Research Article On the Validity of Flat-Plate Finite Strip …downloads.hindawi.com/archive/2014/543794.pdf · 2019-07-31 · Research Article On the Validity of Flat-Plate Finite

Research ArticleOn the Validity of Flat-Plate Finite StripApproximation of Circular Cylindrical Shells

Jackson Kong

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

Correspondence should be addressed to Jackson Kong; [email protected]

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, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The validity of modeling curved shell panels using flat-plate finite strips has been demonstrated in the past by comparing finite stripnumerical results with analytical solutions of a few benchmark problems; to date, nomathematical exact solutions of themethod orits explicit forms of error terms have been rigorously derived to demonstrate analytically its validity. Using a unitary transformationapproach (abbreviated as the U-transformation herein), an attempt is made in this paper to derive mathematical exact solutions offlat-plate finite strips in cylindrical shell vibration analysis. Unlike the conventional finite strip method which involves assemblyof the global system of matrix equation and its numerical solution, the U-transformation method makes use of the inherent cyclicsymmetry of cylindrical shells to decouple the global matrix equation into one involving only a few unknowns, thus renderingexplicit form of solutions for the flat-shell finite strip to be derived. Such explicit solutions can be subsequently expanded intoTaylor’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 analogousproblem in flat plates, since the structural deformationsin a curved shell depend not only on the rotation andradial displacements but also on the coupled tangentialdisplacement caused by the curvature of the shell. To caterfor such couplings, curved elements for modeling practicalstructures with curvature in one or two dimensions havebeen developed since the seventies; some examples of thesedevelopments can be found in [1–4]. Despite such extensiveresearch works, curved beam and shell-type structures arestill very often in practice modeled as an assemblage ofstraight beam elements or flat plate elements, due to theirconceptual simplicity, ease of modeling, and availability ofsuch elements inmost commercial softwares.The validity andefficacy of such a simple and practical approach, however,rely on whether the solution so obtained does convergerapidly to the true solution as the number of elements usedincreases. In this respect, Fumio [5], by making use of thefinite element approximation theory and the idea of partialapproximation, showed that straight Bernoulli beam element

does converge to the exact solution for a uniform plane archwith constant curvature and clamped supports under staticload.The rate of convergence asmeasured by the energy normwas also discussed in the paper. Convergence of flat platefinite elements in modelling cylindrical shells was analyzedusing the same approach in [6]. Following along the sameline, Bernadou et al. [7, 8] analysed the convergence of generalarches using straight beam elements and that of cylindricalshells using flat plate elements. In addition to static problems,convergence for natural vibration of clamped arch usingstraight beams and lumped mass approaches was studied byIshihara [9, 10].

Unlike the finite element approximation theory adoptedby the aforesaid researchers in studying convergence [5–10], a different approach, namely, the U-transformation wasadopted in this work. The method was originally developedby Chan et al. [11] for the exact analysis of structures withperiodicity and subsequently extended by Cai et al. [12]to systems with bi-periodicity. The success of the methodthat relies on a complex unitary matrix U, when appliedto the transformation of a circulant matrix, completelydiagonalizes the latter. Such circulant matrices exist in many

Hindawi Publishing CorporationJournal of Computational EngineeringVolume 2014, Article ID 543794, 13 pageshttp://dx.doi.org/10.1155/2014/543794

2 Journal of Computational Engineering

branches of science and engineering and in the context ofstructural engineering; they correspond to the stiffnessmatri-ces of structures with cyclic symmetry. In essence, the U-transformation provides a mathematical tool for decouplinga cyclic symmetric structural system and making it possiblefor obtaining the solution in an explicit form.Themethod hasbeen successfully applied to study the convergence of finitestrips for rectangular plates [13, 14]. In this paper, the methodis further extended to study the vibration of cylindricalshells using the thin flat-plate finite strips; the latter wasoriginally developed by Cheung [15] and remains as one ofthemost effective elements for analyzing prismatic structures[16]. By making use of the inherent cyclic symmetry of acylindrical shell and the U-transformation method, explicitsolutions for the cylindrical shell vibration problem usingflat-shell finite strips were derived. Such explicit solutions canbe subsequently expanded into Taylor’s series whose resultsreveal directly the error terms and the corresponding rate ofconvergence. The converged solutions can be compared withexact solutions.

This paper is organized as follows: the U-transformationis first introduced by deriving the in-plane vibration solutionof a ring using straight beam elements. The solutions arethen compared directly with the exact solutions. The sameapproach is then extended to the vibration of cylindricalshells using the classical flat-plate finite strips, and thederivation of the explicit solutions and the rate of convergencefollow.

2. U-Transformation

Given 𝑁 entries 𝐾1,𝑗, 𝑗 = 1, . . . 𝑁, a circulant matrix K is a

matrix whose rows are of the following form:

K =

[[[[[[[[[

[

𝐾1,1

𝐾1,2

𝐾1,3

⋅ ⋅ ⋅ 𝐾1,𝑁

𝐾1,𝑁

𝐾1,1

𝐾1,2

⋅ ⋅ ⋅ 𝐾1,𝑁−1

𝐾1,𝑁−1

𝐾1,𝑁

𝐾1,1

⋅ ⋅ ⋅ 𝐾1,𝑁−2

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

𝐾1,2

𝐾1,3

𝐾1,4

⋅ ⋅ ⋅ 𝐾1,1

]]]]]]]]]

]

. (1)

Circulant matrices have a very important property. Theunitary matrix U of order𝑁 which is given by

U ≡ [𝑈1, 𝑈2, . . . , 𝑈

𝑁] , (2a)

where

𝑈𝑚=

1

√𝑁

[1, 𝑒𝑖𝑚𝜑

, 𝑒𝑖2𝑚𝜑

, . . . , 𝑒𝑖(𝑁−1)𝑚𝜑

]𝑇

(2b)

𝜑 =2𝜋

𝑁, 𝑖 = √−1 (2c)

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

UTKU = diag (𝑘1, 𝑘2, . . . , 𝑘

𝑁) , (3a)

where

𝑘𝑟=

𝑁

∑

𝑗=1

𝐾1,𝑗𝑒𝑖(𝑗−1)𝑟𝜑

, (3b)

UTU = I. (3c)

3. In-Plane Vibration of a Ring

A ring with radius 𝑅, flexural stiffness 𝐸𝐼, and mass per unitlength 𝜌𝐴 is shown in Figure 1. The ring is divided into 𝑁

numbers of identical, straight beam elements, whose axialand lateral displacements are approximated by the conven-tional linear and cubic Hermite shape functions, respectively.With reference to local coordinates 𝑥-𝑦, the stiffness andconsistent mass matrices of the straight, 2-node, Bernoullibeam element can be partitioned as [17]:

k = [

[

k11

k12

k𝑇12

k22

]

]

,

m = [

[

m11

m12

m𝑇12

m22

]

]

,

(4a)

where

k11=

[[[[[[[

[

𝐸𝐴

𝐿0 0

012𝐸𝐼

𝐿3

6𝐸𝐼

𝐿2

06𝐸𝐼

𝐿2

4𝐸𝐼

𝐿

]]]]]]]

]

,

k12=

[[[[[[[

[

−𝐸𝐴

𝐿0 0

0 −12𝐸𝐼

𝐿3

6𝐸𝐼

𝐿2

0 −6𝐸𝐼

𝐿2

2𝐸𝐼

𝐿

]]]]]]]

]

,

k22=

[[[[[[[

[

𝐸𝐴

𝐿0 0

012𝐸𝐼

𝐿3−6𝐸𝐼

𝐿2

06𝐸𝐼

𝐿2

4𝐸𝐼

𝐿

]]]]]]]

]

,

(4b)

m11= [

[

140 0 0

0 156 22𝐿

0 22𝐿 4𝐿2

]

]

,

m12= [

[

70 0 0

0 54 −13𝐿

0 13𝐿 −3𝐿2

]

]

,

m22= [

[

140 0 0

0 156 −22𝐿

0 22𝐿 4𝐿2

]

]

,

(4c)

and 𝐿 = 2𝑅 sin(𝜋/𝑁).

Journal of Computational Engineering 3

y

x

1

2

3

N

i

R

Radial

Tangential

X

Y

𝛼

𝛼/2N − 1

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

Instead of transforming to the global Cartesian 𝑋-𝑌axes in the conventional manner, the local stiffness andmass matrices are transformed into the tangential-radialcoordinates (Figure 1) prior to assembly. The transformationmatrix for each element is identical and is given by

T = [

[

t 03𝑥3

03𝑥3

t𝑇]

]

, (5a)

where

t =[[[[[

[

cos 𝜋𝑁

− sin 𝜋

𝑁0

sin 𝜋

𝑁cos 𝜋

𝑁0

0 0 1

]]]]]

]

. (5b)

The transformed element stiffness matrix is then assem-bled to form the stiffness matrix of the 𝑁-sided polygon,denoted by K:

K =[[[

[

k𝑎

k𝑏

0 ⋅ ⋅ ⋅ k𝑐

k𝑐k𝑎

k𝑏⋅ ⋅ ⋅

0 k𝑐

k𝑎

k𝑏⋅ ⋅ ⋅

k𝑏

0 ⋅ ⋅ ⋅ k𝑐

k𝑎

]]]

]

, (6a)

where

k𝑎= t𝑇k11t + tk22t𝑇,

k𝑏= t𝑇k12t𝑇,

k𝑐= k𝑇𝑏,

(6b)

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

Applying the U-transformation to the cyclic symmetricsystem, we can write the deflection variables at node 𝑗 interms of the generalized coordinates 𝑞

𝑟and the associated

symmetry modes 𝑒𝑖(𝑗−1)𝑟𝜑:

(

𝑢

𝑤

𝜃

)

𝑗

=

𝑁

∑

𝑟=1

1

√𝑁

𝑒𝑖(𝑗−1)𝑟𝜏

(

𝑞1

𝑞2

𝑞3

)

𝑟

, (7)

where 𝜏 = 2𝜋/𝑁.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

k𝑟=

𝑁

∑

𝑗=1

K1𝑗𝑒𝑖(𝑗−1)𝑟𝜏

= k𝑎𝑒𝑖(1−1)𝑟𝜏

+ k𝑏𝑒𝑖(2−1)𝑟𝜏

+ k𝑐𝑒𝑖(𝑁−1)𝑟𝜏

= k𝑎+ k𝑏𝑒𝑖𝑟𝜏+ k𝑐𝑒−𝑖𝑟𝜏

.

(8)

Similarly, we have the diagonalized mass matrix

m𝑟=

𝑁

∑

𝑗=1

M1𝑗𝑒𝑖(𝑗−1)𝑟𝜏

= m𝑎+m𝑏𝑒𝑖𝑟𝜏+m𝑐𝑒−𝑖𝑟𝜏

.

(9)

Accordingly, the decoupled eigenproblem of order 3 canbe written as

[(k𝑎+ k𝑏𝑒𝑖𝑟𝜏+ k𝑐𝑒−𝑖𝑟𝜏

)

−𝜔2(m𝑎+m𝑏𝑒𝑖𝑟𝜏+m𝑐𝑒−𝑖𝑟𝜏

)](

𝑞1

𝑞2

𝑞3

)

𝑟

= {0} .(10)

By expanding the determinant, the natural frequenciescan be found from the roots of the cubic characteristicpolynomial:

𝐶1𝜔6+ 𝐶2𝜔4+ 𝐶3𝜔2+ 𝐶4= 0, (11)

whose coefficients can be determined explicitly using Math-ematica. To find the converged finite element solution as 𝑁approaches infinity and the associated rate of convergence,the coefficients are expanded into Taylor’s series of ℎ = 1/𝑁.It can be shown that

𝐶1= 𝛼1ℎ4+ 𝛼2ℎ6+ 𝑂 (ℎ

8) ,

𝐶2= 𝛽1+ 𝛽2ℎ2+ 𝛽3ℎ4+ 𝛽4ℎ6+ 𝑂 (ℎ

8) ,

𝐶3= 𝛾1+ 𝛾2ℎ2+ 𝛾3ℎ4+ 𝛾4ℎ6+ 𝑂 (ℎ

8) ,

𝐶4= 𝛿1+ 𝛿2ℎ2+ 𝛿3ℎ4+ 𝛿4ℎ6+ 𝑂 (ℎ

8) ,

(12)

where 𝛼𝑖, 𝛽𝑖, 𝛾𝑖, 𝛿𝑖are functions of the mode number 𝑟, the

bending and axial stiffness, density, and radius of the ringonly.


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

𝜔2= 𝜆1+ 𝜆2ℎ2+ 𝑂 (ℎ

4) . (13)

Substituting (13) and (12) into (11) and collecting terms of ℎ0and ℎ2, we have

𝛽1𝜆2

1+ 𝛾1𝜆1+ 𝛿1= 0,

𝛽2𝜆2

1+ 2𝛽1𝜆1𝜆2+ 𝛾2𝜆1+ 𝛾1𝜆2+ 𝛿2= 0.

(14)

The explicit form of the first equation is given by

𝐸2𝐼𝑟2(𝑟2− 1)2

− (𝐸𝐼𝜌𝑟2(𝑟2+ 1)𝑅

2+ 𝐸𝐴𝜌 (𝑟

2+ 1)𝑅

4) 𝜆1

+ 𝐴𝜌2𝑅6𝜆2

1= 0.

(15)

Thus, by denoting the radius of gyration as 𝑑 = √𝐼/𝐴, thetwo roots can be expressed as

𝜆1−=

𝐸

2𝜌𝑅2(𝑟2+ 1)(

𝑟2𝑑2

𝑅2+ 1)

×(1 − √1 −

4 (𝑟2𝑑2/𝑅2) (𝑟2− 1)2

(𝑟2 + 1)2

((𝑟2𝑑2/𝑅2) + 1)2),

𝜆1+=

𝐸

2𝜌𝑅2(𝑟2+ 1)(

𝑟2𝑑2

𝑅2+ 1)

×(1 + √1 −

4 (𝑟2𝑑2/𝑅2) (𝑟2− 1)2

(𝑟2 + 1)2

((𝑟2𝑑2/𝑅2) + 1)2),

(16)

which agrees exactly with the analytical solution given inSoedel [18], thus confirming the convergence of the straightbeam finite element solution to the exact analytical solution.It is also apparent from (13) that the straight beam eigenvaluesolution converges to the exact solution at an asymptotic rateof 𝑁−2. For the explicit form of the error term 𝜆

2, it can be

found from (14) that

𝜆2=

𝜋2(−𝐸2(𝑟2𝑑2/𝑅2) (𝑟2− 1)2

(3𝑟2− 1) + 𝐸𝜆

1𝜌𝑅2((𝑟2𝑑2/𝑅2) (4𝑟4+ 𝑟2+ 1) + (3𝑟

4+ 5𝑟2+ 2)) − 4𝜆

2

1𝜌2(𝑟2+ 1) 𝑅

4)

3𝜌𝑅2 (𝐸 (1 + 𝑟2) ((𝑟2𝑑2/𝑅2) + 1) − 2𝜆1𝜌𝑅2)

.

(17)

By substituting the roots 𝜆1into 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 tothe vibration of ring in the previous section, the same conceptis extended to the vibration of a thin cylindrical shell of radius𝑅, thickness 𝑡 (where 𝑡/𝑅 ≪ 1), density 𝜌, Young’s modulus𝐸, and Poisson’s ratio ]. Assume that the circular cylinderis divided into 𝑁 number of identical flat-plate finite stripsaround the circumference, see Figure 2(a). By neglectingtransverse normal and shear deformation, the displacementfunctions of a thin-plate finite strip with thickness 𝑡 andwidth 𝑎 (Figure 2(b)) can bewritten in terms of trigonometricseries longitudinally whilst linear and cubic Hermite shapefunctions, denoted by M(𝑦) and N(𝑦), respectively, are usedacross the strip; therefore we have

𝑢 (𝑥, 𝑦) =

𝑚

∑

𝑠=1

M (𝑦) cos 𝑠𝜋𝑥𝑙{𝑢1, 𝑢2}𝑇

𝑠,

V (𝑥, 𝑦) =𝑚

∑

𝑠=1

M (𝑦) sin 𝑠𝜋𝑥

𝑙{V1, V2}𝑇

𝑠,

𝑤 (𝑥, 𝑦) =

𝑚

∑

𝑠=1

N (𝑦) sin 𝑠𝜋𝑥

𝑙{𝑤1, 𝜃1,𝑤2, 𝜃2}𝑇

𝑠,

(18)

where the four displacement variables on each nodal lineare represented by 𝑢

𝑖, V𝑖, 𝑤𝑖, 𝜃𝑖, 𝑖 = 1, 2 . . .. The displace-

ment functions in (4), originally developed by Cheung andTham [16], satisfy the shear diaphragm end conditions (orconventionally known as the simply-supported conditions)of a cylinder with finite length 𝑙. On the other hand, for aninfinitely long cylinder, 𝑙/𝑠 is the half wavelength of the 𝑠thvibration modes along the length.

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

k𝑠,𝑠=∭B𝑇

𝑠DB𝑠𝑑𝑧 𝑑𝑦𝑑𝑥, (19a)

where

D =𝐸𝑡3

12 (1 − ]2)[[

[

1 ] 0

] 1 0

0 01 − ]2

]]

]

,

k𝑠,𝑡= 0 if 𝑠 = 𝑡.

(19b)


Table 1: (a) Comparison of exact flat-plate finite strip results with analytical results [19] for the first 25 circumferential modes and longitudinalmode 𝑠 = 1 (𝑅/𝑡 = 100, 𝐿/𝑅 = 10, 𝐸 = 1, 𝜌 = 1, and 𝜇 = 0.3). (b) Comparison of exact flat-plate finite strip results with analytical results [19]for the first 25 circumferential modes and longitudinal mode 𝑠 = 2 (𝑅/𝑡 = 100, 𝐿/𝑅 = 10, 𝐸 = 1, 𝜌 = 1, and 𝜇 = 0.3). (c) Comparison of exactflat-plate finite strip results with analytical results [19] for the first 25 circumferential modes and longitudinal mode 𝑠 = 3 (𝑅/𝑡 = 100, 𝐿/𝑅 =10, 𝐸 = 1, 𝜌 = 1, and 𝜇 = 0.3). (d) Comparison of exact flat-plate finite strip results with analytical results [19] for the first 25 circumferentialmodes and longitudinal mode 𝑠 = 1 (𝑅/𝑡 = 100, 𝐿/𝑅 = 2, 𝐸 = 1, 𝜌 = 1, and 𝜇 = 0.3). (e) Comparison of exact flat-plate finite strip resultswith analytical results [19] for the first 25 circumferential modes and longitudinal mode 𝑠 = 2 (𝑅/𝑡 = 100, 𝐿/𝑅 = 2, 𝐸 = 1, 𝜌 = 1, and 𝜇 = 0.3).(f) Comparison of exact flat-plate finite strip results with analytical results [19] for the first 25 circumferential modes and longitudinal mode𝑠 = 3 (𝑅/𝑡 = 100, 𝐿/𝑅 = 2, 𝐸 = 1, 𝜌 = 1, and 𝜇 = 0.3).

(a)

Mode Reference [19] Finite strip % ErrorRoot 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 0.00082 0.00053 1.60297 5.576039 0.000529 1.602897 5.57615 −0.0584 −0.0046 0.0023 0.000644 3.513856 11.08322 0.000644 3.51365 11.08351 −0.0644 −0.0059 0.00264 0.001998 6.200774 18.78118 0.001997 6.200375 18.78173 −0.0205 −0.0064 0.00295 0.005126 9.659683 28.67419 0.005125 9.659031 28.67507 −0.0061 −0.0067 0.00316 0.01098 13.88911 40.76376 0.01098 13.88815 40.76505 −0.0009 −0.0069 0.00327 0.020762 18.88844 55.0505 0.020762 18.88711 55.05228 0.0013 −0.007 0.00328 0.035896 24.65741 71.53469 0.035897 24.65566 71.53703 0.0024 −0.0071 0.00339 0.058031 31.19588 90.21647 0.058033 31.19365 90.21944 0.0029 −0.0072 0.003310 0.089033 38.50377 111.0959 0.089036 38.501 111.0996 0.0033 −0.0072 0.003311 0.130991 46.58103 134.1731 0.130996 46.57767 134.1776 0.0035 −0.0072 0.003312 0.186212 55.42764 159.448 0.186218 55.42363 159.4533 0.0036 −0.0072 0.003313 0.257222 65.04358 186.9207 0.257232 65.03886 186.9269 0.0037 −0.0073 0.003414 0.34677 75.42884 216.5911 0.346783 75.42335 216.5984 0.0038 −0.0073 0.003415 0.457822 86.58342 248.4593 0.457839 86.5771 248.4677 0.0038 −0.0073 0.003416 0.593564 98.5073 282.5253 0.593586 98.5001 282.5348 0.0038 −0.0073 0.003417 0.757403 111.2005 318.7891 0.757432 111.1923 318.7998 0.0039 −0.0073 0.003418 0.952964 124.663 357.2506 0.953001 124.6538 357.2627 0.0039 −0.0073 0.003419 1.184095 138.8947 397.91 1.184141 138.8845 397.9235 0.0039 −0.0073 0.003420 1.45486 153.8958 440.7672 1.454917 153.8845 440.7821 0.0039 −0.0074 0.003421 1.769544 169.6662 485.8221 1.769614 169.6537 485.8386 0.0039 −0.0074 0.003422 2.132654 186.2059 533.0749 2.132738 186.1921 533.0929 0.004 −0.0074 0.003423 2.548913 203.5148 582.5254 2.549015 203.4998 582.5452 0.004 −0.0074 0.003424 3.023267 221.5931 634.1737 3.023388 221.5767 634.1953 0.004 −0.0074 0.003425 3.56088 240.4406 688.0199 3.561022 240.4228 688.0433 0.004 −0.0074 0.0034

(b)


1 0.038162 0.714701 2.415242 0.038162 0.714689 2.415267 0.0025 −0.0016 0.0012 0.00641 1.786224 5.826179 0.006409 1.786145 5.8263 −0.0173 −0.0044 0.00213 0.002154 3.667229 11.36764 0.002152 3.667019 11.36794 −0.0788 −0.0057 0.00264 0.002579 6.339256 19.08146 0.002577 6.338854 19.08202 −0.072 −0.0063 0.00295 0.005463 9.790132 28.98279 0.005461 9.789479 28.98368 −0.0335 −0.0067 0.00316 0.011266 14.01483 41.07719 0.011265 14.01387 41.07849 −0.0147 −0.0069 0.00327 0.02107 19.01119 55.36697 0.021068 19.00986 55.36875 −0.0062 −0.007 0.00328 0.036261 24.77816 71.85317 0.03626 24.77641 71.85552 −0.002 −0.0071 0.00339 0.058475 31.31524 90.53636 0.058475 31.31301 90.53934 0.0002 −0.0071 0.003310 0.089573 38.62211 111.4168 0.089574 38.61935 111.4205 0.0015 −0.0072 0.003311 0.13164 46.69862 134.4948 0.131643 46.69526 134.4992 0.0023 −0.0072 0.003312 0.186982 55.54465 159.7702 0.186987 55.54064 159.7756 0.0027 −0.0072 0.003313 0.258126 65.16014 187.2434 0.258134 65.15542 187.2496 0.0031 −0.0073 0.0033


(b) Continued.


14 0.347819 75.54504 216.9142 0.347831 75.53955 216.9214 0.0033 −0.0073 0.003415 0.459027 86.69932 248.7827 0.459043 86.69301 248.791 0.0035 −0.0073 0.003416 0.594936 98.62296 282.8489 0.594958 98.61576 282.8584 0.0036 −0.0073 0.003417 0.758954 111.3159 319.1129 0.758982 111.3078 319.1236 0.0037 −0.0073 0.003418 0.954705 124.7783 357.5746 0.95474 124.7691 357.5867 0.0037 −0.0073 0.003419 1.186036 139.0099 398.2341 1.186081 138.9997 398.2476 0.0038 −0.0073 0.003420 1.457012 154.0109 441.0914 1.457068 153.9995 441.1063 0.0038 −0.0074 0.003421 1.771919 169.7811 486.1464 1.771987 169.7686 486.1629 0.0039 −0.0074 0.003422 2.135261 186.3207 533.3993 2.135344 186.307 533.4174 0.0039 −0.0074 0.003423 2.551765 203.6296 582.8499 2.551864 203.6146 582.8697 0.0039 −0.0074 0.003424 3.026373 221.7078 634.4983 3.026492 221.6914 634.5199 0.0039 −0.0074 0.003425 3.564252 240.5553 688.3445 3.564392 240.5375 688.3679 0.0039 −0.0074 0.0034

(c)


1 0.117569 1.030781 2.751855 0.117572 1.030763 2.751891 0.0026 −0.0018 0.00132 0.0259 2.065125 6.259916 0.025898 2.06504 6.260051 −0.0083 −0.0041 0.00223 0.007829 3.914095 11.84728 0.007825 3.91388 11.84759 −0.0483 −0.0055 0.00264 0.004751 6.566632 19.58415 0.004747 6.566226 19.58472 −0.0902 −0.0062 0.00295 0.006564 10.00599 29.49815 0.006559 10.00533 29.49906 −0.0669 −0.0066 0.00316 0.012017 14.22358 41.60012 0.012013 14.22261 41.60143 −0.0356 −0.0068 0.00327 0.021736 19.21531 55.89471 0.021732 19.21398 55.8965 −0.0182 −0.0069 0.00328 0.036963 24.97915 72.38416 0.036959 24.97739 72.38651 −0.0092 −0.007 0.00339 0.059275 31.514 91.06963 0.059273 31.51176 91.07262 −0.0043 −0.0071 0.003310 0.090513 38.81925 111.9518 0.090512 38.81648 111.9555 −0.0015 −0.0071 0.003311 0.13275 46.89453 135.0309 0.132751 46.89116 135.0354 0.0002 −0.0072 0.003312 0.188288 55.73962 160.3074 0.188291 55.7356 160.3127 0.0013 −0.0072 0.003313 0.25965 65.35437 187.7812 0.259655 65.34964 187.7875 0.002 −0.0072 0.003314 0.349581 75.73868 217.4526 0.34959 75.73318 217.4599 0.0025 −0.0073 0.003415 0.461047 86.89248 249.3216 0.46106 86.88616 249.33 0.0029 −0.0073 0.003416 0.597234 98.81572 283.3882 0.597253 98.80852 283.3978 0.0031 −0.0073 0.003417 0.761547 111.5084 319.6525 0.761572 111.5002 319.6633 0.0033 −0.0073 0.003418 0.957613 124.9704 358.1146 0.957646 124.9613 358.1267 0.0034 −0.0073 0.003419 1.189277 139.2018 398.7743 1.189319 139.1916 398.7878 0.0035 −0.0073 0.003420 1.460605 154.2026 441.6318 1.460658 154.1913 441.6467 0.0036 −0.0073 0.003421 1.775882 169.9727 486.687 1.775947 169.9602 486.7035 0.0037 −0.0074 0.003422 2.139612 186.5121 533.94 2.139692 186.4984 533.9581 0.0037 −0.0074 0.003423 2.556522 203.8209 583.3908 2.556619 203.8058 583.4106 0.0038 −0.0074 0.003424 3.031555 221.8989 635.0393 3.031671 221.8825 635.0609 0.0038 −0.0074 0.003425 3.569876 240.7463 688.8856 3.570014 240.7285 688.909 0.0039 −0.0074 0.0034

(d)


1 0.360519 1.711392 4.171047 0.36053 1.711367 4.171102 0.0032 −0.0015 0.00132 0.117813 2.821872 7.754094 0.117809 2.821778 7.754261 −0.0035 −0.0033 0.00213 0.042117 4.649301 13.42077 0.042108 4.649077 13.42111 −0.0236 −0.0048 0.00264 0.019279 7.270339 21.2091 0.019267 7.269925 21.2097 −0.0624 −0.0057 0.0028


(d) Continued.


5 0.01385 10.68516 31.15515 0.013838 10.68449 31.15608 −0.092 −0.0062 0.0036 0.016449 14.88542 43.27761 0.016436 14.88445 43.27896 −0.0782 −0.0065 0.00317 0.025049 19.86499 57.58586 0.025036 19.86365 57.58769 −0.0504 −0.0067 0.00328 0.039939 25.62014 74.08476 0.039927 25.61838 74.08715 −0.0303 −0.0069 0.00329 0.062317 32.14864 92.777 0.062306 32.14641 92.78003 −0.0181 −0.007 0.003310 0.093854 39.44914 113.6641 0.093844 39.44637 113.6679 −0.0107 −0.007 0.003311 0.136544 47.5208 136.7471 0.136536 47.51743 136.7516 −0.0062 −0.0071 0.003312 0.192647 56.36306 162.0265 0.192641 56.35903 162.0318 −0.0032 −0.0071 0.003313 0.264666 65.97556 189.5026 0.264663 65.97083 189.5089 −0.0013 −0.0072 0.003314 0.355332 76.35806 219.1759 0.355333 76.35256 219.1832 4𝐸 − 05 −0.0072 0.003315 0.467605 87.51038 251.0464 0.46761 87.50406 251.0548 0.001 −0.0072 0.003416 0.604665 99.43241 285.1143 0.604675 99.4252 285.1238 0.0016 −0.0072 0.003417 0.769916 112.124 321.3796 0.769932 112.1159 321.3904 0.0021 −0.0073 0.003418 0.966981 125.5852 359.8425 0.967005 125.5761 359.8546 0.0025 −0.0073 0.003419 1.199705 139.8159 400.503 1.199739 139.8057 400.5165 0.0028 −0.0073 0.003420 1.472153 154.816 443.3611 1.472198 154.8047 443.3761 0.003 −0.0073 0.003421 1.78861 170.5856 488.4169 1.788667 170.5731 488.4334 0.0032 −0.0073 0.003422 2.153579 187.1245 535.6703 2.153651 187.1108 535.6885 0.0033 −0.0073 0.003423 2.571786 204.4329 585.1215 2.571875 204.4178 585.1413 0.0034 −0.0074 0.003424 3.048175 222.5106 636.7704 3.048283 222.4942 636.792 0.0035 −0.0074 0.003425 3.587911 241.3577 690.617 3.58804 241.3398 690.6405 0.0036 −0.0074 0.0034

(e)


1 0.785423 4.335855 12.10394 0.785472 4.335863 12.10402 0.0062 0.0002 0.00062 0.46872 5.6709 15.53683 0.468725 5.670845 15.53704 0.001 −0.001 0.00143 0.256867 7.642798 21.19587 0.256843 7.642607 21.19629 −0.0096 −0.0025 0.0024 0.142906 10.30723 29.03288 0.142865 10.30684 29.03358 −0.0287 −0.0038 0.00245 0.086917 13.71786 39.03489 0.086867 13.71722 39.03594 −0.0566 −0.0047 0.00276 0.062743 17.89803 51.20571 0.06269 17.89707 51.20718 −0.0844 −0.0054 0.00297 0.057383 22.85481 65.55247 0.057328 22.85348 65.55443 −0.0948 −0.0058 0.0038 0.065245 28.58905 82.08135 0.065191 28.58729 82.08388 −0.0836 −0.0061 0.00319 0.084499 35.09969 100.7969 0.084445 35.09745 100.8 −0.0638 −0.0064 0.003110 0.115233 42.38534 121.7022 0.11518 42.38257 121.7061 −0.0457 −0.0066 0.003211 0.158594 50.44476 144.7996 0.158543 50.44139 144.8043 −0.0321 −0.0067 0.003212 0.21637 59.27693 170.0905 0.216321 59.27291 170.096 −0.0224 −0.0068 0.003213 0.29079 68.88109 197.576 0.290744 68.87635 197.5825 −0.0157 −0.0069 0.003314 0.384424 79.25664 227.2569 0.384382 79.25113 227.2644 −0.0109 −0.0069 0.003315 0.500133 90.40314 259.1337 0.500095 90.3968 259.1423 −0.0074 −0.007 0.003316 0.641033 102.3202 293.2069 0.641001 102.313 293.2166 −0.0049 −0.0071 0.003317 0.810488 115.0077 329.4767 0.810463 114.9995 329.4876 −0.0031 −0.0071 0.003318 1.012094 128.4653 367.9434 1.012077 128.4561 367.9556 −0.0017 −0.0071 0.003319 1.249678 142.6929 408.6071 1.249671 142.6826 408.6207 −0.0006 −0.0072 0.003320 1.527292 157.6903 451.468 1.527295 157.679 451.4831 0.0002 −0.0072 0.003421 1.849211 173.4575 496.5262 1.849227 173.445 496.5429 0.0009 −0.0072 0.003422 2.219934 189.9945 543.7818 2.219965 189.9807 543.8001 0.0014 −0.0072 0.003423 2.64418 207.301 593.2348 2.644228 207.2859 593.2548 0.0018 −0.0073 0.003424 3.126891 225.3771 644.8854 3.126958 225.3607 644.9071 0.0022 −0.0073 0.003425 3.673229 244.2228 698.7335 3.673318 244.2049 698.7571 0.0024 −0.0073 0.0034


(f)


1 0.908799 8.995999 25.62642 0.908914 8.996064 25.62654 0.0126 0.0007 0.00042 0.704499 10.29172 28.98691 0.704552 10.29174 28.98716 0.0075 0.0002 0.00093 0.497109 12.32846 34.57777 0.4971 12.32836 34.57824 −0.0018 −0.0008 0.00144 0.338551 15.07368 42.38017 0.338496 15.07339 42.38095 −0.0163 −0.0019 0.00185 0.233578 18.5403 52.37722 0.233492 18.53974 52.37837 −0.0368 −0.003 0.00226 0.171066 22.74951 64.55981 0.170961 22.74864 64.56141 −0.0613 −0.0038 0.00257 0.139392 27.71679 78.92533 0.139277 27.71553 78.92744 −0.0831 −0.0045 0.00278 0.130515 33.45085 95.47451 0.130393 33.44916 95.4772 −0.0932 −0.0051 0.00289 0.13989 39.9559 114.2094 0.139765 39.95372 114.2127 −0.0888 −0.0055 0.002910 0.165521 47.23361 135.1321 0.165396 47.23087 135.1361 −0.0753 −0.0058 0.00311 0.207115 55.28438 158.2446 0.206991 55.28104 158.2494 −0.0597 −0.006 0.003112 0.265506 64.10807 183.5485 0.265385 64.10407 183.5543 −0.0458 −0.0062 0.003113 0.342301 73.70429 211.0453 0.342182 73.69957 211.0519 −0.0347 −0.0064 0.003214 0.439659 84.07261 240.7358 0.439544 84.06712 240.7434 −0.0261 −0.0065 0.003215 0.560166 95.21262 272.6208 0.560057 95.2063 272.6296 −0.0196 −0.0066 0.003216 0.706759 107.1239 306.7011 0.706655 107.1167 306.711 −0.0147 −0.0067 0.003217 0.882675 119.8063 342.977 0.882578 119.7981 342.9882 −0.011 −0.0068 0.003318 1.091425 133.2593 381.449 1.091336 133.2501 381.4615 −0.0081 −0.0069 0.003319 1.336774 147.4828 422.1174 1.336695 147.4726 422.1313 −0.0059 −0.0069 0.003320 1.622731 162.4767 464.9824 1.622664 162.4653 464.9977 −0.0041 −0.007 0.003321 1.953542 178.2407 510.0441 1.953488 178.2282 510.061 −0.0028 −0.007 0.003322 2.333682 194.7748 557.3029 2.333643 194.761 557.3214 −0.0017 −0.0071 0.003323 2.767854 212.0787 606.7587 2.767833 212.0637 606.7789 −0.0008 −0.0071 0.003324 3.260987 230.1526 658.4117 3.260985 230.1361 658.4337 −5𝐸 − 05 −0.0071 0.003325 3.818234 248.9961 712.2621 3.818254 248.9783 712.2859 0.0005 −0.0072 0.0033

Explicit forms of the strain-displacement relation B𝑠and

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

k𝑠,𝑠= [

k11

k12

k21

k22

]

𝑠,𝑠

. (20)

By the same token, we can partition the strip mass matrix𝑚𝑠,𝑠:

m𝑠,𝑠= [

m11

m12

m21

m22

]

𝑠,𝑠

, (21)

where the subscripts of submatrices k𝑖𝑗and m

𝑖𝑗correspond

to nodal lines 𝑖 and 𝑗. To ensure compatibility of deformationalong a nodal line between two adjacent strips, the localstiffness andmass matrices of a strip are transformed into theradial-tangential coordinates (Figure 2(a)) prior to assembly.The transformation matrix for each strip is identical and isgiven by

T = [

[

t 04𝑥4

04𝑥4

t𝑇]

]

, (22)

where

t =

[[[[[[[[[

[

1 0 0 0

0 cos 𝜋𝑁

− sin 𝜋

𝑁0

0 sin 𝜋

𝑁cos 𝜋

𝑁0

0 0 0 1

]]]]]]]]]

]

. (23)

The transformed strip stiffness matrix can be assembledto form the stiffness matrix of the 𝑁-sided polygonal tube,denoted by K

𝑠,𝑠for longitudinal mode 𝑠:

K𝑠,𝑠=[[[

[

k𝑎

k𝑏

0 ⋅ ⋅ ⋅ k𝑐

k𝑐k𝑎

k𝑏⋅ ⋅ ⋅

0 k𝑐

k𝑎

k𝑏⋅ ⋅ ⋅

k𝑏

0 ⋅ ⋅ ⋅ k𝑐

k𝑎

]]]

]𝑠,𝑠

, (24a)

where, by dropping the subscript 𝑠 for simplicity, we have

k𝑎= t𝑇k11t + tk22t𝑇,

k𝑏= t𝑇k12t𝑇,

k𝑐= k𝑇𝑏,

(24b)


Strip i

x

yz

Radial

Tangential

(a)

Y

X

a

Z

l

(b)

Figure 2: (a) Division of a cylindrical shell into flat-plate finitestrips. (b) Local coordinate system of a flat-plate finite strip withnodal line 1 and 2 along 𝑌 = 0 and 𝑌 = 𝑎, respectively.

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

Applying the U-transformation to the cyclic symmetricsystem, the deflection variables at nodal line 𝑗 are writtenin terms of the generalized coordinates 𝑞

𝑟and its associated

symmetry modes 𝑒𝑖(𝑗−1)𝑟𝜑:

(

𝑢

V𝑤

𝜃

)

𝑗

=

𝑁

∑

𝑟=1

1

√𝑁

𝑒𝑖(𝑗−1)𝑟𝜏

(

𝑞1

𝑞2

𝑞3

𝑞4

)

𝑟

, (25)

where 𝜏 = 2𝜋/𝑁.Using (3a), (3b), and (3c) and (24a) and (24b) the stiffness

matrix K𝑠,𝑠of longitudinal mode 𝑠 can be diagonalized as

k𝑟=

𝑁

∑

𝑗=1

K1𝑗𝑒𝑖(𝑗−1)𝑟𝜏

= k𝑎𝑒𝑖(1−1)𝑟𝜏

+ k𝑏𝑒𝑖(2−1)𝑟𝜏

+ k𝑐𝑒𝑖(𝑁−1)𝑟𝜏

= k𝑎+ k𝑏𝑒𝑖𝑟𝜏+ k𝑐𝑒−𝑖𝑟𝜏

.

(26)

Similarly, we have the diagonalized mass matrix

m𝑟=

𝑁

∑

𝑗=1

M1𝑗𝑒𝑖(𝑗−1)𝑟𝜏

= m𝑎+m𝑏𝑒𝑖𝑟𝜏+m𝑐𝑒−𝑖𝑟𝜏

.

(27)

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

[(k𝑎+ k𝑏𝑒𝑖𝑟𝜏+ k𝑐𝑒−𝑖𝑟𝜏

)

−𝜔2(m𝑎+m𝑏𝑒𝑖𝑟𝜏+m𝑐𝑒−𝑖𝑟𝜏

)](

𝑞1

𝑞2

𝑞3

𝑞4

)

𝑟

= {0} .(28)

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

𝐶1𝜔8+ 𝐶2𝜔6+ 𝐶3𝜔4+ 𝐶4𝜔2+ 𝐶5= 0, (29)

whose coefficients are found explicitly usingMathematica. Tofind the converged solution as𝑁 approaches infinity and theassociated rate of convergence, the coefficients are expandedinto Taylor’s series of ℎ = 1/𝑁. It can be shown that

𝐶1= 𝛼1ℎ4+ 𝛼2ℎ6+ 𝑂 (ℎ

8) ,

𝐶2= 𝛽1+ 𝛽2ℎ2+ 𝛽3ℎ4+ 𝛽4ℎ6+ 𝑂 (ℎ

8) ,

𝐶3= 𝛾1+ 𝛾2ℎ2+ 𝛾3ℎ4+ 𝛾4ℎ6+ 𝑂 (ℎ

8) ,

𝐶4= 𝛿1+ 𝛿2ℎ2+ 𝛿3ℎ4+ 𝛿4ℎ6+ 𝑂 (ℎ

8) ,

𝐶5= 𝜁1+ 𝜁2ℎ2+ 𝜁3ℎ4+ 𝜁4ℎ6+ 𝑂 (ℎ

8) .

(30)

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

𝜔2= 𝜆1+ 𝜆2ℎ2+ 𝑂 (ℎ

4) . (31)

Substituting (31) and (30) into (29) and collecting termsof ℎ0 and ℎ2, we have the following cubic equation for solving𝜆1, that is, the converged finite-strip solution:

𝛽1𝜆3

1+ 𝛾1𝜆2

1+ 𝛿1𝜆1+ 𝜁1= 0, (32)

where

𝛽1=

𝐸𝜌3(]2 − 1)

2

𝑙4𝑅4

4,


Table 2: (a) Comparison of exact flat-plate finite strip results with analytical results [19] for the first 25 circumferentialmodes and longitudinalmode 𝑠 = 1 (𝑅/𝑡 = 20, 𝐿/𝑅 = 10, 𝐸 = 1, 𝜌 = 1, and 𝜇 = 0.3). (b) Comparison of exact flat-plate finite strip results with analytical results [19]for the first 25 circumferential modes and longitudinal mode 𝑠 = 1 (𝑅/𝑡 = 20, 𝐿/𝑅 = 2, 𝐸 = 1, 𝜌 = 1, and 𝜇 = 0.3).

(a)


1 0.003843 0.477163 2.248342 0.003839 0.477163 2.248369 −0.0935 −4𝐸 − 05 0.0012

2 0.0022 1.602974 5.57898 0.00218 1.602975 5.579023 −0.9331 4𝐸 − 05 0.0008

3 0.013578 3.513777 11.09063 0.013552 3.513777 11.09068 −0.1928 2𝐸 − 05 0.0004

4 0.04908 6.200527 18.79498 0.049052 6.200528 18.79503 −0.0582 6𝐸 − 06 0.0003

5 0.127713 9.659199 28.69627 0.127683 9.659199 28.69632 −0.0233 3𝐸 − 06 0.0002

6 0.274152 13.88832 40.79602 0.274122 13.88832 40.79607 −0.0111 2𝐸 − 06 0.0001

7 0.518586 18.8873 55.09486 0.518555 18.8873 55.09492 −0.006 9𝐸 − 07 1𝐸 − 04

8 0.896697 24.65585 71.59311 0.896665 24.65585 71.59316 −0.0035 5𝐸 − 07 8𝐸 − 05

9 1.449656 31.19384 90.29093 1.449624 31.19384 90.29098 −0.0022 3𝐸 − 07 6𝐸 − 05

10 2.224125 38.5012 111.1884 2.224092 38.5012 111.1885 −0.0015 2𝐸 − 07 5𝐸 − 05

11 3.272254 46.57787 134.2857 3.272221 46.57787 134.2858 −0.001 2𝐸 − 07 4𝐸 − 05

12 4.651681 55.42383 159.5829 4.651648 55.42383 159.583 −0.0007 1𝐸 − 07 4𝐸 − 05

13 6.425534 65.03907 187.08 6.425501 65.03907 187.0801 −0.0005 8𝐸 − 08 3𝐸 − 05

14 8.662426 75.42357 216.7772 8.662392 75.42357 216.7772 −0.0004 6𝐸 − 08 3𝐸 − 05

15 11.43646 86.57732 248.6744 11.43642 86.57732 248.6745 −0.0003 5𝐸 − 08 2𝐸 − 05

16 14.82722 98.50032 282.7718 14.82718 98.50032 282.7719 −0.0002 4𝐸 − 08 2𝐸 − 05

17 18.91978 111.1926 319.0696 18.91974 111.1926 319.0696 −0.0002 3𝐸 − 08 2𝐸 − 05

18 23.8047 124.6541 357.5677 23.80466 124.6541 357.5678 −0.0002 3𝐸 − 08 2𝐸 − 05

19 29.57802 138.8848 398.2663 29.57798 138.8848 398.2664 −0.0001 2𝐸 − 08 2𝐸 − 05

20 36.34127 153.8848 441.1656 36.34124 153.8848 441.1656 −0.0001 2𝐸 − 08 1𝐸 − 05

21 44.20147 169.654 486.2656 44.20143 169.654 486.2657 −9𝐸 − 05 2𝐸 − 08 1𝐸 − 05

22 53.27109 186.1924 533.5665 53.27105 186.1924 533.5666 −7𝐸 − 05 1𝐸 − 08 1𝐸 − 05

23 63.66811 203.5001 583.0685 63.66807 203.5001 583.0686 −6𝐸 − 05 1𝐸 − 08 1𝐸 − 05

24 75.51597 221.577 634.7718 75.51593 221.577 634.7719 −5𝐸 − 05 1𝐸 − 08 1𝐸 − 05

25 88.94361 240.4231 688.6765 88.94357 240.4231 688.6766 −5𝐸 − 05 9𝐸 − 09 9𝐸 − 06

(b)


1 0.361681 1.71255 4.171796 0.36151 1.712972 4.172137 −0.0471 0.0247 0.00822 0.123212 2.823816 7.757059 0.122714 2.824171 7.757795 −0.4045 0.0126 0.00953 0.063523 4.651966 13.42785 0.06297 4.652242 13.42882 −0.8711 0.0059 0.00724 0.081789 7.273435 21.22231 0.081068 7.273642 21.22342 −0.8823 0.0028 0.00525 0.161011 10.68842 31.17646 0.160256 10.68858 31.17765 −0.4689 0.0015 0.00386 0.315485 14.88867 43.30898 0.314711 14.88879 43.31023 −0.2454 0.0008 0.00297 0.572148 19.86809 57.62925 0.57136 19.86818 57.63054 −0.1376 0.0005 0.00228 0.965526 25.62299 74.14215 0.964728 25.62306 74.14347 −0.0826 0.0003 0.00189 1.536331 32.15115 92.85039 1.535524 32.15121 92.85173 −0.0525 0.0002 0.001410 2.331023 39.45124 113.7556 2.330207 39.4513 113.7569 −0.035 0.0001 0.001211 3.401653 47.52242 136.8586 3.400828 47.52247 136.86 −0.0243 9𝐸 − 05 0.00112 4.805812 56.36415 162.1602 4.804977 56.36419 162.1616 −0.0174 7𝐸 − 05 0.000913 6.606596 65.97606 189.6609 6.605752 65.9761 189.6623 −0.0128 5𝐸 − 05 0.000714 8.872605 76.35792 219.3608 8.87175 76.35795 219.3622 −0.0096 4𝐸 − 05 0.000615 11.67793 87.50955 251.2603 11.67706 87.50958 251.2618 −0.0074 3𝐸 − 05 0.000616 15.10215 99.43084 285.3596 15.10127 99.43086 285.3611 −0.0058 2𝐸 − 05 0.0005


(b) Continued.


17 19.23033 112.1217 321.6589 19.22944 112.1217 321.6604 −0.0046 2𝐸 − 05 0.000518 24.15303 125.582 360.1584 24.15213 125.5821 360.1599 −0.0037 2𝐸 − 05 0.000419 29.96629 139.8118 400.8581 29.96537 139.8119 400.8596 −0.0031 1𝐸 − 05 0.000420 36.77163 154.811 443.7584 36.7707 154.8111 443.7599 −0.0025 1𝐸 − 05 0.000321 44.67606 170.5796 488.8592 44.67511 170.5797 488.8607 −0.0021 1𝐸 − 05 0.000322 53.79206 187.1176 536.1609 53.79109 187.1176 536.1624 −0.0018 8𝐸 − 06 0.000323 64.2376 204.4249 585.6635 64.23661 204.425 585.6651 −0.0015 7𝐸 − 06 0.000324 76.13612 222.5016 637.3673 76.13511 222.5016 637.3689 −0.0013 6𝐸 − 06 0.000225 89.61654 241.3476 691.2726 89.61551 241.3476 691.2742 −0.0011 6𝐸 − 06 0.0002

𝛾1

=

𝐸2𝜌2(]2 − 1)

48

× {𝑙4𝑅2[𝑟2(𝑟2+ 1)

𝑡2

𝑅2− 6 (𝑟

2(] − 3) − 2)]

+ 2𝑠2𝜋2𝑅4𝑙2

×[𝑡2

𝑅2(𝑟2− ] + 1) − 3 (] − 3)] + 𝑠4𝜋4𝑡2𝑅4} ,

𝛿1= −

𝐸3𝜌

576𝑙2

× {6𝑟2𝑙6[12 (𝑟

2+ 1) (] − 1)

+𝑡2

𝑅2((] − 3) 𝑟4 + (] + 3) 𝑟2 − 2)]

+ 6𝑠2𝜋2𝑅2𝑙4

× [12 (] − 1) (2𝑟2 + 2] + 3)

+𝑡2

𝑅2(3 (] − 3) 𝑟4 − 2 (]2 − 2) 𝑟2

+ 4 (] − 1))]

+ 𝑠4𝜋4𝑅4𝑙2[72 (] − 1)

+ 6𝑡2

𝑅2(3 (] − 3) 𝑟2 + 4 (] − 1))

+𝑡4

𝑅4𝑟2(]2 − 1)]

+2𝑠6𝜋6𝑅4𝑡2(3 (] − 3) +

𝑡2

𝑅2(] − 1))} ,

𝜁1= −

𝐸4

1152𝑙4𝑅4 (] + 1)

× {12𝑡2𝑙8𝑟4(𝑟2− 1)2

+ 48𝑠2𝜋2𝑅2𝑡2𝑙6𝑟2(𝑟2− 1)2

+ 𝑠4𝜋4𝑅6𝑙4

× [ − 144 (]2 − 1)

+ 24𝑡2

𝑅2(3𝑟4+ (]2 − 4) 𝑟2 − 2]2 + 2)

−𝑟4 𝑡4

𝑅4(]2 − 1)]

+ 4𝑠6𝜋6𝑅6𝑟2𝑡2𝑙2(12 +

𝑡2

𝑅2)

+4𝑠8𝜋8𝑅8𝑡2(3 +

𝑡2

𝑅2)} .

(33)

Once the roots are found, the corresponding coefficientof the error term 𝜆

2can be obtained from

𝛽2𝜆3

1+ 3𝛽1𝜆2

1𝜆2+ 𝛾2𝜆2

1+ 2𝛾1𝜆1𝜆2

+ 𝜆1𝛿2+ 𝜆2𝛿1+ 𝜁2= 0.

(34)

Explicit form of the three roots is too lengthy to comparedirectly with analytical solutions. Instead, to compare theconverged flat-plate finite strip solutions with the analyticalcylindrical shell solutions, governing equation [19] is solvedfor shells with different thickness-to-radius ratios (𝑡/𝑅 =

100 and 20) and radius-to-length ratios (𝑙/𝑅 = 2 and 10).Numerical results of the three roots obtained by solving(32) are compared with the analytical solutions in Tables1(a)–1(f) for the first three longitudinal modes (i.e., 𝑠 =

1, 2, 3) and up to the first 25 circumferential modes of a very


First root

0.0 0.5 1.0 1.5 2.0 2.5 3.0

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

Log N

Slope = −1.9937

Loge

(a)

Second root

0.0 0.5 1.0 1.5 2.0 2.5 3.0Log N

−6

−5

−4

−3

−2

−1

0

Loge

Slope= −2

(b)

Third root

0

Log e

Slope= −1.9998

0.0 0.5 1.0 1.5 2.0 2.5 3.0Log N

−8

−7

−6

−5

−4

−3

−2

−1

(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 𝑠 = 1

(𝑅/𝑡 = 100, 𝐿/𝑅 = 10, 𝐸 = 1, 𝜌 = 1, and 𝜇 = 0.3).

thin shell (𝑡/𝑅 = 100); it is apparent from Table 1 that theflat-plate finite strip solutions do agree with, almost up tocomputer precision, the cylindrical shell solutions. The first25 circumferential modes are also computed for the case of amoderately thick shell (𝑡/𝑅 = 20) and summarized in Tables2(a) and 2(b); excellent agreement can be observed among thethree sets of results for both the short and long shells.

In addition, to verify the asymptotic rate of convergence(𝑁−2) as given in (31), higher order terms in (31) are ignored;hence

𝜔2= 𝜆1+ 𝜆2ℎ2. (35)


Substituting ℎ = 1/𝑁 into it, we have

Log (𝜔2 − 𝜆1) = Log 𝜆

2− 2Log𝑁,

Log 𝑒 = Log 𝜆2− 2Log𝑁.

(36)

The convergence of the finite strip solution is plotted ona log scale against the number of strips used, see Figure 3.Apparently, the slope of the straight lines indicated thereonreveals that the finite-strip eigenvalue does converge to theexact solution at the rate of𝑁−2.

5. Conclusions

Vibration of a cylindrical shell using thin flat-plate finitestrips and U-transformation is presented in this paper.Unlike standard finite element approximation theory adoptedby many other researchers, explicit solutions for the saidproblems were derived directly by making use of theinherent cyclic symmetry of the said structures and theU-transformation method. Such explicit solutions can besubsequently expanded into Taylor’s series whose results canbe compared directly with the corresponding exact solutions.By ignoring higher order terms in Taylor’s series, the rateof convergence can also be found directly. The method isnow being extended to study the convergence of finite stripsin the analysis of general symmetric structures with variousboundary conditions.

Conflict of Interests

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

References

[1] A. Krishnan and Y. J. Suresh, “A simple cubic linear element forstatic and free vibration analyses of curved beams,” Computersand Structures, vol. 68, no. 5, pp. 473–489, 1998.

[2] G. R. Liu and T. Y. Wu, “In-plane vibration analyses ofcircular arches by the generalized differential quadrature rule,”International Journal of Mechanical Sciences, vol. 43, no. 11, pp.2597–2611, 2001.

[3] J.-S. Wu and L.-K. Chiang, “Free vibration analysis of archesusing curved beam elements,” International Journal for Numer-ical Methods in Engineering, vol. 58, no. 13, pp. 1907–1936, 2003.

[4] F. Yang, R. Sedaghati, and E. Esmailzadeh, “Free in-planevibration of general curved beams using finite elementmethod,”Journal of Sound and Vibration, vol. 318, no. 4-5, pp. 850–867,2008.

[5] K. Fumio, “On the validity of the finite element analysis ofcircular arches represented by an assemblage of beam elements,”ComputerMethods in AppliedMechanics and Engineering, vol. 5,no. 3, pp. 253–276, 1975.

[6] F. Kikuchi, “Error analysis of flat plate element approximationof circular cylindrical shells,”Theoretical andAppliedMechanics,vol. 32, pp. 469–484, 1984.

[7] M. Bernadou and Y. Ducatel, “Approximation of general archproblems by straight beam elements,” Numerische Mathematik,vol. 40, no. 1, pp. 1–29, 1982.

[8] M. Bernadou, Y. Ducatel, and P. Trouve, “Approximation ofa circular cylindrical shell by Clough-Johnson flat plate finiteelements,” Numerische Mathematik, vol. 52, no. 2, pp. 187–217,1988.

[9] K. Ishihara, “A finite element lumped mass scheme for solvingeigenvalue problems of circular arches,” Numerische Mathe-matik, vol. 36, no. 3, pp. 267–290, 1980.

[10] K. Ishihara, “A finite element lumped mass scheme for solv-ing eigenvalue problems of circular arches—II. Numericalexperiments with double precision arithmetic,” NumerischeMathematik, vol. 46, no. 4, pp. 499–504, 1985.

[11] H. C. Chan, C. W. Cai, and Y. K. Cheung, Exact Analysisof Structures With Periodicity Using U-Transformation, WorldScientific Publishing, Singapore, 1998.

[12] C.W.Cai, J. K. Liu, andH.C. Chan,Exact Analysis of Bi-PeriodicStructures, World Scientific Publishing, Singapore, 2002.

[13] J. Kong and D. Thung, “Convergence and exact solutionsof spline finite strip method using unitary transformationapproach,” Applied Mathematics and Mechanics, vol. 32, no. 11,pp. 1407–1422, 2011.

[14] Y. Li,The U-transformation and the Hamiltonian techniques forthe finite strip method [Ph.D. thesis], The University of HongKong, 1996.

[15] Y. K. Cheung, “Finite strip method in the analysis of elasticplates with two opposite ends simply supported,” ProceedingsInstitution of Civil Engineers, vol. 40, pp. 1–7, 1968.

[16] Y. K. Cheung and L. G. Tham, Finite Strip Method, CRC Press,Boca Raton, Fla, USA, 1998.

[17] D. Cook, D. S. Malkus, and M. E. Plesha, Concepts andApplications of Finite Element Analysis, Wiley, New York, NY,USA, 4th edition, 2002.

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[19] A. Leissa, Vibration of Shells, Acoustical Society of America,1993.

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Research Article On the Validity of Flat-Plate Finite Strip …downloads.hindawi.com/archive/2014/543794.pdf · 2019-07-31 · Research Article On the Validity of Flat-Plate Finite