Nonlinear response and buckling analysis of eccentrically stiffened FGM toroidal shell segments in thermal environment

Accepted Manuscript Nonlinear response and buckling analysis of eccentrically stiffened FGM toroidal shell segments in thermal environment

Pham Minh Vuong, Nguyen Dinh Duc

PII: DOI: Reference:

S1270-9638(17)31661-9 https://doi.org/10.1016/j.ast.2018.05.058 AESCTE 4611

To appear in:

Aerospace Science and Technology

Received date: Revised date: Accepted date:

9 September 2017 17 February 2018 30 May 2018

Please cite this article in press as: P.M. Vuong, N.D. Duc, Nonlinear response and buckling analysis of eccentrically stiffened FGM toroidal shell segments in thermal environment, Aerosp. Sci. Technol. (2018), https://doi.org/10.1016/j.ast.2018.05.058

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Nonlinear response and buckling analysis of eccentrically stiffened FGM toroidal shell segments in thermal environment Pham Minh Vuong1, Nguyen Dinh Duc2,3,4* 1

Faculty of Civil and Industrial, National University of Civil Engineering, 55 Giai Phong Street, Hai Ba Trung District, Hanoi, Vietnam

2

3

4

Advanced Materials and Structures Laboratory, VNU Hanoi -University of Engineering and Technology, 144 Xuan Thӫy Street, Cau Giay District, Hanoi, Vietnam

Infrastructure Engineering Program -VNU-Hanoi, Vietnam-Japan University (VJU), My Dinh 1 – Tu Liem – Hanoi – Vietnam

National Research Laboratory, Department of Civil and Environmental Engineering, Sejong University, 209 Neungdong-ro, Gwangjin-gu, Seoul 05006, Korea *

Corresponding author: Duc ND., Tel: +84-24-37547978, E-mail: [email protected]

Abstract. This paper presents an analytical approach to study nonlinear response and buckling analysis of FGM toroidal shell segments reinforced by FGM stiffeners surrounded by elastic foundations in thermal environment and under external pressure. The formulations are based on Reddy’s third-order shear deformation shell theory (TSDT) with von Karman nonlinearity, Pasternak type elastic foundations and smeared stiffener technique. By applying Galerkin’s method and using stress function, closed-form expressions for determining the static critical external pressure load and postbuckling load–deflection curves are determined. Finally, the influences of geometrical parameters, volume fraction index, elastic foundations, and the effectiveness of stiffeners on the stability of shells are considered. Keywords: Eccentrically stiffened FGM toroidal shell segment; nonlinear response and buckling analysis; TSDT; thermal environment; elastic foundations.

1. Introduction The material has variable mechanical property with international name Functionally Graded Material and often abbreviated FGM is a type of new generation composite, intelligent composite, appears as a result of actual demands for a material that can overcome the disadvantages of traditional metals and laminated normal composites. This functionally graded material is formed from two component materials of ceramic and metal in which the volume ratio of each composition varies smoothly and continuously from this side to the other side according to the structure wall thickness in order to be suitable for the characteristic strength of the component materials. This material combines the metal’s advantage of high durability and the ceramic’s advantage of high temperature durability, so FGM structures have many practical applications. Therefore, the problems of stability and fluctuations of FGM structures such as FGM plates and shells have

attracted scientists’ attention. Batra presented torsion of a functionally graded cylinder [1], Wang et al. proposed exact solution and transient behavior for torsional vibration of functionally graded finite hollow cylinders [2]. Shen studied torsional buckling and postbuckling of FGM cylindrical shells in thermal environments [3]. Huang and Han considered nonlinear buckling of torsion-loaded functionally graded cylindrical shells in thermal environment [4]. They also considered nonlinear elastic buckling and postbuckling of axially compressed functionally graded cylindrical shells in [5], nonlinear buckling and postbuckling of heated functionally graded cylindrical shells under combined axial compression and radial pressure in [6] and nonlinear postbuckling of FGM cylindrical shells under radial load in [7], nonlinear dynamic buckling of functionally graded cylindrical shells subjected to a time-dependent axial load in [8]. Sofiyev et al. investigated the stability of functionally graded cylindrical shells under linearly increasing dynamic torsional loading [9] and thermoelastic stability of freely supported functionally graded conical shells within the shear deformation theory in [10], thermoelastic buckling of FGM conical shells under nonlinear temperature rise in the framework of the shear deformation theory in [11]. Najafizadeh et al. considered mechanical stability of functionally graded stiffened cylindrical shells [12]. Akbari et al. studied thermal buckling of temperature-dependent FGM conical shells with arbitrary edge supports [13]. Torabi investigated thermal buckling analysis of truncated hybrid FGM conical shells [14]. Jam et al. considered buckling of pressurized functionally graded carbon nanotube reinforced conical shells in [15]. Mirzae and Kiani presented thermal buckling of temperature dependent FG-CNT reinforced composite conical shells in [16]. Stein considered buckling of segments of toroidal shells in [17]. Hutchinson studied initial postbuckling behavior of toroidal shell segments in [18]. Asratyan investigated mixed boundary-value problems of thermoelasticity anisotropic-in-plane inhomogeneous toroidal shells in [19]. Tornabene et al. presented static analysis of functionally graded doubly-curved shells and panels of revolution in [20]. Recently, Duc published the monograph about nonlinear static and dynamic Stability of FGM plates and shells [21]. However, in order to increase the ability of the panels and shells in responding positively to different loads, they are usually reinforced with reinforcing stiffeners, and structures are relied on elastic medium. Therefore, static and dynamic stability of reinforcing structures is very interested to the scientific community. Shen et al. considered thermal vibration, buckling and dynamic stability of functionally graded cylindrical shells embedded in an elastic medium in [22], postbuckling of shear deformable FGM cylindrical shells surrounded by an elastic medium in [23] and postbuckling of internal pressure loaded FGM cylindrical shells surrounded by an elastic

2

medium in [24]. Sofiyev et al. presented the stability of cylindrical shells containing a FGM layer subjected to axial load on the Pasternak foundation in [25] and torsional vibration and buckling of the cylindrical shell with functionally graded coatings surrounded by an elastic medium in [26]. Bagherizadeh et al. considered mechanical buckling [27] and thermal buckling [28] for functionally graded material cylindrical shells on elastic foundations. Najafov et al. [29] studied torsional vibration and stability of functionally graded orthotropic cylindrical shells on elastic foundations. Dung et al. investigated nonlinear torsional buckling and postbuckling of eccentrically stiffened functionally graded thin circular cylindrical shells in [30] and of eccentrically stiffened functionally graded cylindrical shells under external pressure in [31]. McElman studied eccentrically stiffened shallow shells of double curvature in [32]. Bich et al considered non-linear buckling analysis of FGM toroidal shell segments filled inside by an elastic medium under external pressure loads including temperature effects in [33], post-buckling of Sigmoid-functionally graded material toroidal shell segment surrounded by an elastic foundation under thermo-mechanical loads in [34], nonlinear buckling of eccentrically stiffened functionally graded toroidal shell segments under torsional load surrounded by elastic foundation in thermal environment in [35], nonlinear dynamical analyses of eccentrically stiffened functionally graded toroidal shell segments surrounded by elastic foundation in thermal environment in [36]. When studying plate, thin structures, classical shell theory is often used. However, for thick shells, classical shell theory is no longer appropriate; instead, higher-order shear theories such as first or third order shear theory results has to be used in order to bring out more accurate analysis. Therefore, using TSDT to solve static and dynamic stability problems of FGM panels and shells is interesting to many authors. There are two main approaches to solve these problems of which one traditional way is using Finite element method and the other is the analytical approach which is used in recent years. The advantage of the analytical method is that critical load and nonlinear responses are presented explicitly by the geometric parameters of the structure and material properties, as well as the extrinsic force and temperature, so we can design and modify these parameters properly to actively control the behavior of the structure. Based on the analytical approach using TSDT to study nonlinear static stability, dynamic response for FGM structures was investigated and solved by Duc and Cong [37, 38] for the FGM plates, by Duc and Quan [39,40] for the FGM double curved shallow shells and FGM cylindrical panels [41,42], by Duc and Thang [43,44,45] for the FGM circular cylindrical shells. It is notice able that the results of research groups of Bich et al., Dung et al. and Duc et al., different from those of other authors, derives from using the analytical methods.

3

Recently, nanocomposite and CNT-reinforced composite have gra- dually gained interestes: R. Kolahchi and A. M. Moniri Bidgoli [46] used the differential quadrature method (DQM) in conjunction with the Bolotin method to calculate the dynamic instability region (DIR) of embedded single-walled carbon nanotubes SWCNTs. Reza Kolahchi and Ali Cheraghbak [47] calculated resonance frequency and dynamic instability region of embedded microplates reinforced by singlewalled carbon nanotubes (SWCNTs) using the Navier method in conjunction with the Bolotin’s methods. Reza Kolahchi et al. [48] investigated the nonlinear dynamic stability of embedded temperature-dependent viscoelastic plates reinforced by single-walled carbon nanotubes (SWCNTs). Maryam Shokravi [49] studied buckling of embedded laminated plates with agglomerated CNT-reinforced composite layers. By the same author, the buckling problem of sandwich plates with FG-CNT-reinforced layers was investigated [50]. Mohammad Hadi Hajmohammad et al. [51] used the differential quadrature method to solve the problem on the dynamic buckling behavior of a sandwich plate composed of laminated viscoelastic nanocomposite layers integrated with viscoelastic piezoelectric layers. Behrooz Keshtegar et al. [52] analyzed the optimization of embedded piezoelectric sandwich nanocomposite plates for dynamic buckling analysis based on Grey Wolf algorithm. Reza Kolahchi et al. [53] studied wave propagation of embedded viscoelastic FG-CNT-reinforced sandwich plates integrated with sensor and actuator based on refined zigzag theory. Ali Jafarian Arani and Reza Kolahchi [54] presented buckling analysis of embedded concrete columns armed with carbon nanotubes. The nonlinear buckling of straight concrete columns armed with single-walled carbon nanotubes (SWCNTs) resting on foundation is investigated by Bilouei et al. [55]. Mehdi Zamanian et al. [56] investigated the agglomeration effects on the buckling behavior of embedded concrete columns reinforced with SiO2 nano – particles. Mohammad Sharif Zarei et al. [57] studied the seismic response of the fluidconveying concrete pipes reinforced with SiO2 nanoparticles and fiber reinforced polymer (FRP) layer. Maryam Shokravi [58] presented vibration analysis of silica nanoparticles-reinforced concrete beams considering agglomeration effects. Reza Kolahchi et al. [59 – 60] considered viscononlocal-refined Zigzag theories for dynamic buckling of laminated nanoplates and embedded piezoelectric nanoplates using differential cubature-Bolotin methods. Reza Kolahchi [61] presented a comparative study on the bending, vibration and buckling of viscoelastic sandwich nano-plates based on different nonlocal theories using DC, HDQ and DQ methods. M. Shokravi [62] considered dynamic pull-in and pull-out analysis of viscoelastic nano-plates under electrostatic and Casimir forces via sinusoidal shear deformation theory. Vahid Baseri et al. [63] presented analytical solution for buckling of embedded laminated plates based on higher order shear deformation plate theory. 4

Toroidal shell segments have been proposed for being applied in various engineering applications such as missiles, spacecraft, submarines and nuclear reactors. Dung et al. studied nonlinear analysis on dynamic buckling of eccentrically stiffened functionally graded material toroidal shell segment surrounded by elastic foundations in thermal environment and under timedependent torsional loads [64], buckling and postbuckling of FGM toroidal shell segment surrounded by elastic foundation in thermal environment and under external pressure using Reddy’s third-order shear deformation theory (TSDT) [65], but without stiffenres. In fact, to reinforce the load-bearing capacity of the structure, the plates and shells are usually reinforced by stiffeners. These stiffeners may be isotropic materials as metal or ceramic, or possibly made of FGM materials. Due to its high thermal stability as well as special shape, FGM toroidal shell has been applied in practical ¿elds such as rocket fuel tanks, fusion reactor vessels, diver's oxygen tanks, satellite support structures, and underwater toroidal pressure hull. In case of thick shells, for high accuracy, higher order shear deformation theories must be used. It can be seen that the study of stability of thick FGM shells reinforced FGM stiffeners using TSDT is necessary and has practical significance. Developing the results achieved in [65], this work uses analytical approach, TSDT, von Karman geometrical nonlinearity and smeared stiffener techniques to study nonlinear response and buckling analysis of eccentrically stiffened FGM toroidal shell segments reinforced by FGM stiffeners. The toroidal shell is reinforced by closely spaced circular rings and longitudinal stringers attached to inside. The analytical expressions to determine the critical buckling load and the postbuckling mechanical load–deflection curves are obtained in this work. The influences of various parameters such as temperature field, stiffeners, elastic foundations, dimensional parameters and volume fraction index of materials on the nonlinear stability of FGM toroidal shell segments are considered in detail. Comparisons with results from open literature are carried out to validate the present approach. 2. Eccentrically stiffened functionally graded material (ES-FGM) toroidal shell segment surrounded by elastic foundations Consider the FGM toroidal shell segments with thickness h, equator radius R and length L formed by the rotation of a plane circular arc of radius a around an axis in the plane of the curve as shown in Figs. 1. The shell which is represented in Figs. 1a, c is called convex shell and one which 5

is represented in Figs. 1b, d is called concave shell. The geometry and coordinate system are described for the middle surface of the shell. In this paper, the FGM toroidal shell segments are free and are simply supported at two end edges and subjected to external pressure. Assume that the shell is reinforced by closely spaced circular rings and longitudinal stringers attached to inside of the shell. In addition the stiffeners and skin are made of functionally graded materials with Young’s modulus, thermal expansion coefficient and thermal conductivity coefficient varying continuously through the thickness direction of the shell with power laws as follows [12, 21, 30, 64, 65]: For shell k ­ h h § 2z + h · ° Esh = Em + Ecm ¨ ¸ , Ecm = Ec − Em , k ≥ 0, − ≤ z ≤ , 2 h 2 2 ° © ¹ ® k h h ° § 2z + h · °α sh = α m + α cm ¨ 2h ¸ , α cm = α c − α m , k ≥ 0, − 2 ≤ z ≤ 2 . © ¹ ¯

(1)

For FGM stringers k ­ § 2z − h · 2 h h ° Es = Ec + Emc ¨ ¸ , Emc = Em − Ec , k2 ≥ 0, ≤ z ≤ + hs , 2 2 ° © 2hs ¹ ® k2 ° § 2z − h · h h °α s = α c + α mc ¨ ¸ , α mc = α m − α c , k2 ≥ 0, ≤ z ≤ + hs . 2 2 © 2hs ¹ ¯

(2)

For FGM rings k ­ § 2z − h · 3 h h ° Er = Ec + Emc ¨ ¸ , k3 ≥ 0, ≤ z ≤ + hr , 2 h 2 2 ° r ¹ © ® k ° § 2z − h · 3 h h α α α = + ° r ¸ , α mc = α m − α c , k3 ≥ 0, ≤ z ≤ + hr , c mc ¨ 2 h 2 2 r ¹ © ¯

(3)

where k, k2 and k3 are volume fractions indexes of the shell, stringer and ring, respectively and the subscripts c, m, sh, s and r denote ceramic, metal, shell, stringers and ring, respectively. Note that h, hs , hr denote the thickness of shell, stringer and ring, respectively; Ec , Em ,α c ,α m are the Young’s

modulus

and

thermal

expansion

coefficients

of

ceramic

and

metal,

respectively,

Esh , Es , Er ,α sh , α s , α r are Young’s modulus and thermal expansion coefficients of the shell,

stringer and ring a distance z from the middle surface, respectively. As can be seen that, from Eqs.

6

(1), (2) and (3) the continuity between the shell and stiffeners is satisfied. In this study, the Poisson's ratio ν is assumed to be constant. The shell–foundation interaction is represented by Pasternak model as [21, 67] qsf = K1w − K 2∇w,

(4)

where qsf is the interaction force between the foundation and the shell, where qsf is the interaction force between the foundations and the shell, ∇ =

∂2 ∂2 , w is the deflection of the FGM shell, + ∂x 2 ∂y 2

K1 and K 2 are Winkler foundation stiffness and shear layer stiffness of Pasternak foundation,

respectively. R x

x

a

L/2

a

z

L/2

L/2 z L/2

R

(b)

(a)

(c)

7

(d) Fig.1 Modelling of the shells reinforced by stiffeners and surrounded by elastic foundations

3. Constitutive relations and governing equations According to Reddy’s third-order shear deformation shell theory, the displacement fields can be written as [66]: 4 3 § · ¨ u + zφ x − 2 z (φx + w, x ) ¸ 3h §u · ¨ ¸ ¨ ¸ ¨ 4 3 ¸ ¨ v ¸ = ¨ v + zφ y − 2 z φ y + w, y ¸ , 3h ¨ ¸ ¨ ¸ ¨ w¸ w © ¹ ¨ ¸ ¨ ¸ © ¹

(

)

(5)

where u, v and w are the displacement components at the middle surface and φx , φ y are the slope rotations in the (x,z) and (y,z) planes. Uses the Kármán geometrical nonlinearity and Stein and McElman [17] assumption, the strain components at the middle surface of toroidal shell segment are related to the displacements u, v and w in the x, y and z coordinate directions as [17] w § · u B + w,2x / 2 ¸ § ε x0 · ¨ , x a ¸ ¨ ¸ ¨ w ¨ ε 0y ¸ = ¨ v, y − + w,2y / 2 ¸ , ¸ R ¨ ¸ ¨ 0 ¸ ¨ ¸ ¨ γ xy © ¹ ¨ u, y + v, x + w, x w, y ¸ ¨ ¸ © ¹

(6)

0 where ε x0 and ε y0 are normal strains, γ xy is the shear strain at the middle surface of shell.

In Eq. (6) and in the equations that follow, the convention is used that, when there is a double sign, the upper sign applies to the convex shell and the lower to the concave shell.

8

The strains across the shell thickness at a distance z from the middle surface are as § k (1) · § k ( 3) · § ε x · § ε x0 · x ¨ ¸ ¨ x ¸ ¨ ¸ ¨ 0 ¸ 1 ( ) ( 3) 3 ¨ε y ¸ = ¨ε y ¸ + z ¨ ky ¸ + z ¨ ky ¸, ¨ ¸ ¨ ¸ ¨¨ ¸¸ ¨ 0 ¸ 1) ¸ 3) ¸ ( ( ¨ ¨ γ ¨ ¸ γ © xy ¹ © xy ¹ © k xy ¹ © k xy ¹

( 2) 0 § γ xz · § γ xz · 2 § k xz · ¨ ¸, ¨ ¸ ¨¨ ¸¸ = 0 + z ¨ k ( 2) ¸ © γ yz ¹ ©¨ γ yz ¹¸ © yz ¹

(7)

where ( 3) § k (1) · § φ · § kx · § · φx , x + w, xx x, x ¨ x ¸ ¨ ¨ ¸ ¸ ¨ ¸ 1 3 ( ) ( ) ¨k ¸ = ¨ φ φ y , y + w, yy ¸ , ¨ k y ¸ = −c ¨ ¸, y y, y ¨ ¸ ¨ ¸ ¸ ¨ ¨¨ ¸ ¨ k (1) ¸ ©¨ φ x , y + φ y , x ¹¸ ¨ k ( 3) ¸ φ x, y + φ y , x + 2w, xy ¹¸ © xy xy © ¹ © ¹ 0 § k ( 2) · § φ + w, x · § γ xz · § φ x + w, x · ¨ xz ¸ = −3c ¨ x ¸= , ¨ , ¨ φ y + w, y ¸¸ ¨ γ 0 ¸ ¨¨ φ y + w, y ¸¸ ¨ k ( 2) ¸ © ¹ © ¹ yz © ¹ © yz ¹

in which c =

(8)

4 , ε x , ε y are normal strains, γ xy is the in-plane shear strain, γ xz , γ yz are the transverse 3h 2

shear deformations and φx , φ y are the rotations of normal to the middle surface of the shell with respect to y and x axes, respectively. The geometrical compatibility equation for a toroidal shell segment is written as [36, 64, 65] 0 ε x0, yy + ε y0, xx − γ xy , xy = −

1 1 w, xx B w, yy + w, xy R a

(

)

2

− w, xx w, yy .

(9)

Hooke law taking into account the thermal element of shell and stiffeners, are defined as follows: For shell

§ σ xsh ¨ ¨ sh ©σ y

· E § ε x + νε y − (1 + ν )α sh ΔT · ¸ = sh 2 ¨ ¸, ¸ 1 − ν ¨ ε y + νε x − (1 + ν )α sh ΔT ¸ © ¹ ¹

sh · § σ xy § γ xy · ¨ ¸ ¨ ¸ E sh sh ¨ σ xz ¸ = ¨ γ xz ¸ . ¨ ¸ 2 (1 + ν ) ¨ ¸ sh ¨ σ yz ¸ © γ yz ¹ © ¹

(10)

For FGM stiffeners § σ xg · § Es ε x − Esα s ΔT · ¨ ¸=¨ ¸. ¨ σ yg ¸ ¨© Er ε y − Erα r ΔT ¸¹ © ¹

9

(11)

sh is shearing where σ xsh ,σ ysh are normal stress in x, y direction of un-stiffened shell, respectively, σ xy sh stress, σ xzsh ,σ yz are the transverse shear stresses of un-stiffened shell, σ xg ,σ yg are normal stress of

stringer and ring stiffeners, respectively. ΔT is the variability of temperature in the environment containing the shell. The resultant forces and moments per unit length of the shell can be figured out as follows: h /2

( Ni , M i , Pi ) = ³ σ ish (1, z, z 3 ) dz + ( Nis , M is , Pi s ), i = x, y, − h /2

h /2

( Qi , Ri ) = ³ σ izsh (1, z 2 ) dz + ( Qis , Ris ),

i = x, y ,

(12)

− h /2

h /2

( N xy , M xy , Pxy ) = ³

− h /2

(

)

sh σ xy 1, z , z 3 dz ,

where N is , M is , Pi s , Qis , Ris with i = x, y , are respective quantities for stiffeners. Substitution of Eqs. (6), (7), (8), (10) and Eq. (11) into Eq. (12) and using the Lekhnitskii smeared stiffener technique [21, 30, 35], yields the constitutive relations as * * * * * * * * * ε x0 = a11 N x + a12 N y + a13 φx , x + a14 φ y , y + a15 w, xx + a16 w, yy + a17 φ1 + a18 φ1s + a19 φ1r , * * * * * * * * * ε 0y = a21 N x + a22 N y + a23 w, xx + a26 w, yy + a27 φx , x + a24 φ y , y + a25 φ1 + a28 φ1s + a29 φ1r , 0 γ xy

=

* a31 N xy

* + a32 φx, y

* + a33 φ y,x

(13)

* + a34 w, xy .

* * * * * * M x = b11 N x + b12 N y + b13 φx, x + b14 φ y , y + b15 w, xx + b16 w, yy * * * + b17 φ1 + b18 φ1s + b19 φ1r + b17φ2 + b18φ2 s , * * * * * * M y = b21 N x + b22 N y + b23 w, xx + b26 w, yy φx, x + b24 φ y , y + b25

(14)

* * * + b27 φ1 + b28 φ1s + b29 φ1r + b27φ2 + b28φ2 r , * * * * φ y , x + b34 w, xy , M xy = b31 N xy + b32 φx, y + b33

* * * * * * Px = c11 N x + c12 N y + c13 φx , x + c14 φ y , y + c15 w, xx + c16 w, yy * * * + c17 φ1 + c18 φ1s + c19 φ1r + c17φ4 + c18φ4 s , * * * * * * Py = c21 N x + c22 N y + c23 φx, x + c24 φ y , y + c25 w, xx + c26 w, yy * * * + c27 φ1 + c28 φ1s + c29 φ1r + c27φ4 + c28φ4 r , * * * * φ y , x + c34 w, xy , Pxy = c31 N xy + c32 φx, y + c33

10

(15)

* * Qx = d11 φx + d12 w, x , * * Q y = d 21 φ y + d 22 w, y ,

(16)

* * Rx = e11 φ x + e12 w, x , * * R y = e21 φ y + e22 w, y ,

where φ1 , φ1s , φ1r , φ2 , φ2 s , φ2r , φ4 , φ4s , φ4r are given by h /2

(φ1 ,φ2 ,φ4 ) = ³

(

)

Esh ( z )α sh ( z ) ΔT 1, z , z 3 dz ,

− h /2 h /2 + hs

(

)

(

)

(φ1s ,φ2 s ,φ4 s ) = ³

Es ( z )α s ( z ) ΔT 1, z , z 3 dz ,

(φ1r ,φ2r ,φ4r ) = ³

Er ( z )α r ( z ) ΔT 1, z , z 3 dz ,

h /2 h /2 + hr h /2

(17)

and the coefficients aij* , bij* , cij* , dij* , eij* can be found in Appendix A and Appendix B. The nonlinear equilibrium equations of a toroidal shell segment surrounded on elastic foundations based on Reddy’s third order shear deformation theory are [66]:

(

N x, x + N xy , y = 0,

(18)

N xy , x + N y , y = 0,

(19)

) (

)

Qx , x + Qy , y − 3c Rx, x + Ry , y + c Px, xx + 2 Pxy , xy + Py , yy +

(

1 1 N y ± Nx R a

)

+ N x w, xx + 2 N xy w, xy + N y w, yy + q − K1w + K 2 w, xx + w, yy = 0,

(20)

(

)

(21)

(

)

(22)

M x , x + M xy , y − Qx + 3cRx − c Px , x + Pxy , y = 0, M y , y + M xy , x − Qy + 3cR y − c Py , y + Pxy , x = 0,

The stress function F(x, y) is defined as N x = F, yy , N y = F, xx , N xy = − F, xy .

(23)

Substituting Eqs. (13), (14), (15), (16) and (23) into Eqs. (9), (20), (21) and (22), after some calculations, yields

11

g11F, xxxx + g12 F, yyyy + g13 F, xxyy + g14 w, xxxx + g15 w, yyyy + g16 w, xxyy + g17φx, xxx + g18φ y , yyy + g19φx, xyy + g110φ y , yxx + g111φ x, x + g112φ y , y + g113 w, xx + g114 w, yy 1 1 + F, xx ± F, yy + F, yy w, xx − 2 F, xy w, xy + F, xx w, yy + q − K1w = 0, R a

(24)

g 21F, xxx + g 22 F, xyy + g 23 w, xxx + g 24 w, xyy + g 25φ x, xx + g 26φ x, yy + g 27φ y , xy + g 28φx + g 29 w, x = 0,

(25)

g31 F, yyy + g32 F, xxy + g33 w, yyy + g34 w, xxy + g35φ y , yy + g36φ y , xx + g37φx, xy + g38φ y + g39 w, y = 0.

(26)

g 41w, xxxx + g 42 w, yyyy + g 43 w, xxyy + g 44 F, yyyy + g 45 F, xxxx + g 46 F, xxyy + g 47φ x , xxx + g 48φ x , xyy + g 49φ y , yyy + g 410φ y , yxx =

1 −1 w, xx B w, yy + w,2xy − w, xx w, yy . R a

(27)

where the coefficients gij can be found in Appendix C. Eqs. (24), (25), (26) and (27) are governing equations using TSTD to study buckling and postbuckling of eccentrically stiffened FGM toroidal shells segment under external pressure surrounded by elastic foundations. 4. Nonlinear response and buckling analysis An eccentrically stiffened FGM toroidal shell segment is assumed to be simply supported and subjected to uniformly distribute external pressure of intensity q. Thus the boundary conditions are [4] w = 0, M x = 0, N x = 0, N xy = 0, φ y = 0 at x = 0; x = L.

(28)

The mentioned conditions (28) can be satisfied on the average sense if the deflection function w is chosen by expressing the three terms as follows [4]: w = f 0 + f1 sin Mx sin Ny + f 2 sin 2 Mx,

(29)

where f 0 ispre-buckling uniform amplitude, f1 is linear amplitude, f 2 is nonlinear amplitude, M=

mπ n , N = , m is a number of half wave in axial direction and n is a number of wave in L R

circumferential direction of the shell. Substitution of (29) into Eqs. (25), (26) and Eq. (27) yields the equations as

12

v11 F, xxxx + v12 F, xxyy + v13φx, xxx + v14φx, xyy + v15φ y , yxx + v16φx, x

(30)

= G11 cos 2 Mx + G12 sin Mx sin Ny,

v21F, yyyy + v22 F, xxyy + v23φ y , yyy + v24φ y , yxx + v25φx, xyy + v26φ y , y

(31)

= G21 sin Mx sin Ny, v31 F, xxxx + v32 F, yyyy + v33 F, xxyy + v34φx, xxx + v35φ y , yyy + v36φx, xyy + v37φ y , yxx

(32)

= G31 cos 2Mx + G32 cos 2 Ny + G33 sin Mx sin Ny + G34 sin 3Mx sin Ny,

where

(

)

* * * * * * * * * * v11 = b12 − cc12 , v12 = b11 − b31 − cc11 + cc31 , v13 = b13 − cc13 , v14 = b32 − cc32 , * * * * * * * * * * * * v15 = b14 + b33 − cc14 − cc33 , v16 = − d11 + 3ce11 , v21 = b21 − cc21 , v22 = b22 − b31 − cc22 + cc31 , * * * * * * * * * * * v23 = b24 − cc24 , v24 = b33 − cc33 , v25 = b23 + b32 − cc23 − cc32 , v26 = − d 21 + 3ce21 , v31 = a22 , v32 * * * * * * * * * v33 = a12 + a21 + a31 , v34 = a23 , v35 = a14 , v36 = a13 − a32 , v37 = a24 − a33 ,

(

)

( ( (

)

* = a11 ,

* * * * G11 = ª b15 − cc15 8M 4 + d12 − 3ce12 2 M 2 º f 2 = G111 f 2 , ¬ ¼ * * 4 * * * * * * G12 = − ª b15 − cc15 M + b16 + b34 − cc16 − cc34 M 2 N 2 + d12 M 2 º f1 = G121 f1 , − 3ce12 ¬ ¼ * * 4 * * * * 2 2 * * 2 G21 = − ª b26 − cc26 N + b25 + b34 − cc25 − cc34 M N + d 22 − 3ce22 N º f1 = G211 f1 , ¬ ¼

( (

) )

) )

( (

) )

§ −2M 2 M 2N 2 2 * · + 8M 4 a25 G31 = ¨¨ f1 = G311 f 2 + G312 f12 , ¸¸ f 2 + R 2 © ¹ M 2N 2 2 f1 = G321 f12 , 2 ªM 2 N2 º * * * * * ± − a25 + a26 − a34 G33 = « M 4 − a16 N 4 − a15 M 2 N 2 » f1 − M 2 N 2 f1 f 2 = G331 f1 + G332 f1 f 2 , a ¬ R ¼ G32 =

(

)

G34 = M 2 N 2 f1 f 2 = G341 f1 f 2 .

(33)

Three equations (30), (31) and (32) form a system of equations containing three hidden functions F ,φx, x and φ y , y . The general solution can be found in the form:

F = B1 sin Mx sin Ny + B2 sin 3Mx sin Ny + B3 cos 2 Mx + B4 cos 2 Ny +

φx , x = C1 cos 2 Mx + C2 sin Mx sin Ny + C3 sin 3Mx sin Ny, φ y , y = D1 cos 2 Ny + D2 sin Mx sin Ny + D3 sin 3Mx sin Ny ,

where σ 0 y is the negative average circumferential stress and 13

σ0yh 2

x2 ,

(34)

B1 = A11 f 2 + A12 f12 , B2 = A21 f12 , B3 = A31 f1 + A32 f1 f 2 , B4 = A41 f1 f 2 , C1 = A51 f 2 + A52 f12 , C2 = A61 f1 + A62 f1 f 2 , C3 = A71 f1 f 2 , D1 =

A81 f12 ,

(35)

D2 = A91 f1 + A92 f1 f 2 , D3 = A101 f1 f 2 ,

the coefficients Aij can be found in Appendix D. Replacing Eq. (29) and Eq. (34) into Eq. (24), and then applying Galerkin’s method for the remaining equation in the ranges 0 ≤ y ≤ 2π R and 0 ≤ x ≤ L to the resulting equations yield σ 0yh R

1 · § − K1 ¨ f 0 + f 2 ¸ + q = 0, 2 ¹ ©

(36)

H1 f12 + H 2 f12 f 2 + H 3 f 2 = 0,

(37)

H 4 f1 + H 5 f1 f 2 + H 6 f1 f 22 + H 7 f13 − σ 0 y hN 2 f1 = 0,

(38)

where § 4M 2 · * * 2 * * 2 2 H1 = ¨¨ 16cc12 M4 − ¸¸ A12 + −4cc13 M + d11 − 3ce11 A52 − M N A31 , R © ¹

(

)

H 2 = M 2 N 2 ( A41 − A32 ) , § 4M 2 · * * 2 * * * 4 H 3 = ¨¨16cc12 M4 − ¸¸ A11 + −4cc13 M + d11 − 3ce11 A51 − 8cc15 M R © ¹ 1 * * + 2 d12 − 3ce12 + K 2 M 2 + K1 , 2

(

(

)

)

ª * 4 M 2 N2 º * * * * H 4 = «cc12 M + cc21 N 4 + c c11 M 2N 2 − + c22 − 2c31 B » A31 R a ¼ ¬ * * * * * º M 2 − c c23 N 2 + d11 A + ª −cc13 + 2c32 − 3ce11 ¬ ¼ 61 * * * * * º + ª −cc24 + 2c33 − 3ce21 N 2 − c c14 M 2 + d 21 A ¬ ¼ 91

(

( (

(

)

) )

)

* * * * * M 4 + cc26 N 4 + c c16 M 2N 2 + cc15 + c25 + 2c34

−

(

* d12

* − 3ce12

)

2

+ K2 M −

(

* d 22

* − 3ce22

14

)

+ K 2 N 2 − K1 ,

ª * 4 M 2 N2 º * * * * H 5 = «cc12 M + cc21 N 4 + c c11 M 2N2 − + c22 − 2c31 B » A32 R a ¼ ¬ * * * * * º M 2 − c c23 N 2 + d11 A + ª −cc13 + 2c32 − 3ce11 ¬ ¼ 62 * * * * * º N 2 − c c14 M 2 + d 21 A + M 2 N 2 ( A31 − 2 A11 ) , + ª −cc24 + 2c33 − 3ce21 ¬ ¼ 92

(

( (

)

) )

H 6 = M 2 N 2 ( A32 − A41 ) , H 7 = −2 M 2 N 2 ( A21 + A12 ) .

(39)

Moreover, the shells have to also satisfy the circumferential closed condition [4, 5, 6, 64, 65]. So we have: 2π R L

2π R L

0 0

0 0

³

³ v, y dxdy =

§

w

·

³ ³ ¨© ε y + R − w, y / 2 ¸¹ dxdy = 0. 0

2

(40)

Using Eqs. (13), (23), (29) and Eq. (34), this integral leads to * * * * a22 σ 0 y h + a27 φ1 + a28 φ1s + a29 φ1r +

1§ 1 · N2 2 f f2 ¸ − f1 = 0. + ¨ 0 R© 2 ¹ 8

(41)

In this study, environment temperature is assumed to be uniformly raised from initial value T0 at which the shell is thermal stress free to final one T and temperature change ΔT = T − T0 is constant and independent to thickness variable. Therefore, from Eq. (17), we have ­ E α + Ecmα m Ecmα cm º ª + = φ10 ΔT , °φ1 = hΔT « Emα m + m cm 2k + 1 »¼ k +1 ¬ ° ° ª Ecα mc + Emcα c Emcα mc º ° + ®φ1s = hs ΔT « Ecα c + » = φ10 s ΔT , 2k2 + 1 ¼ k2 + 1 ¬ ° ° °φ1r = hr ΔT ª« Ecα c + Ecα mc + Emcα c + Emcα mc º» = φ10 r ΔT . °¯ 2k3 + 1 ¼ k3 + 1 ¬

(42)

Substituting Eq. (42) into Eq. (41) leads to

(

)

* * * * a22 σ 0 y h + a27 φ10 + a28 φ10 s + a29 φ10 r ΔT +

Solving Eqs. (36), (37) and (43) yields:

15

1§ 1 · N2 2 f f2 ¸ − f1 = 0. + ¨ 0 R© 2 ¹ 8

(43)

f12 =

f0 = − −

−H3 f2 . H1 + H 2 f 2

(44)

* § − D3 f 2 · a22 K1R 2 RN 2 1 f2 + q+ ¨ ¸ * 2 * 2 2 K1 1 + a22 K1R 8 1 + a22 K1R © D1 + D2 f 2 ¹

(

R * K1R 1 + a22

σ0yh = − −

)

(

* a27 φ10 2

(

* φ10 s + a28

)

* φ10 r + a29

) Δ T.

§ − D3 f 2 · K1 R 2 N 2 R q + ¨ ¸ * 2 * 1 + a22 K1 R K1R 2 © D1 + D2 f 2 ¹ 8 1 + a22

(

K1R 2 * K1R 2 1 + a22

(a

* 27φ10

)

(45)

(46)

)

* * + a28 φ10 s + a29 φ10 r Δ T.

Substituting Eq. (44) and Eq. (46) into Eq. (38) leads to q=

H H § −H3 f2 · −H 4 H5 − f 2 − 6 f 22 + 9 ¨ ¸ H8 H8 H8 H 8 © H1 + H 2 f 2 ¹ − K1 R

(

* φ10 a27

* φ10 s + a28

+

* φ10 r a29

(47)

) ΔT ,

where H8 =

K1R 2 N 4 RN 2 2 2 = + + H M N A A , 2 . ( ) 9 12 21 * * K1R 2 1 + a22 K1R 2 8 1 + a22

(

)

(48)

Take the expression limit in equation (47) when let f2 go up to zero, we obtain the expression used to determine the critical load qcr =

−H 4 * * * φ10 + a28 φ10 s + a29 φ10 r ΔT . − K1R a27 H8

(

)

(49)

The upper critical load is found by minimizing Eq. (49) according to the variables m and n. From Eq. (29), the maximum deflection of the shell is determined as follows Wmax = f0 + f1 + f 2 ,

Combining Eqs. (44), (45) and (50) we have

16

(50)

Wmax = −

* a22 RH 8

N2 H8 N

(

1/2

q+

* a27 φ10 2

1 1 § − H3 f2 · § − H3 f2 · f2 + H8 ¨ ¸+¨ ¸ 2 8 © H1 + H 2 f 2 ¹ © H1 + H 2 f 2 ¹ * + a28 φ10 s

* + a29 φ10 r

(51)

) ΔT .

The load–maximal deflection curves of the shells can be determined based on Eqs. (47) and (51). 5. Numerical results and discussions 5.1. Validation of the present study To verify the accuracy of present approach, three comparisons are considered below. First comparison: FGM cylindrical shells are considered with the following geometric and material properties

as R = 0.4 m, Z = 500, L = RhZ , Ec = 323.11× 109 Pa, Em = 207.79 × 109 Pa, ν = 0.3, ΔT =0 K,

K1 = 0 N/m, K 2 = 0N/m3 . The critical buckling loads calculated for thick shells (R/h = 40) and thin

shells (R/h = 400) are listed in Table 1 to compare with the results given by Shen et al. [24] using the singular perturbation technique and the higher-order shear deformation shell theory. Second comparison of the static postbuckling load–maximum deflection curve for a FGM cylindrical shell without foundation under external pressure is presented d in Fig. 2 (using Eq. (47) and Eq. (51) with a ĺ ’, circular cylindrical shell is a special case of the toroidal shell with the results of Huang and Han [7]. Third comparison: a FGM toroidal shell segment is considered; the static post buckling load– maximum deflection curve is compared in Fig. 3 with the results of Bich et al. [33] using the classical theory. From the above three comparisons, it is possible to conclude that: our results are in good agreement with references considered. The input parameters are taken as Ec = 370 × 109 Pa, Em = 70 × 109 Pa, k = 1, ν = 0.3, R = 0.5m, a / R = 4, L / R = 2.

Table 1. Comparison of upper buckling loads

( R = 0.4m, Z =500, L = Volume fraction indexes k=0 k =0.2 k =1

qcr kPa

for FGM cylindrical shell

RhZ , Ec = 323.11GPa, Em = 207.79GPa,ν = 0.3, ΔT =0 K, K1 = 0 N/m, K 2 = 0N/m3

Thick shells (R/h = 40) Shen [24] Present 9112.24 (1, 4)* 8447.11 (1, 4) 7391.73 (1, 4)

9237.57 (1, 4) 8559.82 (1, 4) 7484.40 (1, 4)

% Error 1.37% 1.33% 1.25%

Volume fraction indexes k =0 k =0.2 k =1

17

)

Thin shells (R/h = 400) Shen [24] Present 87.4899 (1, 11) 81.3248 (1, 11) 71.1508 (1, 11)

88.5202 (1, 11) 82.2487 (1, 11) 71.9031 (1, 11)

% Error 1.18% 1.14% 1.06%

k =2 7032.91 (1, 4) k =5 6690.62 (1, 4) * buckling mode (m, n)

7119.08 (1, 4) 6771.01 (1, 4)

1.22% 1.2%

k =2 k =5

67.3886 (1, 11) 63.7561 (1, 11)

68.0830 (1, 11) 64.3981 (1, 11)

1.03% 1.00%

Fig. 2. Comparison with result of [7] for un-stiffened

Fig. 3. Comparison with result of [33] for un-stiffened

FGM cylindrical shell.

FGM toroidal shell segment.

5.2. Nonlinear resonse and buckling of eccentrically stiffened FGM toroidal shell segments In this section, we propose that the shell with geometric and material properties as follows Ec = 370 × 109 Pa, Em = 70 × 109 Pa, α c = 5.4 × 10−6 K -1 , α m = 22.2 × 10−6 K -1 , k = 1, ν = 0.3, R = 0.5m, R / h = 80, a / R = 4, L / R = 2, K1 = 1.5 × 107 N/ m3 , K 2 = 1.5 × 105 N/ m.

5.2.1. Effect of temperature Table 2, Fig.4 and Fig. 5 show the effect of temperature field on the buckling load and postbuckling paths (q–Wmax/h) of the shell. It is found that the increase in the temperature leads to a decrease in buckling strength. Table 2. Effects of temperature on critical buckling load 7 3 5 ( k = 1, R = 0.5 m, h = R / 80, a / R = 4, L / R = 2, ns = nr = 20, hs = hr = bs = br = h, K1 = 1.5 × 10 N/ m , K 2 = 1.5 × 10 N/ m)

ΔT K

convex shell qcr MPa

concave shell qcr MPa

0

8.8399(1,8)*

2.1285(1,3)

800

8.7687(1,8)

2.0573(1,3)

1600

8.6975(1,8)

1.9861(1,3)

2400

8.6262(1,8)

1.9149(1,3)

* buckling mode (m, n)

18

Fig. 4. Effects of temperature field on nonlinear response

Fig. 5. Effects of temperature field on nonlinear response

of FGM convex shell

of FGM concave shell

5.2.2. Effects of stiffeners The effect of reinforcement stiffener on buckling load and q–Wmax/h curves are shown in Table 3, Fig. 6 and Fig. 7. As can be seen that the buckling loads of un-stiffened FGM toroidal shell segment are the smallest, the buckling loads of ring stiffened FGM shell are higher than that of stringer stiffened shell and the buckling loads of stringer-ring stiffened one are the greatest. This behavior is common to both convex and concave shells. Thus the stiffeners enhance the load carrying of the shell. This is reasonable because the reinforcement stiffeners make the shells to become stiffer, so its load carrying capacity is better. Table 3. Effects of stiffeners on critical buckling load (k = 1, R = 0.5m, h = R / 80, L / R = 2, K1 = 1.5 × 107 N/ m3 , K 2 = 1.5 × 105 N/ m) qcr MPa ΔT = 0 K ΔT = 800 K

nr = 0

nr = 0

nr = 20

nr = 20

nr = 30

ns = 0

ns = 20

ns = 0

ns = 20

ns = 30

Convex shell

5.8942(1,9)*

5.9725(1,9)

8.7425(1,8)

8.8399(1,8)

9.9953(1,8)

Concave shell

1.8275(1,3)

1.8296(1,3)

2.1257(1,3)

2.1285(1,3)

2.2587(1,3)

Convex shell

5.8229(1,9)

5.9013(1,9)

8.6713(1,8)

8.7687(1,8)

9.9241(1,8)

Concave shell

1.7563(1,3)

1.7584(1,3)

2.0544(1,3)

2.0573(1,3)

2.1874(1,3)

*buckling mode (m, n)

19

Fig. 6. Effects of stiffeners on nonlinear response of FGM

Fig. 7. Effects of stiffeners on nonlinear response of FGM

convex shell.

concave shell.

5.2.3. Effects of volume fraction indexes Effects of volume fraction indexes (metal-ceramic ratio) in the shell are represented in Table 4, Fig. 8 and Fig. 9. It is found that the increase in the volume fraction index k leads to a decrease in buckling loads. Because when k increases, the metal ratio is increased and elastic module of metal is lower than ceramic. Table 4. Effects of volume fraction indexes on critical buckling load ( R = 0.5m, h = R / 80, L / R = 2, hs = hr = bs = br = h, ns = nr = 20, K1 = 1.5 × 107 N/ m3 , K 2 = 1.5 × 105 N/ m) qcr MPa

ΔT = 0 K

Convex shell

Concave shell Convex ΔT = 800 K shell Concave shell * buckling mode (m, n)

k =0

k = 0.5

k =1

k =5

k =∞

17.6967(1,7)*

11.3468(1,8)

8.8399(1,8)

5.2365(1,8)

3.6529(1,8)

3.2302(1,3)

2.4251(1,3)

2.1285(1,3)

1.7696(1,3)

1.5979(1,3)

17,6643(1,7)

11.2875(1,8)

8.7687(1,8)

5.1352(1,8)

3.5197(1,8)

3.1978(1,3)

2.3657(1,3)

2.0573(1,3)

1.6683(1,3)

1.4647(1,3)

20

Fig. 8. Effects of volume fraction indexes on nonlinear

Fig. 9. Effects of volume fraction indexes on nonlinear

response of FGM convex shell.

response of FGM concave shell.

5.2.4. Effects of the ratio R/h The effects of R/h ratio on q–Wmax/h curves (nonlinear response) of stiffened FGM convex and concave toroidal shell segment on elastic foundation are illustrated in Figs. 10 and 11. Table 5 illustrates the influences of ratio R/h on the buckling loads of both convex and concave FGM toroidal shells surrounded by elastic foundation. As can be seen that, critical buckling load is very sensitive with the change of the R/h ratio, buckling loaddecreases markedly with the increase of this ratio. Table 5. Effects of the ratio R/h on critical buckling load (k = 1, R = 0.5m, L / R = 2, a = 4 R, hs = hr = bs = br = h, ns = nr = 20, K1 = 1.5 × 107 N/ m3 , K 2 = 1.5 × 105 N/ m) qcr ( MPa ) ΔT = 0 K ΔT = 800 K

R / h = 50

R / h = 60

R / h = 80

R / h = 100

Convex shell

26.7362(1,6)*

17.2603(1,7)

8.8399(1,8)

5.3740(1,9)

Concave shell

5.3036(1,3)

3.4680(1,3)

2.1285(1,3)

1.6894(1,3)

Convex shell

26.6650(1,6)

17.1891(1,7)

8.7687(1,8)

5.3027(1,9)

Concave shell

5.2324(1,3)

3.3968(1,3)

2.0573(1,3)

1.1681(1,3)

*buckling mode (m, n)

21

Fig.10. Effects of the ratio R/h on nonlinear response of

Fig. 11. Effects of the ratio R/h on nonlinear response of

FGM convex toroidal shell.

FGM concave toroidal shell.

5.2.5. Effects of the ratio L/R Effects of ratio L/R on the mechanical behavior of the shell are represented on Table 6, Fig. 12 and Fig. 13. As can be observed that when L/R ratio increases, the critical buckling loads decrease for both stiffened FGM convex and concave shells, but convex shells work better. The loadcarrying of longer shells is lower than that of shorter ones. For instance, at ΔT = 0 K the critical buckling load decreases about 14.86% for stiffened FGM convex shell and about 10.83% for stiffened FGM concave shell from L/R = 3 to L/R = 6. Table 6 Effects of the ratio L/R on critical buckling loads (k = 1, R = 0.5m, h = R / 80, hs = hr = bs = br = h, ns = nr = 20, K1 = 1.5 × 107 N/ m3 , K2 = 1.5 × 105 N/ m) qcr MPa ΔT = 0 K ΔT = 800 K

L/R=3

L/R=4

L/R=5

L/R=6

Convex shell

7.5999(1,8)*

7.0211(1,8)

6.6872(1,8)

6.4702(1,8)

Concave shell

2.1715(2,4)

1.9869(2,3)

1.9819(3,4)

1.9364(3,3)

Convex shell

7.5287(1,8)

6.9499(1,8)

6.6160(1,8)

6.3989(1,8)

Concave shell

2.1003(2,4)

1.9157(2,3)

1.9106(3,4)

1.8651(3,3)

*buckling mode (m, n)

22

Fig. 12. Effects of the ratio L/R on nonlinear response of

Fig. 13. Effects of the ratio L/R on nonlinear response of

FGM convex shell.

FGM concave shell.

5.2.6. Effects of the ratio a/R Effects of ratio a/R on the nonlinear response and critical buckling loads of shell are represented in Fig. 14 and Fig. 15, Table 7. As can be observed that the increase of a/R ratio leads to a decrease markedly in buckling strength. This behavior is common to both convex and concave shells. Table 7. Effects of the ratio a/R on critical buckling loads

qcr MPa

(k = 1, R = 0.5m, L = 2 R, h = R / 80, hs = hr = bs = br = h, ns = nr = 20, K1 = 1.5 × 107 N/ m3 , K 2 = 1.5 × 105 N/ m) qcr MPa

a/R=5

Convex shell 7.6465(1,7)* Concave shell 2.3408(1,4) * buckling mode (m, n)

a/R=6

a/R=7

a/ R=8

6.7587(1,7) 2.0898(1,4)

6.1985(1,7) 2.0956(1,4)

5.8187(1,7) 2.2012(1,4)

Fig. 14. Effects of the ratio a/R on nonlinear response of

Fig. 15. Effects of the ratio a/R on nonlinear response of

FGM convex shell.

FGM concave shell.

23

5.2.7. Effects of elastic foundation parameters The effects of elastic foundations on critical buckling load and nonlinear respons (q–Wmax/h curves) of FGM toroidal shells are analyzed in Table 8, Fig. 16 and Fig. 17. As can be seen, the buckling loads of convex and concave shells are remarkably enhanced due to support of elastic foundations. Furthermore, the shear layer stiffness K2 of Pasternak model has fewer influences in comparison with foundation modulus K1 of Winkler model. Table 8 Effects of elastic foundation parameters on critical buckling load of FGM convex shell and concave shell. qcr MPa

K1 = 0 N/m3 K 2 = 0 N/m

K1 = 2.5 × 108 N/m3

K1 = 0 N/m3

K1 = 2.5 × 108 N/m3 5

K 2 = 0 N/m

K 2 = 1.5 × 10 N/m

K 2 = 1.5 × 105 N/m

Convex shell

8.3903(1,8)*

10.6359(1,9)

8.7019(1,8)

10.9571(1,9)

Concave shell

0.9079(1,3)

6.9391(2,7)

1.2919(1,3)

7.3137(2,7)

* buckling mode (m, n)

Fig. 16. Effects of elastic foundation parameters on

Fig. 17. Effects of elastic foundation parameters on

nonlinear response of FGM convex shell.

nonlinear response of FGM concave shell.

6. Concluding remarks This work presents an analytical investigation of nonlinear response and buckling and postbuckling analysis of eccentrically stiffened FGM toroidal shell segments surrounded by elastic foundation in thermal environment and subjected to external pressure. The formulations are based on Reddy’s third-order shear deformation shell theory with von Karman nonlinearity, Pasternak type elastic foundations and smeared stiffener technique. The new contributions of this work differ from other publications about FGM toroidal shell segments in the way that the shells are reinforced by FGM stiffeners and using TSDT with the solution that has the three terms as in (29). By applying Galerkin’s method and stress function, the critical buckling load and the postbuckling

24

load–deflection curves are determined. Numerical results are given for evaluating effects of temperature, stiffeners, volume fraction index, material properties, foundation stiffness and geometric shapes on nonlinear buckling behaviors of stiffened FGM toroidal shell segments. Especially, the rings stiffeners and stringers have very different influence on critical load of the eccentrically FGM toroidal shell segments. It’s very important in design and manufacturing of eccentrically stiffened FGM toroidal shells. Acknowledgements This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 107.02-2015.03. Appendix A h /2

( E1 , E2 , E3 , E4 , E5 , E7 ) = ³

− h /2

(1, z, z , z , z , z )E 2

3

4

6

sh

( z ) dz,

h /2 + h j

( E1 j , E2 j , E3 j , E4 j , E5 j , E7 j ) = ³ (1, z, z 2 , z3 , z 4 , z 6 )E j ( z ) dz, j = s, r. h /2

§ E bs E1s bE bE · Eν E2 , a12 = 1 2 , a13 = + s 2s − c ¨ 4 2 + s 4s ¸ , 2 ds ds ds ¹ 1 −ν 1 −ν 1 −ν © 1 −ν § E bE · −b Eν Eν Eν −1 a14 = 2 2 − c 4 2 , a15 = −c ¨ 4 2 + s 4 s ¸ , a16 = −c 4 2 , a17 = , a18 = s , ds ¹ ds 1 −ν 1 −ν 1 −ν 1 −ν © 1 −ν a11 =

2

+

§ E br E1r Eν Eν E2 bE bE + r 2r − c ¨ 4 2 + r 4r , a23 = 2 2 − c 4 2 , a24 = 2 dr dr dr − 1 −ν 1 −ν 1 −ν 1 −ν 1 −ν 1 ν © § E Eν bE · −b −1 = −c 4 2 , a26 = −c ¨ 4 2 + r 4 r ¸ , a27 = , a28 = r , dr ¹ dr 1 −ν 1 −ν © 1 −ν

a21 = a25

E1

a31 =

b11 =

b14 =

b21 =

E1ν

2

E1

, a22 =

2

+

· ¸, ¹

E1 E2 E4 E , a32 = −c , a33 = a32 , a34 = −c 4 , 2 (1 + ν ) 2 (1 + ν ) 2 (1 + ν ) 1 +ν

E2 1 −ν

+

2

−c

2

, b22 =

E3ν 1 −ν

E2ν 1 −ν

§ E bs E2 s E bE bE · Eν , b12 = 2 2 , b13 = 3 2 + s 3s − c ¨ 5 2 + s 5 s ¸ , ds ds ds ¹ 1 −ν 1 −ν © 1 −ν

2

§ E bE · Eν −b −1 , b18 = s , , b15 = −c ¨ 5 2 + s 5 s ¸ , b16 = −c 5 2 , b17 = 1 −ν ds ¹ ds 1 −ν 1 −ν © 1 −ν E5ν

2

E2 1 −ν

2

+

§ E E3 bE bE br E2 r Eν Eν , b23 = 3 2 − c 5 2 , b24 = + r 3r − c ¨ 5 2 + r 5 r 2 dr d dr 1 −ν 1 −ν 1 −ν © 1 −ν r

25

· ¸, ¹

b25 = −c

b31 =

§ E bE , b26 = −c ¨ 5 2 + r 5r dr 1 −ν © 1 −ν

E5ν

2

· −b −1 , b28 = r , ¸ , b27 = dr 1 −ν ¹

E3 E5 E2 E , b32 = −c , b33 = b32 , b34 = −c 5 , 2 (1 + ν ) 2 (1 + ν ) 2 (1 + ν ) 1 +ν

§ E bs E4 s E5 bE bE · Eν , c12 = 4 2 , c13 = + s 5s − c ¨ 7 2 + s 7 s ¸ , 2 ds ds ds ¹ 1 −ν 1 −ν 1 −ν © 1 −ν § E Eν Eν bE · Eν −b −1 c14 = 5 2 − c 7 2 , c15 = −c ¨ 7 2 + s 7 s ¸ , c16 = −c 7 2 , c17 = , c18 = s , ds ¹ ds 1 −ν 1 −ν 1 −ν 1 −ν © 1 −ν

c11 =

c21 =

E4

2

E4ν 1 −ν

c25 = −c

2

+

, c22 =

E4 1 −ν

2

+

§ E Eν Eν E5 bE bE · br E4 r + r 5r − c ¨ 7 2 + r 7 r ¸ , , c23 = 5 2 − c 7 2 , c24 = 2 dr dr dr ¹ 1 −ν 1 −ν 1 −ν © 1 −ν

§ E bE , c26 = −c ¨ 7 2 + r 7 r dr − 1 −ν 1 ν © E7ν

2

· −b −1 , c28 = r , ¸ , c27 = − 1 ν dr ¹

c31 =

E5 E7 E E4 , c32 = −c , c33 = c32 , c34 = −c 7 . 2 (1 + ν ) 2 (1 + ν ) 2 (1 + ν ) 1 +ν

d11 =

bs E1s bs E3s · E1 −4 § E3 + + , d12 = d13 = 2 ¨ ¸, 2 (1 + ν ) 2d s (1 + ν ) h ¨© 2 (1 + ν ) 2d s (1 + ν ) ¸¹

d 21 =

br E3r · E1 br E1r −4 § E3 , d 22 = d 23 = 2 ¨ + + ¸, ¨ 2 (1 + ν ) 2d r (1 + ν ) h © 2 (1 + ν ) 2d r (1 + ν ) ¸¹

e11 =

E3 bs E3s bs E5 s · −4 § E5 , e12 = e13 = 2 ¨ + + ¸, ¨ 2 (1 + ν ) 2d s (1 + ν ) h © 2 (1 + ν ) 2d s (1 + ν ) ¸¹

e21 =

E3 br E3r br E5r · −4 § E5 + + , e22 = e23 = 2 ¨¨ ¸. 2 (1 + ν ) 2d r (1 + ν ) h © 2 (1 + ν ) 2d r (1 + ν ) ¸¹

Appendix B Δ* = a11a22 − a12 a21 , * a11 =

a22 * −a12 * a12 a23 − a22 a13 * a12 a24 − a22 a14 * a12 a25 − a22 a15 , a12 = , a13 = , a14 = , a15 = , Δ* Δ* Δ* Δ* Δ*

* a16 =

a12 a26 − a22 a16 a a − a22 a17 * −a22 a18 * a12 a28 * , a17 = 12 27 , a18 = , a19 = , Δ* Δ* Δ* Δ*

* a21 =

a a −a a a a −a a −a21 * a a a −a a * * * , a22 = 11 , a23 = 21 13 11 23 , a24 = 21 14 11 24 , a25 = 21 15 11 25 , Δ* Δ* Δ* Δ* Δ*

26

* a26 =

a21a16 − a11a26 * a a −a a a a −a a * * , a27 = 21 17 11 27 , a28 = 21 18 , a29 = 11 28 , Δ* Δ* Δ* Δ*

* a31 =

−a −a −a 1 * * * , a32 = 32 , a33 = 33 , a34 = 34 , a31 a31 a31 a31

* * * * * * * * * * * * = b11a11 + b12 a21 = b11a12 + b12 a22 = b11a13 + b12 a23 + b13 , b14 = b11a14 + b12 a24 + b14 , b11 , b12 , b13 * * * * * * * * * = b11a15 + b12 a25 + b15 , b16 = b11a16 + b12 a26 + b16 , b17 = b11a17 + b12 a27 b15 , * * * * * * = b11a18 + b12 a28 = b11a19 + b12 a29 b18 , b19 , * * * * * * * * * * * * = b21a11 + b22 a21 = b21a12 + b22 a22 = b21a13 + b22 a23 + b 23 , b24 = b21a14 + b22 a24 + b 24 , b21 , b22 , b23 * * * * * * * * * = b21a15 + b22 a25 + b 25 , b26 = b21a16 + b22 a26 + b 26 , b27 = b21a17 + b22 a27 b25 * * * * * * = b21a18 + b22 a28 = b21a19 + b22 a29 b28 , b29 , * * * * * * * * b31 = b31a31 , b32 = b31a32 + b32 , b33 = b31a33 + b33 , b34 = b31a34 + b34 ,

* * * * * * * * * * * * = c11a11 + c12 a21 = c11a12 + c12 a22 = c11a13 + c12 a23 + c13 , c14 = c11a14 + c12 a24 + c14 , c11 , c12 , c13 * * * * * * * * * = c11a15 + c12 a25 + c15 , c16 = c11a16 + c12 a26 + c16 , c17 = c11a17 + c12 a27 c15 , * * * * * * = c11a18 + c12 a28 = c11a19 + c12 a29 c18 , c19 , * * * * * * * * * * * * = c21a11 + c22 a21 = c21a12 + c22 a22 = c21a13 + c22 a23 + c23 , c24 = c21a14 + c22 a24 + c24 , c21 , c22 , c23 * * * * * * * * * = c21a15 + c22 a25 + c25 , c26 = c21a16 + c22 a26 + c26 , c27 = c21a17 + c22 a27 c25 , * * * * * * = c21a18 + c22 a28 = c21a19 + c22 a29 c28 , c29 , * * * * * * * * c31 = c31a31 , c32 = c31a32 + c32 , c33 = c31a33 + c33 , c34 = c31a34 + c34 ,

* * * * d11 = d11 + d12 , d12 = d11 + d13 , d 21 = d 21 + d 22 , d 22 = d 21 + d 23 , * * * e11 = e11 + e12 , e12 = e11 + e13 ,e*21 = e21 + e22 , e22 = e21 + e23 ,

Appendix C

( ) = c ( c + 2c ) , g = c ( c + 2c ) , g = ( d − 3ce + K ) , g = ( d − 3ce

(

)

* * * * * * * * * * g11 = cc12 , g12 = cc21 , g13 = c c11 + c22 − 2c31 , g14 = cc15 , g15 = cc26 , g16 = c c16 + c25 + 2c34 , * g17 = cc13 , g18

* = cc24 , g19

(

)

* * g112 = d 21 − 3ce21 , g113

* 23

* 32

* 12

( ) ( = ( b − cc ) , g = ( b = ( − d + 3ce )

* 12

* 14

110 2

* 33

* 22

114

)

(

111

* 22

)

=

(

* d11

+ K2

* − 3ce11

) ( = ( −d

),

* * * * * * * * * * * * g 21 = b12 − cc12 , g 22 = b11 − b31 − cc11 + cc31 , g 23 = b15 − cc15 , g 24 = b16 + b34 − cc16 − cc34

g 25 g 29

* 13

* 13

* 12

26

* 32

)

(

)

* * * * * − cc32 , g 27 = b14 + b33 − cc14 − cc33 , g 28

* 12

27

* 11

* + 3ce11

)

)

( = (b

) ( ) , g = (b

)

(

) ( ) , g = ( −d

* * * * * * * * * * * * g31 = b21 − cc21 , g32 = b22 − b31 − cc22 + cc31 , g33 = b26 − cc26 , g34 = b25 + b34 − cc25 − cc34

g35

* 24

* − cc24

* 33

36

)

(

* * * * * − cc33 , g37 = b23 + b32 − cc23 − cc32

(

)

38

* 21

)

(

* * * + 3ce21 , g39 = − d 22 + 3ce22

(

)

* * * * * * * * * * * g 41 = a25 , g 42 = a16 , g 43 = a15 + a26 − a34 , g 44 = a11 , g 45 = a22 , g 46 = a12 + a21 + a31 , g 47 = a23

(

)

(

* * * * * g 48 = a13 − a32 , g 49 = a14 , g 410 = a24 − a33

)

Appendix D

(

)

−4 M 2 v34G111 + 4 M 2 v13 − v16 G311

A11 =

16 M

4

( 4M v v (4N v (4N v v 2

16 N 4

2

− 4 M v11v34 − v16 v31

13 31 2

A21 =

2

23

23 32

)

− v26 G321

− 4 N 2 v21v35 − v26 v32

§ − N 2 v25 det ¨ ¨ −M 2v − N 2v 34 36 © A31 =

)

)

, A12 =

( 4M

16 M

4

( 4M

,

v26 − M 2 v24 − N 2 v23 · ¸ − M 2 v37 − N 2 v35 ¸¹ G

121

Δ2

§ v − M 2 v13 − N 2 v14 det ¨ 16 ¨ − M 2v − N 2v 34 36 © − Δ2

· ¸ − M 2 v37 − N 2 v35 ¸¹ G − M 2 v15

211

§ v − M 2 v13 − N 2 v14 · − M 2 v15 det ¨ 16 ¸ ¨ − N 2 v25 v26 − M 2 v24 − N 2 v23 ¸¹ © + G331 , Δ2 § v − M 2 v13 − N 2 v14 · − M 2 v15 det ¨ 16 ¸ ¨ − N 2 v25 v26 − M 2 v24 − N 2 v23 ¸¹ © A32 = G332 , Δ2 § v − 9 M 2 v13 − N 2 v14 · −9 M 2 v15 det ¨ 16 ¸ ¨ − N 2 v25 v26 − 9 M 2 v24 − N 2 v23 ¸¹ © A41 = G341 , Δ1

A51 = A52 =

−v31G111 + v11G311 2

4 M v13v31 − 4 M 2 v11v34 − v16 v31 v11G312 2

4 M v13v31 − 4 M 2 v11v34 − v16 v31

, ,

28

2

2

)

)

v13 − v16 G312

v13v31 − 4 M 2 v11v34 − v16 v31

)

,

)

§ M 2 N 2 v22 + N 4 v21 det ¨ ¨ M 4v + M 2 N 2v + N 4v 31 33 32 © A61 = − Δ2 § M 4 v11 + M 2 N 2 v12 det ¨ ¨ M 4v + M 2 N 2v + N 4v 31 33 32 © + Δ2

v26 − M 2 v24 − N 2 v23 · ¸ − M 2 v37 − N 2 v35 ¸¹ G

121

· ¸ − M v37 − N v35 ¸¹ G − M 2 v15

2

2

211

§ M 4 v11 + M 2 N 2 v12 · − M 2 v15 det ¨ ¸ 2 2 4 2 2 ¨M N v + N v ¸ 22 21 v26 − M v24 − N v23 ¹ © − G331 , Δ2 § M 4 v11 + M 2 N 2 v12 · − M 2 v15 det ¨ ¸ 2 2 4 2 2 ¨M N v + N v ¸ 22 21 v26 − M v24 − N v23 ¹ © A62 = − G332 , Δ2

§ 9M 2 N 2 v12 + 81M 4 v11 · −9 M 2 v15 det ¨ ¸ 2 2 4 2 2 ¨ 9M N v + N v v26 − 9M v24 − N v23 ¸¹ 22 21 © A71 = − G341 , Δ1 A81 =

v21G321 2

4 N v23v32 − 4 N 2 v21v35 − v26 v32

,

§ M 2 N 2 v22 + N 4 v21 det ¨ ¨ M 4v + M 2 N 2v + N 4v 31 33 32 © A91 = Δ2 § M 4 v11 + M 2 N 2 v12 det ¨ ¨ M 4v + M 2 N 2v + N 4v 31 33 32 © − Δ2

· ¸ − M v34 − N v36 ¸¹ G − N 2 v25

2

2

121

v16 − M 2 v13 − N 2 v14 · ¸ − M 2 v34 − N 2 v36 ¸¹ G

211

§ M 4 v11 + M 2 N 2 v12 v16 − M 2 v13 − N 2 v14 · det ¨ ¸ ¨ M 2 N 2v + N 4v ¸ − N 2 v25 22 21 © ¹G , + 331 Δ2 § M 4 v11 + M 2 N 2 v12 v16 − M 2 v13 − N 2 v14 · det ¨ ¸ ¨ M 2 N 2v + N 4v ¸ − N 2 v25 22 21 © ¹G , A92 = 332 Δ2 § 9 M 2 N 2 v12 + 81M 4 v11 v16 − 9 M 2 v13 − N 2 v14 · det ¨ ¸ ¨ 9M 2 N 2v + N 4v ¸ − N 2 v25 22 21 © ¹G , A101 = 341 Δ1

29

§ 9 M 2 N 2 v12 + 81M 4 v11 ¨ Δ1 = det ¨ 9 M 2 N 2 v22 + N 4 v21 ¨ ¨ 81M 4 v31 + 9 M 2 N 2 v33 + N 4 v32 © § M 4 v11 + M 2 N 2 v12 ¨ Δ 2 = det ¨ M 2 N 2 v22 + N 4 v21 ¨ 4 ¨ M v31 + M 2 N 2 v33 + N 4 v32 ©

v16 − 9M 2 v13 − N 2 v14 − N 2 v25 −9 M 2 v34 − N 2 v36

v16 − M 2 v13 − N 2 v14 2

· ¸ v26 − 9M 2 v24 − N 2 v23 ¸ , ¸ −9 M 2 v37 − N 2 v35 ¸¹ −9M 2 v15

· ¸ v26 − M v24 − N v23 ¸ . ¸ − M 2 v37 − N 2 v35 ¸¹ − M 2 v15 2

− N v25 − M 2 v34 − N 2 v36

2

References [1] R.C. Batra, Torsion of a functionally graded cylinder, AIAA J. 44 (2006) 1363–1365. [2] H.M. Wang, C.B. Liu, H.J. Ding, Exact solution and transient behavior for torsional vibration of functionally graded finite hollow cylinders, Acta Mech. Sin. 25 (2009) 555–563. [3] H.S. Shen, Torsional buckling and postbuckling of FGM cylindrical shells in thermal environments, Int. J. Non. Mech. 44 (2009) 644–657. [4] H. Huang, Q. Han, Nonlinear buckling of torsion-loaded functionally graded cylindrical shells in thermal environment, Euro. J. Mech. ASolids 29 (2010) 42–48. [5] H. Huang, Q. Han, Nonlinear elastic buckling and postbuckling of axially compressed functionally graded cylindrical shells,Int. J. Mech. Sci. 51(2009) 500–507. [6] H. Huang, Q. Han, Nonlinear buckling and postbuckling of heated functionally graded cylindrical shells under combined axial compression and radial pressure, Int. J. Non-Linear Mech. 44 (2009) 209–218. [7] H. Huang, Q. Han, Research on nonlinear postbuckling of FGM cylindrical shells under radial loads, Composite Structures 92 (2010) 1352–1357. [8] H. Huang, Q. Han, Nonlinear dynamic buckling of functionally graded cylindrical shells subjected to a time-dependent axial load, Composite Structures 92 (2010) 593–598. [9] A. H. Sofiyev, H. E. Schnack, The stability of functionally graded cylindrical shells under linearly increasing dynamic torsional loading, Engineering Structures 26 (2004) 1321–1331. [10] A. H. Sofiyev, Thermoelastic stability of freely supported functionally graded conical shells within the shear deformation theory, Composite Structures 152 (2016) 74–84.

30

[11] A. H. Sofiyev, Z. Zerin, N. Kuruoglu, Thermoelastic buckling of FGM conical shells under non-linear temperature rise in the framework of the shear deformation theory, Composites Part B: Engineering 108 (2017) 279–290. [12] M. M. Najafizadeh, A. P. Hasani, Khazaeinejad, Mechanical stability of functionally graded stiffened cylindrical shells, Applied Mathematical Modelling 33 (2009) 1151–1157. [13] M. Akbari, Y. Kiani, M. R. Eslami, Thermal buckling of temperature-dependent FGM conical shells with arbitrary edge supports, Acta. Mech. 226 (2015) 897–915. [14] J. Torabi, Y. Kiani, M. R. Eslami, Linear thermal buckling analysis of truncated hybrid FGM conical shells, Composites Part B: Engineering 50 (2013) 265–272. [15] J. E. Jam, Y. Kiani, Buckling of pressurized functionally graded carbon nanotube reinforced conical shells, Composite Structures 125 (2015) 586–595. [16] M. Mirzaei, Y. Kiani, Thermal buckling of temperature dependent FG-CNT reinforced composite conical shells, Aerospace Science and Technology 47 (2015) 42–53. [17] M. Stein, J. A. McElman, Buckling of segments of toroidal shells, AIAA. J. 3 (1965) 1704– 1709. [18] J. W. Hutchinson, Initial postbuckling behavior of toroidal shell segments, Int. J. Solids. Struct. 3 (1967) 97–115. [19] M. G. Asratyan, R. S. Gevorgyan, Mixed boundary-value problems of thermoelasticity anisotropic-in-plane inhomogeneous toroidal shells, J. Appl Math and Mech. 74 (2010) 306– 312. [20] F. Tornabene, E. Viola, Static analysis of functionally graded doubly-curved shells and panels of revolution, Meccanica 48 (2013) 901–930. [21] N.D. Duc, Nonlinear Static and Dynamic Stability of Functionally Graded Plates and Shells, Vietnam National University Press, Hanoi, 2014. [22] G. G. Sheng, X. Wang, Thermal vibration, buckling and dynamic stability of functionally graded cylindrical shells embedded in an elastic medium, J Reinforced Plastic. Composite 27 (2008) 117–134. [23] H. S. Shen, Postbuckling of shear deformable FGM cylindrical shells surrounded by an elastic medium,Int. J. Mech. Sci. 51(2009) 372–383.

31

[24] H. S. Shen, J. Yang, S. Kitipornchai, Postbuckling of internal pressure loaded FGM cylindrical shells surrounded by an elastic medium, Eur. J. Mech – A/Solids 29 (2010) 448–460. [25] A. H. Sofiyev, M. Avcar, The stability of cylindrical shells containing a FGM layer subjected to axial load on the Pasternak foundation, Engineering 2 (2010) 228–236. [26] A. H. Sofiyev, N. Kuruoglu, Torsional vibration and buckling of the cylindrical shell with functionally graded coatings surrounded by an elastic medium, Composites Part B 45 (2013) 1133–1142. [27] E. Bagherizadeh, Y. Kiani, M. R. Eslami, Mechanical buckling of functionally graded material cylindrical shells surrounded by Pasternak elastic foundation, Composite Structures 93 (2011) 3063–3071. [28] E. Bagherizadeh, Y. Kiani, M. R. Eslami, Thermal Buckling of Functionally Graded Material Cylindrical Shells on Elastic Foundation, AIAA Journal. 50(2) (2012) 500–503. [29] A. M. Najafov, A. H. Sofiyev, N. Kuruoglu, Torsional vibration and stability of functionally graded orthotropic cylindrical shells on elastic foundations, Meccanica 48 (2013) 829–840. [30] D. V. Dung, L. K. Hoa, Research on nonlinear torsional buckling and postbuckling of eccentrically stiffened functionally graded thin circular cylindrical shells, Composites Part B 51 (2013) 300–309. [31] D. V. Dung, L. K. Hoa, Nonlinea buckling and postbuckling analysis of eccentrically stiffened functionally graded cylindrical shells under external pressure, Thin-Walled Structure 63 (2013) 117–124. [32] J. A. McElman, Eccentrically stiffened shallow shells of double curvature, NASA technical note (1967) D–3826. [33] D. H. Bich, D. G. Ninh, T. I. Thinh, Non-linear buckling analysis of FGM toroidal shell segments filled inside by an elastic medium under external pressure loads including temperature effects, Composites Part B 87 (2016) 75–919. [34] D.H. Bich, D.G. Ninh, Post-buckling of sigmoid-functionally graded material toroidal shell segment surrounded by an elastic foundation under thermo-mechanical loads, Composite Structures 138 (2016) 253–263.

32

[35] D. H. Bich, D. G. Ninh, Nonlinear buckling of eccentrically stiffened functionally graded toroidal shell segments under torsional load surrounded by elastic foundation in thermal environment, Mechanics Research Communications 72 (2016) 1–15. [36] D. H. Bich, D. G. Ninh, T. I. Thinh, H. David, Nonlinear dynamical analyses of eccentrically stiffened functionally graded toroidal shell segments surrounded by elastic foundation in thermal environment, Composites Part B 95 (2016) 355–373. [37] N. D. Duc, P. H. Cong, V. D. Quang, Thermal stability of eccentrically stiffened FGM plate on elastic foundation based on Reddy’s third-order shear deformation plate theory, Thermal Stresses 39(7) (2016) 772–794. [38] N. D. Duc, P. H. Cong, V. D. Quang, Nonlinear dynamic and vibration analysis of piezoelectric eccentrically stiffened FGM plates in thermal environment, Int. J. Mech. Sci. 115– 116 (2016) 711–722. [39] T. Q. Quan, N. D. Duc, Nonlinear vibration and dynamic response of shear deformable imperfect functionally graded double curved shallow shells resting on elastic foundations in thermal environments, Thermal Stresses 39 (4) (2016) 437–459. [40] N. D. Duc, T. Q. Quan, V. D. Luat, Nonlinear dynamic analysis and vibration of shear deformable piezoelectric FGM double curved shallow shells under damping-thermo-electromechanical loads, Composite Structures 125 (2015) 29–40. [41] T. Q. Quan, P. Tran, N. D. Tuan, N. D. Duc, Nonlinear dynamic analysis and vibration of shear deformable eccentrically stiffened S-FGM cylindrical panels with metal-ceramic-metal layers resting on elastic foundations, Composite Structures 126 (2015) 16–33. [42] N. D. Duc, N. D. Tuan, P. Tran, T. Q. Quan, N. V. Thanh, Nonlinear dynamic response and vibration of imperfect eccentrically stiffened sandwich third-order shear deformable FGM cylindrical panels in thermal environments, Journal of Sandwich Structures and Materials http://journals.sagepub.com/doi/10.1177/1099636217725251, (2017). [43] N. D. Duc, P. T Thang, N. T. Dao, H. V. Tac, Nonlinear buckling of higher deformable SFGM thick circular cylindrical shells with metal-ceramic-metal layers surrounded on elastic foundations in thermal environment, Composite Structures 121 (2015) 134–141. [44] N. D. Duc, Nonlinear thermal dynamic analysis of eccentrically stiffened S-FGM circular cylindrical shells surrounded on elastic foundations using the Reddy’s third-order shear deformation shell theory, European Journal of Mechanics – A/Solids 58 (2015) 0–30.

33

[45] N. D. Duc, Nonlinear thermo-electro-mechanical dynamic response of shear deformable piezoelectric Sigmoid functionally graded sandwich circular cylindrical shells on elastic foundations, Journal of Sandwich Structures and Materials DOI: 10.1177/1099636216653266 (2016). [46] R. Kolahchi, A. M. M. Bidgoli, Size-dependent sinusoidal beam model for dynamic instability of single-walled carbon nanotubes, Applied Mathematics and Mechanics (English Edition) 37(2) (2016) 265–274. [47] R. Kolahchi, A. Cheraghbak, Agglomeration effects on the dynamic buckling of viscoelastic microplates reinforced with SWCNTs using Bolotin method, Nonlinear Dynamics, 90(1) (2017) 479–492. [48] R. Kolahchi, M. Safari, M. Esmailpour, Dynamic stability analysis of temperature-dependent functionally graded CNT-reinforced visco-plates resting on orthotropic elastomeric medium, Composite Structures 150 (2016) 255–265. [49] M. Shokravi, Buckling analysis of embedded laminated plates with agglomerated CNTreinforced composite layers using FSDT and DQM, Geomechanics and Engineering 12(2) (2017) 327–346. [50] M. Shokravi, Buckling of sandwich plates with FG-CNT-reinforced layers resting on orthotropic elastic medium using Reddy plate theory, Steel and Composite Structures 23(6) (2017) 623–631. [51] M. H. Hajmohammad, M. S. Zarei, A. Nouri, R. Kolahchi, Dynamic buckling of sensor/functionally graded-carbon nanotubes reinforced laminated plates/actuator based on sinusoidal-viscopiezoelasticity theories, Journal of Sandwich Structures and Materials (2017) In press; doi.org/10.1177/1099636217720373. [52] B. Keshtegar, R. Kolahchi, M. H. Fakhar, Optimization of dynamic buckling for sandwich nanocomposite plates with sensor and actuator layer based on sinusoidalvisco-piezoelasticity theories using Grey Wolf algorithm, Journal of Sandwich Structures and Materials (2017) In press; doi.org/10.1177/1099636217731071. [53] R. Kolahchi, M. S. Zarei, M. H. Hajmohammad, A. Nouri, Wave propagation of embedded viscoelastic FG-CNT-reinforced sandwich plates integrated with sensor and actuator based on refined zigzag theory, International Journal of Mechanical Sciences 130 (2017) 534–545.

34

[54] A. J. Arani, R. Kolahchi, Buckling analysis of embedded concrete columns armed with carbon nanotubes, Computers and Concrete 17(5) (2016) 567–578. [55] B. S. Bilouei, R. Kolahchi, M. R. Bidgoli, Buckling of concrete columns retrofitted with Nano-Fiber Reinforced Polymer (NFRP), Computers and Concrete 18(5) (2016) 1053–1063. [56] M. Zamanian, R. Kolahchi, M. R. Bidgoli, Agglomeration effects on the buckling behavior of embedded concrete columns reinforced with SiO2 nano – particles, Wind and Structures 24(1) (2017) 43–57. [57] M. S. Zarei, R. Kolahchi, M. H. Hajmohammad, M. Maleki, Seismic response of underwater fluid-conveying concrete pipes reinforced with SiO2 nanoparticles and fiber reinforced polymer (FRP) layer, Soil Dynamics and Earthquake Engineering 103 (2017) 76–85. [58] M. Shokravi, Vibration analysis of silica nanoparticles-reinforced concrete beams considering agglomeration effects, Computers and Concrete 19(3) (2017) 333–338. [59] R. Kolahchi, M. S. Zarei, M. H. Hajmohammad, A. N. Oskouei, Visco-nonlocal-refined Zigzag theories for dynamic buckling of laminated nanoplates using differential cubatureBolotin methods, Thin-Walled Structures 113 (2017) 162–169. [60] R. Kolahchi, H. Hosseini, M. Esmailpour, Differential cubature and quadrature-Bolotin methods for dynamic stability of embedded piezoelectric nanoplates based on visco-nonlocalpiezoelasticity theories, Composite Structures 157 (2016) 174–186. [61] R. Kolahchi, A comparative study on the bending, vibration and buckling of viscoelastic sandwich nano-plates based on different nonlocal theories using DC, HDQ and DQ methods, Aerospace Science and Technology 66 (2017) 235–248. [62] M. Shokravi, Dynamic pull-in and pull-out analysis of viscoelastic nanoplates under electrostatic and Casimir forces via sinusoidal shear deformation theory, Microelectronics Reliability 71 (2017) 17–28. [63] V. Baseri, G. S. Jafari, R. Kolahchi, Analytical solution for buckling of embedded laminated plates based on higher order shear deformation plate theory, Steel and Composite Structures 22(4) (2016) 889–913. [64] D.V. Dung, P.M. Vuong, Nonlinear analysis on dynamic buckling of eccentrically stiffened functionally graded material toroidal shell segment surrounded by elastic foundations in thermal environment and under time-dependent torsional loads, Applied Mathematics and Mechanics (English Edition) 37(7) (2016) 835–860. 35

[65] D.V. Dung, P.M. Vuong, Analytical investigation on buckling and postbuckling of FGM toroidal shell segment surrounded by elastic foundation in thermal environment and under external pressure using TSDT, Acta. Mech. 228 (2017) 3511–3531. [66] J.N. Reddy, Mechanics of laminated composite plates and shells, Theory and Analysis, Boca Raton: CRS Press 2004. [67] P.L. Pasternak, On a new method of analysis of an elastic foundation by mean of two foundation constan, GosIzd Lit Po StroitArkh. Moscow, URSS 1954.

36