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 .
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)
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.
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)
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.