Based on the Flügge’s shell theory, the vibration characteristics and stability of submerged
functionally graded (FG) cylindrical shell under hydrostatic pressure is examined. By means of conversion
switch on axial wave number, the coupled frequency of submerged FG cylindrical shell with various
boundary conditions is obtained, using wave propagation method and Newton method. Then the critical
pressure of FG cylindrical shells is given by applying linear fitting method. Results are compared to known
solutions, where these solutions exist. The natural frequency and critical pressure of FG cylindrical shell are illustrated. The effects of constituent materials, volume fraction, boundary condition and dimensions on the natural frequencies and critical pressures of submerged FG cylindrical shell are illustrated by examples

Li, R.

Liang, B.

Noda, N.A.

Xu, H.Y.

Kyushu Institute of Technology of Academic Repository

Kyushu Institute of Technology Academic Repository
九州工業大学学術機関リポジトリ
Title Study on vibration and stability of functionally gradedcylindrical shells subjected to hydrostatic pressure
Author(s)Liang, B.; Li, R.; Noda, N.A.; Xu, H.Y.
Issue Date2013
URL http://hdl.handle.net/10228/5417
Rights
International Journal of Fracture Fatigue and Wear, Volume 1 
24 
 
Proceedings of the 2nd International Symposium on  
Engineering Mechanics and its Applications, pp. 24- 30, 2013 
STUDY ON VIBRATION AND STABILITY OF FUNCTIONALLY GRADED 
CYLINDRICAL SHELLS SUBJECTED TO HYDROSTATIC PRESSURE 
B. Liang1, R. Li1, N.A. Noda2 and H.Y.Xu1 
1 Henan University of Science and Technology, Luoyang, China 
2
 Kyushu Institute of Technology, Kitakyushu, Japan 
 
Abstract: Based on the Flügge’s shell theory, the vibration characteristics and stability of submerged 
functionally graded (FG) cylindrical shell under hydrostatic pressure is examined. By means of conversion 
switch on axial wave number, the coupled frequency of submerged FG cylindrical shell with various 
boundary conditions is obtained, using wave propagation method and Newton method. Then the critical 
pressure of FG cylindrical shells is given by applying linear fitting method. Results are compared to known 
solutions, where these solutions exist. The natural frequency and critical pressure of FG cylindrical shell are 
illustrated. The effects of constituent materials, volume fraction, boundary condition and dimensions on the 
natural frequencies and critical pressures of submerged FG cylindrical shell are illustrated by examples. 
 
Keywords: Natural Frequency; Critical Pressure; Functionally Graded Material; Cylindrical Shell; 
Hydrostatic Pressure
1 INTRODUCTION 
Functionally graded materials, FGM for short, is a new kind of compound material structure with component 
and structure graded distribution along thickness. By using the new kind of functionally graded material, the 
requirements of special extreme environment such as ultra-temperature, larger temperature gradient and 
the strong thermal shock on FG cylindrical shell are satisfied. The liquid medium and the material properties 
of functionally graded material have significant impact on the vibration characteristics of cylindrical shell. And 
the structural analyses need to be carried out in the presence of critical pressure. Since the pioneer work of 
Junger [1] was published, a lot of theoretical investigations have appeared. Zhang et al. [2] used wave 
propagation method to consider the vibration characteristics of cylindrical shell with the impact of the fluid. 
Sheng and Wang [3] presented the report of an investigation into the vibration of FG cylindrical shells with 
flowing fluid by employing the first-order shear deformation theory. The input power flow for an infinite 
ring-stiffened cylindrical shell submerged in fluid was investigated by Liu et al. [4]. The hydrostatic buckling 
of shells with various boundary conditions is studied by Pinna and Ronalds [5].  
Based on the Flügge shell theory, wave propagation method and Newton method, the coupled frequency of 
submerged FG cylindrical shell with various boundary conditions is derived. Then the critical pressure of FG 
cylindrical shells is given by applying linear fitting method. The effects of constituent materials, volume 
fraction, boundary condition and dimensions on the natural frequencies and critical pressures of submerged 
FG cylindrical shell are illustrated by examples. In numerical calculations, the functionally graded material 
has ceramic on its inner surface and has metal on its outer surface. 
2 MECHANICAL MODEL 
A FG cylindrical shell is considered, as shown in Fig.1. The shell is 
characterized by its inner radius iR , outer radius 0R , represented 
radius R , length L  and thickness h . The x , θ and z are the axial 
coordinates, circumferential coordinates and radial coordinates, 
respectively.  
3 FUNCTIONALLY GRADED MATERIALS 
The material property P of functionally graded material is the function of temperature and volume fraction, 
which is controlled by the volume fractions of material. The function can be defined as follows 
Fig.1. Geometry of a FG cylindrical 
International Journal of Fracture Fatigue and Wear, Volume 1 
25 
 
1 2 3
0 1 1 2 3( 1 )P P P T PT PT PT−−= + + + +                                                          (1) 
where 0P , 1P− , 1P , 2P , 3P are the coefficients of temperature (K)T and are unique to the constituent materials. 
For cylindrical shell with thickness h , the volume percentages can be given as 
0
2 1
N
o i
i
z hV V
R R
 +
= = − 
− 
                                                                   (2) 
where iV and oV  are the volume percentages of the internal and external surfaces of the functionally 
graded material, respectively. N is the power-law exponent( 0 N≤ ≤ ∞ ).So along the thickness of the 
shell, the Young’s modulus E , Poisson ratio µ and the mass density ρ  can be expressed as 
0
0
0
2( )
2( )
2( )
N
o i i
i
N
o i i
i
N
o i i
i
z hE E E E
R R
z h
R R
z h
R R
µ µ µ µ
ρ ρ ρ ρ
  +
 = − + 
−  

 +
= − +  
− 

 +
= − + 
− 
                                                             (3) 
4 FORMULATION 
By using Flügge’s [6] theory, the equations of motion for a cylindrical shell subjected to hydrostatic pressure 
are obtained. 
2 2
1 2
1
2 2
1 1 1- 1-( - )
2 2 2 2
(1- )( - ) - 0
1 1 3(1- ) 3-( - )
2 2 2 2
(1- )2( ) - 0
1 3-
- + - (1 )
2 2
xx x x xxx x
xx x tt
x xx xx xx xx
tt
x xxx x xx
u u v w K u w w
RT u T u w u
E
u v v w K v w T v
RT v w v
E
u Ku K u v K v K w K
θθ θ θθ θθ
θθ
θ θθ θ θ
θθ θ
θθ θ θ
µ µ µ µµ
ρ µ
µ µ µ µ
ρ µ
µ µµ
− +
+ + + + +
+ +
+ −
+ + + + +
+ +
−
+ + + +
＝
＝
2 2 2 2
1 2
+2
(1- ) (1 )
+ +2 - - ( - + )+
xxxx xx
xx x tt
w Kw
R RKw Kw T w T u v w w
E Eh
θθ
θθθθ θθ θ θθ ψ
ρ µ µ















−


＝-
                         (4) 
where 21 (1- )2 o
RT P
Eh
µ＝ , 22 (1- ) o
RT P
Eh
µ＝ , ()()x R
x
∂
∂
＝ ,
()()θ θ
∂
∂
＝ ,
()()t t
∂
∂
＝ ,
2
212
hK
R
= , oP  is the 
hydrostatic pressure and ψ  is the acoustic pressure. 
The displacements of the cylindrical shell can be expressed in the form of wave propagation. 
( )
( )
( )
( , , ) cos( )
( , , ) sin( )
( , , ) cos( )
m
m
m
i t ik x
m
i t ik x
m
i t ik x
m
u x t U n e
v x t V n e
w x t W n e
ω
ω
ω
θ θ
θ θ
θ θ
−
−
−
 =

=

=
                                                          (5) 
where mU , mV , mW are the wave amplitudes in the x ,θ and z directions, ω is the natural angular frequency. 
The fluid of the cylindrical shell is assumed non-viscous which satisfy the acoustic wave equation. The 
equation of motion of the fluid can be written in the cylindrical co-ordinate system ( , , )x rθ as: 
2 2 2
2 2 2 2 2
1 1 1
r
r r r r x c t
ψ ψ ψ ψ
θ
∂ ∂ ∂ ∂ ∂ 
+ + = ∂ ∂ ∂ ∂ ∂ 
                                                    (6) 
International Journal of Fracture Fatigue and Wear, Volume 1 
26 
 
where t  is the time and c is the sound speed of the fluid. The associated form of the acoustic pressure field 
exterior of the shell, which satisfies the acoustic wave Eq.(6), is given as 
( )(2)cos( ) ( ) m
n
i t ik x
m rn H k r e
ωψ ψ θ −= , 2 (2)'/ ( )
m f r n r mk H k R Wψ ω ρ =                             (7) 
where (2) ()
n
H is the Hankel function of the second kind and order n , the prime on the (2) ()
n
H denotes 
differentiation with respect to the argument 
r
k R . The relationship between radial wave number 
r
k  and 
axial wave number 
m
k is applied to 2 2 2 2( ) ( / ) ( )
r L F mk R C C k R= Ω − . 
In the function, Ω is the non-dimensional frequency, LC and FC are the sound speed of the shell and fluid 
respectively. The fluid radial displacement and shell radial displacement must be equal at the interface of the 
shell inner wall and the fluid.  
Substituting Eq.(5) into Eq.(4), which consideration of acoustic pressure on the shell and coupling Eq.(7), the 
equations of motion of coupled system in matrix form can be obtained. 
11 12 13
12 22 23
31 32 33
0
0
+ 0
m
m
L m
L L L U
L L L V
L L L F W
     
    
=    
         
                                                         (8) 
where ( , 1,2,3)ijL i j = are the parameters operated with the x andθ , LF  is the fluid loading term due to 
the presence of the fluid acoustic field. 
2 1 (2) (2) '( / )( / )( ) ( ) / ( )
nL f r r n rF R h k R H k R H k Rρ ρ
−  = Ω                                          (9) 
By substituting a boundary condition into
r
k , the solution of Eq.(8) can be derived. 
According to the classical theory, the function of elastic critical hydrostatic pressure is given in Шиманский 
[7]. However, the function can only analysis the elastic critical pressure of cylindrical shell for 1m=  under 
simply supported (SS-SS) boundary condition. Because of linear relation between the frequency squared 
and hydrostatic pressure (Abramovich [8]), the critical hydrostatic pressure can be derived while the natural 
frequency is assumed to be zero. Therefore, the critical pressure can be obtained by using linear fitting 
method. 
5 NUMERICAL RESULTS AND DISCUSSION 
5.1 Vibration of FG cylindrical shell 
In this paper, the metal materials are Stainless steel and Ti-6Al-4V, while the ceramic materials are Si3N4 
and Zirconia. The material properties are taken into consideration the temperature dependency for the 
temperature of 300T K= from Shariyat [9] and Kim [10]. 
The natural frequencies of FG cylindrical shell with simply supported ends are listed in Table 1. The 
functionally gradient material considered is composed of Stainless steel and nickel. The validity and 
feasibility of the study are verified by comparing results with those in Ref. [11].The geometric parameters of 
the shell are defined as: 11 22.05098 10 / mNE N= × , 0.31Nµ = , 38900 / mN kgρ = , 
/ 20L R = , / 0.002h R = , 1m = . 
 
Table 1. Comparison of natural frequencies of FG cylindrical shell 
 0N →
 
1N =
 
N →∞
 
n
 
Present Loy Present Loy Present Loy 
1 13.548 13.548 13.211 13.211 12.894 12.894 
2 4.5916 4.5920 4.4768 4.480 4.3687 4.3690 
3 4.2628 4.2633 4.1523 4.1569 4.0484 4.0489 
4 7.2247 7.2250 7.0354 7.0384 6.8574 6.8577 
5 11.542 11.542 11.239 11.241 10.954 10.955 
 
 
International Journal of Fracture Fatigue and Wear, Volume 1 
27 
 
 
As an example, critical pressures for a simply supported ends isotropic cylindrical shell are studies in Table 
2. Because of linear relation between the natural frequency squared and hydrostatic pressure, the equations 
of cylindrical shells for different conditions can be obtained by using linear fitting method, and then the critical 
pressure can be derived. Compared with results from Eq.(15), the feasibility of the study is proved. Some 
parameters are selected as: 11 22.1 10 / mE N= × , 0.3µ = , 37850 / mkgρ = , 0.01mh= , 20mL= , 
1mR= , 1m= , 2n= .  
 
Table 2. Comparison of critical pressures [ MPa ] of simply supported submerged isotropic cylindrical shell 
/h R
 
/L R
 
2 5 10 
Present Eq(15) Present Eq(15) Present Eq(15) 
0.005 36.448 36.115 2.6929 2.6586 0.2079 0.2072 
0.01 73.049 72.333 5.4377 5.3692 0.4610 0.4599 
0.02 147.10 145.48 11.287 11.154 1.2832 1.2831 
 
Because of the structure will create instability when the hydrostatic pressure is larger than critical pressure, 
so in this paper, the hydrostatic pressures should aim for under critical pressures. The natural frequencies of 
FG cylindrical shells under hydrostatic pressure for different volume fractions, axial half wave numbers and 
boundary conditions are studied in this paper, see Fig.2-4. Results given in these figures are obtained by 
setting / 20L R = , / 0.01h R = , 1mR = , 1N = , 1m= , 2n = and 0z = . 
Fig.2 describes the variations of natural frequencies of clamped-free (C-F) FG cylindrical shells for some 
different values of volume fraction, and the functionally graded material is made up of Stainless steel and 
Si3N4. It is shown in these figure that with the increasing of volume fraction, the natural frequency of FG 
cylindrical shell increases gradually. The effects of power-law exponent on natural frequencies are mainly 
reflected in the cases of low exponent.  
 
Fig.2. Variation of frequencies of submerged FG cylindrical shells for different values of volume fraction. 
 
Fig.3 describes the change curves of natural frequencies of FG cylindrical shells for different support 
conditions. It is shown that the coupled frequencies for clamped-clamped (C-C) boundary condition are 
higher than those for other boundary conditions, the frequencies for free-sliding (F-S) boundary condition are 
lower than those for other boundary conditions. As the critical pressure approached, for clamped-free ends 
and free-sliding ends, the decline in natural frequency accelerated. 
International Journal of Fracture Fatigue and Wear, Volume 1 
28 
 
 
Fig.3. Variation of frequencies of submerged FG shells associated with various boundary conditions. 
 
The variations of natural frequencies of clamped-free FG cylindrical shells for some different axial half wave 
numbers m under hydrostatic pressure are shown in Fig.4. It is shown that the natural frequency of FG 
cylindrical shell increased with the increasing of axial half wave number. With the increasing of axial half 
wave number, the range ability of the curves of natural frequency becomes smaller. Inferred by former 
results, with the increasing of axial half wave number, the critical pressure grows exponentially. When 
m comes to a certain degree, the change of the natural frequency is not significant. 
 
Fig.4. Variation of frequencies of submerged FG shells for some different axial half wave numbers. 
5.2 Stability of FG cylindrical shell 
As examples, based on the linear fitting method, critical pressures with different /h R , /L R , various 
boundary conditions, constituent materials and volume fractions are studied, see Fig.5 and Fig.6. 
 
Fig.5 shows the curves of critical pressures of FG cylindrical shells for various values of power-law 
exponent. It is shown in the figure that the effects of power-law exponent on critical pressure are evident. 
The influences of power-law exponent on critical pressures are mainly reflected in the cases of small 
power-law exponent. With the increasing of power-law exponent, the functionally graded material is from a 
pure metal material to a pure ceramic material. This is due to the FG cylindrical shell is reduced to an 
isotropic cylindrical shell. Since critical pressure is material property dependent, use the FG cylindrical shell 
which composed of Stain steel and Zirconia as an example. Because of the critical pressure of pure Stain 
steel cylindrical shell is larger than pure Zirconia cylindrical shell, the critical pressure decreases with the 
increasing of power-law exponent. 
International Journal of Fracture Fatigue and Wear, Volume 1 
29 
 
 
Fig.5. Variation of critical pressures of submerged shell for different values of power-law exponent. 
 
The critical pressures of simply supported FG cylindrical shells for different /h R are shown in Fig.6. It is 
shown that with the increasing of /h R , the critical pressure increases rapidly. The influences of constituent 
material on critical pressure are mainly reflected in the cases of large /h R . 
 
Fig.6. Variation of critical pressures of submerged cylindrical shell with different /L R  ratios. 
6 CONCLUSIONS 
Based on the Flügge’s shell theory, the natural frequencies and critical pressures of submerged FG 
cylindrical shells under hydrostatic pressure are studied. By means of conversion switch on axial wave 
number, the coupled frequency of submerged FG cylindrical shell with various boundary conditions is 
obtained, using wave propagation method and Newton method. Then the critical pressure of FG cylindrical 
shells is given by applying linear fitting method. The effects of the hydrostatic pressure, dimensions, 
constituent material, volume fraction and boundary condition on natural frequencies and critical pressure 
were investigated. Especially for the axial half wave number, the influence is obvious. In some cases, the 
change rules of natural frequency are similar to critical pressure. The characteristics of natural frequency 
and critical pressure will certainly provide guidance to the practical use and risk averse. 
ACKNOWLEDGEMENTS 
The authors wish to express their gratitude to the National Natural Science Foundation of China (No. 
51105132) and Natural Science Foundation of Henan Province (No. 122300410112) that have supported 
this work.. 
 
International Journal of Fracture Fatigue and Wear, Volume 1 
30 
 
7 REFERENCES 
[1] M.C.Junger, Vibrations of elastic shells in a fluid medium and the associated radiation of sound, J, Appl. 
Mech,19(3), 439-445,1952. 
[2] X.M. Zhang,. Frequency analysis of submerged cylindrical shells with the wave propagation approach. J. 
International Journal of Mechanical Sciences. 44 (7), 1259-1273, 2002. 
[3] G.G. Sheng, X.Wang, Thermomechanical vibration analysis of a functionally graded shell with flowing 
fluid , European Journal of Mechanics -A /Solids. 27(6), 1075-1087, 2008. 
[4] Z.Z. Liu, T.Y. Li, X. Zhu, J.J. Zhang, Effect of Hydrostatic Pressure on Input Power Flow in Submerged 
Ring-stiffened Cylindrical Shells, Journal of Ship Mechanics. 15 (3):301-312,2011. 
[5] R. Pinna, B.F. Ronalds, Hydrostatic buckling of shells with various boundary conditions, Constr. Stell. 
Res. 56 (1), 1–16, 2000. 
[6] W. Flügge, Stresses in Shells, Second edition, Springer-Verlag, New York, 1973.  
[7] Ю. А.Шиманский ,. Строительная Механика ПодводныхЛодок, Судпромгиз,1948. 
[8] H. Abramovich, J.Singer, T. Weller, Repeated buckling and its influence on the geometrical imperfections 
of stiffened cylindrical shells under combined loading. International Journal of Non-Linear Mechanics, 
37(4-5), 577–588, 2002. 
[9] M. Shariyat, Dynamic buckling of suddenly loaded imperfect hybrid FGM cylindrical shells with 
temperature-dependent material properties under thermo-electro-mechanical loads, International Journal of   
Mechanical Sciences. 50(12), 1561-1571, 2008. 
[10] Y.W. Kim,. Temperature dependent vibration analysis of functionally graded rectangular plates, Journal 
of Sound and Vibration. 284(3-5), 531-549, 2005. 
[11] C.T Loy , K.Y. Lam, J.N. Reddy, Vibration of functionally graded cylindrical shells. J. International Journal 
of Mechanical Sciences. 41(3), 309-324, 1999. 


Study on vibration and stability of functionally graded cylindrical shells subjected to hydrostatic pressure

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Study on vibration and stability of functionally graded cylindrical shells subjected to hydrostatic pressure

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