Journal of Computational Engineering

Volume 2014 (2014), Article ID 543794, 13 pages

http://dx.doi.org/10.1155/2014/543794

Research Article

On the Validity of Flat-Plate Finite Strip Approximation of Circular Cylindrical Shells

Jackson Kong

Division of Building Science and Technology, City University of Hong Kong, Hong Kong

Received 28 April 2013; Accepted 19 December 2013; Published 11 February 2014

Academic Editor: George Labeas

Copyright © 2014 Jackson Kong. 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 validity of modeling curved shell panels using flat-plate finite strips has been demonstrated in the past by comparing finite strip numerical results with analytical solutions of a few benchmark problems; to date, no mathematical exact solutions of the method or its explicit forms of error terms have been rigorously derived to demonstrate analytically its validity. Using a unitary transformation approach (abbreviated as the U-transformation herein), an attempt is made in this paper to derive mathematical exact solutions of flat-plate finite strips in cylindrical shell vibration analysis. Unlike the conventional finite strip method which involves assembly of the global system of matrix equation and its numerical solution, the U-transformation method makes use of the inherent cyclic symmetry of cylindrical shells to decouple the global matrix equation into one involving only a few unknowns, thus rendering explicit form of solutions for the flat-shell finite strip to be derived. Such explicit solutions can be subsequently expanded into Taylor's series whose results reveal directly their convergence to the exact solutions and the corresponding rate of convergence.

1. Introduction

Analysis of curved shells is more complex than the analogous problem in flat plates, since the structural deformations in a curved shell depend not only on the rotation and radial displacements but also on the coupled tangential displacement caused by the curvature of the shell. To cater for such couplings, curved elements for modeling practical structures with curvature in one or two dimensions have been developed since the seventies; some examples of these developments can be found in [1–4]. Despite such extensive research works, curved beam and shell-type structures are still very often in practice modeled as an assemblage of straight beam elements or flat plate elements, due to their conceptual simplicity, ease of modeling, and availability of such elements in most commercial softwares. The validity and efficacy of such a simple and practical approach, however, rely on whether the solution so obtained does converge rapidly to the true solution as the number of elements used increases. In this respect, Fumio [5], by making use of the finite element approximation theory and the idea of partial approximation, showed that straight Bernoulli beam element does converge to the exact solution for a uniform plane arch with constant curvature and clamped supports under static load. The rate of convergence as measured by the energy norm was also discussed in the paper. Convergence of flat plate finite elements in modelling cylindrical shells was analyzed using the same approach in [6]. Following along the same line, Bernadou et al. [7, 8] analysed the convergence of general arches using straight beam elements and that of cylindrical shells using flat plate elements. In addition to static problems, convergence for natural vibration of clamped arch using straight beams and lumped mass approaches was studied by Ishihara [9, 10].

Unlike the finite element approximation theory adopted by the aforesaid researchers in studying convergence [5–10], a different approach, namely, the U-transformation was adopted in this work. The method was originally developed by Chan et al. [11] for the exact analysis of structures with periodicity and subsequently extended by Cai et al. [12] to systems with bi-periodicity. The success of the method that relies on a complex unitary matrix , when applied to the transformation of a circulant matrix, completely diagonalizes the latter. Such circulant matrices exist in many branches of science and engineering and in the context of structural engineering; they correspond to the stiffness matrices of structures with cyclic symmetry. In essence, the U-transformation provides a mathematical tool for decoupling a cyclic symmetric structural system and making it possible for obtaining the solution in an explicit form. The method has been successfully applied to study the convergence of finite strips for rectangular plates [13, 14]. In this paper, the method is further extended to study the vibration of cylindrical shells using the thin flat-plate finite strips; the latter was originally developed by Cheung [15] and remains as one of the most effective elements for analyzing prismatic structures [16]. By making use of the inherent cyclic symmetry of a cylindrical shell and the U-transformation method, explicit solutions for the cylindrical shell vibration problem using flat-shell finite strips were derived. Such explicit solutions can be subsequently expanded into Taylor’s series whose results reveal directly the error terms and the corresponding rate of convergence. The converged solutions can be compared with exact solutions.

This paper is organized as follows: the U-transformation is first introduced by deriving the in-plane vibration solution of a ring using straight beam elements. The solutions are then compared directly with the exact solutions. The same approach is then extended to the vibration of cylindrical shells using the classical flat-plate finite strips, and the derivation of the explicit solutions and the rate of convergence follow.

2. U-Transformation

Given entries , , a circulant matrix is a matrix whose rows are of the following form:

Circulant matrices have a very important property. The unitary matrix of order which is given by

where

diagonalizes the entire circulant matrix. Denoting as the complex conjugate of , we have

where

3. In-Plane Vibration of a Ring

A ring with radius , flexural stiffness , and mass per unit length is shown in Figure 1. The ring is divided into numbers of identical, straight beam elements, whose axial and lateral displacements are approximated by the conventional linear and cubic Hermite shape functions, respectively. With reference to local coordinates , the stiffness and consistent mass matrices of the straight, 2-node, Bernoulli beam element can be partitioned as [17]:

where

and .

Figure 1: Division of a ring into straight beam elements.

Instead of transforming to the global Cartesian axes in the conventional manner, the local stiffness and mass matrices are transformed into the tangential-radial coordinates (Figure 1) prior to assembly. The transformation matrix for each element is identical and is given by

where

The transformed element stiffness matrix is then assembled to form the stiffness matrix of the -sided polygon, denoted by :

where

which reveal its resemblance with a circulant matrix. The mass matrix can be assembled in exactly the same manner.

Applying the U-transformation to the cyclic symmetric system, we can write the deflection variables at node in terms of the generalized coordinates and the associated symmetry modes :

where .

Using (3a), (3b), and (3c) and (6a) and (6b) with each entry of the circulant stiffness matrix being a 2 × 2 submatrix, we can write the diagonalized stiffness matrix as

Similarly, we have the diagonalized mass matrix

Accordingly, the decoupled eigenproblem of order 3 can be written as

By expanding the determinant, the natural frequencies can be found from the roots of the cubic characteristic polynomial:

whose coefficients can be determined explicitly using Mathematica. To find the converged finite element solution as approaches infinity and the associated rate of convergence, the coefficients are expanded into Taylor’s series of . It can be shown that

where , , , are functions of the mode number , the bending and axial stiffness, density, and radius of the ring only.

From (11) and (12), it can be observed that

Substituting (13) and (12) into (11) and collecting terms of and , we have

The explicit form of the first equation is given by

Thus, by denoting the radius of gyration as , the two roots can be expressed as

which agrees exactly with the analytical solution given in Soedel [18], thus confirming the convergence of the straight beam finite element solution to the exact analytical solution. It is also apparent from (13) that the straight beam eigenvalue solution converges to the exact solution at an asymptotic rate of . For the explicit form of the error term , it can be found from (14) that

By substituting the roots into the above expression, we can obtain the explicit form of the error terms directly.

4. Classical Flat-Plate Finite Strip Formulation

Having demonstrated the application of U-transformation to the vibration of ring in the previous section, the same concept is extended to the vibration of a thin cylindrical shell of radius , thickness (where ), density , Young’s modulus , and Poisson’s ratio . Assume that the circular cylinder is divided into number of identical flat-plate finite strips around the circumference, see Figure 2(a). By neglecting transverse normal and shear deformation, the displacement functions of a thin-plate finite strip with thickness and width (Figure 2(b)) can be written in terms of trigonometric series longitudinally whilst linear and cubic Hermite shape functions, denoted by and , respectively, are used across the strip; therefore we have

where the four displacement variables on each nodal line are represented by , , , , . The displacement functions in (18), originally developed by Cheung and Tham [16], satisfy the shear diaphragm end conditions (or conventionally known as the simply-supported conditions) of a cylinder with finite length . On the other hand, for an infinitely long cylinder, is the half wavelength of the th vibration modes along the length.

(a)

(b)

Figure 2: (a) Division of a cylindrical shell into flat-plate finite strips. (b) Local coordinate system of a flat-plate finite strip with nodal line 1 and 2 along and , respectively.

Due to the orthogonality of the trigonometric series [16], the strip stiffness matrix for each longitudinal mode , which is of order 8 × 8, is given by

where

Explicit forms of the strain-displacement relation and the stiffness matrix can be found in Cheung and Tham [16] and they are omitted herein for brevity.

Partitioning the stiffness matrix into four submatrices (each of order 4 × 4), we have

By the same token, we can partition the strip mass matrix :

where the subscripts of submatrices and correspond to nodal lines and . To ensure compatibility of deformation along a nodal line between two adjacent strips, the local stiffness and mass matrices of a strip are transformed into the radial-tangential coordinates (Figure 2(a)) prior to assembly. The transformation matrix for each strip is identical and is given by

where

The transformed strip stiffness matrix can be assembled to form the stiffness matrix of the -sided polygonal tube, denoted by for longitudinal mode :

where, by dropping the subscript for simplicity, we have

which reveal its resemblance with (6b) except that order of matrices in (4a), (4b), and (4c) is of order 4 × 4. The transformed global mass matrix can be formed in exactly the same manner.

Applying the U-transformation to the cyclic symmetric system, the deflection variables at nodal line are written in terms of the generalized coordinates and its associated symmetry modes :

where .

Using (3a), (3b), and (3c) and (24a) and (24b) the stiffness matrix of longitudinal mode can be diagonalized as

Similarly, we have the diagonalized mass matrix

Accordingly, the decoupled eigenproblem (of order 4 × 4) for longitudinal mode can be written as

Following the same argument as in previous section, the determinant is expanded into a quadric characteristic polynomial:

whose coefficients are found explicitly using Mathematica. To find the converged solution as approaches infinity and the associated rate of convergence, the coefficients are expanded into Taylor’s series of . It can be shown that

From (29) and (30), it can be observed that

Substituting (31) and (30) into (29) and collecting terms of and , we have the following cubic equation for solving , that is, the converged finite-strip solution:

where

Once the roots are found, the corresponding coefficient of the error term can be obtained from

Explicit form of the three roots is too lengthy to compare directly with analytical solutions. Instead, to compare the converged flat-plate finite strip solutions with the analytical cylindrical shell solutions, governing equation [19] is solved for shells with different thickness-to-radius ratios ( and 20) and radius-to-length ratios ( and 10). Numerical results of the three roots obtained by solving (32) are compared with the analytical solutions in Tables 1(a)–1(f) for the first three longitudinal modes (i.e., ) and up to the first 25 circumferential modes of a very thin shell (); it is apparent from Table 1 that the flat-plate finite strip solutions do agree with, almost up to computer precision, the cylindrical shell solutions. The first 25 circumferential modes are also computed for the case of a moderately thick shell () and summarized in Tables 2(a) and 2(b); excellent agreement can be observed among the three sets of results for both the short and long shells.

Table 1: (a) Comparison of exact flat-plate finite strip results with analytical results [19] for the first 25 circumferential modes and longitudinal mode ( = 100, = 10, , = 1, and = 0.3). (b) Comparison of exact flat-plate finite strip results with analytical results [19] for the first 25 circumferential modes and longitudinal mode ( = 100, = 10, , = 1, and = 0.3). (c) Comparison of exact flat-plate finite strip results with analytical results [19] for the first 25 circumferential modes and longitudinal mode ( = 100, = 10, , , and ). (d) Comparison of exact flat-plate finite strip results with analytical results [19] for the first 25 circumferential modes and longitudinal mode ( = 100, = 2, , , and ). (e) Comparison of exact flat-plate finite strip results with analytical results [19] for the first 25 circumferential modes and longitudinal mode ( = 100, = 2, , , and ). (f) Comparison of exact flat-plate finite strip results with analytical results [19] for the first 25 circumferential modes and longitudinal mode ( = 100, = 2, , , and = 0.3).

(a)  

Mode
Reference [19]
Finite strip
% Error

Root 1
Root 2
Root 3
Root 1
Root 2
Root 3
Root 1
Root 2
Root 3

1
0.003838
0.477161
2.247847
0.003839
0.477154
2.247864
0.0024
−0.0014
0.0008

2
0.00053
1.60297
5.576039
0.000529
1.602897
5.57615
−0.0584
−0.0046
0.002

3
0.000644
3.513856
11.08322
0.000644
3.51365
11.08351
−0.0644
−0.0059
0.0026

4
0.001998
6.200774
18.78118
0.001997
6.200375
18.78173
−0.0205
−0.0064
0.0029

5
0.005126
9.659683
28.67419
0.005125
9.659031
28.67507
−0.0061
−0.0067
0.0031

6
0.01098
13.88911
40.76376
0.01098
13.88815
40.76505
−0.0009
−0.0069
0.0032

7
0.020762
18.88844
55.0505
0.020762
18.88711
55.05228
0.0013
−0.007
0.0032

8
0.035896
24.65741
71.53469
0.035897
24.65566
71.53703
0.0024
−0.0071
0.0033

9
0.058031
31.19588
90.21647
0.058033
31.19365
90.21944
0.0029
−0.0072
0.0033

10
0.089033
38.50377
111.0959
0.089036
38.501
111.0996
0.0033
−0.0072
0.0033

11
0.130991
46.58103
134.1731
0.130996
46.57767
134.1776
0.0035
−0.0072
0.0033

12
0.186212
55.42764
159.448
0.186218
55.42363
159.4533
0.0036
−0.0072
0.0033

13
0.257222
65.04358
186.9207
0.257232
65.03886
186.9269
0.0037
−0.0073
0.0034

14
0.34677
75.42884
216.5911
0.346783
75.42335
216.5984
0.0038
−0.0073
0.0034

15
0.457822
86.58342
248.4593
0.457839
86.5771
248.4677
0.0038
−0.0073
0.0034

16
0.593564
98.5073
282.5253
0.593586
98.5001
282.5348
0.0038
−0.0073
0.0034

17
0.757403
111.2005
318.7891
0.757432
111.1923
318.7998
0.0039
−0.0073
0.0034

18
0.952964
124.663
357.2506
0.953001
124.6538
357.2627
0.0039
−0.0073
0.0034

19
1.184095
138.8947
397.91
1.184141
138.8845
397.9235
0.0039
−0.0073
0.0034

20
1.45486
153.8958
440.7672
1.454917
153.8845
440.7821
0.0039
−0.0074
0.0034

21
1.769544
169.6662
485.8221
1.769614
169.6537
485.8386
0.0039
−0.0074
0.0034

22
2.132654
186.2059
533.0749
2.132738
186.1921
533.0929
0.004
−0.0074
0.0034

23
2.548913
203.5148
582.5254
2.549015
203.4998
582.5452
0.004
−0.0074
0.0034

24
3.023267
221.5931
634.1737
3.023388
221.5767
634.1953
0.004
−0.0074
0.0034

25
3.56088
240.4406
688.0199
3.561022
240.4228
688.0433
0.004
−0.0074
0.0034

(b)  

Mode
Reference [19]
Finite strip
% Error

Root 1
Root 2
Root 3
Root 1
Root 2
Root 3
Root 1
Root 2
Root 3

1
0.038162
0.714701
2.415242
0.038162
0.714689
2.415267
0.0025
−0.0016
0.001

2
0.00641
1.786224
5.826179
0.006409
1.786145
5.8263
−0.0173
−0.0044
0.0021

3
0.002154
3.667229
11.36764
0.002152
3.667019
11.36794
−0.0788
−0.0057
0.0026

4
0.002579
6.339256
19.08146
0.002577
6.338854
19.08202
−0.072
−0.0063
0.0029

5
0.005463
9.790132
28.98279
0.005461
9.789479
28.98368
−0.0335
−0.0067
0.0031

6
0.011266
14.01483
41.07719
0.011265
14.01387
41.07849
−0.0147
−0.0069
0.0032

7
0.02107
19.01119
55.36697
0.021068
19.00986
55.36875
−0.0062
−0.007
0.0032

8
0.036261
24.77816
71.85317
0.03626
24.77641
71.85552
−0.002
−0.0071
0.0033

9
0.058475
31.31524
90.53636
0.058475
31.31301
90.53934
0.0002
−0.0071
0.0033

10
0.089573
38.62211
111.4168
0.089574
38.61935
111.4205
0.0015
−0.0072
0.0033

11
0.13164
46.69862
134.4948
0.131643
46.69526
134.4992
0.0023
−0.0072
0.0033

12
0.186982
55.54465
159.7702
0.186987
55.54064
159.7756
0.0027
−0.0072
0.0033

13
0.258126
65.16014
187.2434
0.258134
65.15542
187.2496
0.0031
−0.0073
0.0033

14
0.347819
75.54504
216.9142
0.347831
75.53955
216.9214
0.0033
−0.0073
0.0034

15
0.459027
86.69932
248.7827
0.459043
86.69301
248.791
0.0035
−0.0073
0.0034

16
0.594936
98.62296
282.8489
0.594958
98.61576
282.8584
0.0036
−0.0073
0.0034

17
0.758954
111.3159
319.1129
0.758982
111.3078
319.1236
0.0037
−0.0073
0.0034

18
0.954705
124.7783
357.5746
0.95474
124.7691
357.5867
0.0037
−0.0073
0.0034

19
1.186036
139.0099
398.2341
1.186081
138.9997
398.2476
0.0038
−0.0073
0.0034

20
1.457012
154.0109
441.0914
1.457068
153.9995
441.1063
0.0038
−0.0074
0.0034

21
1.771919
169.7811
486.1464
1.771987
169.7686
486.1629
0.0039
−0.0074
0.0034

22
2.135261
186.3207
533.3993
2.135344
186.307
533.4174
0.0039
−0.0074
0.0034

23
2.551765
203.6296
582.8499
2.551864
203.6146
582.8697
0.0039
−0.0074
0.0034

24
3.026373
221.7078
634.4983
3.026492
221.6914
634.5199
0.0039
−0.0074
0.0034

25
3.564252
240.5553
688.3445
3.564392
240.5375
688.3679
0.0039
−0.0074
0.0034

(c)  

Mode
Reference [19]
Finite strip
% Error

Root 1
Root 2
Root 3
Root 1
Root 2
Root 3
Root 1
Root 2
Root 3

1
0.117569
1.030781
2.751855
0.117572
1.030763
2.751891
0.0026
−0.0018
0.0013

2
0.0259
2.065125
6.259916
0.025898
2.06504
6.260051
−0.0083
−0.0041
0.0022

3
0.007829
3.914095
11.84728
0.007825
3.91388
11.84759
−0.0483
−0.0055
0.0026

4
0.004751
6.566632
19.58415
0.004747
6.566226
19.58472
−0.0902
−0.0062
0.0029

5
0.006564
10.00599
29.49815
0.006559
10.00533
29.49906
−0.0669
−0.0066
0.0031

6
0.012017
14.22358
41.60012
0.012013
14.22261
41.60143
−0.0356
−0.0068
0.0032

7
0.021736
19.21531
55.89471
0.021732
19.21398
55.8965
−0.0182
−0.0069
0.0032

8
0.036963
24.97915
72.38416
0.036959
24.97739
72.38651
−0.0092
−0.007
0.0033

9
0.059275
31.514
91.06963
0.059273
31.51176
91.07262
−0.0043
−0.0071
0.0033

10
0.090513
38.81925
111.9518
0.090512
38.81648
111.9555
−0.0015
−0.0071
0.0033

11
0.13275
46.89453
135.0309
0.132751
46.89116
135.0354
0.0002
−0.0072
0.0033

12
0.188288
55.73962
160.3074
0.188291
55.7356
160.3127
0.0013
−0.0072
0.0033

13
0.25965
65.35437
187.7812
0.259655
65.34964
187.7875
0.002
−0.0072
0.0033

14
0.349581
75.73868
217.4526
0.34959
75.73318
217.4599
0.0025
−0.0073
0.0034

15
0.461047
86.89248
249.3216
0.46106
86.88616
249.33
0.0029
−0.0073
0.0034

16
0.597234
98.81572
283.3882
0.597253
98.80852
283.3978
0.0031
−0.0073
0.0034

17
0.761547
111.5084
319.6525
0.761572
111.5002
319.6633
0.0033
−0.0073
0.0034

18
0.957613
124.9704
358.1146
0.957646
124.9613
358.1267
0.0034
−0.0073
0.0034

19
1.189277
139.2018
398.7743
1.189319
139.1916
398.7878
0.0035
−0.0073
0.0034

20
1.460605
154.2026
441.6318
1.460658
154.1913
441.6467
0.0036
−0.0073
0.0034

21
1.775882
169.9727
486.687
1.775947
169.9602
486.7035
0.0037
−0.0074
0.0034

22
2.139612
186.5121
533.94
2.139692
186.4984
533.9581
0.0037
−0.0074
0.0034

23
2.556522
203.8209
583.3908
2.556619
203.8058
583.4106
0.0038
−0.0074
0.0034

24
3.031555
221.8989
635.0393
3.031671
221.8825
635.0609
0.0038
−0.0074
0.0034

25
3.569876
240.7463
688.8856
3.570014
240.7285
688.909
0.0039
−0.0074
0.0034

(d)  

Mode
Reference [19]
Finite strip
% Error

Root 1
Root 2
Root 3
Root 1
Root 2
Root 3
Root 1
Root 2
Root 3

1
0.360519
1.711392
4.171047
0.36053
1.711367
4.171102
0.0032
−0.0015
0.0013

2
0.117813
2.821872
7.754094
0.117809
2.821778
7.754261
−0.0035
−0.0033
0.0021

3
0.042117
4.649301
13.42077
0.042108
4.649077
13.42111
−0.0236
−0.0048
0.0026

4
0.019279
7.270339
21.2091
0.019267
7.269925
21.2097
−0.0624
−0.0057
0.0028

5
0.01385
10.68516
31.15515
0.013838
10.68449
31.15608
−0.092
−0.0062
0.003

6
0.016449
14.88542
43.27761
0.016436
14.88445
43.27896
−0.0782
−0.0065
0.0031

7
0.025049
19.86499
57.58586
0.025036
19.86365
57.58769
−0.0504
−0.0067
0.0032

8
0.039939
25.62014
74.08476
0.039927
25.61838
74.08715
−0.0303
−0.0069
0.0032

9
0.062317
32.14864
92.777
0.062306
32.14641
92.78003
−0.0181
−0.007
0.0033

10
0.093854
39.44914
113.6641
0.093844
39.44637
113.6679
−0.0107
−0.007
0.0033

11
0.136544
47.5208
136.7471
0.136536
47.51743
136.7516
−0.0062
−0.0071
0.0033

12
0.192647
56.36306
162.0265
0.192641
56.35903
162.0318
−0.0032
−0.0071
0.0033

13
0.264666
65.97556
189.5026
0.264663
65.97083
189.5089
−0.0013
−0.0072
0.0033

14
0.355332
76.35806
219.1759
0.355333
76.35256
219.1832

−0.0072
0.0033

15
0.467605
87.51038
251.0464
0.46761
87.50406
251.0548
0.001
−0.0072
0.0034

16
0.604665
99.43241
285.1143
0.604675
99.4252
285.1238
0.0016
−0.0072
0.0034

17
0.769916
112.124
321.3796
0.769932
112.1159
321.3904
0.0021
−0.0073
0.0034

18
0.966981
125.5852
359.8425
0.967005
125.5761
359.8546
0.0025
−0.0073
0.0034

19
1.199705
139.8159
400.503
1.199739
139.8057
400.5165
0.0028
−0.0073
0.0034

20
1.472153
154.816
443.3611
1.472198
154.8047
443.3761
0.003
−0.0073
0.0034

21
1.78861
170.5856
488.4169
1.788667
170.5731
488.4334
0.0032
−0.0073
0.0034

22
2.153579
187.1245
535.6703
2.153651
187.1108
535.6885
0.0033
−0.0073
0.0034

23
2.571786
204.4329
585.1215
2.571875
204.4178
585.1413
0.0034
−0.0074
0.0034

24
3.048175
222.5106
636.7704
3.048283
222.4942
636.792
0.0035
−0.0074
0.0034

25
3.587911
241.3577
690.617
3.58804
241.3398
690.6405
0.0036
−0.0074
0.0034

(e)  

Mode
Reference [19]
Finite strip
% Error

Root 1
Root 2
Root 3
Root 1
Root 2
Root 3
Root 1
Root 2
Root 3

1
0.785423
4.335855
12.10394
0.785472
4.335863
12.10402
0.0062
0.0002
0.0006

2
0.46872
5.6709
15.53683
0.468725
5.670845
15.53704
0.001
−0.001
0.0014

3
0.256867
7.642798
21.19587
0.256843
7.642607
21.19629
−0.0096
−0.0025
0.002

4
0.142906
10.30723
29.03288
0.142865
10.30684
29.03358
−0.0287
−0.0038
0.0024

5
0.086917
13.71786
39.03489
0.086867
13.71722
39.03594
−0.0566
−0.0047
0.0027

6
0.062743
17.89803
51.20571
0.06269
17.89707
51.20718
−0.0844
−0.0054
0.0029

7
0.057383
22.85481
65.55247
0.057328
22.85348
65.55443
−0.0948
−0.0058
0.003

8
0.065245
28.58905
82.08135
0.065191
28.58729
82.08388
−0.0836
−0.0061
0.0031

9
0.084499
35.09969
100.7969
0.084445
35.09745
100.8
−0.0638
−0.0064
0.0031

10
0.115233
42.38534
121.7022
0.11518
42.38257
121.7061
−0.0457
−0.0066
0.0032

11
0.158594
50.44476
144.7996
0.158543
50.44139
144.8043
−0.0321
−0.0067
0.0032

12
0.21637
59.27693
170.0905
0.216321
59.27291
170.096
−0.0224
−0.0068
0.0032

13
0.29079
68.88109
197.576
0.290744
68.87635
197.5825
−0.0157
−0.0069
0.0033

14
0.384424
79.25664
227.2569
0.384382
79.25113
227.2644
−0.0109
−0.0069
0.0033

15
0.500133
90.40314
259.1337
0.500095
90.3968
259.1423
−0.0074
−0.007
0.0033

16
0.641033
102.3202
293.2069
0.641001
102.313
293.2166
−0.0049
−0.0071
0.0033

17
0.810488
115.0077
329.4767
0.810463
114.9995
329.4876
−0.0031
−0.0071
0.0033

18
1.012094
128.4653
367.9434
1.012077
128.4561
367.9556
−0.0017
−0.0071
0.0033

19
1.249678
142.6929
408.6071
1.249671
142.6826
408.6207
−0.0006
−0.0072
0.0033

20
1.527292
157.6903
451.468
1.527295
157.679
451.4831
0.0002
−0.0072
0.0034

21
1.849211
173.4575
496.5262
1.849227
173.445
496.5429
0.0009
−0.0072
0.0034

22
2.219934
189.9945
543.7818
2.219965
189.9807
543.8001
0.0014
−0.0072
0.0034

23
2.64418
207.301
593.2348
2.644228
207.2859
593.2548
0.0018
−0.0073
0.0034

24
3.126891
225.3771
644.8854
3.126958
225.3607
644.9071
0.0022
−0.0073
0.0034

25
3.673229
244.2228
698.7335
3.673318
244.2049
698.7571
0.0024
−0.0073
0.0034

(f)  

Mode
Reference [19]
Finite strip
% Error

Root 1
Root 2
Root 3
Root 1
Root 2
Root 3
Root 1
Root 2
Root 3

1
0.908799
8.995999
25.62642
0.908914
8.996064
25.62654
0.0126
0.0007
0.0004

2
0.704499
10.29172
28.98691
0.704552
10.29174
28.98716
0.0075
0.0002
0.0009

3
0.497109
12.32846
34.57777
0.4971
12.32836
34.57824
−0.0018
−0.0008
0.0014

4
0.338551
15.07368
42.38017
0.338496
15.07339
42.38095
−0.0163
−0.0019
0.0018

5
0.233578
18.5403
52.37722
0.233492
18.53974
52.37837
−0.0368
−0.003
0.0022

6
0.171066
22.74951
64.55981
0.170961
22.74864
64.56141
−0.0613
−0.0038
0.0025

7
0.139392
27.71679
78.92533
0.139277
27.71553
78.92744
−0.0831
−0.0045
0.0027

8
0.130515
33.45085
95.47451
0.130393
33.44916
95.4772
−0.0932
−0.0051
0.0028

9
0.13989
39.9559
114.2094
0.139765
39.95372
114.2127
−0.0888
−0.0055
0.0029

10
0.165521
47.23361
135.1321
0.165396
47.23087
135.1361
−0.0753
−0.0058
0.003

11
0.207115
55.28438
158.2446
0.206991
55.28104
158.2494
−0.0597
−0.006
0.0031

12
0.265506
64.10807
183.5485
0.265385
64.10407
183.5543
−0.0458
−0.0062
0.0031

13
0.342301
73.70429
211.0453
0.342182
73.69957
211.0519
−0.0347
−0.0064
0.0032

14
0.439659
84.07261
240.7358
0.439544
84.06712
240.7434
−0.0261
−0.0065
0.0032

15
0.560166
95.21262
272.6208
0.560057
95.2063
272.6296
−0.0196
−0.0066
0.0032

16
0.706759
107.1239
306.7011
0.706655
107.1167
306.711
−0.0147
−0.0067
0.0032

17
0.882675
119.8063
342.977
0.882578
119.7981
342.9882
−0.011
−0.0068
0.0033

18
1.091425
133.2593
381.449
1.091336
133.2501
381.4615
−0.0081
−0.0069
0.0033

19
1.336774
147.4828
422.1174
1.336695
147.4726
422.1313
−0.0059
−0.0069
0.0033

20
1.622731
162.4767
464.9824
1.622664
162.4653
464.9977
−0.0041
−0.007
0.0033

21
1.953542
178.2407
510.0441
1.953488
178.2282
510.061
−0.0028
−0.007
0.0033

22
2.333682
194.7748
557.3029
2.333643
194.761
557.3214
−0.0017
−0.0071
0.0033

23
2.767854
212.0787
606.7587
2.767833
212.0637
606.7789
−0.0008
−0.0071
0.0033

24
3.260987
230.1526
658.4117
3.260985
230.1361
658.4337

−0.0071
0.0033

25
3.818234
248.9961
712.2621
3.818254
248.9783
712.2859
0.0005
−0.0072
0.0033

Table 2: (a) Comparison of exact flat-plate finite strip results with analytical results [19] for the first 25 circumferential modes and longitudinal mode ( = 20, = 10, , , and ). (b) Comparison of exact flat-plate finite strip results with analytical results [19] for the first 25 circumferential modes and longitudinal mode ( = 20, = 2, , , and ).

(a)  

Mode
Reference [19]
Finite strip
% Error

Root 1
Root 2
Root 3
Root 1
Root 2
Root 3
Root 1
Root 2
Root 3

1
0.003843
0.477163
2.248342
0.003839
0.477163
2.248369
−0.0935

2
0.0022
1.602974
5.57898
0.00218
1.602975
5.579023
−0.9331

3
0.013578
3.513777
11.09063
0.013552
3.513777
11.09068
−0.1928

4
0.04908
6.200527
18.79498
0.049052
6.200528
18.79503
−0.0582

5
0.127713
9.659199
28.69627
0.127683
9.659199
28.69632
−0.0233

6
0.274152
13.88832
40.79602
0.274122
13.88832
40.79607
−0.0111

7
0.518586
18.8873
55.09486
0.518555
18.8873
55.09492
−0.006

8
0.896697
24.65585
71.59311
0.896665
24.65585
71.59316
−0.0035

9
1.449656
31.19384
90.29093
1.449624
31.19384
90.29098
−0.0022

10
2.224125
38.5012
111.1884
2.224092
38.5012
111.1885
−0.0015

11
3.272254
46.57787
134.2857
3.272221
46.57787
134.2858
−0.001

12
4.651681
55.42383
159.5829
4.651648
55.42383
159.583
−0.0007

13
6.425534
65.03907
187.08
6.425501
65.03907
187.0801
−0.0005

14
8.662426
75.42357
216.7772
8.662392
75.42357
216.7772
−0.0004

15
11.43646
86.57732
248.6744
11.43642
86.57732
248.6745
−0.0003

16
14.82722
98.50032
282.7718
14.82718
98.50032
282.7719
−0.0002

17
18.91978
111.1926
319.0696
18.91974
111.1926
319.0696
−0.0002

18
23.8047
124.6541
357.5677
23.80466
124.6541
357.5678
−0.0002

19
29.57802
138.8848
398.2663
29.57798
138.8848
398.2664
−0.0001

20
36.34127
153.8848
441.1656
36.34124
153.8848
441.1656
−0.0001

21
44.20147
169.654
486.2656
44.20143
169.654
486.2657

22
53.27109
186.1924
533.5665
53.27105
186.1924
533.5666

23
63.66811
203.5001
583.0685
63.66807
203.5001
583.0686

24
75.51597
221.577
634.7718
75.51593
221.577
634.7719

25
88.94361
240.4231
688.6765
88.94357
240.4231
688.6766

(b)  

Mode
Reference [19]
Finite strip
% Error

Root 1
Root 2
Root 3
Root 1
Root 2
Root 3
Root 1
Root 2
Root 3

1
0.361681
1.71255
4.171796
0.36151
1.712972
4.172137
−0.0471
0.0247
0.0082

2
0.123212
2.823816
7.757059
0.122714
2.824171
7.757795
−0.4045
0.0126
0.0095

3
0.063523
4.651966
13.42785
0.06297
4.652242
13.42882
−0.8711
0.0059
0.0072

4
0.081789
7.273435
21.22231
0.081068
7.273642
21.22342
−0.8823
0.0028
0.0052

5
0.161011
10.68842
31.17646
0.160256
10.68858
31.17765
−0.4689
0.0015
0.0038

6
0.315485
14.88867
43.30898
0.314711
14.88879
43.31023
−0.2454
0.0008
0.0029

7
0.572148
19.86809
57.62925
0.57136
19.86818
57.63054
−0.1376
0.0005
0.0022

8
0.965526
25.62299
74.14215
0.964728
25.62306
74.14347
−0.0826
0.0003
0.0018

9
1.536331
32.15115
92.85039
1.535524
32.15121
92.85173
−0.0525
0.0002
0.0014

10
2.331023
39.45124
113.7556
2.330207
39.4513
113.7569
−0.035
0.0001
0.0012

11
3.401653
47.52242
136.8586
3.400828
47.52247
136.86
−0.0243

0.001

12
4.805812
56.36415
162.1602
4.804977
56.36419
162.1616
−0.0174

0.0009

13
6.606596
65.97606
189.6609
6.605752
65.9761
189.6623
−0.0128

0.0007

14
8.872605
76.35792
219.3608
8.87175
76.35795
219.3622
−0.0096

0.0006

15
11.67793
87.50955
251.2603
11.67706
87.50958
251.2618
−0.0074

0.0006

16
15.10215
99.43084
285.3596
15.10127
99.43086
285.3611
−0.0058

0.0005

17
19.23033
112.1217
321.6589
19.22944
112.1217
321.6604
−0.0046

0.0005

18
24.15303
125.582
360.1584
24.15213
125.5821
360.1599
−0.0037

0.0004

19
29.96629
139.8118
400.8581
29.96537
139.8119
400.8596
−0.0031

0.0004

20
36.77163
154.811
443.7584
36.7707
154.8111
443.7599
−0.0025

0.0003

21
44.67606
170.5796
488.8592
44.67511
170.5797
488.8607
−0.0021

0.0003

22
53.79206
187.1176
536.1609
53.79109
187.1176
536.1624
−0.0018

0.0003

23
64.2376
204.4249
585.6635
64.23661
204.425
585.6651
−0.0015

0.0003

24
76.13612
222.5016
637.3673
76.13511
222.5016
637.3689
−0.0013

0.0002

25
89.61654
241.3476
691.2726
89.61551
241.3476
691.2742
−0.0011

0.0002

In addition, to verify the asymptotic rate of convergence () as given in (31), higher order terms in (31) are ignored; hence

Substituting into it, we have

The convergence of the finite strip solution is plotted on a log scale against the number of strips used, see Figure 3. Apparently, the slope of the straight lines indicated thereon reveals that the finite-strip eigenvalue does converge to the exact solution at the rate of .

(a)

(b)

(c)

Figure 3: Convergence rate of flat-plate finite strip results, that is, first, second, and third root of (32) for the first longitudinal mode (, , , , and ).

5. Conclusions

Vibration of a cylindrical shell using thin flat-plate finite strips and U-transformation is presented in this paper. Unlike standard finite element approximation theory adopted by many other researchers, explicit solutions for the said problems were derived directly by making use of the inherent cyclic symmetry of the said structures and the U-transformation method. Such explicit solutions can be subsequently expanded into Taylor’s series whose results can be compared directly with the corresponding exact solutions. By ignoring higher order terms in Taylor’s series, the rate of convergence can also be found directly. The method is now being extended to study the convergence of finite strips in the analysis of general symmetric structures with various boundary conditions.

Conflict of Interests

The author declared that there is no conflict of interests regarding the publication of this paper.

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