• Bulletin of the Society of Naval Architects of Korea
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• v.21 no.2
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• pp.19-27
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• 1984

An extension of the finite element method to the stability analysis of shells of revolution under static axisymmetric loading is presented in this paper. A systematic procedure for the formulation of the problem is based upon the principle of virtual work. This procedure results in an eigenvalue problem. For solution, the shell of revolution is discretized into a series of conical frusta. The buckling mode in the circumferential direction is assumed, this assumption makes the problem economical for the computing time. The present method is applied to a number of shells of revolution, under axial compression or lateral pressure, and comparision are made with other theoretical results. The results show good agreement each other. The effects of aspect ratio, boundary conditions and buckling modes on the buckling strength of shells of revolution are studied. Also the optimum shape of cylindrical shell under uniform axial compression is obtained from the view point of structural stability.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.15 no.4 s.97
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• pp.457-464
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• 2005

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, complete (not truncated) conical shells of revolution, Unlike conventional shell theories, which are mathematically two-dimensional (2-D). the present method is based upon the 3-D dynamic equations of elasticity. Displacement components $u_{r},\;u_{z},\;and\;u_{\theta}$ in the radial, axial, and circumferential directions, respectively, are taken to be sinusoidal in time, periodic in , and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the conical shells are formulated, the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of theconical shells. Novel numerical results are presented for thick, complete conical shells of revolution based upon the 3-D theory. Comparisons are also made between the frequencies from the present 3-D Ritz method and a 2-D thin shell theory.

• Journal of KSNVE
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• v.11 no.1
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• pp.156-166
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• 2001

This work uses tensor calculus to derive a complete set of three-dimensional field equations well-suited for determining the behavior of thick shells of revolution having arbitrary curvature and variable thickness. The material is assumed to be homogeneous, isotropic and linearly elastic. The equations are expressed in terms of coordinates tangent and normal to the shell middle surface. The relationships are combined to yield equations of motion in terms of orthogonal displacement components taken in the meridional, normal and circumferential directions. Strain energy and kinetic energy functionals are also presented. The equations of motion and energy functionals may be used to determine the static or dynamic displacements and stresses in shells of revolution, including free and forced vibration and wave propagation.

• Computational Structural Engineering
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• v.8 no.3
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• pp.135-141
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• 1995

Any arbitrarily shaped laminated composite shells of revolution can be sum of the conical shell elements. Therefore, finite element model of conical shell element will be developed in this study. To verify consistency and validity of this model, natural vibrations of this model is compared with the analytical solution of cylindrical shell. Herein, an extensive parametric study is presented to assess the modeling capability of this model in class of laminated composite cylinders. It is seen that the proposed model provides highly accurate results with analytical solution. Once development of this conical shell element is done, any arbitrarily shaped composite shells of revolution can be easily analyzed.

• Journal of Korean Association for Spatial Structures
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• v.4 no.4 s.14
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• pp.113-128
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• 2004

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of solid paraboloidal and complete (that is, without a top opening) paraboloidal shells of revolution with variable wall thickness. Unlike conventional shell theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. The ends of the shell may be free or may be subjected to any degree of constraint. Displacement components $u_r,\;u_{\theta},\;and\;u_z$ in the radial, circumferential, and axial directions, respectively, are taken to be sinusoidal in time, periodic in ${\theta}$, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the paraboloidal shells of revolution are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four digit exactitude is demonstrated for the first five frequencies of the complete, shallow and deep paraboloidal shells of revolution with variable thickness. Numerical results are presented for a variety of paraboloidal shells having uniform or variable thickness, and being either shallow or deep. Frequencies for five solid paraboloids of different depth are also given. Comparisons are made between the frequencies from the present 3-D Ritz method and a 2-D thin shell theory.

• Structural Engineering and Mechanics
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• v.7 no.3
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• pp.277-288
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• 1999

This paper presents a fuzzy optimum design of axisymmetrically loaded thin shells of revolution. This paper consists of two parts, namely: an elastic analysis using the new curved element for finite element analysis developed in this study for axisymmetrically loaded thin shells of revolution, and the volume optimization on the basis of results evaluated from the elastic analysis. The curved element to meridian direction is used to develop the computer program. The results obtained from the computer program are compared by exact solution of each analytic example. The fuzzy optimizations of thin shells of revolution are done using [Model 2] which is in the form of a conventional crisp objective function and constraints with non-membership function, and nonlinear optimum GINO (General Interactive Optimizer) programming. In this paper, design examples show that the fuzzy optimum designs of the steel water tank and the steel dome roof could provide significant cost savings.

• Structural Engineering and Mechanics
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• v.19 no.3
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• pp.321-336
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• 2005

The structural feasibility of a variety of non-conventional sludge digesters, in the form of thin shells of revolution constructed in concrete, has formed the subject of investigation of a recent programme of research at the University of Cape Town. Such forms are usually known in the literature as "egg-shaped", and the advantages of these over conventional digesters of the wide-cylindrical type are now well-recognised: superior mixing efficiency, less accumulation of deposits at the bottom, easier removal of bottom deposits and surface crust, reduced heat losses, and so forth. With the aim of exploring the structural feasibility of various non-conventional forms for concrete sludge digesters, and making available usable analytical data and practical guidelines for the design of such thin shell structures, a number of theoretical studies have recently been undertaken, and these have covered conical assemblies, spherical assemblies and parabolic ogival configurations. The purpose of the present paper is to bring together the different analytical approaches employed in each of these studies, summarise the main findings in each case, draw comparisons among the various studied configurations with regard to structural efficiency and functional suitability, and make appropriate conclusions and recommendations.

• Structural Engineering and Mechanics
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• v.55 no.2
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• pp.245-261
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• 2015

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies of eccentric hemi-spherical shells of revolution with variable thickness. Unlike conventional shell theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. Displacement components $u_r$, $u_{\Theta}$, and $u_z$ in the radial, circumferential, and axial directions, respectively, are taken to be periodic in ${\theta}$ and in time, and algebraic polynomials in the r and z directions. Potential and kinetic energies of eccentric hemi-spherical shells with variable thickness are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to three or four-digit exactitude is demonstrated for the first five frequencies of the shells. Numerical results are presented for a variety of eccentric hemi-spherical shells with variable thickness.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.16 no.4
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• pp.419-429
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• 2003

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies of thick, hyperboloidal shells of revolution. Unlike conventional shell theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. Displacement components u/sub r/, u/sub θ/, u/sub z/ in the radial, circumferential, and axial directions, respectively, we taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the r and z directions. Potential(strain) and kinetic energies of the hyperboloidal shells are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four digit exactitude is demonstrated for the first five frequencies of the hyperboloidal shells of revolution. Numerical results are tabulated for eighteen configurations of completely free hyperboloidal shells of revolution having two different shell thickness ratios, three variant axis ratios, and three types of shell height ratios. Poisson's ratio (ν) is fixed at 0.3. Comparisons we made among the frequencies for these hyperboloidal shells and ones which ate cylindrical or nearly cylindrical( small meridional curvature. ) The method is applicable to thin hyperboloidal shells, as well as thick and very thick ones.

• Structural Engineering and Mechanics
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• v.58 no.1
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• pp.59-91
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• 2016

Laminated composite shells are commonly used in various engineering applications including aerospace and marine structures. In this paper, using semi-analytical finite strip method, the buckling behavior of laminated composite deep as well as thick shells of revolution under follower forces which remain normal to the shell is investigated. The stiffness caused by pressure is calculated for the follower forces subjected to external fibers in thick shells. The shell is divided into several closed strips with alignment of their nodal lines in the circumferential direction. The governing equations are derived based on first-order shear deformation theory which accounts for through thickness-shear flexibility. Displacements and rotations in the middle surface of shell are approximated by combining polynomial functions in the meridional direction as well as truncated Fourier series with an appropriate number of harmonic terms in the circumferential direction. The load stiffness matrix which accounts for variation of loads direction will be derived for each strip of the shell. Assembling of these matrices results in global load stiffness matrix which may be un-symmetric. Upon forming linear elastic stiffness matrix called constitutive stiffness matrix, geometric stiffness matrix and load stiffness matrix, the required elements for the second step analysis which is an eigenvalue problem are provided. In this study, different parameter effects are investigated including shell geometry, material properties, and different boundary conditions. Afterwards, the outcomes are compared with other researches. By considering the results of this article, it can be concluded that the deformation-dependent pressure assumption can entail to decrease the calculated buckling load in shells. This characteristic is studied for different examples.