On the stability of functionally graded truncated conical shells reinforced by functionally graded stiffeners and surrounded by an elastic medium

Accepted Manuscript On the stability of functionally graded truncated conical shells reinforced by functionally graded stiffeners and surrounded by an elastic medium Dao Van Dung, Le Kha Hoa, Nguyen Thi Nga PII: DOI: Reference:

S0263-8223(13)00449-2 http://dx.doi.org/10.1016/j.compstruct.2013.09.002 COST 5334

To appear in:

Composite Structures

Please cite this article as: Dung, D.V., Hoa, L.K., Nga, N.T., On the stability of functionally graded truncated conical shells reinforced by functionally graded stiffeners and surrounded by an elastic medium, Composite Structures (2013), doi: http://dx.doi.org/10.1016/j.compstruct.2013.09.002

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On the stability of functionally graded truncated conical shells reinforced by functionally graded stiffeners and surrounded by an elastic medium Dao Van Dung, Le Kha Hoa*, Nguyen Thi Nga Vietnam National University, Hanoi, Viet Nam *

Corresponding author: Tel.: +84 989358315. E-mail address: [email protected]

Abstract: This paper presents an analytical approach to investigate the mechanical buckling load of eccentrically stiffened functionally graded truncated conical shells surrounded by elastic medium and subjected to axial compressive load and external uniform pressure. Shells are reinforced by stringers and rings in which material properties of shell and stiffeners are graded in the thickness direction according to a volume fraction power-law distribution. The elastic medium is assumed as two- parameter elastic foundation model proposed by Pasternak. The change of spacing between stringers in the meridional direction is taken into account. The equilibrium and linearized stability equations for stiffened shells are derived based on the classical shell theory and smeared stiffeners technique. The resulting equations which they are the couple set of three variable coefficient partial differential equations in terms of displacement components are investigated by Galerkin method and the closed-form expression for determining the buckling load is obtained. Four cases of stiffener arrangement are analyzed. Carrying out some computations, effects of foundation, stiffener and input factors on stability of shell have been studied. The effectiveness

of FGM stiffeners in enhancing the stability of cylindrical shells comparing with homogenuos stiffener is shown. Keywords: Stiffened truncated conical shell; Stiffener; Elastic medium; Functionally graded material; Critical buckling load.

1. Introduction Functionally graded shells involving conical shells, in recent years, are widely used in space vehicles, aircrafts, nuclear power plants and many other engineering applications. These structures are usually laid on or placed in a soil medium modeled as an elastic foundation. To increase the resistance of shells to buckling, they are strengthened by stiffeners and thus the critical load can be increased considerably with only a little addition of material. As a results stability and vibration analysis of those strutures are very important problems and have attracted increasing research effort. In static analysis of conical shells without foundation and stiffener, many studies have been focused on the buckling behavior analysis of shells under mechanic and thermal loading. Seide [1] analyzed the buckling of conical shells

1

under the axial loading. Singer [2] investigated the buckling of conical shells subjected to the axisymmetrical external pressure. Chang and Lu [3] examined the themoelastic buckling of conical shells based on nonlinear analyses. They used Galerkin method for integrating the equilibrium equation. Tani and Yamaki [4] obtained the results of truncated conical shells under axial compression. Using the Donnell-type shell theory, the linear buckling analysis of laminated conical shells, with orthotropic stretching-bending coupling, under axial compressive load and external pressure, are studied by Tong and Wang [5]. Wu and Chiu [6] studied a three-dimensional solution for the thermal buckling of the laminated composite conical shells. For shells resting on elastic foundations, many significant results on the vibration and dynamic buckling of isotropic and anitropic cylindrical shells are obtained. Paliwal et al.[7] studied free vibration of circular cylindrical shell on Winkler and Pasternak foundations. Amabili and Dalpiaz [8] investigated free vibration of cylindrical shells with non-asymmetric mass ditribution on elastic bed. Ng and Lam [9] considered effects of elastic foundations on the dynamic stability of cylindrical shells. The same authors [10] analyzed free vibration of rotating circular cylindrical shell on an elastic foundation. Naili and Oddou [12] investigated the buckling of short cylindrical shells surrounded by an elastic medium while Fok [12] studied the buckling of long cylindrical shells embedded in an elastic medium using the energy method. Effects of elastic foundations on the vibration of laminated non-homogenous orthotropic circular cylindrical shells is reported by Sofiyev et al. [13]. Geometrically nonlinear dynamic analysis of doubly curved isotropic shells resting on elastic foundation by a combination of hamonic differential quadrature-finite difference method is presented by Civalek [14]. Solution of axisymmetric dynamic problems for cylindrical shells on an elastic foundation is considered by Golovko et al. [15]. Sheng and Wang [16] considered the effect of thermal load on buckling, vibration and dynamic buckling of FGM cylindrical shells embedded in a linear elastic medium based on the firstorder shear deformation theory (FSDT) taking into account the rotary inertia and transverse shear strains. The post-buckling analysis of tensionless Pasternak FGM cylindrical shells surrounded by an elastic medium under the lateral pressure and axial load are studied by Shen [17] and Shen et al. [18] using the singular perturbation technique and the higher-order shear deformation shell theory (HDST). The mechanical buckling of FGM cylindrical shells surrounded by Pasternak elastic foundation is studied by Bagherizadeh et al. [19] in which the equilibrium and stability equations are derived based on the higher-order shear 2

deformation shell theory. The stability and vibration analysis of FGM cylindrical shells resting on the Pasternak elastic foundation have been published by Sofiyev et al. [20, 21] using Galerkin method to determine buckling load and frequency of shell. In recent years, because functionally graded material (FGM) conical shells are widely used in modern engineering, so the stability and vibration behaviors of these structures have attracted attention of a lot of scientists. Among those available, Sofiyev [22-24] investigated the linear stability and vibration of unstiffened FGM truncated conical shells with different boundary conditions. The same author [25] presented the nonlinear buckling behavior and nonlinear vibration [26] of FGM truncated conical shells, and considered [27-31] the buckling of FGM truncated conical shells subjected to axial compressive load and resting on Winkler-Pasternak foundations. For linear analysis, the general characteristics in his works is that the modified Donnell-type equations are used and Galerkin method is applied to obtain closed-form relations of bifurcation type buckling load or to find expressions of fundamental frequencies, whereas for nonlinear analysis, the large deflection theory with von Karman - Donnell type of kinetic nonlinearity is used. Based on the first-order shell theory by Love-Kirchhoff and the Sanders nonlinear kinetic equations, the thermal and mechanical instability of FGM un-stiffened truncated conical shells also is investigated by Naj et al. [32]. Bich et al. [33] presented results on the buckling of un-stiffened FGM conical panels under mechanical loads. The linearized stability equations in terms of displacement components are derived by using the classical shell theory. Galerkin method is applied to obtained explicit expression of buckling load. As can be seen that the above introduced works only relate to unstiffened structures. However, in practice, plates and shells including conical shells, usually reinforced by stiffeners system to provide the benefit of added load carrying capability with a relatively small additional weight. Thus, the study on static and dynamic behavior of theses structures are significant practical problem. Weingarten [34] conducted a free vibration analysis for a ring-stiffened simply supported conical shell by considering an equivalent orthotropic shell and using Galerkin method. He also carried out experimental investigations. Crenwelge and Muster [35] applied an energy approach to find the resonant frequencies of simply supported ring-stiffened, and ring and stringer-stiffened conical shells. Mustaffa and Ali [36] studied the free vibration characteristics of

3

stiffened cylindrical and conical shells by applying structural symmetry techniques. Some significant results on vibration of FGM conical shells, cylindrical shells and annular plate structures with a four-parameter power-law distribution based on the first-order shear deformation theory are analyzed by Tornabene [37] and Tornabene et al. [38]. Srinivasan and Krishnan [39] obtained the results on the dynamic response analysis of stiffened conical shell panels in which the effect of eccentricity is taken into account. The integral equation for the space domain and mode superposition for the time domain are used in their work. Based on the DonnellMushtari thin shell theory and the stiffeners smeared technique, Mecitoglu [40] studied the vibration characteristics of a stiffened truncated conical shell by the collocation method. The minimum weight design of axially loaded simply supported stiffened conical shells with natural frequency constraints is considered by Rao and Reddy [41]. The influence of placing the stiffeners inside as well as outside the conical shell on the optimum design is studied. The expressions for the critical axial (buckling) load and natural frequency of vibration of conical shell also are derived. In 2009, Najafizadeh et al. [42] with the linearized stability equations in terms of displacements studied buckling of FGM cylindrical shell reinforced by rings and stringers under axial compression. The stiffeners and skin, in their work, are assumed to be made of functionally graded materials and its properties vary continuously through the thickness direction. Following this direction, Dung and Hoa [43, 44] obtained the results on the static nonlinear buckling and postbuckling analysis of eccentrically stiffened FGM circular cylindrical shells under external pressure and torsional load. The material properties of shell and stiffeners are assumed to be continuously graded in the thickness direction. Galerkin method was used to obtain closed-form expressions to determine critical buckling loads. By considering homogenous stiffeners, Bich et al. [45] presented an analytical approach to investigated the nonlinear post-buckling of eccentrically stiffened FGM plates and shallow shells based on the classical shell theory in which the stiffeners are assumed to be homogeneous. Bich et al. [46] obtained the results on the nonlinear dynamic analysis of eccentrically stiffened FGM cylindrical panels. The governing equations of motion were derived by using the smeared stiffeners technique and the classical shell theory with von Karman geometrical nonlinearity. The same authors [47] investigated the nonlinear vibration dynamic buckling of eccentrically stiffened imperfect FGM

4

doubly curved thin shallow shells based on the classical shell theory. The nonlinear critical dynamic buckling load is found according to the BudianskyRoth criterion. Dung et al. [48] studied a mechanical buckling of eccentrically stiffened functionally graded (ES-FGM) thin truncated conical shells subjected to axial compressive load and uniform external pressure load based on the smeared stiffeners technique and the classical shell theory and considering homogenous stiffeners. The objective of this study is to extend mechanical buckling results of the work [48] for stiffened FGM thin truncated conical shells surrounded by an elastic medium. The present novelty is that an analytical approach to investigate the buckling load of stiffened functionally graded truncated conical shells surrounded by elastic foundations is presented. Shells under combined load are reinforced by rings and stringers in which their material properties are graded in the thickness direction according to a volume fraction power-law distribution. The change of spacing between stringers in the meridional direction is taken into account. The theoretical formulations based on the smeared stiffeners technique and the classical shell theory, are derived. The resulting equations which they are the couple set of three variable coefficient partial differential equations in terms of displacement components are solved by Galerkin method. The closed-form expressions to determine critical buckling loads are obtained. Four cases of stiffener arrangement are investigated. The influences of various parameters such as foundation, stiffener, dimensional parameters and volume fraction index of materials on the stability of shell are considered. The effectiveness of FGM stiffeners in enhancing the stability of cylindrical shells comparing with homogenuos stiffener is shown. 2. ES-FGM truncated conical shell and derivations 2.1. FGM truncated conical shell Consider a thin truncated conical shell of thickness h and semi-vertex angle α. The geometry of shell is shown in Fig. 1, where L is the length and R is its small base radius. 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, θ is in the circumferential direction and the axes z being perpendicular to the plane ( x, θ ) , lies in the outwards normal direction of the cone. Also, x0 indicates the distance from

5

the vertex to small base, and u, v, and w denote the displacement components of a point in the middle surface in the direction x, θ and z, respectively. Further, assume that the shell is stiffened by closely spaced circular rings and longitudinal stringers and the stiffeners and skin are made of functionally graded materials varying continuously through the thickness direction of the shell with the power law. Four cases are investigated in this work. z O α

x0

R

R

x K2

R

h L d2

K1 d1

θ x bs

Stringer es

d1

hr

hs

Ring er

br

h d2

h

Fig. 1. Geometry of stiffened truncated conical shell

Case 1: Conical shell with ceramic outside surface and metal inside surface and outside stiffener Case 2: Conical shell with ceramic outside surface and metal inside surface and inside stiffener Case 3: Conical shell with metal outside surface and ceramic inside surface and outside stiffener 6

Case 4: Conical shell with metal outside surface and ceramic inside surface and inside stiffener For case 1, Young modulii of FGM shell and FGM stiffeners are given by [42, 43] k

h h  2z + h  Esh = Em + Ecm   , Ecm = Ec − Em , k ≥ 0 , − ≤ z ≤ , 2 2  2h 

(1)

k

 2z − h  2 h h Es = Ec + Emc   , Emc = Em − Ec , k2 ≥ 0 , ≤ z ≤ + hs , 2 2  2hs 

(2)

k

 2z − h  3 h h Er = Ec + Emc   , k3 ≥ 0 , ≤ z ≤ + hr , k2 = k3 = 1/ k , 2 2  2hr 

(3)

where ν sh = ν s = ν r = ν = const , k, k2 and k3 are volume fractions indexes of shell, stringer and ring, respectively and subscripts c, m, sh, s and r denote ceramic, metal, shell, longitudinal stringers and circular ring, respectively. It is evident that, from Eqs. (1)-(3), a continuity between the shell and stiffeners is satisfied. Note that the thickness of the stringer and the ring are respectively denoted by hs , and hr ;and

Ec , Em are Young’s modulus of the ceramic and metal, and E sh , Es and Er are Young moduli of shell, of stiffener in the x-direction and θ -direction, respectively. The coefficient ν is Poison’s ratio. Young’s modulus for remaining cases are given in Appendix I. 2.2. Derivations

Based on the classical shell theory with the geometrical nonlinearity in von Karman sense and smeared stiffeners technique the governing equations are derived in this work. The normal and shear strains at distance z from the reference surface of shell are expressed by [49]

ε x = ε xm + zk x , εθ = εθ m + zkθ , γ xθ = γ xθ m + 2 zk xθ ,

(4)

where ε xm and ε θ m are the normal strains and γ xθ m 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 [49] 1 2

ε xm = u, x + w,2x ,

7

εθm =

1 u w 1 v ,θ + + cot α + 2 2 w,2θ , x sin α x x 2 x sin α

γ xθ m =

1 v 1 u,θ − + v, x + w, x w,θ x sin α x x sin α

k x = − w, xx , kθ = − k xθ = −

(5)

1 1 w,θθ − w, x , 2 x x sin α 2

1 1 w, xθ + 2 w,θ . x sin α x sin α (6)

The stress-strain relations for the FGM conical shell are

σ xsh =

Esh E Esh ε + νε θ ) , σ θsh = sh 2 (ε θ + νε x ) , σ xshθ = γ xθ , 2 ( x 2 (1 + ν ) 1 −ν 1 −ν

(7)

and for FGM stiffeners

σ xst = Esε x ,

σ θst = Er εθ ,

(8)

where the subscripts sh and st denote shell and stiffeners, respectively. Taking into account the contribution of stiffeners by the smeared stiffener technique and omitting the twist of stiffeners because these torsion constants are smaller more than the moments of inertia [49]. In addition, the change of spacing between stringers in the meridional direction also is taken into account. Integrating the above stress-strain equations and their moments through the thickness of the shell, we obtain the expressions for force and moment resultants of an eccentrically stiffened FGM conical shells

 E b  N x =  A11 + 1s s  ε xm + A12ε θ m +  B11 + C1 ( x ) k x + B12 kθ , d1 ( x )  

 E b  Nθ = A12ε xm +  A22 + 1r r  ε θ m + B12 k x + [ B22 + C2 ] kθ , d2  

N xθ = A66γ xθ m + 2 B66k xθ ,

(9)

 E b  M x =  B11 + C1 ( x ) ε xm + B12εθ m +  D11 + 3s s  k x + D12kθ , d1 ( x )  

 E b  M θ = B12ε xm + [B22 + C2 ]ε θ m + D12k x +  D22 + 3r r  kθ , d2   M xθ = B66γ xθ m + 2 D66 k xθ ,

(10)

8

where C1 ( x ) , C2 , d1 ( x ) , d 2 , E1s , E2 s , E3 s , E1r , E2 r , E3 r , Aij , Bij , Cij can be found in Appendix I. Using the classical shell theory, the nonlinear equilibrium equations of truncated conical shells surronded by elastic foundation, are given as follows [49] 1 xN x, x + N xθ ,θ + N x − Nθ = 0 , sin α 1 Nθ ,θ + xN xθ , x + 2 N xθ = 0 , sin α 2  1 1  xM x, xx + 2M x, x + M θ ,θθ − M θ , x − Nθ cot α  M xθ , xθ + M xθ ,θ  + 2 sin α  x  x sin α 1 1  1    N xθ w,θ  + Nθ w,θ  − qx − xK1w +  xN x w, x +  N xθ w, x + sin α x sin α  , x sin α  ,θ . 1 1   + xK 2  w, xx + w, x + 2 2 w,θθ  = 0. x x sin α   (11)

where K1 (N/m 3 ) is modulus of subgrade reaction for foundation and K 2 (N/m)- the shear modulus of the subgrade and q is uniform external pressure. The stability equations of conical shell are derived using the adjacent equilibrium criterion [49]. Assume that the equilibrium state of ES-FGM conical shell under mechanical loads is defined in terms of the displacement components

u0 , v0 and w0 . We give an arbitrarily small increments u1 , v1 and w1 to the displacement variables, so the total displacement components of a neighboring state are

u = u0 + u1 , v = v0 + v1 , w = w0 + w1 .

(12)

Similarly, the force and moment resultants of a neighboring state may be related to the state of equilibrium as

N x = N x 0 + N x1 ,

Nθ = Nθ 0 + Nθ 1 ,

N xθ = N xθ 0 + N xθ 1 ,

M x = M x 0 + M x1 , M θ = M θ 0 + M θ 1 , M xθ = M xθ 0 + M xθ 1 ,

(13)

where terms with 0 subscripts correspond to the u0 , v0 , w0 displacements and those with 1 subscripts represents the portions of increments of force and moment resultants that are linear in u1 , v1 and w1 . The stability equations may be obtained by substituting Eqs. (12) and (13) into Eqs. (11) and note that the terms in the resulting equations with subscript 0 satisfy the equilibrium equations and therefore drop out

9

of the equations. In addition, the nonlinear terms with subscript 1 are ignored because they are small compared to the linear terms. The remaining terms form the stability equations as follows xN x1, x +

1 N xθ 1,θ + N x1 − Nθ 1 = 0 , sin α

1 Nθ 1,θ + xN xθ 1, x + 2 N xθ 1 = 0 , sin α 2  1 1  xM x1, xx + 2M x1, x + M θ 1,θθ − M θ 1, x − Nθ 1 cot α  M xθ 1, xθ + M xθ 1,θ  + 2 sin α  x  x sin α 1 1  1    N xθ 0 w1,θ  + Nθ 0 w1,θ  − xK1w +  xN x 0 w1, x +  N xθ 0 w1, x + sin α x sin α  , x sin α  ,θ 1 1  + xK 2  w, xx + w, x + 2 2 w,θθ x x sin α 

 =0  (14)

where the force and moment resultants for the state of stability are given by

  E b  C0  N x1 =  A11 + 1s s  ε xm1 + A12ε θ m1 +  B11 + 1  k x1 + B12kθ 1 , λ0 x  x      C10  E3sbs  M x1 =  B11 +  ε xm1 + B12εθ m1 +  D11 +  k + D12 kθ 1 , etc λ0 x  x1 x   

(15)

and the linear form of the strains and the change of curvatures in terms of the displacement components are

ε xm1 = u1, x , εθ m1 =

1 u w v 1 v1,θ + 1 + 1 cot α , γ xθ m1 = v1, x − 1 + u1,θ , x sin α x x x x sin α

1 1 w1,θθ − w1, x , 2 x x sin α 1 1 k xθ 1 = − w1, xθ + 2 w1,θ . (16) x sin α x sin α Also assume that a shell only subjected to the axial compressive load P (N) and external uniform pressure q (Pa). Thus, the pre-buckling force resultants of shell are found by solving the membrane form of the equilibrium equations (11), as [49] k x1 = − w1, xx , kθ 1 = −

2

q P N x0 = − x tan α − , Nθ 0 = − qx tan α , N xθ 0 = 0 . 2 π x sin 2α

(17)

Substituting Eqs. (15-17) into Eqs (14), the stability equations in terms of the displacement components are of the form

10

F11 (u1 ) + F12 ( v1 ) + F13 ( w1 ) = 0 ,

(18)

F21 (u1 ) + F22 ( v1 ) + F23 ( w1 ) = 0 ,

(19)

F31 (u1 ) + F32 ( v1 ) + F33 ( w1 ) + qF34 ( w1 ) + PF35 ( w1 ) = 0 ,

(20)

where Fij are variable coefficient differential operators and are defined in Appendix II.

The system of Eqs. (18-20) is used to analyze the stability and find the critical buckling load of ES-FGM truncated conical shells. It is difficult that these equations are a couple set of three variable coefficient partial differential equations. This problem will be got over below. 3. Buckling analysis of ES-FGM truncated conical shell surrounded by elastic foundation

An analytical approach is given, in this section, to investigate the stability of ESFGM truncated conical shells. Assume that a shell is simply supported at both ends. Thus the boundary conditions are expressed by v1 = w1 = 0, M x1 = 0 at

x = x0 , x0 + L .

(21)

The approximate solution satisfying the above mentioned boundary conditions may be found in the form [33, 48] u1 = A cos

mπ ( x − x0 ) nθ sin , L 2

v1 = B sin

mπ ( x − x0 ) nθ cos , L 2

w1 = C sin

mπ ( x − x0 ) nθ sin , L 2

(22)

where m is the number of half-waves along a generatrix and n is the number of fullwaves along a parallel circle, and A, B and C are constant coefficients. Due to

x0 ≤ x ≤ x0 + L ; 0 ≤ θ ≤ 2π and for sake of convenience in integration, Eqs. (18), (19) are multiplied by x and Eq. (20) by x 2 , and applying Galerkin method for the resulting equations, that are

11

x0 + L 2π

∫ ∫ Φ1x cos

x0

0

x0 + L 2π

∫ ∫ Φ2 x sin

x0

0

x0 + L 2π

∫

x0

∫ Φ3 x sin 0

mπ ( x − x0 ) nθ sin sin α dθ dx = 0 , L 2 mπ ( x − x0 ) nθ cos sin α dθ dx = 0 , L 2 mπ ( x − x0 ) nθ sin sin α dθ dx = 0 , L 2

(23)

where Φ1 = x  F11 (u1 ) + F12 ( v1 ) + F13 ( w1 ) ,

Φ 2 = x  F21 (u1 ) + F22 ( v1 ) + F23 ( w1 ) , Φ 3 = x 2  F31 (u1 ) + F32 ( v1 ) + F33 ( w1 ) + qF34 ( w1 ) + PF35 ( w1 ) .

(24)

Substituting expressions (22) and (24) into Eqs. (23), after integrating longer and some rearrangements, may be written in the following form

s11 A + s12 B + s13C = 0 , s21 A + s22 B + s23C = 0 ,

s31 A + s32 B + ( s33 + qs34 + Ps35 + s36 K1 + s37 K 2 ) C = 0 ,

(25)

where the coefficients sij are defined in Appendix III. Because the solutions of Eq. (25) are nontrivial, the determinant of coefficient matrix of this system must be zero. Developing that determinant and solving resulting equation for combination of P and q, leads to

s34 q + s35 P =

s31 ( s12 s23 − s13s22 ) + s32 ( s13s21 − s11s23 ) − ( s33 + s36 K1 + s37 K 2 ) (26) s21s12 − s11s22

Eq. (26) is used to determine the critical buckling load of ES-FGM conical shells surrounded by elastic foundation and under axial compressive load and uniform pressure load. The buckling loads P and q still depend on values of m and n, therefore must minimize these expressions with respect to m and n, we obtain the critical values of P and q respectively.

12

4. Numerical results and discussion 4.1. Comparison results

To validate the present study, firstly, Table 1 compares the results of this paper for an un-stiffened isotropic truncated conical shell under external pressure with the results given in the monograph of Brush and Almroth [49, pp 217]. The data base in this case is taken as: k = 0 , h = 0.01m , R = 100 × h , ν = 0.3 , P=0, q* = 10 4 Rqcr / ( Eh ) , where qcr is found from Eq. (26). Next, Table 2 compares the present results with those of Naj et al. [32] and Baruch et al. [50] for a pure isotropic un-stiffened truncated conical shell. The input parameters are: k = 0 , h = 0.01m , R = 100 × h , ν = 0.3 , q =0, P* = Pcr / Pcl with 2π Eh 2 cos 2 α

Pcl =

(

3 1 −ν

2

)

[32], where Pcr is found from Eq. (26).

Table 1 Comparisons with results of Brush and Almroth [49] for un-stiffened isotropic conical shells

q* =

104 qcr R Eh

[49] Present [49] 30 0 Present [49] 50 0 Present [49] 70 0 Present [49] 85 0 Present a : Buckling mode (m,n)

10 0

L/R=1/2

L/R=1

L/R=2

19.40 19.3867 (1,22)a 14.55 14.4517 (1,23) 8.81 8.6783 (1,21) 3.50 3.4120 (1,17) 0.591 0.5756 (1,9)

8.57 8.5293 (1,17) 5.84 5.7808 (1,19) 3.28 3.2094 (1,19) 1.201 1.1510 (1,16) 0.1710 0.1589 (1,10)

3.74 3.7572 (1,14) 2.24 2.2594 (1,16) 1.164 1.1403 (1,17) 0.399 0.3747 (1,16) 0.0498 0.0445 (1,10)

Table 2 Comparisons with results of [32] and [50] for un-stiffened isotropic truncated conical shells.

R. Naj et al. [32]

L/R=0.2 Baruch et al.[50]

10

1.005 (7)

0

α

L/R=0.5 Baruch et al. [50]

1.005 (7)

1.0002 (1,12)a

1.0017 (8)

1.002 (8)

1.0001 (2,17)

Present

1.006 (7)

1.006 (7)

1.0001 (1,12)

1.001 (8)

1.002 (8)

1.0002 (2,17)

0

1.007 (7)

1.007 (7)

1.0002 (1,12)

1.000 (8)

1.002 (8)

1.0005 (2,17)

30 0

1.0171 (5)

1.017 (5)

1.0017 (1,7)

0.987 (7)

1.001 (7)

1.0023 (2,15)

60

0

1.148 (0)

1.144 (0)

1.1299 (1,1)

1.045 (7)

1.044 (7)

1.0150 (1,14)

80

0

2.492 (0)

2.477 (0)

2.5091 (1,1)

1.004 (5)

1.015 (5)

1.0266 (1,4)

5

10

a

Present

R. Naj et al.[32]

Buckling mode (m,n)

13

Finaly, Table 3 compares the present results with those of Seide [1] and Sofiyev [27] for a pure isotropic un-stiffned truncated conical shell (without foundation). The input parameters are: Em = Ec = E = 210 GPa, ν = 0.3 , h = 0.0005 m,

R ' = 400 × h , L =

1 Pcr R 'sin α , R = R '− L sin α , q=0, with σ cr = 4 π ( 2R + L sin α ) h cosα

[1, 27], where Pcr is found from Eq. (26). Table 3 Comparisons σ cr (MPa) with results of Seide [1] and Sofiyev [27] for un-stiffened isotropic truncated conical shells

α

Seide [1]

Sofiyev [27]

Present

30 0

284.05

283.44 (1,15)

284.2331 (1,30)

45 0

239.66

238.25 (1,15)

240.1864 (1,29)

60

0

175.31

172.52 (1.12)

176.2382 (1,24)

75

0

93.096

89.754 (1,8)

93.8722 (1,17)

As can be seen that good agreements are obtained in these three comparisons.

4.2. ES-FGM truncated conical shell with and without elastic foundation In the following subsections, the materials used are Alumina with Ec = 380 GPa and Aluminum with Em = 70 GPa and ν = 0.3 . The geometrical parameters are taken as h=0.005m, R/h=150, L=2R, hs = hr = 0.005 m, bs = br = 0.0025 m. The stiffener is made of FGM material.

a. Effect of foundation (without stiffener) Using Eq. (29) with q=0, α = 30 0 , k=1, the critical axial compressive load of

FGM truncated conical shell may be found in Table 4. It shows that when q = 0 the foundation parameters K1 and K 2 affect strongly for the critical load P. Table 5 presents effects of foundation on critical pressures qcr (P=0). As can be observed that the critical buckling load corresponding to the presence of the both foundation parameters K1 = 5 × 107 N/m3 and K 2 = 5 × 105 N/m is the biggest. The critical

buckling load of shell without foundation is the smallest. For example, the critical

14

buckling load of shell without foundation Pcr =14.75584 MN, m=7, n=26 when compared in Table 4 with Pcr =18.92762 MN, m=11, n=1, incerases about 28.3%. Table 4: Effects of foundation (without stiffeners) on critical compression load Pcr (q=0) h=5×10-3m , R/h=150, L=2R, α=300, k=1

Pcr (MN)

K2 = 0 N/m

K2 =1 ×105 N/m K2 =2.5 ×105 N/m K2 =5 ×105 N/m

K1= 0 N/m3

14.75584 (7,26)

15.57943 (10,14)

16.60486 (11,1)

18.29228 (11,1)

K1=1 × 107 N/m3

14.99299 (9,21)

15.71948 (11,1)

16.73193 (11,1)

18.41935 (11,1)

K1=2.5 × 107 N/m3 15.23496 (10,15)

15.91008 (11,1)

16.92253 (11,1)

18.60995 (11,1)

K1=5 × 107 N/m3

16.22775 (11,1)

17.24020 (11,1)

18.92762 (11,1)

15.55279 (11,1)

Table 5: Effects of foundation (without stiffeners) on critical external pressure q cr (P=0) h=5×10-3m , R/h=150, L=2R, α=300, k=1

qcr (kPa)

K2 = 0 N/m

K2 =1 ×105 N/m K2 =2.5 × 105 N/m K2 =5 × 105 N/m

K1= 0 N/m3

155.0628 (1,19)

232.1134 (1,19)

347.6894 (1,19)

540.3161 (1,19)

K1=1 ×107 N/m3

254.0953 (1,21)

330.7570 (1,21)

445.7496 (1,21)

637.4040 (1,21)

K1=2.5 ×107 N/m3

368.2026 (1,24)

444.4471 (1,24)

558.6617 (1,25)

749.0034 (1,25)

K1=5 ×107 N/m3

510.0027 (1,28)

585.8807 (1,28)

699.6976 (1,28)

889.2318 (1,29)

b. Effect of stiffener (without foundation).

The effects of stiffeners (without foundation) on critical compression load P and critical external pressure load q are given in Tables 6 and 7. Table 6 shows that in the case of only compression load P, with the same stiffener numbers, the critical buckling load of stiffened shell by orthogonal is the biggest, the second is stiffened shell by stringers and the critical load of the stiffened shell by ring stiffeners is the smallest. In additrion, the critical compressive load increases with the increase of stiffeners number. The effect of pre-loaded axial compression P on critical pressure q is illustrated in Table 7. As can be seen that the value of qcr decreases in the increase of axial load P. Table 7 also shows that the critical pressure q (with P ≠ 0) of stiffened shell by ring is the biggest.

15

Table 6: Effect of stiffeners on critical compression load Pcr MN, q=0, case 2, k=1 h=5×10-3m , R/h=150, L=2R, α=300, K1=0 N/m3, K2=0 N/m, hs=hr=5×10-3m, bs=br=2.5×10-3m

Number of stiffeners (nst)

nst =10

20

30

40

50

Stringer ns=nst

14.86580 (5,27)

14.93088 (4,26)

14.96462 (4,26)

14.99822 (4,26)

15.03168 (4,26)

Ring nr=nst

14.81156 (9,20)

14.81922 (9,20)

14.82689 (9,20)

14.83455 (9,20)

14.84221 (9,20)

Orthogonal ns=nr=nst/2

14.92844 (8,24)

15.08314 (8,23)

15.23399 (8,23)

15.38450 (8,23)

15.53467 (8,23)

Table 7: Effect of stiffeners on critical external pressure q cr , case 2, k=1 h=5×10-3m , R/h=150, L=2R, α=300, K1=0 N/m3, K2=0 N/m, hs=hr=5×10-3m, bs=br=2.5×10-3m

qcr (kPa)

Orthogonal

Un-Stiffened

28 Stringer

28 Ring

P=0

155.0628 (1,19)

155.0942 (1,19)

248.9277 (1,17)

205.0938 (1,18)

P=2.5 MN

135.0946 (1,18)

135.1474 (1,18)

225.9031 (1,17)

183.4218 (1,17)

P=5 MN

114.4528 (1,18)

114.5056 (1,18)

202.5697 (1,16)

160.3973 (1,17)

P=7.5 MN

93.0345 (1,17)

93.1216 (1,17)

176.7332 (1,16)

137.3727 (1,17)

P=10 MN

70.0099 (1,17)

70.0970 (1,17)

150.8968 (1,16)

112.9970 (1,16)

ns=nr=14

c. Effect of stiffener and foundation

Table 8 and 9 illustrate the effects of stiffeners and foundation on critical compression load P and critical external pressure load q . Fig. 2 and Table 8 show that when q = 0 with the same stiffener numbers, the critical buckling load of stiffened shell by stringer is the biggest ( Pxcr =18.27572MN reached at m=9, n=19), the second is stiffened shell by orthogonal stiffeners and the critical load of ring stiffened shell is the smallest. Fig. 3 and Table 9 ( with P=0) show that the critical buckling load of stiffened shell by ring is the biggest ( q xcr =709.4372 kPa for m=1, n=21).

16

Table 8: Effect of stiffeners and foundation on critical compression load Pcr (q=0) h=5×10-3m , R/h=150, L=2R, α=300, hs=hr=5×10-3m, bs=br=2.5×10-3m, k=1

Pcr (MN)

28 Stringer

28 Ring

Orthogonal

ns=nr=14

K1=0 N/m3

Case 1

17.07078 (10,11)

16.94029 (11,1)

17.01734 (11,1)

K2=2.5 ×105 N/m

Case 2

17.74375 (8,23)

16.78028 (10,15)

17.31264 (9,19)

K1=2.5 ×107 N/m3

Case 1

15.64863 (9,20)

15.57054 (11,1)

15.64759 (11,1)

K2=0 N/m

Case 2

16.05840 (7,26)

15.29980 (9,20)

15.78614 (9,21)

K1=2.5 × 107 N/m3

Case 1

17.41028 (11,1)

17.25796 (11,1)

17.33501 (11,1)

K2=2.5 × 105 N/m

Case 2

18.27572 (9,19)

17.16463 (10,15)

17.76068 (10,15)

Table 9: Effect of stiffeners and foundation on critical external pressure q cr (P=0,case 2) h=5×10-3m , R/h=150, L=2R, α=300, hs=hr=5×10-3m, bs=br=2.5×10-3m, k=1

28 Stringer

28 Ring

Orthogonal ns=nr=14

K1=0 N/m3 K2=2.5 ×105 N/m

347.7208 (1,19)

442.8681 (1,17)

398.3257 (1,18)

K1=2.5 ×107 N/m3 K2=0 N/m

368.2049 (1,24)

517.7829 (1,21)

449.8862 (1,22)

K1=2.5 × 107 N/m3 K2=2.5 ×105 N/m

558.6638 (1,25)

709.4372 (1,21)

641.0072 (1,23)

q cr (kPa)

Buckling load P (MN)

(a): K1=2.5×107N/m3, K2=2.5×105N/m (b): K1=107N/m3, K2=105 N/m

20

15 (1a) (2a)

10

5 120

Case 2: (inside stiffeners) hs =hr=0.005m bs =br=0.0025m 140

160

180

(1b) (2b)

200

(3a)

Buckling load q (kPa)

1500

1: Stringer, ns=28 2: Ring, nr=28 3: Orthogonal, ns=nr=14

R=0.75m L=1.5m α=300

25

240

260

(3a) (2a)

500

(3b) (2b) (1b)

Case 2: hs=hr=0.005m bs=br=0.0025m 0 80

280

(a): K1=2.5×107N/m3, K2 =2.5×105 N/m (b): K1=107N/m3, K2=105N/m

1000

(3b)

220

1: Stringer, ns=28 2: Ring, nr=28 3: Orthogonal, ns=nr=14

R=0.75m L=1.5m α=300

100

120

140

160

180

200

(1a)

220

240

260

280

R/h

R/h

Fig. 3. Effects of stiffener and foundation on

Fig. 2. Effects of stiffener and foundation on

critical load q cr (P=0, k=1)

critical load Pcr (q=0, k=1)

17

300

d. Effect of k-volume fraction index

The effects of index volume k on the critical buckling loads and post-buckling behavior are given in Fig. 4, Fig. 5 and Table 10 for stiffened FGM truncated conical shell. In Figs. 4 and 5, the lines 1 and 2 correspond to stiffened shell with α = 100 and

α = 30 0 respectively, while lines 3 and 4 correspond to un-stiffened shell. It is found that curves of stiffened shell always lie over curves of un-stiffened shell. In addition, the critical loads Pcr and qcr of ES-FGM shells decrease with the increase of k. The buckling strength of FGM truncated conical shell is more than fully metal shell and less than that of fully ceramic shell. This property appropriate to the real characteristic of material, because the higher value of k corresponds to a metalricher shell which usually has less stiffness than a ceramic-richer one. Table 10: Effect of volume fraction index k on critical load Pcr (MN) and q cr (kPa) h=5×10-3m, R/h=150, L=2R, α=300, K1=2.5×107N/m3, K1=2.5×105N/m, hs=hr=5×10-3 m, bs=br=2.5×10-3m

Pcr (q=0)

qcr (P=0)

k

Orthogonal

Un-stiffened

ns=nr=14

Un-stiffened

0

29.56545 (10,9)

30.36106 (9,18)

747.5401 (1,22)

825.5335 (1,21)

0.5

20.91305 (11,1)

21.75104 (9,19)

618.8278 (1,23)

700.9229 (1,22)

1

16.92253 (11,1)

17.76068 (10,15)

558.6617 (1,25)

641.0072 (1,23)

5

10.98383 (10,1)

11.48683 (10,9)

483.3974 (1,27)

542.7049 (1,25)

10

9.70820 (10,1)

10.10403 (10,1)

470.1232 (1,27)

516.2677 (1,25)

∞

7.11055 (11,1)

7.33836 (11,1)

409.5113 (1,31)

437.5025 (1,29)

ns=nr=14

e. Effect of width-to-thickness ratio R/h

The effects of the radius-to-thickness ratios R / h and angle α on critical buckling load Pcr and qcr of stiffened FGM truncated conical shells are illustrated in Figs. 6 and 7 in which the values of α are equal to 100, 200, 300 and 400. It is observed that the both critical loads P and q decrease markedly with the increase of R / h ratio

18

and α . This result agrees with the actual property of structure i.e. the shell is thinner the value of critical load is smaller.

40

3

30

2 4

K1 =2.5×107N/m3 K2 =2.5×105N/m

25 20 15 10 5

Case 2: (inside stiffeners) hs=hr=0.005m bs=br=0.0025m ns=nr=14 0.01

0.1

1

10

100

1

1000

3

K1=2.5×107N/m3 K2=2.5×105N/m

800 2

Case 2: 600 (inside stiffeners) hs =hr=0.005m bs =br=0.0025m ns =nr=14 400 -2 10

500

10

4

-1

10

k

30

1: 2: 3: 4:

K1 =2.5×107 N/m3 K2 =2.5×105N/m

(1) (2)

Case 2: (inside stiffeners) hs=hr=0.005m bs=br=0.0025m ns=nr=14

25 20

(3)

15

(4)

3000

α=100 α=200 α=300 α=400

Buckling load q (kPa)

Buckling load P (MN)

35

1

2

10

10

k

R=0.75m L=1.5m

40

0

Fig. 5. Effects of k on critical load qcr (P=0)

Fig. 4. Effects of k on critical load Pcr (q=0)

45

0

1: Stiffened, α=10 0 2: Stiffened, α=30 3: Unstiffened, α=100 4: Unstiffened, α=300

R=0.75m L=1.5m

1200

Buckling load q (kPa)

35

Buckling load P (MN)

1300

0

1: Stiffened, α=10 0 2: Stiffened, α=30 3: Unstiffened, α=100 0 4: Unstiffened, α=30

R=0.75m L=1.5m

1

10

0

2500

R=0.75m L=1.5m

2000

K1=2.5×107N/m3 K2=2.5×105N/m Case 2: (inside stiffeners) hs=hr=0.005m bs=br=0.0025m ns=nr=14

1500 1000 500

1: α=10 2: α=200 3: α=300 4: α=400

(1) (2) (3) (4)

5 0 100

150

200

250

300

350

400

450

0 70 100

500

150

200

250

300

350

400

450

500

R/h

R/h

Fig. 6. Effects of R/h on critical load Pcr

Fig. 7. Effects of R/h on critical load q cr

(q=0, k=1)

(P=0, k=1)

f. Effect of semi-vertex angle α Table 11 and 12 illustrate the effect of semi-vertex angle α on critical axial

compressive load Pcr and critical pressures q cr , respectively. As can be seen that the critical buckling load of truncated conical shell strongly decreases when semivertex angle increases. For example for an orthogonally stiffened shell in Table 12, when the semi-vertex angle varies the values from 50 to 700 , the critical pressure

qcr decreases from 1708.488(kPa) to 279.4340(kPa). This remark also has been

19

pointed out in Ref. [49]. Graphically, the effects of semi-vertex angle α on critical axial compressive load Pcr and critical pressures qcr are plotted in Figs. 8 and 9. They also show that critical axial compressive load Pcr and critical pressures q cr decrease when α increases and the critical load- angle α curve for an orthogonally stiffened shell is the highest.

25

Buckling load q (kPa)

Buckling load P (MN)

(3) (2) (1)

20

1600

1: K1=0, K2=0 2: K1=107, K2=10 5 3: K1=2.5×10 7, K2=2.5×10 5 4: K1=5×10 7, K2=5×10 5

(4)

15

10

R=0.75m L=1.5m

5

Case 2: (inside stiffeners) hs =hr=0.005m bs =br=0.0025m ns =nr=14

0 5

10

20

30

40

α

1: K1=0, K2=0 2: K1=107, K2=105 3: K1=2.5×10 7, K2=2.5×105 4: K1=5×10 7, K2=5×105

R=0.75m L=1.5m

1200

(4)

800

Case 2: hs=hr=0.005 m bs=br=0.0025 m ns=nr=14

(3) (2)

400 (1)

50

60

70

0 5

75

deg (deg)

10

20

30

40

50

60

70

75

α (deg)

Fig. 8. Effects of α on critical load Pcr

Fig. 9. Effects of α on critical load q cr

(q=0, k=1)

(P=0, k=1)

g. Comparison of FGM stiffeners with homogeneous stiffeners

To compare the effects of functionally graded stiffeners (FGMS) with homogeneous stiffeners (HS) to critical compression load P and critical external pressure q, Table 11 and 12 are presented. These are obtained when the value of R/h ratio is equal to 100, 150, 200, 300, 400 and the value of α varies. It is observed that the critical load of FGMS stiffened truncated conical shell is greater than HS. For example for R/h=400 in Table 11, when the semi-vertex angle α = 700 , the critical compression load Pcr = 1.61512 (11,1) of FGMS stiffened shell is bigger than load Pcr = 1.49916 (11,1) of HS stiffened shell about 7.7%. This point shows that the

bearing capacity of FGMS stiffened truncated conical shell is better than HS stiffened truncated conical shell.

20

Table 11: Comparison of FGMS with HS critical compression load P (MN) with q=0, k=1 K1=2.5×107 N/m3, K2=2.5×105 N/m, R=0.75m, hs=hr=5×10-3m, bs=br=2.5×10-3m, ns= nr = 14 α Stiffeners R/h=100 R/h=150 R/h=200 R/h=300 R/h=400 a

FGMS1 46.70287 (9,16) 22.20535 (11,19) 13.64141 (13,21) 45.90148 (11,8) 21.39950 (14,4) 12.81246 (16,1) HS2 45.85692 (9,16) 21.85755 (11,19) 13.47238 (13,20) FGMS 100 45.10968 (10,12) 210.9118 (13,8) 12.67674 (15,9) HS 42.29094 (8,17) 20.31633 (10,19) 12.61495 (12,20) FGMS 200 41.65696 (9,12) 19.64004 (12,4) 11.91593 (14,1) HS 36.57313 (8,13) 17.76068 (10,15) 11.15334 (11,19) FGMS 300 36.04913 (9,1) 17.18818 (11,1) 10.58787 (13,1) HS 29.31671 (7,14) 14.44802 (9,13) 9.22832 (11,9) FGMS 400 28.87903 (8,1) 14.01395 (10,1) 8.75861 (11,1) HS 21.42247 (7,2) 10.78003 (8,1) 7.01462 (10,1) FGMS 500 21.04626 (7,1) 10.47777 (9,1) 6.68777 (10,1) HS 13.72815 (6,1) 7.16700 (7,6) 4.80483 (9,1) FGMS 600 13.50997 (6,1) 6.98730 (7,1) 4.60428 (9,1) HS 7.26295 (5,1) 4.00738 (6,1) 2.81773 (7,1) FGMS 700 7.16134 (5,1) 3.91936 (6,1) 2.72219 (8,1) HS a Buckling mode (m,n); FGMS1 : Functionally graded stiffeners (case 2, k2=k3=k=1) HS2: Homogeneous stiffeners (Inside metal stiffeners) 50

7.56639 (15,24)

5.47073 (16,26)

6.68835 (19,9)

4.55861 (22.1)

7.50991 (15,24)

5.45140 (16,25)

6.66960 (19,1)

4.57465 (21,1)

7.14096 (14,23)

5.24885 (16,23)

6.39524 (17,1)

4.46978 (19,1)

6.44903 (13,21)

4.81212 (15,20)

5.82525 (15,1)

4.16711 (17,1)

5.47490 (13,11)

4.16966 (14,14)

4.99504 (14,1)

3.67236 (16,1)

4.31402 (12,1)

3.36845 (13,1)

39.8356 (12,1)

3.02373 (14,1)

3.09660 (10,1)

2.49362 (12,1)

28.9136 (11,1)

2.27966 (13,1)

1.92909 (9,1)

1.61512 (11,1)

1.82780 (10,1)

1.49916 (11,1)

Table 12: Comparison of FGMS with HS critical external pressure q (kPa) with P=0, k=1 K1=2.5×107 N/m3, K2=2.5×105 N/m, R=0.75m, hs=hr=5×10-3m, bs=br=2.5×10-3m, ns= nr = 14 α Stiffeners R/h=100 R/h=150 R/h=200 R/h=300 R/h=400 1708.488 (1,14) FGMS 1603.120 (1,14) HS 1543.411 (1,15) FGMS 100 1449.154 (1,15) HS 1253.214 (1,17) FGMS 200 1177.637 (1,17) HS 1005.569 (1,19) FGMS 300 944.8480 (1,19) HS 789.1037 (1,20) FGMS 400 744.3302 (1,21) HS 600.2686 (1,21) FGMS 500 567.2216 (1,22) HS 432.1910 (1,22) FGMS 600 409.5415 (1,22) HS 279.4340 (1,22) FGMS 700 265.2870 (1,23) HS a Buckling mode (m,n);

50

1056.465 (1,16)

832.4607 (1,19)

669.7420 (1,22)

610.0932 (1,24)

956.0809 (1,17)

730.6964 (1,20)

560.1511 (1,26)

496.3230 (1,30)

959.4907 (1,18)

758.5432 (1,20)

611.6043 (1,23)

557.1671 (1,25)

869.3618 (1,19)

665.8078 (1,22)

511.2357 (1,28)

453.0235 (1,32)

788.7737 (1,20)

626.4536 (1,23)

506.2019 (1,27)

461.5887 (1,29)

715.6451 (1,21)

550.1280 (1,25)

423.0058 (1,32)

374.9547 (1,37)

641.0072 (1,23)

511.1179 (1,26)

414.0485 (1,30)

377.6974 (1,33)

581.5947 (1,24)

449.0422 (1,29)

345.8001 (1,36)

304.8117 (2,43)

510.4749 (1,25)

408.8812 (1,28)

331.9219 (1,33)

300.2042 (2,38)

463.9726 (1,26)

359.4385 (1,31)

276.0050 (2,42)

242.1326 (2,48)

393.5753 (1,26)

316.6164 (1,30)

254.8275 (2,38)

230.7953 (2,41)

358.5221 (1,28)

278.6545 (1,34)

212.3814 (2,45)

186.5573 (2,52)

287.1395 (1,28)

231.6102 (1,32)

185.0459 (2,40)

167.9370 (2,43)

262.0718 (1,29)

204.1844 (1,36)

154.7798 (2,48)

136.0490 (2,55)

187.7371 (1,28)

150.5049 (2,35)

120.6159 (2,41)

109.5797 (2,45)

171.6812 (1,30)

133.6375 (2,39)

101.2173 (2,50)

88.9827 (2,57)

21

h. Effect of stiffener attached to shell with different types and number of stiffener Table 13 and 14 show the effect of stiffener attached case and number of stiffener

on critical axial compressive load Pcr and critical pressure qcr , respectively. As can be seen that the critical compressive load and critical pressure load in case 3 is the biggest, the second is case 2, the third is case 4 and the critical load in case 1 is the smallest. The obtained results in both tables show that the critical load increases when the number of stiffeners increases and inversely. This increase is considerable. For example, in case 1, comparing

Pcr =17.33501(MN) ( ns = nr = 14 ) with

Pcr =17.74570(MN) ( ns = nr = 28 ) in Table 13, the value of critical compressive load increases 410.69kN. Table 13:

Effect of stiffener attached case and stiffener number on critical axial compressive load Pcr (q=0)

Pcr (MN)

Case 1

Case 2

Case 3

Case 4

0

16.92253 (11,1)

16.92253 (11,1)

16.98603 (11,1)

16.98603 (11,1)

14

17.26526 (11,1) 17.57700 (10,15) 17.60355 (11,1)

17.36990 (10,15)

20

17.33713 (11,1) 17.59782 (10,15) 17.67579 (11,1)

17.40190 (10,15)

28

17.43297 (11,1) 17.62289 (10,16) 17.77210 (11,1)

17.43964 (10,16)

14

17.33501 (11,1) 17.76068 (10,15) 17.78242 (11,1)

17.44975 (10,15)

20

17.40688 (11,1) 17.78145 (10,15) 17.85466 (11,1)

17.48171 (10,15)

28

17.50271 (11,1)

17.79968 (9,19)

17.95097 (11,1)

17.51858 (10,16)

14

17.43936 (11,1)

18.00238 (9,19)

18.04997 (11,1)

17.56919 (10,15)

20

17.51123 (11,1)

18.00775 (9,19)

18.12221 (11,1)

17.60080 (10,16)

28

17.60707 (11,1)

18.01490 (9,19)

18.21852 (11,1)

17.63665 (10,16)

14

17.57799 (11,1)

18.28819 (9,19) 18.36725 (10,10) 17.72780 (10,15)

20

17.64987 (11,1)

18.29352 (9,19)

18.46545 (10,9)

17.75045 (9,20)

28

17.74570 (11,1)

18.30061 (9,19)

18.57383 (11,1)

17.76480 (9,20)

Stringer

Ring

0

10

14

20

28

22

Table 14:

Effect of stiffener attached case and stiffener number on critical pressure q cr , P=0, k=1.

qcr (kPa)

Case 1

Case 2

Case 3

Case 4

0

558.6617 (1,25)

558.6617 (1,25)

568.6891 (1,24)

568.6891 (1,24)

10

582.5873 (1,24)

618.7533 (1,23)

624.4312 (1,23)

595.1930 (1,24)

14

591.8895 (1,24)

641.0067 (1,23)

644.8159 (1,22)

605.5940 (1,24)

20

604.6160 (1,23)

671.3195 (1,22)

672.5120 (1,22)

619.9127 (1,23)

10

582.6096 (1,24)

618.7537 (1,23)

624.4699 (1,23)

595.1945 (1,24)

14

591.9120 (1,24)

641.0072 (1,23)

644.8622 (1,22)

605.5955 (1,24)

20

604.6431 (1,23)

671.3201 (1,22)

672.5589 (1,22)

619.9148 (1,23)

10

582.6431 (1,24)

618.7543 (1,23)

624.5279 (1,23)

595.1968 (1,24)

14

591.9457 (1,24)

641.0078 (1,23)

644.9316 (1,22)

605.5977 (1,24)

20

604.6837 (1,23)

671.3210 (1,22)

672.6292 (1,22)

619.9180 (1,23)

10

582.6877 (1,24)

618.7552 (1,23)

624.6053 (1,23)

595.1999 (1,24)

14

591.9907 (1,24)

641.0086 (1,23)

645.0241 (1,22)

605.6007 (1,24)

20

604.7379 (1,23)

671.3223 (1,22)

672.7229 (1,22)

619.9222 (1,23)

Stringer

Ring

0

10

14

20

28

5. Conclusions

An analytical solution is presented, in this paper, to investigate the linear buckling of eccentrically stiffened FGM truncated conical surrounded by an elastic medium and subjected to axial compressive load and uniform pressure load. The material properties of shell and stiffeners are graded in the thickness direction according to a volume fraction power-law distribution. The change of spacing between stringers in the meridional direction also is taken into account. Equilibrium and stability equations based on the smeared stiffeners technique and classical shell theory, are derived. The couple set of three variable coefficient partial differential equations is investigated by Galerkin method and the closed-form expression for determining the critical buckling load is obtained. Four cases of stiffener arrangement are analyzed. Numerical results showing the effect of foundation, stiffener, volume fraction index and geometrical parameters on the buckling

23

response of FGM truncated conical shells are obtained. The present results show some remarks as i. Foundation parameters K1 , K2 affects strongly for the critical buckling loads. Especially, the critical buckling load corresponding to the presence of the both foundation parameters K1 and K 2 is biggest. ii. Presence of stiffeners enhances the stability of truncated conical shells in which the critical buckling loads of truncated conical shells with the orthogonal stiffeners are greatest. iii. Critical load increases when the number of stiffeners increases and inversely. iv. Loading carrying capacity of ES-FGM truncated conical shell is reduced considerably when R/h ratio or volume fraction index k increases. v. Critical loads Pcr and qcr of ES-FGM truncated conical shells decrease when the semi-vertex angle increases. vi. Critical pressure q cr decreases in the increase of pre-loaded axial compression

P and critical axial compression Pcr decreases when pre-loaded pressure q increases. vii. Bearing capacity of FGMS stiffened truncated conical shell is better than HS stiffened truncated conical shell. Acknowledgements This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 107.01-2012.02. Appendix I

In Eqs. (9) and (10) C2 =

E2 r br , d1 ( x ) = λ0 x , d2

C1 ( x ) =

C10 , x

C10 =

E2 sbs

λ0

,

d2 =

L , nr

es =

h + hs , 2

λ0 =

2π sin α , ns

er =

h + hr , 2

A11 = A22 =

E1 , 1 −ν 2

A12 =

ν E1 , 1 −ν 2

A66 =

E1 , 2 (1 + ν )

B11 = B22 =

E2 , 1 −ν 2

B12 =

ν E2 , 1 −ν 2

B66 =

E2 , 2 (1 + ν )

24

D11 = D22 =

E3 , 1 −ν 2

D12 =

ν E3 , 1 −ν 2

D66 =

E3 , 2 (1 + ν )

in which ns , nr are the number of stringer and ring respectively; hs and bs are the thickness and width of stringer (x-direction); hr and br are the thickness and width of ring ( θ -direction). Also, d1 = d1 ( x ) and d 2 are the distance between two stringers and two rings, respectively. The quantities es , er represent the eccentricities of stiffeners with respect to the middle surface of shell (Fig. 1).

1) Conical shell with ceramic outside surface and metal inside surface h/2

h/2

E h kEcm h 2 E1 = ∫ Esh dz = Em h + cm , E2 = ∫ zEsh dz = , k +1 2(k + 1)(k + 2) −h / 2 −h / 2 h/2

 1 1 1  Em h3 E3 = ∫ z Esh dz = + Ecm h3  − + . 12  4(k + 1) k + 2 k + 3  −h / 2 Case 1: Outside stiffener 2

h / 2+ hs

E1s =

∫

Es dz = Ec hs + Emc

∫

zEs dz =

h/2 h / 2+ hs

E2 s =

h/2 h / 2+hs

∫

E3s =

 1 hs Ec 1  h  hhs  s + 1 + Emc hs h  + , 2 + 2 2 + 2 k h k h   2  2

 1 Ec 3  3 h2 3 h  h2  1 h 1 3 , z Esdz = hs  2 + +1 + Emchs  + + 2 3  4 hs 2 hs   k2 + 3 k2 + 2 hs 4 ( k2 + 1) hs  2

h/ 2 h / 2 +hr

E1r =

∫

Er dz = Ec hr + Emc

∫

zEr dz =

h/ 2 h / 2 + hr

E2 r =

hs , k2 + 1

h/2

hr , k3 + 1

 1 hr Ec 1  h  hhr  r + 1 + Emc hr h  + , 2 h   k3 + 2 h 2k3 + 2 

 1 Ec 3  3 h2 3 h  1 h 1 h2  3 . h + + 1 + E h + +    r mc r 2 ∫ 3  4 hr2 2 hr   k3 + 3 k3 + 2 hr 4( k3 +1) hr  h/2 Case 2: Inside stiffener h / 2+hr

E3r =

z 2 Er dz =

E1s = Em hs + Ecm

hs , k2 + 1

 hs 2 hs h + hs 2 hs h  E2 s = − Em − Ecm  + , 2  k 2 + 2 2 k2 + 2 

25

E3 s = Em

 h3 3hs h 2 + 6hs 2 h + 4hs 3 h 2h h h2  + Ecm  s + s + s , 12  k2 + 3 k2 + 2 4k 2 + 4 

E1r = Em hr + Ecm

hr , k3 + 1

 hr 2 hr h + hr 2 hh  E2r = − Em − Ecm  + r , 2  k3 + 2 2k3 + 2  E3r = Em

 h3 3hr h2 + 6hr 2h + 4hr 3 h 2h h h2  + Ecm  r + r + r , 12  k3 + 3 k3 + 2 4k3 + 4 

k 2 = k3 = k . 2) Conical shell with metal outside surface and ceramic inside surface

E1 = Em h + Ecm E2 =

h , k +1

−kh2 Ecm , 2 ( k + 1)( k + 2 )

h3 1 1   1 E3 = Em + Ecm h 3  − + . 12  k + 3 k + 2 4k + 4 

Case 3: Outside stiffener E1s = Em hs + Ecm

hs , k2 + 1

E2 s = Em

 h2 hs h + hs 2 hs h  + Ecm  s + , 2 + 2 2 + 2 k k 2 2  

E3s = Em

 h3 3hs h2 + 6hs 2h + 4hs 3 h 2h h h2  + Ecm  s + s + s , 12  k 2 + 3 k 2 + 2 4k 2 + 4 

E1r = Em hr + Ecm

hr , k3 + 1

E2r = Em

 h2 hr h + hr 2 hh  + Ecm  r + r , 2  k3 + 2 2k3 + 2 

E3r = Em

 h3 3hr h 2 + 6 hr 2 h + 4hr 3 h 2h h h2  + Ecm  r + r + r , 12 + 3 + 2 4 + 4 k k k 3 3 3  

k 2 = k3 = k . 26

Case 4: Inside stiffener E1s = Ec hs + Emc

hs , k2 + 1

 hs 2 hs h + hs 2 hs h  E2 s = − Ec − Emc  + , 2  k2 + 2 2k 2 + 2  E3 s = Ec

 h3 3hs h 2 + 6 hs 2 h + 4hs 3 h 2h h h2  + Emc  s + s + s , 12  k 2 + 3 k 2 + 2 4 k2 + 4 

E1r = Ec hr + Emc E2r = − Ec

hr , k3 + 1

 h2 hr h + hr 2 hh  − Emc  r + r , 2  k3 + 2 2k3 + 2 

 hr 3 3hr h 2 + 6hr 2 h + 4hr 3 hr 2 h hr h 2  E3r = Ec + Emc  + + , 12  k3 + 3 k3 + 2 4 k3 + 4  k2 = k3 = 1/ k .

Appendix II

In Eqs. (18)-(20)

 1 E1s bs  ∂ 2 ∂2 ∂  E b F11 =  A11 x + A66 2 + A11 −  A22 + 1r r  2+ 2 d2 λ0  ∂x x sin α ∂x  ∂θ  F12 =

1  , x

∂2 1 1  E1r br  ∂ − , ( A12 + A66 )  A22 + A66 +  ∂x∂θ x sin α  sin α d 2  ∂θ

(

F13 = − B11 x

+ C10

)

 E b  ∂3 1 ∂3 1 − − cot α  A22 + 1r r  ( B12 + 2 B66 ) 3 2 2 x d2  ∂x x sin α ∂x∂θ 

∂2 ∂2  1 1  ∂ + 2 2 ( B12 + 2 B66 + B22 + C2 ) 2 − B11 2 +  ( B22 + C2 ) + A12 cot α  , x sin α ∂θ ∂x  x  ∂x

∂2 1 1  E1r br F21 = + ( A12 + A66 )  A22 + A66 + ∂x∂θ x sin α  sin α d2 F22 =

 ∂ ,   ∂θ

1  E1r br  ∂ 2 ∂2 ∂ 1 A xA + + + A66 − A66 ,  22  2 66 2 2 d2  ∂θ ∂x x x sin α  ∂x

27

∂3 ∂3 1 1 F23 = − ( B12 + 2 B66 ) 2 − 2 3 ( B22 + C2 ) 3 sin α ∂x ∂θ x sin α ∂θ

−

 ∂2 1 1 E b ( B22 + C2 ) cot α  A22 + 1r r + ∂x∂θ x sin α x sin α d2 

(

F31 = B11 x + C10

) ∂∂x

3 3

+

 ∂  ,  ∂θ

∂3 ∂2 ∂2 1 1 B + 2 B + 2 B + B + C ( ) ( ) 12 66 11 22 2 x sin 2 α ∂x∂θ 2 ∂x 2 x 2 sin 2 α ∂θ 2

 E b 1 1 1   ∂ −  A12 cot α + ( B22 + C2 ) + 2 ( B22 + C2 ) − cot α  A22 + 1r r x x d2   ∂x x 

 , 

∂3 ∂3 ∂2 1 1 1 B + 2 B + ( B + C ) − B + C ( 12 ( 22 2 ) 66 ) 22 2 sin α ∂x∂θ ∂x 2∂θ x 2 sin 3 α ∂θ 3 x sin α  1  1 E b  ∂ cot α  A22 + 1r r   , + 2 ( B22 + C2 ) − x sin α d2   ∂θ   x sin α

F32 =

 E b  ∂4 1 F33 = −  D11 x + 3 s s  4 − 3 4 λ0  ∂x x sin α  +

 E3r br  ∂ 4 2 ∂4 D + − D + 2 D ( )  22  12 66 d 2  ∂θ 4 x sin 2 α ∂x 2 ∂θ 2 

 ∂2 ∂3 ∂3  1  2 E3r br  D + 2 D − 2 D + D + + 2 B cot + xK α ( )   2   12 66 11 22 12 2 d2  x 2 sin 2 α ∂x∂θ 2 ∂x3  x   ∂x

 2 2 +  2 2 cot α ( B22 + C2 ) − 3 2 x sin α  x sin α

 E3r br  D12 + 2 D66 + D22 + d2 

 K2  ∂2 +  2  2  x sin α  ∂θ

∂ 1  E b  ∂ E b  1 1 − 2  D22 + 3r r  + 2 cot α ( B22 + C2 ) − cot 2 α  A22 + 1r r  − xK1 , d 2  ∂x x x d2  ∂x x    x2 ∂2 ∂ 1 ∂2  F34 = − tan α  + x + , 2 ∂x sin 2 α ∂θ 2   2 ∂x + K2

∂2 1 F35 = − . π sin 2α ∂x 2

Appendix III

In Eq. (25)  ( x + L )4 − x 4 3L3 ( 2 x + L )  π 3 m2 E b  ( x + L )3 − x3 L3  0 0 0 0 0 1s s     s11 = − sin α A + − sin α + 11 4 3 2 L2 4 m2π 2  2 L2 λ0 2m 2 π 2     

π 3 m2

−

π n2 16sin α

A66 L ( 2 x0 + L ) −

π

E1r br  π  A22 +  sin α L ( 2 x0 + L ) + A11 (sin α ) L ( 2 x0 + L ) , 4 d2  4

28

 ( x + L )3 − x3  E1r br L3  L2 n  0 − s12 = − + + A66  , ( A12 + A66 )  0  A22 + 2 2 4L 3 d2 2m π  8 m    

π 2 mn

 ( x + L ) − x3 π 2 mn2 π 2m π 2m L3  0 0 sin α ( B22 + C2 )( 2 x0 + L ) + sin α .cot α . A12  + 2 2 ( B12 + 2B66 )(2x0 + L) + 16sin α 4 L 6  4m π  3

s13 =

 4 3 4 3 3 4 3    ( x0 + L ) − x0 3L ( 2 x0 + L )  π m ( x + L ) − x0 π m L3   + sin α B11  + + + (sin α ) C10  0 3 2 2 3 4 3 2L 4m π 2 m 2π 2    2L      4

−

s21 = −

s22 = −

3

π 2m

B11 sin α ( 2 x0 + L ) +

4

π 2 mn 12 L

( A12 + A66 ) ( x0 + L )3 − x03  +

π n2  E1r br  A22 + 16sin α  d2

−

π 3m 2 2

8L

E1r br  L2   A22 +  cot α .sin α 4m  d2 

,

 nL2 nL2  E b ( A12 + A66 ) −  A22 + 1r r + A66  , 8m 8m  d2 

 π  L ( 2 x0 + L ) − A66 (sin α ) L ( 2 x0 + L ) 2 

3π 4 A66 (sin α ) ( x0 + L ) − x04  + A (sin α ) L ( 2 x0 + L ) ,   8 66

 ( x + L ) − x3 π n 3 B22 + C2 π 3 m2 n L3  0 0   L + B + 2 B − ( ) 12 66 16 sin 2 α 6 2 L2 4 m 2π 2    3

s23 =

+ s31 = −

πn

E1r br  π nL ( B22 + C2 ) ,  A22 +  ( cot α ) L ( 2 x0 + L ) + 8  d2  8

π 3m2 L2

(

)

 L 3 L4  n 2 B22 + C2 2 B22 + C2 3 B11 sin α  x03 − ( x0 + L ) + L − (sin α ) L2 + 3 3  2 m π m 16 sin α 4 m 4 m π  

2  ( x + L )5 − x 5  E b  L ( 2 x0 + L ) π 4 m3 L2 3L5  3 0  +  A22 + 1r r  (cot α )sin α + 3 (sin α ) B11  0 + x03 − ( x0 + L ) + 2 2 d2  4m L 10 2m π 4m 4π 4    

(

+

+

 ( x + L )4 − x 4 0 0

π 4 m3

(sin α ) C10   

L3

8

−

)

3 3L3 ( 2 x0 + L )  π 2 mn 2 ( B12 + 2 B66 )  ( x0 + L ) − x03 L3     + . − 2 2 4L sin α 6   8m π 4 m 2π 2    

 ( x + L )4 − x 4 3L3 ( 2 x + L )   ( x + L )3 − x 3 π 2m L3  0 0 0 0  , A12  − (cot α ) sin α + (sin α ) B + C − ( 22 2 )  0 2 2 L 8 L 6    8m π 4 m 2π 2     

π 2m

s32 = −

πn

+

s33 =

+

4

( B22 + C2 ) L ( 2 x0 + L ) +

π 3 m2 n 2 L2

 ( x + L )4 − x 4 0 0

( B12 + 2 B66 )  

E1r br π n3 B22 + C2 πn L ( 2 x0 + L ) +  A22 + 2 32 sin α 2  d2

8

−

3L3 ( 2 x0 + L )   8 m 2π 2 

3   ( x0 + L ) − x03 L3    cot α , −  6 4m 2π 2    

)

(

 L  E b  π n 2 L D12 + 2 D66 π 4 m3 3L4  π L 3 + 3 D11 sin α  x03 − ( x0 + L ) + + sin α  D22 + 3r r  3 3 8 sin α d2  L 4m π  4   2mπ

π 4

( B22 + C2 ) L ( 2 x0 + L ) cot α sin α −

π 5 m4 L4

 ( x + L )5 − x5 L2 3L5  3 0 0  sin α .D11  + x03 − ( x0 + L ) + 2 2 10  2m π 4 m 4π 4   

(

29

)



 ( x + L ) − x 4 3L3 ( 2 x + L )  E3r br  π 5 m 4 E3s bs π n4 L  0 0  0 − sin α −  D22 +  4 2 2 λ0 8 d2  L  8m π  32 sin 3 α  4

−





3  ( x + L )3 − x 3 E3 r br  π 3 m2 n 2 D12 + 2 D66  ( x0 + L ) − x0 L3  π 3 m2  L3  0 0    − − − 2  D22 + −  sin α  2 2 2 sin α 6 d2  6   2L 4m π  L  4 m 2π 2  3



+

−





 E3 r br  E1r br π n2 L   D12 + 2 D66 + D22 +  − π  A22 + 4sin α  d2  d2 

π 3 m 2 B12 2

4L

(cos α ) ( x0 + L) 4 − x04  +

3   ( x0 + L ) − x03 L3  2  cot α sin α −  6 4 m 2π 2    

2 3πL B12 (cosα )(2 x0 + L) − π n ( B22 + C2 )(cot α ) L(2 x0 + L) 8sin α 4

4 3  L 3L4 ( 2x0 + L )  π n2 tan α  ( x0 + L ) − x0 3L ( 2x0 + L)  π 2 m 4 4   − − tan α sin α x − x + L +   ( ) 0 0 4sin α  8 8m2π 2  2L 2m3π 3   2mπ 4

s34 =





  ( x + L )6 − x 6 5 L2 4  0 0 tan α sin α + x04 − ( x0 + L )  2 2 2 12 2L  8 m π  

s35 =

(

π 3m 2

+

)

(



 15 L5 ( 2 x + L )  0 + ,  8 m 4π 4  

)

 ( x + L )4 − x 4 3 L3 ( 2 x + L )  0 0  0 . − 8 2 L2 cos α  8m 2π 2  

π 2 m2

 (x + L )5 − x05 L2 3L5  3 3 s36 = −π sin α  0 + x − ( x + L ) + . 0 0 10 2 m 2π 2 4m 4π 4  

(

s37 =

)

 L 3L4  3 sin α  x03 − (x0 + L ) +  2L 4m 3π 3   2 mπ

π 2m

−

(

)

π 3m 2 L2

 (x + L )5 − x05 L2 3 L5  3 3 sin α  0 + x − ( x + L ) + . 0 0 10 2 m 2π 2 4m 4π 4  

(

)

References

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