Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

51, 1, pp. 143-158, Warsaw 2013

THREE-DIMENSIONAL ASYMMETRIC THERMO-ELASTIC ANALYSIS OF
A FUNCTIONALLY GRADED ROTATING CYLINDRICAL SHELL

R. Akbari Alashti, M. Khorsand, M.H. Tarahhomi

Babol University of Technology, Mechanical Engineering Department, Babol, Iran

e-mail: raalashti@nit.ac.ir

In this paper, asymmetric deformation and stress analysis of a functionally graded hollow
cylindrical shell under the effect of thermo-mechanical loadsusing thedifferential quadrature
method is carried out. Without losing the generality, material properties of the cylindrical
shell are assumed to be graded in the radial direction obeying a power law,while thePoisson
ratio is assumed to be constant. The governing partial differential equations are expressed
in terms of displacement and thermal fields in series forms with the help of two versions of
differential quadrature methods, namely the polynomial and Fourier quadrature methods.
The cylindrical shell is considered under both axisymmetric and asymmetric loading con-
ditions. Numerical results for the axisymmetric loading condition of the cylindrical shell
graded according to a power law function are obtained and compared with exact solutions
which are found to be in very good agreement. Asymmetric thermo-elastic analysis of the
shell rotating at a constant angular velocity ismade and the effect of the gradingparameter,
angular velocity, temperature difference and geometry on stresses, radial displacement and
temperature fields are presented.

Key words: functionally graded material, rotating cylindrical shell, differential quadrature,
thermoelasticity

1. Introduction

A functionally graded material (FGM) is an inhomogeneous composite material at microsco-
pic scale with the composition function that is designed to vary continuously within the solid.
The mechanical and thermal responses of materials with spatial gradients in composition are
of considerable interests in numerous industrial applications such as tribology, biomechanics,
nanotechnology and high temperature technologies. Thesematerials are usuallymixtures ofme-
tal and ceramic which exhibit excellent thermal resistance with low levels of thermal stresses.
Thesematerials were first introduced by some Japanese Scientists in 1984 as ultra light tempe-
rature resistant materials for space vehicles (1993). Suresh andMortensen (1998) presented the
fundamentals of FGmaterials.
Two-dimensional higher-order deformation theory for the evaluation of displacements and

stresses inFGplates subjected to thermal andmechanical loadingswas presented byMatsunaga
(2009). He employed the method of power series expansion of displacement components and
several sets of truncated approximations to solve the static boundary value problem. Wu et
al. (2005) investigated thermal buckling in the axial direction of cylindrical shells made of FG
materials based on Donnell’s shell theory and obtained their closed form solutions. Sofiyev
(2007) presented closed form solutions of the thermal buckling loads of truncated conical shells
made of an FG material, using the modified Donnell type dynamic stability and Galerkin’s
method. Pelletier and Vel (2006) analyzed the steady state response of an FG thick cylindrical
shell subjected to thermal and mechanical loads and simply supported boundary conditions
employing the power series method to solve the ordinary differential equations. Arciniega and
Reddy (2007) presented geometrically nonlinear analysis of anFGshell using the first-order shell



144 R. Akbari Alashti et al.

theory and high-order Lagrangian interpolation functions. Bahtui and Eslami (2007) employed
the second-order shear deformation shell theory, the Galerkin finite element formulation in the
spacedomain and theLaplace transform in the timedomain to formulate andobtain the thermo-
elastic response of an FG circular cylindrical shell subjected to thermal shock loading. Jabbari
et al. (2002) presented a directmethod of solution of theNavier equation for the general thermo-
elastic analysis of a hollow thick FG cylinder under the effect of one-dimensional steady state
temperaturedistributionas a function of the radial coordinate. ZhaoandLiew (2011) carried out
free vibration analysis of a conical shell with material properties varying continuously through
the thickness according with a power-law distribution function, using the element-free kp-Ritz
method.

Peng andLi (2010) presented an analyticalmethod to investigate steady state thermal stres-
ses and radial displacement in FG disks by reducing the associated boundary value problem to
a Fredholm integral equation. Afsar and Go (2010) formulated a thermo-elastic axisymmetric
problemof a thinFGcircular disk, leading to a secondorder ordinarydifferential equationwhich
was solvedby thefinite elementmethod.Hojjati andJafari (2008) employed homotopyperturba-
tion andAdomian’s decompositionmethods to obtain distributions of stresses anddisplacements
in rotating annular elastic disks with uniform and variable thicknesses and densities. Bayat et
al. (2007) carried out thermo-elastic analysis of an FG rotating disk with axisymmetric bending
and steady-state thermal loading, using the first-order shear deformationMindlin plate and von
Karman theories. Shahzamanian et al. (2010) presented a finite element thermo-elastic analysis
of the contact problem of a rotating brake disk made of a material functionally graded accor-
ding to a power law function.Akbari alashti andKhorsand (2011) carried out three-dimensional
thermo-elastic analysis of an FG cylindrical shell with piezoelectric layers under the effect of
asymmetric thermo-electro-mechanical loads, using two versions of the differential quadrature
methods, i.e. the polynomial and Fourier quadrature methods.

In this paper, three-dimensional asymmetric stress and deformation analysis of an FG hol-
low circular cylindrical shell under the effect of thermo-mechanical loads are carried out with
the help of the Fourier and polynomial differential quadraturemethods. Numerical solutions of
stress, displacement and temperature fields in an FG cylindrical shell rotating at a constant
angular velocity are obtained. Numerical results are compared with exact solutions for material
properties defined by the power law which are found to be in good agreement. The effect of
variation of material grading parameters, temperature variation, angular velocity and the geo-
metry of the cylindrical shell on the distribution of stress, displacement and temperature fields
are investigated.

2. Governing equations

Acylindrical shell made of an FGmaterial with the inner radius of a, and the outer radius of b,
is considered. A schematic view of the shell under study and the corresponding cylindrical coor-
dinate system, i.e. r, θ and z-coordinates are shown in Fig. 1. The cylindrical shell is assumed
to be isotropic and rotating at a constant angular velocity ω and subjected to mechanical and
thermal loadings.Material properties of the shell, i.e. thermal and elastic constants are assumed
to be independent of temperature and vary along the radial direction according to the following
general formula

Y =Y (r) (2.1)

where Y (r) represents a material parameter of the shell such as the mass density ρ, Young’s
modulus E, Poisson’s ratio ν, thermal expansion coefficient α and thermal conductivity coeffi-
cient K.



Three-dimensional asymmetric thermo-elastic analysis... 145

Fig. 1. Geometry and coordinates of the cylindrical shell

In this work, a three-dimensional thermo-elastic analysis of a cylindrical shell is carried out.
The governing equations of the shell in three-dimensions and in presence of body forces are
defined as

∂σr
∂r
+
1

r

∂τrθ
∂θ
+
∂τrz
∂z
+
σr−σθ
r
+Fr =0

∂τrθ
∂r
+
1

r

∂σθ
∂θ
+
∂τθz
∂z
+2
τrθ
r
+Fθ =0

1

r

∂τθz
∂θ
+
∂σz
∂z
+
∂τrz
∂r
+
1

r
τrz+Fz =0

(2.2)

And the linear strain displacement relations are

εrr =
∂u

∂r
εθθ =

u

r
+
1

r

∂v

∂θ
εzz =

∂w

∂z

εrθ =
1

2

(1

r

∂u

∂θ
+
∂v

∂r
−
v

r

)

εzθ =
1

2

(∂v

∂z
+
1

r

∂w

∂θ

)

εrz =
1

2

(∂u

∂z
+
∂w

∂r

)

(2.3)

The thermo-elastic constitutive equations describing the relation between the strain and the
stress fields take the following form

σij =
E

1+ν
εij +

Eν

(1+ν)(1−2ν)
εkkδij −

EαT

1−2ν
δij (2.4)

By substituting Eq. (2.3) into Eq. (2.4), components of the stress field are defined in terms of
the displacement and temperature fields, and further substitution of stress components into Eq.
(2.2) lead to the governing equations of equilibrium in terms of displacement and temperature
fields of the shell. The corresponding differential equations in the three-direction form are found
to be

∂E

(1+ν)(1−2ν)∂r

[

(1−ν)
∂u

∂r
+ν
(u

r
+
1

r

∂v

∂θ

)

+ν
∂w

∂z

]

−
∂E

(1−2ν)∂r
αT

+
E

(1+ν)(1−2ν)

[

(1−ν)
∂2u

∂r2
+ν
(1

r

∂u

∂r
−
u

r2
+
1

r

∂2v

∂r∂θ
−
1

r2
∂v

∂θ

)

+ν
∂2w

∂r∂z

]

−
Eα

1−2ν

∂T

∂r
−

∂α

(1−2ν)∂r
ET +

E

2(1+ν)r

(1

r

∂2u

∂θ2
+
∂2v

∂r∂θ
−
1

r

∂v

∂θ

)



146 R. Akbari Alashti et al.

+
E

2(1+ν)

(∂2u

∂z2
+
∂2w

∂r∂z

)

+
E

(1+ν)(1−2ν)ri

{[

(1−ν)
∂u

∂r
+ν
(u

r
+
1

r

∂v

∂θ

)

+ν
∂w

∂z

]

−
[

(1−ν)
(u

r
+
1

r

∂v

∂θ

)

+ν
∂u

∂r
+ν
∂w

∂z

]}

+ρrω2 =0

∂E

2(1+ν)∂r

(1

r

∂u

∂θ
+
∂v

∂r
−
v

r

)

+
E

2(1+ν)

(1

r

∂2u

∂r∂θ
−
1

r2
∂u

∂θ
+
∂2v

∂r2
−
1

r

∂v

∂r
+
v

r2

)

(2.5)

+
E

(1+ν)(1−2ν)r

[

(1−ν)
(1

r

∂u

∂θ
+
1

r

∂2v

∂θ2

)

+ν
∂2u

∂r∂θ
+ν
∂2w

∂θ∂z

]

+
E

2(1+ν)

(∂2v

∂z2
+
1

r

∂2w

∂θ∂z

)

+
E

(1+ν)r

(1

r

∂u

∂θ
+
∂v

∂r
−
v

r

)

=0

∂E

2(1+ν)∂r

(∂u

∂z
+
∂w

∂r

)

+
E

2(1+ν)

( ∂2u

∂r∂z
+
∂2w

∂r2

)

+
E

2(1+ν)r

( ∂2v

∂θ∂z
+
1

r

∂2w

∂θ2

)

+
E

(1+ν)(1−2ν)

[

(1−ν)
∂2w

∂z2
+ν
(1

r

∂u

∂z
+
1

r

∂2v

∂θ∂z

)

+ν
∂2u

∂r∂z

]

+
E

2(1+ν)r

(∂u

∂z
+
∂w

∂r

)

=0

In the present work, the heat conduction problem without the heat source is considered, and
the heat balance equation for the corresponding steady state is defined as:

1

r

∂

∂r

(

rK
∂T

∂r

)

+
∂

∂z

(

K
∂T

∂z

)

=0 (2.6)

It is noted that in the axisymmetric case, all field variables are independentof the circumferential
direction θ, and there are only the radial and axial components of the displacement field, i.e. u
and w, to be considered.

3. Thermo-mechanical boundary conditions

Mechanical and thermal boundary conditions of the shell for two cases of simply supported and
clamped ends are described by the following relations:

Clamped: at z=0,L : u= v=w=T =0

Simply supported: at z=0,L : u= v=σz =T =0
(3.1)

where L is the length of the shell.

The inner and outer thermal and mechanical boundary conditions corresponding to the
inner and outer radii of the cylindrical shell are defined according to the following relations,
respectively

at r= a : σr =P(θ) τrz = τrθ =0 T =0

at r= b : σr =0 τrz = τrθ =0 T =Tb
(3.2)

where for the axisymmetric problem, the radial stresses at the internal boundary are assumed to
be zero, i.e. P(θ)= 0, and for the asymmetric case the radial stress is assumed to be a function
of the circumferential coordinate, i.e. P(θ)=P0cosθ. As it is observed fromEqs. (3.2), for both
cases the inner and outer boundaries are kept at constant temperatures of T =0 and T = Tb,
respectively.



Three-dimensional asymmetric thermo-elastic analysis... 147

4. Differential quadrature formulation

In the present study, two types of differential quadratures are employed to approximate the first
and the second order derivatives of the function; the polynomial expansion based differential
quadrature for the radial and the longitudinal directions and the Fourier expansion based for
the circumferential direction. The governing partial differential equations are discretized into
the series form, using the differential quadratures.

4.1. Fourier expansion based differential quadrature (FDQ)

The desired function, i.e. f(x), is approximated by a Fourier series expansion of the form

f(x)= c0+

N/2
∑

k=1

(ck coskx+dk sinkx) (4.1)

where the function f(x) constitutes an N+1dimensional linear vector space VN+1. Using these
sets of base vectors, explicit formulations to compute the weighting coefficients of the first and
the second order derivatives of the following forms are obtained (Shu, 2000)

aij =
q(xi)

2q(xj)sin
xi−xj
2

q(xi)=
N
∏

k=0

sin
xi−xk
2

(i 6= j) aii =−
N
∑

j=0

i6=j

aij

bij = aij
(

2aii− cot
xi−xj
2

)

(i 6= j) bii =−
N
∑

j=0

i6=j

bij

(4.2)

where aij and bij denote the weighting coefficients of the first and the second order derivatives
of the function f(x), respectively.

4.2. Polynomial based differential quadrature (PDQ)

Several polynomial differential quadrature methods are developed by researchers. In this
work, the one that uses the following Lagrange interpolation polynomials as a test function is
used (Shu, 2000)

fk(x)=
M(x)

(x−xk)M
(1)(xk)

k=1,2, . . . ,N (4.3)

where

M(x)= (x−x1)(x−x2) · · ·(x−xN) M
(1)(xi)=

N
∏

k=1

k 6=i

(xi−xk) (4.4)

By applying the above equation at N grid points, the following algebraic formulations to com-
pute the weighting coefficients are developed (Shu, 2000)

aij =
1

xj −xi

N
∏

k=1

k 6=i,j

xi−xk
xj −xk

(i 6= j) aii =
N
∑

k=1

k 6=i

1

xi−xk
bij =

N
∑

k=1

aikakj (4.5)



148 R. Akbari Alashti et al.

4.3. Three-dimensional differential quadratures

The above differential quadrature approximation corresponds to a one-dimensional formu-
lation. The differential quadrature approximation can be extended from the above formulation
to other coordinates. The first order derivatives in the three-dimensional formulations can be
approximated by

∂f

∂r

∣

∣

∣

∣

ijk

≈

N
∑

l=1

A
(1)
il fljk

∂f

∂θ

∣

∣

∣

∣

ijk

≈

M
∑

m=1

B
(1)
jmfimk

∂f

∂z

∣

∣

∣

∣

ijk

≈

P
∑

n=1

C
(1)
knfijn (4.6)

And the second order derivatives are defined as

∂2f

∂r2

∣

∣

∣

∣

ijk

≈
N
∑

l=1

A
(2)
il fljk

∂2f

∂θ2

∣

∣

∣

∣

ijk

≈
M
∑

m=1

B
(2)
jmfimk

∂2f

∂z2

∣

∣

∣

∣

ijk

≈
P
∑

n=1

C
(2)
knfijn

∂2f

∂r∂θ

∣

∣

∣

∣

ijk

≈
N
∑

l=1

A
(1)
il

M
∑

m=1

B
(1)
jmflmk

∂2f

∂r∂z

∣

∣

∣

∣

ijk

≈
N
∑

l=1

A
(1)
il

P
∑

n=1

C
(1)
knfljn

∂2f
∂θ∂z

∣

∣

∣

∣

ijk

≈
∑M
m=1B

(1)
jm

∑P
n=1C

(1)
knfimn

(4.7)

where A(1), B(1), C(1) and A(2), B(2), C(2) denote the weighting coefficients of the first and
the second order derivatives of the function f(r,θ,z) with respect to the r, θ and z-directions
respectively. N, M and P are the number of grid points chosen in the r, θ and z-directions,
respectively. Using the differential quadraturemethod, the first governing equation in the radial
direction is discretized, as follows

∂E

(1+ν)(1−2ν)∂r

[

(1−ν)
N
∑

l=1

A
(1)
il uljk+ν

(

uijk
ri
+
1

ri

M
∑

m=1

B
(1)
jmvimk

)

+ν
P
∑

n=1

C
(1)
knwijn

]

−
∂E

(1−2ν)∂r
αTijk+

E

(1+ν)(1−2ν)

[

(1−ν)
N
∑

l=1

A
(2)
il uljk+ν

(

1

ri

N
∑

l=1

A
(1)
il uljk−

uijk
r2i

+
1

ri

N
∑

l=1

M
∑

m=1

A
(1)
il B

(1)
jmvlmk−

1

r2i

M
∑

m=1

B
(1)
jmvimk

)

+ν
N
∑

l=1

P
∑

n=1

A
(1)
il C

(1)
knwljnk

]

−
Eα

1−2ν

N
∑

l=1

A
(1)
il Tljk−

∂α

(1−2ν)∂r
ETijk+

E

2(1+ν)ri

(

1

ri

M
∑

m=1

B
(2)
jmuimk

+
N
∑

l=1

M
∑

m=1

A
(1)
il B

(1)
jmvlmk −

1

ri

M
∑

m=1

B
(1)
jmvimk

)

+
E

2(1+ν)

( P
∑

n=1

C
(2)
knuijn

+
N
∑

l=1

P
∑

n=1

A
(1)
il C

(1)
knwljnk

)

+
E

(1+ν)(1−2ν)r

{[

(1−ν)
N
∑

l=1

A
(1)
il uljk

+ν

(

uijk
ri
+
1

ri

M
∑

m=1

B
(1)
jmvimk

)

+ν
P
∑

n=1

C
(1)
knwijn

]

−

[

(1−ν)

(

uijk
ri
+
1

ri

M
∑

m=1

B
(1)
jmvimk

)

+ν
N
∑

l=1

A
(1)
il uljk+ν

P
∑

n=1

C
(1)
knwijn

]}

+ρriω
2 =0

(4.8)

Derivatives of the elastic constant E and the thermal expansion coefficient α are defined accor-
ding to their respective functions. The steady state temperature field is defined as

K
N
∑

l=1

A
(2)
il Tljk+

K

ri

N
∑

l=1

A
(1)
il Tljk+

∂K

∂r

N
∑

l=1

A
(1)
il Tljk+K

P
∑

n=1

C
(2)
knTijn =0 (4.9)



Three-dimensional asymmetric thermo-elastic analysis... 149

where the derivative of K is obtained according to its respective function.

The stress boundary conditions are defined as

E

(1+ν)(1−2ν)

[

(1−ν)
N
∑

l=1

A
(1)
il uljk+ν

(

uijk
ri
+
1

ri

M
∑

m=1

B
(1)
jmvimk

)

+ν
∂w

∂z

]

−
E

(1−2ν)
αTijk−P(θ)= 0

E

2(1+ν)

(

1

ri

M
∑

m=1

B
(1)
jmuimk+

N
∑

l=1

A
(1)
il vljk−

1

ri
vijk

)

=0

E

2(1+ν)

( P
∑

n=1

C
(1)
knuijn+

N
∑

l=1

A
(1)
il wljk

)

=0

(4.10)

In order to proceed with numerical computations, sampling points in the radial and axial direc-
tions are defined using theChebyshev polynomials, while for circumferential direction a uniform
distribution is used. The coordinates of grid points used in the r, θ and z-directions are defined
as follow

rk1 = a+
b−a

2

[

1− cos
(k1−1

N−1
π
)]

k1 =1, . . . ,N

θk2 =2π
k2−1

M
k2 =1, . . . ,M

zk3 =
L

2

[

1− cos
(k3−1

P −1
π
)]

k3 =1, . . . ,P

(4.11)

5. Numerical results and discussions

At first, the results obtained are compared with the results reported for a thick hollow cylinder
by Jabbari et al. (2002). It is assumed that the cylinder is made of materials graded along the
r-coordinate. Variation of the modulus of elasticity E, the coefficient of thermal expansion α,
and the thermal conduction coefficient K, are assumed to be obeying the following power law
formulation

E(r)=Eir
m α(r)=αir

m K(r)=Kir
m (5.1)

where Ei,αi and Ki are the modulus of elasticity, the coefficient of thermal expansion and the
thermal conduction coefficient at the inner radius.

The inner and the outer radii of the cylinder are assumed to be a = 1m and b = 1.2m,
respectively and the grading parameter m is chosen as 3. Poisson’s ratio is assumed to be 0.3
and the modulus of elasticity and the thermal coefficient of expansion at the inner radius are
Ei =200GPa and αi =1.2 ·10

−6/◦C, respectively. Temperature of the hollow cylinder at the
inner and outer radii are assumed to be T(a)= 100◦Cand T(b)= 0◦C, respectively. Thehollow
cylinder is under the effect of an internal pressure of 50MPa, while the external pressure is zero,
i.e. σrr(a)=−50MPa and σrr(b)= 0MPa.

Results of the convergence study for different grid numbers are shown in Figs. 2a,b. It
is observed from Fig. 2a that by increasing the number of grid points, the results converge
rapidly and the rate of convergence decreases as the number of grid points increases. Hence, an
appropriate grid number of 20 is chosen. The radial distribution of the hoop stress and the radial
displacement fields calculated by the presentmethod are compared with the results reported by
Jabbari et al. (2002) which indicate very good agreement, as shown in Figs. 2a and 2b.



150 R. Akbari Alashti et al.

Fig. 2. Radial distribution of hoop stress (a), radial distribution of radial displacement (b) for an FGM
cylindrical shell

5.1. Axisymmetric problem

In this section, numerical calculations for thehollow cylindrical shellmadeof anFGmaterial,
i.e. theouter surfacemadeofZirconia (ceramic) and the inner surfacemadeofAluminum(metal)
are carried out.Properties of thesematerials are listed inTable 1 (Bayat et al. (2007)). Formany
engineering materials, the Poisson ratio is within the range of 0.25 to 0.35 and varies slightly.
Hence, for simplicity, it is assumed that the Poisson ratio is constant as depicted in Table 1.

Table 1.Material properties

Material E [GPa] ν α [10−6/◦C] K [W/m◦C] ρ [kg/m3]

Aluminum 70 0.3 23 209 2700

Zirconia 151 0.3 10 2 5700

A power law formulation is used to define the variation of material properties in the radial
direction

Y (r)=Yi
[

1−
(r−a

b−a

)β]

+Yo
(r−a

b−a

)β
(5.2)

where Yi and Yo denote the material properties at the inner and the outer surfaces of the
hollow cylindrical shell, respectively and β is the grading parameter describing the change of
the volume fraction of two constituents. These properties include the elastic constant E, the
coefficient of thermal expansion α, the thermal conduction coefficient K and themassdensity ρ.
Dimensionless results are obtained bydefining the following dimensionless stresses, displacement
and temperature fields

σij =
1−ν

E0α0T0+(1−ν)ρ0ω
2
0b
2
σij T =

T

To

u=
E0

(1+ν)[E0α0T0+(1−ν)ρ0ω
2
0b
2]b
u

(5.3)

It should be noted that all numerical results are calculated and presented for the midpoint, i.e.
z=L/2, of the cylindrical shell with clampedboundary conditions at two ends, as defined inEq.
(3.1)1. For the axisymmetric case, variations of parameters i.e. the stresses, radial displacement
and temperature of the midpoint along the radial direction are shownwhile for the asymmetric
case, variations of the parameters related to mid-plane along the circumferential direction are
presented.



Three-dimensional asymmetric thermo-elastic analysis... 151

An FG shell rotating at ω = 500rad/s and subjected to a thermal loading of T(a) = 0◦C
and T(b) = 100◦C at the inner and outer boundaries respectively, is considered. The effect of
β on the variation of stresses, radial displacement and temperature distributions are examined
first and the results are shown in Fig. 3. It is observed fromFig. 3a that the dimensionless radial
stress reaches zero at the inner and outer surfaces and themaximumdimensionless radial stress
decreases as β increases. Moreover, maximum values occur at certain internal positions which
shift toward the outer surface as β increases. Such a trend is also found with the value of the
dimensionless circumferential stress, as depicted in Fig. 3b. It is observed from Fig. 3c that the
dimensionless axial stress reduces as β increases. For the case of β =10, the rate of change of
the dimensionless circumferential and axial stresses at the interior sections of the structure is
lower than at the sections adjacent to the inner and the outer boundaries. It is revealed from
Fig. 3d that thedimensionless shear stress reaches zero at the inner andouter surfaces, satisfying
the boundary conditions. As β increases, values of the dimensionless shear stress decrease in
magnitude. The influence of β on the dimensionless radial displacement is shown in Fig. 3e
indicating a monotonous decrease with the increase of β. The maxima of the dimensionless
radial displacement for linearly varying material properties occur at the outer surface. As it is
expected fromFig. 3f, the distribution of temperature in the shell is nonlinear, due to the power
law variation of the thermal conductivity across the thickness.

Fig. 3. Effects of β on the (a) radial stress, (b) circumferential stress, (c) axial stress, (d) shear stress,
(e) radial displacement, (f) temperature distribution



152 R. Akbari Alashti et al.

The effects of temperaturedifference at two surfaces of the cylindrical shell on the dimension-
less stresses and radial displacement are demonstrated in Figs. 4a-4e. For the sake of simplicity,
numerical calculations are carried out with the temperature of the inner surface being fixed,
i.e. T(a) = 0◦C, with the temperature at the outer surface of the cylindrical shell varying as
a parameter. The material properties are assumed to vary linearly in the radial direction, i.e.
β = 1 and the cylindrical shell is assumed to be rotating at a constant angular velocity of
ω =500rad/s. It is observed from Fig. 4a that the dimensionless radial stress increases with a
slow rate as the temperature difference increases and the location of its maxima shifts toward
the outer surface of the cylinder. It is expected that at low temperature difference, the induced
stresses and deformations are mainly due to the centrifugal force and not the temperature dif-
ference. Variations of the dimensionless circumferential and axial stresses are shown in Figs. 4b
and 4c respectively, which indicate that these stresses tend towards the compressive region and
their absolute values increase as the temperature difference increases. It is also realized from
Fig. 4d that the absolute value of the dimensionless shear stress increases as the temperature
difference increases. Variation of the dimensionless radial displacement with temperature dif-
ference at boundaries is displayed in Fig. 4e. It is also revealed from Figs. 4a to 4e that the
dimensionless stresses and the radial displacement are sensitive to the applied thermal loading.

Fig. 4. Effects of Tb on the (a) radial stress, (b) circumferential stress, (c) axial stress, (d) shear stress,
(e) radial displacement



Three-dimensional asymmetric thermo-elastic analysis... 153

As the last case of an axisymmetric problem, the effect of boundary condition, i.e. simply
supported or clamped at two ends, is investigated and the results are plotted in Figs. 5a-5e. In
this section the cylindrical shell with linearly varyingmaterial properties rotating at a constant
angular speed of ω=500rad/s and subjected to a thermal boundary condition of T(a) = 0◦C
and T(b) = 100◦C is considered. It is revealed from Figs. 5a-5d that the pattern of dimen-
sionless stresses are similar for both boundary conditions, however the maximum values of the
dimensionless stresses for the simply supported boundary condition is higher than those of the
clamped one. It is noted fromFig. 5e that for simply supportedboundary conditions at z=0,L,
the variation amplitude of the curve of the radial displacement becomes higher than that the
clamped condition.

Fig. 5. Effects of the boundary condition on the (a) radial stress, (b) circumferential stress, (c) axial
stress, (d) shear stress, (e) radial displacement

5.2. Asymmetric problem

For the asymmetric problem, it is assumed that the cylindrical shell is subjected to a cosine
type of internal pressure loading, i.e. P(θ) = P cos(θ). Effects of ω on the distributions of
the dimensionless stresses and the radial displacement of the mid-plane of the cylindrical shell
with linearly varying material properties and thermal boundary conditions of T(a) = 0◦C
and T(b) = 100◦C are presented in Figs. 6a to 6e. The dimensionless radial, circumferential



154 R. Akbari Alashti et al.

and axial stresses increase with the increase in ω showing high dependency on the angular
velocity; however, the dimensionless shear stress remains nearly constant. It is also noted that
the dimensionless radial displacement increases as ω increases.

Fig. 6. Effects of ω on the (a) radial stress, (b) circumferential stress, (c) axial stress, (d) shear stress,
(e) radial displacement

The effects of temperature difference on the stresses and radial displacement at the mid-
plane of the shell are calculated and shown in Figs. 7a-7e. The temperature of the inner surface
is assumed to be constant (i.e. T(a)= 0◦C) and the temperature at the outer surface varying as
a parameter, as in the case of the axisymmetric problem. Linearly varying material properties
and ω = 500rad/s are assumed. As the temperature difference increases, the dimensionless
radial stress and displacement increase, however the dimensionless circumferential and shear
stresses decrease. It is observed that variations of the dimensionless shear stress are negligible
indicating that the temperature difference has higher effect on the normal stresses in the r, θ
and z-direction and has negligibly small effect on the shear stress.

Finally, the effect of the thickness on the dimensionless stresses and the radial displace-
ment are investigated. In this calculation, parameters are chosen as β = 1, T(a) = 0◦C and
T(b)= 100◦C and ω=500rad/s. The dimensionless stresses, radial displacement and tempera-
turedistributions for different ratios of the outer radius to the inner radius of the cylindrical shell
equal to b/a=3, 6, 9, 12 are shown in Figs. 8a-8e. As it is expected, values of the dimensionless
stresses and radial displacement become lower as the hollow cylindrical shell becomes thicker.



Three-dimensional asymmetric thermo-elastic analysis... 155

Fig. 7. Effects of Tb on the (a) radial stress, (b) circumferential stress, (c) axial stress, (d) shear stress,
(e) radial displacement

6. Conclusions

In this paper, a numericalmethod to solve the thermo-elastic problemof a rotating hollow cylin-
drical shellmade of anFGmaterial is presented.Governing differential equationswere rewritten
with the help of theFourier andpolynomial differential quadratures and solved numericallywith
complete satisfaction of all boundary conditions. The distributions of dimensionless stresses, ra-
dial displacement and temperature fields of the rotating shell for various boundary conditions,
material properties, angular velocities and temperature differences are calculated andpresented.

From the present study, the following conclusions are obtained:

A – Axisymmetric loading and boundary conditions

• The dimensionless stresses and radial displacement decrease and the dimensionless
temperature distribution increases in magnitude as the gradient parameter β incre-
ases.

• The maximum value of the dimensionless radial stress increases as the temperature
difference increases.



156 R. Akbari Alashti et al.

Fig. 8. Effects of b/a on the (a) radial stress, (b) circumferential stress, (c) axial stress, (d) shear stress,
(e) radial displacement

• The maximum values of stresses and displacement in shells with simply supported
end boundary conditions are higher than those with clamped boundary conditions.

• In an FG shell with a power law variation of properties, the distribution of tempera-
ture through thickness of the shell is nonlinear.

B – Asymmetric loading and boundary conditions

• The dimensionless radial, circumferential and axial stresses and radial displacement
at the mid-plane along the circumference increase as ω increases, indicating strong
dependency of these stresses on the angular velocity, however the dimensionless shear
stress is almost constant.

• The dimensionless radial stress and displacement at themid-plane along the circum-
ference increase, while the dimensionless circumferential and axial stresses decrease
as the temperature difference increases.

• As the hollow cylindrical shell becomes thicker, values of the dimensionless stresses
and radial displacement at the mid-plane along the circumference become smaller.



Three-dimensional asymmetric thermo-elastic analysis... 157

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158 R. Akbari Alashti et al.

Trójwymiarowa analiza asymetrycznie termosprężystego zagadnienia wirującej powłoki
cylindrycznej wykonanej z materiału gradientowego

Streszczenie

W pracy zaprezentowano analizę asymetrycznej deformacji i naprężeń w otwartej powłoce cylin-
drycznej wykonanej z materiału gradientowego i poddanej obciążeniom termosprężystym.Do badańwy-
korzystanometodę kwadratury różniczkowej. Nie tracąc na ogólności rozważań, założono, że właściwości
materiału zmieniają się stopniowo w kierunku promieniowym zgodnie z przyjętym prawem potęgowym.
Przyjęto ponadto, że współczynnik Poissona jest stały. Cząstkowe równania różniczkowemodelu wyrażo-
no w funkcji przemieszczenia i pola temperatur ujętych w formie szeregów uzyskanych za pomocą dwóch
wersji metody kwadratury różniczkowej, tj. kwadratury Fouriera i wielomianowej. Powłokę cylindrycz-
ną badano dla osiowo-symetrycznego i asymetrycznego stanu obciążenia. Wyniki symulacji numerycz-
nych dla stanu osiowo-symetrycznego porównano z rozwiązaniami ścisłymi, stwierdzając bardzo dobrą
zgodność. Przedstawiono takżewyniki analizy termosprężystości dla przypadku asymetrycznego powłoki
wirującej ze stałą prędkością kątową. Przedyskutowano ponadto wpływ wykładnika potęgowego rozkła-
du gradientowegowłaściwości materiału, wpływ prędkości wirowania, różnic temperatury i geometrii na
przemieszczenia promieniowe powłoki i kształt pola cieplnego.

Manuscript received December 12, 2011; accepted for print April 15, 2012