Vibrations of completely free shallow shells of rectangular planform

Leissa, A. W.; Narita, Yoshihiro

doi:10.1016/0022-460x(84)90579-0

Cited by 63 publications

(44 citation statements)

References 4 publications

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“…Deep shells will be analysed assembling several elements, but the commonly accepted limit of shallow shells theory, a/R 0.5 [13], is respected in each element. This theory of shallow shells is based upon the assumption that the squares and products of ∂w i (x, y) ∂x and ∂w i (x, y) ∂y are small [11].…”

Section: Finite Element Modelmentioning

confidence: 99%

“…The values are compared with results from Bardell et al [6], who used both a commercial finite element software and a thin shell p-version element, and from Leissa and Narita (Table 1 of reference [13]), who employed the Rayleigh-Ritz method and thin shell theory. The lower six eigenvalues computed with the present approach are not shown in the table, but they are very close to zero, value they should have in the case of free boundaries.…”

Section: Linear Natural Frequenciesmentioning

confidence: 99%

“…A shallow shell is a shell whose smallest radius of curvature is larger than its greatest length measured along the middle surface [9][10][11][12][13], so that the raise is small in comparison with the span. In this case, the displacement field is only slightly more complex than the one of plates.…”

Section: Introductionmentioning

confidence: 99%

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Non-linear Vibrations of Deep Cylindrical Shells by thep-Version Finite Element Method

Ribeiro

Cochelin

Bellizzi

2010

Shock and Vibration

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Abstract.A p-version shell finite element based on the so-called shallow shell theory is for the first time employed to study vibrations of deep cylindrical shells. The finite element formulation for deep shells is presented and the linear natural frequencies of different shells, with various boundary conditions, are computed. These linear natural frequencies are compared with published results and with results obtained using a commercial software finite element package; good agreement is found. External forces are applied and the displacements in the geometrically non-linear regime computed with the p-model are found to be close to the ones computed using a commercial FE package. In all numerical tests the p-FE model requires far fewer degrees of freedom than the regular FE models. A numerical study on the dynamic behaviour of deep shells is finally carried out.

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Section: Finite Element Modelmentioning

confidence: 99%

Section: Linear Natural Frequenciesmentioning

confidence: 99%

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Non-linear Vibrations of Deep Cylindrical Shells by thep-Version Finite Element Method

Ribeiro

Cochelin

Bellizzi

2010

Shock and Vibration

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“…A variety of computational methods have been successfully employed for such analysis. Earlier success in the vibration analysis of plates includes methods of finite strip [2,3], spline finite strip [4], Ritz variational methods [5,6], Rayleigh methods [7], Galerkin approaches [8], and series expansions [9]. Recently, the least squares technique [10], meshless methods [11], Rayleigh-Ritz methods [12][13][14][15][16][17], and finite element methods [18,19] have been introduced to the plate vibration analysis.…”

Section: Introductionmentioning

confidence: 99%

Matched interface and boundary (MIB) method for the vibration analysis of plates

Xiang

Wei

2008

Commun. Numer. Meth. Engng.

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SUMMARYThis paper proposes a novel approach, the matched interface and boundary (MIB) method, for the vibration analysis of rectangular plates with simply supported, clamped and free edges, and their arbitrary combinations. In previous work, the MIB method was developed for three-dimensional elliptic equations with arbitrarily complex material interfaces and geometric shapes. The present work generalizes the MIB method for eigenvalue problems in structural analysis with complex boundary conditions. The MIB method utilizes both uniform and non-uniform Cartesian grids. Fictitious values are utilized to facilitate the central finite difference schemes throughout the entire computational domain. Boundary conditions are enforced with fictitious values-a common practice used in the previous discrete singular convolution algorithm. An essential idea of the MIB method is to repeatedly use the boundary conditions to achieve arbitrarily high-order accuracy. A new feature in the proposed approach is the implementation of the cross derivatives in the free boundary conditions. The proposed method has a banded matrix. Nine different plates, particularly those with free edges and free corners, are employed to validate the proposed method. The performance of the proposed method is compared with that of other established methods. Convergence and comparison studies indicate that the proposed MIB method works very well for the vibration analysis of plates. In particular, modal bending moments and shear forces predicted by the proposed method vanish at boundaries for free edges.

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“…In subsequent decades many studies of the vibration of OCCSs have been carried out. Early research mainly used numerical approaches such as the Rayleigh-Ritz method (Sewall, 1967;Leissa and Narita, 1984), the finite element method (FEM) (Cantin and Clough, 1968;Lakis and Selmane, 2000), and the finite strip method (Cheung et al, 1989). Exact solutions for determining the natural frequencies of OCCSs were presented in (Suzuki and Leissa, 1986;Lim and Liew, 1995;Yu et al, 1995;Price et al, 1998;Ye et al, 2014a).…”

Section: Introductionmentioning

confidence: 99%

Exact free vibration analysis of open circular cylindrical shells by the method of reverberation-ray matrix

Yao

Tang

Pang

et al. 2016

J. Zhejiang Univ. Sci. A

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Abstract:This paper is concerned with the free vibration analysis of open circular cylindrical shells with either the two straight edges or the two curved edges simply supported and the remaining two edges supported by arbitrary classical boundary conditions. Based on the Donnell-Mushtari-Vlasov thin shell theory, an analytical solution of the traveling wave form along the simply supported edges and the modal wave form along the remaining two edges is obtained. With such a unidirectional traveling wave form solution, the method of the reverberation-ray matrix is introduced to derive the equation of natural frequencies of the shell with different classical boundary conditions. The exact solutions for natural frequencies of the open circular cylindrical shell are obtained with the employment of a golden section search algorithm. The calculation results are compared with those obtained by the finite element method and the methods in the available literature. The influence of length, thickness, radius, included angle, and the boundary conditions of the open circular cylindrical shell on the natural frequencies is investigated. The exact calculation results can be used as benchmark values for researchers to check their numerical methods and for engineers to design structures with thin shell components.

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Vibrations of completely free shallow shells of rectangular planform

Cited by 63 publications

References 4 publications

Non-linear Vibrations of Deep Cylindrical Shells by thep-Version Finite Element Method

Non-linear Vibrations of Deep Cylindrical Shells by thep-Version Finite Element Method

Matched interface and boundary (MIB) method for the vibration analysis of plates

Exact free vibration analysis of open circular cylindrical shells by the method of reverberation-ray matrix

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