2w i,j,k+ G 1 N a 1) i,l w l,j,k+gz) l=1 n=1 N a 1) i,l c1) k,m v l,j,m+ Gz) 2 b 2) j,n w i,n,k+g 1 N l=1 +σx 0 b 2) j,n w i,n,k+σ 0 2 θθ n=1 2 c 1) k,m v i,j,m+σ 0 1 θθ N N c 1) l=1 + Gz) a 2) i,l v l,j,k+ G 1 N l=1 c 1) k,m a1) i,l w l,j,m+ G 3 2 a 1) i,l v l,j,k Gz) 2 v i,j,k+ G 2 2 l=1 +σx 0 b 2) j,n v i,n,k σ 0 2 θθ n=1 2 c 1) k,m w i,j,m+σθθ 0 N a 2) i,l u l,j,k+g 2 b 2) j,n v N i,n,k+g 1 l=1 n=1 l=1n=1 + Gz) 2 c 2) k,m u i,j,m+ G 1 +σx 0 b 2) j,n u i,n,k σ 0 1 θθ n=1 2 c 1) k,m v i,j,m+ G 1 c 2) k,m u i,j,m=0 l=1n=1 c 2) k,m w i,j,m+ G 3 2 2B i,j,k σθθ c 2) k,m v i,j,m+ G 1 a 1) i,l b1) j,n u l,n,k k,m w i,j,m+gz) n=1 1 2v i,j,k σθθ b 1) j,n a1) i,l w l,n,k+ Gz) n=1 b 1) j,n w i,n,k N c 1) k,m v i,j,m c 2) k,m w i,j,m=0 n=1 b 2) j,n v i,n,k c 1) k,m b1) j,n u i,n,m l=1 c 2) k,m v i,j,m=0 a 1) i,l u l,j,k 3.5) whee G 1 =Gz)+λz) G 2 =2Gz)+λz) G 3 =3Gz)+λz) anda k) ij,bk) ij andc k) ij denotetheweightingcoefficientsofthek-thodedeivativeinthe,θ andx-diection,espectively;n,qandmaegidpointnumbesinthe,θandx-diection, espectively. The citical value of the buckling load is obtained by solving the set of equations pesented in the matix fom as [ ] d b [ ] BB B D u 0 0 D B D D v =σ D BG D DG w wheethesub-maticesb B,B D andd BG,D D,D B,D DG aefoundfomtheboundayconditions and govening equations, espectively. Equation3.6) is tansfomed into the standad eigenvalue equation, as 3.6) D BG B 1 B B D+D DG ) 1 D B B 1 B B D+D D )[u v w] T σi[u v w] T =0 3.7) fomwhich,theeigenvaluesofσcanbefound.thesmallestvalueofσisfoundtobethebuckling load.

6 140 S.A. Ahmadi, H. Poushahsavai 4.1. Unifom tempeatue ise 4. Themalloading The tempeatue changes unifomly though the thickness and emains constant in the longitudinal and cicumfeential diections of the panel. This themal vaiations induces only nomal stess,andthepaameteφisdefinedas and σ= N h Φ= h 2 h 2 N= Φ 1 ν [E m +E cm 2z+h 2h ) K 2z+h ) K ] ][α m +α cm Tx,θ,z)dz 2h Φ= E c α c h+ [E cα m α c )+α c E m E c )]h K+1 + α m α c )E m E c )h) T c 2K+1 Substituting buckling stess obtained by numeical solution into Eq.4.1) and4.2), helps us to obtainthethemalbucklingload T c Non-unifom tempeatue ise in the axial diection In this case, the assumed tempeatue vaies in the longitudinal diection accoding to the following fomula x ) n+tm T= T T=T c T m n>0 4.3) L wheet m isthetempeatueatthemetalsufaceofthepanels.accodingtotheaboveequations, axial stesses caused by the tempeatue ise have the same vaiation in this diection. The citical stesses ae obtained by consideing the effects of this loading in the discetized govening equations and then, the buckling tempeatues ae achieved using equations4.1) and4.2) Non-unifom tempeatue ise in the adial diection The functionally gaded mateials ae designed in ode to esist against high tempeatue isebyceamic,sothetempeatuechangewillbequitediffeentatthetwosidesoffgm stuctues. The tempeatue distibution acoss the thickness is a function of the z coodinate as follows T= T z ) q+tm h +1 h 2 2 <z<h T=T c T m 4.4) 2 ThepaameteΦisdefinedas Φ= h 2 h 2 [E m +E cm 2z+h 2h 4.1) 4.2) ) K 2z+h ) K ][ ][α m +α cm T z ) q+tm ] 2h h +1 dz 4.5) 2 The buckling tempeatue ise will be obtained using equation4.1). Fo example, fo q = 1, thepaameteφisgivenas Φ= E c α c h+ [E cα m α c )+α c E m E c )]h + α m α c )E m E c )h) T m K+1 2K+1 h { + E c α c K K+1)K+2)2K+1) 2 E cα c K 2 4.6) [ Em E c +α c K2 +E c )α m α c )+ 2E m E c )+ 7 } 4 2 E m )]+E m α m T c

7 Thee-dimensional themal buckling analysis Numeical esults and discussion In ode to illustate the esults of the pesented method fo an inhomogeneous shell, a functionally gaded cylindical shell made of aluminum and alumina is consideed. Young s modulus is assumed to be tempeatue independent and vay smoothly in the adial diection accoding to a powe law distibution of the volume faction of the constituent mateials. Young s modulusfoaluminaattheinnesufaceandfoaluminumattheoutesufaceisassumedtobe E c =380GPaandE m =70GPa,espectively.ItisalsoassumedthatPossion satiosofthe constituent mateials ae constant and equal to 0.3. At the fist step, the buckling tempeatues calculated by the pesent study ae validated against the esults epoted in the liteatue. Figues2and3plottheciticaltempeatuechangesofthecompleteshellwithL=a=1 againsttheatioofthethicknesstomidsufaceadiusoftheshellh/afotheunifomtempeatue ise loading. The esults ae compaed to the finite element esults and those epoted by Beivik1997). Fig. 2. Compaisons of the citical tempeatue of the complete shell unde unifom tempeatue ise, a) aluminum,b) alumina,c) functionally gaded shell K = 1 Fig. 3. Buckling mode shapes of complete shells made of aluminum unde unifom tempeatue ise Itisevidentthattheesultsofthepesentedmethodaeingoodageementwiththefinite elementesultsandthoseofbeivik1997).itcanbeseenfomthesefiguesthatthecitical

8 142 S.A. Ahmadi, H. Poushahsavai buckling tempeatue inceases linealy as the atio of h/a inceases, and also the diffeence between the esults inceases when the elative thickness gows. It is because of the fact that Beivik1997) used Donnell s theoy to obtain the buckling equation of the thin shell. This equation ceates an oveestimation in the pediction of buckling load fo a thick shell. The effectsofpanelangleonthebucklingtempeatueaeshowninfig.4.theesultsaecompaed totheesultsepotedbywuetal.2005).theesultsaeobtainedfothepanelcase1)unde unifom and non-unifom tempeatue ise in the adial diection, and the panel is assumed to bemadeofaluminumwitha=1m,l/a=1andh/a=0.02. Fig. 4. Compaisons of the citical tempeatue of the homogeneous panelcase 1) made of aluminum withdiffeentanglest m =0) Fig.5.EffectofthevolumefactionindexKonthebucklingtempeatueofthepanelcase1), non-unifomtempeatueisein:a)axialdiection,b)adialdiection,c)combinedloadingt m =0) To make calculations following Wu et al.2005), the citical stess is obtained fist though thegivenfomulaandthensubstitutedintoeq.4.6).itcanbeinfeedfomfig.4thatthe buckling tempeatue changes decease when the panel angel inceases, and fo highe angles the

9 Thee-dimensional themal buckling analysis esults appoach constant values. The esults obtained based on Donnell s theoy and obtained bywuetal.2005),showlowvaiationofthebucklingloadvesuspanelangles. Next,vaiationoftheciticalbucklingtempeatuefothepanelcase1)withL=a=1m, h=0.01mandβ=1adundenon-unifomtempeatueloadingvesusmateialgadient index K, ae pesented. ItisobviousfomFig.5thatasthemateialgadientindexKinceasesfom1to10,the citical buckling tempeatue gows apidly and, fo highe values, the esults appoach constant values. It is also evident that the citical buckling tempeatues incease as the value of K inceases.themaineasonfosuchaninceaseisthefactthatahighevalueofkcoesponds to a ceamic-iche panel, which usually has a highe themal stength than a metal-iche one. Figue6showsthebucklingtempeatuevesustheatioofh/afothepanelcase1)with L=1mandβ=1adfotheetypesofloading,i.e.unifomtempeatueiseandnon-unifom tempeatue ise in the axial and adial diection. Buckling modes obtained though the finite element pogam fo the unifom tempeatue ise ae illustated in Fig. 7. Fig.6.Bucklingtempeatueofthepanelcase1)vesush/aT m =0) Fig.7.BucklingmodeshapesoftheinhomogeneousK=1)panelcase1) ItcanbeobsevedinFig.7thatwhenthethemalloadinghasalineavaiation,thebuckling tempeatue ises significantly. Then, the citical buckling tempeatues with espect to the panel ae plotted fo the cylindical panel with diffeent tempeatues at the oute suface unde the action of non-unifom tempeatue ise in the adial diection. Assuming that the panel has

10 144 S.A. Ahmadi, H. Poushahsavai L=a=1m,h=0.01mandβ=1ad,itisfoundthatthevaiationoftheoutesuface tempeatue has a significant effect on the buckling tempeatue of the panel. Fig.8.Bucklingtempeatueofthepanelcase1)vesusβfovaiousvaluesofT m Figue 9 demonstates vaiation of the buckling tempeatue change vesus aspect atio h/a of the panelcase 2). The esults obtained though the pesent method ae compaed with the esults obtained though the given govening equations in Beivik1997). It should be noted that the buckling equations pesented by Beivik1997) ae only solved fo the shellcase 1), andheeweesolveitfothepanelcase2)usingequation4.1). Fig. 9. Compaisons of the citical tempeatue of the panelcase 2) with diffeent gadient indices undeunifomtempeatueiseloading,l=a=1m,β=1adt m =0) It is obvious that the esults of the pesented numeical method ae in good ageement with the esults issuing fom Donnell s shell theoy. As concluded above, the diffeence between the pesent esults and those obtained based on Donnell s theoy incease as the thickness of the panel gows. Buckling tempeatue changes against the thickness atio fo the panelcase 2) unde the action of vaious loading conditions ae illustated in Figs. 10 and 11. ItcanbefoundfomFigs.10and11thatthebucklingtempeatueiseslinealywhen thickness of the panel inceases. It is also clea that the citical buckling tempeatues incease as the volume factions of the ceamic incease. To study the effects of themal loading vaiation insevealdiectionsandbucklingtempeatueisesfothepanelcase2)withl=a=1m, β=0.8ad,h=0.005mvesustheindexkundecombinedtempeatueloadingsaegivenin Table 1.

11 Thee-dimensional themal buckling analysis Fig. 10. Buckling tempeatue of panelcase 2) vesus h/a fo unifom tempeatue ise and non-unifomloadingintheadialdiection,β=0.8ad,l/a=1 Fig. 11. Buckling tempeatue of the panelcase 2) vesus h/a fo non-unifom tempeatue ise in the axialandadialdiection,β=0.8ad,l/a=1 Table 1. Buckling tempeatue of the panelcase 2) vesus K fo combined load conditions K n=0 n=0.5 n=1 q=0 q=1 q=0 q=1 q=0 q= Conclusion In this pape, buckling analysis of FG cylindical panels unde the action of themal loading is caied out. Mateial popeties ae assumed to be tempeatue-independent and gaded though the simple powe law distibution in tems of the volume factions of the constituents. The Diffeential Quadatue Method is used to discetize and solve buckling equations. The buckling analysis of such panels unde the action of thee types of themal loadings, i.e. unifom tempeatue ise and non-unifom tempeatue ise in the axial and adial diection consideing two

12 146 S.A. Ahmadi, H. Poushahsavai types of bounday conditions, is caied out. Fom the pesent study, the following conclusions ae dawn: Detemination the citical loads by the use of equations extacted fom Donnell s theoy cause an oveestimation when thickness of the panel inceases. The esults obtained by the thee-dimensional buckling equations, as pesented in this wok, ae moe accuate in compaison with the esults based on the Donnell shell theoy. TheciticalbucklingtempeatueT c inceaseslinealywithaninceaseinthethickness to mid-suface adius atio h/a. Fo functionally gaded cylindical panels unde vaious themal loads, an incease in the volume faction of the ceamic constituent inceases the citical load. Refeences 1. Akbai Alashti R., Ahmadi S.A., 2014, Buckling of impefect thick cylindical shells and cuved panels with diffeent bounday conditions unde extenal pessue, Jounal of Theoetical and Applied Mechanics, 52, 1, Alibeigloo A., Kani A.M., 2010, 3D fee vibation analysis of laminated cylindical shell integated piezoelectic layes using diffeential quadatue method, Applied Mathematical Modeling, 34, Bellman R.E., Casti J., 1971, Diffeential quadatue and long tem integation, Jounal of Mathematical Analysis and Applications, 34, 1, Beivik N.L., 1997, Themal and mechanical esponse of cuved composite panels, Dissetation submitted to the Faculty of the Viginia Polytechnic Institute and State Univesity in patial fulfillment of the equiements fo the degee of docto of Philosophy in engineeing mechanics 5. Chang J.S., Chui W.C., 1991, Themal buckling analysis of antisymmetic laminated cylindical shell panels, Intenational Jounal of Solids and Stuctues, 27, 1, Haftchenai H., Davizeh M., Davizeh A., Ansai R., Shama C.B., 2007, Dynamic analysis of composite cylindical shells using Diffeential Quadatue MethodDQM), Composite Stuctues, 78, Jang S.K., Bet C.W., Stiz A.G., 1989, Application of diffeential quadatue to static analysis of stuctue components, Intenational Jounal fo Numeical Methods in Engineeing, 28, Javahei R., Eslami M.R., 2002a, Themal buckling of functionally gaded plates based on highe ode theoy, Jounal of Themal Stesses, 25, Javahei R., Eslami M.R., 2002b, Themal buckling of functionally gaded plates, AIAA Jounal, 40, 1, Koizumi M., 1993, The concept of FGM, Ceamic Tansactions, Functionally Gadient Mateials, 34, Mahayni M.A., 1966, Themal buckling of shallow shells, Intenational Jounal of Solids and Stuctues, 2, Mifakhaei P., and Redekop D., 1998, Buckling of cicula cylindical shells by the diffeential quadatue method, Intenational Jounal of Pessue Vessels and Piping, 75, Muphy K.D., Feeia D., 2001, Themal buckling of ectangula plates, Intenational Jounal of Solids and Stuctues, 38, 22/23, Shahsiah R., Eslami M.R., 2003, Themal buckling of functionally gaded cylindical shell, Jounal of Themal Stesses, 26, Shu C., 2000, Diffeential Quadatue and its Application in Engineeing, Spinge-Velag, London, UK

13 Thee-dimensional themal buckling analysis Thonton E.A., 1993, Themal buckling of plates and shells, Applied Mechanics Review, 46, 10, Wu L.H., 2004, Themal buckling of a simply suppoted modeately thick ectangula FGM plate, Composite Stuctues, 64, 2, Wu L., Jiang Z., Liu J., 2005, Themoelastic stability of functionally gaded cylindical shells, Composite Stuctues, 70, Zhao X., Liew K.M., 2010, A mesh-fee method fo analysis of the themal and mechanical buckling of functionally gaded cylindical shell panels, Computational Mechanics, 45, Zhao X., Yang Y., Liew K.M., 2007, Geometically nonlinea analysis of cylindical shells using the element-fee kp-ritz method, Engineeing Analysis with Bounday Elements, 31, Manuscipt eceived May 13, 2014; accepted fo pint July 15, 2015