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ISRN Mechanical Engineering

Volume 2012 (2012), Article ID 232498, 13 pages

http://dx.doi.org/10.5402/2012/232498

## Vibration Characteristics of Ring-Stiffened Functionally Graded Circular Cylindrical Shells

^{1}Department of Mathematics, G. C. University Faisalabad, Faisalabad 38000, Pakistan^{2}Department of Mathematics, Government Post Graduate College Jhang, Jhang 35200, Pakistan^{3}Department of Civil Engineering, Iqra National University Peshawar, Peshawar 25000, Pakistan^{4}Department Mathematics, The Islamia University Bahawalpur, Bahawalpur 63100, Pakistan

Received 31 May 2012; Accepted 25 July 2012

Academic Editors: C. C. Huang, M. Lefik, and A. Tounsi

Copyright © 2012 Muhmmad Nawaz Naeem et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The vibration characteristics of ring stiffened cylindrical shells are analyzed. These shells are assumed to be structured from functionally graded materials (FGM) and are stiffened with isotropic rings. The problem is formulated by coupling the expressions for strain and kinetic energies of a circular cylindrical shell with those for rings. The Lagrangian function is framed by taking difference of strain and kinetic energies. The Rayleigh-Ritz approach is employed to obtain shell dynamical equations. The axial model dependence is approximated by characteristic beam functions that satisfy the boundary conditions. The validity and efficiency of the present technique are verified by comparisons of present results with the previous ones determined by other researchers.

#### 1. Introduction

Circular cylindrical shells stiffened by rings are widely used in many structural applications such as airplanes, marine crafts, pressure vessels, silos, core barrels of pressurized water reactors, submarine hulls, offshore drilling rings, and construction buildings. Usually cylinders are stiffened by rings or strings to increase the stiffness and strength, reduce the weight structure to be designed. In designing these shells, it is vital to know their resonant frequencies because excessive vibrations can lead to fatigue rupture.

First proper shell theory was proposed by Love [1]. This theory was based on Rayleigh [2] approximations for plates. Arnold and Warburton [3, 4] solved the shell problem for various physical parameters and interpreted the dip in the frequency curve on the basis of shell energy variations. They used Lagrange equations with strain and kinetic energy expressions to derive these equations. Forsberg [5] studied shell equations to scrutinize the effect of boundary conditions on vibration characteristics of circular cylindrical shells. Exponential axial model dependence was measured in this work. Bonger and Archer [6] showed that axisymmetric modes of a general shell revolution are orthogonal for clamped, simply supported, and free end. Tso [7] illustrated that the vibrational modes are orthogonal if the displacements or their corresponding natural forces vanish on the shell by using the Hamilton’s principle. Sewall and Naumann [8] studied that stiffened cylindrical shells problem experimentally and analytically. They approximated the shell displacement deformations by the beam functions and used Rayleigh-Ritz method to derive shell frequency equation in the eigenvalue form. The shells are stiffened periodically to increase the stability and efficiency. Wang et al. [9] transform vibration analysis of shell eigen-frequency equation in the general eigenvalue problem. They used Ritz polynomial functions for the axial deformation displacements by considering boundary condition equations. They designed three types of shells by the locations of isotropic rings. Sharma and Johns [10] applied Rayleigh-Ritz method for theoretical analysis of vibrating clamped-free and clamped-stiffened shell using Flugge’s shell theory for a variety of choices of axial model shape. Swaddiwudhipong et al. [11] presented an excellent study on vibrations of cylindrical shells with intermediate supports. They adopted an automated Rayleigh-Ritz method to evaluate the natural frequencies and the mode shapes of shells. Loy and Lam [12] studied the vibration of thin cylindrical shell with ring supports. The study was carried out using Sander’s shell theory. The governing equations were obtained using energy functional with the Ritz method.

There is bulk of studies on isotropic homogeneous materials and the studies on new invented composite materials such as functionally graded materials (FGMs) have been carried out by many researchers. In this material compositions and functions are varying continuously from one side to the other side. For example, one side may have high mechanical strength and the other side may have high thermal resistant property; thus, there are “two aspects” in one material. In FGMs, change of compositions is continuous and it does not come out from simply “bonding individual substances.” This generates boundaries among the bonded ones. They are all considered by mechanically and chemically. Koizumi [13] and other Japanese materialists gave the idea of fabrication of this material. Benachour et al. [14] gave refined plate theory with four variables for the study of free vibrations of functionally graded plates with arbitrary gradient. Loy et al. [15] have analyzed frequency spectrum of FGM cylindrical shells for simply supported boundary conditions. An FGM cylindrical shell composed of stainless steel and nickel as constituent materials was considered. The material properties are graded in the thickness direction in accordance with a volume fraction law. Ng et al. [16] introduced a formulation for the dynamic stability analysis of functionally graded material (FGM) cylindrical shells under harmonic axial loading. Li and Batra [17] have studied bucking aspect of three layer circular cylindrical shells under axial compressive load by simply supported boundary condition. The middle layer of the cylindrical shell was assumed to be of functionally graded material. Benyoucef et al. [18] investigated the bending behavior of thick functionally graded plates resting on Winkler-Pasternak elastic foundations.

Although an extensive amount of research work has been carried out to study the vibration characteristics of isotropic as well as composite cylindrical shells, there is no evidence of work on vibration of functionally graded cylindrical shells stiffener with ring support. The present study is concerned with analysis of the vibration characteristics of functionally graded cylindrical shell stiffened with isotropic ring on the outer surface of the shell. The Rayleigh-Ritz method is used to formulate the shell eigen-frequency equation.

#### 2. Theoretical Consideration

Consider a circular thin shell of uniform thickness , length *,* radius , mass density , modulus of elasticity , Poisson’s ratio , and shear and effective modulus . The shell is reinforced by ring-stiffeners of either equal or unequal sizes or spacing. The kth stiffener located at measured from one end of the shell and its rectangular cross-section has a depth and width as shown in Figure 1. The stiffeners may be constructed from materials used for shell or different from shell material, with denoting the stiffener mass density, its young’s modulus, the shear modulus, and the poisson’s ratio.

For a thin cylindrical shell, plane stress condition is assumed and the constitutive relation for a thin cylindrical shell is given by where is stress vector, is strain vector, and is reduced stiffness matrix.

Stress vector and strain vector are defined as where and are stresses in and directions is shear stress on the plane, , are strains in and directions, and is shear strain on plane. The reduced stiffness matrix is defined as For the isotropic materials, the reduced stiffness are given as The components of the strain vector are defined as the linear functions of thickness coordinates and are given as where , and are reference surface strains, and , and are the surface curvatures. The strain energy, , of a cylindrical shell can be written as where where where , , are the matrices of the extensional stiffness , coupling stiffness and bending stiffness , respectively. Stiffnesses , and are defined as where the coupling stiffness reduces to zero for isotropic cylindrical shell and nonzero for FGM shells. For FGM shells, the sign of depends upon the order of constituent materials in the FGM. They are positive for a FGM configuration and negative if the order of the constituent material is reversed. This arises because of material properties asymmetry about the mid plane. are function of for FGMs. Substituting the expression for and in (6)At time “” the expression for kinetic energy “” of the vibrating shell is given in the form where is the mass density per unit length defined as where is the mass density per unit length. Adopting Sander’s thin shell theory for the surface strains [20], the strain energy is given by The strain energy of the th ring-stiffener is given by Galletly [17] aswhere is the eccentricity of the ring stiffener and is given by The second moment of areas , , cross-sectional area , and the torsional constant are furnished by The kinetic energy of the th stiffener is given by: From geometric considerations, the relationships between the displacements of the kth stiffener and the displacements of the shell at the position of the stiffener at are given by Combining the strain and kinetic energies of the cylindrical shell and the th stiffener, the Lagrangian functional is formulated in the following form:

#### 3. Solution Procedure

The Rayleigh-Ritz approach is employed to analyze vibrational behaviour of stiffened functionally graded circular cylindrical shells. The use of Ritz method provides a rapid convergence and excellent accuracy. The following displacement functions are adopted to separate the spatial variables and time variable in the following form: where is the number of circumferential modes; is the circular frequency of the vibrating shell; and , and represents the axial modal dependence in the axial, circumferential, and radial directions, respectively. are the vibrating amplitudes.

Using the expressions for the functions , , and given in and their partial derivatives in the expressions for given in (11), (13), (14), and (17), respectively, and employing the principle of minimum energy, we obtain the following form of the Lagrangian functional: where and are the sum of maximum kinetic and strain energies, respectively, of stiffened functionally graded cylindrical shell. To derive frequency equation, the Lagrangian functional is extremized with respect to the amplitude coefficients , and . This yields a set of three homogeneous simultaneous equations Rearranging the terms in (22), we get the following eigenvalue equation to find the shell natural frequencies and mode shapes: or where where and are stiffness and mass matrices of the cylindrical shell, respectively, and and are the corresponding matrices of the th ring stiffener, and The expression for the matrices and are given in the appendix. This is an eigenvalue problem and is solved by using Matlab software package. Equation (24) is true for both isotropic as well as FGM cylindrical shells. For isotropic cylindrical shells the coupling stiffness is zero whereas it is non-zero for FGM cylindrical shells. Eigenvalues correspond to the natural frequencies and eigenvectors to the corresponding mode shapes of shells respectively.

#### 4. Functionally Graded Materials

The FGMs are much advanced materials and are used in engineering science and technology. Material properties of an FGM are the functions of the temperature and the position. These properties of a constituent material are managed by a volume fraction. If represents a material property of the th constituent material of an FGM consisting of constituent materials, then the effective materials property of the FGM is written as where is the volume fraction of the th constituent material. Also, the sum of volume fractions of the constituent materials is equal to 1. That is, The volume fraction depends upon the thickness variable and is defined as for a cylindrical shell. and denote inner and outer radii of the shell, respectively, and is the thickness variable in the radial direction. is known as the power law exponent. It is a nonnegative real number and lies between zero and infinity. For a cylindrical shell, the volume fraction is assumed as where is the shell uniform thickness. When the shell is considered to consist of two materials, the effective Young’s modulus , the poisson ratio and the mass density are given by

#### 5. Results and Discussion

Numerical technique known as the Rayleigh-Ritz method has been employed to study the vibration characteristics of functionally graded circular cylindrical shells with ring- stiffeners. To confirm the efficiency and validity of the present method, the frequencies of cylindrical shells with and without ring-stiffeners are compared with those values found in the literature.

In Table 1, a comparison of frequency parameters calculated by Loy et al. [15] is done with those values obtained by the present procedure using the Rayleigh-Ritz method for an isotropic cylindrical shell simply supported at both ends. Shell parameters are given in Table 1. Two sets of frequency parameters are very close to each other and a good agreement is observed between them. The minimum frequency parameter occurs at and is less than, by 0.061%, its corresponding value in Loy et al. [15]. The frequency parameters for are 0.006%, 0.01%, and 0.002% lower and the frequency parameters for circumferential wave numbers are 0.0007%, 0.001%, 0.0009%, and 0.001% greater than the corresponding results performed by Loy et al. [15]. The frequency parameters for are the same for both cases. This shows the validity of the present method.

##### 5.1. Simply Supported Cylindrical Shells with Ring Stiffeners

The often-cited Galletly [21] S-S shells with 14 evenly spaced eccentric/concentric ring stiffeners of various depth-to-width ratios are analyzed. In Tables 2(a) and 2(b), the nondimensionalized frequency parameters are listed for isotropic cylindrical shells with ring-stiffeners.

Frequency study is analyzed for 14 evenly spaced eccentric ring stiffeners externally and internally in Tables 2(a) and 2(b), respectively. They are compared with those values evaluated by Swaddiwudhipong et al. [11]. Shell parameters are listed in the tables. The axial wave mode is . The circumferential wave numbers are 2, 3, 4, and 5. It is seen that as the depth-to-width ratio is increased, the frequency increases. Also the comparison of the results shows that the present results are slight higher than those in Swaddiwudhipong et al. [11]. The difference may be due to the axial modal dependence measured by the different sets of functions. In Swaddiwudhipong et al. [11], Ritz polynomials have been used whereas in the present study, characteristics beam functions are utilized.

##### 5.2. Comparisons of Results of FGM Cylindrical Shells without Ring Stiffeners

The functionally graded material considered here is composed of stainless steel and nickel. The variations of natural frequencies of a functionally graded cylindrical shell are compared with those determined by Naeem et al. [19]. The influence of the constituent volume fractions is studied by varying the volume fractions of the Stainless Steel and Nickel. This is carried out by varying the value of power law exponent . The effects of the FGM configuration are studied by studying the frequencies of two functionally graded cylindrical shells. Type-I functionally graded cylindrical shell has Nickel on its inner surface and Stainless Steel on its outer surface and Type-II functionally graded cylindrical shell has Stainless Steel on its inner surface and Nickel on its outer surface. The material properties for Stainless Steel and Nickel, calculated at K, are presented in Table 3.

Natural frequencies (Hz) determined by the present method for a simply supported functionally graded cylindrical shell are presented in Tables 4(a) and 4(b) and are compared with the corresponding results evaluated by Naeem et al. [19]. In Table 4(a), a comparison of natural frequencies for the shell is given for the power law exponent *, *1*,* and 15. The respective frequencies determined by two techniques are the same for the circumferential wave number and for greater than 4 against power law exponent . The frequencies determined by the present method are slightly greater than those in the study of Naeem et al. [19] for the circumferential wave number between 2 and 10 against the power law exponent and for , 3, and 4 against power law exponent . The frequencies determined by the present method are slightly lower for the circumferential wave number to against the power law exponent . The minimum frequency corresponds to and decreases with and corresponding decrement is 0.0218 per cent.

In Table 4(a), a comparison of of natural frequencies for the shell is given for the power law exponent , 1*,* and 15. The respective frequencies determined by two techniques are the same for the circumferential wae number and for greater than 4 against power law exponent . The frequencies determined by the present method are slightly lower than those in the study of Naeem et al. [19] for the circumferential wave number between 2 and 10 against the power law exponent and for , 3, and 4 against power law exponent . The minimum frequency corresponds to and decreases with and corresponding decrement is 0.0143 per cent.

##### 5.3. Functionally Graded Circular Cylindrical Shells with Ring-Stiffeners

###### 5.3.1. Variations of Natural Frequencies with Circumferential Wave Number ()

Tables 5 and 6 list the variations of natural frequencies (Hz) with the circumferential wave number “” for a type I FG cylindrical shell with the Nickel ring-stiffeners. The axial mode “” is taken to be unity. is kept equal to 20 where , 20 in Tables 5 and 6, respectively. The columns and show the natural frequencies (Hz) for a Stainless Steel cylindrical shell and Nickel cylindrical shell, respectively. The influence of the value of , which affects the constituent volume fractions, is analyzed. It is seen that when the value of is increased, the natural frequencies decreased. The decrease in the natural frequencies in Table 5, from to is about 1.6% at and about 1.5% at . When is small, the natural frequencies approached those of and when is large they approached those of . Hence the natural frequencies for greater than zero fell between those of and for a given circumferential wave number .

Tables 7 and 8 show the variation of the natural frequencies (Hz) versus the circumferential wave number “” for a Type II FGM cylindrical shell with Nickel ring-stiffeners. The influence of or constituent volume fraction on the natural frequencies is the opposite of a Type I FGM cylindrical shell. Unlike a Type I FGM cylindrical shell where the natural frequencies (Hz) decreased with , the natural frequencies (Hz) for a Type II FGM cylindrical shell increased with . The increase in the natural frequencies (Hz) from to is about 1.64% at and about 1.61% at . Thus the influence of the constituent volume fractions for a Type II FGM cylindrical shell is different from that of a Type I FGM cylindrical shell.

Comparing the frequencies in Tables 5 and 6 with those in Tables 7 and 8, it can be seen that for , the natural frequencies of a Type II FG cylindrical shell with Nickel ring-stiffeners are higher than those a Type I FG cylindrical shell with Nickel ring-stiffeners.

On the other hand, for , the frequencies for a Type I FGM cylindrical shell with Nickel ring-stiffeners are higher than a Type II FG cylindrical shell with Nickel ring-stiffeners. For example, for at and , the natural frequencies (Hz) for a Type II FG cylindrical shell with Nickel ring-stiffeners are about 3.23% higher than a Type I FG cylindrical shell with Nickel ring-stiffeners. For at and , the frequencies for a Type I FGM cylindrical shell with Nickel ring-stiffeners are 1.26% higher than a Type II FG cylindrical shell with Nickel ring-stiffeners. Thus the natural frequencies are affected by the configuration of the constituent materials in the functionally graded cylindrical shells.

Tables 9 and 10 show the variation of natural frequencies against circumferential wave number *,* for Type I FG cylindrical shell with Stainless Steel ring-stiffeners. The influence of the value of , which affects the constituent volume fraction, can be seen from the tables. It is seen that when the value of is increased, the natural frequencies decreased. The decrease in the natural frequencies in Table 9, from to is about 1.60% at and about 1.56% at . When is small, the natural frequencies approached those of and when is large, they approached those of . Hence the natural frequencies for greater than zero fell between those of and for a given circumferential wave number *. *

Tables 11 and 12 show the variation of natural frequencies against circumferential wave number *,* for Type II FG cylindrical shell with Stainless Steel ring-stiffeners. The influence of or the constituent volume fraction on the natural frequencies is opposite of a Type I FGM cylindrical shell. Here, the natural frequencies for a Type II FGM cylindrical shell increase with increasing the value of . The increase in the natural frequencies from to is about 1.67% at and about 1.63% at . Thus the influence of the constituent volume fractions for a Type II FGM cylindrical shell is different from that of a Type I FGM cylindrical shell.

Comparing the frequencies in Tables 9 and 10 with those in Tables 11 and 12, it can be seen that for , the natural frequencies of a Type II FGM cylindrical shell with Stainless Steel ring-stiffeners are higher than those of a Type I FGM cylindrical shell with Stainless Steel ring-stiffeners. On the other hand, for , the frequencies for a Type I FGM cylindrical shell with Stainless Steel ring-stiffeners are higher than those for a Type II FGM cylindrical shell with Stainless Steel ring-stiffeners. For example, for at and , the natural frequencies for a Type II FGM with Stainless Steel ring-stiffeners is about 3.27% higher than a Type I FGM cylindrical shell with Stainless Steel ring-stiffeners. For at and , the natural frequency for a Type I FGM cylindrical shell with Stainless Steel ring-stiffeners is 1.2% higher than a Type II FGM cylindrical shell with Stainless Steel ring-stiffeners. Thus the natural frequencies are affected by the configuration of the constituent materials in the functionally graded cylindrical shells.

###### 5.3.2. Variation of Minimum Frequency with

Tables 13 and 14 show the variations of the fundamental natural frequencies (Hz) with the ratio for a Type I and Type II FGM cylindrical shells with Stainless Steel ring-stiffeners. For a Type I FGM cylindrical shell with Stainless Steel ring-stiffeners, the fundamental frequencies decreased with , and for a Type II FGM cylindrical shell with Stainless Steel ring-stiffeners, the fundamental frequencies increased with *. *The difference in the fundamental frequencies between and is about 1.78% for Type I and 1.87% for Type II FGM cylindrical shells with Stainless steel ring-stiffeners. The fundamental natural frequencies for Type I and Type II FGM cylindrical shells with Stainless Steel ring-stiffeners occur at the same circumferential wave numbers. For all values of , the fundamental natural frequencies fall between those for and .

###### 5.3.3. Variation of Minimum Frequency with

Tables 15 and 16 show the variations of the fundamental natural frequencies (Hz) with the ratio for a Type I and Type II FGM cylindrical shells with Stainless Steel ring-stiffeners. For a Type I FGM cylindrical shell with Stainless Steel ring-stiffeners, the fundamental frequencies decreased with , and for a Type II FGM cylindrical shell with Stainless Steel ring-stiffeners, the fundamental frequencies increased with *.* For all values of , the fundamental natural frequencies lie between those for and .

The frequency characteristics of FGM cylindrical shells with ring-stiffeners are similar to those for homogeneous isotropic cylindrical shells. Other interesting frequency characteristics are also observed in the FGM cylindrical shells. These characteristics arise when the constituent volume fractions and the configurations of the constituent materials in the functionally graded cylindrical shells are varied in the thickness direction.

#### 6. Conclusion

In this study, the Rayleigh-Ritz approach has been employed to analyze the vibration characteristics of functionally graded circular cylindrical shells with ring-stiffeners of different materials. The axial model dependence has been approximated by the characteristic beam functions. Sander’s thin shell theory of first order has been used to perform the vibration analysis. From the vibration results of cylindrical shells with identical and evenly spaced ring-stiffeners, it is found that stiffeners placed eccentrically are more effective than concentric ones. The study is carried out for isotropic as well as two types of functionally graded cylindrical shell with and without ring stiffeners where the configurations of the constituent materials in the functionally graded cylindrical shells are varied by the volume fraction law. One is termed as Type I FG cylindrical shell and has properties that vary continuously from Nickel on its inner surface and Stainless Steel on its outer surface. The other is termed as a Type II FG cylindrical shell and has properties that vary continuously from Stainless Steel on its inner surface and Nickel on its outer surface. A validation of the analysis has been carried out by comparing results with those found in literature and a good agreement has been observed among the results evaluated by different shell theories and numerical approaches. It is seen that the variations of natural frequency of FGM circular cylinders are similar to that of isotropic ones. The frequency is influenced by the volume fraction law exponents. It decreases or increases with N depending upon the order of constituent material in FGM shells. For the Type I and Type II FG cylindrical shells the natural frequencies for all values of N lie between those for a Stainless Steel and Nickel cylindrical shells. For the natural frequencies for Type I FG cylindrical shells are higher than those for Type II FG cylindrical shells and for the natural frequencies for Type II FG cylindrical shells are higher than those for Type I FG cylindrical shells. Thus the constituent volume fractions and the configurations of the constituent materials affect the natural frequencies. This work can be extended to analyze vibrations of cylindrical shells by varying material composition by the interchange of isotropic and functionally graded layers in the radial direction.

#### Appendix

#### Stiffness and Mass Matrices

One has where

Here

#### References

- A. E. H. Love, “On the small free vibrations and deformations of thin elastic shell,”
*Philosophical Transactions of the Royal Society of London A*, vol. 179, pp. 491–549, 1888. View at Google Scholar - J. W. S. Rayleigh,
*Theory of Sounds*, vol. 1, Macmillan, London, UK, 1882. - R. N. Arnold and G. B. Warburton, “Flexural vibrations of the walls of thin cylindrical shells having freely supported ends,”
*Proceedings of the Royal Society of London A*, vol. 197, pp. 238–256, 1949. View at Google Scholar - R. N. Arnold and G. B. Warburton, “The flexural vibrations of thin cylinders,”
*Proceedings of the Institution of Mechanical Engineers A*, vol. 167, pp. 62–80, 1953. View at Google Scholar - K. Forsberg, “Influence of boundary conditions on modal characteristics of cylindrical shells,”
*AIAA Journal*, vol. 2, pp. 182–189, 1964. View at Google Scholar - F. K. Bogner and R.P. Archer, “On the orthogonality condition of axisymmetric vibration modes for shells of revolution,”
*Journal of Applied Mechanics*, vol. 32, pp. 447–448, 1965. View at Google Scholar - W. K. Tso, “Orthogonality condition for the vibrational modes of elastic shells,”
*Journal of Applied Mechanics*, vol. 34, pp. 782–793, 1967. View at Google Scholar - J. L. Sewall, E. C. Naumann, et al., “An experimental and analytical vibration study of thin cylindrical shells with and without longitudinal stiffeners,”
*NASA*TN D-4705, Langley Research Centre Langley Station, Hampton, Va, USA, 1968. View at Google Scholar - C. M. Wang, S. Swaddiwudhipong, and J. Tian, “Ritz method for vibration analysis of cylindrical shells with ring stiffeners,”
*Journal of Engineering Mechanics*, vol. 123, no. 2, pp. 134–142, 1997. View at Google Scholar · View at Scopus - C. B. Sharma and D. J. Johns, “Vibration characteristics of a clamped-free and clamped-ring-stiffened circular cylindrical shell,”
*Journal of Sound and Vibration*, vol. 14, no. 4, pp. 459–474, 1971. View at Google Scholar · View at Scopus - S. Swaddiwudhipong, J. Tian, and C. M. Wang, “Vibrations of cylindrical shells with intermediate supports,”
*Journal of Sound and Vibration*, vol. 187, no. 1, pp. 69–93, 1995. View at Publisher · View at Google Scholar · View at Scopus - C. T. Loy and K. Y. Lam, “Vibration of cylindrical shells with ring support,”
*International Journal of Mechanical Sciences*, vol. 39, no. 4, pp. 455–471, 1997. View at Google Scholar · View at Scopus - M. Koizumi, “FGM activities in Japan,”
*Composites Part B*, vol. 28, no. 1-2, pp. 1–4, 1997. View at Google Scholar · View at Scopus - A. Benachour, H. D. Tahar, H. A. Atmane, A. Tounsi, and M. S. Ahmed, “A four variable refined plate theory for free vibrations of functionally graded plates with arbitrary gradient,”
*Composites Part B*, vol. 42, no. 6, pp. 1386–1394, 2011. View at Publisher · View at Google Scholar · View at Scopus - C. T. Loy, K. Y. Lam, and J. N. Reddy, “Vibration of functionally graded cylindrical shells,”
*International Journal of Mechanical Sciences*, vol. 41, no. 3, pp. 309–324, 1999. View at Google Scholar · View at Scopus - T. Y. Ng, K. Y. Lam, K. M. Liew, and J. N. Reddy, “Vibration of functionally graded cylindrical shells under periodic axial loading,”
*International Journal of Computational Engineering Science*, vol. 38, pp. 1295–1309, 2001. View at Google Scholar - S. R. Li and R. C. Batra, “Buckling of axially compressed thin cylindrical shells with functionally graded middle layer,”
*Thin-Walled Structures*, vol. 44, no. 10, pp. 1039–1047, 2007. View at Publisher · View at Google Scholar · View at Scopus - S. Benyoucef, I. Mechab, A. Tounsi, A. Fekrar, H. Ait Atmane, and E. A. Adda Bedia, “Bending of thick functionally graded plates resting on Winkler-Pasternak elastic foundations,”
*Mechanics of Composite Materials*, vol. 46, no. 4, pp. 425–434, 2010. View at Publisher · View at Google Scholar · View at Scopus - M. N. Naeem, S. H. Arshad, and C. B. Sharma, “The Ritz formulation applied to the study of the vibration frequency characteristics of functionally graded circular cylindrical shells,”
*Proceedings of the Institution of Mechanical Engineers Part C*, vol. 224, no. 1, pp. 43–54, 2010. View at Publisher · View at Google Scholar · View at Scopus - J. R. J. L. Sanders, “An improved first order approximation theory of thin shell,” Tech. Rep. R-24, NASA, Washington, DC, USA, 1959. View at Google Scholar
- G. D. Galletly, “On the in-vacuo vibrations of simply supported ring-stiffened cylindrical shells,” in
*Proceedings of the 2nd National Congress of Applied Mechanics Processing*, pp. 225–231, 1954.