Free vibration analysis of FGM cylindrical shells surrounded by Pasternak elastic foundation in thermal environment considering fluid-structure interaction

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Free vibration analysis of FGM cylindrical shells surrounded by Pasternak elastic foundation in thermal environment considering fluid-structure interaction Abdolhossein Baghlani Associate Professor , Majid Khayat , Seyed Mehdi Dehghan Associate Professor PII: DOI: Reference:

S0307-904X(19)30611-0 https://doi.org/10.1016/j.apm.2019.10.023 APM 13084

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Applied Mathematical Modelling

Received date: Revised date: Accepted date:

16 March 2019 10 September 2019 8 October 2019

Please cite this article as: Abdolhossein Baghlani Associate Professor , Majid Khayat , Seyed Mehdi Dehghan Associate Professor , Free vibration analysis of FGM cylindrical shells surrounded by Pasternak elastic foundation in thermal environment considering fluid-structure interaction, Applied Mathematical Modelling (2019), doi: https://doi.org/10.1016/j.apm.2019.10.023

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Highlights 

The developed formulation is based on a higher order shear deformation theory and the smeared stiffeners.



Linear potential flow theory is applied to describe the fluid–structure interaction.



The solution method is based on truncated Fourier series.



The governing equations of liquid motion are derived using a finite strip element.

Free vibration analysis of FGM cylindrical shells surrounded by Pasternak elastic foundation in thermal environment considering fluid-structure interaction Abdolhossein Baghlani Associate Professor, Department of Civil Engineering and Environment, Shiraz University of Technology, Shiraz, Iran

Majid Khayat* Department of Civil Engineering and Environment, Shiraz University of Technology, Shiraz, Iran

Seyed Mehdi Dehghan Associate Professor, Department of Civil Engineering and Environment, Shiraz University of Technology, Shiraz, Iran

Abstract This paper presents an investigation on partially fluid-filled cylindrical shells made of functionally graded materials (FGM) surrounded by elastic foundations (Pasternak elastic foundation) in thermal environment. Material properties are assumed to be temperature dependent and radially variable in terms of volume fraction of ceramic and metal according to a simple power law distribution. The shells are reinforced by stiffeners attached to their inside and outside in which the material properties of shell and the stiffeners are assumed to be continuously graded in the thickness direction. The formulations are derived based on smeared stiffeners technique and classical shell theory using higher-order shear deformation theory which accounts for shear flexibility through shell’s thickness. Displacements and rotations of the shell middle surface are approximated by combining polynomial functions in the meridian direction and truncated Fourier series with an appropriate number of harmonic terms in the circumferential direction. The governing equations of liquid motion are derived using a finite strip element formulation of incompressible inviscid potential flow. The dynamic pressure of the fluid is expanded as a power series in the radial direction. Moreover, the quiescent liquid free surface is modeled by concentric annular rings. A detailed numerical study is carried out to investigate the effects of power-law index of functional graded material, fluid depth, stiffeners, boundary conditions, temperature and geometry of the shell on the natural frequency of eccentrically stiffened functionally graded shell surrounded by Pasternak foundations.

Keywords: Functionally Graded Materials, Free Vibration, Semi-Analytical Method, Pasternak Elastic Foundation, Fluid-Structure Interaction, Stiffeners 1. Introduction Structures made of shell are widely used in mechanical, civil and aerospace industries and have become more popular due to development of new materials and technologies. Meanwhile, a major concern regarding the use of such structures is their complex dynamic behavior. As a consequence, the dynamic performance of such structures, especially shell structures for liquid storage, has attracted many researchers and engineers till now (e.g. [1-3]). Functionally Graded Material (FGM) belongs to a class of advanced material characterized by variation in properties as the dimension varies. Currently, FGMs have been widely used in aerospace and nuclear industries. The overall properties of FGM are unique and different from any of the individual material that forms it. There is a wide range of applications for FGM which is expected to increase as the cost of material processing and fabrication processes are reduced by improving production techniques (Mahamood et al. [4]). The new class of engineering materials, FGM, has been also subjected to various types of studies including, free vibration, linear and nonlinear buckling and thermo-mechanical analyses ([5-18]). Generally, detailed study of dynamic behavior of structures calls for recognizing the influential parameters affecting the structure frequencies. For instance, the presence of fluid in shell structures leads to significant decrease in structure’s natural frequencies which can consequently increase the probability of resonance. Furthermore, valid and complete structural health monitoring is performed based on the complete recognition of dynamic parameters of the healthy structure. Various aspects of dynamic behavior of composite materials such as laminated composite materials and FGM have been investigated in many studies. Patel et al. [19] studied free vibration characteristics of functionally graded elliptical cylindrical shells using a higher-order theory through the thickness approximations of both in-plane and transverse displacements. The finite element method employed in their study was based on fieldconsistency approach and free from shear and membrane locking problems. Ganapathi [20] explored the dynamic stability behavior of FGM spherical shells subjected to external pressure load. The material

properties were graded in the thickness direction according to the power-law distribution in terms of volume fractions of the constituents of the material. The structural model was based on shear deformation theory and geometric non-linearity was considered in the formulation by using von Karman’s assumptions. Nanda and Sahu [21] analyzed the free vibration response of laminated composite shells with delamination using the finite element method based on first-order shear deformation theory. The employed shell theory used was based on the extension of a dynamic, shear deformable theory according to Sanders’ first approximation for doubly curved shells which can be reduced to Love and Donnell’s theories by means of tracers. Tornabene et al. [22] presented a numerical procedure based on the Generalized Differential Quadrature (GDQ) method to solve the strong form of the differential equations that govern the free vibration problem of some structural elements. In semi-analytical finite strip method, Fourier series expansion was taken into account to approximate displacements and rotations in the direction in which geometry and material properties were invariant. In the other directions, the structure was discretized into several finite elements which can be of low or higher order types. The detailed study of the dynamic behavior of fluid-filled shells or shells exposed to fluid flow is of particular importance because of their widespread use in engineering sciences. The thermo-electroelastic behaviors of a fluid-filled functionally graded piezoelectric material cylindrical thin-shell under the combination of mechanical, thermal and electrical loads have been studied by Dai et al. [23]. Paliwal and Pandey [24] investigated the effect of foundation on the natural frequency of a thin circular cylindrical by using the first order shell theory of Sanders and eigen-frequencies. They reported chiefly effects of the foundation modulus on the radial mode eigen-frequency as well as torsional and longitudinal modes. Chen et al. [25] presented free vibrations of cylindrical shells with non-uniform boundary constraints by using improved Fourier series method. Tj et al. [26] studied the free vibration characteristics of fluidfilled cylindrical shells on elastic foundations where semi-analytical finite element method was applied to obtain required relations for Eigen-value analysis. Xi et al. [27] discussed the coupling between symmetric and anti-symmetric modes in the formulation of free vibrations of partially fluid-filled laminated composite cylindrical shells by using semi-analytical approach. Jeongt and Lee [28] proposed

an analytical method to determine the natural frequencies and modes of either a partially liquid-filled or a partially liquid surrounded cylindrical shell with various classical boundary conditions. Few researchers have also studied the effect of elastic foundation on free vibration of partial-filled fluid shells, e.g. ([2932]). However, very little attention has been paid to FGM shells surrounded by elastic foundation under thermal environment. A free vibration analysis was performed by Park and Kim [33] for a cylindrical shell with an oblique edge elastic foundation subjected to a flowing fluid with electric loads. The fluid was described by the classical potential flow theory. Chen et al. [34] used the state-space method to study the coupled vibration of an inhomogeneous orthotropic piezoelectric hollow cylinder filled with internal compressible fluid based on the three dimensional equations of piezoelasticity. Sheng and wang [13] developed a method to predict the nonlinear dynamic behavior of a fluid-conveying functionally graded cylindrical shell. It was shown that the nonlinear behavior of the fluid-conveying FG cylindrical shell was more evident as the thermal load and nonlinear behavior became small, as the volume fraction exponent was increased. Shen et al. [35] and Daneshjou et al. [36] investigated wave propagation and transient response of a fluid-filled FGM by different methods of finite elements. Other studies have also been conducted to investigate the effect of fluid flows on other materials such as sandwich FGM; e.g. Bich et al. [37]. Saeidifar and Ohadi [38] used Rayleigh-Ritz method to study free vibration of partially fluidfilled functionally graded material cylindrical shells and examined the depth of liquid and boundary conditions on the fundamental frequency of shells. The materials in their study were independent of temperature. As it can be observed, most of the aforementioned studies focus on dynamic behaviors and natural frequencies of shells without elastic foundations and in the absence of stiffeners and liquid simultaneously. However, the elastic foundation significantly affects the dynamic behavior of shells. Consequently, many studies have been conducted on this subject to explore various aspects of such effects. Duc et al. [39] studied mechanical and thermal stability behavior of eccentrically stiffened functionally graded conical shell panels resting on elastic foundations by using finite-element analysis based on classical shell theory. They also studied the effect of elastic foundation on stability of the shell.

Nonlinear vibration of eccentrically stiffened functionally graded double curved shallow shells resting on elastic foundation was investigated by Bich [40]. It was shown that elastic foundation had a strong effect on the nonlinear dynamic response of FGM shells. Bahmyari et al. [41] studied free vibration analysis of thin plates resting on Pasternak foundations by using element free Galerkin method. Additionally, dynamic analysis of shells surrounded by an elastic medium using the finite element method has also been studied by ([35, 41-47]). Investigating the dynamic behavior of shells with stiffeners is of paramount importance in engineering structures. Attaching stiffeners to any structure, especially to shell structures, increases structure’s strength and stability and consequently results in enhancing local behavior at critical points. Thus, true understanding of the dynamic behavior of structures reinforced by various stiffeners can remarkably improve fundamentals of the design of such structures. For this reason, many studies have been made on the dynamic behavior of shells stiffened by various materials in the recent decade (e.g. [41, 48-50]). In the previous studies, the influence of stiffeners on the dynamic behavior of shells with various geometry and materials was confirmed. Geometry, the distance between stiffeners and their positions have been reported as the parameters affecting the natural frequency of shells. Shell structures made of FGM in thermal environment with all kinds of composite materials, attached stiffeners and complicated boundary conditions, including elastic supports, are encountered in many engineering applications. Since FGMs can withstand high thermal gradient, this makes it appropriate for use in jet engine components, storage tanks, pressure vessels, water pipes, pipe lines, casing pipes, processing equipment of chemical planet, deep sea pressure vessels and structures and space plane body that some of these structures contain various liquid. Due to making use of thin walled elements in their structure, stringer and ring stiffeners are usually used to prevent local or global instability of shells [51]. However, from the present literature review, it appears that most of the existing works are limited to isotropic shells partially filled with fluid and imposing simple classical boundary conditions. Moreover, most of the previous studies on mechanical behavior of the stiffened FGM shell structures have employed the classical shell theory to construct the governing equations which does not take into account the

important effects of shear deformation and rotatory inertia [52] as well as temperature. To develop more accurate mathematical models for vibration analysis of stiffened FGM shells on elastic foundations under thermal environment, the present investigation employs the higher order shear deformation theory (HSDT) covering the effects of shear deformation and rotatory inertia to derive thermo–elastic vibration response of the cylindrical shells. Under temperature variation, both the FGM shells and the stiffeners are deformed; therefore, determination of the response of free vibration response of FGM shells with stiffeners becomes more difficult. Hence, this study aims to examine the dynamic behavior of stiffened functionally graded circular cylindrical shells partially filled with fluid with higher order shear deformation theory surrounded on Pasternak elastic foundation using semi-analytical finite strip method. One of the advantages of the finite strip method in shell analysis, compared to other numerical methods, is the fact that less number of degrees of freedom is required to achieve the same accuracy; hence, the computational cost is reduced and the speed of analysis is also increased. The results of this study, give a better insight to engineers for understanding the dynamic behavior of stiffened FGM cylindrical shells. In the present work, the following assumptions and techniques are used to carry out the study: (1) Material properties are assumed to be temperature dependent and continuously vary in the thickness direction in terms of the volume fraction according to a simple power law distribution. (2) Higher order shear deformation theory and the smeared stiffeners are used to establish governing equations and to determine the vibration of stiffened FGM shells. (3) Displacements and rotations of the shell middle surface are approximated by combining polynomial functions in the meridian direction and truncated Fourier series with an appropriate number of harmonic terms in the circumferential direction. (4) Linear potential flow theory is applied to describe the fluid–structure interaction. The governing equations of liquid motion are derived using a finite strip element formulation of incompressible inviscid potential flow.

(5) The dynamic pressure of the fluid is expanded as a power series in the radial direction. Moreover, the quiescent liquid free surface is modeled by concentric annular rings. (6) Pasternak’s reaction–deflection relation is assumed for foundation modeling. The structure of the paper is organized as follows: In section 2 the governing equations are presented and discretization of the equations based on finite strip method is derived. In section 3 numerical results are presented by first verifying the model through solving three benchmark problems and comparing the results with those obtained by previous researches and by finite element method. The effect of various parameters on the natural frequencies of the shell is then presented and discussed. Finally, section 4 concludes the study.

2. Governing equations 2.1 Functionally graded shells surrounded by elastic foundations A typical functionally graded ceramic-metal cylindrical shell is defined in the coordinate system (r, θ, z). The position of a shell point is given by: θ as the circumferential coordinate, s as the meridional coordinate, and z as the coordinate normal to the middle surface. Both ring and stringer stiffeners are attached to the shell and it is assumed that the FGM shell is resting on a Winkler- Pasternak elastic foundation (Fig. 1).

(a)

(b)

(c)

Fig. 1: Geometry and coordinate system of stiffened FGM (a): shell and liquid (b): Stringers and elastic foundation (c): Rings and elastic foundation

2.2 Theoretical formulation Fourier series expansion is employed in order to approximate displacements and rotations in the direction in which geometry and material properties do not vary. In the other directions, the structure is discretized into several finite elements which can be of low or higher order types. In present analysis the shell is divided into several closed strips with their nodal lines located in the circumferential direction. The circumferential variables of the global displacements (u, v, w, s , and  ) can be described by a suitable Fourier series expansion which generally consists of both symmetric and anti-symmetric terms as: NH

u  s, , t   (u co (s)    u cn  s  cos  n   u sn  s  sin(n ) ) e  it n 1 NH

v  s, , t   (v co (s)    v cn  s  cos  n   v sn  s  sin(n) ) e  it n 1

NH

w  s, , t   (w (s)    w cn  s  cos  n   w sn  s  sin(n) ) e  it co

(1)

n 1

NH

s  s, , t   (sco  s    scn  s  cos  n   ssn  s  sin  n   ) e  it n 1

NH

  s, , t   (co (s)   cn  s  cos  n   sn  s  sin(n) ) e  it n 1

where, u, v and w are displacements along s,

and z axes, respectively, and s and  are the

rotations along the s and  axis, respectively. Moreover, n is the circumferential wave number, NH is the number of harmonic terms in the truncated series, and c n and s n are coefficients of Fourier series. Winkler type elastic foundation is the simplest model to describe the mechanical behavior of elastic supports. In this model interaction between lateral springs is ignored. Pasternak foundations are characterized by two independent elastic constants which are derived by extension of the Winkler’s model (Bahmyaria et al. [41]) (Fig 1-b,c). The reaction–deflection relation of Pasternak foundations is given by:

USp 

1 w 2 w 2 (K w w 2 K p (( ) ( ) )ds 2 s R

(2)

where K w (N / m3 ) denotes the radial stiffness of the spring layer and K p (N / m) denotes the transverse stiffness of the shear layer showing the effect of the shear interactions of the vertical elements. In this study both radial stiffness and transverse stiffness are considered [53].

2.3 Constitutive equations It is assumed that the FGM shell is made of a mixture of a metal phase (denoted by m) and a ceramic phase (denoted by c), with the material composition varying smoothly along its thickness direction only. Thus, the materials properties of FGMs can be expressed as (Sofiyev [54]-Torki et al. [55]): Feff (z)  Fm Vm (z)  Fc Vc (z)

(3)

where Feff is the effective mechanical or physical property and Fm and Fc are the material properties of the metal and ceramic, respectively, and they may be expressed as a function of temperature: 2 3 Feff (T)  F0 (F1T1  FT 1  F2 T  F3T )

(4)

in which F0 , F1 , F1 , F2 and F3 are the coefficients of temperature T (K) expressed in Kelvin and are unique to the constituent materials. Moreover, Vm and Vc stand for the volume fraction of metal and ceramic, respectively and they are related by: Vm (z)  Vc (z)  1

(5)

The ceramic phase has greater elasticity modulus and lower density and Poisson's ratio compared to the metal phase. Vm can be expressed by the power law as:

N

 z 1 Vm     , N  0 h 2

(6)

where N is the power law exponent which is a critical parameter to control the distribution of the constituents. From above relations, the equations used to estimate the effective material properties of the shell are based on the power law distribution are as follow:

z 1 E eff  (E c  E m )(  ) N  E m h 2 z 1 N eff  (c  m )(  )   m h 2 z 1  eff  ( c   m )(  ) N   m h 2 z 1 G eff  (G c  G m )(  ) N  G m h 2 z 1 N  eff  ( c   m )(  )   m h 2

(7)

In this study, the higher order shear deformation theory and the smeared stiffeners are used to establish governing equations and to determine the vibration of stiffened FGM shells. The displacement field corresponding to the higher order shear deformation theory is given by (Szekrenyes [56]): w(s, , t) ) s w(s, , t) v(s, , z, t)  v(s, , t)  z  (s, , t)  c1 z 3 ( (s, , t)  )  w(s, , z, t)  w(s, , t)

u(s, , z, t)  u(s, , t)  z s (s, , t)  c1 z 3 (s (s, , t) 

(8)

The strains across the shell thickness at a distance z from the mid-surface according to Sanders are given as: 0 1 3ss  ss  ss  ss     0   1  3 3         z     z       0   1   3  s    s    s    s  0 2   sz    sz  2   sz      0 z  2    z    z    z 

where

(9)

 u   s  0 ss     0   v         0       s   u v     s     s   s       1              1      s   s      s    1 ss

 s   s  3 ss     3         c1     3      s   s    s     

(10)

w     0   sz   s s   0     z    w      w   s  s    sz2   2   c 2     w    z     

where c1  4 / 3h 2 and c2  3c1 . These coefficients are determined using the condition that the transverse shear stresses vanish on the shell’s top and bottom surfaces, or the corresponding strains be zero on these surfaces. The temperature variation can be gradually or uniformly distributed along the thickness of the shell [10, 57-60]. In the study, we assume that the temperature changes between top and bottom surfaces ( T ) is uniform across the thickness. Hook’s law for a stiffened FGM shell under temperature variation is

defined as:  ss  E  ss        (1  )ET 2     1      ss   s    s  E      sz   2(1  )   sz       z   z 

(11)

and for stiffeners  sss   ss  1 ET  s   Es       1     

(12)

where the second terms in RHS of Eqs. (11) and (12) are due to thermal effects. If the ceramic to metal elastic modulus’s ratio is taken more than 10, the shear correction factor obtained based on inhomogeneous shell assumption can differ remarkably from moderately thick shells. In the current study the elastic modulus’s ratio is less than 7; thus, isotropic shear correction factor has been used. Unlike FSDT, in the higher-order shear deformation theory, the transverse shear stresses are more correctly approximated throughout the thickness and consequently no transversal shear correction factors are needed [61]. The stress resultants are related to the strains by the following relations: 1  s  s  ss dA ss   s A ss  1    s dA sr   s r Asr       N ss  ss   0      1   N   s   z  dA    N s   ss s  s  s   s s A ss     0    [A] [B] [D]   M ss  h / 2 zss      1    1  s M     z  dz    z dA sr   [B] [D] [F]     M   h / 2  z   s 2 A sr  [E] [F] [H]  3       s   s       0 3 Pss   z ss       3  1  3 s P   z     z ss dA ss  P   3   s s A ss   s   z  s     1 z 3 s dA s  r  s 2 As  r   0      sz  Qsz  0    h / 2  [A] [D]   Qz   z       2  dz     [D] [F]   2  R sz   h / 2 z sz  R z   2  z z 

where

(13)

h/2

(Aij , Bij , Dij , Eij , Fij , Hij ) 



Sij (1, z, z 2 , z3 , z 4 , z 6 ) dz i, j  1, 2, 6

h / 2 h/2

(Aij , Dij , Fij ) 



Sij (1, z 2 , z 4 ) dz i, j  4,5

(14)

h / 2

The equations for calculating A, B, D, E, F and H are given in Appendix A. The equations of motion of the higher-order theory are derived using the dynamic version of principle of virtual displacements. The virtual strain energy δU , the virtual kinetic energy δK and the virtual potential energy δV are given by (Reddy [62]):



U 

h/2

{



0 [ss (ss0  z1ss  c1z 3 3ss )    (   z1  c1z 3 3 )

 (s,  )  h / 2 0 s ( s0  z1s  c1z3  3s )  sz ( sz  z 2  sz2 )   z ( 0z  z 2  2z )]dz}ds d 





(15)

(N ss ss0  M ss 1ss  c1Pss 3ss  N   0  M  1  c1P 3

 (s,  ) 0 N s  s0  M s 1s  c1Ps  s3  Qs  sz  c 2 R sz  sz2  Qz  0z  c 2 R z  2z )ds d

(16)

V    q w ds d  s

K 



h/2

{



 (s,  )  h / 2

eff [(

  u s s u  z s  c1z 3 s )( z  c1z 3 ) t t t t t t

  v   v v v (  z   c1z 3  )( z  c1z 3 ) ]dz}dsd t t t t t t t t   u   s u u   ((Io  I1 s  c1 I3 s )  (I1  I 2 s  c1 I 4 s ) t t t t t t t t 

(17)

  s   v u v  I 4 s  c1 I6 s )  (Io  I1   c1 I3  ) t t t t t t t t       v u (I1  I2  c1 I 4 )  c1 (I3  I4  c1 I 6 ) )ds d t t t t t t t t c1 (I3

2.4 Fluid equations and boundary conditions The fluid is described by the potential flow theory. For the irrotational flow of an incompressible inviscid liquid, the velocity potential (r,s, , t) satisfies the Laplace equation: 2  

 2 (r, , z, t) (r, , z, t)  2 (r, , z, t)  2 (r, , z, t)    0 rr r 2 r 2 2 z 2

The velocity potential must satisfy the following boundary conditions:

(18)

a) at the rigid tank bottom: v.n 

w(, z, t) (R, , z, t)  t r

(19)

b) at the tank : v.n  0z  H 

(r, , H, t) 0 z

(20)

where v is velocity and n is normal vector. c) at the free surface (for small amplitude waves):  (r, , 0, t) (r, , 0, t)   z t   (r,  , 0, t)   g L (r, , 0, t)  0 L t 

 2 (r, , 0, t) (r, , 0, t) g 0  2 z  t   2  1  Pdy  Pdy  0  g t 2 z

(21)

where Pdy is hydrodynamic pressure and L is density of liquid. The liquid velocity potential function (r,s, , t) can be expressed as:

n     jn r  jn z   r z      r (r, , z, t)   Aon (t)     Ain (t) I n ( i ) cos( i )   B jn (t) J n ( ) cos( )   cos(n) H H  j1  R R    R  i 1  n 1  

(22)

The coefficients A on , Ain and B jn can be obtained by imposing the aforementioned boundary conditions (See Appendix B). Functional representation of equations (11) to (13) by using variational principle (Hamilton principle) can be expressed as follows (Haroun [63]). t2

t2

t1 V

t1 S1

I  f   (.) dV dt  f   (

2   w    g) ds dt  f   ( ) ds dt t t t t1 S2

t

(23)

where  is the free surface displacements measured from the quiescent liquid free surface which can be expressed as: 

(r, , t)   n (r, t) cos (n)

(24)

n 1

The quiescent liquid free surface is divided to concentric annular rings. Thus, the free surface displacement is defined in terms of nodal displacements of finite elements (Fig 1a).

2.5 General formulation Governing equations of the higher-order theory are derived using dynamic version of the principle of virtual displacements: t

 (U  V  K  I)  0

(25)

0

We first substitute δU , δV , δI and δK from Eqs. 15, 16, 17 and 24 into the virtual work statement in Eq. 25 by noting that the virtual strains can be written in terms of the generalized displacements. Then, integrating by parts to relieve the virtual generalized displacements, δu , δv , δw , δs and δ and using the fundamental lemma of calculus of variations yields to the following Euler-Lagrange equations:   K ss     0

  Mss1 0  0 0   2         0 0 K ff    0

0  Mss2  0  Mfs

Msf    d       0 Mff     

(26)

where d is  u v w s   , K ss and Mss are the global stiffness matrix and the global mass matrix of shell, T

1

respectively, K ff is the free surface stiffness matrix, M ss is added mass matrix of the liquid-shell system, 2

M ff is the free surface mass matrix, and M fs and Msf is the coupling mass matrix which are all given in

Appendix C.

3. Results and discussion In this section, the numerical results of free vibration of FGM cylindrical shells reinforced by stiffeners using the aforementioned formulation and numerical finite strips approach are presented. A MATLAB code has been prepared to find eigenstructure of the shell based on the new developed formulation for considering fluid-structure interaction, presence of stiffeners, thermal effects and elastic foundation simultaneously. Firstly, the model is verified by solving three benchmark problems and comparing the results with those obtained in previous studies and by finite element method (FEM). Secondly, the effects of various parameters including volume fraction index, fluid level, stiffeners, elastic foundation, temperature and boundary condition on natural frequencies of the shell are investigated.

3.1 Model verification For verification of the developed numerical method presented in this study, three problems are considered as follows. The results of the first two problems are compared with those obtained by other researchers and the results of the third problem are compared with those obtained by traditional finite element method (FEM). 3.1.1 Simply supported FGM shell in absence of fluid In the first problem, we compare the natural frequencies of functionally graded simply supported cylindrical shells in the absence of fluid with the previous results. The properties of constituents, including the Young’s modulus, Poisson’s ratio, density and geometry are given in Table 1. It was assumed that FGM shells were made of a mixture of nickel and stainless steel. Table 2 presents the results of this study and those obtained by Kim [30] and Loy [64]. Table 1: Properties of the functionally graded materials

Material Nickel stainless steel

Young’s modulus (GPa) 205.098 207.788

Geometry

R = 1m

L/R = 20

Poisson’s ratio (  ) 0.31 0.24

Density ( kg / m3 ) 8900 2370

m=1

Table 2: Comparison of natural frequencies (Hz) for an empty FGM cylindrical shell

N=0.5 h/R

0.002

0.05

n 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

Loy et al. [64]

Kim [30]

13.321 4.5169 4.1911 7.0973 11.336 16.594 22.826 30.022 38.180 47.300 13.345 32.655 90.982 173.63 279.48 407.83 557.99 729.18 920.61 1131.4

13.321 4.5168 4.1911 7.0972 11.336 16.594 22.826 30.023 30.181 47.301 13.345 32.702 91.319 174.83 282.57 414.39 570.25 750.12 953.99 1181.9

N=5 Present study 13.179 4.412 4.123 6.955 11.237 16.225 22.597 29.455 37.931 46.354 13.125 31.681 88.651 170.354 277.719 396.766 541.886 721.859 909.885 1120.086

Loy et al. [64]

Kim [30]

12.998 4.4068 4.0892 6.9251 11.061 16.192 22.273 29.295 37.255 46.152 13.021 31.863 88.778 169.43 272.72 397.98 544.52 711.60 898.44 1104.2

12.998 4.4068 4.0891 6.9251 11.061 16.192 22.273 29.296 37.257 46.155 13.021 31.910 88.109 170.60 275.73 404.36 556.45 731.97 930.90 1153.3

Present study 12.738 4.306 4.064 6.837 10.840 15.759 21.831 28.449 36.130 44.774 12.878 31.586 86.234 167.927 267.947 386.053 542.750 700.727 875.979 1083.110

It is clear from Table 2 that the results of present method are in good agreement with those of Kim [30] and Loy et al. [64]. The maximum difference is about 3.02%.

3.1.2 Simply supported fluid-filled shell The second comparative study is carried out for a simply supported fluid-filled isotropic cylindrical shell. The shell has the following characteristics: R  25 m L  30 m H  21.6 m h  0.03m s  7850 Kg / m3 E  206 Gpa   0.3 f  1000 Kg / m3

The first five natural frequencies for two modes of Bulging and Sloshing are compared with the results of Kondo et al. [65], Amabili [66] and Saeidifar [38] in Table 3. Table 3: Comparison of bulging and sloshing frequencies (rad/s) for the simply supported fluid-filled isotropic cylindrical shell

Bulging modes Mode 1 2 3 4 5

Present Study 21.731 42.796 55.074 64.972 74.227

Sloshing modes

Kondo [65]

Amabili [66]

Saeidifar [38]

22.096 43.762 56.829 66.888 75.347

22.244 44.010 57.191 67.291 75.850

22.227 43.997 57.171 67.255 75.800

Present study 1.187 1.588 1.916 2.197 2.481

Kondo [65]

Amabili [66]

Saeidifar [38]

1.2238 1.6582 1.9969 2.2853 2.5409

1.2244 1.6591 1.9980 2.2865 2.5422

1.2244 1.6591 1.9980 2.2866 2.5424

It can be seen from Table 3 that the approach proposed in current study gives satisfactory results close to those found by other researchers. The maximum discrepancy is limited to 3.7% for bulging modes and 4.3% for sloshing modes.

3.1.3 Stiffened FGM cylindrical shell For the third model verification, a new problem involving fluid-structure interaction, elastic foundation, stiffeners and thermal effects is defined in this section and the results are compared with those obtained by FEM. Consequently, these results can be used for verification of other numerical approaches in the future. For this purpose, a ceramic-metal FGM shell which consists of Silicon nitride (ceramic) and Stainless steel (metal) with the properties reported in Table 4 is considered.

Table 4. Temperature dependent coefficients for materials [67]

Silicon nitride ( Si3 N4 )

Coefficients

F0 F1 F1 F2 F3

Stainless steel ( SUS304 )

Ec (Pa)

c

c (kg/m3 )

m

E m (Pa)

m

m (kg/m3 )

c

348.43e+9

0.24

2370

5.8723e-6

201.04e+9

0.3262

8166

12.330e-6

0

0

0

0

0

0

0

0

-3.070e-4

0

0

9.095e-4

3.079e-4

-2.002e-4

0

8.086e-4

-2.160e-7

0

0

0

-6.534e-7

3.797e-7

0

0

-8.946e-11

0

0

0

0

0

0

0

The support is assumed to be fixed-free and the stiffeners are located outside the surface. Geometrical properties of the shell and the stiffeners and elastic foundation parameters are: h s  0.01m

d s  0.0025m

n s  30

h r  0.01 m

d r  0.0025m

n r  30

R  0.5 m

L  0.75 m

h  0.00625

Tm  Tc  300 K K w  0.5 107 N / m3

K p  0.5 105 N / m

For verification purposes, a numerical model based on finite element method (FEM) for the aforementioned fluid-shell system is developed by utilizing ABAQUS and the results are compared with those acquired by the presented method in this study. In order to analyze the structure using threedimensional element of the software, shell and stiffeners are first discretized to appropriate number of elements in three longitudinal, circumferential and thickness directions (Wattanasakulpong and Chaikittiratana [52]). In the FE analysis, the stiffened FGM container is modelled by about 39500 eightnodded C3D8R solid elements with linear geometric order, reduced integration and hourglass control. In the direction of thickness, the shell and stiffeners are divided into at least 1800 solid elements so that constant properties can be attributed to each element. They could therefore be simulated correctly by changing FGM mechanical properties through thickness direction. The Connectivity types of springs are standard. The liquid region is discretized by acoustic elements (AC3D8), which are eight-node brick acoustic elements with linear interpolation functions. Acoustic three-dimensional finite elements based on linear wave theory are used to represent the hydrodynamics of fluid. The location of each node on the constrained surfaces of liquid coincides exactly to the location of corresponding node of structure. Along

the interface between liquid and shell, the fluid surface is tied to the shell surface in normal direction to satisfy compatibility conditions. The liquid is divided into 45730 elements of AC3D8 type. A finer grid for the problem has negligible effect on the results. Table 5: Comparison of bulging frequencies (rad/s) for the filled stiffened FGM cylindrical shell with FEM

n (Circumferential wave number) FEM

N=0

N=4

N=16

The proposed method

H/L

1

2

3

4

1

2

3

4

0.2

1152.90

642.41

411.21

372.64

1129.84

638.94

407.51

370.03

0.4

944.28

586.02

390.30

359.00

934.56

573.98

384.96

352.79

0.6

687.36

455.77

321.68

303.42

681.17

442.87

318.27

297.54

0.8

536.09

345.13

244.58

234.73

532.18

335.12

241.69

229.89

1.0

434.75

265.68

184.47

178.55

425.14

264.62

180.80

174.89

0.2

642.90

358.64

233.39

221.26

636.98

350.24

231.99

214.62

0.4

601.32

345.20

228.13

217.70

589.09

344.51

227.40

216.62

0.6

493.61

301.88

206.94

200.65

484.08

295.72

205.08

199.16

0.8

400.72

247.07

172.68

170.32

395.35

241.34

169.95

168.49

1.0

332.27

199.33

138.68

138.59

328.78

197.18

136.97

137.22

0.2

586.75

327.86

214.93

207.41

574.31

325.56

208.68

201.07

0.4

553.29

316.73

210.52

204.37

542.62

307.20

208.83

198.32

0.6

461.34

279.95

192.46

189.58

447.27

279.39

190.56

184.78

0.8

377.15

231.38

162.26

162.49

373.23

226.08

159.60

159.30

1.0

313.87

187.92

131.40

133.41

312.17

187.35

128.69

130.63

The maximum difference of about 3% shows that the proposed method can precisely predict the model properties of the stiffened FGM container surrounded by Pasternak elastic foundation in thermal environment considering fluid-structure interaction.

3.2 Parametric study After model verification, a comprehensive parametric study is carried out to investigate the effects of influential parameters on natural frequencies of the shell as follows.

3.2.1 The effect of volume fraction index (N) Fig. 2 shows the effect of volume fraction index (N) on natural frequencies of the shell. The natural frequencies for different circumferential modes and the first longitudinal mode for various N values are plotted. 950 N=0 N=2 N=4 N=8 N=16

Natural frequency (Hz)

850

750

H/L=0.4 m=1

650 550 450 350 250 150 1

2

3 4 Circumferential wave number, n

5

6

Fig 2: The effect of volume fraction index N on natural frequencies of fluid-filled cylindrical shell for various circumferential wave numbers

In Fig. 3, the natural frequency for the third circumferential mode and the four first longitudinal modes for various N values are shown.

1150

Natural frequency (Hz)

950

750 550 H/L=0.4 n=4

350

150 1

2

3

N=0 N=2 N=4 N=8 N=16

4

longitudinal wave number, m Fig 3: The effect of volume fraction index N on natural frequencies of fluid-filled cylindrical shell for various longitudinal wave numbers

According to Figs 2 and 3, it can be found that the natural frequency of the shell reduces by increasing N value for all circumferential and longitudinal modes. Moreover, it can be seen that by increasing values of N from N=0 (Isotropic material) to N=2, the natural frequency is very abruptly reduced. Further increase in N decreases the natural frequency more slowly.

3.2.2 The effect of fluid level In this section, the effect of fluid height (H) on the natural frequency of stiffened shell is investigated. In Figs. 4 to 6, the fluid height and the natural frequency of cylindrical shell for different N and H/L values are shown.

1350

H/L=0 H/L=0.2 H/L=0.4 H/L=0.6 H/L=0.8 H/L=1

Natural frequency (Hz)

1200 1050 900

N=0

750 600 450 300 150 1

2

3

4

5

6

Circumferential wave number, n (a)

Natural frequency (Hz)

700 H/L=0 H/L=0.2 H/L=0.4 H/L=0.6 H/L=0.8 H/L=1

550

400

N=4

250

100 1

2

3

4

Circumferential wave number, n (b)

5

6

Natural frequency (Hz)

600

H/L=0 H/L=0.2 H/L=0.4 H/L=0.6 H/L=0.8 H/L=1

500 400

N=16

300 200 100 1

2

3

4

5

6

Circumferential wave number, n (c)

600

H/L=0 H/L=0.2 H/L=0.4 H/L=0.6 H/L=0.8 H/L=1

Natural frequency (Hz)

500 400

N=64

300 200 100 1

2

3 4 Circumferential wave number, n

5

6

(d)

Fig 4: The effect of fluid level on natural frequencies of the shell for m=1 (a): N=0, (b): N=4, (c): 16 and (d): N=64

As it is predictable and can be also seen in Fig. 4, increasing fluid height leads to reduction of the natural frequency of the structure, for all circumferential modes and for both isotropic and FGM materials. The natural frequency reduces slowly by increasing fluid height up to value H/L=0.4, while after this value the reduction of natural frequency becomes more rapid. Moreover, Fig. 4 shows that the reduction of frequency with increasing fluid height for different circumferential modes has almost the same trend.

3.2.3 The effect of stiffeners In this section the effect of various types of stiffeners, location of stiffeners, and spacing between stiffeners on natural frequencies of the cylindrical FGM shell is studied. In Fig. 5, the natural frequencies of fluid-filled shell are shown for various circumferential modes for the case of orthogonal stiffeners. According to Fig. 5, the trend of frequency changes before and after the fundamental frequency mode (n=4) is reversed with increasing number of stiffeners. Actually, the natural frequency decreases with increasing the number of stiffeners before the fundamental mode and oppositely after the fundamental mode. This trend is the same for all three considered fluid levels. However, for lower H/L ratios the difference in frequencies for the modes before the fundamental mode is more obvious. This trend is also valid for increasing stiffener in the cases of stringers (Fig 6) and ring (Fig 7). 500

ns=45 ns=30 ns=15 ns=5

Natural frequency (Hz)

450 400

nr=45 nr=30 nr=15 nr=5

H/L=0.6

350 300 250 200 150 1

2

3

4

circumferential wave number, n (a)

5

6

350

ns=45 ns=30 ns=15 ns=5

Natural frequency (Hz)

300

nr=45 nr=30 nr=15 nr=5

H/L=1

250 200 150 100 1

2

3 4 circumferential wave number, n

5

6

(b)

Fig 5: Variation of natural frequencies of cylindrical shell reinforced by orthogonal stiffeners with respect to circumferential wave number for m=1 (a): H/L=0.6 and b: H/L=1

500 ns=0 ns=0 ns=0 ns=0

450

H/L=0.6

400 Natural frequency (Hz)

nr=45 nr=30 nr=15 nr=5

350 300 250 200 150 1

2

3 4 circumferential wave number, n (a)

5

6

350

ns=0 ns=0 ns=0 ns=0

Natural frequency (Hz)

300

nr=45 nr=30 nr=15 nr=5

H/L=1

250

200

150

100 1

2

3

4

5

6

circumferential wave number, n (b)

Fig 6: Variation of natural frequencies of cylindrical shell reinforced by ring stiffeners with respect to circumferential wave number for m=1 (a): H/L=0.6 and b: H/L=1

Natural frequency (Hz)

550

ns=45 ns=30 ns=15 ns=5

450

nr=0 nr=0 nr=0 nr=0

H/L=0.6

350

250

150

1

2

3

4

circumferential wave number, n (a)

5

6

Natural frequency (Hz)

350

ns=45 ns=30 ns=15 ns=5

300

nr=0 nr=0 nr=0 nr=0

H/L=1

250 200 150 100 1

2

3

4

5

6

circumferential wave number, n (b)

Fig 7: Variation of natural frequencies of cylindrical shell reinforced by stringer stiffeners with respect to circumferential wave number for m=1 (a): H/L=0.6 and b: H/L=1

In order to compare natural frequencies of the shell for various types of stiffeners and for two different fluid levels, Fig 8 is presented. As Fig. 8 indicates, among the three arrangement types (orthogonal, stringer and ring), with the same number of stiffeners for different modes and both fluid heights, the stringer type has the lowest and the ring type has the highest natural frequency. In addition, the trend of frequency change with fluid level alteration is almost the same. 700 Orthogonal ns=15 nr=15 Stringer ns=30

Natural frequency (Hz)

600

Ring nr=30

H/L=0.6

500 400 300 200 1

2

3 4 cicumferential wave number, n (a)

5

6

500 Orthogonal ns=15 nr=15

Natural frequency (Hz)

Stringer ns=30 Ring nr=30

H/L=1

400

300

200 1

2

3 4 cicumferential wave number, n

5

6

(b)

Fig 8: Variation of natural frequencies of cylindrical shell reinforced by three types of stiffeners with respect to circumferential wave number for m=1 (a): H/L=0.6 and b: H/L=1

In order to examine the effect of stiffeners’ location on the natural frequencies of the shell, Fig. 9 is prepared. The stiffeners are assumed to be pure-metal if they are located inside the surface of the shell and pure-ceramic if they are located outside the surface of the shell. 700 Inside stiffeners

Natural frequency (Hz)

Outside stiffeners

600

H/L=0.5

500

400

300 1

2

3

4

circumferential wave number, n (a)

5

6

500 Inside stiffeners Outside stiffeners

Natural frequency (Hz)

450

H/L=1

400

350

300

250 1

2

3

4

5

6

circumferential wave number, n (b)

Fig 9: The effect of stiffeners location (inside or outside) on natural frequencies of the shell for various circumferential wave numbers and m=1 (a): H/L=0.5 and (b): H/L=1

According to Fig. 9 it can be seen that the placement of the stiffeners on the outer surface of the shell increases the natural frequency in various circumferential modes. Furthermore, the stiffeners’ location does not change the number of the fundamental modes of the fluid-filled shell. In Fig. 10, the effect of the stiffeners’ location on the natural frequencies of the shell is plotted for various longitudinal modes. Fig. 10 indicates that in longitudinal modes, the placement of stiffeners in the outer surface of the shell slightly increases the natural frequency of the fluid-filled shell similar to behavior seen for circumferential modes (Fig. 9).

1250

Natural frequency (Hz)

1050 850 650 H/L=0.5

450

Inside stiffeners Outside stiffeners

250 1

2

3 longitudinal wave number, n

4

5

(a)

1000

Natural frequency (Hz)

850 700 550 H/L=1

400

Inside stiffeners Outside stiffeners

250 1

2

3 longitudinal wave number

4

5

(b)

Fig 10: The effect of stiffeners location (inside or outside) on natural frequencies of the shell for various longitudinal wave numbers and n=5 (a): H/L=0.5 and (b): H/L=1

The influence of spacing between stiffeners on natural frequencies of the FGM cylindrical shell is presented in Table 5 in different modes for N=8 and two values of H/L.

Table 5: Computed values of natural frequency for various spacing of stiffeners and two different values of H/L Ss

H/L=0.6 n

Sr

1

2

3

4

5

6

7

8

0.15

0.05

351.558

196.591

132.533

145.347

202.303

269.115

348.249

446.954

0.1

0.05

350.901

196.462

132.484

144.900

201.678

268.758

347.714

446.020

0.05

0.05

349.063

196.066

132.327

143.691

199.904

267.652

346.174

443.414

0.05

0.15

351.547

198.140

132.327

143.691

199.904

267.652

346.174

443.414

0.05

0.1

350.948

197.624

129.357

130.489

176.904

238.301

306.660

389.592

0.05

0.05

349.063

196.066

128.229

125.259

167.342

225.884

290.602

367.939

0.15

0.05

231.227

125.958

86.691

98.076

146.011

215.450

302.503

406.971

0.1

0.05

231.227

125.192

86.042

98.055

145.746

214.956

301.732

405.830

0.05

0.05

231.186

124.920

85.815

97.988

144.954

213.482

299.439

402.453

0.05

0.15

231.549

126.033

86.691

97.988

144.954

213.482

299.439

402.453

0.05

0.1

231.469

126.022

83.983

87.995

125.859

183.999

257.659

346.063

0.05

0.05

231.186

125.958

82.991

84.137

118.260

172.148

240.809

323.299

H/L=1

As it can be also seen in Table 5, the values of natural frequencies increase with increasing the distance between stiffeners (i.e. decreasing the number of stiffeners) for different modes. 3.2.4 The effect of surrounded elastic foundations The effect of elastic foundations on natural frequencies of the FGM shell is shown in Fig. 11. The value of natural frequencies increases when the modulus values of the Pasternak foundation ( K p ) and the Winkler elastic foundation ( K w ) increase. Furthermore, the Winkler elastic foundation has larger effects on the natural frequencies of stiffened functionally graded circular cylindrical shell partially filled with fluid than the Pasternak foundation.

550

Kw=0 Kw=1.0 E7 Kw=3.0 E7 Kw=5.0 E7 Kw=7.0 E7

500

frequency (Hz)

450

400

Kp=0 Kp=0 Kp=0 Kp=0 Kp=0

350 300 250 200 150 1

2

3 4 circumferential wave number, n

5

6

(a)

550

Kw=0 Kw=0 Kw=0 Kw=0 Kw=0

500

frequency (Hz)

450

Kp=0 Kp=1.0 E5 Kp=2.0 E5 Kp=3.0 E5 Kp=4.0 E5

400 350 300 250 200 150 1

2

3 4 circumferential wave number, n (b)

5

6

550

Kw=0 Kw=1.0 E7 Kw=2.0 E7 Kw=1.0 E7 Kw=2.0 E7 Kw=3.0 E7

500

frequency (Hz)

450

Kp=0 Kp=1.0 E5 Kp=1.0 E5 Kp=2.0 E5 Kp=2.0 E5 Kp=3.0 E5

400 350 300 250 200 150 1

2

3

4

5

6

circumferential wave number, n (c)

Fig 11: The effect of properties of elastic foundation on natural frequencies of the shell for various circumferential wave numbers for ns  n r  25 and m=1 by change of (a): K w (b): K p (c): K w and K p

Fig. 12 shows the effect of the elastic foundation on natural frequencies of FGM shell for various longitudinal wave numbers. It is clear that the natural frequency increases as the module of Winkler foundation increases. The modulus parameter of the Pasternak foundation has a similar behavior as well. 950 850

frequency (Hz)

750 650 550 450 350 250

Kw=0

Kp=0

Kw=1.0 E7

Kp=0

Kw=3.0 E7

Kp=0

Kw=5.0 E7

Kp=0

Kw=7.0 E7

Kp=0

150 1

2

3 circumferential wave number, n

4

5

(a)

950 850

frequency (Hz)

750 650 550 450 Kw=0 Kw=0 Kw=0 Kw=0 Kw=0

350 250

Kp=0 Kp=1.0 E5 Kp=2.0 E5 Kp=3.0 E5 Kp=4.0 E5

150 1

2

3

4

5

circumferential wave number, n (b)

950 850

frequency (Hz)

750 650 550 450 350

Kw=0 Kw=1.0 E7 Kw=2.0 E7 Kw=2.0 E7 Kw=2.0 E7 Kw=3.0 E7

250 150

1

2

3

4

Kp=0 Kp=1.0 E5 Kp=1.0 E5 Kp=2.0 E5 Kp=2.0 E5 Kp=3.0 E5

5

circumferential wave number, n (c)

Fig 12: The effect of properties of elastic foundation on natural frequencies of the shell for various longitudinal wave numbers for ns  n r  25 and n=4 by change of (a): K w (b): K p (c): K w and K p

3.2.5 The effect of temperature In the problems investigated so far, the temperature on the top and bottom of the shells was considered to be identical. In this section, the effect of temperature difference between the top and bottom of the FGM shells is taken into account. For this purpose, the temperature of iron is considered to be equal to 300 ºK while for carbon it is assumed to be equal to 300, 700 and 1100 ºK, respectively. Fig. 13 shows, the natural frequency curves of the functionally graded stiffened shell under different thermal environmental conditions for N = 8 and two different values of H/L. Fig. 14 shows that with increasing temperature difference in the inner and outer sides of the shell, the structure’s natural frequency decreases which is more visible for lower circumferential modes. This trend is similar for both fluid levels. 550 ΔT=0 ΔT=400 ΔT=800

frequency (Hz)

450

H/L=0.5

350

250

150 1

2

3 4 circumferential wave number, n (a)

5

6

350

ΔT=0 ΔT=400

frequency (Hz)

300

ΔT=800 H/L=1

250 200 150 100 1

2

3 4 circumferential wave number, n

5

6

(b)

Fig 13: The effect of temperature difference on natural frequencies of the shell for various circumferential wave numbers for ns  n r  20 and m=1 (a): H/L=0.5 and (b): H/L=1

3.2.6 The effect of boundary conditions Fig. 14 shows the effect of imposing various boundary conditions on natural frequencies of a partially fluid-filled stiffened FGM cylindrical shell for N=8. It is clear from Figs. 14(a) and (b) that the boundary conditions remarkably affect natural frequencies of both circumferential and longitudinal waves. Indeed, the fixed-free support has the lowest and the fixed-fixed support has the highest natural frequencies. As a result, the more degree of freedom of the supports is, the higher the natural frequencies of the stiffened shells are. The variation trend of the frequency with changing the mode number for the three out of four supports (i.e. fixed-fixed, pinned-free, and pinned-pinned supports) is almost the same and close to each other. However, the variation trend is different for the fixed-free support before and after the fundamental mode, in such a way that, the intensity of the changes is more at the beginning. As Fig. 14(a) indicates, the frequency varies more rapidly with changing the circumferential mode for the fixed-free support than the other three supports. Fig. 14(b) shows that the aforementioned behavior of variations of natural frequencies with the circumferential modes is also valid for longitudinal modes. Fig. 14(c) depicts the effect of imposing various boundary conditions on the natural frequencies of the shell for different values of H/L. As the figure shows, the variations of natural frequency of the stiffened cylinder shell with

increasing fluid height are approximately identical for the three out of four supports (i.e. fixed-fixed, pinned-free, and pinned-pinned supports). But this trend is not linear, so that it is more intense for the fluid percentage of more than 40%. The variation of the fixed-free support with changing the fluid height is closer to linear trend than the other supports. 400 Fixed-Free

Natural frequency (Hz)

Fixed-Fixed

330

Pined-Pined Fixed-Pined H/L=1 m=1

260

190

120

50 1

2

3 4 cicumferential waves number, n

5

6

(a)

Natural frequency (Hz)

800

650

500

350

H/L=1 n=3 Fixed-Free Fixed-Fixed Pined-Pined Fixed-Pined

200

50 1

2

3 longitudinal waves number (b)

4

5

400 Fixed-Free

Natural frequency (Hz)

350

Fixed-Fixed Pined-Pined

300

Fixed-Pined n=3 m=1

250 200 150 100 50 0 0

0.2

0.4

0.6

0.8

1

H/L (c)

Fig 14: The natural frequencies of partially fluid-filled with different boundary conditions for ns  n r  20 versus (a): circumferential wave number, (b): longitudinal waves number, (c):H/L ratio

4. Conclusion In this study, the dynamic behavior of partially fluid-filled cylindrical shells made of functionally graded materials surrounded by elastic foundation under thermal environment is investigated. The higher order shear deformation theory and the smeared stiffeners are used to establish governing equations and to determine the natural frequencies of stiffened FGM shells under the aforementioned assumptions. The dynamic pressure of the fluid is expanded as a power series in the radial direction. Moreover, the quiescent liquid free surface is modeled by concentric annular rings. Pasternak’s reaction–deflection relationship is assumed to model the foundation. The effect of various parameters on the natural frequency of the structure is explored. The numerical results support the following conclusions: 

The assumptions used in this paper result in satisfactory accuracy for modeling free vibration of partially fluid-filled circular cylindrical shells made of functionally graded materials.



By increasing volume fraction index (N) value and the fluid level (H), the magnitude of the natural frequency of the stiffened shell is reduced. For different N values, by moving from isotropic (N=0) mode to composite mode, the variations of the frequency are more severe at the

beginning; however, for the fluid height, the reduction is first moderate and then it becomes more severe for ratios above H/L=0.5. 

Stiffeners significantly affect the dynamic behavior of FGM cylindrical shells. Increasing the number of stiffeners (stringers and rings) results in reducing the natural frequency. However, the trend of variations is not the same for different arrangements (stringers, rings, and orthogonal).



The positions of stiffeners (inside or outside the shell) have large effect on the natural frequencies of FGM shells.



The value of the natural frequency increases when the modulus parameter values of the Pasternak foundation and the Winkler elastic foundation increase. Furthermore, the Winkler elastic foundation has greater effect on natural frequencies of the shell than the Pasternak foundation.



The type of supports of fluid-filled shells has great influence on their natural frequency. Among the supports examined (fixed-fixed, pinned-free, pinned-pinned, and fixed-free supports), the fixed-free support has the lowest and the fixed-fixed support has the highest values of natural frequency.



By increasing the temperature difference of the inner and outer sides of the shell, the natural frequency of the structure decreases and this is more perceptible for lower environmental modes. This trend is also valid for various fluid levels.

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Appendix A h/2



A11 

S11dz 

h / 2 h/2



A 22 

S22 dz 

h / 2 h/2



B11 

S11zdz 

h / 2

Es A r sr



S66 dz

h / 2 h/2

B12 



S12 zdz

h / 2

S11z 3 dz 

E s A s z s3 E s d s z s h s3  Ss 4Ss

E12 

E s A r z 3r E s d r z r h 3r  Sr 4Sr

E 66 

h/2

h / 2 h/2



E 22 

h/2

A 66 

h/2 Es A r z r B66   S66 zdz Sr h / 2





S12 dz

h / 2

S22 zdz 

h / 2

E11 



A12 

Es As zs Ss

h/2

B22 

h/2

Es As Ss

S22 z 3 dz 

h / 2

h/2



h/2



h/2

D12 



h/2



F11 

S11z 4 dz 

h / 2

S12 z 3dz

h / 2

h/2

E A z2 E d h3 D 22   S22 z 3 dz  s r r  s r r Sr 12Sr h / 2

S66 z 3dz

h / 2

h/2

E A z2 E d h3 D11   S11z 2 dz  s s s  s s s Ss 12Ss h / 2

S12 z 3 dz

h / 2

h/2

D 66 



S66 z 3 dz

h / 2

E s A s z s4 E s d s h s3 z s2 E s d s h s5   Ss 2Ss 80Ss

h/2

F12 



h / 2

h/2

E A z4 E d h 3z 2 E d h 5 F22   S22 z 4 dz  s r r  s r r r  s r r Sr 2Sr 80Sr h / 2 h/2



H11 

S11z 6 dz  E s A s z s6 

h / 2

S12 z 4 dz h/2

F66 



S66 z 4 dz

h / 2

15E s d s h 3s z s4 15E s d s h 5s z s E s d s h s7   12 80 448

h/2



H12 

S12 z 6 dz

h / 2 h/2



H 22 

S22 z 6 dz   E s A r z 6r 

h / 2

15E s d r h 3r z r4 15E s d r h 5r z r E s d r h 7r   12 80 448

h/2



H 66 

S66 z 6 dz

h / 2 h/2



A 44 

h/2

S44 dz

A 55 

h / 2 h/2

D 44 



h / 2



D55 

h / 2



S55 dz

h/2

S44 z 2 dz

S55 z 2 dz

h / 2

h/2

F44 



h / 2

h/2

S44 z 4 dz

F55 



S55 z 4 dz

h / 2

E(z, t) (z, t)E(z, t) S11  S12  1  (z, t) 2 1  (z, t) 2 E(z, t) S44  S55  2(1  (z, t))

S66 

E(z, t) 2(1  (z, t))

Appendix B R w(z, t) dz nH 0 t H

A on 

z 2 w(z, t) cos( i ) dz  R t H  i (I n ( i ) /  r) r  R 0 H R (r, t)  jn r 2 r Jn ( ) dr t R 0 B jn   jn H n2  jn R sin h( )(1  2 )J n 2 ( jn ) R  jn H

A in 

Appendix C H

Mss2   0

NEH e  (s (e  1) Le ) s  2 w (s, t)  2 w (s, t) cos( i ) ds    N(s) cos( i ) ds 2 2 H R t  t e 1 0 L

R

NER R e

0

e 1 0

K ff  l g  r(r, t) dr  l g  H

Mff   r 0

  S(r)

NER  jn r  2 (r, t) Jn ( ) dr   2 R t e 1

H

Mfs  Maf   w cosh( 0

 jn s R

) ds 

T

(r  (e 1) R e )S(r)dr

Re

 (r  (e  1) R

e

)S(r)J n (

0

NEH Le

  d N(s) cosh( e 1 0

where N and S are shape functions.

e

 jn (r  (e  1) R e )  2 (r, t) ) dr R t 2

 jn (s (e  1) Le R

) ds