Vibration Characteristics of Ring-Stiffened Functionally Graded Circular Cylindrical Shells

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 ℎ, length 𝐿, radius 𝑅, mass density 𝜌, modulus of elasticity 𝐸, Poisson’s ratio 𝜈, and shear and effective modulus 𝐺=𝐸/2(1+𝜈). The shell is reinforced by 𝑞 ring-stiffeners of either equal or unequal sizes or spacing. The kth stiffener located at 𝑎_𝑘𝐿 measured from one end of the shell and its rectangular cross-section has a depth 𝑑_𝑘 and width 𝑏_𝑘 as shown in Figure 1. The stiffeners may be constructed from materials used for shell or different from shell material, with 𝜌_𝑘 denoting the stiffener mass density, 𝐸_𝑘 its young’s modulus, 𝐺_𝑘 the shear modulus, and 𝜈_𝑘 the poisson’s ratio.

Figure 1 

Cylindrical shell with Ring-Stiffeners.

For a thin cylindrical shell, plane stress condition is assumed and the constitutive relation for a thin cylindrical shell is given by {𝜎}=[𝑄]{𝑒}, (1) where {𝜎} is stress vector, {𝑒} is strain vector, and [𝑄] is reduced stiffness matrix.

Stress vector and strain vector are defined as {𝜎}^𝑇=𝜎_𝑥𝜎_𝜃𝜎_𝑥𝜃,{𝑒}^𝑇=𝑒_𝑥𝑒_𝜃𝑒_𝑥𝜃, (2) where 𝜎_𝑥 and 𝜎_𝜃 are stresses in 𝑥 and 𝜃 directions 𝜎_𝑥𝜃 is shear stress on the 𝑥𝜃 plane, 𝑒_𝑥, 𝑒_𝜃 are strains in 𝑥 and 𝜃 directions, and 𝑒_𝑥𝜃 is shear strain on 𝑥𝜃 plane. The reduced stiffness matrix is defined as [𝑄]=𝑄₁₁𝑄₁₂0𝑄₁₂𝑄₂₂000𝑄₆₆. (3) For the isotropic materials, the reduced stiffness 𝑄_𝑖𝑗(𝑖,𝑗=1,2,and6) are given as 𝑄₁₁=𝑄₂₂=𝐸1−𝜈²,𝑄₁₂=𝜈𝐸1−𝜈²,𝑄₆₆=𝐸2().1+𝜈 (4) The components of the strain vector {𝑒} are defined as the linear functions of thickness coordinates 𝑧 and are given as 𝑒_𝑥=𝑒₁+𝑧𝑘₁,𝑒_𝜃=𝑒₂+𝑧𝑘₂,𝑒_𝑥𝜃=𝛾+2𝑧𝜏, (5) where 𝑒₁,𝑒₂, and 𝛾 are reference surface strains, and 𝑘₁,𝑘₂, and 𝜏 are the surface curvatures. The strain energy, 𝑆, of a cylindrical shell can be written as 1𝑆=2^𝐿₀₀^2𝜋{𝜀}^𝑇[𝑇]{𝜀}𝑅𝑑𝜃𝑑𝑥, (6) where {𝜀}^𝑇=𝑒₁,𝑒₂,𝛾,𝑘₁,𝑘₂,2𝜏,[𝑇]=𝐴𝐵𝐵𝐷, (7) where 𝐴𝐴=¹¹𝐴₁₂0𝐴₁₂𝐴₂₂000𝐴₆₆,𝐵𝐵=¹¹𝐵₁₂0𝐵₁₂𝐵₂₂000𝐵₆₆,𝐷𝐷=¹¹𝐷₁₂0𝐷₁₂𝐷₂₂000𝐷₆₆, (8) where 𝐴, 𝐵, 𝐷 are the matrices of the extensional stiffness 𝐴_𝑖𝑗, coupling stiffness 𝐵_𝑖𝑗 and bending stiffness 𝐷_𝑖𝑗(𝑖,𝑗=1,2,and6), respectively. Stiffnesses 𝐴_𝑖𝑗,𝐵_𝑖𝑗, and 𝐷_𝑖𝑗 are defined as 𝐴_𝑖𝑗,𝐵_𝑖𝑗,𝐷_𝑖𝑗=^ℎ/2_−ℎ/2𝑄_𝑖𝑗1,𝑧,𝑧²𝑑𝑧𝑖,𝑗=1,2,and6, (9) where the coupling stiffness 𝐵_𝑖𝑗 reduces to zero for isotropic cylindrical shell and nonzero for FGM shells. For FGM shells, the sign of 𝐵_𝑖𝑗 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. 𝑄_𝑖𝑗 are function of 𝑧 for FGMs. Substituting the expression for {𝜀}^𝑇 and {𝑇} in (6)𝑅𝑆=2^𝐿₀₀^2𝜋𝐴₁₁𝑒²₁₁+2𝑒₁𝑒₂𝐴₁₂+2𝑒₁𝑘₁𝐵₁₁+2𝑒₁𝑘₂𝐵₁₂+𝑒₂²𝐴₂₂+2𝑒₂𝑘₁𝐵₁₂+2𝑒₂𝑘₂𝐵₂₂+𝛾²𝐴₆₆+4𝜏𝛾𝐵₆₆+𝑘₁²𝐷₁₁+2𝑘₁𝑘₂𝐷₁₂+𝑘₂²𝐷₂₂+4𝜏²𝐷₆₆𝑅𝑑𝜃𝑑𝑥. (10)At time “𝑡” the expression for kinetic energy “𝐾” of the vibrating shell is given in the form 1𝐾=2^𝐿₀₀^2𝜋𝜌_𝑡𝜕𝑢𝜕𝑡²+𝜕𝑣𝜕𝑡²+𝜕𝑤𝜕𝑡²𝑅𝑑𝜃𝑑𝑥, (11) where 𝜌_𝑡 is the mass density per unit length defined as 𝜌_𝑡=^ℎ/2_−ℎ/2𝜌𝑑𝑧, (12) where 𝜌 is the mass density per unit length. Adopting Sander’s thin shell theory for the surface strains [20], the strain energy 𝑆 is given by 𝑅𝑆=2^𝐿₀₀^2𝜋𝜕²𝑤𝜕𝑥²𝐴₁₁𝜕𝑢𝜕𝑥²+𝐴₂₂𝑅²𝜕𝑣𝜕𝜃−𝑤²+2𝐴₁₂𝑅𝜕𝑢𝜕𝑥𝜕𝑣𝜕𝜃−𝑤+𝐴₆₆𝜕𝑣+1𝜕𝑥𝑅𝜕𝑢𝜕𝑥²+2𝐵₁₁𝜕𝑢𝜕𝜕𝑥²𝑤𝜕𝑥²+2𝐵₁₂𝑅²𝜕²𝑤𝜕𝜃²+𝑤𝜕𝑢+𝜕𝑥2𝐵₁₂𝑅𝜕²𝑤𝜕𝑥²𝜕𝑣+𝜕𝜃−𝑤2𝐵₂₂𝑅³𝜕𝑣𝜕𝜕𝜃−𝑤²𝑤𝜕𝜃²+𝑤+4𝐵₆₆𝜕²𝑤+3𝜕𝑥𝜕𝜃4𝜕𝑣−1𝜕𝑥4𝑅𝜕𝑢𝜕𝜃𝜕𝑣+1𝜕𝑥𝑅𝜕𝑢𝜕𝜃+𝐷₁₁𝜕²𝑤𝜕𝑥²²+𝐷₂₂𝑅⁴𝜕²𝑤𝜕𝜃²+𝑤²+2𝐷₁₂𝑅²𝜕²𝑤𝜕𝑥²𝜕²𝑤𝜕𝜃²+𝑤+4𝐷₆₆𝜕²𝑤+3𝜕𝑥𝜕𝜃4𝜕𝑣−1𝜕𝑥4𝑅𝜕𝑢𝜕𝜃²×𝑅𝑑𝜃𝑑𝑥. (13)The strain energy 𝑆_𝑘 of the 𝑘th ring-stiffener is given by Galletly [17] as𝑆_𝑘=𝐸_𝑘𝐼_𝑧𝑘2𝑅+𝑒_𝑘𝜕𝑤_𝑘+1𝜕𝑥𝑅+𝑒_𝑘𝜕²𝑢_𝑘𝜕𝜃²²+𝐸_𝑘𝐼_𝑥𝑘2𝑅+𝑒_𝑘³𝑤_𝑘+𝜕²𝑤_𝑘𝜕𝜃²²+𝐸_𝑘𝐴_𝑘2𝑅+𝑒_𝑘𝜕𝑣_𝑘𝜕𝜃−𝑤_𝑘²+𝐺_𝑘𝐽_𝑘2𝑅+𝑒_𝑘−𝜕²𝑤_𝑘+1𝜕𝑥𝜕𝜃𝑅+𝑒_𝑘𝜕𝑢_𝑘𝜕𝜃²𝑑𝜃, (14)where 𝑒_𝑘 is the eccentricity of the ring stiffener and is given by 𝑒_𝑘=ℎ+𝑑_𝑘2𝑒forexternallyeccentricstiffener,_𝑘𝑒=0forconcentricstiffener,_𝑘=−ℎ+𝑑_𝑘2forinternallyeccentricstiffener. (15) The second moment of areas 𝐼_𝑧𝑘, 𝐼_𝑥𝑘, cross-sectional area 𝐴_𝑘, and the torsional constant 𝐽_𝑘 are furnished by 𝐼_𝑧𝑘=𝑏_𝑘³𝑑_𝑘12,𝐼_𝑥𝑘=𝑏_𝑘𝑑_𝑘³12,𝐴_𝑘=𝑏_𝑘𝑑_𝑘,𝐽_𝑘=131−192𝑏_𝑘𝜋⁵𝑑_𝑘^∞_𝑛=1,3,51𝑛⁵tanh𝑛𝜋𝑑_𝑘2𝑏_𝑘𝑏_𝑘³𝑑_𝑘. (16) The kinetic energy 𝐾_𝑘 of the 𝑘th stiffener is given by:𝐾_𝑘=𝜌_𝑘2₀^2𝜋𝐴_𝑘𝜕𝑢_𝑘𝜕𝑡²+𝜕𝑣_𝑘𝜕𝑡²+𝜕𝑤_𝑘𝜕𝑡²+𝐼_𝑥𝑘+𝐼_𝑧𝑘𝜕²𝑤_𝑘𝜕𝑡𝜕𝑥²𝑅+𝑒_𝑘𝑑𝜃. (17) From geometric considerations, the relationships between the displacements (𝑢_𝑘,𝑣_𝑘,𝑤_𝑘) of the kth stiffener and the displacements (𝑢,𝑣,𝑤) of the shell at the position of the stiffener at 𝑥=𝑎_𝑘𝐿 are given by 𝑢_𝑘=𝑢+𝑒_𝑘𝜕𝑤_𝑘,𝑣𝜕𝑥_𝑘𝑒=𝑣1+^𝑘𝑅+𝑒_𝑘𝑅𝜕𝑤,𝑤𝜕𝜃_𝑘=𝑤. (18) Combining the strain and kinetic energies of the cylindrical shell and the 𝑘th stiffener, the Lagrangian functional is formulated in the following form: =𝐾+^𝑞_𝑘=1𝐾_𝑘−𝑆+^𝑞_𝑘=1𝑆_𝑘. (19)

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 𝑢,𝑣,𝑤 are adopted to separate the spatial variables 𝑥,𝜃 and time variable 𝑡 in the following form: 𝑢(𝑥,𝜃,𝑡)=𝐴_𝑚𝑈(𝑥)()(),𝑣()cos𝑛𝜃sin𝜔𝑡𝑥,𝜃,𝑡=𝐵_𝑚𝑉(𝑥)()(),𝑤()sin𝑛𝜃sin𝜔𝑡𝑥,𝜃,𝑡=𝐶_𝑚𝑊(𝑥)cos(𝑛𝜃)sin(𝜔𝑡), (20) where 𝑛 is the number of circumferential modes; 𝜔 is the circular frequency of the vibrating shell; and 𝑈(𝑥),𝑉(𝑥), and 𝑊(𝑥) represents the axial modal dependence in the axial, circumferential, and radial directions, respectively. 𝐴_𝑚,𝐵_𝑚,𝐶_𝑚 are the vibrating amplitudes.

Using the expressions for the functions 𝑢, 𝑣, and 𝑤 given in and their partial derivatives in the expressions for 𝐾,𝑆,𝑆_𝑘,𝐾_𝑘 given in (11), (13), (14), and (17), respectively, and employing the principle of minimum energy, we obtain the following form of the Lagrangian functional: =𝐾_max−𝑆_max, (21) where 𝐾_max and 𝑆_max 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 𝐴_𝑚,𝐵_𝑚, and 𝐶_𝑚. This yields a set of three homogeneous simultaneous equations 𝜕∏𝜕𝐴_𝑚=𝜕∏𝜕𝐵_𝑚=𝜕∏𝜕𝐶_𝑚=0. (22) Rearranging the terms in (22), we get the following eigenvalue equation to find the shell natural frequencies and mode shapes: [𝑆]+𝑆_𝑘−𝜔²[𝐾]+𝐾_𝑘𝑋=[0]. (23) or [𝐶]−𝜔²[𝐷]𝑋=[0], (24) where [𝐶]=[𝑆]+𝑆_𝑘,[𝐷]=[𝐾]+𝐾_𝑘, (25) where [𝑆] and [𝐾] are stiffness and mass matrices of the cylindrical shell, respectively, and [𝑆_𝑘] and [𝐾_𝑘] are the corresponding matrices of the 𝑘th ring stiffener, and 𝑋=[^𝐴_𝑚^𝐵_𝑚^𝐶_𝑚]^𝑇 The expression for the matrices [𝐶] and [𝐷] 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 𝐵_𝑖𝑗 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 𝑃_𝑖 represents a material property of the 𝑖th constituent material of an FGM consisting of 𝑘 constituent materials, then the effective materials property 𝑃 of the FGM is written as 𝑃=^𝑘_𝑖=1𝑃_𝑖𝑉_𝑖, (26) where 𝑉_𝑖 is the volume fraction of the 𝑖th constituent material. Also, the sum of volume fractions of the constituent materials is equal to 1. That is, ^𝑘_𝑖=1𝑉_𝑖=1. (27) The volume fraction depends upon the thickness variable and is defined as 𝑉_𝑖=𝑍−𝑅_𝑖𝑅_𝑜−𝑅_𝑖^𝑁 (28) for a cylindrical shell. 𝑅_𝑖 and 𝑅_𝑜 denote inner and outer radii of the shell, respectively, and 𝑧 is the thickness variable in the radial direction. 𝑁 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 𝑉_𝑖=𝑍+0.5ℎℎ^𝑁, (29) where ℎ is the shell uniform thickness. When the shell is considered to consist of two materials, the effective Young’s modulus 𝐸, the poisson ratio 𝜐 and the mass density 𝜌 are given by 𝐸=𝐸₁−𝐸₂Z+0.5ℎℎ^𝑁+𝐸₂,𝑛𝜐=𝑛𝜐₁−𝑛𝜐₂Z+0.5ℎℎ^𝑁+𝑛𝜐₂,𝜌=𝜌₁−𝜌₂Z+0.5ℎℎ^𝑁+𝜌₂. (30)

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 Ω=𝜔𝑅(1−𝜈²)𝜌/𝐸 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 𝑛=2 and is less than, by 0.061%, its corresponding value in Loy et al. [15]. The frequency parameters for 𝑛=1,3,and4 are 0.006%, 0.01%, and 0.002% lower and the frequency parameters for circumferential wave numbers 𝑛=7,8,9,and10 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 𝑛=5,6 are the same for both cases. This shows the validity of the present method.

Table 1 

Comparison of frequency parameters Ω=𝜔𝑅(1−𝜈²)𝜌/𝐸 simply supported isotropic cylindrical shell. (𝑚=1,𝐿/𝑅=20,ℎ/𝑅=0.01,𝐸=30𝑒+06𝑙𝑏𝑓𝑖𝑛⁻²,𝜈=0.3,𝜌=7.35𝑒⁻⁰⁴𝑙𝑏𝑓𝑠²).

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 𝑑/𝑏 are analyzed. In Tables 2(a) and 2(b), the nondimensionalized frequency parameters Ω are listed for isotropic cylindrical shells with ring-stiffeners.

Table 2 

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 𝑚=1. The circumferential wave numbers are 2, 3, 4, and 5. It is seen that as the depth-to-width ratio (𝑑/𝑏) 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 𝑁. 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 𝑇=300 K, are presented in Table 3.

Table 3 

Properties of materials.

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 𝑁=0.5, 1, and 15. The respective frequencies determined by two techniques are the same for the circumferential wave number 𝑛=1 and for 𝑛 greater than 4 against power law exponent 𝑁=1. 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 𝑛 between 2 and 10 against the power law exponent 𝑁=0.5 and for 𝑛=2, 3, and 4 against power law exponent 𝑁=1. The frequencies determined by the present method are slightly lower for the circumferential wave number 𝑛=3 to 𝑛=10 against the power law exponent 𝑁=15. The minimum frequency corresponds to 𝑛=4 and decreases with 𝑁=15 and corresponding decrement is 0.0218 per cent.

Table 4 

In Table 4(a), a comparison of of natural frequencies for the shell is given for the power law exponent 𝑁=0.5, 1, and 15. The respective frequencies determined by two techniques are the same for the circumferential wae number 𝑛=1 and for 𝑛 greater than 4 against power law exponent 𝑁=1. 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 𝑛 between 2 and 10 against the power law exponent 𝑁=0.5 and for 𝑛=2, 3, and 4 against power law exponent 𝑁=1. The minimum frequency corresponds to 𝑛=4 and decreases with 𝑁=0.5 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 (𝑛)

Tables 5 and 6 list the variations of natural frequencies (Hz) with the circumferential wave number “𝑛” for a type I FG cylindrical shell with the Nickel ring-stiffeners. The axial mode “𝑚” is taken to be unity. 𝐿/𝑅 is kept equal to 20 where 𝑅/ℎ=500, 20 in Tables 5 and 6, respectively. The columns 𝑁^𝑆𝑆 and 𝑁^𝑁 show the natural frequencies (Hz) for a Stainless Steel cylindrical shell and Nickel cylindrical shell, respectively. The influence of the value of 𝑁, which affects the constituent volume fractions, is analyzed. It is seen that when the value of 𝑁 is increased, the natural frequencies decreased. The decrease in the natural frequencies in Table 5, from 𝑁=1 to 𝑁=15 is about 1.6% at 𝑛=1 and about 1.5% at 𝑛=10. When 𝑁is small, the natural frequencies approached those of 𝑁^𝑆𝑆 and when 𝑁 is large they approached those of 𝑁^𝑁. Hence the natural frequencies for 𝑁greater than zero fell between those of 𝑁^𝑆𝑆 and 𝑁^𝑁 for a given circumferential wave number 𝑛.

Table 5 

Variation of natural frequencies against circumferential wave number n, type I FG cylindrical shell with nickel ring stiffeners (𝑚=1, ℎ/𝑅=0.002, 𝐿/𝑅=20, 𝑑/𝑏=1.334).

Table 6 

Variation of natural frequencies against circumferential wave number n, type I FG cylindrical shell with nickel ring stiffeners (𝑚=1, ℎ/𝑅=0.05, 𝐿/𝑅=20, 𝑑/𝑏=1.334).

Tables 7 and 8 show the variation of the natural frequencies (Hz) versus the circumferential wave number “𝑛” for a Type II FGM cylindrical shell with Nickel ring-stiffeners. The influence of 𝑁 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 𝑁, the natural frequencies (Hz) for a Type II FGM cylindrical shell increased with 𝑁. The increase in the natural frequencies (Hz) from 𝑁=1 to 𝑁=15 is about 1.64% at 𝑛=1 and about 1.61% at 𝑛=10. 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.

Table 7 

Variation of natural frequencies against circumferential wave number 𝑛, type II FG cylindrical shell with nickel ring stiffeners (𝑚=1, ℎ/𝑅=0.002, 𝐿/𝑅=20, 𝑑/𝑏=1.334).

Table 8 

Variation of natural frequencies against circumferential wave number 𝑛, type II FG cylindrical shell with nickel ring stiffeners (𝑚=1, ℎ/𝑅=0.05, 𝐿/𝑅=20, 𝑑/𝑏=1.334).

Comparing the frequencies in Tables 5 and 6 with those in Tables 7 and 8, it can be seen that for 𝑁>1, 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 𝑁<1, 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 𝑁=15 at 𝑛=10 and ℎ/𝑅=0.002, 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 𝑁=0.5 at 𝑛=10 and ℎ/𝑅=0.002, 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 𝑛, for Type I FG cylindrical shell with Stainless Steel ring-stiffeners. The influence of the value of 𝑁, which affects the constituent volume fraction, can be seen from the tables. It is seen that when the value of 𝑁 is increased, the natural frequencies decreased. The decrease in the natural frequencies in Table 9, from 𝑁=1 to 𝑁=15 is about 1.60% at 𝑛=1 and about 1.56% at 𝑛=10. When 𝑁 is small, the natural frequencies approached those of 𝑁^𝑆𝑆 and when 𝑁is large, they approached those of 𝑁^𝑁. Hence the natural frequencies for 𝑁 greater than zero fell between those of 𝑁^𝑆𝑆 and 𝑁^𝑁 for a given circumferential wave number 𝑛.

Table 9 

Variation of natural frequencies against circumferential wave number 𝑛, type I FG cylindrical shell with stainless steel ring stiffeners (𝑚=1, ℎ/𝑅=0.002, 𝐿/𝑅=20, 𝑑/𝑏=1.334).

Table 10 

Variation of natural frequencies against circumferential wave number 𝑛, type I FG cylindrical shell with stainless steel ring stiffeners (𝑚=1, ℎ/𝑅=0.05, 𝐿/𝑅=20, 𝑑/𝑏=1.334).

Tables 11 and 12 show the variation of natural frequencies against circumferential wave number 𝑛, for Type II FG cylindrical shell with Stainless Steel ring-stiffeners. The influence of 𝑁 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 𝑁. The increase in the natural frequencies from 𝑁=1 to 𝑁=15 is about 1.67% at 𝑛=1 and about 1.63% at 𝑛=10. 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.

Table 11 

Variation of natural frequencies against circumferential wave number 𝑛, type II FG cylindrical shell with stainless steel ring stiffeners (𝑚=1, ℎ/𝑅=0.002, 𝐿/𝑅=20, 𝑑/𝑏=1.334).

Table 12 

Variation of natural frequencies against circumferential wave number 𝑛, type II FG cylindrical shell with stainless steel ring stiffeners (𝑚=1, ℎ/𝑅=0.05, 𝐿/𝑅=20, 𝑑/𝑏=1.334).

Comparing the frequencies in Tables 9 and 10 with those in Tables 11 and 12, it can be seen that for 𝑁>1, 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 𝑁<1, 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 𝑁=15 at 𝑛=10 and ℎ/𝑅=0.002, 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 𝑁=0.5 at 𝑛=10 and ℎ/𝑅=0.002, 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 𝐿/𝑅

Tables 13 and 14 show the variations of the fundamental natural frequencies (Hz) with the 𝐿/𝑅ratio 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 𝑁, and for a Type II FGM cylindrical shell with Stainless Steel ring-stiffeners, the fundamental frequencies increased with 𝑁. The difference in the fundamental frequencies between 𝑁=1 and 𝑁=15 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 𝑁, the fundamental natural frequencies fall between those for 𝑁^𝑆𝑆 and 𝑁^𝑁.

Table 13 

Variation of natural frequencies against 𝐿/𝑅, type I FG cylindrical shell with stainless steel ring stiffeners (𝑚=1, ℎ/𝑅=0.002, 𝑑/𝑏=1.334).

Table 14 

Variation of natural frequencies against L/R, type II FG cylindrical shell with stainless steel ring stiffeners (𝑚=1, ℎ/𝑅=0.002, 𝑑/𝑏=1.334).

5.3.3. Variation of Minimum Frequency with ℎ/𝑅

Tables 15 and 16 show the variations of the fundamental natural frequencies (Hz) with the ℎ/𝑅 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 𝑁, and for a Type II FGM cylindrical shell with Stainless Steel ring-stiffeners, the fundamental frequencies increased with 𝑁. For all values of 𝑁, the fundamental natural frequencies lie between those for 𝑁^𝑆𝑆 and 𝑁^𝑁.

Table 15 

Variation of fundamental natural frequencies against h/R, type I FG cylindrical shell with stainless steel ring stiffeners (𝑚=1, 𝐿/𝑅=20, 𝑑/𝑏=1.334).

Table 16 

Variation of fundamental natural frequencies against h/R, type II FG cylindrical shell with stainless steel ring stiffeners (𝑚=1, 𝐿/𝑅=20, 𝑑/𝑏=1.334).

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 𝑁<1 the natural frequencies for Type I FG cylindrical shells are higher than those for Type II FG cylindrical shells and for 𝑁>1 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 [𝐶]=𝑐₁₁𝑐₁₂𝑐₁₃𝑐₂₁𝑐₂₂𝑐₂₃𝑐₃₁𝑐₃₂𝑐₃₃,[𝐷]=𝑑₁₁𝑑₁₂𝑑₁₃𝑑₂₁𝑑₂₂𝑑₂₃𝑑₃₁𝑑₃₂𝑑₃₃, (A.1) where 𝑐₁₁=𝑅𝐴₁₁^𝐿₀𝑑𝑈𝑑𝑥²+𝑛𝑑𝑥₂𝑅²𝐴₆₆−𝐵₆₆+14𝐵₆₆^𝐿₀𝑈²+𝑛𝑑𝑥₂𝑅+𝑒_𝑘³𝐺_𝑘𝐽_𝑘+𝑛²𝐸_𝑘𝐼_𝑧𝑘𝑈²,𝑐₁₂𝑛=𝑅𝑅𝐴₁₂^𝐿₀𝑉𝑑𝑈+𝑛𝑑𝑥𝑑𝑥𝑅−𝐴₆₆−𝐵₆₆+34𝐷₆₆^𝐿₀𝑈𝑑𝑉,𝑐𝑑𝑥𝑑𝑥₁₃=𝑅−𝐴₁₂𝑅+1−𝑛²𝑅²𝐵₁₂^𝐿₀𝑊𝑑𝑈𝑑𝑥𝑑𝑥+𝐵₁₁^𝐿₀𝑑²𝑊𝑑𝑥²𝑑𝑈+𝑛𝑑𝑥𝑑𝑥₂𝑅2𝐵₆₆−𝐷₆₆^𝐿₀𝑈𝑑𝑊+𝑛𝑑𝑥𝑑𝑥₂𝐸_𝑘𝐼_𝑧𝑘𝑅+𝑒_𝑘²𝑛²𝑒_𝑘𝑅+𝑒_𝑘−1𝑈𝑑𝑊+𝑛𝑑𝑥₂𝐺_𝑘𝐽_𝑘𝑅+𝑒_𝑘²𝑒_𝑘𝑅+𝑒_𝑘−1𝑈𝑑𝑊,𝑐𝑑𝑥₂₂𝑛=𝑅²𝑅²𝐴₂₂^𝐿₀𝑉²+𝐴𝑑𝑥₆₆+3𝐵₆₆+94𝐷₆₆^𝐿₀𝑑𝑉𝑑𝑥²+𝑛𝑑𝑥₂𝐸_𝑘𝐴_𝑘𝑅+𝑒_𝑘𝑒1+^𝑘𝑅²𝑉²,𝑐₂₃𝑛=𝑅𝑅²−𝐴₂₂+1−𝑛²𝑅𝐵₂₂^𝐿₀𝑉𝑊𝑑𝑥−𝑛2𝐵₆₆+3𝐷₆₆^𝐿₀𝑑𝑉𝑑𝑥𝑑𝑊+𝑛𝑑𝑥𝑑𝑥𝑅𝐵₁₂^𝐿₀𝑉𝑑²𝑊𝑑𝑥²−𝑑𝑥𝑛𝐸_𝑘𝐴_𝑘𝑅+𝑒_𝑘𝑒1+^𝑘𝑅𝑛1+²𝑒_𝑘𝑅𝑐𝑉𝑊,₃₃1=𝑅𝑅²𝐴₂₂−21−𝑛²𝑅𝐵₂₂+1−𝑛²²𝑅²𝐷₂₂×^𝐿₀𝑊²𝑑𝑥+𝐷₁₁^𝐿₀𝑑²𝑊𝑑𝑥²²×2𝑑𝑥𝑅²−𝑅𝐵₁₂+1−𝑛²𝐷₁₂^𝐿₀𝑊𝑑²𝑊𝑑𝑥𝑑𝑥+4𝑛²𝐷₆₆^𝐿₀𝑑𝑊+𝐸𝑑𝑥𝑑𝑥_𝑘𝐼_𝑧𝑘𝑅+𝑒_𝑘𝑛1−²𝑒_𝑘𝑅+𝑒_𝑘²+𝑛²𝐺_𝑘𝐽_𝑘𝑅+𝑒_𝑘𝑒1−^𝑘𝑅+𝑒_𝑘²𝑑𝑊𝑑𝑥²+𝐸_𝑘𝐼_𝑥𝑘𝑅+𝑒_𝑘³1−𝑛²²+𝑛1+²𝑒_𝑘𝑅²𝐸_𝑘𝐴_𝑘𝑅+𝑒_𝑘𝑊²,𝑑₁₁=𝑅ℎ𝜌^𝐿₀𝑈²𝑑𝑥+𝜌_𝑘𝐴_𝑘𝑅+𝑒_𝑘𝑈²,𝑑₁₂𝑑=0,₁₃=𝜌_𝑘𝐴_𝑘𝑒_𝑘𝑅+𝑒_𝑘𝑈𝑑𝑤,𝑑𝑑𝑥₂₂=𝑅ℎ𝜌^𝐿₀𝑉²𝑑𝑥+𝜌_𝑘𝐴_𝑘𝐴+𝑒_𝑘𝑒1+^𝑘𝑅²𝑉²,𝑑₂₃=−𝑛𝑒_𝑘𝜌_𝑘𝐴_𝑘𝑅+𝑒_𝑘𝑒1+^𝑘𝑅1𝑅𝑑𝑉𝑊,₃₃=𝑅𝜌ℎ^𝐿₀𝑊²𝑑𝑥+𝜌_𝑘𝑅+𝑒_𝑘𝑒_𝑘²𝐴_𝑘+𝐼_𝑥𝑘+𝐼_𝑧𝑘𝑑𝑤𝑑𝑥²+𝜌_𝑘𝐴_𝑘𝑅+𝑒_𝑘𝑛1+²𝑒_𝑘²𝑅²𝑊². (A.2)

Here (𝑥)=(𝑥)𝑈=𝑈𝑑𝑉,(𝑥)𝑑𝑥𝑉=𝑉=sin𝑚𝜋𝑥𝐿,𝑊=𝑊(𝑥)=𝑉_𝐿0(𝑥),𝑉²(𝑥)𝐿𝑑𝑥=2_𝐿0,𝑈²(𝑥)()𝑑𝑥=𝑚𝜋²,2𝐿_𝐿0𝑑𝑈(𝑥)𝑑𝑥²()𝑑𝑥=𝑚𝜋⁴2𝐿³_𝐿0,𝑑²𝑉(𝑥)𝑑𝑥²𝑉(𝑥)()𝑑𝑥=−𝑚𝜋².2𝐿 (A.3)