Accepted Manuscript
Nonlinear thermal stability of eccentrically stiffened functionally graded truncated
conical shells surrounded on elastic foundations
Nguyen Dinh Duc , Pham Hong Cong
PII:

S0997-7538(14)00167-3

DOI:

10.1016/j.euromechsol.2014.11.006

Reference:

EJMSOL 3141

To appear in:

European Journal of Mechanics / A Solids

Received Date: 28 February 2014
Revised Date:

1 November 2014

Accepted Date: 7 November 2014

Please cite this article as: Duc, N.D., Cong, P.H., Nonlinear thermal stability of eccentrically stiffened
functionally graded truncated conical shells surrounded on elastic foundations, European Journal of
Mechanics / A Solids (2014), doi: 10.1016/j.euromechsol.2014.11.006.
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to


closed-form expression for determining the thermal buckling load is obtained. The numerical
results show that the critical thermal load in the case of the uniform temperature rise is
smaller than one of the linear temperature distribution through the thickness of the shell, and
the critical thermal load increases when increasing the coefficient of stiffeners and vice versa.
The paper also analyzes and discussed the significant effects of material and geometrical

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properties, elastic foundations on the thermal buckling capacity of the eccentrically stiffened
FGM truncated conical shell in thermal environment. The obtained results are validated by
comparing with those in the literature.

Keywords: Thermal stability, Eccentrically stiffened truncated conical shell, Functionally

1. Introduction

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graded materials, elastic foundations.

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The idea of the construction of functionally graded meterials (FGM) was first
introduced in 1984 by a group of Japanese materials scientists (Koizumi, 1997).
Due to high performance heat resistance capacity and excellent characteristics of
FGM in comparison with conventional composites, functionally graded shells
involving conical shells are widely used in exhaust nozzle of solid rocket engine,

functionally graded truncated conical shells (Sofiyev, 2007), the buckling of thin
truncated conical shells made of FGMs subjected to hydrostatic pressure, uniform
external pressure and uniform axial compressive load (Sofiyev et al. 2004, 2009,
2010a). The shell structures supported by an elastic foundations have been widely
used in many applications such as in aircraft, reusable space transportation vehicles

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and civil engineering. Therefore, studies on the effects of elastic foundations on
behavior and loading capacity of the shells are highly important. The nonlinear
buckling of the truncated conical shell made of FGMs was surrounded by an elastic
medium and Winkler–Pasternak type elastic foundation using the large deformation

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theory with von Karman–Donnell-type of kinematic nonlinearity (Sofiyev, 2010b;
Sofiyev and Kuruoglu, 2013). Najafov and Sofiyev (2013) obtained the nonlinear

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dynamic analysis of FG truncated conical shells surrounded by an elastic medium
using the large deformation theory with von Karman–Donnell-type of kinematic nonlinearity.

Pratically, the composite plates and shells usually are reinforced by stiffening

components to provide the benefits of added load-carrying static and dynamic
capability with a relatively small additional weight. There have had some publications
on the buckling of composite shells reinforced by stiffeneres: a free vibration analysis


From the above review, to the best of our knowledge, it has showed that there is
no publiation about buckling of FGM conical shell with stiffeneres in thermal
environment. Under temperature, both of the FGM shell as well as the stiffeners are

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deformed, therefore, the calculation on the thermal mechanism of FGM shells and
stiffeners has become more difficult. Recently, Duc and Quan (2013) researched the
nonlinear postbuckling for imperfect eccentrically stiffened FGM double curved thin
shallow shells on elastic foundation using a simple power-law distribution in thermal

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environments. Duc and Cong (2014) also investigated the nonlinear postbuckling of

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imperfect eccentrically stiffened thin FGM plates under temperature.
This paper studied the stability of an eccentrically stiffened functionally graded
truncated conical shells surrounded on elastic foundations under thermal loads with
both FGM shell and stiffeners having temperature-dependent properties. Addionally,
the paper analyzed and discussed the effects of material and geometrical properties,
temperature, elastic foundations and eccentrically stiffeners on the buckling and
postbuckling loading capacity of the functionally graded truncated conical shells in
thermal environments.

3

and b2 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. z1 , z2 represent the

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eccentricities of stiffeners with respect to the middle surface of shell.
The effective properties of the FGM truncated conical shell (the elastic
modulus E , the Poisson ratioν , the thermal expansion coefficient α ) can be written as
2014):

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follows (Bich et al., 2011; Dung et al., 2013; Duc and Quan, 2013; Duc and Cong,

2z + h 
( E ,α ) = ( Em ,α m ) + ( Ecm ,α cm ) 
 ,
 2h 
N

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(1)

where Ecm = Ec − Em , α cm = α c − α m , the volume fraction index N is a nonnegative
number that defines the material distribution and can be chosen to optimize the
structural response, and subscripts m and c stand for the metal and ceramic
constituents, respectively. And the Poisson ratio is assumed to be constant ν = const .

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Fig. 1. Eccentrically stiffened FGM truncated conical shell surrounded by an elastic
foundations.

A material property Pr of both FGM truncated conical shell and stiffeners, such

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as the elastic modulus E , the Poisson ratioν , the thermal expansion coefficient α can
be expressed as a nonlinear function of temperature (Touloukian, 1967):
−1
3
Pr = P0 ( P−1T −1 + 1 + PT
+ P2T 2 + PT
),
1
3

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(2)

in which T = T0 + ∆T ( z ) and T0 = 300 K (room temperature); P−1 , P0 , P1 , P2 , P3 are
coefficients characterizing of the constituent materials. ∆T is temperature rise from
stress free initial state, and more generally, ∆T = ∆T ( z ) . In short, we will use T-D
(temperature dependent) for the cases in which the material properties depend on
temperature. Otherwise, we use T-ID for the temperature independent cases. The
material properties for the later one have been determined by Eq. (2) at room

(3)

stiffeness and K 2 (in N / m ) is the shear subgrade modulus of the Pasternak

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foundation model.

3. Eccentrically stiffened FGM truncated conical shell under temperature

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The present study uses the classical shell theory with the geometrical
nonlinearity in von Karman sense and smeared stiffeners technique to establish the
governing equations. Thus, the normal and shear strains at distance z from the middle
surface of shell are (Brush and Almroth, 1975):

ε x = ε x0 + zk x , εθ = εθ0 + zkθ , γ xθ = γ x0θ + 2 zk xθ ,

(4)

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in which ε x0 and ε θ0 are the normal strains and γ x0θ is the shear strain at the middle
surface of the shell, and k x , kθ and k xθ are the change of curvatures and twist,
respectively. They are related to the displacement components as (Brush and Almroth,



kx = −

2

(5)

∂2w
1
∂ 2 w 1 ∂w
,
k
=
−
−
,
θ
∂x 2
x 2 sin 2 β ∂θ 2 x ∂x

1
∂ 2w
1
∂w
k xθ = −
+ 2
.
x sin β ∂x∂θ x sin β ∂θ
Hooke law for an FGM truncated conical shell with temperature-dependent
6


Est
α st ∆T ( z ) ,
1 − 2vst

Er
σ θ = Er εθ −
α r ∆T ( z ) ,
1 − 2vr

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σ xs = Est ε x −

(7)

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s

here, Est = Est (T ) , vst = vst (T ) , α st = α st ( T ) are the Young’s modulus, Poisson ratio
and thermal expantion coefficient of the stiffener in the x -direction, respectively. And

Er = Er (T ) , vr = vr (T ) ,α r = α r (T ) are the Young’s modulus, Poisson ratio and
thermal expantion coefficient of the stiffener in the θ -direction, respectively. To

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account. Integrating the above stress–strain equations and their moments through the

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thickness of the shell, we obtain the expressions for force and moment resultants of an
eccentrically stiffened FGM conical shells (Dung et al., 2013, 2014).

N xθ = A66γ x0θ + 2 B66 k xθ ,

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
Est A1T  0
0
N x =  A11 +
 ε x + A12ε θ +  B11 + C1 ( x )  k x + B12 kθ + Φ a ,
d1 ( x ) 


Er A2T  0
0
Nθ = A12ε x +  A22 +
 ε θ + B12 k x + [ B22 + C2 ] kθ + Φ a ,
d2 



(8a)

(8b)


ACCEPTED MANUSCRIPT
E1 = E m h +
E3 =

E cm h
1 
 1
, E 2 = E cm h 2 
−
,
N +1
 N + 2 2N + 2 

1
1
1 
 1
E m h 3 + E cm h 3 
−
+
,
12
 N + 3 N + 2 4N + 4 


− h/2

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h/2

∫

(9)

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h/2

Φb = −

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A1T = b1T h1T , A2T = b2T h2T , z1T =

2
2
1 T T 3
1 T T 3
b1 ( h1 ) + A1T ( z1T ) , I 2T =
b2 ( h2 ) + A2T ( z 2T ) ,
12

E3
vE 3
E3
, D12 =
, D66 =
,
2
2
1− v
1− v
2 (1 + v )

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B11 = B22 =

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After the thermal deformation process, the geometric shapes of stiffeners can be
determined as follows:

h1T = h1 (1 + α 0 ∆T ( z ) ) , h2T = h2 (1 + α 0 ∆T ( z ) ) ,

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z1T = z1 (1 + α 0 ∆T ( z ) ) , z2T = z2 (1 + α 0 ∆T ( z ) ) ,

b1T = b1 (1 + α 0 ∆T ( z ) ) , b2T = b2 (1 + α 0 ∆T ( z ) ) ,

(11a)

∂N
1 ∂Nθ
+ x xθ + 2 N xθ = 0,
sin α ∂θ
∂x

(11b)

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2  ∂ 2 M xθ 1 ∂M xθ 
∂2M x
∂M x
2
+
+
+

+
sin α  ∂x∂θ
x ∂θ 
∂x 2
∂x
1
1
∂ 2 M θ ∂M θ
∂w
∂w 

1 
∂w
1
∂w 
+
Nθ
 N xθ
 − xK1w + xK 2 ∆w = 0,
sin α 
∂x x sin α
∂θ ,θ

The stability equations of conical shell are derived using the adjacent

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equilibrium criterion (Brush and Almroth, 1975; Naj et al., 2008). Assume that the
equilibrium state of ES-FGM conical shell under thermal 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

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neighboring state are:

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

(12)

sin α ∂θ

(14a)

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x

∂N
1 ∂Nθ 1
+ x xθ 1 + 2 N xθ 1 = 0,
sin α ∂θ
∂x

(14b)

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∂ 2 M x1
∂M x1
2  ∂ 2 M xθ 1 1 ∂M xθ 1 
x
+2
+
+

+
∂x 2
∂x

∂w1 
+
Nθ 0
 N xθ 0
 − xK1w + xK 2 ∆w = 0.
sin α 
∂x x sin α
∂θ ,θ

where the force and moment resultants for the state of stability are given by (Naj et al.,
2008):

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

E AT 
C0 
N x1 =  A11 + st 1  ε x01 + A12ε θ01 +  B11 + 1  k x1 + B12 kθ 1 ,
λ0 x 
x 



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
E AT 
Nθ 1 = A12ε x01 +  A22 + r 2  ε θ01 + B12 k x1 + [ B22 + C2 ] kθ 1 ,

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0
1

T
st 1


Er I 2T
M θ 1 = B ε + [ B22 + C2 ] ε + D12 k x1 +  D22 +
d2

0
12 x1

0
θ1


 kθ 1 ,


M xθ 1 = B66γ x0θ 1 + 2 D66 k xθ 1 ,
and the linear form of the strains and curvatures in terms of the displacement
components are:
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x sin β ∂θ 2 x ∂x
1 ∂ 2 w1
1
∂w1
+ 2
.
x sin β ∂x∂θ x sin β ∂θ

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For simplicity, the membrane solution of the equilibrium equations are
considered (Meyers and Hyer, 1991; Naj et al., 2008). For this aim, all the moment

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and rotation terms must be set equal to zero in the equilibrium equations. By solving
the membrane form of equilibrium equations, it is found that

x +L
=− 0
∆T ( z )E ( z ) α ( z ) dz,
x − h∫/2
h /2

N x0

Nθ 0 = 0.
N xθ 0 = 0.


4. Thermal buckling analysis of ES-FGM truncated conical shell
In this section, an analytical approach is given to investigate the thermal stability
of ES-FGM truncated conical shells. Assume that a shell is simply supported at both
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ACCEPTED MANUSCRIPT
ends. The boundary conditions in this case, are expressed by (Dung et al., 2013, 2014)

v1 = w1 = 0, M x1 = 0 at x = x0 , x0 + L.

(19)

The approximate solution Eqs. (18) satisfying the boundary conditions (19) may
be described as

mπ ( x − x0 )
nθ
sin ,
2
L
mπ ( x − x0 )
nθ
v1 = Y sin
cos ,
2
L
mπ ( x − x0 )
nθ

0

∫ ∫∆

sin

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

x0

3

sin

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

0

x0 + L 2π



∆1 = x C11 ( u1 ) + C12 ( v 2 ) + C13 ( w1 )  ,
∆ 2 = x C21 ( u1 ) + C22 ( v 2 ) + C23 ( w1 )  ,
∆ 3 = x 2 C31 ( u1 ) + C32 ( v1 ) + C33 ( w1 ) + C34 N x 0 ( w1 )  .

(22)

Substituting expressions (20) and (22) into Eq. (21), after integrating longer and

some rearrangements, may be written in the following form

d11 X + d12Y + d13 Z = 0,

(23a)

d 21 X + d 22Y + d 23 Z = 0,

(23b)
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d31 X + d32Y + ( d33 + d34 N x' 0 + d35 K1 + d36 K 2 ) Z = 0,

(23c)

in which

N


d13
d 23

d31

d32

d33 + d35 K1 + d36 K 2 + d34 N ' x 0

N 'x0 =

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Eq. (25) may be expressed as

= 0.

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To derive the thermal buckling force for the conical shell, the coefficient matrix

d31 ( d12 d 23 − d 22 d13 ) − d32 ( d11d 23 − d 21d13 ) d33 + d 35 K1 + d36 K 2
−
.
d34 ( d 21d12 − d11d 22 )
d34

(25)

ACCEPTED MANUSCRIPT
(26) ∆T2 . This iterative procedure will stop at the k th - step if ∆Tk satisfies the
condition | ∆T − ∆Tk |≤ ξ . Here, ∆T is a desired solution for the temperature and ξ is
a tolerance used in the iterative steps.

4.1. Uniform temperature rise

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Consider a conical shell under uniform temperature rise, temperature was
increased steadily from the first value to the last value, the difference in temperature

∆T = T f − Ti is a constant and does not consider the transfering of heat in conical
shell. After substituting ∆T in Eq. (24) the prebuckling force is obtained as

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N
N

 2z + h   
 2z + h  
= − ( L + x0 ) ∫ ∆T  Em + Ecm 
  ×  α m + α cm 
  dz =

 2h   
 2h  
− h /2

∆T = −

d31 ( d12 d 23 − d 22 d13 ) − d32 ( d11d 23 − d 21d13 ) d33 + d35 K1 + d36 K 2
+
d34 ( d 21d12 − d11d 22 ) ( L + x0 ) P1
d 34 ( L + x0 ) P1

(28)

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

1
1 
 1
in which P1 = h  Emα m +
Emα m + 
+
 Ecmα cm  .
N +1
 N + 1 2N + 1 


Eq. (28) gives the buckling temperature difference for a conical shell under

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uniform thermal rise. The minimum value of with respect to m and n is called the


 2 z + h  
 2z + h  
= − ( L + x0 ) ∫  ∆T +
α + α cm 
 
 dz. (30)
 ×  Em + Ecm 
 m
h
2  
 2h  
 2h  
− h /2 
h /2

N

'
x0

Eq. (30) may be written as

By considering Tb = 0 , Eq. (31) reduces to

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1

1 

1

N x' 0 = −∆Th ( L + x0 )  Emα m +
Emα cm +
Ecmα m +
Ecmα cm 
N +2
N +2
2N + 2
2

1
1
1


Emα cm +
Ecmα m +
Ecmα cm  .
−Ta h ( L + x0 )  Emα m +
N +1
N +1
2N + 1



(32)

Setting Eq. (32) equal to Eq. (26), Ta is derived where


In Eq. (33), Ta is used to obtain the buckling temperature difference. The

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minimum value of Ta with respect to m and n is obtained and called the critical
temperature difference ∆Tcr .

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5. Results and discussion
5.1. Comparison results

To evaluate the reliability of the method used in the paper. Consider an FGM
truncated conical shells un-stiffened and not resting on elastic foundation with the
geometric parameters and materials were as follows:

Em = 200GPa,α m = 11.7 × 10−6 1 / 0 C ,
Ec = 380GPa,α m = 7.4 × 10−6 1 / 0 C ,

(34)

v = 0.3,
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ACCEPTED MANUSCRIPT
h = 0.01m, β = 100 , H / R = 1, K1 = 0, K 2 = 0.
Tables 1, 2 compare the present result with those of Naj et al. (2008) for unstiffened FGM truncated conical shells. The result in the two tables compared shows


2.44(1,17)

2.43

1.20(11,18)

1.24

1

2.22(9,13)

2.22

1.07(11,18)

1.08

5

1.95(8,1)

1.92

0.97(11,13)

0.99

∞


present

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α c ∆Tcr × 103

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conical shells under linear load.

R / h = 200 ,
Ref. (Naj et al.,
2008)

R / h = 400 ,
present

R / h = 400 ,
Ref. (Naj et al.,
2008)

Ta = 0

4.16(4,26)(a)

4.17

2.09(2,27)



Si3 N 4

0

348.43 × 109

−3.070 × 10−4

2.160 × 10−7

−8.946 × 10−11

SUS 304

0

201.04 × 109

3.079 × 10−4

−6.534 × 10−7

0

Si3 N 4

0

5.8723 × 10−6


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Table 3. Material properties of the constituent materials of the considered FGM shells

8.086 × 10−4

5.2.1. Effect of stiffener arrangement and stiffener number

The parameters for the stiffeners and the geometric parameters were chosen as

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below:

h = 0.012m, R1 = 1.27 m, L = 2 R1 , h1 = 0.01375m, h2 = 0.01m, b1 = 0.0127 m,
b2 = 0.0127 m, N = 1, β = 30o.

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Table 4 shows the critical temperature value ( ∆Tcr ) in two temperature field
uniform linear temperature rise and temperature distribution through the thickness.

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The value of the critical temperature in the case of stiffeners inside smaller than in the
case of external stiffeners. With the same number of stiffeners ( ns = 30 ) , the critical


205(8,9)

214(3,20)

215(8,17)

220(7,15)

Outside

356(9,1)

386(3,21)

397(7,16)

404(6,18)

Inside

356 (9,1)

378(3,20)

379(9,1)

388(6,18)

Buckling mode (m,n).


Stiffener number

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Uniform

Outside

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( nst = nr )

Linear temperature
distribution

Inside

Outside

Inside

214(8,13)

211(8,5)

376(6,18)

370(6,19)


50

245(8,6)

231(8,8)

431(6,18)

405(6,18)

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10 ( nr = 5, nst = 5 )

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ACCEPTED MANUSCRIPT
Table 6. Effect of stiffener number on critical thermal load ∆Tcr ( K ) , (outside
stiffener).
Uniform

Linear temperature distribution

Stiffener

Stringer


386(3,21)

397(7,16)

40

221(3,20)

225(9,8)

391(3,20)

399(8,11)

Stringer

Ring (nr = ns )

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(nst = ns )

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Ring (nr = ns )

5.2.2. Effect of semi-vertex angle β

load)


600

750

345

271

218

173

116

99

51

(3,17)

(4,19)

(3,19)

(3,21)

(3,21)

(3,19)


(8,11)

(7,12)

(6,4)

(6,5)

(4,8)

417

365

287

229

181

122

105

53

(10,8)

(9,13)

example, a orthogonal stiffened shell in Table 7, when the semi-vertex angle varies

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ACCEPTED MANUSCRIPT
the values from 50 to 750 , the critical thermal load ∆Tcr decreases from 417K to 53K,
about 87.3%.
Graphically, the effect of semi-vertex angle β on critical temperature ∆Tcr is
plotted in Fig. 2. They also show that critical load ∆Tcr decrease when β increase and

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the critcal load ∆Tcr - semi-vertex angle β curve for a orthogonal stiffened shell is
the highest.
450

1: Stringer, nst=30
2: Ring, nr=30

3
400

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2
1

3: Orthogonal, nst=nr=15

10

15

20

25

30

35

40

β (Degree)

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Fig. 2. Variation of the critical thermal difference versus β for ES-FGM conical
shells under uniform thermal load.

5.2.3. Effect of elastic foundations

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Effect of foundation on critical thermal load of ES-FGM conical shells under
uniform thermal load are show in Tables 8, 9 and Fig. 4.
Table 8 analyzes the influence of background factors K1 and K 2 on critical
thermal load (without stiffeners). We find that when we increase the value of the

Table 9 shows the value of the ground coefficient K 2 which affects the critical
greater than the ground coefficient

K1 = 0 N / m3 , K 2 = 2.5 × 105 N / m

(Stringers)

results

K1 = 2.5 × 107 N / m3 , K 2 = 0 N / m (Stringers) results

K1 . For example

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∆Tcr

thermal load

∆Tcr = 240 K

and

∆Tcr = 237 K . And Table 9

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shows the critical thermal load of stiffened shell by stringers is the biggest, the
stiffend shell by orthogonal is the second and the critical thermal load of the stiffend


2 × 105

210(8,5)

214(9,4)

216(8,5)

220(8,8)

3.5 × 105

212(8,9)

215(8,9)

218(8,7)

222(9,4)

217(8,5)

221(9,6)

223(8,5)

228(8,5)

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K1 = 0 N / m3 , K 2 = 2.5 × 105 N / m

240(8,5)

224(9,6)

232(8,6)

K1 = 2.5 × 107 N / m3 , K 2 = 2.5 × 105 N / m

244(8,6)

229(8,4)

237(8,7)

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ACCEPTED MANUSCRIPT
K1 = 2.5 × 107 N / m3 , K 2 = 0 N / m

237(7,12)

222(9,4)

233(7,15)

5.2.4. Effect of the volume fraction index N
The parameters for the stiffeners and the geometric parameters were chosen as

101

5

102

5

557(8,16)

525(8,15) 420(8,16) 381(8,16) 369(8,15) 358(8,16) 341(8,15)

10

494(9,14)

464(8,15) 369(8,16) 334(8,15) 324(9,13) 313(9,12) 298(9,12)

30

320(8,8)

300(8,7)

203(8,10)

196(8,9)

1: β=5o
2: β=10o


10-1

400
350
300

h=0.0127m, R1/h=100, L=2R1

250
200
150

0

2

4

6
N

23

8

10

186(8,2)



TE
D

then make the ability of heat load low. The relationship between the ratio R1 / h and
the critical temperature in both stiffined case and un-stiffined case is also shown in
Fig. 4. It can be seen that in the absence of un-stiffined case will make the curve

AC
C

EP

becomes lower than stiffined case.

24