DSpace at VNU: Nonlinear thermal stability of eccentrically stiffened functionally graded truncated conical shells surrounded on elastic foundations

DSpace at VNU: Nonlinear thermal stability of eccentrically stiffened functionally graded truncated conical shells surro...

Accepted Manuscript Nonlinear thermal stability of eccentrically stiffened functionally graded truncated conical shells surrounded on elastic foundations Nguyen Dinh Duc , Pham Hong Cong PII: S0997-7538(14)00167-3 DOI: 10.1016/j.euromechsol.2014.11.006 Reference: EJMSOL 3141 To appear in: European Journal of Mechanics / A Solids Received Date: 28 February 2014 Revised Date: November 2014 Accepted Date: November 2014 Please cite this article as: Duc, N.D., Cong, P.H., Nonlinear thermal stability of eccentrically stiffened functionally graded truncated conical shells surrounded on elastic foundations, European Journal of Mechanics / A Solids (2014), doi: 10.1016/j.euromechsol.2014.11.006 This is a PDF file of an unedited manuscript that has been accepted for publication As a service to our customers we are providing this early version of the manuscript The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain ACCEPTED MANUSCRIPT Nonlinear thermal stability of eccentrically stiffened functionally graded truncated conical shells surrounded on elastic foundations Nguyen Dinh Duc*, Pham Hong Cong Vietnam National University, Hanoi -144 Xuan Thuy-Cau Giay- Hanoi-Vietnam RI PT Abstract This paper studies the thermal stability of an eccentrically stiffened functionally graded truncated conical shells in thermal environment and surrounded on elastic foundations Both of the FGM shell as well as the stiffeners are deformed under temperature The formulations SC are based on the classical shell theory taking into account geometrical nonlinearity, initial geometrical imperfection, temperature-dependent properties and the Lekhnitsky smeared stiffeners technique with Pasternak type elastic foundation By applying Galerkin method, the M AN U closed-form expression for determining the thermal buckling load is obtained The numerical results show that the critical thermal load in the case of the uniform temperature rise is smaller than one of the linear temperature distribution through the thickness of the shell, and the critical thermal load increases when increasing the coefficient of stiffeners and vice versa The paper also analyzes and discussed the significant effects of material and geometrical TE D properties, elastic foundations on the thermal buckling capacity of the eccentrically stiffened FGM truncated conical shell in thermal environment The obtained results are validated by comparing with those in the literature Keywords: Thermal stability, Eccentrically stiffened truncated conical shell, Functionally Introduction EP graded materials, elastic foundations AC C The idea of the construction of functionally graded meterials (FGM) was first introduced in 1984 by a group of Japanese materials scientists (Koizumi, 1997) Due to high performance heat resistance capacity and excellent characteristics of FGM in comparison with conventional composites, functionally graded shells involving conical shells are widely used in exhaust nozzle of solid rocket engine, some important details of space vehicles, aircrafts, nuclear power plants and many other engineering applications As a results stability analysis of those strutures are very important problems and have attracted increasing research effort ACCEPTED MANUSCRIPT The static stability of conical shells has been studied by many researchers in the recent years However, most of them have focused on buckling behaviors and determining the critical loads for the shells without elastic foundations and stiffeners: The buckling of conical shells under the axial compression (Seide 1956, 1961), the thermal buckling of concial shells (Lu and Chang, 1967), solution of buckling for RI PT truncated conical shells under combined pressure and heating (Tani, 1984), the thermoelastic buckling of laminated composite conical shells (Wu and Chiu, 2001), thermal and mechanical instability of truncated conical shells made of FGM (Naj et al 2008), an analytical approach to investigate the linear buckling of truncated conical SC panels made of FGM and subjected to axial compression, external pressure and the combination of these loads (Bich et al, 2012), the thermoelastic stability of M AN U functionally graded truncated conical shells (Sofiyev, 2007), the buckling of thin truncated conical shells made of FGMs subjected to hydrostatic pressure, uniform external pressure and uniform axial compressive load (Sofiyev et al 2004, 2009, 2010a) The shell structures supported by an elastic foundations have been widely used in many applications such as in aircraft, reusable space transportation vehicles TE D and civil engineering Therefore, studies on the effects of elastic foundations on behavior and loading capacity of the shells are highly important The nonlinear buckling of the truncated conical shell made of FGMs was surrounded by an elastic medium and Winkler–Pasternak type elastic foundation using the large deformation EP theory with von Karman–Donnell-type of kinematic nonlinearity (Sofiyev, 2010b; Sofiyev and Kuruoglu, 2013) Najafov and Sofiyev (2013) obtained the nonlinear AC C dynamic analysis of FG truncated conical shells surrounded by an elastic medium using the large deformation theory with von Karman–Donnell-type of kinematic nonlinearity Pratically, the composite plates and shells usually are reinforced by stiffening components to provide the benefits of added load-carrying static and dynamic capability with a relatively small additional weight There have had some publications on the buckling of composite shells reinforced by stiffeneres: a free vibration analysis for a ring-stiffened simply supported conical shell by considering an equivalent ACCEPTED MANUSCRIPT orthotropic shell and using Galerkin method and carried out experimental investigations (Weingarten, 1965), an energy approach to find the resonant frequencies of simply supported ring-stiffened, and ring and stringer-stiffened conical shells (Crenwelge and Muster, 1969), the nonlinear static buckling and post-buckling for imperfect eccentrically stiffened functionally graded thin circular cylindrical shells RI PT surrounded on elastic foundation (Duc and Thang, 2014), a semi-analytical approach to investigate the nonlinear dynamic of imperfect eccentrically stiffened FG shallow shells taking into account the damping subjected to mechanical loads (Bich et al., 2013), the study of instability of eccentrically stiffened functionally graded truncated SC conical shells under mechanical loads and shells are reinforced by stringers and rings (Dung et al 2013), the stability of functionally graded truncated conical shells M AN U reinforced by functionally graded stiffeners, surrounded by an elastic medium and under mechanical loads (Dung et al., 2014) From the above review, to the best of our knowledge, it has showed that there is no publiation about buckling of FGM conical shell with stiffeneres in thermal environment Under temperature, both of the FGM shell as well as the stiffeners are TE D deformed, therefore, the calculation on the thermal mechanism of FGM shells and stiffeners has become more difficult Recently, Duc and Quan (2013) researched the nonlinear postbuckling for imperfect eccentrically stiffened FGM double curved thin shallow shells on elastic foundation using a simple power-law distribution in thermal EP environments Duc and Cong (2014) also investigated the nonlinear postbuckling of AC C imperfect eccentrically stiffened thin FGM plates under temperature This paper studied the stability of an eccentrically stiffened functionally graded truncated conical shells surrounded on elastic foundations under thermal loads with both FGM shell and stiffeners having temperature-dependent properties Addionally, the paper analyzed and discussed the effects of material and geometrical properties, temperature, elastic foundations and eccentrically stiffeners on the buckling and postbuckling loading capacity of the functionally graded truncated conical shells in thermal environments ACCEPTED MANUSCRIPT Eccentrically stiffened functionally graded (ES-FGM) truncated conical shell surrounded on elastic foundations Consider a thin truncated conical shell of thickness h and semi-vertex angle β , the geometry of shell is shown in Fig 1, in which L is the length, R1 is its small RI PT base radius and R, H geometrical parameters as shown in Fig The truncated cone is referred to a curvilinear coordinate system ( x,θ , z ) whose the origin is located in the middle surface of the shell, x is in the generatrix direction measured from the vertex of conical shell, h is in the circumferential direction and the axes z being SC perpendicular to the plane ( x, h ) , lies in the outwards normal direction of the cone Also, x0 indicates the distance from the vertex to small base, and u , v and w denote M AN U the displacement components of a point in the middle surface in the direction x, h and z , respectively; h1 and b1 are the thickness and width of stringer ( x -direction); h2 and b2 are the thickness and width of ring ( θ -direction) Also, d1 = d1 ( x ) and d are the distance between two stringers and two rings, respectively z1 , z2 represent the TE D eccentricities of stiffeners with respect to the middle surface of shell The effective properties of the FGM truncated conical shell (the elastic modulus E , the Poisson ratioν , the thermal expansion coefficient α ) can be written as 2014): EP follows (Bich et al., 2011; Dung et al., 2013; Duc and Quan, 2013; Duc and Cong, 2z + h  ( E ,α ) = ( Em ,α m ) + ( Ecm ,α cm )   ,  2h  N AC C (1) where Ecm = Ec − Em , α cm = α c − α m , the volume fraction index N is a nonnegative number that defines the material distribution and can be chosen to optimize the structural response, and subscripts m and c stand for the metal and ceramic constituents, respectively And the Poisson ratio is assumed to be constant ν = const From Eq (1) we have: E = Em at z = −h / (metal-rich) and E = Ec at z = h / (ceramic-rich) ACCEPTED MANUSCRIPT β RI PT β M AN U SC θ x0 = R1 sin β , H = L cos β , R = R1 + L sin β TE D Fig Eccentrically stiffened FGM truncated conical shell surrounded by an elastic foundations A material property Pr of both FGM truncated conical shell and stiffeners, such EP as the elastic modulus E , the Poisson ratioν , the thermal expansion coefficient α can be expressed as a nonlinear function of temperature (Touloukian, 1967): −1 Pr = P0 ( P−1T −1 + + PT + P2T + PT ), AC C (2) in which T = T0 + ∆T ( z ) and T0 = 300 K (room temperature); P−1 , P0 , P1 , P2 , P3 are coefficients characterizing of the constituent materials ∆T is temperature rise from stress free initial state, and more generally, ∆T = ∆T ( z ) In short, we will use T-D (temperature dependent) for the cases in which the material properties depend on temperature Otherwise, we use T-ID for the temperature independent cases The material properties for the later one have been determined by Eq (2) at room temperature, i.e T0 = 300 K ACCEPTED MANUSCRIPT The FGM truncated conical shell is surrounded by an elastic foundations (Fig 1) Pasternak model is used to describe the reaction of the elastic foundations on the conical shell If the effects of damping and inertia force in the foundations are neglected, the foundation interface pressure (Sofiyev, 2013; Najafov, 2013): q ( x,θ ) = K1w − K ∆w, ∂ w ∂w ∂2w + + , K1 (in N / m3 ) is the Winkler foundation 2 2 ∂x x ∂x x sin α ∂θ RI PT where ∆w = (3) stiffeness and K (in N / m ) is the shear subgrade modulus of the Pasternak SC foundation model Eccentrically stiffened FGM truncated conical shell under temperature M AN U The present study uses the classical shell theory with the geometrical nonlinearity in von Karman sense and smeared stiffeners technique to establish the governing equations Thus, the normal and shear strains at distance z from the middle surface of shell are (Brush and Almroth, 1975): ε x = ε x0 + zk x , εθ = εθ0 + zkθ , γ xθ = γ x0θ + zk xθ , (4) TE D in which ε x0 and ε θ0 are the normal strains and γ x0θ is the shear strain at the middle surface of the shell, and k x , kθ and k xθ are the change of curvatures and twist, respectively They are related to the displacement components as (Brush and Almroth, EP 1975): ∂u  ∂w  ε = +   , ∂x  ∂x  AC C x ∂v u w  ∂w  εθ = + + cot β + 2   , x sin β ∂θ x x x sin β  ∂θ  ∂u v ∂v ∂w ∂w γ x0θ = − + + , x sin β ∂θ x ∂x x sin β ∂x ∂θ kx = − (5) ∂2w ∂ w ∂w , k = − − , θ ∂x x sin β ∂θ x ∂x ∂ 2w ∂w k xθ = − + x sin β ∂x∂θ x sin β ∂θ Hooke law for an FGM truncated conical shell with temperature-dependent ACCEPTED MANUSCRIPT properties is defined as E ε x + vε θ − (1 + v ) α∆T ( z )  , − v2  E = ε θ + vε x − (1 + v ) α∆T ( z )  , − v2  E γ xθ = (1 + v ) σ xsh = σ xshθ (6) RI PT σ θsh For stiffeners in thermal environments, we have proposed its form adapted from (Duc and Quan, 2013; Duc and Cong, 2014) as the follows: Est α st ∆T ( z ) , − 2vst Er σ θ = Er εθ − α r ∆T ( z ) , − 2vr SC σ xs = Est ε x − (7) M AN U s here, Est = Est (T ) , vst = vst (T ) , α st = α st ( T ) are the Young’s modulus, Poisson ratio and thermal expantion coefficient of the stiffener in the x -direction, respectively And Er = Er (T ) , vr = vr (T ) ,α r = α r (T ) are the Young’s modulus, Poisson ratio and thermal expantion coefficient of the stiffener in the θ -direction, respectively To TE D guarantee the continuity between the stiffener and shell, the stiffener is taken to be pure-metal if it is located at metal-rich side and is pure-ceramic if it is located at ceramic-rich side (this assumption was proposed by Bich in (Bich et al., 2011)) In EP order to investigate the FGM truncated conical shell with stiffeners in thermal environment, we have not only taken into account the materials muduli with AC C temperature-dependent properties but also assumed that all elastic moduli of FGM truncated conical shell and stiffener are temperature dependence and they are deformed in the presence of temperature Hence, the geometric parameters, the shell’s shape and stiffeners are varied through the deforming process due to the temperature change We have assumed that the thermal stress of stiffeners is subtle which distributes uniformly through the whole shell structure, therefore, we can ignore it because the size of stiffeners is great smaller than the plates and the gap between every stiffeners is not tight (this assumption first was proposed by Duc et al (2013) and has been used in Duc and Cong (2014)) ACCEPTED MANUSCRIPT Taking into account the contribution of stiffeners by the smeared stiffener technique and omitting the twist of stiffeners because of these torsion constants are smaller than the moments of inertia (Brush and Almroth, 1975) In addition, the change of spacing between stringers in the meridional direction is also taken into account Integrating the above stress–strain equations and their moments through the RI PT thickness of the shell, we obtain the expressions for force and moment resultants of an eccentrically stiffened FGM conical shells (Dung et al., 2013, 2014) N xθ = A66γ x0θ + B66 k xθ , M AN U SC  Est A1T  0 N x =  A11 +  ε x + A12ε θ +  B11 + C1 ( x )  k x + B12 kθ + Φ a , d1 ( x )    Er A2T  0 Nθ = A12ε x +  A22 +  ε θ + B12 k x + [ B22 + C2 ] kθ + Φ a , d2    Est I1T  M x =  B11 + C1 ( x )  ε + B12ε θ +  D11 +  k x + D12 kθ + Φ b , d1 ( x )    E IT  M θ = B12ε x0 + [ B22 + C2 ] ε θ0 + D12 k x +  D22 + r  kθ + Φ b , d2   M xθ = B66γ x0θ + D66 k xθ , AC C EP in which TE D x (8a) (8b) ACCEPTED MANUSCRIPT E1 = E m h + E3 = E cm h   , E = E cm h  − , N +1  N + 2N +  1   E m h + E cm h  − + , 12  N + N + 4N +  h T + h1T T h T + h2T , z2 = , 2 E AT z T L 2π sin β C = ± r 2T , d1 ( x ) = λ0 x , d = , λ0 = , d2 nr n st C1 ( x ) = C10 E AT z T , C1 = ± st 1 , x λ0 E ( z )α ( z ) ∫ − v ∆ T ( z ) dz , − h/2 Φa = − E ( z )α ( z ) ∆ T ( z ) zdz , 1− v − h/2 SC h/2 ∫ (9) M AN U h/2 Φb = − RI PT A1T = b1T h1T , A2T = b2T h2T , z1T = 2 T T T T b1 ( h1 ) + A1T ( z1T ) , I 2T = b2 ( h2 ) + A2T ( z 2T ) , 12 12 E1 vE1 E1 A11 = A22 = , A12 = , A66 = , 2 1− v 1− v (1 + v ) I1T = E2 vE E2 , B12 = , B66 = , 2 1− v 1− v (1 + v ) D11 = D22 = E3 vE E3 , D12 = , D66 = , 2 1− v 1− v (1 + v ) TE D B11 = B22 = EP After the thermal deformation process, the geometric shapes of stiffeners can be determined as follows: h1T = h1 (1 + α ∆T ( z ) ) , h2T = h2 (1 + α ∆T ( z ) ) , AC C z1T = z1 (1 + α ∆T ( z ) ) , z2T = z2 (1 + α ∆T ( z ) ) , b1T = b1 (1 + α ∆T ( z ) ) , b2T = b2 (1 + α ∆T ( z ) ) , (10) d1T = d1 (1 + α ∆T ( z ) ) , d 2T = d (1 + α ∆T ( z ) ) , in which α is thermal expantion coefficient of the stiffener; nst , nr are the number of stringer and ring respectively The quantities A1 , A2 are the cross-section areas of stiffeners and I1 , I are the second moments of inertia of the stiffener cross sections related to the shell middle surface Although the stiffeners are deformed by ACCEPTED MANUSCRIPT of critical thermal load increase Without elastic foundation ( K1 = K = ) the value of critical thermal load is the smallest and K1 = × 107 N / m3 , K = × 107 N / m is the biggest Table shows the value of the ground coefficient K which affects the critical greater than the ground coefficient K1 = N / m3 , K = 2.5 × 105 N / m (Stringers) results K1 = 2.5 × 107 N / m3 , K = N / m (Stringers) results K1 For example RI PT ∆Tcr thermal load ∆Tcr = 240 K and ∆Tcr = 237 K And Table SC shows the critical thermal load of stiffened shell by stringers is the biggest, the stiffend shell by orthogonal is the second and the critical thermal load of the stiffend shell by ring stiffeners is the smallest M AN U Table Effect of foundation (without stiffeners) on critical thermal load ∆Tcr ( K ) , ( h = 0.0127m, R / h = 100, L = 2R ,α = 30 , N = 1) 1 K1 ( N / m3 ) ∆Tcr ( K ) 3.5 × 107 × 107 205(8,9) 209(8,8) 212(8,6) 216(9,4) × 105 210(8,5) 214(9,4) 216(8,5) 220(8,8) 3.5 × 105 212(8,9) 215(8,9) 218(8,7) 222(9,4) 217(8,5) 221(9,6) 223(8,5) 228(8,5) EP × 105 TE D K2 ( N / m) × 107 AC C Table Effect of stiffeners and foundations on critical thermal load ∆Tcr ( K ) , ( h = 0.0127m, R1 = 1.27m, L = R1 , h1 = 0.01375m, h2 = 0.01m, b1 = 0.0127m, b2 = 0.0127m ) Orthogonal ∆Tcr ( K ) 28 Stringer 28 Ring nst = nr = 14 K1 = N / m3 , K = 2.5 × 105 N / m 240(8,5) 224(9,6) 232(8,6) K1 = 2.5 × 107 N / m3 , K = 2.5 × 105 N / m 244(8,6) 229(8,4) 237(8,7) 22 ACCEPTED MANUSCRIPT K1 = 2.5 × 107 N / m3 , K = N / m 237(7,12) 222(9,4) 233(7,15) 5.2.4 Effect of the volume fraction index N The parameters for the stiffeners and the geometric parameters were chosen as RI PT below: h = 0.0127, R1 = 1.27 m, L = 2.54m, h1 = 0.03175m, h2 = 0.01m, b1 = b2 = 0.0127 m, nst = nr = 15, K1 = 2.5 × 107 , K = Table 10 Effect of the volume fraction index ( N ) and the semi-vertex angle β on SC critical temperature ∆Tcr ( K ) and FGM truncated conical shells under uniform M AN U thermal load N 10-2 100 101 102 557(8,16) 525(8,15) 420(8,16) 381(8,16) 369(8,15) 358(8,16) 341(8,15) 10 494(9,14) 464(8,15) 369(8,16) 334(8,15) 324(9,13) 313(9,12) 298(9,12) 30 320(8,8) 300(8,7) 203(8,10) 196(8,9) 1: β=5o 2: β=10o Outside stiffeners h1=0.01375m, h2=0.01m 3: β=30o b1=0.0127m, b2=0.0127m AC C 450 210(8,4) EP 550 500 234(8,10) TE D 600 ∆ Tcr(K) β 10-1 400 350 300 h=0.0127m, R1/h=100, L=2R1 250 200 150 N 23 10 186(8,2) ACCEPTED MANUSCRIPT Fig Effect of the volume fraction index N on critical thermal load ∆Tcr Table 10 shows the changes in the values of the critical thermal loads when changing volume ratio N Figure shows the variation of contact curve between the critical heat load values - the coefficient of volumetric percentage in three cases RI PT β = ( 50 ,100 ,300 ) It can be seen that when the value N is increasing it makes the critical temperature decreases This is expected because the elastic modulus E of the ceramic is much larger than the metal while the volume ratio of ceramic components in the shell decreases when increasing N Moreover, Figure also shows the SC relationship curved between the critical temperature value - volume ratio coefficient will be lowered if the semi - vertex angle β increases M AN U 5.2.5 Effect of the ratio R1 / h Figure shows the effect of radius ratio on the shell thickness ( R1 / h ) on the critical temperature value and shells under the effect of temperature rising We found that if the ratio R1 / h is increasing, it will reduce the value of the critical temperature This is appropriate because the increasing ratio R1 / h will reduce h ( thinner shell) TE D then make the ability of heat load low The relationship between the ratio R1 / h and the critical temperature in both stiffined case and un-stiffined case is also shown in Fig It can be seen that in the absence of un-stiffined case will make the curve AC C EP becomes lower than stiffined case 24 ACCEPTED MANUSCRIPT 450 Outside stiffeners h1=h2=0.005m 350 b1=b2=0.0025m 3: Stiffened, β=10° 4: Stiffened, β=30° nst=nr=14 300 ∆ Tcr(K) 1: Unstiffened, β=10° 2: Unstiffened, β=30° RI PT 400 250 200 150 R=0.75m, L=1.5m, N=1, 50 50 K1=2.5x10 N/m , K2=2.5x10 N/m 100 150 200 250 300 R1/h 350 400 450 500 M AN U 100 SC Fig Effect of R1 / h ratio on critical thermal loads ∆Tcr ( K ) 5.2.6 Effect of the ratio L / R1 Table 10 and Fig illustrate the effect of L / R1 ratios on critical thermal load in TE D two cases which are uniform temperature rise and linear temperature distribution through the thickness It can be observed, the critical temperature load ∆Tcr decrease with the increase of length-to-radius ratio L / R1 And critical load value in the linear AC C rise EP temperature distribution through the thickness is bigger than the uniform temperature 25 ACCEPTED MANUSCRIPT 650 1: 2: 3: 4: 600 550 Uniform, (a) Uniform, (b) Linear, (a) Linear, (b) 400 RI PT 450 h1=0.03175m, h2=0.01m b1=0.0127m, b2=0.0127m 350 (a): K 1=0, K2=0 (b): K 1=2.5x107N/m3, K2=2.5x105N/m 300 R=1.27m, h=0.0127m, N=1,β=300 250 0.5 1.5 M AN U SC ∆ Tcr(K) 500 L/R1 Fig Effect of L / R1 ratio on critical loads ∆Tcr ( K ) 5.2.7 The influence of the material properties depending on temperature ∆Tcr ( K ) Table 11 Comparing the value of ∆Tcr in the two cases T-D and T-ID, o = 2.5 × 107 , K = 2.5 × 105 , nst = 15, nr = 15 ) TE D ( β = 30 , K T-D 328(8,7) 238(8,5) 214(8,6) 207(8,6) 404(8,8) 277(7,17) 244(7,13) 235(7,12) AC C T-ID EP N Table 11 shows the value of ∆Tcr when comparing between T-D case and T-ID case with diffirence in component ratios Obviously, this table shows once more that when N increase, the value of ∆Tcr decrease and the material properties significantly are effected on critical thermal loads and also the critical loads decrease if the shell have temperature-dependent properties Therefore, when the FGM shells under highly changed temperature environment, we need to consider the temperature-dependent properties of the shell 26 ACCEPTED MANUSCRIPT Conclusion This paper studied the thermal stability of an eccentrically stiffened functionally graded truncated conical shells with temperature-dependent properties in thermal environment surrounded on elastic foundations We summarize the following main findings: RI PT • The value of the critical temperature increases when we increase the coefficient of stiffeners and vice versa • The value of the critical temperature in case of the uniform temperature rise is smaller than one of the linear temperature distribution through the thickness SC • Heat load carrying capacity of ES-FGM truncated conical shell is reduced considerably when R1 / h ratio or L / R1 ratio or volume fraction index N M AN U increases • Critical thermal loads ∆Tcr of ES-FGM truncated conical shells decrease when the semi-vertex angle β increases • Buckling strength of ES-FGM truncated conical shell with fully ceramic shell is better than it with fully metal shell TE D • Foundation parameters K1 and K affect strongly on the critical thermal loads Furthermore, the value of the ground coefficient K affects the critical thermal load ∆Tcr greater than the ground coefficient K1 EP • The dependence of the material properties (T-D ) significantly influences the AC C critical temperature value Critical temperature value ∆Tcr will decreases if the effects of temperature on the material properties is mentioned Acknowledgment This work was supported by Grant “Nonlinear analysis on stability and dynamics of functionally graded shells with special shapes” of Vietnam National University, Hanoi The authors are grateful for this support 27 ACCEPTED MANUSCRIPT Appendix I  E AT C11 =  A11 x + st λ0   ∂2 ∂2 ∂  Er A2T  + A + A − A +   22  , 66 11 2 ∂ x x sin x ∂ θ ∂ x d x    ∂2 − ( A12 + A66 ) sin β ∂x∂θ x sin β C12 =  Er A2T + + A A  22 66 d2   ∂  ,  ∂θ RI PT ∂3 ∂3 C13 = − ( B11 x + C ) − ( B12 + B66 ) ∂x x sin β ∂x∂θ  + 2 ,  x sin β SC  Er A2T − cot β  A22 + x d2  ∂2  Er A2T + A + A + ( A12 + A66 )  22 66 sin β ∂x∂θ x sin β  d2 C22 =  Er A2T A +  22 x sin β  d2 C23 = −  ∂2 ∂2 ∂ xA + + A66 − A66 ,  66 ∂x ∂x x  ∂θ ∂3 ∂3 ( B12 + B66 ) − ( B22 + C2 ) sin β ∂x ∂θ x sin β ∂θ  1 ∂2 E AT  ∂ + cot β  A22 + r  , ( B22 + C2 ) x sin β ∂x∂θ x sin β d  ∂θ  TE D −  ∂  ,  ∂θ M AN U C21 = ∂3 ∂3 ∂2 C31 = ( B11 x + C ) + + B11 ( B11 + B66 ) ∂x x sin β ∂x∂θ ∂x EP ∂2   ∂ + 2 ( B22 + C ) −  A12 cot β + ( B22 + C2 )  x sin β ∂θ x   ∂x AC C  1 Er A2T  + ( B22 + C ) − cot β  A22 + , x x d2   C32 = ∂3 ∂3 ( B12 + B66 ) + ( B22 + C2 ) sin β ∂x ∂θ x sin β ∂θ   ∂2 Er A2T   ∂ − + cot β  A22 + , ( B22 + C2 ) ( B22 + C2 ) −  x sin β ∂x∂θ  x sin β x sin β d   ∂θ  28 ACCEPTED MANUSCRIPT   E I T  ∂4 E I T  ∂4 C33 = −  D11 x + st  −  D22 + r  λ0  ∂x x sin β  d  ∂θ  2 ∂4 ∂3 − ( D12 + D66 ) 2 + 2 ( D12 + D66 ) x sin β x sin β ∂x ∂θ ∂x∂θ Er I 2T ∂3   + D +   22 d2 ∂x  x   ∂2  β + B cot + xK  12 2   ∂x RI PT −2 D11 SC   2 E IT  K2  ∂2 +  2 cot β ( B22 + C2 ) −  D12 + D66 + D22 + r  +  x sin β  d  x sin β  ∂θ  x sin β 1 1  ∂ Er I 2T  ∂ Er I 2T  + K −  D22 +  + cot β ( B22 + C2 ) − cot β  A22 +  − xK1 , d  ∂x x x d2  ∂x x   M AN U C34 = xw1, xx Appendix II m 2π ( sin β ) A11  ( x0 + L ) − x04 3L3 ( x0 + L )  π A11 ( sin β ) L ( x0 + L ) d11 = − +  + 2 π L2 4 m   − π ( sin β ) L ( x0 + L )  TE D  ( x0 + L )3 − x03 m 2π Est A1T L3  − × ( sin β )  +  L2 λ0 2m 2π   Er A2T A +  22 d2  π mn ( A12 + A66 )  ( x0 + L ) − x03   EP 4L  L3  nL2  Er A2T + + A66  , −  A22 + 2 2m π  8m  d2  AC C d12 = −  n 2π A66 L ( x0 + L ) , − 16sin β  29 ACCEPTED MANUSCRIPT d13 = mn 2π ( B12 + B66 )( x0 + L ) mπ ( sin β )( B22 + C ) ( x0 + L ) + 16sin β π m ( sin β )( cot β ) A12  ( x0 + L ) − x03 +   L + L3   4m 2π   ( x0 + L )4 − x04 3L3 ( x0 + L )  sin β B11  + +  L3 4m 2π   π m3 ( sin β ) C10  ( x0 + L ) − x03 L3  +   L3 2m 2π   π mB11 ( sin β ) ( x0 + L ) L2 ( cot β )( sin β )  Er A2T  − +  A22 + , 4m d2   SC + RI PT π m3 M AN U mnπ ( A12 + A66 )  nL2 ( A12 + A66 ) nL2   Er A2T 3 d 21 = − x0 + L ) − x0 + − A22 + + A66  , (    12 L 8m 8m  d2  n 2π L ( x0 + L )  Er A2T  π A66 ( sin β ) L ( x0 + L ) d 22 = −  A22 + − 16sin β d2   3π A66 ( sin β ) L ( x0 + L ) m 2π A66 sin β  − x0 + L ) − x04  + , (   8L EP nπ ( cot β ) L ( x0 + L )  Er A2T A +  22 d2   nπ L ( B22 + C2 ) , +  AC C + TE D n3π L ( B22 + C2 ) m nπ ( B12 + B66 )  ( x0 + L ) − x03 L3  d 23 = + −   16sin β L2 4m 2π   30 ACCEPTED MANUSCRIPT m 2π B11 ( sin β )  L 3L4  3 d31 = − x x L − + + ( )  2mπ L2 4m3π   ( )  Er A2  L ( x0 + L ) cot β sin β +  A22 +  d2  4m  ( ) RI PT n ( B22 + C2 ) L2 ( B22 + C2 )( sin β ) L2 + − + 16m sin β 4m m3π sin β C10 + L3  ( x0 + L ) − x04 3L3 ( x0 + L )  −   2 8 m π   M AN U mn 2π ( B12 + B66 )  ( x0 + L ) − x03 L3  + −   L sin β 4m 2π   SC 3   m3π sin β B11  ( x0 + L ) − x0 L x0 − ( x0 + L ) 3L5  + × + +  L3 10 2m 2π 4m 4π     ( x0 + L ) − x04 3L3 ( x0 + L )  −   8m 2π   mπ ( sin β )( B22 + C2 )  ( x0 + L ) − x03 L3  + −  , 2 π L m   TE D mπ A12 cot β sin β + L π n ( B22 + C2 ) L ( x0 + L ) π n3 ( B22 + C2 ) L ( x0 + L ) d32 = − + 32sin β Er A2T A +  22 d2    ( x0 + L ) − x03 L3  −  ,  2 m π    AC C + π n cot β  EP m nπ ( B12 + B66 )  ( x0 + L ) − x04 3L3 ( x0 + L )  + −   L2 8m 2π   31 ACCEPTED MANUSCRIPT ( )   π n L ( D12 + D66 ) π m3 D11 sin β  L x0 − ( x0 + L ) 3L4  + × + d33 =  L3 8sin β 2mπ 4m3π    π L sin β  Er I  π ( B22 + C2 ) L ( x0 + L ) cot β sin β π m sin β D11 + −  D22 + + 4 d2  L4   ( x0 + L )5 − x05 3L5  L2 3 × + x0 − ( x0 + L ) +  10 2m 2π 4m 4π   π m Es I1 sin β − L4 λ0  ( x0 + L ) − x04 3L ( x0 + L )  −   8m 2π   π n4 L  Er I 2T  D +  22  32sin β  d2  SC − ) π 3m n ( D12 + D66 )  ( x0 + L ) − x03 L3  − −   L2 sin β 4m 2π   3 Er I   ( x0 + L ) − x0 L3  −   D22 +  d   4m 2π   π 3m sin β  L2 M AN U − RI PT ( TE D π n2 L  Er I 2T  +  D12 + D66 + D22 +  d2  4sin β    ( x0 + L ) − x03 L3  −    4m 2π    3π L π 3m B12 cos β  − x0 + L ) − x04  + B12 cos β ( x0 + L ) (   4L π n ( B22 + C2 )( cot β ) L ( x0 + L ) − , 8sin β  E AT −π cot β sin β  A22 + r d2  AC C EP m 2π sin β d34 = − L2  ( x0 + L )4 − x04 L2 x02 − ( x0 + L ) +  2 8mπ  ( (  L2 x03 − ( x0 + L ) x + L − x ( ) 0 d35 = −π sin β  +  10 2m 2π  32 )  ) ,   3L5  + , 4m 4π   ACCEPTED MANUSCRIPT ( )   π m sin β  L x0 − ( x0 + L ) 3L4  + d36 =  2L 2mπ 4m3π    ( ) 3   L x − x + L ( ) 0 x L x + − ) 3L5  π m sin β  ( 0 − + +  L2 10 2m 2π 4m 4π    RI PT References Bich, D.H., Phuong, N.T., Tung, H.V., 2012 Buckling of 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composite conical shells AC C EP TE D M AN U SC J Therm Stresses 24(9), 881-901 35 ACCEPTED MANUSCRIPT Highlights • The thermal stability of an eccentrically stiffened functionally graded truncated conical shells with temperature-dependent properties • The shells are in thermal environment and surrounded on elastic foundations Both RI PT shells and stiffeners are deformed simultaneously due to temperature • Using Galerkin method, the closed-form expression for determining the thermal buckling load is obtained SC • Effects of temperature, material and geometrical properties, elastic foundations and eccentrically outside stiffeners on the critical thermal buckling loads of the FGM truncated conical shells are analyzed and discussed AC C EP TE D M AN U • The obtained results are validated by comparing with those in the literature ... MANUSCRIPT Nonlinear thermal stability of eccentrically stiffened functionally graded truncated conical shells surrounded on elastic foundations Nguyen Dinh Duc*, Pham Hong Cong Vietnam National University,... paper studies the thermal stability of an eccentrically stiffened functionally graded truncated conical shells in thermal environment and surrounded on elastic foundations Both of the FGM shell... stability, Eccentrically stiffened truncated conical shell, Functionally Introduction EP graded materials, elastic foundations AC C The idea of the construction of functionally graded meterials (FGM)