ISRN Mechanical EngineeringVolume 2012 (2012), Article ID 232498, 13 pagesdoi:10.5402/2012/232498

Research Article

Vibration Characteristics of Ring-Stiffened Functionally Graded Circular Cylindrical Shells

Muhmmad Nawaz Naeem,¹ Shazia Kanwal,¹ Abdul Ghafar Shah,² Shahid Hussain Arshad,³ and Tahir Mahmood⁴

¹Department of Mathematics, G. C. University Faisalabad, Faisalabad 38000, Pakistan²Department of Mathematics, Government Post Graduate College Jhang, Jhang 35200, Pakistan³Department of Civil Engineering, Iqra National University Peshawar, Peshawar 25000, Pakistan⁴Department 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.

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 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.

Table 1: Comparison of frequency parameters simply supported isotropic cylindrical shell. .

Loy et al. [15]
Present

1
0.016102
0.016101

2
0.009387
0.009381

3
0.022108
0.022105

4
0.042096
0.042095

5
0.068008
0.068008

6
0.099730
0.099730

7
0.137239
0.137240

8
0.180527
0.180529

9
0.229594
0.229596

10
0.284435
0.284439

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.

Table 2

(a) Comparison studies of frequency parameter for simply-supported cylindrical shell with 14 evenly spaced, externally eccentric ring stiffeners. ,

Stiffener's depth-to-width ratio
Circumferential wave no.
Swaddiwudhipong et al. [11]
Present analysis

1.3314
2
0.08472
0.08503

1.3314
3
0.06711
0.07049

1.3314
4
0.1072
0.11493

1.3314
5
0.1714
0.18394

2.6628
2
0.08372
0.08591

2.6628
3
0.1095
0.12269

2.6628
4
0.2037
0.2296

2.6628
5
0.3271
0.37226

3.9942
2
0.08788
0.09371

3.9942
3
0.1592
0.18444

3.9942
4
0.3049
0.35343

3.9942
5
0.4598
0.57304

(b) Comparison studies of frequency parameter for simply-supported cylindrical shell with 14 evenly spaced, internally eccentric ring stiffeners. ,

Stiffener's depth-to-width ratio
Circumferential wave no.
Swaddiwudhipong et al. [11]
Present analysis

1.3314
2
0.08282
0.0876

1.3314
3
0.05992
0.0771

1.3314
4
0.08641
0.1221

1.3314
5
0.1346
0.1918

2.6628
2
0.08141
0.0933

2.6628
3
0.08826
0.1380

2.6628
4
0.1536
0.2499

2.6628
5
0.2459
0.4000

3.9942
2
0.08425
0.1076

3.9942
3
0.1280
0.2128

3.9942
4
0.2363
0.3970

3.9942
5
0.3807
0.6377

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.

Table 3: Properties of materials.

Coefficients
Stainless steel
Nickel

 (N/m²)

 (kg/m³)
(N/m²)

(N/m²)

0.3262
8166

0.3100
8900

0
0
0
0
0
0

0

0
0

0

0
0

0
0
0
0
0
0

8166

0.3100
8900

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.

Table 4

(a) Comparison of natural frequencies (Hz) for type I FG cylindrical shell with simply-supported boundary conditions. (, , )

Naeem et al. [19]
Present

1
13.321
13.211
12.933
13.321
13.211
12.932

2
4.5162
4.4794
4.3829
4.5164
4.4796
4.3830

3
4.1903
4.1562
4.0646
4.1909
4.1564
4.0641

4
7.0967
7.0379
6.8851
7.0976
7.0381
6.8836

5
11.335
11.241
10.998
11.336
11.241
10.996

6
16.594
16.455
16.101
16.595
16.455
16.097

7
22.826
22.635
22.148
22.828
22.635
22.143

8
30.023
29.771
29.132
30.025
29.771
29.125

9
38.181
37.862
37.048
38.185
37.862
37.039

10
47.301
46.905
45.897
47.305
46.905
45.886

(b) Comparison of natural frequencies (Hz) for type II FG cylindrical shell with simply-supported boundary conditions. (, , )

Naeem et al. [19]
Present

1
13.103
13.211
13.505
13.103
13.211
13.505

2
4.4382
4.4742
4.5759
4.4378
4.4739
4.5757

3
4.1152
4.1486
4.2451
4.1144
4.1481
4.2454

4
6.9754
7.0330
7.1943
6.9744
7.0327
7.1956

5
11.145
11.238
11.494
11.143
11.237
11.496

6
16.317
16.453
16.453
16.315
16.452
16.830

7
22.447
22.633
23.147
22.444
22.633
23.152

8
29.524
29.770
30.446
29.521
29.770
30.453

9
37.548
37.861
38.720
37.544
37.861
38.729

10
46.517
46.904
47.968
46.511
46.904
47.979

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 .

Table 5: Variation of natural frequencies against circumferential wave number n, type I FG cylindrical shell with nickel ring stiffeners (, , , ).

5
0

1
26.831
25.864
26.497
26.421
26.335
26.176
26.018
25.922
25.894

2
30.397
29.289
30.014
29.926
29.828
29.645
29.466
29.355
29.323

3
47.192
45.494
46.603
46.468
46.317
46.037
45.762
45.594
45.545

4
82.416
79.471
81.393
81.158
80.896
80.411
79.935
79.643
79.559

5
131.10
126.42
129.47
129.10
128.68
127.91
127.16
126.69
126.56

6
191.52
184.69
189.14
188.60
187.99
186.86
185.76
185.09
184.89

7
263.21
253.82
259.59
259.20
258.37
256.82
255.30
254.37
254.11

8
346.02
333.69
341.74
340.75
339.66
337.62
335.63
334.41
334.o6

9
439.88
424.20
434.43
433.18
431.79
429.21
426.68
425.12
424.68

10
544.74
525.34
538.01
536.46
534.73
531.53
528.40
526.48
525.93

Table 6: Variation of natural frequencies against circumferential wave number n, type I FG cylindrical shell with nickel ring stiffeners (, , , ).

5
0

1
14.715
14.025
14.436
14.421
14.360
14.246
14.134
14.066
14.046

2
35.517
33.764
34.836
34.768
34.618
34.342
34.068
33.885
33.828

3
96.256
91.453
94.364
94.184
93.772
93.017
92.275
91.782
91.626

4
183.17
174.01
179.56
179.21
178.43
176.99
175.57
174.64
174.34

5
295.50
280.72
289.66
289.11
287.84
285.52
283.24
281.73
281.25

6
433.06
411.40
424.51
423.70
421.84
418.44
415.10
412.88
412.18

7
595.76
565.96
584.00
582.89
580.33
575.65
571.05
568.00
567.04

8
783.53
744.35
768.06
766.60
763.24
757.08
751.04
747.03
745.76

9
996.33
946.51
976.66
974.80
970.53
962.70
955.02
949.92
948.31

10
1234.1
1172.4
1209.7
1207.46
1202.17
1192.5
1182.9
1176.6
1174.6

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.

Table 7: Variation of natural frequencies against circumferential wave number , type II FG cylindrical shell with nickel ring stiffeners (, , , ).

5
0

1
26.831
25.864
26.176
26.251
26.335
26.498
26.663
26.768
26.798

2
30.397
29.289
29.647
29.732
29.830
30.016
30.204
30.324
30.359

3
47.192
45.494
46.044
46.176
46.325
46.611
46.897
47.082
47.135

4
82.416
79.471
80.426
80.655
80.914
81.408
81.904
82.225
82.317

5
131.10
126.42
127.94
128.30
128.71
129.50
130.28
130.80
130.94

6
191.52
184.69
186.90
187.43
188.03
189.18
190.33
191.08
191.29

7
263.21
253.82
256.87
257.60
258.42
260.00
261.58
262.60
262.90

8
346.02
333.69
337.69
338.65
339.73
341.80
343.88
345.22
345.61

9
439.88
424.20
429.29
430.51
431.89
434.52
437.15
438.87
439.36

10
544.74
525.34
531.64
533.15
534.85
538.11
541.37
543.49
544.10

Table 8: Variation of natural frequencies against circumferential wave number , type II FG cylindrical shell with nickel ring stiffeners (, , , ).

5
0

1
14.714
14.025
14.245
14.298
14.359
14.475
14.594
14.669
14.691

2
35.517
33.764
34.329
34.461
34.610
34.891
35.178
35.383
35.446

3
96.256
91.453
93.018
93.383
93.790
94.559
95.344
95.892
96.063

4
183.17
174.01
177.00
177.70
178.47
179.94
181.43
182.48
182.80

5
295.50
280.72
285.54
286.66
287.92
290.28
292.70
294.38
294.90

6
433.06
411.40
418.47
420.12
421.96
425.42
428.96
431.42
432.19

7
595.76
565.96
575.69
577.96
580.49
585.25
590.12
593.50
594.57

8
783.53
744.35
757.14
760.12
763.44
769.71
776.11
780.56
781.96

9
996.33
946.51
962.78
966.56
970.79
978.76
986.89
992.55
994.33

10
1234.1
1172.4
1192.5
1197.2
1202.5
1212.3
1222.4
1229.4
1231.6

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 .

Table 9: Variation of natural frequencies against circumferential wave number , type I FG cylindrical shell with stainless steel ring stiffeners (, , , ).

5
0

1
25.274
24.348
24.955
24.881
24.799
24.646
24.496
24.403
24.377

2
29.526
28.439
29.151
29.064
28.968
28.788
28.613
28.504
28.473

3
45.740
44.073
45.161
45.029
44.881
44.605
44.337
44.171
44.124

4
79.445
76.570
78.446
78.217
77.962
77.487
77.024
76.739
76.658

5
126.21
121.65
124.63
124.26
123.86
123.10
122.37
121.92
121.79

6
184.32
177.66
182.00
181.47
180.88
179.78
178.71
178.05
177.86

7
253.29
244.15
250.11
249.38
248.57
247.06
245.59
244.68
244.42

8
332.97
320.96
328.79
327.83
326.77
324.79
322.85
321.66
321.32

9
423.29
408.03
417.98
416.77
415.41
412.90
410.43
408.92
408.49

10
524.21
505.31
517.64
516.14
514.46
511.34
508.29
506.42
505.88

Table 10: Variation of natural frequencies against circumferential wave number , type I FG cylindrical shell with stainless steel ring stiffeners (, , , ).

5
0

1
14.482
13.800
14.206
14.192
14.131
14.019
13.908
13.841
13.821

2
35.176
33.434
34.499
34.433
34.283
34.009
33.737
33.555
33.498

3
95.883
91.090
93.995
93.817
93.405
92.652
91.911
91.419
91.263

4
182.65
173.50
179.04
178.70
177.91
176.48
175.07
174.13
173.83

5
294.72
279.96
288.89
288.35
287.08
284.76
282.48
280.97
280.49

6
431.95
410.31
423.41
422.60
420.75
417.35
414.01
411.79
411.09

7
594.24
564.47
582.49
581.38
578.83
574.15
569.56
566.51
565.55

8
781.54
742.40
766.08
764.63
761.27
755.12
749.08
745.07
743.81

9
993.81
944.04
974.15
972.31
968.04
960.22
952.54
947.45
945.84

10
1231.0
1169.4
1206.6
1204.4
1199.1
1189.4
1179.9
1173.6
1171.6

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.

Table 11: Variation of natural frequencies against circumferential wave number , type II FG cylindrical shell with stainless steel ring stiffeners (, , , ).

5
0

1
25.274
24.348
24.646
24.718
24.799
24.955
25.113
25.214
25.243

2
29.526
28.439
28.790
28.874
28.970
29.152
29.338
29.455
29.490

3
45.740
44.073
44.613
44.742
44.888
45.168
45.451
45.632
45.684

4
79.445
76.570
77.502
77.726
77.978
78.461
78.945
79.259
79.349

5
126.21
121.65
123.12
123.48
123.88
124.65
125.42
125.91
126.06

6
184.32
177.66
179.81
180.34
180.92
182.04
183.16
183.89
184.09

7
253.29
244.15
247.11
247.82
248.63
250.16
251.70
252.70
252.99

8
332.97
320.96
324.85
325.79
326.85
328.86
330.88
332.20
332.57

9
423.29
408.03
412.97
414.16
415.51
418.07
420.64
422.30
422.78

10
524.21
505.31
511.44
512.91
514.57
517.74
520.92
522.99
523.58

Table 12: Variation of natural frequencies against circumferential wave number , type II FG cylindrical shell with stainless steel ring stiffeners (, , , ).

5
0

1
14.482
13.800
14.018
14.071
14.130
14.245
14.362
14.436
14.458

2
35.176
33.434
33.995
34.126
34.274
34.553
34.839
35.043
35.105

3
95.883
91.090
92.651
93.015
93.421
94.188
94.972
95.519
95.691

4
182.65
173.50
176.49
177.18
177.96
179.42
180.91
181.95
182.28

5
294.72
279.96
284.78
285.90
287.15
289.51
291.92
293.60
294.13

6
431.95
410.31
417.37
419.02
420.85
424.31
427.85
430.31
431.08

7
594.24
564.47
574.19
576.45
578.98
583.74
588.60
591.98
593.05

8
781.54
742.40
755.17
758.14
761.47
767.73
774.12
778.57
779.97

9
993.81
944.04
960.28
964.06
968.29
976.24
984.38
990.03
991.81

10
1231.0
1169.4
1189.5
1194.2
1199.4
1209.3
1219.3
1226.3
1228.5

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 .

Table 13: Variation of natural frequencies against , type I FG cylindrical shell with stainless steel ring stiffeners (, , ).

5
0

0.2
20
2875.5
2866.8
2872.5
2871.8
2871.1
2869.6
2868.2
2867.3
2867.1

0.5
15
1614.0
1602.9
1610.2
1609.4
1608.4
1606.5
1604.7
1603.5
1603.3

1.0
11
900.77
890.59
897.30
896.49
895.59
893.89
892.22
891.20
890.90

2.0
8
488.92
480.57
486.06
485.40
484.66
483.27
481.91
481.07
480.82

5.0
5
181.84
177.15
180.23
179.85
179.44
178.66
177.90
177.43
177.29

10
4
100.74
97.538
99.630
99.380
99.095
98.566
98.046
97.727
97.636

20
3
45.740
44.073
45.162
45.029
44.881
44.605
44.336
44.171
44.124

50
2
13.408
12.874
13.222
13.179
13.132
13.044
12.958
12.905
12.890

100
1
2.5497
2.4455
2.5135
2.5052
2.4960
2.4788
2.4620
2.4516
2.4486

Table 14: Variation of natural frequencies against L/R, type II FG cylindrical shell with stainless steel ring stiffeners (, , ).

5
0

0.2
20
2875.5
2866.8
2869.3
2870.5
2871.2
2872.7
2874.1
2875.0
2875.2

0.5
15
1614.0
1602.9
1606.3
1607.6
1608.5
1610.4
1612.2
1613.4
1613.7

1.0
11
900.77
890.59
893.66
894.81
895.71
897.41
899.07
900.15
900.45

2.0
8
488.92
480.57
483.07
484.00
484.74
486.14
487.51
488.40
488.65

5.0
5
181.84
177.15
178.52
179.06
179.47
180.26
181.04
181.54
181.68

10
4
100.74
97.538
98.430
98.833
99.115
99.652
100.19
100.54
100.64

20
3
45.740
44.073
44.505
44.742
44.888
45.168
45.450
45.631
45.684

50
2
13.408
12.874
12.994
13.086
13.133
13.223
13.314
13.373
13.390

100
1
2.5497
2.4455
2.4788
2.4868
2.4959
2.5135
2.5314
2.5427
2.5461

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 .

Table 15: Variation of fundamental natural frequencies against h/R, type I FG cylindrical shell with stainless steel ring stiffeners (, , ).

5

0.001
3
52.567
50.896
51.992
51.859
51.711
51.434
51.163
50.995

0.005
2
24.993
23.985
24.643
24.563
24.473
24.307
24.145
24.045

0.007
2
23.114
22.162
22.783
22.707
22.623
22.466
22.313
22.218

0.01
2
21.393
20.493
21.081
21.000
20.929
20.782
20.637
20.547

0.02
1
15.812
15.090
15.561
15.504
15.440
15.321
15.204
15.133

0.03
1
15.096
14.396
15.060
14.798
14.736
14.621
14.507
14.438

0.04
1
14.717
14.028
14.472
14.424
14.362
14.249
14.137
14.069

0.05
1
14.482
13.800
14.245
14.191
14.131
14.019
13.908
13.841

Table 16: Variation of fundamental natural frequencies against h/R, type II FG cylindrical shell with stainless steel ring stiffeners (, , ).

5

0.001
3
52.567
50.896
51.439
51.569
51.716
51.996
52.279
52.459

0.005
2
24.993
23.985
24.309
24.387
24.475
24.645
24.816
24.927

0.007
2
23.114
22.162
22.467
22.541
22.624
22.784
22.945
23.050

0.01
2
21.393
20.493
20.782
20.851
20.930
21.081
21.232
21.333

0.02
1
15.812
15.090
15.321
15.376
15.440
15.561
15.685
15.764

0.03
1
15.096
14.396
14.620
14.674
14.735
14.853
14.974
15.050

0.04
1
14.717
14.028
14.248
14.301
14.361
14.477
14.596
14.671

0.05
1
14.482
13.800
14.018
14.070
14.130
14.245
14.362
14.436

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

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