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

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

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

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.

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Nonlinear thermal stability of eccentrically stiffened functionally graded

truncated conical shells surrounded on elastic foundations

Nguyen Dinh Duc*, Pham Hong Cong

Vietnam National University, Hanoi -144 Xuan Thuy-Cau Giay- Hanoi-Vietnam

Keywords: Thermal stability, Eccentrically stiffened truncated conical shell, Functionally graded materials, elastic foundations.

1 Introduction

The idea of the construction of functionally graded meterials (FGM) was first introduced in 1984 by a group of Japanese materials scientists (Koizumi, 1997) Due to high performance heat resistance capacity and excellent characteristics of FGM in comparison with conventional composites, functionally graded shells involving conical shells are widely used in exhaust nozzle of solid rocket engine, some important details of space vehicles, aircrafts, nuclear power plants and many other engineering applications As a results stability analysis of those strutures are very important problems and have attracted increasing research effort

Pratically, the composite plates and shells usually are reinforced by stiffening components to provide the benefits of added load-carrying static and dynamic capability with a relatively small additional weight There have had some publications

on the buckling of composite shells reinforced by stiffeneres: a free vibration analysis for a ring-stiffened simply supported conical shell by considering an equivalent

to investigate the nonlinear dynamic of imperfect eccentrically stiffened FG shallow shells taking into account the damping subjected to mechanical loads (Bich et al., 2013), the study of instability of eccentrically stiffened functionally graded truncated conical shells under mechanical loads and shells are reinforced by stringers and rings (Dung et al 2013), the stability of functionally graded truncated conical shells reinforced by functionally graded stiffeners, surrounded by an elastic medium and under mechanical loads (Dung et al., 2014)

From the above review, to the best of our knowledge, it has showed that there is

no publiation about buckling of FGM conical shell with stiffeneres in thermal environment Under temperature, both of the FGM shell as well as the stiffeners are 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 environments Duc and Cong (2014) also investigated the nonlinear postbuckling of 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

2 Eccentrically stiffened functionally graded (ES-FGM) truncated conical shell

surrounded on elastic foundations

Consider a thin truncated conical shell of thickness h and semi-vertex angle

β, the geometry of shell is shown in Fig 1, in which L is the length, R is its small 1

base radius and R H geometrical parameters as shown in Fig 1 The truncated cone ,

is referred to a curvilinear coordinate system (x, ,θ z) whose the origin is located in the middle surface of the shell, x is in the generatrix direction measured from the

vertex of conical shell, h is in the circumferential direction and the axes z being

perpendicular to the plane ( )x h , lies in the outwards normal direction of the cone ,Also, x indicates the distance from the vertex to small base, and ,0 u v and w denote

the displacement components of a point in the middle surface in the direction x h and ,

z, respectively; h and 1 b are the thickness and width of stringer ( x -direction); 1 h 2

and b are the thickness and width of ring (2 θ-direction) Also, d1 =d x1( )and d are 2

the distance between two stringers and two rings, respectively z z represent the 1, 2

eccentricities of stiffeners with respect to the middle surface of shell

The effective properties of the FGM truncated conical shell (the elastic modulusE, the Poisson ratioν , the thermal expansion coefficient α) can be written as follows (Bich et al., 2011; Dung et al., 2013; Duc and Quan, 2013; Duc and Cong, 2014):

constituents, respectively And the Poisson ratio is assumed to be constant ν =const From Eq (1) we have: E=E m at z= −h/ 2 (metal-rich) and E=E c at z=h/ 2

(ceramic-rich)

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

as the elastic modulusE, the Poisson ratioν, the thermal expansion coefficient α can

be expressed as a nonlinear function of temperature (Touloukian, 1967):

in which T = + ∆T0 T z( ) and T0 = 300K (room temperature); P−1, , , , P P P P0 1 2 3 are

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

3 Eccentrically stiffened FGM truncated conical shell under temperature

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):

2 0

2 0

thermal expantion coefficient of the stiffener in the θ -direction, respectively To guarantee the continuity between the stiffener and shell, the stiffener is taken to be pure-metal if it is located at metal-rich side and is pure-ceramic if it is located at ceramic-rich side (this assumption was proposed by Bich in (Bich et al., 2011)) In order to investigate the FGM truncated conical shell with stiffeners in thermal environment, we have not only taken into account the materials muduli with temperature-dependent properties but also assumed that all elastic moduli of FGM truncated conical shell and stiffener are temperature dependence and they are deformed in the presence of temperature Hence, the geometric parameters, the shell’s shape and stiffeners are varied through the deforming process due to the temperature change We have assumed that the thermal stress of stiffeners is subtle which distributes uniformly through the whole shell structure, therefore, we can ignore it because the size of stiffeners is great smaller than the plates and the gap between every stiffeners is not tight (this assumption first was proposed by Duc et al (2013) and has been used in Duc and Cong (2014))

T r

T r

E z z

T z zdz v

in which α0 is thermal expantion coefficient of the stiffener; n , st n are the number of r

stringer and ring respectively The quantities A A are the cross-section areas of 1, 2stiffeners and I I are the second moments of inertia of the stiffener cross sections 1, 2related to the shell middle surface Although the stiffeners are deformed by

0,sin

1, 1

u v and w to the displacement variables, so the total displacement components of a 1

neighboring state are:

0 1, 0 1, 0 1

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

to the state of equilibrium as:

Eqs (12) and (13) into Eqs (11) and note that the terms in the resulting equations with

subscript 0 statisfy the equilibrium equations and therefore drop out 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 (Dung et al., 2014) :

1

0,sin

E I C

( ) ( ) ( )

/2 0

0

/2 0

0

,0

0

h x

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

0 1

0 1

where m is the number of half-waves along a generatrix and n is the number of

ful-waves along a parallel circle, and X Y Z are constant coefficients Due to , ,

0

2

0 2

0

2

0 3

x d dx L

x d dx L

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

of algebraic Eqs (23) must be set equal to zero as

Once the temperature distribution of the shell is obtained, Eq (24) is integrated

By equating Eqs (24) and (26) the value of the buckling temperature difference is obtained The mininum value of the buckling temperature difference for different values of m and n is called the critical temperature difference

In case of T-D, the two hand sides of Eq (26) are temperature dependence which makes it very difficult to solve Fortunately, we have applied a numerical technique using the iterative algorithm to determine the buckling loads as well as the deflection – load relations in the postbuckling period of the FGM shell More details, given the material parameter N , the geometrical parameter (R / h,1 L / R1) and the value β, we can use these to determine ∆T in (26) as the follows: we choose an initial step for ∆T1 on the right hand side in Eq (26) with ∆ =T 0 (since

0

T=T =300K, the initial room temperature) In the next iterative step, we replace the known value of ∆T1 found in the previous step to determine the right hand side of Eq

a tolerance used in the iterative steps

4.1 Uniform temperature rise

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

critical temperature difference

4.2 Linear temperature distribution through the thickness

If the conical shell is thin enough, a linear temperature distribution across the shell thickness is the first approximation to the solution of the heat conduction equation of the FGM conical shell Thus, we assume (Naj et al., 2008):

2

T T z

In Eq (33), T is used to obtain the buckling temperature difference The a

minimum value of T with respect to m and n is obtained and called the critical a

un-Table 1 Comparisons with result of Naj et al., (2008) for un-stiffened FGM truncated

conical shells under uniform load

N

310

(a) Buckling mode (m,n)

Table 2 Comparisons with result of Naj et al., (2008) for un-stiffened FGM truncated

conical shells under linear load

310

c T cr

R h= , present

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

R h= , present

R h= , Ref (Naj et al., 2008)0

In this section, the components of the material are silicon nitride Si N (ceramic) 3 4

and SUS304 stainless steel (metal) The material properties P in the formula Eq (2) r

is shown in Table 3, Poisson ratio is chosen to be v=0.3

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

(Reddy and Chin, 1998)

5.2.1 Effect of stiffener arrangement and stiffener number

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

is the second, stringer stiffened shell is the third and the critical load values in the stiffened case is smaller than stiffened case In the case of uniform temperature rise, the critical thermal load value is smaller than the case of linear temperature distribution through the thickness

Un-Stringer (n st =30) Ring (n r =30) Orthogonal

(n st =n r =15) Uniform

Table 5 Effect of stiffener number on critical thermal loads ∆T cr( )K

distribution Stiffener number

5.2.2 Effect of semi-vertex angle β

Table 7 Effect of semi-vertex angle β (outside stiffeners and under uniform thermal load)

345 (4,19)

271 (3,19)

218 (3,21)

173 (3,21)

116 (3,19)

99 (3,18)

51 (3,10) Ring

(n r =30)

387 (11,4)

353 (10,15)

274 (9,8)

221 (8,11)

176 (7,12)

118 (6,4)

101 (6,5)

52 (4,8) Orthogonal

(n st =n r =15) 417

(10,8)

365 (9,13)

287 (8,14)

229 (6,19)

181 (6,16)

122 (6,5)

105 (5,4)

53 (4,5)

Table 7 and Fig 2 illustrate the effect of semi-vertex angle β on critical temperature load ∆T cr It can be seen that the critical thermal buckling load of truncated conical shell strongly decreases when semi-vertex angle β increases For example, a orthogonal stiffened shell in Table 7, when the semi-vertex angle varies

150 200 250 300 350 400 450

Outside stiffeners

h1=0.01375m, h2=0.01m

b1=0.0127m, b2=0.0127m

123

h=0.0127m, R1=1.27m, L=2R1, N=1, K1=0, K2=0

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

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 K and 1 K on critical 2

thermal load (without stiffeners) We find that when we increase the value of the